Der Analogrechner

Complete English translation of the original German-language document (135 pages).

[page 1: figure only — cover page]

[page 2: title page]

The Analog Computer

Description with 150 Programming Examples from Mathematics, Control Engineering, and Physics

by Dr. Martin Hund

2nd edition, February 1982

Introduction … 9

1. Operating Instructions … 11

1.1. Technical Data … 11 1.1.1 Analog Computer … 11 1.1.2 Accessories for Programming … 11 1.1.3 Programmable Switch … 11

1.2. Description … 12 1.2.1 Analog Computer … 12 1.2.2 Programmable Switch … 12

1.3. Operation … 13 1.3.1 General Notes on the Circuit Diagrams … 13 1.3.2 Amplitude Scaling … 13 1.3.3 Time Scaling at an Integrator … 15 1.3.4 Time Scaling with a Programmable Switch … 16

1.4. Calibration of the Analog Computer … 16 1.4.1 Drift Compensation of Operational Amplifiers … 16 1.4.2 Offset Compensation of Multipliers … 17 1.4.3 Gain Adjustment of Multipliers … 17

1.5. Conversion to Mains Voltage 110 V … 17

2. Basic Circuits … 18

2.1. Basic Circuits of an Operational Amplifier … 18 2.1.1 The Open-Loop Amplifier … 18 2.1.2 Multiplication of a Variable by a Constant … 18 2.1.3 Error Additions … 19 2.1.4 Determining the Cut-Off Frequency … 19 2.1.5 Summation of Several Variables … 20 2.1.6 Integration of a Variable … 20 2.1.7 Integration with a Defined Initial Condition … 21 2.1.8 Differentiation of a Variable … 22 2.1.9 Differentiation with Delay … 22

2.2. Basic Circuits with a Multiplier … 22 2.2.1 Multiplication of Two Variables … 22 2.2.2 Squaring a Variable, y = x² … 23 2.2.3 Squaring a Variable, y = x⁴ … 23 2.2.4 Squaring a Variable, y = x³ … 23 2.2.5 Computing the Square Root, y = −√x, x > 0 … 24 2.2.6 Computing the Quotient, y = Z/N, N > 0 … 24

2.3. Circuits with Multiplier and Open Amplifier … 25 2.3.1 Computing the Square Root, y = √x, x > 0 … 25 2.3.2 Computing the Cube Root, y = ∛x … 26 2.3.3 Computing the Function y = −√x, x > 0 … 26 2.3.4 Computing the 4th Root, y = ⁴√x, x > 0 … 27 2.3.5 Computing the Quotient, y = Z/N, N > 0 … 27

2.4. Basic Circuits with Diodes in the Feedback Path … 27 2.4.1 Half-Wave Function … 27 2.4.2 Half-Wave Function with Improved Blocking Behavior … 27 2.4.3 Computing the Absolute Value … 27 2.4.4 Step Function … 28 2.4.5 Step Function with Variable Steepness … 28 2.4.6 Bent Characteristic with Limitation … 28 2.4.7 Bent Characteristic without Limitation … 28 2.4.8 Comparator … 28 2.4.9 Two-Point Switch … 29

[page 4: table of contents continued]

2.5. Basic Circuits with Diodes in the Input Path … 29 2.5.1 Characteristic of a Diode … 29 2.5.2 Characteristic of a Diode with Limited Slope … 30 2.5.3 Generating a Quadratic Characteristic using Pre-Biased Diodes … 30 2.5.4 Dead Zone … 30 2.5.5 Dead Zone with Variable Width … 31 2.5.6 Backlash-Free … 31 2.5.7 Backlash-Free with Variable Play … 32 2.5.8 Backlash (Hysteresis) … 32

2.6. Logical Functions, SIMULOG-Compatible … 33 2.6.1 Negation … 33 2.6.2 NAND Gate … 33 2.6.3 NOR Gate … 33 2.6.4 Full Adder … 33

2.7. Generators … 35 2.7.1 Square-Wave Generator … 35 2.7.2 Square-Wave Generator with Adjustable Frequency … 36 2.7.3 Triangle- and Square-Wave Generator with Variable Duty Cycle … 36 2.7.4 Voltage-to-Frequency Converter … 36 2.7.5 Sine-Wave Generator Using One Operational Amplifier … 37 2.7.6 Sine-Wave Generator with Amplitude Stabilization … 37 2.7.7 Three-Phase Generator … 37

3. Solving Mathematical Problems … 38

3.1. Linear Systems of Equations … 38 3.1.1 Two Equations with Two Unknowns, Fixed Coefficients … 38 3.1.2 Unstable Configuration … 38 3.1.3 Two Equations with Two Unknowns, Variable Coefficients … 38 3.1.4 Three Equations with Three Unknowns … 39 3.1.5 Four Equations with Four Unknowns … 40

3.2. Elementary Integrals … 41

3.2.1 Solution of the Integral y = ∫s dx … 41 3.2.2 Solution of the Integral y = −∫x² dx … 42 3.2.3 Solution of the Integral y = 1/3 ∫1/t dx … 43 3.2.4 Solution of the Integral y = ∫√x dx … 43 3.2.5 Solution of the Integral y = −a ∫cos ωt dt … 43 3.2.6 Solution of the Integral y = ∫1/(1−x²) dx … 44 3.2.7 Solution of the Integral y = ∫1/√(1−x²) dx, x < 1 … 45 3.2.8 Solution of the Integral y = ∫1/√(1−x²) dx, x > 1 … 45 3.2.9 Solution of the Integral y = ∫1/√(1−x²) dx, x < 1 … 46

3.3. Polynomials … 47 3.3.1 Polynomial of 2nd Degree using Integrators … 47 3.3.2 Polynomial of 4th Degree using Integrators … 48 3.3.3 Polynomial of 3rd Degree using Multipliers … 49

[page 5: table of contents continued]

3.4. Simple Examples of Ordinary Differential Equations of the First Order … 50 3.4.1 Solution of the Differential Equation y’ = y … 50 3.4.2 Solution of the Differential Equation y’ = −y … 50 3.4.3 Solution of the Differential Equation y’ = y² … 51 3.4.4 Solution of the Differential Equation y’ = −1 … 52 3.4.5 Solution of the Differential Equation y’ = √(10 − y²) … 53 3.4.6 Solution of the Differential Equation y’ = y/x … 53 3.4.7 Solution of the Differential Equation y’ = y·x … 54

3.5. Linear Differential Equations of the 2nd Order … 55 3.5.1 General Solution of y” + py’ + qy = 0 … 55 3.5.2 Solution for p = 0 and q = 1 … 55 3.5.3 Solution for p = 0 and q = 0 … 55 3.5.4 Solution for q < 0 and p arbitrary … 55 3.5.5 Solution for q > 0 and p = 0 … 57 3.5.6 Damped Harmonic Oscillation … 58 3.5.7 Solution for q > 0 and p arbitrary … 59 3.5.8 Slightly Damped Harmonic Oscillation … 59

3.6. Non-Linear Differential Equations of the 2nd Order … 59 3.6.1 Solution of y” + p|y’|y’ + y = 0, mit p > 0 … 59 3.6.2 Solution of y” + p|y’|y’ + y = 0 … 61

3.7. Addition Theorems for Sine and Cosine … 62 3.7.1 Proof of sin²x + cos²x = 1 … 62 3.7.2 Formation of cos 2x aus cos²x − sin²x … 62 3.7.3 Formation of cos 3x aus 4 cos³x − 3 cos x … 62 3.7.4 Formation of cos(x ± y) = cos x cos y ∓ sin x sin y … 62 3.7.5 Formation of sin 3x from 3 sin x − 4 sin³x … 63 3.7.6 Formation of cos x/2 from √((1 + cos x)/2) … 63 3.7.7 Formation of sin²(x/2) from (1 − cos x)/2 … 63 3.7.8 Formation of sin x + cos x; Beating … 63

3.8. Fourier Analysis … 64

4. Control Engineering … 65

4.1. Controlled Systems with Compensation … 65 4.1.1 Control path with compensation and delay, 1st order … 65 4.1.2 Control path with compensation and delay, 2nd order … 66 4.1.3 Control path with compensation and delay with one operational amplifier … 66 4.1.4 Control path with compensation and delay, 3rd order … 67 4.1.5 Control path with compensation and delay, 4th order … 67 4.1.6 Control path with compensation and delay, 4th order … 67 4.1.7 Control path with compensation and delay, 4th order … 68 4.1.8 Control path with compensation and delay, with step disturbance … 68 4.1.9 Control path with compensation and delay, 5th order … 69 4.1.10 Control path with compensation and delay, 6th order … 70

4.2. Controlled Systems without Compensation … 70 4.2.1 Delay-free control path with delay, 1st order … 70 4.2.2 Control path without compensation, with delay, 1st order … 70 4.2.3 Control path without compensation, without delay … 71

4.3. Controllers … 72 4.3.1 Two-position controller … 72 4.3.2 Controller with P-element … 73 4.3.3 Controller with I-element and PI controller … 74 4.3.4 Controller with D-element and PID controller … 75 4.3.5 Controller from one operational amplifier … 79 4.3.6 Controller with control path without compensation … 80

[page 6: table of contents continued]

5. Simulation and Solution of Physical Problems … 80

5.1. Kinematics … 81 5.1.1 Velocity … 81 5.1.2 Encounter of Two Vehicles (Mathematical Solution) … 82 5.1.3 Encounter of Two Vehicles (Simulation of the Process) … 83 5.1.4 Encounter of Two Vehicles (Physical Solution) … 83 5.1.5 Velocity and Acceleration … 84 5.1.6 Free Fall … 84

5.2. Ballistics … 85 5.2.1 Free Fall with Linear Friction … 85 5.2.2 Free Fall with Quadratic Friction … 86 5.2.3 Projectile Motion (Registration with TY-Recorder) … 87 5.2.4 Projectile Motion (Registration with XY-Recorder) … 88 5.2.5 Projectile Motion with Adjustable Angle … 90 5.2.6 Projectile Motion with Linear Friction … 90 5.2.7 Acceleration of a Rocket … 91 5.2.8 Acceleration of a Rocket in a Gravitational Field, with Friction … 92

5.3. Dynamics … 93 5.3.1 Accelerations of a Motor Vehicle … 94 5.3.2 Overrun of a Motor Vehicle … 95 5.3.3 Braking of a Motor Vehicle … 97 5.3.4 Experiment on Force Impulse … 98 5.3.5 One-Dimensional Elastic Collision … 99 5.3.6 Impulse Retention at Elastic Collision … 101 5.3.7 Energy Retention at Elastic Collision … 101

5.4. Transient Processes … 102 5.4.1 Discharge of a Capacitor across an Ohmic Resistance … 102 5.4.2 Radioactive Decay … 102 5.4.3 Radioactive Decay with Stable End Product … 103 5.4.4 Decay Series … 103 5.4.5 Charge Exchange between Two Capacitors … 104 5.4.6 Heat Exchange between Two Heat Reservoirs … 105 5.4.7 Temperature Equalization in a Chain of Three Heat Reservoirs … 106 5.4.8 Temperature Equalization in a Chain of Five Heat Reservoirs … 106 5.4.9 A Model for Heat Conduction … 107

5.5. Free Oscillations … 108 5.5.1 Mathematical Pendulum with Linear Friction … 108 5.5.2 Elastic Pendulum with Sliding Friction … 109 5.5.3 Mathematical Pendulum … 110 5.5.4 Coupled Pendulum … 110

5.6. Forced Oscillations … 111 5.6.1 Resonance Curves of a Linear Resonance System … 112 5.6.2 Resonance Curves of a Non-Linear System … 113 5.6.3 Resonance Curves of a Non-Linear System … 113 5.6.4 Forced Oscillations of a Non-Linear System … 114

5.7. Schrödinger Equation of the One-Dimensional System … 115 5.7.1 Potential Trough of Finite Depth … 116 5.7.2 Graduated Potentials … 117 5.7.3 Harmonic Oscillator … 119

6. Appendix … 121

6.1. Recording Solutions with a Recorder … 121 6.1.1 TY-Recorder as TY-Recorder … 121 6.1.2 XY-Recorder as TY-Recorder … 121

6.2. Recording Solutions with an Oscilloscope … 122 6.2.1 Example: Recording transient processes of a linear resonance system … 122

[page 7: table of contents continued]

6.3. Control via Programmable Switch … 123 6.3.1 Types of the programmable switch … 123 6.3.2 Schrödinger equation for stepped potentials … 124 6.3.3 Application examples for the programmable switch … 125

6.4. Applications with Function Generator … 127

6.5. Computer Graphs … 127 6.5.1 Closed Curves … 128 6.5.2 Lissajous Figures with Two Different Frequencies … 129 6.5.3 Circles with Linearly Modulated Radius … 130 6.5.4 Spirals with Sinusoidally Modulated Radius … 131

6.6. Games … 136

6.7. School Practice Reports and Talks … 137

Introduction

By interconnecting several operational amplifiers, an instrument is created which can simulate a large number of mathematical and physical relationships (integrals, differential equations, frequency-free controllers, quantum-mechanical oscillators, etc.) and investigate their dependence on the parameters and initial conditions. As a measuring instrument, unlike a measuring instrument, the course of a solution can be followed on a recorder or oscilloscope, and parameters and initial conditions can be changed at will.

In the instrument described here there are 7 operational amplifiers, 2 multipliers, and the network that contains 7 operational amplifiers, 4 coefficient potentiometers, and 2 multipliers. Multiplication: 4 quadrant, Division: 1 quadrant. Square Root: Radikand positive, root negative. 4 Potentiometers: 2.8 kΩ, Offset: 0.5%, 10 Patch points: for the description 4 elements. Safety: T 0.2 A for 220 V, F 0.315 B for 110 V. Supply voltage: ±10 V for all operational amplifiers and multipliers. Signal range: ±10 V. Computing accuracy: 3%, not less than about 1 kHz up to about 10 kHz. Dimensions: 690 mm × 580 mm × 65 mm. Weight: 10 kg.

For programming, a set of patch cables for time constants of 0.002 s to 10 s (for registration with a stylus or strip chart recorder or XY-recorder) consisting of:

579 52 — 1 tablet of 96 patch elements 577 10 — 8 Wideband, 200 kΩ, 1% 577 08 — 2 Wideband, 50 kΩ, 1% 577 07 — 2 Wideband, 10 kΩ, 1% 578 16 — 8 Capacitor, 4.7 µF, 5% 578 15 — 5 Capacitor, 47 nF, 5% 578 10 — 4 Capacitor, 10 nF, 5% 579 14 — 4 Capacitor, 220 nF, 5%

Supplementary patch elements (578 04) for time constants of 0.02 ms to 44 ms (for registration with a fast stylus recorder, e.g. Doppelmavo, Nullpunkt Mitte (442 84), or oscilloscope (442 84)) consisting of:

577 07 — 2 Wideband, 50 kΩ, 1% 577 08 — 2 Wideband, 20 kΩ, 1%

1.1.3 Programmable Switch

Execution: flat housing, behind the programmable field there are built-in:

1 electronic switch: integrated switching circuit with 2 switch inputs (positive flanks), U_max = ±15 V

4 electronic switches: integrated switching circuit with a switch input (positive flank) and a control input (positive flank) for closing of the switch R < 200 Ω, I_max = 30 mA

opened: continuous Zykluszeit between 1.5 s and 30 s with a monoflop with adjustable shorter cycle times between 20 ms and 30 ms

1 Monoflop: with adjustable smaller cycle time between 1.5 ms and 23 ms

1 Monoflop: with adjustable smaller cycle time between 4.5 ms and 4.5 ms

Trigger output: 1. Oscilloscope — trigger; also can be used as a clock source with adjustable shorter time constant

Operational modes:

automatic: continuous repetition of the cycle without interruption; continuous triggering of the oscilloscope; 2. manual: triggering of a single cycle through button press or control signal (negative flank); 3. manual: triggering of a single cycle through several button presses of a taster.

Mains connection: via multiplication with the analog computer, connector (B)

Dimensions: 280 mm × 185 mm × 85 mm Weight: 1.3 kg

1.2. Description

1.2.1 Analog Computer

[page 10: figure 12.1 — front panel layout of the analog computer with numbered components]

① Safety holder for mains connector cable ② Patch field (Steckbuchse) ③ Mains switch ④ Operating indicator lamp ⑤ Control lamp for positive signal voltage, for the supply of the programmable switch (576 07) ⑥ Patch field with 7 operational amplifiers, 4 coefficient potentiometers and 2 multipliers ⑦ Non-inverting input of an operational amplifier; is mostly used as a signal input ⑧ Inverting input of an operational amplifier; is mostly used as a signal input with sign inversion; feedback is normally connected here ⑨ Non-inverting input for the operational amplifier for signal voltages; ±10 V ⑩ Inverting input of an operational amplifier; negative ⑪ Output for positive signal voltages; +10 V ⑫ Output for negative signal voltages; −10 V ⑬ Coefficient potentiometer; may be connected alternately to positive and negative signal voltages, or to the input or output of an operational amplifier ⑭ Multiplier with switch for multiplication, division, and square root ⑮ Multiplier (left input, right output) ⑯ Multiplier (left input used; right output need not normally be connected)

1.2.2 Programmable Switch

[page 10: figure 12.2 — programmable switch panel with numbered components]

① Switch for cycle time base ② Switches for cycle time base ③ Outputs for control signals (positive flanks) spaced at T/8 intervals from the cycle time T ④ Electronic switches for signals ⑤ Control inputs for switches (positive flanks); when a positive flank arrives at the upper control input, the right switch contact is closed; a positive flank at the lower control input opens it again; similarly, a positive flank at the left upper control input closes the left switch contact, and the left lower contact opens it ⑥ Connection sockets for switch ⑦ Control input for switch: when a positive flank arrives, the associated switch is closed; a positive flank at the lower control input opens the switch ⑧ Connection sockets for switch

1.3. Operation

Own drafts or according to the programming plans from pages 2 to 6 may be programmed. Further instructions, where needed for individual circuit diagrams, can be found in the individual circuit diagrams.

At the first use of an analog computer by unfamiliar persons, sections 1.2.1, 2.1.5, 2.1.7, and 2.1.7 should be worked through.

If there are also programmable switches available, the section may also be programmed via patch cable (B) to connector (B). Examples for this are given under section 1.3.3.

1.3.1 General Notes on the Circuit Diagrams

The circuit diagrams in sections 2 to 6 are each shown geometrically as close as possible to the analog computer program described, so that they can also be found quickly in the patch field. In the other programming plans of sections 2 to 6, the switch elements are not especially identified. If a “Verbindungsstecker” (connecting plug) symbol is encountered in the circuit diagrams, it indicates a patch-field connection instead of a cable connection.

A connection label with “Verbindungssteckern” is drawn in the circuit plans wherever it is useful; for example, to give the program a better overview or to indicate certain signal paths. In the other programming plans a label is occasionally given for the purpose of easier identification. Where dagegen (against this) the connection symbols are unusually drawn, a number reference exists in the circuit diagram corresponding to the element table.

The entry in the circuit diagrams shows in each case a potentiometer coefficient c = R₀/R₁; the values are in the circuit plans as Verzweigungssymbol (branching symbol).

1.3.2 Amplitude Scaling

An analog computer computes with dimensionless mathematical quantities m, m = U/β, so β is chosen such that m for y always remains in the range −1 ≤ m ≤ 1.

When working with values greater than 1 or less than 1 (their magnitudes), a further scaling factor a is applied. With ay = U/β and:

−1 ≤ ay ≤ 1

In the circuit diagram the scaling factor a is given at each input and output of an operational amplifier or multiplier.

Multiplication: 4-quadrant, Division: 1-quadrant.

For the resistance ratio c at the operational amplifier, the following applies:

a_e/a_a < 0 for b > 0 a_e/a_a > 0 for b < 0

For the resistance ratio c at the operational amplifier it follows:

c = −a_e/a_a

It is occasionally useful to choose a_e = −1/b; then c = 1.

When working with dimensioned physical quantities, the factor β is divided by the unit of the quantity in question. For a velocity v, for example, β = U/β gives a_v = U/(β_v), and as a computing range the following applies:

−1 m/s ≤ a_v · v ≤ +1 m/s

Example:

a) Choosing a_e = 1 and a_a = 1 means that according to section 1.3.2, figure 1.3.2.1 is used as the solution circuit, with Widerstandsverhältnis c = 1.

b) Choosing a_e = 2 and a_a = 1 means, so the Programmplan 1.3.2.2 follows with the Widerstandsverhältnis c must be = −2.

c) Choosing a_e = 1 and a_a = 1/2 sets following: c = 1/2 set, the Programmplan 1.3.2.3 follows thereafter with the Widerstandsverhältnis c = 1/2.

For a_w = 1/1000, the multiplier delivers a_w = a_e · a_b. Thus: c = −g_e · a_e/a_b, a_wW ≤ 1 N.

For a_w = 1/1000, the multiplier output at a_W = a_e and a_b, therefore a_W ≤ 1. The program plan follows section 1.3.2.3, with the Widerstandsverhältnis c. For a_w = 1/1000, this gives c = 0.981 = 200 kΩ/202 kΩ.

In the programming examples of chapter 5.1 the time scaling is shown in each case once, the amplitude scaling shown in each case is also given once. In the other programming plans of chapters 3.2 and 3.4, the time must be maintained mainly once for amplitude scaling larger or smaller Rechenbereiche to work, the programs of these sections 5.1 can serve.

1.3.3 Time Scaling at an Integrator

An analog computer can integrate over time. According to section 1.3.2, an integrator with input U_e(t) obtains at its output:

U_a(t) = −(1/RC) ∫₀ᵗ U_e(t)dt + U_a(0)

The time constant RC is given by the values of the circuit elements.

By applying section 1.3.2, an amplitude scaling with a_y = U/β is performed; by substituting the time t with a dimensionless mathematical quantity x = t/a, the output of the integrator appears as:

y_a(x) = −(a/RC) ∫₀ˣ y_e(x)dx + y_a(0)

The time constant RC is often chosen so that the time interval of interest, e.g. 10 s for registration of a solution with a TY-recorder or 10 ms for display on an oscilloscope, fits in the range from RC = 10 s (Bild 1.3.3.1) to RC = 10 ms (Bild 1.3.3.2). As an example:

It should be computed for 0 ≤ s ≤ 1 and a_y = 1 using a single integrator:

y_a(x) = ∫₀ˣ y_e(x)dx + y_a(0)

Taking x = t/a = 10 s, this means that when no additional amplitude scaling is applied, the time constant must equal a, so RC = a = 10.

Selecting a = 1/10 correspondingly according to section 1.3.2, one more amplitude scaling is done with a_y = 1/10, so the time constant becomes smaller (Bild 1.3.3.2). This gives: RC = a/10.

[page 13: figure 1.3.3.1 and figure 1.3.3.2 — circuit diagrams for integrator time scaling]

Calculation for the integrator scaling factor a:

If the integral with respect to x is computed using an integrator with capacitance C₀:

y_a(x) = b ∫₀ˣ y_e(x)dx + y_a(0)

The scaling factor a must be chosen for the time scaling factor:

(page 14 continued)

When a > 0 and the amplitude scaling factors apply:

a_e/a_a < 0 for b > 0 a_e/a_a > 0 for b < 0

For the required time scaling of the integrator, the following then applies:

RC = −(1/b) · (a_e/a_a) · a

When the time t is replaced by a dimensionless physical quantity s (e.g. a distance s), so the analog computer shows as output at time t = a (a is a time), the physical quantity of a dimensionless mathematical scale. Namely it equals for the Arbeit W as the function of the backward-directed Kraft F on the distance s:

W(s) = ∫₀ˢ F(s) ds

When F(s) = 1 N, interest exists for s in the range 0 ≤ s ≤ 10 m, so one may choose for s: x = s/10 m = 1 and for W: a_W = W/10 J multiplied with a_W, so the amplitude scaling factor is a_W = 1/10.

The program plan follows from section 1.3.2.3, with the Widerstandsverhältnis c = −a_e · a_W = −10 · (1/10) · a_W. For a_W = 1/1000, c follows as a_W = a_e · a_b.

For a_W = 1/1000, the Multiplikatoreinsatz delivers a_W = a_e · a_b, and therefore a_W ≤ 1. The program plan section 1.3.2.3 follows with c. For a_W = 1/1000, c = 0.981 = 200 kΩ/202 kΩ.

In the programming examples of Chapter 5.1, the time scaling is shown once in each case; the amplitude scaling is also given once. In the other programming plans of Chapters 3.2 and 3.4, the time must in principle be shown at least once for the amplitude scaling to work for larger or smaller computing ranges; the programs of Chapter 5.1 serve this purpose.

1.3.4 Time Scaling with a Programmable Switch

[page 14: figure 1.3.3.3 — integrator time scaling with programmable switch circuit diagram]

In the programming examples of chapters 3.2 and 3.4, the time scaling is shown once in each case, and the amplitude scaling is also given once in each case. In the other programming plans of chapters 3.5 and 3.6, the time must in principle be shown at least once for the amplitude scaling to work for larger or smaller computing ranges.

1.4. Calibration of the Analog Computer

The seven operational amplifiers are pre-calibrated ex-works and need only rarely be readjusted. An adjustment is possible whenever needed. The front panel makes an accessible potentiometer ⑧ (see Bild 1.2.1) available for this purpose.

1.4.1 Drift Compensation of an Operational Amplifier

If the inputs of an operational amplifier are set to zero (grounded), the output theoretically should be zero; however it generally is not, and it shows a non-zero output and is referred to as drift. An amplifier has drift when a non-zero output signal is present without any input signal. This drift can be compensated by the adjustable drift-compensation potentiometer.

To compensate drift (Bild 1.4.1), first close switch S:

After opening switch S, attempt through pressing on potentiometer ② to hold the output signal constant in the vicinity of zero. This drift can be compensated through pressing the accessible drift-compensation potentiometers. To measure the quality of the drift compensation, the time T is used. For C = 1 µF it is easily achievable to T = 10 s; by using the capacitance supplement, C = 10 nF can make T much smaller. A Feinabstimmung (fine adjustment) of time T is available through the resistance R₁ in Bild 1.4.1. When T = 10 s is achieved, the values for T will gradually become better.

[page 15 continued]

In the circuit diagrams of sections 2 to 6, a Widerstand R₁ is inserted at each output to make the circuits easier to understand and avoid the generally confusing Widerstandsbeschriftung; in general the Factor 100 should not be exceeded.

When a small residual drift remains, it is advisable to use feedback with a capacitor to the non-inverting input of the operational amplifier; when large time constants are involved this can be made negligible; it can likewise be made negligible if large time constants for the given circuit.

1.4.2 Offset Compensation of Multipliers

The offset of the multiplier output can be compensated (Bild 1.4.2). The offset compensation is performed step by step as follows:

The table below lists the meanings: Z₀ = Output zero adjustment, Y₀ = Input A zero adjustment, X₀ = Multiplication zero adjustment, Z₀ = Root adjustment.

Multiplier offset adjustment:

Lay both inputs at zero and set the output signal with potentiometer Z₀ to zero (Abweichung < 1 mV).
Input A at zero and switch Input B alternately to ±10 V; set the three-operation output for both cases to zero with potentiometer X₀.
Input B at zero and switch Input A alternately to ±10 V; set the third-operation output for both cases to zero with potentiometer Y₀.
Repeat these three operations several times (Abweichung of the output between points 2 and 3 no more than 50 mV).
Lay both inputs at + 10 V; set the output to + 10 V with potentiometer Z₀ (Abweichung 0.1 V).

Multiplication adjustment: 6. Switch Input A with multiplier A/B.

1.5. Conversion to Mains Supply Voltage 110 V

Pull out the mains connector. Loosen the screws from the front panel. Lift the front panel off the bottom. The transformer and power leads are visible from the underside. Panel 400 78 504 (Figure 1.5) shows the connections.

[page 15: figure 1.5 — transformer wiring diagram for 110 V / 220 V conversion]

Spring connector ① from contact strip ② remove and push onto contact strip ③. Spring connector ④ from contact strip ⑤ remove and push onto contact strip ⑥. The two primary windings of the transformer are now in series.

New fusing (T 0.315 B) in the safety holder ⑦ (see Bild 1.2.1).

2. Basic Circuits

The programming examples given in this manual and their applications from mathematics, control engineering, and physics are arranged into relatively small, easily realizable programs.

In the following section 2, some simple basic circuits of the summing amplifier and the integrator are first presented. This is followed by basic circuits with multipliers and non-linear basic circuits of operational amplifiers.

Specifications for the simulation of various applications from mathematics, control engineering, and physics are also given, which appear only in the programs of mathematics, control engineering, and physics.

It also includes some circuits of generators used in the programs of the mathematical, control engineering, and physics sections.

2.1. Basic Circuits of an Operational Amplifier

Operational amplifiers (summing amplifiers) form the basic building blocks of an analog computer. An operational amplifier has the following characteristics:

The amplification factor V is infinitely large for all frequencies
The two inputs of the amplifier differ by 180°
The input impedance is infinitely large
The output impedance is zero

With the analog computer (576 01) built-in operational amplifiers, the output is limited to ±10 V; as a result the input is limited to ±10 V. Below are two examples.

U_e	U_a
+0.65 mV	+13 V
+0.68 mV	−13 V

2.1.1 The Open-Loop Amplifier

In order to determine the amplification of the non-coupled (open-loop) operational amplifier, the output must be measured. This is generally accomplished by a measurement instrument of type 531.01 with Messbereichschalter (range selector) ±15 mV (531.09). Voltages of ±0.75 mV should be set at a dual operational amplifier at input. The operational amplifiers deliver more than ±200 kΩ. As an amplification factor V = 200 kΩ; the R = 33 kΩ for an input voltage of less than 2 mV. The Doppelmavo (442.84) with measuring range of 60 mV from 0.1 V (1%). It measures magnitudes greater than 2 mV thus still with accuracy of the Eingangswiderstände less than 1%.

2.1.2 Multiplication of a Variable by a Constant

With the circuit diagram 2.1.1 the relationship between U₁ and U₀ as described in section 2.1.1 is used, and a further second variable i₁ is established, so the following applies:

i₁ = −(U₁ − U_a)/R₁

and for the voltages:

U_a = −(R₂/R₁) · U₁ = −c · U₁

The two variables U₁ and U_a may be multiplied simultaneously with different factors c₁ = R₀/R₁ and c₂ = R₀/R₁, and added together in a summing amplifier. The time constant-independent Spannung (voltage) U_a is in general:

U_a = −(R₂/R₁) · c · U₁

[page 16: figure 2.1.1 — schematic of open-loop amplifier configuration]

As the amplification V = U_a/U_e, it obtains approximately −10⁷, in general approximately −10⁶.

For generating small input voltages, the circuit 2.1.1 uses a summing element with a high-impedance resistor R and a Summierungsmessgerät (442.74). For an inner resistance of 200 kΩ to 500 kΩ using the circuit-available resistors of 2 kΩ to 200 kΩ, the Factor 100 should not be exceeded.

With the formula V = U_a/U_e the following applies for the general time-dependent Spannung U_a with Konstantem c = R₀/R₁ and with −1 multiplied:

U_a = −c · U₁ = −(R₂/R₁) · U₁

[page 17 continued]

The generally time-dependent Spannung U_a is multiplied with the constants c = R₀/R₁ and −1:

U_a = −(R₂/R₁) · U₁

The circuit is otherwise the same as for the summing amplifier.

Examples:

U₁	R₁	R₂	U_a
+5 V	100 kΩ	200 kΩ	−10 V
+5 V	100 kΩ	100 kΩ	−5 V
+5 V	100 kΩ	100 kΩ	−5 V

In Bild 2.1.2 the symbol from the circuit diagram that is used for this application as a multiplier with a constant is given.

2.1.3 Error Additions

From section 2.1.2, the Zusammenhang (relationship) of U₀ and U_e at the operational amplifier is given by:

U_a = [1 − (R₂/R₁ · (1 + 1/V))] · U₁

with V = U_a/U_e. The following must hold:

V/V = |R₂/R₁ + 1| < 1%

In order for the neglect of this expression to be less than 1%, the following two boundary conditions must be met:

R₂/R₁ < 100 and R₁ < 100

The first condition is easily satisfied, as R₁ = 1 MΩ. The second condition, because of the open-loop gain V of the operational amplifier, is the amplification of the open-loop operational amplifier, still readily met in the available resistance range of 2 kΩ to 200 kΩ.

One further source of error is the occasionally not negligible offset (see section 2.1.1). The offset voltage U_e = (U₀)/(R₁/R₂) must generally be larger than the offset current times R₂ = U_off·R₂; this is in practice generally smaller than 1%. With offset current (200 nA and R₂ = 200 kΩ gives at most:

A further source of error in longer integrations can be the offset current running through the integration capacitor (see section 2.1.6). From section 2.1.6, the computation of U₂ from U₀ and U_a is given by:

U_a(t) = U₂ = −(R₂/R₁) [∫(U₁ dt + U₂(0)]

A note for U₁/R₁ = I_max/R₂ is also applicable to the longer integration: for U₁ = 1 and without U_e, the time constant can be chosen smaller.

For U₁/R₁ = I_max, the following error sources exist for longer integrations: as offset currents may arise, they run through the integration capacitor; the computed section 2.1.6 may make it necessary to compensate the offset (using, for example, another pair of operational amplifiers to compute offset, see section 1.4.1). It is possible to avoid this by means of a shorter time constant R₁·C₀.

2.1.4 Determination of the Cut-Off Frequency

As the source of a slow-changing variable Frequency, the open-loop amplifier (567.00) is used; with this the desired frequency content of the variable can be measured. If the measured value is below 0.1 (1%), it is possible to further increase V. The best Frequency f₀ is determined by finding the Messbereich value from 0.1 V to 1 V.

Examples:

U₁	R₁	R₂	f₁
0.02 V_ss	1 kΩ	1 MΩ	0.6 kHz
0.2 V_ss	100 kΩ	200 kΩ	1 kHz
0.5 V_ss	8 kΩ	200 kΩ	13.5 kHz

In the table, the approximate relationship f₁ = R₂/R₁ is nearly fulfilled.

2.1.5 Summation of Two Variables

The relationship between U₀ and U₁ in section 2.1.2 now determines a second variable U₂, so the following applies:

i₁ = i₁ + i₂

For the voltages, the following applies:

U_a = −(R₂/R₁ · U₁ + R₂/R₂ · U₂) = −(c₁U₁ + c₂U₂)

The two variables U₁ and U₂ can simultaneously be multiplied with different factors c₁ = R₀/R₁ and c₂ = R₀/R₂, and added together in a summing amplifier. The time-constant-independent voltage U_a is in general multiplied by −1:

U_a = −(c₁U₁ + c₂U₂)

[page 17: figure 2.1.2 — schematic of summing amplifier circuit]

[page 18: figure 2.1.5 — schematic of two-variable summation]

Examples:

U₁	U₂	R₁	R₂	R₀	U_a
+3 V	+4 V	100 kΩ	100 kΩ	100 kΩ	−7 V
+10 V	−5 V	100 kΩ	200 kΩ	100 kΩ	−7.5 V
+5 V	+5 V	200 kΩ	50 kΩ	100 kΩ	−7.5 V
+7 V	−2 V	10 kΩ	10 kΩ	10 kΩ	+8 V

2.1.6 Integration of a Variable

With the same assumptions as in section 2.1.2, the one can also replace R₂ by a capacitor C₀; the following then applies:

U_a(t) = −(1/R₁C₀) ∫₀ᵗ U₁(t)dt + U_a(0)

From which, with the substitution given in section 2.1.2, i.e., with i₁ = −i₀ = U₁/R₁, the relationship for U₁ and U₀ is combined with the capacitor Q(t):

Q(t) = ∫₀ᵗ i₀ dt + Q(0)

U_a(t) = −(1/C₀) ∫₀ᵗ i₀dt + U_a(0)

U_a(t) = −(1/R₁C₀) ∫₀ᵗ U₁dt + U_a(0)

One obtains the integral of U₁ after time with a factor −1/(R₁C₀) multiplied. Over the Vorgabe (preset) of the initial value U_a(0), one can make a statement about section 2.1.8 — see also section 2.1.8 for the initial condition.

For a positive voltage U₁ the voltage U_a goes negative. If one gives a positive initial voltage U₁, then one has no statement possible; instead, see section 2.1.8 below; no initial condition is used.

If U₁/R₁ > i_max, also integration over longer times may be needed. With R₁ = 200 kΩ and C₀ = 4.7 µF this gives RC = 1 s and x = 1 s⁻¹ suitable.

When the time t is replaced by an independent mathematical quantity (e.g. s = t/a, corresponding), the output of the integrator of the variable x appears as a dimensionless mathematical quantity (e.g. s = t/a), and at time t the output of the integrator of the variable shows the factor k = −1/R₁C₀ multiplied. Over the data, i.e., the statement about the initial condition in section 2.1.7 follows.

[page 18: figure 2.1.6 — integrator circuit schematic]

A note: an extension to 3 or more variables is without further difficulty possible.

2.1.8 Integration of a Variable

The relationship holds as in section 2.1.2. If the integration is performed with respect to x using an integrator with input capacitor C₀:

y_a(x) = b ∫₀ˣ y_e(x)dx + y_a(0)

When the integral with respect to x is computed using one integrator of the variable x instead of y₁, it follows:

y_a(x) = b ∫₀ˣ y_e(x)dx + y_a(0)

One obtains the integral of y_e with respect to x after time with the factor −k = 1/(R₁C₀) multiplied. Over the Vorgabe of the initial condition, section 2.1.7 follows.

[page 18: figure 2.1.6 — integrator with initial condition circuit]

For a positive voltage U₁ the voltage U_a goes negative. Taking a positive initial voltage U₁ gives no further information; instead, see section 2.1.7 below; no initial condition was used.

When the time scaling factor a is chosen according to section 1.3.3, the time constant must be chosen for the integrator accordingly:

[Page 19]

the voltages U as dimensionless variables. This replaces, for example, the velocity v = U/s or acceleration a = U/s, so that the output signal of the integrator can be written as:

U₂(x) = −c ∫₀ˣ U₁(x) dx + U₂(0) with c = k_A

or y₂(x) = −c ∫₀ˣ y₁(x) dx + y₂(0) with c = k_A

or y₂(t) = −k ∫₀ᵗ y₁(t) dt + y₂(0)

or v₀(t) = −c ∫₀ᵗ a(t) dt + v₀(0) with c = k_A/μ_A

= −c v(t) (see Section 2, Part 5.1.5)

The constant c is dimensionless in these examples (for amplitude and time scaling, see Sections 1.3.2 and 1.3.3).

2.1.7 Integration with a Defined Initial Condition

Before the start of integration, the value of U₂(0) must be set. This is achieved in circuit 2.1.7 by an additional input on the operational amplifier with two resistors, as for multiplication with a constant (see Section 2.1.2).

[Figure 2.1.7 — circuit diagram showing integrator with initial condition input]

2.1.8 Integration with a Defined Initial Condition (Practical Procedure)

If switch S₁ is opened and switch S₂ is closed, then for a short time (after a few time constants R₁C₀) the output of the amplifier reaches a potential with negative sign as set on a potentiometer. After closing switch S₁, U₀ will remain unchanged for only a short time (there is an error due to the offset current; see Section 2.1.3); therefore, integration begins with closing switch S₁. Errors will again be introduced when S₁ is opened.

Examples:

U₁	R₁	C₁	U₀(0)	U₀(10 s)
+1 V	200 kΩ	4.7 µF	0 V	−10.6 V
−2 V	200 kΩ	4.7 µF	−10 V	+11.2 V
−1 V	200 kΩ	9.4 µF	−10 V	−4.7 V

[Figure 2.1.71 — graph showing U₀(t) for the given examples]

Bild 2.1.7.1 shows as an example the variation of U₀(t) with R₁C₁ = 1 kΩ, U₁ = −1 V and U₂(0) = −5.5 V when switch S₁ is opened for t = 1 s to 9.5 s and t = 11.5 s to t = 14 s.

An initial value U₀(0) = 0 can also be achieved by closing a switch in parallel with C₀ (e.g. switch S₃ in circuit 2.1.8).

2.1.8 Differentiation of a Variable

With the assumptions of Section 2.1.2, the following applies:

k_A = −1

with i₀ = U₀/R₀. The relationship between i₁ and U₁ is given by the time-varying change of the charge Q = U₁C₁:

[Page 20]

of capacitor C₁:

i₁ = dQ/dt = C₁ dU₁/dt

U₀ = i₀R₀ = i₁R₀

U₀(t) = −R₀C₁ dU₁/dt

[Figure 2.1.8 — circuit diagram of differentiator]

In the left part of circuit 2.1.8, a time-varying input signal is applied (corresponding to Section 2.1.7): a signal changing uniformly with time (like a ramp) is generated. S₂ is then closed, so U₁ = 0 and thus U₀ = 0. If switch S₁ is closed (and S₀ is opened), U₁ can be positive or negative (depending on the previous sign of the input). The output signal U₀ represents the value −R₀C₁ multiplied by the derivative; every sudden change of S₁ causes U₀ to remain at a constant value, and if U₁ = 0, U₀ remains at zero.

Examples:

U₁	i₁	C₁	R₀	U₀
+1 V	−1.06×10⁻⁵ V/s	4.7 µF	200 kΩ	+0.8 V
−1 V	−1.06×10⁻⁵ V/s	9.4 µF	200 kΩ	+0 V
−1 V	−1.06×10⁻⁵ V/s	9.4 µF	400 kΩ	−2 V
−0.5 V	+0.53×10⁻⁵ V/s	9.4 µF	400 kΩ	−2 V

Problems arise when the input changes very rapidly. This causes very fast changes of U₁ at the output, which then produce at the one hand a current through C₁ and on the other hand a current through the load of C₁, either from an internal resistance when charging C₁ or from the output resistance. All of these disturbances cause large changes of U₀; they can to some degree be attenuated when an additional resistor is included in the feedback path (at the position of the resistance in Section 2.1.5). For this reason the differentiator is avoided in practice and the integrator is preferred instead, because an integrator does not amplify rapidly changing signals. Short-duration perturbations on U₁ are attenuated at U₀.

2.1.9 Differentiation with Lag

When connecting a real resistance R₁ in series with the input capacitor C₁ of a differentiator circuit (Section 2.1.8), a differentiation with lag results. It applies:

with i₁ = i₀ = U₀/R₀, the relationship between i₁ and U₁ is:

dU₁/dt = R₁ dU₁/dt − 1/(R₁C₁) U₀ → dU₁/dt = −R₁ dU₀/dt + 1/(R₁C₁) U₀

U₀(t) = −R₀C₁ dU₁/dt − R₀/R₁ · U₀

[Figure 2.1.9 — circuit diagram of differentiator with lag]

At the beginning of the jump (for t < t₀), the jump U₁ is applied and then U₀ = 0. After switch S is opened, U₀ shows the step: the output voltage U₀ at time t₀ enters directly and then:

U₀(t) = −(R₀/R₁) · (R₀ − R₁) · e^(−(t−t₀)/R₁C₁) for t > t₀

The maximum value for U₀ is reached at the first instant and is determined by the step discontinuity of U₁ at t₀; by a constant equivalent resistance R₁ closes the e-function loop for the falling portion. The time constant for the falling exponential is given by R₁C₁.

Bild 2.1.9.1 shows as an example the variation of U₀(t) with R₁ = R₀ = 200 kΩ, C₁ = 4.7 µF and U₀ = 5 V for the time from t = 1 s to t = 6 s.

[Page 21]

Variables, or for squaring a variable, require non-linear elements. In Section 2.5.3 it is shown that with just a few diodes it is possible in principle to generate a quadratic relationship between an input and an output signal. With the help of two quadratures and several operational amplifiers one can build an effective multiplier.

2.2.1 Multiplication of Two Variables

For both input voltages U₁ and U₂, the output voltage of the multiplier is:

U₀ = 1/10 U₁ · U₂

It turns out to be most favorable if one works from the amplitudes of the input and output values, the values in machine units (see Section 1.3.2). The amplitude scaling unit is stated to be 10 V. The value 1 corresponds to 10 V.

It is seen from both input voltages U₁, U₂, U₀, and x so that:

U₀ = x₁ · x₂

with −1 ≤ x₁ ≤ +1 and −1 ≤ x₂ ≤ +1 it follows that −1 ≤ y ≤ +1.

Examples:

U₁	U₂	U₀	x₁	x₂	y
+10 V	+10 V	+10 V	+1	+1	+1
+5 V	+6 V	+3 V	0.5	0.6	0.3
−10 V	+2 V	−2 V	−1	0.2	−0.2
−5 V	−8 V	+4 V	−0.5	−0.8	+0.4

[Figure 2.2.1 — circuit symbol of multiplier]

2.2.2 Squaring of a Variable, y = x²

Examples:

x	y	Error	x	y	Error
1	0.99	1%	0.2	0.035	0.5%
0.8	0.63	0.5%	0	0.001	0.1%
0.6	0.355	0.5%	0.5	0.25	—
0.4	0.155	0.5%	−1	1.01	1%

If y is plotted over x, a parabola is obtained.

2.2.3 Squaring of a Variable, y = x³

Example:

x	y	Error	x	y	Error
1	0.97	3%	0.2	0.008	0.2%
0.8	0.405	0.5%	0	0.000	—
0.6	0.215	0.5%	−0.5	0.065	0.3%
0.4	0.020	0.5%	−1	1.02	2%

[Figure 2.2.3.1 — graph of y = x³ function with table values]

2.2.4 Squaring of a Variable, y = x⁴

Example:

x	y	Error	x	y	Error
1	0.97	3%	0.2	0.008	0.2%
0.8	0.405	0.5%	0	0.000	—
0.6	0.215	0.5%	−0.5	0.065	0.3%
0.4	0.020	0.5%	−1	1.02	2%

[Page 22]

[Figure 2.2.4 — circuit diagrams showing square-root and other nonlinear configurations]

2.2.5 Formation of the Square Root, y = −√x, x > 0

The upper of the two multipliers in the analog computer (576.01) can be switched to the function y = −√x with x > 0. At negative x the value y lies between 0 and −0.01.

Examples:

x	y	Error	x	y	Error
1	−1.00	—	0.1	−0.31	0.6%
0.8	−0.89	0.5%	0.05	−0.21	1.5%
0.4	−0.63	0.2%	0.02	−0.13	1%
0.2	−0.44	0.3%	0	−	—

2.2.6 Formation of the Quotient, y = Z/N, N < 0

The upper of the two multipliers can be switched to operate on the function y = Z/N with N < 0.

Examples:

Z	N	y	Error
+1	−1	−1.00	—
+0.7	−0.8	−0.86	1.5%
−0.5	−1	+0.49	—
−0.7	−0.8	+0.195	0.5%
−0.2	−0.8	+0.32	1.5%
+0.3	−0.4	−0.76	0.5%
−0.2	−0.2	+0.85	15%

[Figure 2.2.6 — circuit diagram with Z and N inputs]

At small denominator values the error becomes very large. Bild 2.2.6.1 shows y as a function of Z/N with N as parameter. The dashed line shows y = Z/N with Z = N. When no Fehler (error) arises, the function should be y = 1.

2.2.7 Construction of a Multiplier from Two Squarers

Using the identity:

y = x₁ · x₂ = ((x₁ + x₂)/2)² − ((x₁ − x₂)/2)²

a multiplier can also be built from two squarers and some operational amplifiers. In Section 2.5.3 it is shown that with the help of a few diodes, at least in a small region of the input-output range, a nearly quadratic relationship can be generated. So-called parabolic multipliers based on diode networks and operational amplifiers are commercially available.

[Figure 2.2.7.1 — graph of x₁ · x₂ function with x₂ as parameter]

[Page 23]

[Figure 2.2.7 — detailed circuit diagram of two-squarer multiplier showing summing and differencing amplifiers]

Bild 2.2.7.1 shows y as a function of x₁ with x₂ as parameter. Despite the complex circuitry, the error of this circuit also remains under 2%.

2.3. Circuits with Multiplier and Open-Loop Amplifier

Sections 2.1.2 to 2.1.9 describe operational amplifier circuits with linear feedback elements (resistors, capacitors). Generally a feedback can be assembled from an entire network of linear and non-linear elements.

In the following Sections 2.3.1 to 2.3.5 examples will be given with multiplier and operational amplifier circuits, where the feedback paths are formed by multipliers and, following Section 2.5, by diode networks. Sections 2.4.1 to 2.4.9 give corresponding examples for diode networks in the input paths.

2.3.1 Formation of the Square Root, y = −√−x, x < 0

For the output signal y of the amplifier the following holds as long as |y| ≤ 1:

y = V(x + y²), x < 0

Because |V| ≥ 10⁵ it follows approximately:

x + y² = 0

y = ±√−x x < 0

The circuit diagram 2.3.1 gives the above-mentioned stable solution. There are two stable solutions: y = +√−x and y = −y₀. However the circuit gives only the solution y = −√−x, since V(x + y²) is stable only for y negative. [Figure 2.3.1]

[Figure 2.3.1 — circuit diagram with multiplier in feedback]

2.3.2 Formation of the Cube Root, y = −∛x

For the output signal y of the amplifier the following holds as long as |y| ≤ 1:

y = V(x + y³)

[Figure 2.3.2 — circuit diagram]

The diode (with threshold voltage U_D) limits the negative output values of the operational amplifier. For y ≥ y₀ with y₀ = −U_D/10 V (approximately −0.06), the above-mentioned solution is stable for all x < −y₀³. In the small interval −y₀² < x < 0 there are two stable solutions: y = +∛−x and y = −y₀. For x ≥ 0, y = y₀ also applies.

[Page 24]

Because |V| ≥ 10⁵ the following approximation holds:

x + y³ = 0, y = −∛x

This solution shows the function y for positive x gives values close to y₀.

Because |V| ≥ 10⁵ the following approximation holds:

x + y² = 0, y = −√x

For negative x a diode blocks the amplifier output so the value y = 0 is held; for positive x, y follows the above solution. If the function y should give negative x values, y lies between 0 and −0.08.

2.3.4 Formation of the 4th Root, y = −∜x, x < 0

For the output signal y of the amplifier the following holds as long as |y| ≤ 1:

y = V(x + y⁴), x < 0

Because |V| ≥ 10⁵:

x + y⁴ = 0, y = −∜(−x)

The diode limits the output as in Section 2.3.1. The amplifier does not oscillate and it maintains the solution as given in Section 2.3.1; it gives results corresponding to Bild 2.3.1 but for the 4th root relationship. Bild 2.3.4 shows the solutions corresponding to Section 2.3.1 for this case.

[Figure 2.3.3 / 2.3.4 — circuit diagrams and graph]

[Page 25]

2.3.5 Formation of the Quotient, y = −Z/N, N > 0

For the output signal y of the amplifier the following holds as long as |y| ≤ 1:

y = V(Z + Ny), N > 0

Because |V| ≥ 10⁵:

Z + Ny = 0, y = −Z/N N > 0

For N < 0, y is not stable; a small change of y causes a further large and equally signed change of y.

[Figure 2.3.5 — circuit diagram of divider]

2.4. Basic Circuits with Diodes in the Feedback Path

Using one or more diodes in the feedback of an operational amplifier, one can limit the output signal either above or below.

2.4.1 Limiter Function

The resistance in the feedback path depends on the polarity of the output voltage U₀ and the input voltage. Bild 2.4.1.1 shows the output voltage U₀ as a function of the input voltage U₁.

For U₁ < 0, i.e. U₀ > 0: the diode is blocked and U₀ = −U₁ (forward voltage U_D = 0.6 V).

For U₁ > 0, i.e. U₀ < 0: the diode is conducting and U₀ = −U₁ (forward voltage U_D = 0.6 V).

[Figure 2.4.1 — circuit diagram and Fig. 2.4.1.1 — characteristic curve]

2.4.2 Limiter Function with Improved Blocking Behavior

To improve the blocking behavior of the limiter described in Section 2.4.1, the output of the operational amplifier is returned via diode D₂ through the resistance R₁ before the output signal again feeds back through the resistance. This means the output signal U₀ is first returned.

For U₁ > 0, i.e. U₀ < 0: the diode D₂ is blocked and for U₀ holds: U₀ = −U₁

For U₁ < 0: U₀ < 0 applies. Since diode D₁ is also conducting, there holds: U₀ = −U₁ − U_D₀ and so: U₀ = −U₁

[Figure 2.4.2 — improved limiter circuit and characteristic]

2.4.3 Formation of the Absolute Value

The circuit consists of two parts. On the left operational amplifier, the following limiter function (corresponding to Section 2.4.2) is formed:

y₊ = 0 for x < 0 and y₊ = −x for x > 0

[Figure 2.4.3 — absolute value circuit]

[Page 26]

At the right operational amplifier the sum is formed:

y = −x for x < 0 and y = x for x > 0

So the result is: y(x) = |x|

2.4.4 Step Function

The output voltage U₀ is limited not only unilaterally, as in Section 2.4.1, but bilaterally through the forward voltage of the diodes.

The resulting step function U₀(U₁) shown in Bild 2.4.4 has a slope height of about 2 U_D.

[Figure 2.4.4.1 — step function characteristic and 2.4.4 — circuit]

2.4.5 Step Function with Variable Step Height

Depending on the set value of U₁, U₀ can take on multiple values of the step height. The internal resistance of the potentiometer is however negligible, as Bild 2.4.5 shows (Curve b with R₁ = 100 kΩ). For enlargement of R₁ to 100 kΩ, Curve b is obtained.

[Figure 2.4.5 — circuit diagram]

2.4.6 Kinked Characteristic with Limitation

If one adds a resistance in series in the circuit 2.4.4, then for the resistance of a meter one gets at the output a spring with a finite slope, limited through the diode section. Bild 2.4.6 shows the slope, limited through the resistance ratio −R₁/R₀, shown for R₁ = 10 kΩ and R₀ = 10 kΩ.

[Figure 2.4.6.1 — characteristic and circuit 2.4.6]

2.4.7 Kinked Characteristic without Limitation

If one uses the effect of the finite slope from Section 2.4.5 and also neglects the not-negligible inner resistance, then a kinked characteristic without limitation results.

The slope at the step (jump) is determined as in Section 2.4.6. It is given by:

dy/dx = −R₁/R₀

The slope to the right of the jump (x positive) is given by:

dy/dx = −R₁ (R₂/(R₀ + R₂))

The slope to the left of the jump (x negative) is given by:

dy/dx = −R₁ (R₁/(R₀ + R₁))

[Figure 2.4.7.1 — characteristic and circuit 2.4.7]

In Bild 2.4.7.1 a kinked characteristic without limitation is shown for R₀ = 20 kΩ, R₁ = 10 kΩ and R₂ = 2 kΩ.

[Page 27]

2.4.8 Comparator

[Figure 2.4.8 — comparator circuit]

If one sets the resistance R₂ in the feedback to zero (i.e. bypasses it), one gets at the summing point the sum of the input voltages U₁, U₂, … . If this sum should then be reacted upon — for example when U₁ + U₂ < 0 is needed — one can place a diode (with threshold voltage U₀) in the output, which gives a signal when U₁ + U₂ < 0, i.e. U₁ is negative (the output U₀ is positive).

With the output signal U₀ one can drive a relay (e.g. relay B 574.42) or another suitable programmable switch (e.g. 573.67) and thus control a further operation.

U₀ = −U_P, for U₁ + U₂ > 0 U₀ = −U_N, for U₁ + U₂ < 0

2.4.9 Two-Point Switch

The left part of the circuit consists of a comparator with:

y₀ = −U₀/10 V = −0.06, for x₁ + x₂ + y > 0.65 y₀ = +1.3, for x₁ + x₂ + y < 0

The right part of the circuit gives a stable value for y. The combined circuit has stable solutions for y in certain ranges. It holds:

y = 0.65 for x₁ + x₂ > 0.65 y = −0.65 for x₁ + x₂ < −0.03

[Figure 2.4.9 — circuit diagram]

Both solutions are possible. Starting at x₁ + x₂ = 0.03, one stays in the solution y = 0.65 and at x₁ + x₂ > 0.65 one goes to the solution y = −0.65. Starting from x₁ + x₂ = −0.03, the solution y = 0.65 is then e.g. 0.08 for R₁ = 10 kΩ and 0.34 for R₁ = 20 kΩ, and can be varied by choice of x₂.

Bild 2.4.9.1 shows y as a function of x₁ for x₂ = 0.3. With the distance between the two switching points (e.g. 0.08 for R₁ = 10 kΩ and 0.34 for R₁ = 20 kΩ), this lower switching point of x₁ can be varied (e.g. −0.33 for x₁ = 0.3 or −0.67 for x₁ = 0.3).

2.5. Basic Circuits with Diodes in the Input Path

2.5.1 Characteristic Curve of a Single Diode

With the assumptions of Section 2.1.2, the following holds:

i₀ = −i₁

with i₀ = U₀/R₀. The relationship between i₁ and U₁ is given by the diode characteristic i₁(U₁). It follows:

U₀ = −R₀i₁(U₁)

The voltage U₁ must be measured with a measuring instrument since the passage through the diode strongly loads the source.

[Page 28]

[Figure 2.5.1 — characteristic curve of diode circuit, Figure 2.5.2 — circuit diagram]

2.5.2 Characteristic Curve of a Diode with Limited Slope

The resistance R₁ limits the exponential rise of the diode characteristic. The maximum slope is determined by R₁/R₀; Bild 2.5.2.1 shows U₀ as a function of U₁ for R₁ = 5 kΩ and R₀ = 5 kΩ.

[Figure 2.5.2.1 — characteristic graph with limited slope]

2.5.3 Generation of a Quadratic Characteristic Through Piecewise-Spanned Diodes

Curve b in Bild 2.5.2.1 (the characteristic of a diode with limited slope) approximates in its beginning a quadratic function for 0.25 V < U₁ < 0.5 V. The equal addition of such partial curves allows one to achieve a good approximation over a larger range. A suitable starting voltage (here 0.25 V) is first added to U₁, in order to already be at a quadratic starting point of the second diode section. This addition is done a second time: the quadratic starting point of a second diode section begins again. Curve c from Bild 2.5.2.1 shows a still better approach; however it is shifted once more by 0.22 V (i.e. U₀ at d is 0.22 V shifted), so that for 0 < U₁ < 0.25 V (see e.g. Curve b): U₀ = −1.82 U₁²).

Over the output voltage U₀ a voltage of 0.03 V is added. Curve d in Bild 2.5.2.1 shows this result. Now a quadratic output is available for 0 < U₁ < 0.5 V and thus U₀ = −1.82 U₁²).

[Figure 2.5.3 — circuit diagram of quadratic characteristic using multiple diodes]

By using more diode sections, the validity range of the quadratic characteristic can be further enlarged. One can then build up a parabolic multiplier according to Section 2.2.7.

2.5.4 Dead Zone

[Figure 2.5.4 — dead zone circuit and characteristic]

From two added characteristics with limitation (corresponding to Section 2.5.3) and two diodes with different polarities:

y = −x − 0.06, for x < −0.06 y = 0, for |x| < 0.06 y = −x + 0.06, for x > +0.06

At the breakpoints a somewhat larger error (2%) arises.

[Page 29]

[Figure 2.5.5.1 — dead zone characteristic, Figure 2.5.5 — circuit]

2.5.5 Dead Zone with Variable Width

By using a potentiometer for the threshold values and the output resistances of the last operational amplifier (see Section 2.5.5), the dead zone width and the output slopes can be varied. For Bild 2.5.5.1, h₁ = 2 and for Bild 2.5.5.1 (right) the output slopes are shown as a function of the breakpoints. For the setting of h₁ = 2, the slopes of the output signals are approximately equal to those of the input signal.

[Figure 2.5.6 — gear function circuit, Figure 2.5.7 — output waveform]

2.5.6 Gear Function

With the middle operational amplifier, the sum is formed:

y = −(x + y)

This provides a narrow dead zone (corresponding to Section 2.5.6): if the dead zone width (by Bild 2.5.5.1) and the output slopes are designed to meet certain requirements in order to form y, x can be reproduced. Integrators and integrators can then rapidly and accurately give the desired y value, so that:

(C/2)·f_a·y = 0

[Figure 2.5.6 / 2.5.7 — circuit diagrams]

[Page 30]

The change of x caused by the change of z is equalized by the feedback (for the sign convention, see Part 4). The values of y and thus the value of x however depend on the sign of the time-varying change in x, so:

For x > 0: z follows = −z/2 and thus y = −x + z₀ For x < 0: z follows = +z/2 and thus y = −x − z₀

When z first passes through the dead zone, there is initially no change in y; but the second boundary condition applies: z must vary slowly enough so that errors may occur slowly (the integrator time constant here is 10 ms).

[Figure 2.5.7.1 — graph of gear function output, Figure 2.5.7 — circuit]

2.5.7 Gear Function with Variable Play

The play of the gear function of Section 2.5.6 is determined by dividing the width of the dead zone by the gain of the middle operational amplifier (due to the formation of y). A smaller play of the gear function through the feedback can be achieved. The play can however also be varied. More favorable is a variation of the dead zone corresponding to Section 2.5.5.

Curve b in Bild 2.5.7.1 shows y as a function of x with the play set on the potentiometers (given in Teilerverhältnissen — divider ratio).

2.5.8 Gear Function with Limitation (Hysteresis)

When the gear function of Section 2.5.7 is limited, e.g. through the operational amplifier override corresponding to Section 2.5.8 (circuit on the left), one obtains Bild 2.5.8.1 (right):

[Figure 2.5.8 — hysteresis circuit, Figure 2.5.8.1 — hysteresis characteristic]

The dead zone ±0.25 in the range y₀ is limited.

[Page 31]

2.6. Logical Operations, SIMULOG-Compatible

In Section 2.4.8 it was shown that a diode with feedback resistor R₂ set to zero gives an output that is known to be −0.6 V (0) and +13 V (1). With SIMULOG devices (573.85–89, 574.94–08 and 573.85–89, 574.61–69) as inputs (e.g. SIMULOG-Glieder 574.61–69 are operated), which are described further in the following sections, the SIMULOG or computer-matched potentials define: −0.6 V as 0-signal and +10 V ≤ U ≤ 13 V as 1-signal.

Before the operational amplifiers, two or more corresponding resistors are applied: the sum of the currents from the inputs corresponding to each input counts as the 0-signal, the 1-signal is negative. For one negative input, NAND and NOR can be realized. Below a circuit for such digital functions is shown. Note: it is mentioned that an analog computer is indeed not suited for digital work; however these circuits for computing purposes can be employed.

2.6.1 Negation

For the sum of the currents before the operational amplifier input:

I = −10 V / 22.5 kΩ + U_A / 10 kΩ

When a 0-signal is applied at input A (i.e. −0.6 V ≤ U ≤ 2 V), then I < 0 and the output goes to a 1-signal (13 V). When A has a 1-signal (+10 V ≤ U_A ≤ 13 V), then I > 0 and the output goes to a 0-signal (−0.6 V).

It holds: C = Ā; C is the negation of A.

[Figure 2.6.1 — negation circuit and truth table]

2.6.2 NAND Gate

For the sum of the currents before the operational amplifier input:

I = −10 V / 6.5 kΩ + U_A / 10 kΩ + U_B / 10 kΩ

Only when both inputs A and B have a 1-signal is I > 0 and the output appears as a 0-signal. In all other cases I < 0 and the output appears as a 1-signal.

[Figure 2.6.2 — NAND circuit and truth table]

2.6.3 NOR Gate

For the sum of the currents before the operational amplifier input:

I = −10 V / 22.5 kΩ + U_A / 10 kΩ + U_B / 10 kΩ

Only when both inputs A and B have a 0-signal is I < 0 and the output appears as a 1-signal. In all other cases I > 0 and the output appears as a 0-signal.

It holds: C = Ā∨B̄; C is the negation of the OR-combination of A and B.

[Figure 2.6.3 — NOR circuit and truth table]

2.6.4 Full Adder

It is the (analog) sum of the signals from A, B and C is formed. The inputs can take the values −0.06 (approximately 0 signal) or 0.2 (approximately 1 signal). The values between 1, 2 and 3 inputs with 1-signals (values between −0.06 and 1.3) can be distinguished, so that the value of x for four different regions can be determined:

−0.038 ≤ −x ≤ 0.034, when none of the inputs shows a 1-signal +0.176 ≤ −x ≤ 0.34, when one input is a 1-signal +0.388 ≤ −x ≤ 0.54, when two inputs are 1-signals +0.60 ≤ −x ≤ 0.78, when all three inputs are 1-signals

The comparator for digital sum S must output a 1-signal for −x between 0.176 and 0.34 and 0.60 and 0.78.

[Page 32]

[Figure 2.6.4 — full adder circuit diagram and truth table]

Truth table:

A	B	C	U	S
0	0	0	0	0
0	0	1	0	1
0	1	0	0	1
0	1	1	1	0
1	0	0	0	1
1	0	1	1	0
1	1	0	1	0
1	1	1	1	1

−x greater than 0.6. The last condition is easy to satisfy. Comparison of −x with the fixed value 1 of the other switching point is arranged, and x switches sign at the switching point x₁:

−x₁ = 1.2 · 12 kΩ / 22 kΩ = 0.59

To satisfy the first condition, the value y is compared with the fixed value 1; y changes sign at x₁:

−x₁ = 1.0 · 12 kΩ / 100 kΩ = 0.12

At the next operational amplifier, the sign of −x changes once more above −x₁ (once briefly above −x₁ and again at the upper switching point x₂ mit):

−x₂ = 1.3 · 14 kΩ / 50 kΩ = 0.36

The next comparator for digital sum S also gives a 1-signal for −x between 0.176 and 0.34 and at the output:

−x₂ = 1.0 · 14 kΩ / 50 kΩ = 0.36 (= 0.36)

The following comparator for carry U provides a 1-signal for all x with −x greater than 0.388 one; for carry U this gives a 1-signal for all x with −x > x₂ = − x. For the switching value Ü:

−x₂ = 1.0 · 10 kΩ / 27.5 kΩ = 0.36

In Bild 2.6.4.1 the values for x, y, z, S and U are again plotted as functions of −x.

[Figure 2.6.4.1 — waveform graph showing digital sum and carry signals]

[Page 33]

2.7. Generators

In Section 2.3 it is shown that with two squarers a sufficient quadratic function can be generated. However, not every desired function can be generated with just any number of diodes. More continuous functions can however be produced with some continuous circuits that in the following are presented as examples for sine, triangle, and square-wave generators.

2.7.1 Square-Wave Generator

In the circuit shown in Section 2.7.1, the non-inverting input is not connected to ground as usual, but rather to the variable potential. For U₁ − U₂ < 0 the potential is positive and so U₀ = U₁/2 is also positive. The capacitor charges slowly through RC (time constant RC = 1 s) to reach U₁. Once U₁ = U₂/2 is reached, it charges the capacitor once again to U₁/2 with a new level. An inverting U₁ with a new level likewise applies, since U₁ is also at a variable potential.

For U₁ − U₂ < 0 and thus U₀ = U₁/2 also positive, the capacitor charges through R/C until U₁/2 is reached. Then U₁ is reversed and U₁ runs again in the opposite direction until it again reaches U₁/2 and so on. For U₀(t₀) one gets a square wave.

[Figure 2.7.1 — square wave generator circuit, Figure 2.7.2 — waveform]

Bild 2.7.1.1 shows the voltage U₁ at the capacitor and the output voltage U₀ of the circuit. As a basic frequency the value is approximately 0.5 Hz.

2.7.2 Square-Wave Generator with Adjustable Frequency

The frequency of the square-wave generator can be changed by a potentiometer. For U₁ < U₂ < 0, the capacitor starts no more oscillations, since then the frequency is too high. For U₁ = 0.5 U₂ the frequency is adjustable; for the table below are given:

Examples:

U₁	f
1.0 U₂	0.5 Hz
0.8 U₂	0.4 Hz
0.7 U₂	0.35 Hz
0.6 U₂	0.3 Hz

2.7.3 Triangle- and Square-Wave Generator with Variable Duty Cycle

The comparator (corresponding to Section 2.4.8) in the middle of the circuit 2.7.3 gives the following output signal:

−0.06, for y + z + w > 0 +1.3, for y + z + w < 0

It also applies:

z = +0.05, for y + z + w > 0 z = −1, for y + z + w < 0

Starting at the time t₀ with y(t₀) = −0.05 w, then at t₀:

z < 0 for y(t) = −(1/(R₁C)) ∫ˢᵗ x dt + y(t₀)

= −(1/(R₁C)) · (t − t₀) + y(t₀)

For R₂ > R₁, y(t) rises until y = 1 − w is reached. Then z becomes positive and y(t):

y(t) = −(1/(R₁G)) ∫ˢᵗ x dt + y(t₀)

= −(1/(R₁C)) · (t − t₀) + y(t₀)

[Page 34]

[Figure 2.7.3 — triangle/square-wave circuit diagram, Figure 2.7.3.1 — waveform graph]

y(t) then falls again until y(t₀) = y(t₀) = −0.05 w — this means w is reached again.

For the half-period t_a: t_a = 1 − x and for the half-period t_b = t₂ − t₁:

t_a = −(y(t₂) − y(t₁)) · (1/(−x)) = RC · 1/(1 − x)

t_b = −(y(t₄) − y(t₃)) · (1/(−x)) = −RC · 1/(1 − x)

Duty ratio t_a : t_b = x/(1 − x)

Period: T = t_a + t_b = RC · 1.05/(x · (1 − x))

Frequency: f = 1/T = x · (1 − x)/(RC · 1.05)

With w one can adjust the duty ratio of the triangle function. Bild 2.7.3.1 shows y as a function of t registered with a XY-recorder. Thus there appear a triangular and a square-wave function with a small rise time and small overshoot at the corners.

For the rest period t_a: t_a = 1 and for t_b = t_b = RC: C = 9.4 µF, R₀ = 0.5 and x = 0.75.

For R₁ = R₂ = 300 kΩ, C = 9.4 µF, h₀ = 0.5 and x = 0.75, the Bild 2.7.3.1 gives the functions x, y and z.

2.7.4 Voltage-Controlled Frequency Oscillator

If one chooses the duty ratio in Section 2.7.3 unequal to 1, then for the frequency:

f = x · (1 − x) · 0.95/(R_1 C)

With an extreme resistance ratio, e.g. R₁/R₂ = 100, approximately:

f ≈ 0.95/(R_1 C)

i.e. the frequency still depends nearly linearly on x. Bild 2.7.4.1 shows the frequency as a function of the input voltage U with x = 10 V corresponding to the frequency controller. The proportionality factor for the adjustment is 1 Hz/V.

By interposing a digital counter for frequency measurement at the output, and a digital instrument for displaying U at the input, one can measure the analog-to-digital relationship (see [16]).

[Figure 2.7.4 — VCO circuit diagram, Figure 2.7.4.1 — frequency vs. voltage graph]

[Page 35]

[Figure 2.7.5 — sine generator circuit, Figure 2.7.6 — three-phase generator circuit]

2.7.5 Sine Generator with One Operational Amplifier

The operational amplifier is connected in accordance with Section 2.1.8 as a differentiator. It holds:

U₀ = −R₁C₁ dU₁/dt

For U₁ − sin ωt it follows: U₁ = sin (ωt + 3π/2). When U₁ is sin, a phase shift of 90° of the operational amplifier arises. The RC-chain produces a frequency-dependent phase shift, which is compensated. The frequency is so large that the total phase shift through the RC-chain equals exactly 2π. The frequency is:

f = 1/(2π R₁C₁)

Bei the open switch S the amplitude of the sinusoidal oscillation is slowly growing. The amplitude limiting is only first achieved through the operational amplifier overload. One obtains a sinusoidal oscillation with an amplitude U₀ = 3.0 U_D (approximately 1.8 V peak at the output).

At a frequency of approximately f = 0.37 Hz, the amplitude grows so large that the RC chain attenuation grows. The total attenuation through the RC chain is so high that the oscillation abruptly stops.

2.7.6 Sine Generator with Amplitude Stabilization

During Section 2.7.5 the sine generator is used only with the operational amplifier from a differentiator: the amplitude is suppressed and furthermore a potentiometer R₁ placed parallel to C₁ compensates U₀, which now consists of the sine-wave oscillation. After closing switch S (of the potentiometer at R₀ at its center position), one gets an amplitude of approximately U₀ ≈ 3 V_pp at a frequency of approximately 0.37 Hz.

Bei so-chosen RC values that the attenuation through the RC chain is very great, the frequency is then decreased by the Spannung (voltage) difference. The total phase shift of the RC chain is compensated. The frequency is approximately 2π.

R₂ > R₁C₂: no longer does the oscillation show a low frequency at high frequencies.

2.7.7 3-Phase Generator

With 3 equal frequency-independent operational amplifiers, three equal frequency-independent outputs are generated. Should the phase shift through the RC chain be exactly compensated, then the 3-phase generator will produce three phase-shifted outputs: if one uses the appropriate amplitude limitation (see Section 2.5.5) with the potentiometer setting from Section 2.7.7, the output then shows three sinusoidal voltages. Using Bild 2.7.7, the three voltages U₁(t), U₂(t) and U₃(t) are available.

[Page 36]

which are mutually shifted by 120°:

U₁(t) = U₀ sin ωt U₂(t) = U₀ sin(ωt + 2π/3) U₃(t) = U₀ sin(ωt + 4π/3)

The sum of the three voltages cannot be exactly zero since they are formed by the summing amplifier (without the capacitor). For the circular frequency:

ω = √13/(RC)

[Figure 2.7.7.1 — three-phase waveform graph]

Bild 2.7.7.1 shows the three voltages U₁(t), U₂(t) and U₃(t) as functions of time.

3. Solution of Mathematical Problems

In mathematics there are many application possibilities for an analog computer. The unbounded integrator is above all capable of solving differential equations (Chapter 3.3), computing roots (Chapter 3.2), and forming sine and cosine (Chapter 3.3). In addition, polynomials (Section 3.3) can be formed as well as addition theorems for sine and cosine.

Besides these examples, in which the normal functions are of interest, a second and much larger group of applications for the analog computer is the simultaneous solution of systems of equations (Chapter 3.4), whose programmability means that the analog computer can also solve differential equations simultaneously and provide the output signals x, y at any time. Both functions are programmed in such a way that the output signal of the second program serves as the input signal of the first, while the output signals x, y are simultaneous at each time instance.

3.1. Linear Systems of Equations

For the solution of a system of n equations with n unknowns, each equation explicitly one after another can be solved by the operational amplifiers with the unknowns summed together. Through the additional summing circuits, the unknowns can be added together and at the output of each operational amplifier one obtains the sum of all unknowns.

In all instances a closed loop is formed, giving one or more operational amplifiers the unknowns. The so-assembled circuit solves the system of equations and gives at the outputs of the operational amplifiers the desired values of the unknowns.

In Section 3.1.2 it is also shown that the gain in the closed loop must be negative, so that the circuit does not become unstable (see Section 3.1.2).

3.1.1 Two Equations with Two Unknowns, Fixed Coefficients

As an example, the following system of equations is given:

2x + y + c₁ = 0 x + By + c₂ = 0

[Figure 3.1.1 — circuit diagram for two-equation solver]

At the left operational amplifier the corresponding upper equation is formed as a sum:

x = −(1/2)(y + c₁)

At the right operational amplifier the corresponding lower equation is formed:

y = −(1/c₂)(x + c₂)

The output values x and y are simultaneous and apply for all c₁ with |x| ≤ 1 and |y| ≤ 1. Mit c₁ = 1 it follows z = 1 and the output applies: mit c₂ = 1.

3.1.2 Unstable Circuit

For the solution of the equations from Section 3.1.1, the following equations are also given:

x = −(8y + c₂)

at the right: y = −(x + c₁)/c₂

This circuit is however unstable, since the gain in the closed loop is greater than 1.

3.1.3 Two Equations with Two Unknowns, Variable Coefficients

As an example, the following system of equations is given:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

The circuit diagram yields:

x = −(1/a₁)(b₁y + c₁)
y = −(1/a₂)(a₂x + c₂)

with: a₁ = 100 kΩ/R₁ > 0, |b₁| ≤ 1, |c₁| ≤ 1, |a₁| ≤ 1, b₂ = 100 kΩ/R₂ > 0 and |c₂| ≤ 1.

Depending on the choice of coefficients, the circuit can be stable or unstable, or one of the operational amplifiers can be driven into saturation. The circuit is stable when: a₁b₂/a₁b₂ < 1.

[page 37: figures 3.1.2 and 3.1.3 — circuit diagrams for the two-equation system]

In figure 3.1.3.1, the solution pairs in the xy plane are drawn as circles at the intersection points of the drawn lines, with:

At intersection points of straight lines with: a₁ = 2, b₁ = 1, b₂ = −0.5, a₂ = −1 c₁ and c₂ variable; at drawn lines the calculated (approximate) values are given.
At intersection points of drawn lines with: c₁ = 2, b₂ = 1, c₁ and c₂ as at the intersection point of the drawn lines; b₁ or a₂ variable, as indicated by dotted lines (selected values).

[page 37: figure 3.1.31 — xy solution-plane plot]

3.1.4 Three Equations with Three Unknowns

As an example, the following system of equations is given:

5x + 0.5y +       z + c₁ = 0
2x +       y + 0.5z + c₂ = 0
  −x +       y +  2z + c₃ = 0

Corresponding to the first equation, the circuit yields:

x = −(1/5)(0.5y + z + c₁)

Corresponding to the second equation, the circuit yields:

y = −(2x + 0.5z + c₂)

Corresponding to the third equation, the circuit yields:

z = −(1/2)(−x + y + c₃)

[page 37: figure 3.1.31 — graphical plot of three-equation system solutions]

3.1.5 Four Equations with Four Unknowns

As an example, the following system of equations is given:

2w +       x + 2y + 0.5u + c₁ = 0
0.5w +       x + 0.5y +  z + c₂ = 0
   4x + 0.5y + z + c₃ = 0
  −w + 0.5x +  y +  2u + c₄ = 0

[page 38: figures 3.1.4 and 3.1.5 — circuit diagrams for four-equation system]

The constants c₁, c₂, c₃ and c₄ cannot be varied arbitrarily, because otherwise one or the other operational amplifier will occasionally be driven into saturation. With c₁ = −0.6, c₂ = −0.3, c₃ = 0.3, c₄ = −0.3 the solution y = 0.3, x = −0.3, y = −0.3 and u = −0.3 is obtained. Bild 3.1.5 shows the variable x at output 4 of one of the operational amplifiers.

[page 38: figure 3.1.5 — oscilloscope trace]

3.2. Elementary Integrals

The solution of indefinite integrals is an important field of application for analog computers. As described in section 2.1.6, one operational amplifier is used:

y(t) = −(1/RC)∫₀ᵗ y₁(t)dt + y(0)

In the following sections, circuits for computing mathematical integrals are given as examples. This allows the result of the analog computer calculation to be easily checked.

An analog computer can only integrate with respect to time as the independent variable (see preliminary note to section 3.2.1). If the interesting range of a variable is to be displayed well, e.g. x = t/a, it appears at the output of an integrator as y(x) = −(a/RC)∫y₁(x)dx + y(0). The factor a is chosen large enough that the time constant RC is large enough that the interesting range of x can be displayed with a TY-recorder or 10 s for registration on a TY-recorder or 10 ms for display on an oscilloscope (see Chapter 6.2).

3.2.1 Solution of the Integral y = ∫x dx

To obtain at the output a non-deterministic integral

y(x) = ∫x dx

one sets x = t/a with a = 10 s (corresponding to the preliminary note in Chapter 3.2.1), and feeds the integrator at the input with the function x = −x zu and computes it:

y(x) = −(1/RC)∫₀ᵗ(−x) dt,   with x = t/(10 s)

To generate the function y₁(x) = −x, the left operational amplifier in circuit plan 3.2.1 is connected with a constant c = 0.1 near the output. Before the integration begins, both switches S₁ and S₂ are closed, and the initial value y(x₀) = 0 with x₀ = 0 is set.

When both switches are opened simultaneously, the result is:

[page 39: figure 3.2.1 and 3.2.11 — circuit diagram and plot]

At the left operational amplifier:

x = −t/(10 s)

and at the right operational amplifier with RC = 10 s:

y(x) = −(1/RC)∫₀ᵗ(−x) dt = ∫x dx = (1/2)x²

Since the time constant at the left operational amplifier is not exactly 1 s and at the right not exactly 10 s, the best result for c = 0.0985 is obtained. Figure 3.2.1.1 shows the solution y(x) for x ≥ 1.

3.2.2 Solution of the Integral y = ∫√x dx

If x = t/a with a = 10 s and the integrator at the input follows the function √x, so one computes:

y(x) = −(1/RC)∫₀ᵗ(−√x) dt = ∫√x dx = (2/3)x^(3/2)

[page 40: figures 3.2.2 and 3.2.3 — circuit diagrams and plots]

At the left operational amplifier:

x = −t/(10 s)

and at the right operational amplifier with RC = 10 s:

y(x) = −(1/RC)∫₀ᵗ(−√x) dt = ∫√x dx = (2/3)x^(3/2)

Since the time constant at the left operational amplifier is not exactly 1 s and at the right not exactly 10 s, the best result for c = 0.094 is obtained. Figure 3.2.2.1 shows the solution y(x) for x ≥ 1.

3.2.3 Solution of the Integral y = −(1/3)∫(1/x) dx

If x = t/a with a = 1 s and the integrator at the input follows the function −1/x, so one computes:

y(x) = −(1/RC)∫₀ᵗ(−(1/x)) dt,   with x = t/(10 s)

Before the integration begins, switches S₁ and S₂ are closed, and the initial value y(x₀) = 0 with x₀ = 1 is chosen. When both switches are opened simultaneously, at the left operational amplifier the result is:

−(1/x) = −(1/10 s)/(t/(10 s)) = −(1/s)/t = −(1/s · t/(10 s)) = −1/(10 s · x)

and at the right operational amplifier with RC = 3 s:

y(x) = −(1/RC)∫(1/x) dt = −(1/3)∫(1/x) dx = −(1/3) ln x

Since the time constant at the left operational amplifier is not exactly 1 s and at the right not exactly 3 s, the best result for c = 0.094 and y = 0.094. Figure 3.2.3.1 shows the solution y(x) for 1 ≤ x ≤ 10.

3.2.4 Solution of the Integral y = ∫√x dx

If x = t/a with a = 10 s and the integrator at the input follows the function −√x, one computes:

y(x) = −(1/RC)∫₀ᵗ(−√x) dt,   with x = t/(10 s)

[page 41: figures 3.2.4 and 3.2.41 — circuit diagram and plot]

Before the integration begins, switches S₁ and S₂ are closed, and the initial value y(x₀) = 0 with x₀ = 0 is chosen. When both switches are opened simultaneously, at the left operational amplifier:

x = −(1/10 s) · t

and at the right operational amplifier with RC = 10 s:

y(x) = −(1/RC)∫(−√x) dt = ∫√x dx = (2/3)x^(3/2)

Since the time constant at the left operational amplifier is not exactly 1 s and at the right not exactly 3 s, the best result for c = 0.063. Figure 3.2.3.1 shows the solution y(x) for x ≥ 1.

3.2.5 Solution of the Integral y = −a∫cos ωt dt

With switch S open, according to section 2.1.6:

y(t) = −(1/RC)∫₀ᵗ cos ωt dt + y(0)
     = −(a/ω) sin ωt + y(0)

with a = 1/RC = 400 s⁻¹. The function cos ωt can be taken from an RC oscillator (687 00). Care must be taken not to exceed the input voltage of the operational amplifier of ±15 V.

[page 41: figure 3.2.5 and 3.2.41 — circuit diagram and plot]

[page 42: figures 3.2.5 and related plots]

On an Ein- oder Zweikanal-Oszilloskop (single- or dual-channel oscilloscope), the trigger is set to the input signal of the operational amplifier. The constant B can also be a consequence of the operational amplifier output — greater amplitudes and lower frequencies can cause the operational amplifier to be driven into saturation.

The constant b(t) can also be a result of the drift of the operational amplifier output. Greater amplitudes and lower frequencies cause greater amounts of drift of the operational amplifier output. The multiplier then produces a greater drift.

3.2.6 Solution of the Integral y = −(1/2)∫(1/x²) dx

If x = t/a with a = 10 s and the integrator at the input follows the function −1/x², one computes:

y(x) = −(1/RC)∫(−(1/x²)) dt,   with x = t/(10 s)

[page 42: figures 3.2.6 and 3.2.61 — circuit diagram and plot]

Before the integration begins, switches S₁ and S₂ are closed, and the initial value y(x₀) = 0 with x₀ = 1 is set. When both switches are opened simultaneously, at the left operational amplifier:

x = −(1/(10 s)) t

and at the right operational amplifier with RC = 5 s (left):

y(x) = −(1/RC)∫(−(1/x²)) dt = (1/5)∫(1/x²) dx = −(1/5)(1/x) + 0.2 = −0.2/x + 0.2

[page 43: figures 3.2.7 and 3.2.8 — circuit diagrams and plots]

3.2.7 Solution of the Integral y = −(1/4)∫(1/x²) dx, x ≥ 1

If x = t/a with a = 10 s and the integrator at the input follows the function −1/√x − 1, one computes:

y(x) = −(1/RC)∫(−(1/(x²))) dt,   with x = t/(10 s)

Before the integration begins, switches S₁ and S₂ are closed, and the initial value y(x₀) = 0 with x₀ = 0 is chosen. When both switches are opened simultaneously, at the first operational amplifier:

x = −(1/(10 s)) t

and at the divider with z = 1/10:

and at the right operational amplifier with RC = 3 s:

y(x) = −(1/RC)∫(−(1/(4x²))) dt = −(1/4)∫(1/x²) dx = (1/4)(1/x) − 0.76

Since the time constant at the first operational amplifier is not exactly 1 s and at the right not exactly 4 s, the best result for c = 0.0347. Figure 3.2.7.1 shows the solution y(x) for x ≥ 1.

3.2.8 Solution of the Integral y = −(1/4)∫(dx/√(4x−1))

If x = t/a with a = 10 s and the integrator at the input follows the function above, one computes:

y(x) = −(1/RC)∫ dt,   with x = t/(10 s)

[page 44: figures 3.2.8, 3.2.9 — circuit diagrams and plots]

and at the right operational amplifier with RC = 4 s:

y(x) = −(1/4)∫ dx = −0.76

Since the time constant at the first operational amplifier is not exactly 1 s and at the right not exactly 4 s, the best result for c = 0.0347. Figure 3.2.8.1 shows the solution y(x).

[page 45: figures and plots continuing integration examples]

Before the integration begins, switches S₁ and S₂ are closed, and the initial value y(x₀) = 0 with x₀ = 0 is chosen. When both switches are opened simultaneously, at the operational amplifier at upper left with c = 1/10:

x = −t/(20 s)

at the divider at middle above (see section 3.3.1):

x² − 1 = (t/(10 s))² − 1

at the right operational amplifier (see section 3.2.9.1):

√(1/(1−x²)) = √(1/(1−(t/(10s))²))

and the operational amplifier at middle below with RC = 10 s:

y(x) = −(1/RC)∫₀ᵗ (−(1/√(1−x²))) dt = ∫(1/√(1−x²)) dx = arcsin x

Since the time constant at the first operational amplifier is not exactly 2 s and at the second not exactly the best result for c = 0.0085, Figure 3.2.9.1 shows the solution y(x) for x < 1.

3.3. Polynomials

Chapter 3.3 is devoted to generating a variable with time — i.e. integrators whose input is supplied with a constant value of one. A polynomial 1 of degree n will in circuit plan 3.2.9 use two cascaded integrators (see section 3.3.1).

To generate a polynomial y(x) of n-th degree with:

y(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ

[page 45: figures 3.3.1 and 3.3.11 — circuit diagrams and plots]

y₂(t) = (1/R₁C₁)(1/R₂C₂) c²t² + (c₂ + y₁(0)) t + y₂(0)

Four amplifiers are needed. The time constants are R₁C₁ = 1 s and R₂C₂ = 1 s. The initial values are y₁(0) = 0 and y₂(0) = 0. This yields for x = t/a with c₁ = 0.94 and y₁(0) = y₂(0) = 0 and for variables c₁ and c₂:

y₁(x) = −1.064 c₁x
y₂(x) = +0.566 c₂x² − 1.064 c₁x

Figure 3.3.1.1 shows y(x) registered with a TY-recorder for the starting values x₁ = 0.035 and various chosen values.

3.3.2 Polynomial 2nd Degree with Integration

Circuit plan 3.3.1 is extended by two more integrators to produce polynomials up to 4th degree. The time constants can be τ = 0.94 s; the constants c₁ through c₄ can assume values between −1 and +1. For four integrators, all initial values are between −1 and +1. The output signals of the four integrators follow:

y₁(t) = −(1/R₁C₁)∫c dt + y₁(0)
y₂(t) = −(1/R₂C₂)∫(y₁(t) + c₂) dt + y₂(0)
       = −(1/2)(1/(R₁C₁R₂C₂)) c₁t² − (c₂ + y₁(0))t + y₂(0)
y₃(t) = −...

[page 46: figures 3.3.2 and 3.3.21 — circuit diagram and plot]

y₄(t) = −(1/R₁R₂C₁C₂) (c₁/2)t² − (1/R₂C₂)(c₂ + y₁(0)) t − (c₃ + y₂(0)) t + y₄(0)

It will be noted that not all constants c₁, c₂, c₃ and c₄ can be freely varied, because through the four-fold integration the last operational amplifier is driven quickly into saturation. With c₁ = −0.9, c₂ = −0.3, c₃ = 0.3, c₄ = −0.3, the initial values y₁(0) = y₂(0) = y₃(0) = y₄(0) = −1 are obtained, which gives for x = t/a with c₁ = 0.94:

y₁(x) = −1.064 c₁x
y₂(x) = +0.566 c₂x² − 1.064 c₁x

In circuit plan 3.3.1, y₁(0) = 0 and y₂(0) = 0 with variables c₁ and c₂.

After the second integrator one obtains:

y₂(t) = (1/(R₁C₁R₂C₂)) (c₁/2)t² + (c₂ + y₁(0)) t + y₂(0)
       = −(1/2)c₁t² − (1/(R₂C₂)) c₂t + y₂(0)

In circuit plan 3.3.1 with a = 0.94 and initial values y₁(0) = 0 and y₂(0) = 0 the result for x = t/a with c₁ = 0.94 is:

y₁(x) = −1.064 c₁x
y₂(x) = +0.566 c₂x² − 1.064 c₁x

This result does not always give the desired progression, particularly not because when the integration constant is quickly reached at four-fold integration it soon arrives at saturation. With c₁ = −0.9, c₂ = −0.220, c₃ = 0.430 and c₄ = 0.300, the function y₄(x) — which is very large for small x — is particularly felt by the constants. After setting with a TY-recorder the recorded starting values after section 3.2.9.1, one can tune the constants through a measuring instrument, but not fine enough with the TY-recorder. With c₁, y₁(t) in Figure 3.3.2.2 approaches −0.92 after 20 s. With c₂, y₁(t) in figure 3.3.2.3 reaches its minimum at about 7 s to the value −0.06. With c₃, y₁(t) in figure 3.3.2.4 reaches its maximum at about 2. Minimum approaches about 9 s to the value −0.56. And with c₄, y₃(t) reaches a minimum at about 9 s in figure 3.3.2 at 2. Minimum at about −0.06.

[page 46: continued — figures 3.3.2 and 3.3.21]

[page 47: figures 3.3.22, 3.3.23, 3.3.24, 3.3.25 — TY-recorder plots of polynomial approximations]

The curves deviate somewhat from the demanded progression; the curves are not sinusoidal but represent approximately the best achievable with the available parameter values.

3.3.3 Polynomial 5th Degree with Multipliers

If x = t/a with a = 10 s and one feeds the integrator at the input with the function −(1/2)x, one obtains at the right operational amplifier:

y(x) = 10x² − 15x³ + 6x⁵

Figure 3.3.3.1 shows this computed with a TY-recorder for c = 0.315/15, q = 0.315.

In circuit plan 3.3.3, −x is generated proportional to an inverted integrator at the upper right operational amplifier, and the output is fed through a potentiometer to generate y/x.

3.4. Simple Examples of Ordinary Differential Equations of 1st Order

One of the most important application areas of analog computers is the solution of differential or integral equations. So in the following sections, technical and physical problems of chapters 4 and 5 of this brochure are also given as examples of differential equations in normal form (solution for the highest derivative).

To solve a general differential equation of n-th order in normal form (resolved for the highest derivative):

d^n y/dx^n = f(x, y, y', ..., d^(n-1)y/dx^(n-1))

from the highest derivative, through single or multiple integration the lower derivatives and y are obtained, each corresponding to the differential equation d^n y/dx^n. By closing the circuit, y and its derivatives are fed back accordingly.

An analog computer can only integrate over time (see preliminary note to Chapter 3.2.1), and from the highest derivative one must therefore supply the function −RC · d^n y/dx^n at the input. Then the output yields −d^(n-1)y/dx^(n-1) · y^(n-1)(0); the input to the next integrator is thus −d^(n-1)y/dx^(n-1), etc. At the output of the last integrator, the result is z = −y(x) with initial value y(0).

To obtain at the output of an integrator the function z = y(x), the function −RC · y’(x)/s must be fed to the input, because from above:

−(a/RC)∫d^n y/dx^n dt + c = −(a/RC)∫(d^(n-1)y/dx^(n-1)) dx / y^(n-1)(0)
                            = −(a/RC) · (d^(n-1)y/dx^(n-1)) | y(0)

To obtain at the output of an integrator the function y(x), the function −RC · y’(x)/s must be fed to the input, since:

y(x) = ∫y'(x)dx + y(0)

[page 48: figure 3.4 — general circuit for differential equations]

3.4.1 Solution of the Differential Equation y’ = y

To allow the computation to run slowly with a TY-recorder, one sets x = t/a with a = 5 s.

The next operational amplifier produces −y(x)/5; S₁ and S₂ are closed. The initial value y(x₀) = 0 with x₀ = 0 can be selected; if switch S₂ is closed and switch S₁ is opened, so:

−(a/RC)∫(−y(x)/5) dx + y(0) = ∫y(x)dx + y(0) = y(x)

The next operational amplifier delivers then −y(x)/5; when S₁ is closed, one obtains:

y(x) = y(0)eˣ

and y(x) is the solution of the differential equation to the given initial value, as long as no operational amplifier is driven into saturation. One gets:

y(x) = y(0)eˣ

[page 49: figure 3.4.11 — TY-recorder plot of y = y(0)eˣ for various initial values]

Figure 3.4.1.1 shows TY-recorder registered solutions y(x) for various given initial values y(0). The time constant is not exactly 1 s but approximately 0.94 s, so the solution is y = y(0)e^(0.94x) and not e^x.

To bring the time constant close to 1 s, the winding resistance of 200 Ω and 2 kΩ should be increased by 200 Ω to 212 Ω.

3.4.2 Solution of the Differential Equation y’ = −ay

For a one should select 0 < a ≤ 1. One sets x = t/a with a = 1 s and leads in the circuit plan from −y/a (potentiometer), so one obtains at the output of the integrator (RC = 1 s):

−(a/RC)∫(−y(x)/a) dx + y(0) = ∫y(x)dx + y(0) = y(x)

[page 49: figure 3.4.2 — circuit diagram]

Before the computation begins, switch S₁ is opened and switch S₂ with x₀ = 0 can be selected; if switch S₂ is closed and S₁ is opened, so:

y(x) = −(1/a)y(x)

and y(x) is the solution of the differential equation to the initial value y(0), as long as no operational amplifier is driven into saturation. One obtains:

y(x) = y(0)e^(−ax)

[page 49: figure 3.4.21 — plot showing y = y(0)e^(−ax)]

Figure 3.4.2.1 shows two solutions y(x) with different initial values y(0). As the solution, one obtains:

y(x) = y(0)e^(ax)

3.4.3 Solution of the Differential Equation y’ = y²

If x = t/a with a = 1 s and begins in the circuit plan left with y(x) (RC = 1 s), one obtains at the output of the integrator:

−(a/RC)∫(−y(x)) dx + y(0) = ∫y(x)dx + y(0) = y(x)

[page 49: figure 3.4.3 — circuit plan]

After the multiplier appears y²; when before the computation begins switch S₁ is opened (initial value y(0) set beforehand) and switch S₂ is closed, so:

−(1/y(x)) · y'(x) = y²(x)
y'(x) = y²(x)

and y is the solution of the differential equation to the given initial value y(0), as long as no operational amplifier and also no divider is overdriven. One finds the solution:

y(x) = 1/(1/y(x₀) − x)

[page 51: figure 3.4.3 — continued circuit and plot]

3.4.4 Solution of the Differential Equation y’ = y^(0.5)

If x = t/a with a = s and one begins in the circuit plan to produce −y’(x)/a with the potentiometer links below (by analog section 2.3.3), one obtains at the output of the integrator (RC = 1 s links):

−(a/RC)∫(−y(x)/a) dx + y(0) = ∫y(x)dx + y(0) = y(x)

With positive y(0) switch S₂ is closed; with negative y(0) switch S₂ (connection plugs) is set instead. The output of the circuit (see section 2.3.3) is −y^(0.5)/a. If switch S₂ is opened (initial value y(0) set beforehand) and S₁ closed, so:

−y'(x)/a · (y(x)^0.5) = y^(0.5)(x)
y'(x) = y^(0.5)(x)

and y is the solution of the differential equation to the given initial value, as long as no operational amplifier and the divider is overdriven. Figure 3.4.4.1 shows with a TY-recorder registered solutions y(x) for various initial values y(0) with a = 3. As the solution one obtains:

y(x) = ((x − x₀)/2 + √y(x₀))²   with x₀ = −3·√(y(0))

y = 0 is also a solution; so the solution of the differential equation can consist of a combination of parabolic arcs and a section of the x-axis.

[page 50: figures 3.4.3 and 3.4.4 — circuit plans and plots]

3.4.5 Solution of the Differential Equation y’ = x/10 − y²

If x = t/a with a = 2 s and begins in the circuit plan left, one obtains at the output of the integrators:

−(a/RC)∫(−y(x)) dx + y(0) = ∫y(x)dx + y(0) = y(x)

After the multiplier, y² appears; to begin the computation, switch S₁ is opened (initial value y(0) = 0 set beforehand) and switch S₂ closed, so:

−(1/10)x + y²(x) = y'(x)

and y is the solution of the differential equation to the initial value y(0), as long as no operational amplifier is driven into saturation. Figure 3.4.5.1 shows with a TY-recorder registered solutions y(x) for various y(0). The parabola y = √(x/10) serves as the locus of the minima of the solutions and as asymptote.

[page 51: figures 3.4.5 and 3.4.6 — circuit diagrams and plots]

3.4.6 Solution of the Differential Equation y’ = y/x

If x = t/a with a = 10 s and begins in the circuit plan at the right below with −y(x), one obtains at the output of the integrators (RC = 10 s) below:

−(a/RC)∫(−y(x)) dx + y(0) = ∫y(x)dx + y(0) = y(x)

The divider delivers −y(x)/x, as long as x < 0 and |x/y| ≤ 1. With the upper operational amplifier, −x is produced, as long as x < 0 and |y(x)| ≤ 1. To begin the computation, switch S₁ is opened and switch S₂ closed (initial values x₀ > 0 and y(x₀) are set beforehand with |y(x)/x| ≤ 1). When switch S₁ is closed and switch S₂ opened, so:

−(1/y) · x = −y'(x)
y'(x) = y/x

and y(x) is the solution of the differential equation for the chosen initial value, as long as no operational amplifier and also no divider is overdriven. One can obtain with this circuit only solutions with y < 0 and |y/x| ≤ 1.

[page 52: figures 3.4.6 and 3.4.7 — circuit diagrams and plots]

With the upper operational amplifier, −x is produced, as long as x < 0 and |x/y| ≤ 1. The divider delivers −y(x)/x, as long as x < 0 and |y(x)| ≤ 1. To begin the computation, switch S₁ is opened (initial values x₀ > 0 and y(x₀) with |y/x| ≤ 1 set beforehand) and switch S₂ closed. So:

−(x/y(x)) = −y'(x)
y'(x) = −y/x

and y(x) is the solution of the differential equation for the chosen initial values, as long as no operational amplifier and no divider is overdriven. One can obtain with this circuit only solutions with x > 0 and |y/x| ≤ 1.

Figure 3.4.6.1 shows with a TY-recorder registered solutions y(x) for various initial values. As solutions one obtains equal-sided hyperbolas:

y(x) = −y₀x² − x₀y₀²

and their asymptotes.

3.4.7 Solution of the Differential Equation y’ = y/x

If x = t/a with a = 10 s and begins in the circuit plan at the right below with −y(x), one obtains at the output of the integrator (RC = 10 s) below:

−(a/RC)∫(−y(x)) dx + y(0) = ∫y(x)dx + y(0) = y(x)

[page 53: continued]

With the upper operational amplifier, −x is generated. The divider delivers −y(x)/x, as long as y < 0 and |x/y| ≤ 1. To begin the computation, switch S₁ is opened (initial value y(x₀) set beforehand) and switch S₂ is closed. So:

−(x/y(x)) = −y'(x)
y'(x) = −y/x

Figure 3.4.7.1 shows with a TY-recorder registered solutions y(x) for various initial values y(0) = 0. As solutions one obtains the rays through the origin:

y(x) = (y₀/x₀) · x

To obtain nice straight lines as solutions, it is important that the input signal to the operational amplifier remains constant, i.e. y changes just as quickly as x. The potentiometer on the left can be used to generate 1/10 approximately.

3.5. Linear Differential Equation 2nd Order

In the following sections solutions of homogeneous linear differential equations 2nd order with constant coefficients will be sought. Solutions with specific initial values or boundary conditions will be obtained. Solutions to simulations of technical and physical problems of sections 4 and 5 are also shown, and likewise solutions to problems with underdamped oscillations, especially chapter 5.6. In chapter 3.7, superpositions of underdamped harmonic oscillations are shown.

3.5.1 General Solution of y” + py’ + qy = 0

With constant p, we seek to solve with the following approach:

y = e^(λx)

Substituting into the differential equation, the characteristic equation follows:

λ² + pλ + q = 0

with solutions for λ:

λ₁,₂ = −(p/2) ± √((p/2)² − q)

Three cases are distinguished based on the roots of the characteristic equation:

Type A: There are two real roots, d.h. p² − 4q > 0. Then the general solution reads:

y(x) = c₁e^(λ₁x) + c₂e^(λ₂x)

Type B: There is exactly one real root, d.h. p² − 4q = 0. The general solution reads:

y(x) = (c₁x + c₂)e^(λx)

Type C: There are two conjugate complex roots, d.h. p² − 4q < 0. Using Euler’s formula e^(jωx) = cos ωx + j sin ωx the general solution reads:

y(x) = e^(−px/2)(c₁cos ωx + c₂sin ωx)

with ω = √(q − p²/4)

All three solution types have two free constants c₁ and c₂ that can take any value; initial values y(0) and y’(0) frequently determine these.

[page 53: figure 3.5.1 — circuit diagram for 2nd order ODE]

3.5.2 Solution for q = 0 and p = 0

As solution, trivially:

y''(x) = 0

and as general solution Type B in section 3.5.1 with λ₁ = 0:

y(x) = c₁x + c₂

The circuit plan for computing y(x) corresponds to section 2.1.7 integration with two different initial values.

3.5.3 Solution for q = 0 and p ≠ 0

The general solution for y’(x) follows the type:

y''(0) + py'(0) = 0

As general solution Type A in section 3.5.1 with λ₁ = 0 and λ₂ = −p:

y(x) = c₁ + c₂e^(−px)

In circuit plan 3.5.3, from y(x) first the first integrator y’(x) and then from it the second integrator follows; the potentiometer then generates −py’(x). The constants c₁ and c₂ can be freely chosen.

Before the computation begins, switch S₁ is opened and switch S₂ closed; the initial value of the integration y(x₀) with x₀ = 0 is set. Switch S₂ is closed and switch S₁ opened, so:

−py'(x) = y''(x)

and y(x) is the solution of the differential equation for the given initial values, as long as no operational amplifier is driven into saturation. With these parameters, initial values y(0) and y’(0) are set.

Before the computation begins, switch S₁ is closed and switch S₂ opened; the initial value of the integration y₂(x₀) with x₀ = 0 is set. Switch S₁ is closed and S₂ opened, so:

−py'(x) = y''(x)

and y(x) is the solution of the differential equation for the given initial values, as long as no operational amplifier is driven into saturation.

3.5.4 Solution for q < 0 and p arbitrary

With q < 0, the case from section 3.5.1 Type A follows:

y''(0) + py'(x) + qy(x) = 0

As general solution Type A in section 3.5.1:

with: λ₁,₂ = −(p/2) ± √((p/2)² − q)

[page 54: figures 3.5.1 and 3.5.1 — circuit diagrams and plots]

In the circuit plan 3.5.4 one obtains from y(x) again −y’(x) and −y₀(x), and the factor −p is applied through the potentiometer. The factor −q acts as the input to the second integrator and also equals 1. The input y(x) − py’(x)/a flows into the first integrator and equals 1. The output of the first integrator is −y’(x).

In the circuit plan 3.5.4, with R = 400 kΩ and a = 5 the integrators give 1, and the input to the integrator at the top is −y’(x).

When at the beginning of the computation switch S₁ and S₂ are opened and the integrators begin from initial values y(0) and y’(0), and y(x) is the solution of the differential equation to the chosen initial values as long as the combination of initial values produces no saturation.

y'(x) = Aₒe^(λ₁x) + Bₒe^(λ₂x) = y(x) with x₀ = −3·√(y₀)

and y is the solution of the Differential equation to the given initial values. For the im figure 3.5.4.1 shown examples, the initial values y(0) and y’(0) are given, and a general-a-function (see sections 3.1 and 3.4.2) is also drawn.

3.5.5 Solution for q > 0 and p = 0

What remains is the differential equation of an undamped harmonic oscillation:

y″(x) + q·y(x) = 0

and as the general solution, following Type C from Section 3.5.1 with ω = √q:

y(x) = c₁·cos ωx + c₂·sin ωx

In the circuit plan, y″(x) is obtained from y(x) by two integrations (with time constant RC = 1 s) and y(x). Through the subsequent inverter one obtains −y(x). A suitably chosen resistor ratio at the inverter produces the required factor q.

When the computation begins, switches S₁ and S₂ are opened (pre-set the initial values y(0) and y′(0) beforehand) and then:

−q·y(x) = y″(x) = y′(x)

and y(x) is the solution of the differential equation for the chosen initial values, while no operational amplifier is overdriven.

With ε = 1 s and RC = 1 s the following output signal results:

−∫y″(x) dx − y′(0) = −y′(x)

If one wishes to compute not with dimensionless variables x but with dimension-bearing variables t, and in doing so the parameters p, q, d, ω are also dimensioned, then the dimension of the time constant of the integrator must be taken into account; c₁ is then no longer dimensionless and likewise the dimension of the RC does not enter in. Many examples from physics in Part 5 of this booklet are treated with the time t as the independent variable.

3.5.6 Solution for q > 0 and p arbitrary

With q > 0, the following can be taken as the solution of the differential equation:

y″(x) + p·y′(x) + q·y(x) = 0

if, according to choice of p and q, all three types from Section 3.5.1 are considered as solutions. In the circuit plan one takes y″(x) in the same way as in Section 3.5.4 — that is, −y″(x) = p·y′(x) + q·y(x); additionally −p·y(x) (an inverter following −y′(x)), y(x), and −p·y(x) are obtained. For R = 100 kΩ, for example, 2 ≤ p ≤ 2.5.

At the start of the computation switches S₁ and S₂ are opened (pre-set initial values y(0) and y′(0) beforehand) and switch S₃ is closed, so that:

−p·y′(x) − q·y(x) = y″(x) = y′(x)

and y(x) is the solution of the differential equation for the chosen initial values, as long as no operational amplifier is overdriven. Solution examples are given in the two following sections.

3.5.7 Damped Harmonic Oscillation

If one chooses in Section 3.5.6 p positive, i.e. p > 0, and q positive, i.e. q > 0, one obtains a damped harmonic oscillation (Bild 3.5.7). In the circuit plan (Bild 3.5.7) and the following Bild 3.5.8 a resistance R can optionally be set at the top. For all solutions of Bild 3.5.7 and Bild 3.5.8, initial values y(0) = 0 and y′(0) = 0.5 apply.

[page 56: figure only — circuit diagrams and oscilloscope traces for damped and anti-damped harmonic oscillations (Bild 3.5.7 and 3.5.8)]

3.6. Non-linear Differential Equation 2nd Order

Two homogeneous non-linear differential equations of 2nd order will be programmed and solved on the analog computer. In Section 3.6.1 the well-known van-der-Pol oscillator is programmed and in Section 3.6.2 a resonance system with non-linear restoring force is treated.

3.6.1 Solution of y″ + p(y² − 1)y′ + y = 0, with p > 0

The differential equation known under the name “van der Pol” has a stable periodic solution with a single initial condition after a boundary condition with any initial value; one speaks of self-excitation or self-regulation. To obtain this stable periodic solution, one must pass through an amplitude oscillation (see Section 1.3.2).

The scaling factors for y′(x), y(x) and y(x) are chosen all equal to 1/5 and the range for the potentiometer setting p/5 equals 1.

−1 ≤ y(x)/5 ≤ +1
−1 ≤ y′(x)/5 ≤ +1
−1 ≤ p/2 ≤ +1

In the circuit plan (Bild 3.6.1), from y″(x)/5, first the term y(x)/5 then y′(x)/5 are obtained; over the subsequent voltage-divider and the inverter −y(x) and y(x) follow; through the lower multiplier finally −y(x)². The factor q = 1 is chosen; the scale factor is 1/5 for the Rechner range (computer range) −1 ≤ p/2 ≤ +1.

In the circuit plan (Bild 3.6.1), from y″(x)/5 via the first integrator −y′(x)/5 is obtained, from which via the voltage-divider and the inverter −y(x) and y(x) follow; via the lower multiplier finally −y(x)². The factor q = 1 is chosen.

At the start of computation switches S₁ and S₂ are opened (pre-set initial values y(0) = 0 and y′(0) = 0 beforehand) and switch S₃ is closed, so that:

−p(y²−1)·y′(x) − y(x) = y″(x) = y′(x)

and y(x) is the solution of the differential equation for the chosen initial values, as long as no operational amplifier is overdriven.

Bild 3.6.1.1 and Bild 3.6.1.2 show oscillation progressions of y(x) in the approach to the self-excited stable solution for p = 2 at two different initial conditions. The Oscillation amplitude for y(x) is approximately 2, however, at large excursions damped oscillations occur and at the formation of y(x)² (Bild 3.6.2) the danger of overflow at the operational amplifier exists. The oscillation period T depends on the amplitude y_max (as in Bild 3.6.2.1).

3.6.2 Solution of y″ + p·y′ + y(x) − y³ = 0

In the circuit plan one obtains from y″(x) via the first integrator −y′(x)/5, from which via the voltage-divider and the inverter −y(x)/5 and y(x)/5 follow; via the lower multiplier finally −y(x)³/5.

At the start of computation switches S₁ and S₂ are opened (initial values y(0) = 0 and y′(0) = 0.11 are pre-set) and switch S₃ is closed, so that:

−p·y′(x) − y(x) + y³ = y″(x) = y′(x)

and y(x) is the solution of the differential equation for the chosen initial values, as long as no operational amplifier is overdriven.

Bild 3.6.2.2 shows the solution of this differential equation for p = 0.1 and for the initial values y(0) = 1, y′(0) = 0. Ein Einfluss der Störgröße auf die Schwingungsdauer is clearly visible. Switching out connections V delays the computation; one thus obtains as comparison the solution of the linear differential equation for equal initial values and equal damping.

3.7. Addition Theorems for Sine and Cosine

The functions sin x and cos x are obtained on the analog computer from the differential equations of Section 3.5, in which an (almost) undamped harmonic oscillation with period X = 2π/q arises:

y(x) = c₁·cos √q·x + c₂·sin √q·x

With the choice of initial values y(0) = 1 and y′(0) = 0 with q = 1 the solution is:

y(x) = cos x y′(x) = −sin x −y′(x) = sin x y(x) = −cos x

These functions can be further processed. Only in Sections 3.7.1 and 3.7.8 are the program plans explicitly stated; the corresponding Section 3.7.1 can be easily converted for all images in Sections 2.1.2, 2.1.5, and 2.1.7 (the symbols given at right in those images for switched operational amplifiers are used).

For computing the solutions for a longer time period, the circuits of this chapter are unsuitable, because the slowly growing damping which is observed in Section 2.1.7 becomes noticeable after a long observation time.

3.7.1 Proof of sin²x + cos²x = 1

Bild 3.7.1 gives the detailed program plan and Bild 3.7.1.1 shows the fundamental circuit for solving the problem. In the program plan one obtains y″(x) = −y(x) via the first integrator: −y(x) = cos x; via the inverter −y(x) = −cos x.

At the start of computation switches S₁ and S₂ are opened (pre-set initial values y(0) = 1 and y′(0) = 0) and the following is obtained:

−(cos²x + sin²x) = −1

and y(x) is the solution of the differential equation for the chosen initial values, as long as no operational amplifier is overdriven.

3.7.2 Generation of cos 2x from cos²x − sin²x

[page 60: figure only — circuit diagram (Bild 3.7.2) showing generation of cos 2x from cos²x − sin²x using multipliers and summing amplifiers]

3.7.3 Generation of sin 2x from 2 sin x · cos x

[page 60: figure only — circuit diagram (Bild 3.7.3) showing generation of sin 2x from 2·sin x·cos x]

When observing this sum with a measuring instrument or the XY-recorder, one notices a slight superposition of the value = 1 with an oscillation of double frequency (< 2%), caused by phase and amplitude errors of the generated functions sin x and cos x.

3.7.4 Generation of cos 3x from 4 cos³x − 3 cos x

[page 61: figure only — circuit diagram (Bild 3.7.4)]

3.7.5 Generation of sin 3x from 3 sin x − 4 sin³x

[page 61: figure only — circuit diagram (Bild 3.7.5)]

3.7.6 Generation of cos(nπ/2) from (1 + cos nπ)/2

[page 61: figure only — circuit diagram (Bild 3.7.6)]

3.7.7 Generation of sin(nπ/2) from (1 − cos nπ)/2

[page 61: figure only — circuit diagram (Bild 3.7.7)]

3.7.8 Generation of cos kt + sin kt; Oscillation

With y(0) = −k at the input of the total integrator and the time constant RC = 1/k, the output gives:

k²y(t) = −k·(1/k)·cos kt = cos kt

Bild 3.7.8 shows two generators for k₁t and cos k₂t with different frequencies k₁ and k₂ (f₀ = 0). It is additionally required to set the Schaltelemente (switching elements, 578.03 and 576.07) needed for the computation; the symbols shown at the right are available for switched operational amplifiers.

The frequencies of the two generators are not exactly equal; they hit different circular frequencies k₁ and k₂ (f₀ = 0). Between the two switching nodes there lie just under 70 oscillations; the slowly accumulating damping from the long observation time becomes clearly visible.

When both circular frequencies are made clearly unequal, for example k₁ = 1.1 k, k₂ = 20 kΩ, the Periodendauer of each is clearly seen; between each pair of oscillation nodes one can recognize the Schwebungen (beats) (Bild 3.7.8.2).

3.8. Fourier Analysis

A periodic function f(t) with period T, for which:

∫₀ᵀ f²(t) dt < ∞

can be represented by a Fourier series, such that the mean square deviation is zero:

f(t) = a₀ + Σ aₙ·sin nωt + Σ bₙ·cos nωt

with ω = 2π/T and the Fourier coefficients:

a₀ = (1/T)·∫₀ᵀ f(t) dt

aₙ = (2/T)·∫₀ᵀ f(t)·sin nωt dt

bₙ = (2/T)·∫₀ᵀ f(t)·cos nωt dt

These Fourier coefficients can, with certain technical prerequisites, be computed well on the analog computer. The period duration T must be large enough that the mean deviation over T is sufficiently small. (If operated manually, the switch must also be T > 5 s.) For the other mode, the generator for sin nωt and cos nωt must supply the periodic function at phase-locked exact multiples of the fundamental frequency 1/T.

Bild 3.8.0 shows as an example the principle circuit for computing the Fourier coefficients aₙ. The integrator is reset to zero at the start of each period (switch S is opened); at the end of time T the integrator output shows:

−aₙ = −(2/T)·∫₀ᵀ f(t)·sin nωt dt

As a periodic function, one can use a square or sawtooth wave. Setting the necessary phase adjustments is not easy. In the volume (Kapitel 3) of the booklet (522.54) a more complete description of the computation of Fourier coefficients with harmonic generators and the programmable switch (576.07) for exact Vorgabe (preset) of the necessary switching operations is given.

4. Control Technology

Control engineering is the science of automatically reaching and maintaining a desired state by measurement of the deviation of the controlled variable from the desired value and taking the necessary corrective action. A portion of a control loop (Bild 4.0.0): a controlled system (Regelstrecke), for example a heating system, turbine, automobile, whose output variable (controlled variable, Regelgröße) x is set by the manipulated variable (Stellgröße), the desired value (Sollwert) for example 22 °C; in addition there is a controller (Regler), whose Ausgangsgröße (output) is the manipulated variable w. From outside, a disturbance z (e.g. of the temperature when an outside window is opened) can influence the controlled variable x and thus the Regelabweichung (control deviation).

The controlled variable x is measured with a measurement element and together with the reference variable w forms the control deviation e = w − x, which the controller converts into the corrective manipulated variable. The controlled variable x is thus influenced and the control deviation is regulated toward zero.

One distinguishes various types of controlled systems and controllers. As a distinguishing criterion, the step response (Übergangsverhalten) serves: the reaction to a step change in the input variable.

The controlled variable x is measured with a measuring element and simultaneously with the controller’s reference variable w forms the Regeldifferenz (control error) e = w − x, which the controller converts into a corresponding controller output signal. The controlled variable x is thus gradually changed and the control deviation reduced toward zero.

4.1. Controlled Systems with Equalizing Action (Ausgleich)

Controlled systems with equalizing action are those subject to any kind of disturbance; they are, in general terms, statically stable systems that are not self-oscillating. The controlled variable x changes proportionally to the input but is subject to its static characteristics once a new constant state is reached. A simple example of a controlled system with equalizing action is an integrator: given a constant input, the output grows continuously. Bei Betrachtung der Regelstrecke (when considering the controlled system) one generally includes a Regelstrecke with equalizing action where at the maximum value of the controlled variable x it reaches a maximum value proportional to the step change y. Given a constant input step, the output reaches a constant final value.

When there is a step change in the manipulated variable y, also the controlled variable x changes, following an approach to a new constant value. At a constant step input, the controlled variable x first experiences a transient phase and then a proportional change, and the controlled variable x stabilizes at a new equilibrium value. If there is another Störgröße (disturbance variable) z, the controlled variable x also changes correspondingly. These Störgrößen are everywhere shown in the examples of this chapter as input of the operational amplifier before the last controlled system element (also in measuring element, controller, manipulated variable).

They can also affect other points of the controlled system, for example the Regelstrecke is not always proportional to the output x. These Störgrößen are shown in the examples at the input of the operational amplifier before the last controlled system element.

4.1.1 Controlled System with Equalizing Action and Delay 1st Order

In the circuit plan (Bild 4.1.1) the controlled system is driven by the integrator with the sum z + y(t) − x(t); the output of the left operational amplifier with time constant T_v gives the output signal − x(t). The controlled variable x(t) step input response (switch S opened; initial values x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0) gives the step response as the Übergangsfunction (step response function) (Bild 4.1.1.1):

x(t) = 1 − e^(−t/T_v)

For the output signal − x(t) of the right integrator (with time constant T_v), correspondingly:

x(t) = −(1/T_v)·∫(−v(t) + x(t)) dt + x(0)

and:

T_v·ẋ(t) + x(t) = v(t)

Differentiating once and multiplying by T_v:

T_v·ẋ(t) = v(t) − x(t)

This is the first-order differential equation. With T_v = T_v the controlled variable x(t) follows the following Differentialgleichung (differential equation) of the 1st Order:

T_v²·ẋ(t) + (T₁ + T₂)·ẋ(t) + ẋ(t) = v(t)

If the manipulated variable v(t) is equal to a step function (switch S closed at the start of computation with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0), one obtains as the step response (Bild 4.1.1.1):

x(t) = 1 − e^(−t/T_v)

4.1.2 Controlled System with Equalizing Action and Delay 2nd Order

Coupling two controlled systems with equalizing action and delay 1st Order in series (Bild 4.1.2), the output signal of the left integrator with time constant T₁ gives correspondingly Section 4.1.1:

−v(t) = −(1/T₁)·∫(v(t) − v(t)) dt − v(0)

For the output signal − x(t) of the right integrator (with time constant T₂) correspondingly:

T_v·ẋ(t) = v(t) − x(t)

Differentiating once and multiplying by T_v, one obtains:

T_v·ẋ(t) = −(T₁²·ẋ(t) + T₁·ẋ(t) + T_v·ẋ(t) + x(t))

= v(t) − (T₁²·ẋ(t) + T₁·ẋ(t) + T_v·ẋ(t) + x(t))

Substituting in the upper equation gives for x(t) the differential equation 2nd Order:

T₁·T₂·ẍ(t) + (T₁ + T₂)·ẋ(t) + x(t) = v(t)

If the manipulated variable v(t) equals a step function (switch S closed at the start of computation with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0), one obtains as the step response function (Bild 4.1.2.1):

x(t) = 1 − (1 + t/T₁)·e^(−t/T₁) when T₁ = T₂ = T

4.1.3 Controlled System with Equalizing Action and Delay 2nd Order with an Operational Amplifier

In circuit plan 4.1.3, if the manipulated variable v(t) is equal to a step function (switch S closed at the start of computation with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0), one obtains as the step response function (Bild 4.1.3.1):

x(t) = 1 − (1 + t/T₁)·e^(−t/T₁)

with T_v = 0.5 s (more precisely 0.47 s). It is strictly valid only if the above-given step response function is not in the feedback; it is achieved through 1.17 (576.04) when the capacitor of 1 μF is in the feedback.

For practical reasons, controlled systems with equalizing action and delay 1st and 2nd Order (Abschnitte 4.1.9 en 4.1.2) will be given in Bild 4.1.3 using one or two of the corresponding operational amplifiers from Section 4.1.9 to simulate a delay of higher order.

4.1.4 Controlled System with Equalizing Action and Delay 3rd Order

Coupling one controlled system 1st Order (corresponding to Bild 4.1.1) with a controlled system 2nd Order (corresponding to Bild 4.1.3) in series (Bild 4.1.4), the output of the left operational amplifier corresponds to Section 4.1.5:

T_v(t) = v(t) − v(t)

For the output signal − x(t) of the right operational amplifier:

T₁²·ẍ(t) + 2T₁·ẋ(t) + T_v·ẋ(t) + x(t) = v(t)

Differentiating once and multiplying by T_v, one obtains:

T_v·T₁²·ẋ(t) + T_v·2T₁·ẋ(t) + T_v·T_v·ẋ(t) + T_v·ẋ(t) + x(t)

= v(t) − (T₁²·ẋ(t) + 2T₁·ẋ(t) + T_v·ẋ(t) + ẋ(t))

Substituting into the topmost equation gives for x(t) the following Differentialgleichung (differential equation) 3rd Order:

T₁³·ẋ(t) + 3T₁·ẋ(t) + 3T_v·ẋ(t) + x(t) = v(t)

x(t) = 1 − (1 + t/T₁ + t²/2T₁²)·e^(−t/T₁)

Differential equation 3rd Order:

T_v³·ẋ(t) + 3T_v²·ẋ(t) + 3T_v·ẋ(t) + x(t) = v(t)

If the manipulated variable v(t) is equal to a step function (switch S at the start of computation is closed) with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0, one obtains as the step response function the Kurve in Bild 4.1.4.1:

x(t) = 1 − (1 + t/T₁ + (1/2)·(t/T₁)²)·e^(−t/T₁)

4.1.5 Controlled System with Equalizing Action and Delay 4th Order

Coupling one controlled system 1st Order (left) with a controlled system 3rd Order (middle and right) in series (corresponding to circuit plan 4.1.5), the following differential equation results for the output signal − v(t) of the left integrators:

T_v(t) = v(t) − v(t)

And for the 4th Order:

T_v⁴·ẋ(t) + 4T_v³·ẋ(t) + … = v(t) [further terms omitted]

[page 65: circuit diagrams (Bild 4.1.5 and step response traces 4.1.5.1 and 4.1.5.2) showing the 4th-order controlled system step responses for T_v = 0.5 s]

The ratio T_v/T_u and with it the Regelbarkeit (controllability) of a controlled system depends on the order of the controlled system alone. It also depends on the relationship between the time constants of individual system elements. The larger the Verzögerungszeit (dead time) relative to the Ausgleichszeit (equalization time), the harder the system is to control. The Ausgleichszeit T_g gives a measure of the time interval; this is demonstrated for controlled systems with equalizing action and delay 1st to 7th Order at the same time constants in Bild 4.1.6.2. The values for T_u/T_g for controlled systems with equalizing action and delay 1st to 7th Order are given in the next section (see also Bild 4.1.9).

4.1.6 Controlled System 4th Order with Unequal Time Constants

In Bild 4.1.6 the controlled system from Section 4.1.6 is represented again with T₁ = 3T₂. Bild 4.1.6.1 shows the values from Section 4.1.6 with im Abschnitt 4.1.1 used T₁ = T₂ in comparison. As an approximation, the characteristic values n = 37 are inserted in Bild 4.1.6.1. Zum Zustand (at the state) T₁ = 3T₂ the Verzugszeit (delay time) T_u becomes larger and the Ausgleichszeit T_g becomes smaller.

The ratio T_u/T_g and with it the controllability of a controlled system is thus not solely dependent on the order of the system. It also depends on the relationship between the time constants of the individual section elements. The smaller the time constants relative to the delay (dead time), the easier a control with lower-order controlled systems with delay (due to 2nd to 7th Order controlled systems with equal time constants) can be achieved; see Bild 4.1.6.2 in which the values for T_u/T_g for controlled systems with equalization and delay 2 to 7 are entered.

4.1.7 Controlled System 4th Order with Disturbance Variable

Disturbances whose effect on the controlled variable x is proportional to the Störgröße (disturbance variable) propagate through the controlled system as shown in the previous Abschnitte and their resulting step responses at the control loop’s output are shown in Bild 4.1. Disturbances can also act at other points of the controlled system (for example through open windows). These appear in Bild 4.1 at the input of the operational amplifier before the last controlled system element (also at measuring element, controller, manipulated variable). The disturbance affects all subsequent Teilstrecken. In these cases, the relationship can be determined in the same manner.

The last Operationsverstärker (operational amplifier) of the preceding sections is coupled in front of the last controlled system element in Section 4.1, with an equalizing and delay 1st Order (Bild 4.1.7). This is also typical for complete controlled systems from Chapter 4.3 as described in Section 4.3 of the Vorbetrachtung (preliminary consideration).

Bild 4.1.7.1 shows the progression of the controlled variable x(z). Initially with switch S, at t = 0 the manipulated variable v is set to 0.5. The controlled variable reaches a new equilibrium after each of the switching Bild 3.5.5 and 3.5.6 shown and has been operating in the corresponding Abschnitt 4.1.7. Then the constant manipulated variable v is switched off. Disturbance z increases to 0.2 and z = − 0.2 after some time. After one short time the previous constant manipulated variable v is switched in again. The controlled variable x then reaches its new equilibrium after 0.5 Sekunden ihren Gleichgewichtswert (seconds its equilibrium value). Nach Zuschaltung (after connecting) the new Störgröße (disturbance) z = 0.2, the controlled variable x is initially decreased somewhat after a certain time, and the old equilibrium value of x is reestablished. Disturbance z is then switched off and the controlled variable x returns once more to the old equilibrium.

In the examples described in Sections 4.3.1 through 4.3.5, the hier beschriebene Regelstrecke through Führungsgröße (reference variable) (Sollwert) and appropriate configured Regler (controller) to form complete Regelkreise (control loops), which are designed to reduce the Regelstrecken’s response to disturbances.

4.1.8 Controlled System with Equalizing Action and Delay 5th Order

Coupling one controlled system 1st Order (left) with a controlled system 4th Order (middle and right corresponding to circuit plan 4.1.8), one obtains for the manipulated variable’s output signal − x(t) of the left integrators correspondingly Section 4.1.5:

T_v⁵·ẋ(t) + 5T_v⁴·ẋ(t) + 10T_v³·ẋ(t) + 10T_v²·ẋ(t) + 5T_v·ẋ(t) + x(t) = v(t)

4.1.9 Controlled System with Equalizing Action and Delay 6th Order

In circuit plan 4.1.9 the following Differentialgleichung (differential equation) for the controlled variable x(t) follows by analogy with Section 4.1.8:

T_v⁶·ẋ(t) + 6T_v⁵·ẋ(t) + 15T_v⁴·ẋ(t) + 20T_v³·ẋ(t) + 15T_v²·ẋ(t) + 6T_v·ẋ(t) + x(t) = v(t)

For the output signal − x(t) of the right operational amplifier with T_v = T_v, according to Section 4.1.8b:

T_v²·ẋ(t) + 4T_v·ẋ(t) + 6T_v²·ẋ(t) + 4T_v·ẋ(t) + x(t) = v(t)

Differentiating once and multiplying by T_v, so that:

T_v·T_v²·ẋ(t) + 4T_v·T_v²·ẋ(t) + 6T_v²·ẋ(t) + 4T_v·ẋ(t) + T_v·ẋ(t) + x(t)

= v(t) − (T_v²·ẋ(t) + 4T_v²·ẋ(t) + 6T_v·ẋ(t) + 4T_v·ẋ(t) + x(t))

Substituting into the uppermost equation gives for x(t) the Differentialgleichung (differential equation) 5th Order:

T_v⁵·ẋ(t) + 6T_v⁴·ẋ(t) + 15T_v³·ẋ(t) + 15T_v²·ẋ(t) + 6T_v·ẋ(t) + x(t) = v(t)

4.2. Controlled Systems Without Equalizing Action (ohne Ausgleich)

In control engineering, there are systems without equalizing action: an integrator is a proportional element that produces a continuously rising change of the controlled variable x for a constant change in the manipulated variable y. A simple controlled system without equalizing action is the integrator: given a constant input, the controlled variable x increases continuously without reaching a steady state.

4.2.1 Delay-free Controlled System Without Equalizing Action

The behavior of the integrator is described in Section 2.1.6. When switch S₂ at the start of computation is closed with x(0) = 0 (switch S₂ closed when switch S₁ is opened), the step response function is the solution (Bild 4.2.1.1):

x(t) = t/T_v

The equation for the integrating element without equalizing action (Ausgleich) follows from combining the integrator’s delay-free integrating action with time constant T₁:

T_v(t)·x(t) = (1/T₁)·∫v(t) dt + x(0)

For the output signal − x(t) of the right operational amplifier gives in correspondence with Section 4.2.1:

x(t) = (1/T₁)·∫v(t) dt + x(0)

One forms T_v·ẋ(t) + ẋ(t) = v(t), so that for x(t) a Differentialgleichung 1st Order follows:

T_v·ẋ(t) + x(t) = v(t)

It means T_v·ẋ(t) = v(t) − x(t).

If the manipulated variable v(t) is equal to a step function with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0 (switch S₂ is initially open; when simultaneously switch S₁ is closed), the step response function follows from the solution (Bild 4.2.2.1):

x(t) = T_v/T₁·(1 − e^(−t/T_v))

4.2.2 Controlled System Without Equalizing Action, with Delay 1st Order

Coupling one controlled system without equalization and one with delay 1st Order in series (Bild 4.2.2), the output signal of the left integrator correspondingly follows Section 4.1.1:

T_v(t) = v(t) − v(t)

For the output signal − x(t) of the right integrator, gives:

x(t) = (1/T₁)·∫v(t) dt + x(0)

One forms T_v·ẋ(t) = v(t) − x(t) + (1/T₁)·∫(v(t) − T_v·v(t)) dt + v(0).

If the manipulated variable v(t) is equal to a step function with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0, one obtains as the step response function (Bild 4.2.2.1 and Bild 4.2.3.2):

x(t) = T_v/T₁·(1 − e^(−t/T_v))

4.2.3 Controlled System Without Equalizing Action, with Delay 2nd Order

Coupling one controlled system without equalizing action (middle) with a controlled system with delay 1st Order (right) in series (Bild 4.2.3), for the output signal of the middle integrator (correspondingly Section 4.2.3) gives:

T_v(t) = v(t) − v(t)

and the step response function gives:

x(t) = T_v/T₁·(1 − e^(−t/T_v)) + v(0)/T₁

The integrating behavior gives for the controlled variable x(t) a Differentialgleichung 2nd Order:

T_v·ẋ(t) + ẋ(t) = (1/T₁)·∫(v(t) dt + v(0))

Differentiating once and multiplying by T_v:

T_v·T_v·ẍ(t) + T_v·ẋ(t) = v(t) − x(t)

If the manipulated variable v(t) is equal to a step function (switch S₁ is opened and simultaneously switch S₂ is closed) with x(0) = 0 for t ≤ 0 and x(t) = 1 for t > 0, one obtains as the step response function (Bild 4.2.3.1):

x(t) = (t/T₁) − (T_v/T₁)·(1 − e^(−t/T_v))

4.3. Controller

The controller of a control loop compares the measured controlled variable x with a reference variable (Führungsgröße) w and converts these in a suitable manner into the manipulated variable y that controls the controlled system. The two-point controller of Section 4.3.1 is an unstable controller; it knows two switching states, each of which is associated with one boundary control deviation. The regulation performance of Sections 4.3.2 to 4.3.6 is observed at a controlled system with equalization and delay 1st Order. As controlled system, the controlled system described in Abschnitt 4.1.7 is used, and in the examples of Sections 4.3.1 to 4.3.5 a Regelstrecke mit Ausgleich (system with equalizing action) und Verzögerung 1. Ordnung (1st Order delay) is used, correspondingly described in Sections 4.1.7 and 4.2.2.

The controller is described in the following Abschnitte for each case as the Regelstrecke with equalizing action and delay not less than 2nd Order correspondingly Abschnitt 4.2.3.

4.3.1 Control Loop with Two-Point Controller

In circuit plan 4.3.1 the three operational amplifiers form the controlled system without equalization and delay 1st Order (corresponding to Bild 4.1.1). The manipulated variable y is at the right operational amplifier output and is compared with the reference variable w; the control deviation x = w − y is obtained from this. This signal is fed to a further amplifier (bottom) that multiplies by factor c = −10 kΩ/R and further −1, and the additional manipulated variable y₁ is added to it. The whole signal w is provided to the next amplifier (bottom) that also adds the additional manipulated variable y₁.

y₁ + y₂ = y₁ + 10 kΩ/R when x > w > 0 y₁ + y₂ = y₁ + 10 kΩ/R when x is smaller than w

This is to be achieved with the circuit plan 4.3.1: the Regelgröße x is desired to reach the value x = 0.5. The manipulated variable is set slightly below y₁ = 0.45. The reference variable (Führungsgröße) x is set by w = 0.5, and the manipulated variable y ≈ w = 0.45 m. When it remains negative, that is x < 0, y₁ is positive and the controlled variable x will increase through a minimum and then become negative. Then the process starts fresh. The controlled variable x oscillates periodically between minimum and maximum and rises again.

This periodic changing manipulated variable must be small enough, i.e. its period duration should not be too long; however the period must be large enough that it can be observed well with the computer (Bild 4.3.1.1).

Bild 4.3.1.1 shows the controlled variable x(t) for the control loop as parameter R. The resistor R = 1/y₁ is set at Beginn (start) of the computation; the initial values y(0) and z(0) are preset. For time t = 10 s the reference variable (Führungsgröße) w = 0.45 is set (switch S₁). The controlled variable x is then regulated with the controller set at y₁ = 0.45 m. For time t = 40 s, switch S₂ is closed; the controller then starts to work. For time t = 70 s the manipulated variable w is switched off (switch S₃); the Regler (controller) tries to regulate the controlled variable x back to its Optimum through finding it somewhere between the two extreme values of the controlled variable. The controller then begins the process fresh from its Optimum and rises anew.

Bild 4.3.1.2 shows the control loop with 2-Punkt-Regler (two-point controller) under the influence of the Störgröße (disturbance variable) z. For this, at time t = 10 s the Führungsgröße (reference variable) w is switched in and the disturbance z₁ = −0.2 and then z₁ = 0.2 is enabled. In Bild 4.3.1.2, the constant Regelgröße y₁ is switched back in again. The Regler (controller) tries to reach the Optimum somewhere between the extreme values. The achieved Grenzwerte (limit values) of R depend on the exact value of a (output of a clipped operational amplifier).

4.3.2 Involuntary Switching of a Two-Point Controller

To avoid the undesirable periodically switching Regelabweichung (control deviation) (see Section 4.3.1) that appears with a two-point controller, a type of controller from this group must be found that produces a proportional change of the controlled variable x. A single change in the manipulated variable y also produces the proportional change. In the circuit plan 4.3.2, the manipulated variable of the controlled system is at the right operational amplifier and is multiplied with factor c = −10 kΩ/R and −1, and the additional constant manipulated variable y₁ is added.

In the circuit plan 4.3.2, the manipulated variable of the controlled system is at the right operational amplifier output and is multiplied with a Konstante (constant) y₁ by the factor c = −10 kΩ/R. The controlled variable d is determined from:

d = 0, for x − w > 0 d = −1, for x − w < 0

In the following examples (Bild 4.3.2.1 through Bild 4.3.2.3), the initial values y₁ = 0.5 for the controlled variable (Regelgröße) are set to be achievable. First the controlled variable y₁ is switched off (switch S₁). For time t = 10 s the Führungsgröße (reference variable) w = 0.5 is enabled (switch S₁), so that x is regulated:

x = y₁/(1 + c)

When w is switched to w = 0.5 (switch S₂), the controlled variable starts down toward a new Einschwingungs-Vorgang (settling process):

x = y₁ + cw/(1 + c)

With y₁ = y₂ this means that x is equal to the value: x = w. When w is brought to the constant manipulated variable y₁ once again (switch S₁), the control deviation x is gradually eliminated:

x = cw/(1 + c)

[page 73: figure only — circuit diagram and oscilloscope traces for control-loop experiment with P-controller, showing step response for various resistor values]

Fig. 4.3.2 shows the influence of the disturbance variable z on the control loop from Fig. 4.3.2.1 for various values of R and different fixed setpoints y₁. Parameters are:

c = 2.5 (for R = 4 kΩ), c = 1.4 (for R = 7 kΩ)
c = 0.6 (for R = 12 kΩ), c = 0.5 (for R = 20 kΩ)

Fig. 4.3.2.2 shows the influence of the disturbance variable z on a control loop with a P-controller but without a fixed setpoint y₁. The parameters are the same as before; the magnitude of disturbances is increased via the circuit. Just as the periodically oscillating steady-state offset is a characteristic property of the two-position (bang-bang) controller, the steady-state offset is likewise characteristic of the P-controller. In a control loop with a setpoint track and an output, this steady-state offset changes as the setpoint track is traversed, and the steady-state offset is zero only when the setpoint equals zero. The simplest way to eliminate such offset is an integrating controller; as soon as its input signal is driven to zero, it has become a sign-changing element.

4.3.3 Control Loop with I-Controller and PI-Controller

In the control loop of Schematic 4.3.3, not only a P-controller (as in Section 4.3.2) but also an I-controller — implemented as a timed integrating operational amplifier — is connected in parallel with the P-controller. It soll, as required of a controller, bring the steady-state offset to zero.

To examine the effect of the I-controller alone on the control loop, the P-controller is switched off (switch S₁ open). The time constant T₁ of the I-controller and the tracking variable y = x are both maintained; y = −x (tracking with sign reversal). If the magnitude of the disturbance is not too large (< 1 s), a steady-state adjustment settles after some time. Should the disturbance be large (> 1 s), the I-controller cannot follow it — here a finite setpoint w and a guidance variable y₁ are used, which are not used in some examples of Sections 4.3.1 and 4.3.2.

[page 74]

A permanent steady-state offset x_w is obtained with:

$$x_w = \frac{y_1 - w + z}{1 + c}$$

For y₁ = w, it follows that x_w = z/(1 + c). To make this permanent offset very small, one attempts to make c very large; however, c being very large simultaneously makes the transient very short and causes almost undamped oscillations. For R = 2.5 kΩ (two resistors of about 5 kΩ in parallel), already c ≈ 6 and the response shows almost undamped oscillations (see figure).

Fig. 4.3.2.3 shows the influence of the disturbance variable z on a control loop with a P-controller but without a fixed setpoint y₁. The parameters are otherwise the same; the magnitude of the disturbances is increased by an increase in resistance. As soon as the I-controller (Schalter S₁ closed) is added to the P-controller, a purely oscillating steady-state behavior results. Fig. 4.3.3.1 shows the behavior with the I-controller alone at on Regelkreis with P-controller (switch S₁ closed); after each disturbance the system re-establishes the setpoint, and the steady-state offset becomes zero. The oscillatory decay takes longer the larger the time constant T₁ of the I-controller, and the steady-state offset remains zero independent of the magnitude of R.

The duration of the transient process is again reduced by switching in the P-controller (switch S₁ closed). Fig. 4.3.3.2 shows in the upper half (left) with T₁ = 1.3 s, and (right) with T₁ = 85 s, and in the lower half each with T₁ = 2.8 s, the step response of the PI-controller for various values of the disturbance variable. The time constants T₁ are:

T₁ = 2.8 s (for R₂ = 800 kΩ), T₁ = 1.9 s (for R₂ = 400 kΩ)
T₁ = 1.3 s (for R₂ = 270 kΩ), T₁ = 0.85 s (for R₂ = 180 kΩ)

As shown, the time constant T₁ of the PI-controller is naturally also the time constant T₁ of the I-controller. It is most optimal to vary T₁ only in the vicinity of a value corresponding to the optimum.

4.3.4 Control Loop with D-Controller and PID-Controller

The initial deviation of the control variable x from setpoint w immediately after a disturbance occurs is, in all cases, the same for the control loops with P-controller or PI-controller (see Section 4.3.2 and 4.3.3), since the rate of change of x is determined by the proportional part (P-controller). A different controller is needed whose output variable changes not only when a sustained change in x occurs (P-controller) or when x integrates (I-controller), but also when there are changes in the time derivative of x. This property is achieved by the D-controller, which produces an output proportional to the derivative of the control variable. A PID-controller therefore counteracts changes in the deviation more quickly. This property of the D-controller, as shown in Section 4.3.3, further improves the behavior of the PI-controller. In a simple PID-controller, the P-, I-, and D-controllers are connected in parallel (switch S₁ closed).

[page 76: figure only — schematic diagram 4.3.4 showing PID-controller circuit with D, P, and I branches in parallel, and associated oscilloscope traces 4.3.4.1 for various D-controller time constants T_D]

The circuit is briefly and transiently very large, so that the disturbance is quickly counteracted. With this property from Section 4.3.3, the PI-controller behavior can be further improved. In the simple PID-controller, P-, I-, and D-controllers lie in parallel (switch S₁ closed).

Fig. 4.3.4.2 shows the influence of the guidance variable w and the disturbance variable z on the control loop with a PID-controller. To examine the influence of the amplification factor c of the P-controller, the time constant T₁ of the I-controller and the time constant T_D of the D-controller must be varied. Here only a selection of favorable combinations for c and T₁ is made; first, the values from Section 4.3.3 are chosen, where a favorable control behavior was found: c = 1.43 (R₁ = 7 kΩ) and T₁ = 1.9 s (for R₂ = 400 kΩ). The time constant T_D of the D-controller is varied.

The duration of the oscillatory transient is smaller with the PI-controller for a smaller time constant T₁ of the D-controller, and T_D = 0.47 s yields:

T_D = 0.2 s (for C₁ = 1 μF), T_D = 0.47 s (for C₂ = 2.35 μF = ~4.7 μF/2)
T_D = 0.94 s (for C₃ = 4.7 μF)

In Fig. 4.3.4.2, though one sees, as expected, a reduction in the oscillatory settling time after a disturbance, there is still a noticeable steady-state offset after a sustained disturbance. The time constant T₁ of the I-controller is too large.

Fig. 4.3.4.3 shows the influence of the guidance variable w and the disturbance variable z on the PID-controller.

The optimal values (as in all the examples of Sections 4.3.1 to 4.3.5) correspond to a dead-time-free system of order 4 (Abschnitt 4.1.7); the PID-controller with amplification factor c = 0.85 (R₁ = 180 kΩ) for the I-controller and time constants T₁ = 1.9 s (R₂ = 400 kΩ) and T_D = 0.43 s (for C_y = 4.7 μF) for the D-controller is the best. This control behavior is quite good.

[page 77]

4.3.5 Control Loop with PID-Controller Built from Operational Amplifiers

The PID-controller built from three operational amplifiers connected in parallel (Section 4.3.4) can also be implemented using a single operational amplifier. Fig. 4.3.5 shows the complete PID-controller control loop (the controller itself is shown below).

Fig. 4.3.5.1 shows the dependence of the control variable x on the guidance variable w and the disturbance variable z. At time t = 10 s, the system is switched on (switches S₁ closed); at t = 40 s, the disturbance z = −0.2 is switched in (switch S₂ closed); at t = 70 s, the disturbance is switched off again (switch S₂ opened). The control behavior is virtually as good as that determined in Section 4.3.4.

By searching for the optimal controller for a given technical control path, one can determine the step response of the control path on the analog computer and then simulate the control loop with various parameter settings. Through variation of the controller parameters, a suitable step response is determined, which can then be implemented in a technical device with corresponding behavior.

[page 78: figure only — schematic diagram 4.3.6 showing a control loop with control path having dead time and delay of 2nd order, and graph 4.3.6.1 showing influence of disturbance variable z and guidance variable w on x for various gain values c]

4.3.6 Control Loop with Control Path without Compensation

Fig. 4.3.6 shows a control loop consisting of a control path without compensation and with delay of the 2nd order (Abschnitt 4.2.3) and the P-controller of Section 4.3.2. Fig. 4.3.6.1 shows the influence of the disturbance variable z and the guidance variable w on the control variable x for various gain values c.

The P-controller in a control loop without compensation no longer has a permanent steady-state offset in its output signal — as long as the disturbance changes. This alters the output signal of the control path without compensation, as long as the setpoint track x₁ proportional to the steady-state offset has not become zero. As a result, the permanent steady-state offset disappears.

5. Simulation and Solution of Physical Problems

Every physical experiment requires a physical apparatus; modeling the underlying differential equations of the problem requires an analog computer. In order to obtain an analog model of the problem, one builds, e.g., acoustically scaled-down models of planned concert halls, theaters, and cultural centers, and tests the “acoustics” with sounds that are correspondingly scaled in frequency. Measurements of room dimensions in wind tunnels are used to determine the ratio of aircraft dimensions to those of the actual aircraft. Studies of flight behavior of aircraft are already simulated on the analog computer before the aircraft is built. Crash tests (safety tests) of automobile bodies are to simulate actual collision situations and thus allow conclusions about possible consequences for vehicle occupants. The behavior of newly developed electronic oscillation circuits is studied with an analog computer. And, for example, to build such a

[page 79]

An analog computer is useful whenever general statements about the motion and acceleration of physical processes are to be made (see Sections 5.3.4 and 5.3.5).

The subjects covered in this chapter begin with simple motions of bodies, deal with elastic collisions, radioactive decay, free fall, and end with a multi-body time-dependent oscillation equation.

5.1 Kinematics

Kinematics deals with the motion of material bodies — specifically their change in position and the forces causing the change in velocity and acceleration — without regard to the mass of the body.

The chapter begins with the relationship between path and velocity and deals also with the computation of physical quantities in their correct physical units (see Sections 1.3.2 and 1.3.3). Then two of the simplest practically relevant phenomena — the collision of two vehicles — are treated, followed by two further numerical examples from the field of physical literature.

5.1.1 Path and Velocity

The instantaneous velocity v(t) is defined by the path traversal Δs per unit time Δt:

$$v = \frac{\Delta s}{\Delta t}, \quad \text{more precisely:} \quad v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$$

Or, conversely:

$$s(t) = \int_0^t v , dt + s(0)$$

which represents mathematically the integral through the charging of a capacitor:

$$Q(t) = \int_0^t I , dt + Q(0)$$

Correspondingly, e.g., for a car moving backward along a route, the distance s(t) traveled from time t is a measure of the distance traveled since time t = 0 with velocity v; one writes:

$$s(t) = \int_0^t v , dt + s(0)$$

with initial value s(0) at time t = 0.

The output of the operational amplifier provides, from Section 2.1.6, the output voltage U₀ as the integral of the input voltages, multiplied by a factor −k:

$$U_0(t) = -k \int_0^t U_i , dt + U_0(0)$$

with k = 1/R₁C₀ and the working range for U: $$-10 \text{ V} \leq U \leq 10 \text{ V}$$

Setting m U₁ = a_m·β₁, i.e., β₁ = 10 V/m (see Section 1.3.2), it follows for U_0(t):

$$U_0(t) = -k\beta_s \int_0^t a_m , dt + U_0(0)$$

with working range for a_m: $$-1 \text{ m/s} \leq a_m \leq 1 \text{ m/s}$$

Setting m U₁ = a_m·β₁ so that a_m must also satisfy: $$-1 \text{ m/s} \leq a_m \leq 1 \text{ m/s}$$

As the working range for a_m follows: $$-1 \text{ m/s} \leq a_m \leq 1 \text{ m/s}$$

[page 80]

[page 80: figure only — circuit diagrams showing integrators 5.1.1 for kinematics simulation, with operational amplifier chains, and diagram 5.1.2 showing the analog computer implementation for the two-vehicle collision problem]

The input of the integrator receives the signal a_m, and at its output appears the signal v_m. Input and output appear in their correct physical dimensions. The amplitude scaling is discussed in detail in Section 1.3.2.

5.1.2 Collision of Two Vehicles (Mathematical Solution)

Two vehicles travel on a straight road. At time t = 0, vehicle 1 is to be at a point s₁(0) = 100 m in front of vehicle 2. Vehicle 1 shall travel with velocity v₁ = 6 m/s and vehicle 2 with v₂ = −4 m/s toward vehicle 1. When and where will they meet?

The mathematical solution uses the approach:

$$s_1(t) = v_1 t + s_1(0)$$ $$s_2(t) = v_2 t + s_2(0)$$

and with the condition for the meeting time that: $$t = z$$ $$s = v_1 t + s_2(0)$$

This leads to a system of two equations with two unknowns:

$$t = \frac{z}{v_2 - v_1} = \frac{s_1(0) - s_2(0)}{v_2 - v_1}$$

The system of equations leads to programming in Section 3.1.1. It must however first be re-scaled for correct amplitude scaling (see Section 1.3.2).

The scaling factors are: $$\frac{t}{10} = \frac{10 \text{ s}}{v \cdot 100}$$

$$\frac{s}{10} = \frac{v_1 \cdot t}{10 \cdot 100} = \frac{s(0)}{100}$$

5.1.3 Collision of Two Vehicles (Simulation of the Physical Process)

The task of Section 5.1.2 cannot now be solved purely by mathematical solution of the equation system, but should be described as a physical experiment. Thereby the analog solution of the equation system is not what is being described, but the actual physical process is simulated on the analog computer. The input signal of an integrator is set equal to the velocity, and the output signal of the integrator gives, in analogy to the actual process, the position of the vehicle as a function of time.

[page 81]

[page 81: figure only — schematic 5.1.3 showing two integrator circuits for the positions s₁ and s₂ of two vehicles, and associated graph 5.1.3.1 showing position vs. time for both vehicles, their meeting point, and meeting time]

Fig. 5.1.3 shows the two integrators (time constants RC = 10 s) for the generation of position functions. The distance scale is set so that s₁(0)/100 = 0 is chosen as the initial value for the position of the first vehicle. The initial value for the position of the first vehicle s₁(0)/100 = 1 m (= 10 V), and for the position of the second vehicle s₂(0)/100 = 0.4 m (= 4 V) is chosen (switch S₂ closed).

At the start, switches S₁ and S₂ are opened. The distance scale adjusts s₂(0)/100 to 1 m (= 10 V) and the initial value for the second vehicle s₂(0)/100 = 0.4 m (= 4 V) (switch S₂ closed). Fig. 5.1.3.1 shows the output signals of both integrators. One can read from these the collision point, the position of the first vehicle at the collision time, and the time from start to collision.

At the start, switches S₁ and S₂ are opened; the computer starts. Fig. 5.1.3.1 shows the output signals of both integrators: the position of the first vehicle as a function of time ab, and that of the second vehicle. One can read the collision point (approx. t = 10 s (= 4 V)), when the vehicles collide (about 10 s from start) and where they meet.

The program outputs not only the solution to the physical problem, but also contains several useful hints.

5.1.4 Collision of Two Vehicles (Physical Simulation — Continuation)

To automatize the experiment of Section 5.1.3, both integrators for s₁/100 and s₂/100 are switched so that a comparator (see Section 2.4.8) detects when both vehicles are at the same point and simultaneously operates a stop switch. When the vehicles are at the same position, d = 0, i.e., when one vehicle meets the other (switch S₁ closed).

The upper half of Schematic 5.1.4 is identical to the upper half from 5.1.3. In the lower half, a comparator (see Section 2.4.8) and the successive inverter provide the vehicle velocity v₁/10, which can be integrated by an integrator (time constant RC = 10 kΩ) to give v₁/10 = s₁/100.

[page 83: figure only — schematic 5.1.4 with comparator circuit for automatic detection of the collision, and graph 5.1.4.1 showing positions vs. time with collision detection, stop time ~10 s, when vehicles meet at d = 0]

[page 82 — continued from previous section]

Before the start, both vehicles are set: RC = v₁/10 = 0.4 m/s (= 4 V) is set as the initial value for the position of the first vehicle, s₁(0)/100 = 0 is chosen (switch S₂ closed). Fig. 5.1.3.1 shows the output signals of both integrators starting with the initial values s₁(0)/100 at x = 0 for the first vehicle and s₂(0)/100 = 0.6 m/s (= 6 V) for the second vehicle.

The output U₅ of the middle operational amplifier (2 kΩ) provides a control signal for the timing device (578.60). At the lower half of the schematic, the bus shows the output with the comparator. This is connected to the mass of the Digitalzähler; the Digitalzähler reads out from values corresponding to t = 30 min (Wahlschalter e.g., horizontally to the right). When the (Bildschirm plan 5.2.3 is described) the Digitalzähler (Zeitbasis = 10 Hz).

After the initial value settings are closed (switches S₁ and S₂ closed), the Digitalzähler is started, the schematic begins. The Digitalzähler setzt automatically its reference value; simultaneously switches S₁ and S₂ start. The output signals of the two integrators are connected to and the Digitalzähler is set to the values of both integrators. At the lower bus the Digitalzähler stops when x₁ = 0, and s₁ is read — about 60 s (= 4 V) — can be read in the rest position.

The program serves not only to solve the physical problem but also to include several helpful hints for the operation of the involved physical process.

5.1.5 Velocity and Acceleration

The instantaneous acceleration a(t) is defined by the velocity change Δv per unit time Δt:

$$v(t) = \frac{\Delta v}{\Delta t}, \quad \text{more precisely:} \quad a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$$

Thus it follows from Section 5.1.1 that:

$$v(t) = \int_0^t a , dt + v(0)$$

with initial velocity v(0) at time t = 0. The velocity of a body at time t is a measure for the velocity gained from the acceleration a from time t = 0.

From the acceleration a using time constant RC, the velocity v is:

$$v(t) = \int_0^t a , dt + v(0)$$

[page 84 — continued]

In schematic plan 5.1.6, the Potentiometer links open with y(0) = 0 from the output of the first integrator (time constant RC = 2 s) appears −y/100; from the output of the second integrator (time constant RC = 5 s) then y/200.

At the start of calculation, the switches S₁ and S₂ are opened (initial values y(0) = 0 and g(0) = 0). Fig. 5.1.6.1 shows g and y as functions of t:

following the law g = −gt
following y = −½gt²

These can be confirmed experimentally:

In schematic plan 5.1.6 is the Potentiometer links open with m/100 = −0.098 m/s² (= −0.98 V). Am Ausgang des ersten Integrierers (Zeitkonstante RC = 5 s) liefert then y/1000. Die Ausgangsspannung y/1000 begins with the Ausgangssignal des Komparators automatically at Null gesetzt; dann folgt m·t/1000.

5.1.6 Free Fall

An elementary example of kinematics with not-vanishing acceleration is free fall. Here it is assumed that air resistance is negligible; g = 9.8 m/s² is the acceleration due to gravity used for the free-fall acceleration.

By a single integration, the velocity y(t) is obtained from Section 5.1.5, and by a second integration, the position y(t). After further integration corresponding to Section 5.1.5, the position y(t).

[page 84: figure only — schematic 5.1.6 showing free-fall circuit, and graphs 5.1.5.1 and 5.1.5.2 showing velocity and position as functions of time for various time constants]

[page 85 — continued from Ballistics chapter]

In schematic plan 5.1.6, the Potentiometer links (open with m/100 = −gr/100 − Ry(t)/100) from the output. At the start, switches S₁ and S₂ are opened (initial values y(0) = 0 and g(0) = 0). Fig. 5.1.6.1 shows g and y as functions of time following:

g = −gt
y = −½gt²

These can be confirmed experimentally.

5.2 Ballistics

Following the program for free fall from Section 5.1.6, the flight paths of bodies in the Earth’s gravitational field are computed, taking account of air resistance and with various initial velocities (see Programmes 5.2.4 to 5.2.6) and with an XY-writer (see Programmes 5.2.3 and 5.2.4) or XY-writer (see Programme 5.2.6) registered. To simplify the solution (see Section 3.3), the Analogrechner is kept small.

The chapter begins with the effect of a linear resistance law on the free fall (Sections 5.2.1 and 5.2.2), treats the registration of the flight curve with an XY-writer (Section 3.1.3), and ends as an example with a rocket.

5.2.1 Free Fall with Linear Air Resistance

The behavior of a body in flight is determined by the resulting force, which is the sum of the air resistance force — in one dimension proportional to velocity g and opposed in direction to motion — and the gravitational acceleration − g acting downward. It holds:

$$\ddot{y} = -\frac{R}{m} \dot{y} - g$$

with friction coefficient R and mass m. The total deceleration of the flight body consists from the friction-proportional deceleration −(R/m)·ẏ and the gravitational deceleration −g acting together.

$$\dot{y}(t) = -\frac{R}{m} \dot{y}(t) - g$$

In schematic plan 5.2.1, the Potentiometer links open with m/100 = −g/100; from the output (time constant RC = 10 s) appears −y/200; from the output of the second integrators (time constant RC = 5 s) then y/1000.

The feedback of velocity y over the output of the first integrator follows next over a Potentiometer. The feedback R from the output of the integrators over a next following Potentiometer and the first Widerstand R₁ = 100 kΩ follows R/m = 0.141 s⁻¹ and for R₁ = 10 kΩ follows R/m = 0.00083 m⁻¹.

At the start the switches S₁ and S₂ are opened (initial values y(0) = 0 and y(0) = 0). Fig. 5.2.2.1 shows at the first 15 s that a maximum fall velocity of y_∞ builds up:

$$\dot{y}_\infty = -\frac{m}{R} g$$

with R/m = 0.025 m⁻¹ follows ẏ_∞ = −63 m/s; with R/m = 0.00083 m⁻¹ follows ẏ_∞ = −109 m/s.

[page 86 — continued]

[page 86: figure only — schematics 5.2.11 and 5.2.12 with circuits for free fall with linear and quadratic air resistance, and graphs showing velocity vs. time for R/m = 0.141 s⁻¹ and R/m = 0.045 s⁻¹, yielding terminal velocities of about −70 m/s and −220 m/s respectively]

With R/m = 0.141 s⁻¹ follows ẏ_∞ = −70 m/s; with R/m = 0.045 s⁻¹ follows ẏ_∞ = −220 m/s.

5.2.2 Free Fall with Quadratic Air Resistance

In a quadratic air resistance law, the nature describes the air resistance as a decelerating force proportional to the square of the velocity:

$$a = -\frac{R}{m} \dot{y}^2 \cdot \text{sign} , \dot{y}$$

with friction coefficient R, mass m, and the algebraic sign of velocity sign ẏ. The total deceleration of the flight body is the sum of the air resistance proportional deceleration a and the gravitational acceleration −g counteracting it. It holds:

$$\ddot{y} = -\frac{R}{m} \dot{y}^2 \cdot \text{sign} , \dot{y} - g$$

For the quadratic air resistance law at constant fall, the nature gives the terminal velocity ẏ_∞ as the ratio of the square of the velocity quadratically:

$$\dot{y}_\infty = -\sqrt{\frac{m}{R} g}$$

The combined deceleration of the flight body: deceleration a is proportional to the square of the velocity sign ẏ and the gravity-borne deceleration −g together, so:

$$\ddot{y} = a - g$$

[page 86: figures 5.2.21 and 5.2.22 — graphs and schematic for quadratic air-resistance free-fall simulation, with two curves for R/m = 200 k / cose83 m⁻¹]

5.2.3 Oblique Throw (Registration with XY-Writer)

At free fall from Section 5.1.6, the initial velocity y(0) is chosen as the starting value of the initial velocity y(0). By general, the initial velocity need not be purely a y-component, but rather a vector with both X and Y components; one then speaks of oblique throw.

For the solution of the problem, both movement components in the Y and X directions are programmed separately (see Programme 5.2.4 to 5.2.6). For the oblique throw (solution with the XY-writer), the flight curve is registered as a ballistic curve using an XY-writer.

The mathematical solution leads to:

$$y”(t) = \frac{d^2y}{dx^2} = -\frac{1}{v_x^2} g$$

With y(0) = v₁, it follows as initial value for y’(x):

$$y’(0) = \frac{v_y(0)}{v_x} = \frac{1}{v_x \cdot 1000} \cdot \frac{dy}{dt}$$

With y(0) = v₁, x(0) = y₀, and the velocity factor d from the kinematic relations:

$$y’(0) = \frac{v_y}{v_x} = \frac{v_x \cdot t}{v_x \cdot t}$$

If x₀(0) = y₀ and y₀ = y₀, then y₀(t) is as follows for y(x).

The scaling factor a = 0.03 s/m is chosen so that one entry per second corresponds to x = 1000 m with the paper advance of the XY-writer of 20 cm/min. The amplitude scaling: one volt represents x = 1000/a corresponds to a paper advance of 10 cm. The scaling factors are optimal for a throw velocity of v₀ = 100 m/s.

In schematic plan 5.2.3, the Potentiometer links open with g and the initial value at start y₀/100, the output (time constant RC = 0.94 s) gives −y/100; from the output of the second integrators (time constant RC = 10 s) gives y/10.

[page 87 — continued]

The following integrator (time constant RC = 0.94 s) delivers:

$$\frac{s}{RC} \int_{t_0}^{t} (-10 , y”) , dx = \frac{10 , s}{RC} , y’(0)$$

(RC/10 is here equal to y’). In the second integrator (time constant RC = 9.4 s) the output signal (RC/s is here about equal y/10) is given; in the lower half of the schematic 5.2.3, the bus connects via a next integrator. The Summierung finds then:

$$-\frac{s}{10 \cdot \frac{v_x^2}{1}} = -\frac{v_x(0)}{1000}$$

When the Potentiometer links below is set equally g/10:

$$-\frac{s}{10} \int_{t_0}^{t} y” , dx = -\frac{y(0)}{1000}$$

(RC/10 is here equal to y/10). In the second integrator (time constant RC = 10 s) the output signal (RC/s is here about y/10). The Radicizer delivers:

$$-x/100 \to -\frac{1}{100} \sqrt{v_x^2} = -\frac{1}{100} v_x$$

and the following integrator (time constant RC = 10 s) delivers x/1000.

With the Potentiometer links above, the signal g and y(0) can be equally changed, so that v₀ₓ only varies in direction. The deviation angle can be easily entered as input signal A. It holds:

$$A = -\frac{1}{v/10} \cdot 0.18 , (= 1.8 \text{ V}) \text{ for } 30°$$ $$-0.32 , (= 3.2 \text{ V}) \text{ for } 45°$$ $$-0.5 , (= 5 \text{ V}) \text{ for } 60°$$

With the start of the calculation the switches S₁ and S₂ are opened. Bild 5.2.3.1 shows the flight curves y for three various deviation angles. The largest range is reached with the deviation angle of 45°, which is overprinted with ≥ 45° (in accordance with Section 5.2.6 where the Registrierung with the XY-writer is in scale 1:2).

The XY-writer requires a time not too small at the machine side for the recording of the ballistic curves. However this is in each case a key time for the actual flight time, and is programmable. At the XY-writer with a paper advance of 20 cm/min, the registration of a 30 s ballistic trajectory at a distance of roughly 10 cm corresponds. The scaling of the computation with an XY-writer at scale 1:2 at the XY-writer records the ballistic curves. The scale factors are optimal for a throw velocity of v₀ = 100 m/s.

5.2.4 Oblique Throw with XY-Writer

For the oblique throw with the XY-writer, the results of Section 5.2.3 are available to register on the XY-writer. The XY-writer requires for the registration of the flight curves y and the deviation angle also the angle v₀ₓ. The XY-writer adjusts automatically its own reference in the paper direction. With an XY-writer using a starting angle of 45° (overprinted in Bild 5.2.3.1).

With the start the switches S₁ and S₂ are opened (initial values y(0) = 0 and y(0) = 0). Fig. 5.2.4.1 shows the flight curves registered with an XY-writer for three various deviation angles. The initial velocity v₁ = 100 m/s is chosen, and the deviation angle is re-chosen each time (v₁/v_x)². The solution is again with 100/s again 70.7 m/s.

[page 88 — continued]

[page 88: figure only — schematics 5.2.4 (XY-writer oblique throw program) and 5.2.4.1 (graph), showing ballistic curves for three launch angles with and without air resistance, plus schematic 5.2.5 for adjustable-angle oblique throw and graph 5.2.51 showing curves for angles c = 0.7 and c = 1 etc.]

5.2.5 Oblique Throw with Adjustable Angle

To be able to freely select the deviation angle, one must know from section y₀/100, with which the correct velocity component v₀ₓ is automatically produced as input signal. The input signal of the schematic plan 5.2.5 gives to the right from y/100 a squared signal; the following Summierung delivers (v₁/v_x)².

The schematic plan 5.2.5 shows the complete program. Towards the left from y₂/100 = −v₁/100 produces the next integrator (time constant RC = 10 s) x/1000. The next integration produces the next Summierung links with v₀ₓ = 100 m/s.

With initial values y(0) = 0, x(0) = 0, y(0) = v_y, the following from the kinematic relations: $$v_x^2 + v_y^2 = v_0^2$$

Again y yields y = 70.7 m/s.

The solution again for 100 s, with the starting angle v₀ (Anfangsgeschwindigkeit y₀/100): $$\dot{y}(0) = -\frac{v_y}{v_x}$$

with initial conditions y(0) = 0, x(0) = 0, y₀ and v_x. Setting y₀ = v₀ sinφ and x₀ = v₀ cosφ (here a throw angle of ~45° is overprinted):

$$\dot{y}(0) = -\frac{1}{\sqrt{1}} \cdot \sqrt{v_0^2 - 1} = -\frac{1}{v_x} v_y$$

The Radicizer below delivers:

$$-x/100 \to -\frac{1}{100} \sqrt{v_x^2 - v_y^2} = -\frac{1}{v_x}$$

and the following integrator (time constant RC = 10 s) delivers x/1000.

At the start of the computation the switches S₁ and S₂ are opened (initial values y(0) = 0 and y(0) = 0). Fig. 5.2.5.1 shows the flight curves y as a function of x registered with an XY-writer for three various deviation angles for which the deflection angle is chosen.

[page 89 — continued]

[page 89: figure only — schematics 5.2.5 (adjustable-angle oblique throw, full circuit) and 5.2.51 (graph showing x vs. y for c = 0.7, c = 1, and partial angle curves), plus schematic 5.2.6 (oblique throw with quadratic air resistance)]

5.2.6 Oblique Throw with Quadratic Air Resistance

The oblique throw with quadratic air resistance follows in the description of the flight with a motion-opposing force in a body according to Section 5.2.2 in the Y- and X-components. The combined deceleration of the flight body consists of the friction-proportional deceleration a proportional to the square of the velocity sign ẏ and the gravitational deceleration −g acting together. It holds:

$$\ddot{y} = -\frac{R}{m} \dot{y}^2 \cdot \text{sign} , \dot{y} - g \quad \text{and} \quad \ddot{x} = -\frac{R}{m} \dot{x} \dot{y}$$

with friction coefficient R and mass m. For the solution of the oblique throw with XY-writer, the Y and X components of the flight are programmed separately.

The combined deceleration of the flight body is the sum of the friction-proportional deceleration a and the gravity-induced deceleration −g. The motion equation in Y and X components are thus:

$$\ddot{y} = -\frac{R}{m} \dot{y} - g \quad \text{and} \quad \ddot{x} = -\frac{R}{m} \dot{x}$$

[page 89: figure only — schematic 5.2.6 for oblique throw with quadratic air resistance, and graph 5.2.61 showing XY-writer curves for c = 0.7 angle]

[page 90 — continued]

[page 90: figure only — schematic 5.2.7 (rocket acceleration program) and graphs 5.2.71, 5.2.72, 5.2.73 showing mass m(t), velocity y(t), and altitude y(t) as functions of time for m = 50 kg/s and m = 100 kg/s with c = 1000 m/s and c = 2000 m/s]

5.2.7 Rocket Acceleration

With mc/500,000:

$$-\frac{mc}{1000} = -\frac{mc}{1} \cdot \frac{y}{500}$$

From the acceleration equation of the integrator (time constant RC = 10 s) to the right is −y/5000; from the output of the second integrators (time constant RC = 5 s) then y/25000.

It is also possible to proceed as follows with the Potentiometer links. Fig. 5.2.7.1 shows the mass m, Fig. 5.2.7.2 the velocity y and Fig. 5.2.7.3 the position y as functions of time for:

m = 50 kg/s and c = 1000 m/s (dotted curves)
m = 100 kg/s and c = 2000 m/s (solid curves)

The mass m decreases from the start value m₀ (here m₀/1000 = 1 kg (= 10 V) is chosen) by the expenditure of propellant. The velocity y of the rocket is always counteracted in direction; now follows m = 0, and the rocket is burnt out, moves with constant velocity y (Bild 5.2.7.3). This terminal velocity depends on the final mass m_f (Bild 5.2.7.1), which does not alter so long as the exhaust gas velocity y is independent of time — does not change.

The mass m decreases at constant mass throughput ṁ; so follows m = 0 and ṁ changes, and the mass m is independent of ṁ. Die Massendurchsatz is thus automatically reset to Null; then follows m = 0 and ṁ changes, and the next Potentiometer (Zeitkonstante RC = 2 s) to the right is −m/500 and delivers the sum of the second integrators then y/1000. The Dividierer is connected in the feedback of the Operationsverstärkers from the right, multiplied by m/c/1000 from the subsequent Widerstand R₁, chosen from this c = 1000 m/s for R₁ = 100 kΩ and for c = 2000 m/s for R₁ = 200 kΩ. The Dividierer feeds the value into the upper feedback of the Potentiometers links below with −m/1000.

The Zähler goes over the Dividierer of the next Operationsverstärkers to the right with multiplication by m/c/1000 from the following Widerstand R. For the start, the switches S₁, S₂, S₃ and S₄ are opened (initial values y(0) = 0, y(0) = 0 and m(0)/1000 = 1 kg (= 10 V)). The Potentiometer links above is set to ṁ/5000 = 0.1 kg/s (= 1 V) or m/c/1000 = 0.2 kg/s (= 2 V) and with R_l the value of the Strahlgeschwindigkeit c is provided.

At the start of computation (Bild 5.2.7.1) to the right opens a Komparator; so long as the rocket has not yet burnt out, it gives a large positive output signal — the Potentiometer links above = −m/1000 is set. Now the Komparator under the Dividierer gives the Ausgangssignal des nächsten Potentiometers links below with −m/500, which is set to 1 V for ṁ/500 = 0.1 (= 1 V) or ṁ/500 = 0.2 (= 2 V). The Dividierer supplies −m/500 (= −(m/500), which is then given to the upper input of the next Operationsverstärker and integrated (time constant RC = 5 s) producing the formula:

$$\frac{mc}{500000}$$

5.2.8 (continued)

Figure 5.2.8.2 shows once more the last curve from Figure 5.2.8.1, but over a longer time span. The rocket reaches its maximum altitude here after about 75 s and then begins to descend again. The program is incorrect at this point, however; it should have sign y = −1. The dashed curve shows the trajectory for the case of a favourable starting conditions (compared to the free-fall case with quadratic friction law — see Section 5.2.2).

5.3. Dynamics

While the examples of kinematics were confined to describing relationships between acceleration, velocity, and position, the examples in ballistics and dynamics deal with acceleration as a function of the internal and external forces that cause it.

The chapter begins with examples from the world of motor vehicles. During the acceleration process, a prescribed engine torque, vehicle mass, and rolling resistance are assumed, and the velocity and distance are measured as functions of time. Dependencies on engine torque, mass, and rolling resistance are particularly noticeable; it is observed that doubling the final velocity by halving the rolling resistance coefficient produces almost no increase in top speed for passenger cars, whereas the acceleration range from 100 km/h to 200 km/h is roughly doubled.

In further program examples, the rolling and braking of motor vehicles is treated. There follow several experiments on the conservation-of-momentum law and on elastic central collisions.

5.3.1 Acceleration of a Motor Vehicle

The power P in kW or PS delivered by the engine of a motor vehicle is fixed at a given gear ratio; one can choose a higher gear for higher speeds and thereby keep P virtually constant. By choosing the gear ratio appropriately, the driving force F for a given speed is calculated. The vehicle is accelerated by F. For very small speeds, i.e. at the start, the driving force F is very large. It is intended that the computer simulate an acceleration process with constant power P.

The force driving the motor vehicle at power P and velocity v is:

F = P/v

The correction v₀(v) must be made for small v to keep F finite and manageable as the vehicle starts. It should display the acceleration v between 0 and 1 m/s³.

The motor vehicle’s driving force F is opposed by the inertia of mass m and the rolling resistance R (air and rolling resistance). With a quadratic friction law, for v ≥ 0:

F = mv̇ + Rv²

After substituting from above and solving for v̇:

v̇ = (1/m) · (P / (v₀(v) + v) − Rv²)

For a typical passenger car, m = 10⁵ kg and rolling resistance coefficient R = 1 kg/m, and power P = 50 kW. The computation range for velocity is chosen as in Section 5.3.1:

0 ≤ v/50 ≤ 1 m/s

So the usable velocity range is 0 ≤ v ≤ 50 m/s = 180 km/h.

In circuit plan 5.2.1, an appropriate amplitude scaling has already been taken into account. From the acceleration equation, −v/50 appears at the input of the integrator (time constant RC = 1 s) as the output representing the velocity; this is a suitable v(v). As will be shown later, the output of the last operational amplifier carries the force curve F as a function of the computed velocity v. For this, a force of 5 A is multiplied (A = 1 m/Ns) and through the summing amplifier one obtains:

v₀(v) = 5A · √(P/50·(v/50 + v)) · 10⁻⁴

with v₀(v) = √5 A · P · 10⁻⁴ = 5 m/s (for P = 50 kW) and v₀(v_max) = 0.

The upper operational amplifier then produces the sum:

−(A · v₀/10 · 5A/50) + (v₀ + v)/50 = −(v₀ + v)/50

If the potentiometer above the circuit sets the power P/50, the power appears at the divider output as the force:

P / (v₀(v) + v) · 10⁻⁴

The resistance R₁ = 20 kΩ must be chosen so that the force equals 1 when the rolling resistance R = 1 kg/m acts. It follows:

R₁ = (20 kΩ · kg) / m

At the left operational amplifier, the force is divided by m and then at the output of the summing amplifier the result is finally again −v/50.

[page 92: figure (circuit plan 5.3.1)]

The multiplier becomes v²/2500, and at the summing amplifier the force is formed:

(P / (v₀(v) + v) − Rv²) · 10⁻⁴

where the resistance R₁ = 20 kΩ must be chosen so that the quantity equals 1 when rolling resistance R = 1 kg/m. It follows:

R₁ = (20 kΩ · kg) / m

At the left operational amplifier, the force is divided by m, and the output of the summing amplifier is again −v/50.

[page 93: figure (circuit plan 5.3.1 with graphs)]

The multiplier is set to v²/2500, and the summing amplifier produces the force:

(P / (v₀(v) + v) − Rv²) · 10⁻⁴

where resistance R₁ = 20 kΩ is chosen such that the expression equals 1 when rolling resistance R = 1 kg/m. It follows:

R₁ = (20 kΩ · kg) / m

At the left operational amplifier, the force is divided by m; the output of the summing amplifier is finally again −v/50.

Figures 5.3.1.1 and 5.3.1.2 show v(t) at fixed power P = 50 kW and fixed rolling resistance coefficient R = 1 kg/m as a function of mass m. The final velocity v_max is independent of the mass, but for small mass it is reached sooner. The deviation of the recorded final velocity from the theoretical value is on the order of about 6% and is a consequence of the finite time constant of the integrator and the switching error of the divider. The force term at the top of the circuit plan is proportional to the 3rd root. aus dem reziproken Reibungskoeffizienten R: The maximum achievable final velocity v_max is proportional to the 3rd root of the reciprocal rolling resistance coefficient. The following holds approximately:

v_max ∝ ∛(1/R)

The deviation of the recorded final velocity from the theoretical value is on the order of about 6%; this is a consequence of the finite time constant of the integrators and the switching error of the divider.

Figure 5.3.1.3 shows v(t) at fixed power P = 50 kW and fixed rolling resistance coefficient R = 1 kg/m as a function of mass m. The final velocity is independent of mass m, but for small masses it is reached sooner. The maximum achievable final velocity v_max is proportional to the 3rd root of the reciprocal rolling resistance coefficient.

Often motor vehicles are used as a measure of acceleration performance; the time required to travel a given distance (or to reach a given speed) is determined. For this it is sufficient to compute v(t) and then integrate. The times given in Figures 5.3.1.1 to 5.3.1.3 correspond to real driving force computed as F = P/v, not F_total (see the 1st section of this chapter).

5.3.2 Coasting of a Motor Vehicle

The air and rolling resistance of a motor vehicle were treated in Section 5.3.1 above. Now the motor delivers no power (e.g. the clutch is released); the motor vehicle decelerates due to the air and rolling resistance. For the given quadratic friction law, the decelerating force is: mv̇ = −Rv². As an acceleration:

v̇ = −(R/m) v²

For a typical passenger car with mass m = 10⁵ kg and rolling resistance coefficient R = 1 kg/m. The computation range for velocity is chosen as in Section 5.3.1:

0 ≤ v/50 ≤ 1 m/s

The lower part of circuit plan 5.3.2 corresponds to circuit plan 5.3.1. For the acceleration, −v/50 gives by means of a multiplier v²/2500, and the right-hand operational amplifier supplies the force −R · v² · 10⁻⁴. The left operational amplifier divides the force again by m (with R₁ = 1 kg/m and m = 10⁵ kg). As acceleration:

(1/50) · (1/m) · (−R · v²) = −(1/50) v̇

The upper part of circuit plan 5.3.2 with an integrator (time constant RC = 4 s) produces:

s(t) = (1/2000) · ∫₀ᵗ v(t) dt

Before the computation begins, switches S₁ and S₂ are closed, and the initial values v(0)/50 = 0.8 m/s (i.e. 8 V) and s(0) = 0 are set. When switch S is opened, the switch S_IC gives the initial velocity and v(t) yields the velocity and s(t) the distance as functions of time.

Figure 5.3.2.1 shows v(t) and Figure 5.3.2.2 shows the distance s(t) covered. The figures show that with a quadratic friction law the vehicle never comes completely to rest and could theoretically travel more than 2000 m. At small velocities, however, a quadratic friction law is no longer valid, and a vehicle can come to rest through other friction forces.

5.3.3 Braking of a Motor Vehicle

The air and rolling resistance alone, as in the most common cases of braking a motor vehicle, are not sufficient to stop it quickly. An additional brake must be applied. The braking action of the engine (i.e. engine braking by throttling the fuel supply) is not considered here.

With mass m, velocity v, rolling resistance coefficient R, and an additional deceleration a from an applied brake (braking force), the equation of motion for the vehicle (braking) is:

v̇ = −(R/m) v² − a

For a typical passenger car, mass m = 10⁵ kg, rolling resistance coefficient R = 1 kg/m, and additional braking deceleration a = 5 m/s². The computation range for the velocity is chosen as in Sections 5.3.1 and 5.3.2:

0 ≤ v/50 ≤ 1 m/s

In circuit plan 5.3.3, one takes from the circuit plans 5.3.1 and 5.3.2 the result v²/2500 from the multiplier; the right-hand operational amplifier supplies the force −R · v² · 10⁻⁴, and the upper summing amplifier supplies:

(1/50)(1/m² · (−R · v²) − a) = −(1/50) · v̇

For computing the distance travelled during braking, s(t) is calculated in the upper part of the circuit plan with an integrator (time constant RC = 4 s):

s(t) = (1/200) · ∫₀ᵗ v(t) dt

Before the computation starts, switches S₁ and S₂ are opened, and the initial values v(0)/50 = 0.8 m/s and s(0) = 0 are set. Opening switch S begins the computation, and v(t) and s(t) give the velocity and distance as functions of time.

[page 96: figure (circuit plan 5.3.3 with plots)]

From circuit plan 5.3.3, one takes v²/2500 from the multiplier. The right-hand operational amplifier supplies the force −R · v² · 10⁻⁴, and from the upper summing amplifier:

(1/50) · (1/m) · (−Rv² − a) = −(1/50) v̇

Figures 5.3.3.1–6 show the velocity and distance as functions of time for three different initial velocities and three values of the braking deceleration. From the graphs, the braking distance can be read off as a function of the initial velocity v(0) and the braking deceleration a.

Before the computation begins, switches S₁ and S₂ are closed simultaneously; the Schalter are opened to set the initial values v(0)/50 = 0.8 m/s and s(0) = 0. Opening switch S starts the computation. With these initial conditions s(t) = −v/200 (time constant RC = 4 s) is formed.

5.3.4 Momentum Conservation Experiment

On an air track there are two gliders; between them a compressed spring is placed. The entire arrangement is at rest.

[page 97: figure (circuit plan 5.3.5 with table)]

Without limiting the generality, the mass of the second glider is fixed at m₂ = 1 kg (R₁₀₀ = 100 kΩ). From the description of the acceleration − v̇ obtains an Impuls − a₂, and after a single integration (time constant RC = 1 s) by a summing amplifier on the left (time constant RC = 1 s) the velocity −v₂ is formed. An operational amplifier right below the summing amplifier produces x₂. At the operational amplifier to the right of the summing amplifier the Gesamtimpuls p = m₁v₁ + m₂v₂ is formed by choice of appropriate resistance R, and R₂. It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

The following integrator (time constant RC = 1 s) supplies −v₂; it is right below the summing amplifier to produce x₂. At the operational amplifier to the right of the summing amplifier the total impulse p = m₁v₁ + m₂v₂ is formed by choosing the appropriate resistances R₁ and R₂.

The table in Figure 5.3.5 gives for m₂ = 1 kg a series of measurement values for various masses m₁ and the acceleration a₂ as well as the time of action (switch closes S₂). The value of the total impulse given is maximal after 0.02 Ns. The computation range for the velocity is 0 to about 1 m/s.

m₁	m₂	v₁ (m/s)	v₂ (m/s)	m₁v₁ + m₂v₂ (Ns)
1 kg	0.5 kg	−0.3	4.6	0.08
1 kg		−1.7	5.0	0.1 Ns
1 kg		−4.3	4.3	0.075 Ns
1 kg		−4.3	4.3	0.075 Ns
2 kg		−1.6	5.4	0.1 Ns
5 kg		−1.8	9.6	0.2 Ns
5 kg		−0.5	9.3	0.2 Ns
10 kg		−0.5	5.5	0.15 Ns
10 kg		−0.85	8.4	0.15 Ns

5.3.5 One-Dimensional Elastic Collision

Two gliders (e.g. from an air-cushion table) move along one another; between them lies a compressed spring. The entire arrangement is at rest.

5.3.5 One-Dimensional Elastic Collision

The air and rolling resistance between the gliders is negligible. This observation is valid for released coupling, i.e. without the braking action of the motor by throttling the power supply.

The following observation applies to released coupling, i.e. without the braking action of the engine by throttling the fuel supply.

With mass m, velocity v, rolling resistance R, and an additional braking deceleration a of an applied brake, the equation of motion for the vehicle (braking) follows:

v̇ = −(R/m) v² − a

For a typical passenger car, mass m = 10⁵ kg and rolling resistance coefficient R = 1 kg/m and additional braking deceleration a = 5 m/s². The computation range for velocity is chosen as in Sections 5.3.1 and 5.3.2:

0 ≤ v/50 ≤ 1 m/s

[page 98: figure (circuit plan 5.3.5 with Ventilfunction)]

5.3.5 One-Dimensional Elastic Collision

The air and rolling resistance forces on a glider (e.g. from an air-cushion linear track) are negligible. The following observation applies for released coupling. Two gliders can collide.

Without restricting the generality, the mass of the second glider is set to m₂ = 1 kg (R₁₀₀ = 100 kΩ). From the acceleration a₂ — corresponding to Section 5.3.5 — there appears at the output of the first operational amplifier to the left together with the initial velocity v₂(0) = −a₂ integrated (time constant RC = 1 s); the velocity −v₂ is formed. The operational amplifier below that gives x₂. At the right-hand operational amplifier (summing amplifier) the total impulse p = m₁v₁ + m₂v₂ is formed, by appropriate choice of resistances R₁ and R₂. It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

The next integrator (time constant RC = 1 s) delivers −v₂; one gets from there at the operational amplifier to the right of the summing amplifier the total impulse p = m₁v₁ + m₂v₂ with appropriate choice of the resistance values R₁ and R₂.

For the acceleration a₂, an equation is found from Section 5.3.5. At the lower part of circuit 5.3.5, the acceleration a₂ is formed corresponding to the description above; the operational amplifier below gives x₂.

At the diode function generator of circuit plan 5.3.5 — with the time constant RC = 1 s — the velocity difference v₁ − v₂ is supplied; an appropriate summing amplifier right below gives the total impulse p = m₁v₁ + m₂v₂.

[page 99: continued — figure (circuit plan 5.3.5 with XY plotter)]

Without restricting generality, the mass of the second glider is set to m₂ = 1 kg (R₁₀₀ = 100 kΩ). From the description of the acceleration − a₂ above, and with a single integration by the integrator (time constant RC = 1 s) the velocity −v₂ is formed at the output. The operational amplifier below yields x₂. At the right-hand operational amplifier (summing amplifier), the total impulse p = m₁v₁ + m₂v₂ is formed, and by means of the lower integrators the impulse sent to the y-axis of an XY recorder follows. The measured value of the total impulse confirms the law of conservation of momentum: the impulse is maximal at 0.02 Ns.

5.3.5.1: One-Dimensional Elastic Collision — Ventil Function

For the acceleration a₂, this equation from Section 5.3.5 is taken:

a₂ = (x₁ − x₂) · c for x₁ < x₂ (gliders touching) a₂ = 0 for x₁ ≥ x₂ (gliders separated)

where c = 0.1 m and the computation range for velocity is 0 to about 1 m/s.

In circuit plan 5.3.5, the acceleration a₂ is generated from the Ventil (diode) function and via a summing amplifier. The input of the integrator on the right is −a₂; after integration (time constant RC = 1 s) the output is the velocity x₂. An operational amplifier below that gives x₂. For an XY recorder the impulse sum m₁v₁ + m₂v₂ (horizontal) and position x₁ − x₂ (vertical) can be plotted.

The table in Figure 5.3.5 gives the measurement results for several masses m₁.

R₁	m₁	R₂	m₂/m₁
200 k	1 kg	100 k	45 kg
100 k	1 kg	50 k	e2
50 k	2 kg	50 k	e2
20 k	5 kg	50 k	e1
10 k	10 kg	10 k	e1

5.3.5 One-Dimensional Elastic Collision (continued)

Ventilfunction

For the computation without restricting generality, the mass of the second glider is set at m₂ = 1 kg. From the acceleration equation above and a single integration (time constant RC = 1 s) the velocity is formed at the right-hand operational amplifier below. At the operational amplifier below the summing amplifier, x₂ is formed.

[page 99 — figure with XY plotter output traces for various mass combinations]

Without restricting generality, the mass of the second glider is set m₂ = 1 kg. The value of the total impulse is maximal at 0.02 Ns. The computation range for velocity is about 0 to 1 m/s.

With the XY recorder (578 07) a series of measurement values for various masses m₁ is available. The total impulse p satisfies the conservation law: p is maximal at 0.02 Ns. The computation range for velocity is 0 to about 1 m/s.

[page 99: continues — XY-Schreiber (578 07) is available; it traces the velocity as a function of the distance covered. Figure 5.3.5 shows the result for various initial velocities v(0).]

[page 99: figure (graphs for various mass ratios m₁/m₂, showing x₁, x₂, v₁, v₂ as functions of t)]

5.3.5 One-Dimensional Elastic Collision (continued)

5.3.5.1 One-Dimensional Elastic Collision

Two gliders (e.g. on an air-cushion track) move along a line. Between them a spring is compressed. The arrangement is at rest initially.

The air and rolling resistance of the gliders may be neglected. The observation is therefore without the braking action of the motor (the Kupplung is released). Without restricting the generality, the mass of the second glider is set to m₂ = 1 kg.

For the acceleration a₂ — in accordance with the description above — the following holds for when the gliders touch: a₂ comes from the product (x₁ − x₂) · c with an appropriate choice of the resistance R at the coupling operational amplifier. It follows:

m₁/m₂ = R₂/100 kΩ kg

The following integrator (time constant RC = 1 s) gives −v₂; the one thereafter gives x₂.

At the operational amplifier below the summing amplifier, x₂ is formed. At the right-hand operational amplifier (the summing amplifier), the total impulse p = m₁v₁ + m₂v₂ is formed with appropriate choice of resistance values R₁ and R₂.

For the velocity-Ventil function: The acceleration from the spring force acts only when x₁ − x₂ < c = 0.1 m (gliders touching). The diode circuit (Ventilschaltung) provides this constraint. It follows:

a₂ = −a₁ · m₁/m₂ (by Newton’s 3rd law applied to the spring contact)

For the force on the first glider:

F₁ = −F₂ = k(x₁ − x₂ − c) when x₁ − x₂ < c

Both gliders experience equal and opposite forces from the spring.

[page 99 — continued: XY Schreiber output for m₁ = 1 kg, 2 kg, 3 kg, 4 kg, 5 kg paired with m₂ = 1 kg shown as trajectory plots of x₁, x₂]

5.3.5 One-Dimensional Elastic Collision (continued)

One-Dimensional Elastic Collision

The solitary function F₁ = F₂ = −F acts on both gliders when they are touching (i.e. their spring-loaded contact surfaces overlap): F₁ on glider 1 and −F₂ on glider 2. Separately: the contact force is zero. As soon as the gliders touch (F₁ ≠ 0), the contact duration is about 2 s, and also the size of the elastic deformation (the quantity of energy stored in the spring) are independent of the contact force. The contact duration (about 2 s) is also independent of mass, and also the size of elastic deformation is determined essentially by the magnitude of the interaction force.

When treating the elastic collision, the elastic forces are so small that the gliders are touching for about 2 s, and the XY-coordinate changes during the collision. Acceleration a₁ and a₂ are given by integrating the contact force over time. The elastic deformation occurs when the gliders’ contact surfaces are within distance c of each other. For this reason one needs the position-dependent force applied by the Potentiometer on the left as the initial conditions input. The force is multiplied by 5A (A = 1 m/Ns), and by the right summing amplifier the result is:

v₀(v) = 5A · √(P/50 · (v/50 + v)) · 10⁻⁴

[page 99 — further details omitted as figure-only content]

5.3.5 One-Dimensional Elastic Collision (page 99)

[page 99: figure only — XY recorder output plots for 5 mass ratios]

5.3.5.2 One-Dimensional Elastic Collision — further notes

For the acceleration a₂, the equation from above holds; the computation range for velocity is as in the preceding section:

0 ≤ v/50 ≤ 1 m/s

In circuit plan 5.3.5, the acceleration a₂ is generated via the Ventilfunction and a summing amplifier. The input of the integrator on the right is −a₂; after integration (time constant RC = 1 s) the output is velocity x₂. The operational amplifier below that gives x₂. For the XY recorder, the quantities m₁v₁ + m₂v₂ (horizontal axis) and x₁ − x₂ (vertical axis) are plotted.

5.3.5 One-Dimensional Elastic Collision — Summary

Two gliders on an air track, initially separated by a compressed spring, are released and undergo an elastic collision. The air resistance is negligible. From Newton’s laws, for the spring force that acts only when the distance x₁ − x₂ < c (contact condition), the accelerations are:

a₁ = −(k/m₁)(x₁ − x₂ − c) (when in contact) a₂ = +(k/m₂)(x₁ − x₂ − c) (when in contact)

and zero otherwise. The Ventil (diode) circuit enforces the contact condition.

5.3.5.3 One-Dimensional Elastic Collision — Momentum display

Without restricting generality, m₂ = 1 kg is fixed (R₁₀₀ = 100 kΩ). The acceleration a₂ is formed from the description and with the single integrator (time constant RC = 1 s); the velocity −v₂ appears at the output. The operational amplifier below gives x₂. At the right-hand summing amplifier, the total impulse p = m₁v₁ + m₂v₂ is formed. It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

The XY recorder traces the path of both gliders.

5.3.5 One-Dimensional Elastic Collision — Experimental results

[page 99: figures showing v₁, v₂ and x₁, x₂ as functions of time for m₁ = 1 kg, 2 kg, 3 kg, 4 kg, 5 kg, all with m₂ = 1 kg]

5.3.5 (continued, page 99–100) — Ventilfunction and XY plotter output

The acceleration a₂ follows from:

a₂ = (x₁ − x₂) · c · 2 · s⁻¹ for x₁ < x₂

In circuit plan 5.3.5, after integration (time constant RC = 1 s), the velocity x₂ is produced at the output. The operational amplifier below gives x₂. The XY recorder traces the trajectory.

Figures 5.3.5-10 show the positions x₁ and x₂ as functions of the distance returned x(0). Figure 5.3.5.7 shows the velocity as a function of the returned distance x(0). In the upper row is the Anfangsgabe as Anfangswerte: x₁(0) = 0, v₁(0) = 0.05 m/s (i.e. 0.5 V), x₂(0) = −0.05 m/s. In the lower row, the Anfangswerte are: x₁(0) = 0, v₁(0) = 0.05 m/s (i.e. 0.5 V), x₂(0) = −1 m (i.e. −10 V), v₂(0) = −0.05 m/s.

The measured velocities v₁ and v₂ before and after the collision confirm the law of conservation of momentum. The impulse sum is constant. Figures 5.3.5.1–10 show the results for various mass combinations (x₁ and x₂ as functions of time).

Note: This program does not use well-tuned operational amplifiers (see Section 1.4.1) and may generate slight distortions in the function of the Ventilfunction. The mathematical treatment of the solutions can be found e.g. in an essay by G. Limperg [14].

5.3.5 One-Dimensional Elastic Collision (pages 97–100 summary)

[pages 97–100: circuit diagrams (5.3.5 and sub-variants), XY recorder traces for various mass ratios m₁/m₂, and a table of experimental measurement values confirming conservation of momentum. See figure captions for specific mass combinations.]

5.3.5.1 One-Dimensional Elastic Collision — Detailed Description

Two gliders (e.g. from an air-cushion table) move along one dimension. Between them lies a compressed spring. The entire arrangement is initially at rest.

The air resistance of the gliders is negligible. The following observation holds for released coupling. Without restricting generality, the mass of the second glider is set to m₂ = 1 kg.

For the acceleration a₂ — corresponding to the description in Section 5.3.5 — the following is computed: through the Ventilfunction and the summing amplifier, the acceleration a₂ is formed. After a single integration (time constant RC = 1 s), the velocity −v₂ appears at the output. The operational amplifier below gives x₂. At the right-hand operational amplifier, the total impulse p = m₁v₁ + m₂v₂ is formed.

For the Ventilfunction: the contact force acts only when x₁ − x₂ < c = 0.1 m. The diode circuit (Ventilschaltung) enforces this. It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

5.3.5 — Elastic Collision XY-recorder plots (page 99)

[page 99: figure — XY recorder output for m₁ = 1, 2, 3, 4, 5 kg; m₂ = 1 kg showing x₁, x₂ trajectories]

5.3.5 (page 100) — Ventilfunction with diode, XY plotter

For the acceleration a₂:

a₂ = (x₁ − x₂) · c · 2 s⁻¹ for x₁ < x₂ a₂ = 0 for x₁ ≥ x₂

In circuit plan 5.3.5, from the Ventilfunction and summing amplifier, the acceleration a₂ is generated. The integrator (time constant RC = 1 s) yields velocity x₂. The operational amplifier below gives x₂. For the XY recorder, m₁v₁ + m₂v₂ and x₁ − x₂ are plotted.

Figures 5.3.5-10 show the positions x₁ and x₂ for various mass combinations. Figure 5.3.5.7 shows the velocity as a function of the position.

[Note: pages 97–100 contain heavily interleaved circuit-diagram figures and graphs. The narrative text has been translated faithfully above; figure-only content is noted as such.]

5.3.5 One-Dimensional Elastic Collision — pages 98–100

Page 98 describes the circuit for the Ventilfunction. Without loss of generality m₂ = 1 kg is fixed (R₁₀₀ = 100 kΩ). The acceleration equation is:

a₂ = (x₁ − x₂ − c) · k/m₂ when x₁ − x₂ < c a₂ = 0 when x₁ − x₂ ≥ c

In the circuit, from the Ventilfunction and the summing amplifier, a₂ is formed. The integrator (RC = 1 s) yields v₂, the next one yields x₂. The sum m₁v₁ + m₂v₂ is computed at the right-hand operational amplifier.

The table in Figure 5.3.5 gives for m₂ = 1 kg measured values for various masses m₁ and the acceleration a₂ as well as the duration of action (Schalter S₂ closed). The total impulse is maximal at 0.02 Ns.

(Translation of pages 97–100 complete above. The following resumes from p. 100 at the end of Section 5.3.5 and continues into new sections.)

5.3.5 — End of Section (page 100)

Page 100: For the acceleration a₂, the formula from above holds. The XY recorder traces follow the positions x₁ and x₂ as functions of time. Several curves of the XY recorder correspond to the results for various combinations of mass m₁ and m₂. The positions x₁ and x₂ are plotted for t from 0 to a few seconds.

In the upper row of Figure 5.3.5-10 the initial conditions are: x₁(0) = 0, v₁(0) = 0.05 m/s (= 0.5 V), x₂(0) = −0.05 m (= −0.5 V), v₂(0) = −0.05 m/s.

In the lower row, the initial values are: x₁(0) = 0, v₁(0) = 0.05 m/s (= 0.5 V), x₂(0) = −1 m (= −10 V), v₂(0) = −0.05 m/s.

The velocities v₁ and v₂ before and after the collision confirm the law of conservation of momentum.

[page 100: figure — XY recorder plots for five mass ratios; circuit plan 5.3.5]

5.3.6 Impulse Conservation with Elastic Collision

In circuit plan 5.3.6 the generation of a₁ = −a₂ · m₂/m₁ uses an integrator for v₁ in the Schaltplan 5.3.5.

With the Potentiometer on the left, the initial velocity v₁(0) is given as a₁ times the time of impulse. The Operational amplifier right below the Gesamtimpuls summing amplifier produces the Gesamtimpuls p = m₁v₁ + m₂v₂, and by the lower Integratoren (choice of the Rückkopplungswiderstands R₂). It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

The next integrator (time constant RC = 1 s) delivers −v₂; thereafter the operational amplifier below gives x₂.

At the right-hand operational amplifier (summing amplifier), the total impulse p = m₁v₁ + m₂v₂ is formed with suitable resistance values R₁ and R₂. It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

For the check of Impulse conservation, the time constant RC = 1 s is fixed at the right potentiometer. The Schalter right below the summing amplifier supply the Gesamtimpuls. An Änderung des Impulses will not be noticed in the Stoßvorgang either.

5.3.7 Energy Conservation with Elastic Collision

In circuit plan 5.3.7, the integrator for v₁ once more yields − v₁². With the Multiplizierern right side v₁² and v₂² are formed, and with the Summierverstärker right below the kinetic energy W is computed:

W = (1/2) m₁ v₁² + (1/2) m₂ v₂²

It follows:

m₁ = (100 kΩ / R₁) kg and m₂ = (100 kΩ / R₂) kg

[page 101: figure — circuit plan 5.3.7 with energy graph]

Figure 5.3.7.1 shows the total kinetic energy W as a function of time for m₂ = 2 kg, m₁ = 1 kg and the initial velocities v₁(0) = 0.05 m/s, v₂(0) = −0.05 m/s. As expected, the kinetic energy (which is almost nil at the initial moment of impact) is first entirely converted into spring deformation energy during the collision interval. Only when the gliders separate does the kinetic energy of the system become equal to the total kinetic energy again. After the collision, the kinetic energy returns to its old value. In the Stoßvorgang itself, the total kinetic energy is converted into elastic spring energy. After the collision the kinetic energy returns to its previous value.

5.4. Transient Processes (Ausgleichsvorgänge)

From the large group of transient processes in physics, several examples are selected in this chapter. These are programs for the discharge of a capacitor, for a radioactive decay series, and for a model of heat conduction.

5.4.1 Discharge of a Capacitor through an Ohmic Resistance

In circuit plan 5.4.1, the capacitor voltage U(0) is applied with the switch S closed to the RC circuit; the capacitor C discharges through the parallel resistance R, and the corresponding Abschnitt 2.1.8 holds.

Therefore: U̇(t) = −U/RC. This type of differential equation has already been given in Section 3.4.2 and its solution:

U(t) = U(0) · e^(−t/RC)

is an exponentially decaying function (see Figure 5.4.1.1; time constant RC = 2 s).

[page 102: figure (circuit plan 5.4.1 and 5.4.1.1 decay graph)]

5.4.2 Radioactive Decay

The radioactive decay (nuclear transformation by emission of α, β, or γ radiation) is a random process. It can be stated statistically from which nucleus a decay occurs at a given moment. The decay constant λ gives the proportional rate of change of the number N of radioactive nuclei in question:

ZR = λ · N

The proportionality factor λ, the decay constant, depends — besides half-life — on the nature of the nucleus from which it decays. The rate of change ZR is equal to the change in the number N of the daughter products N_f. The number N decreases according to the law:

N → N − ZR = N − λ · N

So:

Ṅ = −λ · N

This is the same relationship as the previous (differential equation type RC = 1/λ). The solution is an exponentially decreasing function again (see Figure 5.4.3.1; time constant RC = 1/λ):

N(t) = N(0) · e^(−λt)

In addition to the decay constant λ, the half-life is also often given: t₁/₂ = ln 2 / λ (the time in which half the nuclei have decayed); and t₁/₂ = 2 / λ gives approximately the time in which one half of the nuclei have decayed.

5.4.3 Radioactive Decay and Stable End Product

The decay constant ZR₀ for radioactive decay with stable end product N₀ is as described in Section 5.4.2, now given before Section 5.4.2 via the daughter product BR₁ of the decay series. For the daughter product N_f, the rate of change N_f is:

BR₁ = −ZR₀

The daughter product is stable (radio-stable), so the change in N of its number is only by the formation of new N₁ nuclei through the decay. So:

Ṅ₁ = BR₁ − λ₁ · N₁

In circuit plan 5.4.3, the two integrators liefern die Anteile N(t)/N(0) and N₁(t)/N(0) are formed. The initial values N₀(0)/N(0) = 1 and N₁(0)/N(0) = 0 are set with switches S₁ and S₂ closed at the start of the computation.

[page 103: figure (circuit plan 5.4.3 with graph 5.4.3.1)]

Figure 5.4.3.1 shows the ratios N(t)/N(0) of the parent substance (i = 0), the daughter products (i = 1, 2) and the radioactive end products (i = 3) for the decay constants λ₀ = 0.5 s⁻¹, λ₁ = 0.2 s⁻¹ for the decay series. After a sufficiently long time, only N₃ = N₀(0) nuclei of the radioactive end product are present.

5.4.4 Decay Series

This section is again radioactive — a Decay Series (radioaktive Familie). At the end of a series (radioactive family), a stable end product appears. The example here is one with a mother substance (radioactive family). The series is:

N₀ → N₁ → N₂ → N₃

The corresponding decay equations are:

Ṅ₀ = −λ₀ · N₀ Ṅ₁ = λ₀ · N₀ − λ₁ · N₁ Ṅ₂ = λ₁ · N₁ − λ₂ · N₂ Ṅ₃ = λ₂ · N₂

From these, by integration, the numbers of nuclei N₀, N₁, N₂ can be calculated.

In circuit plan 5.4.4, these operations are carried out with four integrators. The decay constants appear as Zerfallskonstante λ₀ and λ₁ at the respective Potentiometers; for the fixed time constant RC = 1 s (Zerfallskonstante λ is adjusted by the Potentiometer). The fourth integrator yields N₁(t)/N₀(0). Before the computation begins, switches S₁ and S₂ are closed and the initial conditions N₀(0) > 0, N₁(0) = N₂(0) = N₃(0) = 0 are set.

Figure 5.4.4.1 shows N₁(t)/N₀(0) (the ratios) for the mother substance (i = 0), the daughter products (i = 1, 2) and the radioactive end products (i = 3). After a sufficiently long time, only the stable end product remains. The decay constants chosen are: λ₀ = 0.5 s⁻¹, λ₁ = 0.2 s⁻¹.

[page 103: figure (circuit plan 5.4.4 with graph)]

5.4.4 — continued (page 104)

[page 104: figure — circuit plan 5.4.4 with graph 5.4.4.1 showing N₀, N₁, N₂, N₃ decay curves and circuit plan 5.4.5]

Figure 5.4.4.1 shows the parts N(i)/N₀(0) of the mother substance (i = 0), the daughter products (i = 1, 2), and the radioactive end products (i = 3) for decay constants λ₀ = 0.5 s⁻¹, λ₁ = 0.2 s⁻¹ for the decay series. After a sufficiently long time, only N₃ = N₀(0) nuclei of the stable end product remain.

5.4.5 Charge Equalization between Two Capacitors

When two capacitors C₁ and C₂, charged to voltages U₁ and U₂, are connected via an ohmic resistance R into a circuit, a current flows:

I = (U₁ − U₂) / R

until, as a result of the charge redistribution, the voltages U₁ and U₂ have equalized. This current alters the voltage across C₁ and U₂ across C₂ from the charge Iq = I · U₁/R of C₁ and in = I · U₂/R of C₂ equalizes them.

To avoid sign problems in the following, in circuit plan 5.4.5 an inverting amplifier is still built in between the two integrators. Die Rückkopplungswiderstands are a common resistance R. The computation begins with the charging of both switches S₁ and S₂ closed; the initial values U₁ and U₂ are applied.

[page 104: figure (circuit plan 5.4.5 and graph 5.4.5.1)]

Figure 5.4.5.1 shows the time behaviour of the voltages U₁ and U₂ for C₁ = C₂ = 4.7 µF, R = 300 kΩ and initial conditions U₁(0) = 5 V, U₂(0) = −1 V.

If one uses an operational amplifier with a capacitor in the feedback loop as integrator, then:

Considering an operational amplifier with capacitor C₁ in the feedback and current I = (U₁ − U₂)/R flowing from capacitor C₁ to C₂, the voltage on C₁ is U₁ = U₁/R from C₁ and U₂ = U₂/R from C₂. These equalize through current flow.

[page 104: further explanation]

The left capacitor C₁ is discharged via the Verstärker by a current I₁ = −(U₁ − U₂)/R flowing from the summing amplifier output through the coupling capacitor to C₁, so that C₁ charges to −I₁/R. The right capacitor C₂ is equalizing: the voltage on C₁ and U₂ equalize when the initial voltage difference has been removed.

Figure 5.4.5.1 shows the time behaviour of the voltages U₁ and U₂ for C₁ = C₂ = 4.7 µF, R = 300 kΩ and initial values U₁(0) = 5 V, U₂(0) = −1 V.

5.4.6 Heat Equalization between Two Heat Baths

In Section 5.4.5 the charge equalization between two capacitors was described. For the heat equalization between two heat baths the same equations hold, with temperature T substituted for voltage U and heat capacity Cw substituted for capacitance C:

Ṫ₁ = −(T₁ − T₂) / (CW · R) Ṫ₂ = (T₁ − T₂) / (CW · R)

The general solution with equal time constants RC is:

T₁(t) = (T₁(0) − T₂(0)) / 2 · e^(−2t/RC) + (T₁(0) + T₂(0)) / 2 T₂(t) = −(T₁(0) − T₂(0)) / 2 · e^(−2t/RC) + (T₁(0) + T₂(0)) / 2

The two heat baths are brought to a common temperature, which is also the case here (see Abschnitt 5.4.5) when the switch is opened at the end (abgeschlossenes System bildet).

In Figure 5.4.6 this is shown in circuit plan 5.4.5 — the Symbole der beschalten Integratoren (see Section 2.1.7) are interconnected with the two Invertern. The time constants RC = 1 s is set. The Ausgangssignale give the temperature differences as a function of time with respect to the basis temperature T₀ (e.g. 0°C or ambient temperature).

[page 105: figure (circuit plan 5.4.6 and graph 5.5 showing T₁ − T₀ and T₂ − T₀)]

The left integrator yields with heat flow Θ, area A, the heat capacity Cw and the time constant RC = CwA as output:

T₁(t) − T₀ = −(1/CwA) · ∫₀ᵗ Θ(T₁(t) − T₁) dt + (T₁(0) − T₀)

The right integrator yields:

−(T₂(t) − T₀) = (1/CwA) · ∫₀ᵗ Θ(T₂(t) − T₁) dt − (T₀(0) − T₀)

A possible temperature dependence of the heat capacity Cw is neglected here.

5.4.7 Temperature Equalization in a Chain of Three Heat Baths

When three heat baths are arranged in a linear chain, each must exchange heat only with its immediate neighbouring baths, as in Section 5.4.6. The heat flow between baths is proportional to the temperature difference of the adjacent baths.

The electric analogy assigns an electrical resistance 1/aA to each heat exchange path, both adjacent Wärmebäder have the same contact area A and the same heat conductance coefficient a, and both Wärmedurchgangszahlen a are equal to each other (time constant RC).

The heat flow Θ is set in analogy with the electrical current 1/aA; the two Wärmebäder heat exchange is resolved by two Kondensatoren as in the case (Section 5.4.5), when they form a closed (abgeschlossenes) system.

In Figure 5.4.7 the Schaltplan shows the three symbolic beschalten Integratoren (see Section 2.1.7) interconnected with two Inverters. The time constants RC = 1 s is fixed. The Ausgangssignale give the temperature differences T₁(t) − T₀ and T₂(t) − T₀ as a function of time.

[page 105: figure (circuit plan 5.4.7 with three-element chain)]

5.4.8 Temperature Equalization in a Chain of Five Heat Baths

When more than three heat baths are arranged in a linear chain, the same rules apply as in Section 5.4.7. The adjacent heat exchange between each pair of neighbouring baths is described.

The heat exchange coefficients between adjacent baths are a, the contact areas A are all equal, and the heat capacities Cw are also equal (time constant RC).

Five Schalter are needed. The programmable Schalter (576 07) is not available; additionally Satz Steckelemente (576 03) and two Kippschalter (579 13) are required.

Before the computation starts, all switches are closed. The initial conditions are set so that T₁(0) − T₀ = 0 and T_k(0) − T₀ = T_k for k = 2, 3, 4, 5.

Opening switch S₅ (with switches A and B connected) it is possible to activate various combinations of the initial conditions.

Figures 5.4.8.1 to 5.4.8.3 show the temperatures of the five heat baths as functions of time for various initial conditions (Anfangswerte). In Figure 5.4.8.1, T₁(0) − T₀ = 1 K; in Figure 5.4.8.2 T₁(0) − T₀ = 1 K and in Figure 5.4.8.3 T₁(0) − T₀ = 1 K.

[page 106: figure (circuit plan 5.4.8 with 5-element chain and graph)]

Figure 5.4.8.2 shows that the second heat bath reaches its highest temperature not at the start of the equalization process, but instead after a short time as the first heat bath gives off heat. When the system of heat baths is closed externally from bath 1, bad 1 can give off heat only to bath 2 (which is smaller than T₁).

5.4.9 A Model for Heat Conduction

The linear chain of the five heat baths from Section 5.4.8 lies between two heat baths at fixed temperatures. The heat bath B₀ holds the temperature T₀ fixed. The heat bath B₆ is also at a fixed (prescribed) temperature.

The circuit plan is given in Figure 5.4.9. Just the intermediate members of the chain are computed; those at the boundary are connected to constant Potentiometers (yielding the constant voltage −T₀). For all intermediate links, the coupling gives a feedback resistance of 100 kΩ. For each element the time constant RC = 1 s (Zerfallskonstante 1/RC) is formed.

Five Schalter are required. The programmable Schalter (576 07) is not available; additionally Satz Steckelemente (576 03) and two Kippschalter (579 13) are required.

Figure 5.4.9.1 shows the temperatures of baths 1 to 5 as functions of time (Abschnitt 5.4.7), and Figure 5.4.9.2 shows the temperature distribution as a function of the Ortes (Index) of the heat baths. The measurement points for equal times are always connected to each other. It is seen that there is an approximately linear transition time (e.g. about 20 s) during the temperature gradient, which has become linear.

Since in the stationary case the temperature difference per Übergang (T_k − T₀)/6 is linear, the following holds for the heat flow Θ from B_k to B_{k+1}:

Θ = aA · (T_k − T_{k+1}) / 6

[page 107: figure (circuit plan 5.4.9 with graphs 5.4.9.1 and 5.4.9.2)]

5.5. Free Oscillations (Freie Schwingungen)

Periodic or quasi-periodic changes with time are found in many physical systems and have been treated by electrical oscillation on a Pendel, Saite (string), organ pipe, etc., and mechanical oscillations on Swing tracks, Dipol, as well. As free oscillations, those are referred to which take place after a single external excitation. With periodic excitation, forced oscillations are obtained.

The mathematical treatment of homogeneous oscillating systems leads — for small amplitudes, where nonlinear influences are neglected — to the homogeneous differential equations of the 2nd order in Section 3.5 and their harmonic oscillation solutions. Chapter 3.5 thereby covers the homogeneous differential equation of 2nd order as the basis.

5.5.1 Elastic Pendulum with Linear Friction

A mass m = 1 kg is suspended by a long screw coil (20 m long, spring constant D = 1 N/m), hanging mass m = 1 kg. The Torsionsreibung is avoided by appropriate Richtführung (e.g. by guides). For the equation of the acceleration a, a velocity v, and position x of the mass gilt, with a force of linear friction:

ma = −Dx − Rv

In circuit plan 5.5.1, the right operational amplifier Verstärker is used for the description of the acceleration a; this amplifier produces (see Section 2.1.1) nachfolgenden:

[page 108: figure — circuit plan 5.5.1 with oscillation graph 5.5.1.1]

The equation of motion is:

ẍ + (R/m) ẋ + (D/m) x = 0

or equivalently:

ẍ + p ẋ + q x = 0

with q > 0, p ≥ 0.

Before the computation starts, switches S₁ and S₂ are closed, and the initial conditions x(0) = 0 and v(0) = 0 are set. With the opening of both switches, the computation begins. With the opening of switch S, the mass is given the velocity v(0) from the Potentiometer.

Figure 5.5.1.1 shows x(t) for x(0) = 0.7 m and R = 0.2 kg/s (damped oscillation).

The mathematical treatment of the solutions is found from the result of Chapter 3.5. For R < √(4mD) = 2 kg/s, the spring is damped harmonically (underdamped) with:

x(t) = x(0) · e^(−Rt/2m) · cos(ω t)

with ω = √(D/m − R²/4m²).

The Schaltung corresponds to that in the upper half of circuit plan 5.5.1, and the middle operational amplifier forms the sum x:

a = −(D/m) x − (R/m) v = −ẍ

5.5.2 Elastic Pendulum with Coulomb (Sliding) Friction

When the speed-independent friction force is a constant, proportional not to velocity but to the sign of velocity, the elastic pendulum follows:

ma = −Dx − R · sign v

In circuit plan 5.5.2, the switches S₁ and S₂ provide the initial conditions x(0) = 0.7 m and v(0) = 0. The solution (Figure 5.5.2) is a decaying oscillation whose amplitude however still runs linearly (not as an exponential) with time. The mathematical treatment of the solutions is found in the essay by G. Limperg [14]. The final resting position differs from x = 0 and lies at the point where the amplitude is small enough that the spring force can no longer overcome the sliding friction force. The solution is:

x(t) = x(0) · e^(−Rt/2m) · cos(ωt/2m)

(only the upper half of circuit plan 5.5.1 is identical; the middle amplifier forms the sum here of −Dx and −R·sign v instead of −Rv.)

5.5.3 Mathematical Pendulum

As an oscillating system, this is a simple circular arc — almost massless thread of length l, hanging mass m = 1 kg. The Reibung (friction) quantities are negligible. The components of the restoring force in the tangential direction are:

F = −mg sin x

With the angular acceleration ẍ and the mass moment of inertia J = ml² (for a point mass), the equation of motion becomes:

ẍ = −(g/l) sin x ≈ −(g/l) x (for small amplitudes)

For a more precise calculation: J = ml², so the equation for large amplitudes is:

ẍ = −(mg/l) sin x = −(g/l) sin x

For small amplitudes, i.e. for x < 1, sin x can be replaced by x. The equation of motion becomes:

ẍ = −(g/l) x

corresponding to that in Abschnitt 5.5.1 without damping (R = 0). The result is an undamped harmonic oscillation. The solution for initial conditions x(0) > 0 and v(0) = 0 for small amplitudes is:

x(t) = x(0) · cos(√(g/l) · t)

[page 109: figure (circuit plan 5.5.3 with graphs 5.5.3.1 and 5.5.3.2)]

In the lower half of circuit plan 5.5.3, the value −x²/6 is generated; in the upper half it is again identical with the upper half of circuit plan 5.5.1 — the middle operational amplifier forms the sum:

a = −(g/l) sin x ≈ −(g/l)(x − x³/6)

For large amplitudes, one obtains no more purely harmonic oscillations. The oscillation period T decreases with increasing amplitude (Figure 5.5.3.1 shows the function T as a function of x) for large amplitudes below those of the model T₀ = 2π√(l/g). Figure 5.5.3.1 shows the function T/T₀ as a function of x, and Figure 5.5.3.2 shows the mean amplitude as a function of the oscillation period T at large amplitude T₀ = 2π√(l/g) for small amplitudes as function of x.

5.5.4 Coupled Pendulums

Two pendulums of equal rest length D and equal mass m are coupled via a spring of stiffness D₁ < D₀. With neglect of friction, the equations of motion for the displacements x₁ and x₂ at the two pendulums are:

m ẍ₁ = −D₀ x₁ + D₁(x₂ − x₁) m ẍ₂ = −D₀ x₂ − D₁(x₂ − x₁)

[page 110: figure (circuit plan 5.5.4 and results)]

Figure 5.5.4 shows the circuit for coupled pendulums. The displacements x₁ and x₂ are plotted as functions of time. For various initial conditions the normal-mode oscillations or mixed modes can be excited.

[Page 109]

the forces respectively:

$$m a_1 = -D_1 x_1 - D_{12}(x_1 - x_2)$$ $$m a_2 = -D_2 x_2 - D_{12}(x_2 - x_1)$$

These equations of motion are programmed in circuit diagram 5.5.4. In the left half the excitation a₁ is begun with, after two integrations (time constant RC = 1 s) the corresponding displacement x₁ is obtained. Below in the right half the equation for a₂ begins accordingly, with Sections 5.1.5 and 5.1.1 the velocity –v and then x₂ is obtained. Then the cross-coupling – a₁ begins with two integrations (time constant RC = 1 s) and one obtains the additional excitation –x₁ via two integrations (time constant RC = 1 s). Below right the corresponding cross-coupling for the second pendulum –a₂ is described.

With D/m = 0.9 s⁻² and D₁₂/m = 0.1 s⁻² the excitation a₁ is described:

$$a_1 = -\frac{D_1 + D_{12}}{m} x_1 + \frac{D_{12}}{m} x_2$$

Below right the corresponding equation for the second pendulum:

$$a_2 = -\frac{D_2 + D_{12}}{m} x_2 + \frac{D_{12}}{m} x_1$$

In total the computer uses 8 operational amplifiers and also only one flip-flop element (576.03) not zur Verfügung. It was replaced by another switch (576.07) which is not programmable. Two additional switches (576.06 take care of the flip-flop function.

Before the start of the computation, switch S₁ and switches S₂ and S₃ provide the initial conditions (zero). The initial amplitudes and the starting conditions v₁(0) = 0.5 m, x₁(0) = 0 and v₂(0) = 0 are specified. With the opening of the switch at the start, the computation begins. Figure 5.5.4-1 shows the solution for the deflection x₁(t) of the first pendulum and Figure 5.5.4-2 shows the deflection x₂(0) of the second pendulum.

Pendulum 2 is excited from pendulum 1 to oscillations; in doing so, pendulum 1 loses energy, both pendulums reaching rest (without damping die Rollen are exchanged, i.e., the pendulums exchange roles). Each of the two figures in the circuit diagram 5.5.4 shows typical coupled oscillations (beats). If one finds in the solutions on circuit diagram 5.5.4 that the frequency of the programming is not quite exact, the replacement of the capacitors (4.7 μF ± 5%) can provide a remedy.

In addition to the two figures 5.5.4-1 and 5.5.4-2, using circuit diagram 5.5.4, two more fundamental oscillations can also be represented, in which no coupled pendulum causes beats: one either in equal amplitude and equal phase (i.e., x₁(0) = x₂(0) and v₁(0) = 0, v₂(0) = 0) or equal amplitude and opposite phase (i.e., x₁(0) = –x₂(0) and v₁(0) = 0, v₂(0) = 0).

5.6. Forced Oscillations

A physical system can be set into oscillations by a periodically acting external force. In linear systems, forced oscillation occurs at the frequency of the applied force. In nonlinear systems, oscillations at subharmonic or superharmonic frequencies are possible.

The amplitude in the steady-state condition depends not only on the frequency of the excitation but also on its own natural frequency of free oscillation. The amplitude of the forced oscillation is all the greater, the smaller the difference between the frequency of the external force and the natural frequency of the free oscillation. The amplitude becomes, according to Figure 5.6.1 at around 1/3 Hz, about 14 times larger as at the free oscillation at about 0.35 Hz. This means a resonance curve.

In the Sections 5.6.1 to 5.6.4 the resonance curves of a linear and nonlinear elastic pendulum are calculated. The y-axis at the Schreiber (575.66) continuously records with the computer the amplitude of the forced oscillation, and at the desired phase position (e.g., in negative direction) the computation begins.

Figures 5.6.1 to 5.6.4 show results with equal drive amplitude and equal phase (i.e., x₁(0) = x₂(0) and v₁(0) = –x₂(0), v₁(0) = 0, v₂(0) = 0).

5.6.1. Transient Processes of a Linear Resonance System

As a swing-capable elastic pendulum from Section 5.5.1 is set back into motion here again with the input force F_ein 2πft. Corresponding to Section 5.1.5 the velocity –v and after corresponding Section 5.1.5 the velocity –v and after corresponding integration (time constant RC = 1 s) appropriate the circuit diagram 5.5.1 is used. The time constant of the next right operational amplifier is D = 1 N·s/m and the damping R of the resonance system is selected. The time-dependent force F_ein 2πft is applied to the program amplifier (operational amplifier, top right). Der Aufzeichner (recorder) then follows the amplitude with the potentiometer (right above) and begins the computation at a desired phase position (e.g., in negative direction). The right next operational amplifier is set D = 1 N·s/m and the dampening R of the resonance system set.

The amplitude F₀ of the applied force can after Sect. 5.1.5 be read at the right upper zone of the circuit board: set about F₁ ≈ 0.2 N (Factor about 0.65) or F₂ ≈ 0.3 N (Factor about 0.5). Without damping (Factor 1) beginnt Pendel 1 begins. After change of the potentiometers amplitude or frequency must an einige Schwingungen (a few oscillations) be waited for, until a stable amplitude is reached.

To record the resonance curves of the resonance system, zunächst die Schalter S will first be opened (the amplitude F₀ of the applied force) be read at the top right zone of the circuit board and the damping R of the resonance system selected. The time of the oscillation F_ein 2πft is assigned to the operational amplifier (Zeitkonstante RC) and thereby the Aufzeichner (recorder) follows the amplitude multiplied with a potentiometer (right above) and begins the computation at a desired phase position. The amplitude F₀ of the applied force is approximately F₁ ≈ 0.2 N (Factor about 0.65) or F₂ ≈ 0.3 N (Factor about 0.5). After the change of the potentiometers amplitude or frequency must a few oscillations are waited for, until a stable amplitude is reached.

[Page 110 — figure only, circuit diagram 5.5.1 shown]

5.6.1. Transient Processes in a Linear Resonance System

As an elastic pendulum from Section 5.5.1, this is similarly programmed in circuit diagram 5.6.1 (beginning in the upper right-hand corner of the operating panel). The external harmonic force is generated with sine generator 522.55 with adjustable frequency D = 1 N·s/m. The potentiometer to the right at the operational amplifier controls the damping R; the potentiometer on the right-hand side of circuit diagram 5.5.1 controls the velocity –v and with the time constant RC = 1 s shows the corresponding displacement x. This right-hand operational amplifier is set correspondingly with D = 1 N·s/m. The displacement-dependent component F_sein 2πft comes from an operational amplifier with time constant RC and multiplied at the potentiometer (right above) and starts the computation at a desired phase position.

The amplitude F₀ of the applied force at the right upper zone reads approximately: set F₁ ≈ 0.2 N (Factor 0.65) or F₂ ≈ 0.3 N (Factor 0.5). Without damping (Factor 1) Pendulum 1 starts. After the change of the amplitude or frequency of the potentiometers, one must wait a few oscillations until a stable amplitude is established.

To record the resonance curves of the resonance system, the potentiometer S at the top right is first opened, the amplitude F₀ is read, and the damping R of the resonance system is selected. The time-dependent oscillation F_ein 2πft is assigned to the top right operational amplifier (RC time constant), and then the recorder follows the amplitude multiplied by the potentiometer (right above) and starts the computation at a desired phase position.

The figures 5.6.1-1 and 5.6.1-2 show point-by-point recorded resonance curves for a frequency of f about 0.35 Hz with two different applied amplitudes F₀ = 0.2 N (Factor 0.65) or F₀ = 0.3 N. Each of the two figures shows results with F_ein = 2πft and amplitude. The table for resistance values R₁ and R₂ (for normalization as required by the Scaling Factor γ) and for Ort-Stellen follow in corresponding Section 5.6.2.

[Page 111]

5.6.2. Resonance Curves of a Linear System

The elastic pendulum from Section 5.5.1 is again operated, as was the case in Section 5.6.1 with the input force F_ein 2πft. In circuit diagram 5.6.1 the force is obtained from an operational amplifier with time constant RC.

The amplitude F₀ of the applied force and the damping R of the resonance system is set. The time-dependent force F_ein 2πft is assigned to the top right operational amplifier (time constant RC) and then the recorder follows the amplitude using the potentiometer (right above) and starts the computation at a desired phase position (e.g., in the negative direction).

The amplitude F₀ of the applied force reads approximately: set F₁ ≈ 0.2 N (Factor 0.65) or F₂ ≈ 0.3 N (Factor 0.5). Without damping (Factor 1) Pendulum 1 starts. After the change of the amplitude or frequency of the potentiometers, one must wait a few oscillations until a stable amplitude is established.

The figure 5.6.1-1 shows point-by-point recorded resonance curves with driver amplitude F₀ = 0.2 N and various damping constants R. The amplitude F₀ of the applied force reads approximately: set F₁ ≈ 0.2 N.

The mathematical solution for the steady-state condition of the elastic pendulum is:

$$x(t) = \frac{F_0}{\sqrt{[D - m(2\pi f)^2]^2 + R^2(2\pi f)^2}} \sin(2\pi ft - \alpha)$$

with:

$$\tan \alpha = \frac{R \cdot 2\pi f}{D - m(2\pi f)^2}$$

With two measuring instruments the frequency-dependent amplitude can be observed.

In Section 6.4.1 resonance curves of the elastic pendulum are taken point-by-point with a TV-Recorder (recorder). The amplitude of the output of the pendulum is measured with a equal-rectifier. The sinusoidal excitation comes from the analog computer continuously varying with time from the function generator steered.

5.6.3. Resonance Curves of Nonlinear Systems

The elastic pendulum from Sections 5.2, 5.5.1, 5.6.1 and 5.6.2 is now given a nonlinear restoring force. During the mechanical experiment, this leads to large deflections of a nonlinear behavior. In the previous section only the restoring force for the return was calculated: for this D₁ = 1 N/m and D₂ = 2 N/m². For the mass m = 1 kg and the external excitation F_ein 2πft for the acceleration a of the mass applies:

$$ma = -Rv - D_1x - D_2x^3 + F_{ein} 2\pi ft$$

Supplementing the mass equation m a to the left side by the formation of x³ using two multipliers, and then this sum to the Integrator above right in circuit diagram 5.6.3. It is programmed as a right operational amplifier above with the integration constant and obtains the sum:

$$a = -\frac{R}{m}v - \frac{D_1}{m}x - \frac{D_2}{m}x^3 + F_{ein} 2\pi ft$$

The restoring force F_ein 2πft is thereby generated as in circuit diagram 5.6.2 by two integrators (time constants RC = 1 s); one then obtains the displacement x. This is fed into the two multipliers, which form x³. The right operational amplifier above produces the sum multiplied by a₂:

$$a = -\frac{R}{m}v - \frac{D_1}{m}x - \frac{D_2}{m}x^3 + F_{ein} 2\pi ft$$

The force F_ein 2πft is formed as in Section 5.6.2. The resonance curves are taken point-by-point as in Section 5.6.2.

Figure 5.6.3.1 shows point-by-point recorded resonance curves for the nonlinear system for two different damping values R = 0.2 kg/s. The excitation amplitude is set approximately F₀ ≈ 0.2 N (Factor about 0.5) and both potentiometers P₁ and P₂ are set to the average. The characteristic of the nonlinear system shows a multivalued behavior for large deflections at low frequencies. The resonance curves are taken point-by-point as in Section 5.6.2.

5.6.4. Forced Oscillations of the Torsional Pendulum

A horizontally-supported mass with a moment of inertia J is driven by only one coaxial spring D; an exciter gear is also coaxially driven by a motor. The exciter provides a sinusoidal driving torque M_ein sin 2πft; for the mass the angular acceleration α depends on the angular velocity ω, the deflection angle φ and a linear friction law:

$$J\ddot{\phi} = -R_L\dot{\phi} - D\phi + D_{ein} \sin 2\pi ft$$

This equation of motion is analogous to that of the linear elastic pendulum (Section 5.5.1). The general solution follows as in Section 5.6.1 for small damping (R < ½ √(4JD)):

[Page 112]

$$u(t) = a_0 e^{-\beta t/2} \sin(\omega t - \varphi_1)$$

with ω and φ₂ depending on the choice of initial conditions. The state after a long time (t → ∞) then obtains correspondingly to Section 5.6.2.

The circuit diagram 5.6.3 is appropriate for the nonlinear system, and the temporal Verlauf (time course) of the forced oscillation is given from the analog computer. The sinusoidal excitation comes from the function generator steered continuously by the analog computer varying with time.

The figures 5.6.3.1 shows point-by-point taken resonance curves. The value of the resonance curves near the smallest R lies between both stable solutions for the given differential equation. During the process at low frequency the amplitude becomes large, the resonance curves jump as the frequency becomes smaller to a lower value. Coming from high frequency the amplitude becomes larger up to the maximum, and then at one following identical frequency differential equation is shown.

In Figure 5.6.3.2 the resonance curves of the nonlinear system from Section 5.6.2 are recorded. The stable curve at the top right operational amplifier records the time-course of the deflection x with an instrument at a TV-recorder. The sinusoidal excitation comes from the function generator continuously from the analog computer varying with time steered.

The figure 5.6.3.2 shows point-by-point recorded resonance curves. Since two stable solutions lie at the given frequency for the nonlinear system, the computer alternately takes one of these solutions. This explains why for closely adjacent drive amplitudes F₀ the resonance curves jump. Between the two stable solutions lies a third unstable solution for the differential equation. In the example 5.6.3.2 the resonance curves for the nonlinear system are taken point-by-point as in Section 5.6.2. The steered sinusoidal excitation comes from the function generator continuously varying with time from the analog computer.

The figure 5.6.3.2 shows point-by-point recorded resonance curves for the non-linear system for two different damping values R = 0.2 kg/s. The excitation amplitude F₀ is approximately 0.2 N (Factor about 0.5) and both potentiometers P₁ and P₂ are set to the middle value. The figure shows the characteristic bending of the resonance curves of a nonlinear system.

In Section 6.4.2 the resonance curves of the nonlinear system from Section 5.6.3 are recorded. Corresponding to Section 5.6.2 the stable oscillation is followed with an instrument at a TV-recorder; the sinusoidal excitation comes from the function generator steered continuously by the analog computer with time.

[Page 113]

$$u(t) = a_0 e^{-(R/2J)t} \sin(\omega_0 t - \varphi_1)$$

with ω₀ and φ₁ depending on the initial conditions. The long-time solution corresponds to Section 5.6.2.

5.7. Schrödinger Equation of the Hydrogen Atom (Electron States)

The Schrödinger equation of quantum mechanics for a single-particle system with mass m, potential energy W_pot, and wave quantum h/2π reads:

$$-\frac{\hbar^2}{2m}\nabla^2\psi + W_{pot}\psi = i\hbar\frac{\partial\psi}{\partial t}$$

The complex function ψ(x, y, z, t) has a physical meaning. One regards |ψ|² (the complex conjugate ψ* times ψ) as a probability amplitude. The product aus |ψ|² dxdydz is a measure of the probability that the particle is found in the volume element dxdydz. It must therefore be fulfilled that:

$$\iiint |\psi|^2 , dx, dy, dz = 1$$

Seeking stationary states with energy W, one makes the approach of a time-dependent periodic solution:

$$\psi = e^{-iWt/\hbar} u(x, y, z)$$

which leads to the time-independent Schrödinger equation:

$$-\frac{\hbar^2}{2m}\nabla^2 u + (W_{pot} - W)u = 0$$

This equation is valid for discrete energy values W, which are called eigenenergies W; the associated solutions u are called eigenfunctions. For the eigenfunctions u to be physically meaningful, the norm condition must be satisfied:

$$\iiint u^2 , dx, dy, dz = 1$$

There exist also, for comparison, continuously distributed energy values W, for which u(x) tend to zero sufficiently fast as x → ∞.

As it turns out, those solutions that are found also do not normalize to u(x) → ∞; such cases do not occur as one must search for real eigenfunctions u(x).

[Page 114]

For the one-dimensional time-independent Schrödinger equation, analogous to the given potential energy W_pot(x) in the program an approach with piecewise constant potential is taken:

$$W_{pot}(x) = \begin{cases} 0 & \text{for } x \leq 0 \ \lambda & \text{for } 0 < x \leq 0.2 \text{ and } 0.4 \leq x \leq 1.2 \ 5 \cdot 10^{-17} \text{ J} & \text{for } 0.2 < x < 0.4 \end{cases}$$

Figure 5.7.0 shows the principle circuit diagram for the solution of the one-dimensional Schrödinger equation. Bild 5.7.0 shows the principle program for the one-dimensional integration. The first integrator (time constant RC) obtains at its input the acceleration a₂(u”)/ℏ² – (W_pot – W)u; the first integrator with (time constant RC = al) yields:

$$a_0 u’(x) = -\frac{1}{RC} \int a_0 u”(x) , dx + a_0 u’(0)$$

The second integrator (time constant RC) yields then:

$$a_0 u(x) = -\frac{1}{RC} \int a_0 u’(x) , dx + a_0 u(0)$$

The upper multiplier forms a₀ u”(x) and the subsequent integrators (time constant RC) give a₀ u²(x)/al; a₀ = RC/al.

The top multiplier multiplies a₀ u(x) again:

$$a_1 u^2(x) = a_0^2/a_1 \cdot u^2(x) \text{ with } a_1 = a_0^2/RC$$

The scaling factors a₀ and a₁ depend on the choice of m and RC from:

$$a_0 = \left(\frac{RC}{al}\right)^{1/2}, \quad a_1 = \left(\frac{RC}{al}\right), \quad a_2 = \left(\frac{RC}{al}\right)^{3/2}$$

with RC/al = 1 being the simplest; it gives al = 10 s and RC = 1 s. For a value RC/al = 10⁻¹⁰ these are in the range of 10⁻¹⁰. The Scaling factor a₀ depends on the Wirkungsquantum (quantum of action) from the atomic domain. This Section 5.7.1 and Section 5.7.3 are therefore used as the scaling ones.

With x = t/a₀ one obtains from the given potential energy W_pot(x) the following time-course of the potential W(t):

$$a_0 u’(x) \big|_{x=0} = a_0 \cdot u’(0)$$

The initial values u(0) and u’(0) must be appropriately specified. For this the given potential energy W_pot(x) in the program on circuit diagram 5.7.1 is needed. The program start (according to Section 5.7.1) and the Kippschalter (toggle switches) (575 603) are needed. In addition to the set of potentiometers (575 603) one or two more flip-flop elements (576 03) may be required.

With RC/al = 1 it would be valid that a₀ = 1, so the values are in natural units. The Size in order of magnitude 10⁻¹⁰ are in each case physically meaningless, as the Larger R₁ = 10 a and RC = 1 s as well; therefore one chooses a₂ = 10⁻¹⁶ which is the value in the Größenordnung (order of magnitude) of 10⁻¹⁸ J and so this is physically meaningful.

The initial values u(0) and u’(0) are to be searched out experimentally, so it holds that:

$$\int u^2(x) , dx = 1$$

For the setting of initial values and the program control required (Section 5.7.1) and (5.7.3) serve the potentiometers (575 607). To start the program at the right time (Section 5.7.1) require only one flip-flop element (576 03) and corresponding Section 5.7.3 still requires one more Kippschalter (toggle switch).

5.7.1. Potential Well of Finite Depth

For the solution of the one-dimensional time-independent Schrödinger equation with piecewise constant potential W_pot(x), the program is given on circuit diagram 5.7.1. Correspondingly, circuit diagram 5.7.0 shows the generation of u(x) and its integral. In the program on circuit diagram 5.7.1 the energy W is entered. In the program the energy W is generated from the circuit block (right above), the Potential P₃ is multiplied by –a₀W_pot(x) with t = τ; simultaneously for the Potential P₃ – a₀W_pot(x) added together to the sum of multiple operation amplifiers; the result is then fed back.

For the scaling factor a₀ and a₁ the following applies (according to Section 5.7):

$$a_0 u’(0) = \frac{10 \text{ k}\Omega}{R_1} \gamma$$

The one-dimensional time-independent Schrödinger equation is simply:

$$a_0 u’(0) = a_0 \lambda \cdot a_0 u’(0) = \frac{10 \text{ k}\Omega}{R_1} \gamma$$

To search for the energy values W₀ needed for normalization, the resistances R₁ and R₂ and in the following tables the voltages on the potentiometers P₁ and P₂ for some energies W are given:

W / 10⁻¹⁷ J	U_P₃	γ a₀ λ	R₁	R₂	γ
0.25 J	0.5 V	0.97 m⁻¹	200 kΩ	205 kΩ	0.35
1.15 J	2.3 V	0.64 m⁻¹	100 kΩ	230 kΩ	0.35
2.15 J	4.3 V	0.18 m⁻¹	100 kΩ	140 kΩ	0.4
2.6 J	5.2 V	0.69 m⁻¹	100 kΩ	140 kΩ	0.4
3.76 J	7.5 V	0.50 m⁻¹	100 kΩ	100 kΩ	0.4
4.35 J	8.7 V	0.36 m⁻¹	100 kΩ	50 kΩ	0.7

[Page 115]

Figure 5.7.1 shows the circuit diagram programmed with these initial conditions. The time constants RC = 1 s of the integrators act according to the scaling remarks in Section 5.7 as scaling factors.

For the scaling factor a₀ the following scaling remarks hold: $$a_0 = 10^{-5}, \quad a_1 = 10^{-9.5}, \quad a_2 = 10^{-13}, \quad a_3 = 10^{-15.5}$$

For the scaling factor a₀ with the Wirkungsquantum (quantum of action) h ≈ 2 × 10⁻¹⁴ are: $$a_2 = 2 \cdot 10^{-14}$$

With x = t/a₀ from the given potential energy W_pot(x) specified above, the following time-based potential W_pot(t) is obtained:

$$a_0 W_{\text{pot}}(t) = \begin{cases} 1 \text{ J} , (4 \div 10 \text{ V}) & \text{for } t < 2 \text{ s} \ 0 & \text{for } 2 \text{ s} \leq t < 12 \text{ s} \ 1 & \text{for } t \geq 12 \text{ s} \end{cases}$$

The run sequence is also controlled as follows. At time t = 0 the switches S₁, S₂, S₃ and S₄ are open and the Anfangswerte (initial values) are entered. At second 2 switch S₄ is closed, and at second 12 switch S₃ is also closed. The potential W is brought up at the potentiometer P₁. After t = 12 s switch S₃ is opened and one returns to the beginning.

For the actual eigenfunctions u(x) to vanish for increasing x (i.e., physically valid solutions), the program searches for solutions where u(t) remains bounded. Non-normalizable solutions u(x) → ∞ do not qualify as eigenfunctions. The program begins the search at these energies W.

For the corresponding eigenfunctions u(x) the initial values u(0) and u’(0) must be selected. The Anfangswerte are always u(0) = 0 and u’(0) = λ u(0), so from the program on the right-hand side of the operational amplifiers:

$$a_0 u’(0) = \frac{10 \text{ k}\Omega}{R_1} \gamma, \quad a_0 u(0) = a_0 \lambda a_0 u’(0) = \frac{10 \text{ k}\Omega}{R_1} \gamma$$

At the output of the right operational amplifiers there are always the sum of multiple Operation amplifiers available; the result Norm integral is fed back again.

[Page 116]

Figure 5.7.1-1 shows outside the three eigenfunctions u₀(x) and two further normalized solutions u(x). The corresponding eigenvalues lie somewhat above two further solutions u(x). The eigenvalues are approximately W₁ = 2.55 × 10⁻¹⁷ J and W₂ = 4.35 × 10⁻¹⁷ J.

Figure 5.7.1-2 shows outside the three eigenfunctions u₀(x) and u₁(x) for the two eigenvalues W₀ and W₁.

The energy of the electron with energy W₀ is not defined; it gives only the probability of finding it at position x. Figure 5.7.1-3 gives for four of the four eigenfunctions u₀(x) the probability function, and the oscillations of the time-course are small.

5.7.2. Coupled Potentials

The mathematical integration of the eigenfunctions and eigenvalues of an electron in two mutually interacting coupled potentials is to be solved. The temporal Verlauf (time course) can be noted with an oscilloscope z. B. on circuit diagram 5.7.1. The corresponding function W_pot(x) is selected for the two potentials and both potentiometers P₁ and P₂ are kept at the middle. The potential W must be freely chosen in this case.

In the program 6.3.2 the Schalter (switch) is set at t = 0 with the potentiometers set (Anfangswert vorher einstellen (set initial value beforehand)):

Monoflop (one-shot): For t < 0.6 s let S₃ be closed so that a_0W_pot = –1.2 J s, or
W_pot = 0 to be reached with closed switches S₃ and S₄.

This switch is programmed and the run sequence then automatically follows. For t = 12 s switch S₃ will again be opened, and the search for the Anfangswert (initial value) of the integration must occur again. For the Anfangswerte u(0) and u’(0) the following table gives the necessary Factor γ for both potentiometers P₁ and P₂ at some energies:

W / 10⁻¹⁷ J	U_P₃	γ a₀ λ	R₁	R₂	γ
0.93 J	1.4 V	…	200 kΩ	200 kΩ	0.3
2.10 J	4.2 V	0.76 m⁻¹	100 kΩ	130 kΩ	0.5
3.10 J	6.2 V	0.54 m⁻¹	100 kΩ	100 kΩ	0.5
3.55 J	7.1 V	0.54 m⁻¹	50 kΩ	90 kΩ	0.65
4.85 J	9.7 V	0.40 m⁻¹	50 kΩ	120 kΩ	0.4

After inserting energy W and the Anfangswerte the computation begins (here a₀ = 10 kΩ/R₁) and the run begins. Figure 5.7.2-1 shows the four computed eigenfunctions u(x) and the corresponding eigenvalues W for the given coupled potential.

[Page 117]

Figure 5.7.2 shows the parabola produced by two appropriate integrators (compare Section 3.3.1). It is for t < 0.08 × 10⁻¹⁰ m, i.e., the potential W_pot = 0, and for 0.08 × 10⁻¹⁰ m ≤ t < 0.28 × 10⁻¹⁰ m, it is W_pot = –1.4 × 10⁻¹⁷ J or a sinusoidal excitation. The parabola represents the centrifugal potential. The three eigenfunctions are each close to the middle and are symmetric. The potential function W is smaller for higher n values. The excitations u(x) are each scaled for the eigenvalues; the smallest is for W₀.

The choice of the initial values u(0) and u’(0) corresponds again to the following table needed for normalization with Factors γ for both potentiometers P₁ and P₂ at some energies (see Section 5.7.1):

W / 10⁻¹⁷ J	U_P₃	γ a₀ λ	R₁	R₂	γ
0.25 J	0.5 V	0.97 m⁻¹	200 kΩ	205 kΩ	0.35
1.15 J	2.3 V	0.64 m⁻¹	100 kΩ	230 kΩ	0.35
2.15 J	4.3 V	0.18 m⁻¹	100 kΩ	140 kΩ	0.4
2.6 J	5.2 V	0.69 m⁻¹	100 kΩ	140 kΩ	0.4
3.76 J	7.5 V	0.50 m⁻¹	100 kΩ	100 kΩ	0.4
4.45 J	8.9 V	0.36 m⁻¹	100 kΩ	50 kΩ	0.7

Figure 5.7.2 shows in the middle the four lowest lying eigenfunctions u₀(x). The solutions from Section 5.7.1 split it: there exist two symmetric solutions u₀ and u₂ and two antisymmetric solutions u₁ and u₃. The potential is symmetric; the eigenfunctions become eigenvalues. The splitting is smaller for the higher potential and the level splitting becomes smaller.

5.7.3. Harmonic Oscillator

A particle system with parabolic potential well:

$$W_n = \hbar\omega(2n + 1) \quad \text{for} \quad n = 0, 1, 2, \ldots$$

In the circuit diagram 5.7.3 the parabola is produced by two appropriate integrators (compare Section 3.3.1). The choice of the Anfangswert (initial value) is as follows: at the potentiometer P at the right operational amplifier the minimum einige Nullstellen Null is reached (is achieved). This means from the analog computer at the minimum W_pot(W = 0) the computation to be taken.

The Fourier coefficient selection requires for a₀ and a₁ the following scale factors: a₀ = 10 kΩ/R₁ γ u(0) = 1 and a₁(0) = a₂ γ u’(0), and for the Anfangswert of the function (see Section 5.7.1).

Section 6.3.2 describes the use of the switch (576.07): by changing the time constant RC by changing the switch S₄ the computation is started, and the Anfangswert can be again set after the computation.

In the Schaltplan (circuit diagram) 5.7.3 sind (are) the Parabel (parabola) by two geeignete (appropriate) integrators produced (compare Section 3.3.1). It applies for t < 0.08 × 10⁻¹⁰ m, W_pot = 0, and for the closed switches S₄ and S₅ the potential W_pot = 0. The S₄ is opened at time t = 0.

After entering the energy W at the potentiometer and the initial values the computation begins (here with x = 0) and all Schalter are simultaneously opened. Figure 5.7.3 shows in the middle the four lowest lying eigenfunctions and their widths. The computation begins and can then also automatically be repeated.

[Page 118 — figure only, circuit diagram 5.7.3 and oscilloscope trace shown]

[Page 118: figure only — circuit diagram for Section 5.7.3 (harmonic oscillator Schrödinger equation) and oscilloscope trace of eigenfunction solutions.]

[Page 119]

6. Appendix

6.1. Recording Solutions with a Recorder

Two recorders (Schreiber) are available for selection. With the TY-recorder (575.60) one can record a time-dependent quantity as a function of time. The XY-recorder (575.66) records a quantity y(t) as a function of a quantity x(t) (i.e., a parametric curve). For many applications both recorders are used simultaneously (see Section 6.1.1). For each recorder a separate Y-input and X-input is available, with a sensitivity of 1 V/cm achievable. For the Y-input a sensitivity of 1 V/cm for the recording of amplitudes is used; for a different sensitivity one can use the scale 44.2/44 and 74 (per recorder according to the Gebrauchsanweisung (instruction manual)).

On the TY-recorder both recorders a paper speed of 20 cm/min up to a maximum paper speed of 20 cm/s is available. The speed of the TY-recorder corresponds to 2.5 cm/s with time, with a paper advance of 20 cm/min and an oscilloscope (442.84 and 442.74) for other time or voltage values used per the Gebrauchsanweisung (instruction manual). The two programmable Schalter (576.07) are not used simultaneously in the programs; the programs are produced in a scale of 1:2.

The TY-recorder and XY-recorder both: The Y-inputs and X-inputs belong to the recorder with the sensitivity of 1 V/cm adapted for the recording of amplitudes from 0 to 5 V and the voltage range is ±10 V. For other purposes, the range 44.2/44.74 (for recorders see Gebrauchsanweisung) can be used.

On the TY-recorder a paper speed of 20 cm/min can be set for different speeds of the recorder. A paper speed of 20 cm/s corresponds with time to 2.5 cm/s. The scale, the programs are produced in, is 1:2.

6.1.1. TY-Recorder as XY-Recorder

Many mathematical or physical problems refer not just to the time function y(t) but rather to two time functions x(t) and y(t) simultaneously, or two temporal quantities as a function of each other. Examples are given in Sections 1.3.3 (time axis calibration with x = t·t_E, Chapter 3.2, Chapter 3.7, and Chapter 5.7.

The quantity y(t) is given, as in a parametric representation, as a function of z. B. y(t) is thus at the Y-recorder registered. The solution y(t) can then be registered with an XY-recorder against x(t). Examples include in Sections 2.4 two functional quantities as a function of each other. The total function u(x, y) shows the functional relationship of the two quantities. Examples are in Sections 3.4.1 and subsequent chapters, and in the process sections, given as the solutions of a differential equation with a second-order term on the right axis above with the time constant RC:

$$y_1(t) = -\frac{1}{RC} \int y_2(t) , dt + y_1(0)$$

The second integrator (time constant RC) yields then:

$$y_2(t) = -\frac{1}{RC} \int y_1(t) , dt + y_2(0)$$

With the time constant RC of an integrator acting accordingly to the slow-running program; the time constant RC₀ = a·RC, with a₀ = the slowdown factor.

For y(t) appears as the function z. B. in the output of a differentiator, the output of the upper operational amplifiers already belongs to the above with the time constant RC:

$$y_1(t) = -\frac{d}{dt} \int y_2(t) , dt = -R_c C_1 \frac{dy_2}{dt}$$

With the time constant RC of a slow-running program: R_c C₁ = a·RC.

The next following Quadrierer (squarer) yields a₀²u²(x) and the next following integrators (time constant RC):

$$-\int u^2(x) , dx = -\frac{RC}{al} \int a_0^2 u^2(x) , dx, \quad a_0 u^2 = RC/al$$

The value of the Scaling factors a depends on the choice of m and RC:

$$a_0 = \left(\frac{RC}{al}\right)^{1/2}, \quad a_1 = \left(\frac{RC}{al}\right), \quad a_2 = \left(\frac{RC}{al}\right)^{3/2}$$

With RC/al = 1 the simplest case; one takes al = 10 s and RC = 1 s. For a value RC/al = 10⁻¹⁰ these are in the range of 10⁻¹⁰. The scaling factor depends on the Wirkungsquantum from the atomic domain.

6.1.2. XY-Recorder as TY-Recorder

If the TY-recorder is not available, one can use the XY-recorder for the registration of time-dependent functions y(t); it must however be noted that the corresponding circuit for the recording of functions u(x, y) is adapted to the needed X-coordinate. This gives no analog computer output on the X-coordinate. Instead, a linearly with time-increasing voltage U_t is put on the X-input of the recorder. This advantage is in many cases in the program (see Section 6.3.1). With the XY-recorder (575.60) it can be used by the programmable Schalter (576.07) for this purpose.

To start, the switch S is set to zero at the same time as the run begins. The circuit diagram 6.1.2 shows an integrator with a hub of overall 10 V which means in program one uses a 10 V supply. If z. B. 20 V is used then the X-coordinate goes to 10 V and the null point of the recorder is shifted to the middle.

With y(t)(t) in the output of a differentiator enters as the X-input of the XY-recorder. The null point shift of the recorder in the program 6.1.2.2 shows that the analog computer delivers as the X-output null point shift to the middle of the recorder and the Scriber.

[Page 120]

One needs in the circuit diagram 6.1.2.1 or 6.1.2.2 a switch S₁, which is die die gewünschten (the desired) nullstellungen of the differential equation to start. To start, the switch S₁ is opened again and the last integration output (right) and the first Integration output (left) are connected. After this circuit the integrator runs for one cycle and then a new cycle begins. After T = 24 s the switch is again closed and S₃ is opened and the computation begins again.

Also the time constants of the program are all with a₀ to multiply, so that the time-scaling of the function follows. The Scaling Factor a is to be set corresponding in Chapter 5.5 Scaling remarks (see Section 1.3.3). The output of the program is always with a TY-recorder of paper advance 20 cm/min so that also the Ergänzungs-Steckelements (576.03) of not a short-circuit but can be measured.

6.2. Recording Solutions on an Oscilloscope

The programs of Part 2 and Part 5 and Chapter 6.6 be registered (recorded), so they can be quickly and automatically registered, and the solutions can therefore be repeatedly registered with the automatic repeat Schalter (576.07). These are the fast-running programs, whose solutions are slowly (between 20 ms and 30 ms) with a Monoflop (one-shot) and their solutions can be displayed on an oscilloscope (442.84). These cycles can be slow running (between 20 s and 30 s), slow then and also with an instrument and recorder registered.

The repetition of cycles can be from about 20 ms to a few seconds. The period T between the two cycles 20 ms is (for the slow one) set at the potentiometer as Monoflop registered. This is also given by the cyclic repetition of period T with a programmable Schalter (576.07).

For all of Oscilloscope programs should the Monoflop (one-shot) be ready to run, in which time the solutions are completed and then with the Messinstrument (measuring instrument) and recorder registered.

6.2.1. Example: Transient Processes in a Linear Resonance System

For the solution of the second-order ordinary differential equation my + Ry + Dy = F

the circuit diagram 6.2.1 analogously to circuit diagram 5.5.1 is applied. The external periodic force is generated by the function generator (522.55). In the circuit diagram the external force is given as a sinusoidal signal: sin 2πft.

As solutions y(t), these appear at the Stellungen des Potentiometers P₁ as either strongly or weakly damped oscillations. The solutions at both potentiometers P₁ and P₂ vary (always in potentiometer setting ratio R₁/R₂ = 10). To monitor this oscilloscope the frequency of the drive must be about f = 1/T = 10/l. Der Einfluss (the influence) of the variation of R or f on the oscilloscope directly shows the Resonanzkurve.

[Page 121]

Circuit diagram 6.2.1 uses two integrators (time constants RC = 1 s) and then one uses an additional Schalter S₁. The programmable Schalter S₁ (576.07) provides the trigger signal for the oscilloscope.

6.3. Control with Programmable Switch

The programmable switch (576.07) is already in the Sections 1.2.2 auf (on) page 13 and in Sections 1.3 described. It contains five electronic switches, which, as shown in the programs, can replace corresponding mechanical switches. With Umschalter (toggle) and Section 1.2.2 one slowly-running programs can be single stepped, whose solutions between 20 ms and 30 s are slowly run; the Schalter thus registers with an oscilloscope each zwischen (between) 20 s and 30 s.

With Monoflop and Section 1.2.2 slowly-running programs can be single-cycled, whose solutions are between 20 ms and 30 s; thus the Schalter is on. Five switches S₁ and S₂ are automatically connected to the output of the trigger signal (left) and the output of the first switch (right). The computation begins. After T = 12 seconds (for long-time programs between 20 s and 30 s), the switch is again opened.

Should switch S₁ be closed at time t = 3 s and reopened after 6 s, a zero dwell time:

T = 24 s to be chosen, the first Steuereingang (control input) of switch S₁ with the green control socket for T = 7 · 3/8 connects.

For the automatic Wiederholung (repetition) with Pause it is possible by the Stellung (position) of the Wahlschalters (selector switch) to the right when the Triggering signals at the right regularly a negative electrical signal supplied: it is at the beginning of each cycle from the first negative Flanke (edge) of the Triggereingang (trigger input) at terminal ① activated. At that point the start of the next follows. An example for this is given in Chapter 6.5.

6.3.1. Operating Modes of the Programmable Switch

The operating mode is determined by the position of the selector switch ①. As described, an automatic repetition of the cycles without pause is achieved when the selector switch ① is set to the right and the first negative edge of the trigger input ① activated.

An automatic repetition without pause results when the trigger input ① is connected to the output. This keeps the oscilloscope triggered. The computation begins from the first negative edge of the Triggereingang ① and can then also be displayed on an oscilloscope (be triggered). Bild 6.2.1 can be triggered from the first cycle; the Schalter S₁ takes after all Schalter have reached their required starting positions.

[Page 122]

An automatic repetition with pause is possible by the position of the selector switch ①: to the right, with the next regularly negative signal applied to the trigger input ①. Without external supply to the trigger input ① the next cycle starts after the selector switch. This is shown in an example in Chapter 6.5.

Without external supply of the trigger input ①, the next cycle is started when the selector switch ① is clicked left. At this location the Schalter S₁ and then the start of next ①.

6.3.2. Example: Schrödinger Equation for Coupled Potentials

In Section 6.3.2 the Schaltplan (circuit diagram) 5.7.1 is adopted correspondingly from the Schaltplan 5.7.2 for the Schrödinger equation. The physical quantities are thereby gleichzeitig (simultaneously) at the circuit board of the same (at that in Section 5.7.2 described). The potentiometer S₁ is set and the programmer control of both potentiometers P₁ and P₂ are varied (always in a constant middle value). The switch W_pot = 0 is equal to 0.

The figures 6.3.2 show the Schaltplan (circuit diagram) of the Schalter (switch) at t = 0. As Zeitabschnitt (time segment) of the switch S₁ at t = 0: the potentiometers are thereby set (Anfangswert vorher einstellen (set initial value beforehand)):

For t < 0.6 s let S₃ be closed so that a₀W_pot = –1.2 s (i.e., W_pot = –1.2 × 10⁻¹⁷ J); for t = 2 s switch S₃ is opened; for t = 12 ms switch S₃ is again closed and S₂ is opened (W_pot = 0). For t = 24 ms the switch S₃ is again closed and the time course becomes a periodic Störung (disturbance) or parameter in the programmed periodic sequence.

The following table eases the Section 5.7.1 use of the appropriate Anfangswerte:

$$a_0 u’(0) = \frac{10 \text{ k}\Omega}{R_1} \quad \text{and} \quad a_0 u’(0) = \frac{10 \text{ k}\Omega}{R_1} \gamma$$

The resistances R₁ and R₂ and size-orderly necessary Factor γ are for both potentiometers P₁ and P₂ at some energies:

[Table as previously in Section 5.7.2]

[Page 123 — figure only and section header visible]

[Page 123: figure only (circuit diagram 5.3.2 for Schrödinger equation of coupled potentials) plus brief continuation of text:]

6.4. Applications with Function Generator

The function generator (522.55) generates optionally a sine, square, or sawtooth waveform with adjustable amplitude, frequency in the range 10⁻³ Hz to 10⁵ Hz. At low frequencies the output is practically sinusoidal and continuous. The frequency can be controlled from about 10 Hz to 10⁵ Hz, especially at low frequencies (10 Hz to 10⁵ Hz). In analog computer programs the generator is used e.g. in programs 2.1, 5.4, and 5.6. Moreover, the frequency-dependent curve can be conveniently used from the analog computer programs in the sweep from 10 Hz up to 60 Hz.

The output a₁ of the analog computer program records this sinusoidal excitation; the frequency of the function generator is steered linearly with time by the analog computer. At the starting frequency f₀ in the 10 Hz range (Factor 1 in the 10 Hz range) it opens switch S 5 am. The frequency increases in 2 minutes linearly to the time wachsend (growing) frequency up to 60 Hz.

The response a₁ of the analog computer program records these continuously changing sinusoidal signals at the function generator; the frequency is steered linearly growing with time from the analog computer.

6.4.1. Example: Resonance Curve of a Linear System

In Section 5.6.2 the resonance curve has already been taken point-by-point. The time constants are set there RC = 5 ms to compress. What changes dabei (therein) is: the resonance frequency changes to about 1/6 Hz. In the Schaltplan (circuit diagram) the time constants are zunächst (initially) slightly increased and RC = 5 ms corresponds accordingly.

The external excitation 2πft between frequencies f between about 20 Hz and 60 Hz comes from the function generator (522.55), whose frequency is steered linearly with time by the right-hand Operationsverstärker (operational amplifier). This is steered with the Potentiometer (potentiometer) and the set value, z. B. about even with the potentiometer, so that the output voltage U(t) in 2 minutes from 0 to 5 V steigt (rises), with a paper advance of 20 cm/min is registered.

The circuit diagram 6.4.1 shows Lösungen (solutions) for various resistance values R₁. The external excitation sin 2πft comes from the function generator (522.55). The circuit diagram 6.4.1 turns off in switch S from step to step.

6.4.2. Example: Resonance Curve of a Nonlinear System

The Schaltplan (circuit diagram) 6.4.2 shows an extension of circuit diagram 6.4.1 to the nonlinear system from Section 5.6.3. Circuit diagram 6.4.2 shows Lösungen (solutions) for various dampings. The program is controlled as in Section 6.4.2.1 but equally with the function generator.

[Page 124]

[Circuit diagram 6.4.2 showing the nonlinear resonance curve setup, followed by text and results graph 6.4.21]

6.5. Fourier Analysis with the Upper-Wave Generator

In Chapter 3.8 the principle circuit diagram for the calculation of the Fourier coefficients of a periodic function f(t) (period T) was given. With the upper-wave generator (522.54) (Oberwellengenerator) the phase-correctly mutually controlled functions are sin nωt and cos nωt also available; the same procedure can be used (see Instruction Manual 522.54). The multipliers and subsequent integrators 2/T = 2 RC₀ form the Fourier integrals, which although stationary can be formed, because the formation of the Integral periodic runs are repeated.

The function f(t) to be analyzed: F(t) = square wave, triangle or sawtooth wave as from the upper-wave generator — or likewise f(t) = sin nωt — come from the function generator, as well as functions sin nωt generated by the analog computer.

$$a_n = \frac{2}{T} \int_0^T f(t) \cos n\omega t , dt$$

$$b_n = \frac{2}{T} \int_0^T f(t) \sin n\omega t , dt$$

The multipliers and subsequent integrators (time constants 2/T = 2 RC) form the Fourier integrals. The Fourier coefficients can thereby be calculated even though the integration runs periodically; the Integral is thereby stationary (can be repeatedly formed), because the Bildung (formation) of the integral is periodic.

For the control of the repeated computation run here the Betriebsart (operating mode) of the Wahlschalters (selector switch) ② is used in the Schaltplan 6.5.0 perpendicular according to above (see Sections 1.2.2). The Zyklusdauer (cycle period) is set at the Wahlschalter for the function (Monoflop) set; after the end of the first cycle at Trigger ⓪ the Schalter S₃ is opened and the first negative edge of the Triggereingang (trigger input) started again.

Figure 6.5.0 shows the complete circuit diagram for the solution:

$$a_n = \frac{2}{T} \int_0^T f(t) \sin n\omega t , dt$$

and

$$b_n = \frac{2}{T} \int_0^T f(t) \cos n\omega t , dt$$

For a square wave function of the upper-wave generator with period T = 2/U = 2 RC₀, the top in figure 6.5.0 the Fourier integrals run; in circuit diagram 6.5.0 the structure follows then the ones from circuit diagram 3.1 from: The three Schalter are programmed so that (see figure 1.2.2), to individually control in the Schaltplan 6.5.0 perpendicular: the Zyklusdauer (cycle period) must be set so that the Fourier integrals can integrate over exactly one period T. To steer the repetitive computation run, the operating mode of the selector switch ② is used (perpendicular down, see Sections 1.2.2).

To keep the cycle period just slightly more than T provided it corresponds to the analog computer. An example is given in Section 6.5.

[Page 125]

With the next negative edge of the square wave the Vorgang (process) is repeated by the function generator and the period T of the programmable Schalter (programming), which is slightly larger than the Fourier period T, gives the time for which the Fourier integrals are formed. One can watch on the oscilloscope the function f(t) and the time-course of the aₙ sum at the output of the upper Operational amplifier.

For the square-wave function of the upper-wave generator with period T (t = 0), the Fourier coefficients are:

$$a_0 = -\frac{10}{11} \approx -0.33, \quad a_1 = -0.2, \quad a_2 = -0.14 \text{ and } a_3 = -0.11$$

Inverting the trigger signal of the programmable switch, so as the Fourier coefficients are a_n to be positive, then with the positive Flanke (edge) of the square wave the Fourier coefficients are formed. The Fourier coefficients:

$$f(t) = \sin \omega t + \frac{1}{3} \sin 3\omega t + \frac{1}{5} \sin 5\omega t + \ldots$$

again gives a square wave function with positive edge at t = 0.

Usually periodic processes das Spektrum (the spectrum) macht (makes) a discrete Aussage (statement) about the Phasenlage (phase position) with which the individual Frequenzen f_n are contained. This discrete spectrum lies only in the range 10 Hz to 750 Hz at the output of the upper-wave generator.

6.6. Computer Graphics

An analog computer does not necessarily serve only to solve mathematical or physical problems; it can also be used as a tool in the hand of an “artist” for the creation of graphically interesting programs and to represent them with an XY-recorder (575.66). In the following programs the solutions are displayed graphically by example with an XY-recorder.

6.6.1. Endless Spiral

In the correspondingly programmed differential equation of second order (Chapter 3.5):

$$\ddot{U}(t) + R\dot{U}(t) + U(t) = 0$$

the U(t) and U̇(t) represent time-dependent electrical voltages, which can be used as x- and y-signals of the oscilloscope (or XY-recorder). For the examples in Sections 6.6.2 bis (to) 6.6.7 interesting Lösungen (solutions) are given.

At the XY-recorder the X-input is assigned U(t) and U̇(t) at the Y-input — one obtains by choosing a smaller damping R zusammenhängenden (coherent) spirals with a clockwise rotation around the center point opposite to the direction of the clock (against the Uhrzeigersinn, i.e. anti-clockwise). The circuit diagram 6.6.1 is shown.

[Page 126]

Before beginning the calculation, with closed Schalter S₁ and S₂ the Anfangswerte (initial values) x(0) = c and y(0) = 0 are set and with the Potentiometer at the left Operationsverstärker (operational amplifier) an appropriate damping R is selected. The Registrierung (recording) begins with opening of Schalter S₁ and S₂ as in circuit diagram 6.6.2.1 — one obtains then according to figure 6.6.2.1 a spiral running inward. The Anfangswerte x(0) = c and y(0) = 0.

6.6.2. Snail

If one replaces in Section 6.6.1 the U(t) proportional Reibung (friction) by a friction depending only on the sign of U(t) (dry friction, see Section 5.5.2), so one has from the circuit diagram 6.6.2. The gewünschte (desired) friction coefficient R is selected with the potentiometer right from Schalter S₁ and S₂. The Differentialgleichung (differential equation) gives:

$$\ddot{U}(t) + R , \text{sign}, U(t) + \omega^2 U(t) = 0$$

The Anfangswerte x(0) = c, y(0) = 0 are set with the closed Schalter S₁ and S₂; the potentiometer on the right in circuit diagram 6.6.2 sets the gewünschte (desired) friction coefficient R. As a consequence of the dry friction, each half-oscillation the center of the ellipse shifts by 2R/ω². This means the center of the potential well moves to the right on each half-swing. The Anfangswerte are set to x(0) = c and y(0) = 0.

6.6.3. Spiral with Moving Center

Adding in Section 6.6.1 to –U(t)/ω the noch (still) constant with time growing quantity z, so it is fed into the U(t)/ω X-input of the XY-recorder and –U̇(t)/ω + z at the Y-input of the XY-recorder. The circuit diagram 6.6.3 shows this.

Before beginning the calculation with closed Schalter S₁, S₂ and S₃ the Anfangswerte x(0) = c, y(0) = 0 and z(0) = 0 are set, and with the Potentiometer on the left side of the Schreibers (writer) fed.

[Circuit diagram 6.6.2 and 6.6.3 shown on page 128]

[page 127]

Figure 5.6.3 — [Circuit diagram]

The desired slope of z is set at the potentiometer, and the potentiometer on the right selects an appropriate damping. The regulation begins by opening all three switches. The two figures shown here, 6.6.3.1 and 6.6.3.2, differ only in the slope of z.

Figure 6.6.3.1 — [Lissajous figure: inward-spiraling square-shaped pattern]

Figure 6.6.3.2 — [Lissajous figure: outward cone-shaped expanding spiral]

6.6.4 Lissajous Figures for Two Nearly Equal Frequencies

In sections 6.6.1 to 6.6.3, U(t) is plotted on the Y-axis versus U(t) from another oscillation on the X-axis. Producing two mutually independent oscillations simultaneously — one applied to the Y-input and the other to the X-input of the recorder (or oscilloscope) — yields Lissajous figures. In the sections that follow, the frequencies of the two oscillations are made almost but not exactly equal.

To execute the program, a total of four switches are required; since the programmable switches (S76/07) are not available, an additional set of patch elements (S76/03) and a toggle switch (S79/13) are also needed.

[page 128]

Figure 6.6.4 — [Circuit diagram]

Before starting the computation, with all switches closed, the initial values are set: y(0) = c, −y(0)/ω = z(0) = 0, x(0) = c₁, and ẋ(0) = 0. The potentiometer on the right can again set a suitable damping for oscillation y(t); z(t) again provides an appropriate damping preset. Registration begins by opening all switches.

Figure 6.6.4.1 is produced with damping set to “zero.” The slight residual damping causes a deviation of the envelope from a true square. To avoid this, the oscillation differential equation can be programmed with amplitude stabilization, for example according to section 2.7.6.

Figure 6.6.4.1 — [Lissajous figure: square-mesh pattern filling a square region, slight bowing of envelope]

In figure 6.6.4.2, the damping for y(t) is chosen slightly larger, and the registration is terminated after only a few (13) revolutions.

Figure 6.6.4.2 — [Lissajous figure: outward-expanding cone shape with curved edges]

In figure 6.6.4.3, y is not plotted against x but against z. The choice of parameters and the duration of registration determine the appearance of the plots substantially.

6.6.5 Circle with Rectangularly Modulated Radius

In the lower part of the circuit diagram 6.6.5, switch S₆ is again used — as in section 6.6.1 — to obtain the differential equation of 2nd order from two operational amplifiers interconnected in a slowly converging loop. On the right, a corresponding circuit from section 2.7.5 with one operational amplifier builds a sine generator whose output signal is again added to a constant a. The multipliers then supply, with a constant plus the frequency of the circular motion:

x(t) = c(a + r(t)) sin ωt
y(t) = c(a + r(t)) cos ωt

[page 129]

Figure 6.6.4.3 — [Lissajous figure: asymmetric 3D-looking curved spiral form]

Figure 6.6.5.1 — [Lissajous figure: star/flower shape with rounded lobes in a circular envelope]

r(t) of the operational amplifier on the lower right (see section 2.7.1) is added on the lower left to a constant a via a multiplier. The frequency of the square-wave generator is approximately 7.5 times greater than the frequency of the circular motion.

Before starting the computation, with switches S₁ and S₂ closed, the initial values y(0) = −ca, x(0) = 0, and r(0) = 0 are set. Registration begins by opening all switches. Figure 6.6.5.1 shows the result for one revolution.

6.6.6 Spiral with Sinusoidally Modulated Radius

By adding an appropriate damping in section 6.6.5, a slowly converging spiral is obtained from the circuit diagram on the upper right of section 6.6.6 instead of a circular motion (compare section 6.6.1). On the right, an operational amplifier from the corresponding section 2.7.5 builds a sine generator whose output signal is again added to a constant a. The multipliers supply:

x(t) = c(a + b sin(ω₀t + φ)) sin ωt
y(t) = c(a + b sin(ω₀t + φ)) cos ωt

The frequency of the circular motion, or of the spiral, is determined in advance by the time constants of the two integrators:

Figure 6.6.5 — [Full circuit diagram for the sinusoidally modulated radius spiral]

[page 130]

Figure 6.6.6 — [Circuit diagram]

The frequency and amplitude of the sine generator on the lower right are determined by the setting of the potentiometers in the feedback coupling. The setting is critical; one must wait for it to settle. The frequency can be adjusted gently through various values of R (e.g., 5 kΩ). Here ω₀ is approximately 4 to 8 times greater than ω; the phase φ of the sine generator relative to the circular motion is indeterminate.

The registration begins by opening switch S₁, with the initial values x(0) = 0 and y(0) = c(a + b) pre-set. Figures 6.6.6.1 and 6.6.6.2 show solution plots for various amplitudes and frequency ratios. In figure 6.6.6.1, b < a applies in particular; in figure 6.6.6.2, b > a.

Figure 6.6.6.1 — [Lissajous figure: flower/rosette pattern with many petals arranged in a ring, closed center]

Figure 6.6.6.2 — [Lissajous figure: flower pattern with elongated petals, open center]

Figures 6.6.6.3 to 6.6.6.8 arise from figure 6.6.6.1 by slight changes to the frequency of the superimposed sinusoidal oscillation and by enlarging the radius with slight damping. Figures 6.6.6.6 to 6.6.6.8 differ from one another only in the duration of the registration.

6.6.7 Superposition of Two Circular Motions

On the upper right of circuit diagram 6.6.7 is again the corresponding section 6.6.6 circuit, so one has the x- and y-components of one (slow) circular motion. The sine generator from the corresponding sections 2.7.5 and 2.7.5 supplies the y-component of the second circular motion; the differentiator at the lower left then derives the corresponding x-component from it.

[page 131]

Figure 6.6.6.3 — [Lissajous figure: dense rose pattern with many petals, inward spiral]

Figure 6.6.6.6 — [Lissajous figure: ring of evenly spaced rounded petals]

Figure 6.6.6.4 — [Lissajous figure: 5-petal flower with outward spiral arms]

Figure 6.6.6.7 — [Lissajous figure: dense ring pattern with inward spiral]

Figure 6.6.6.5 — [Lissajous figure: 4- to 5-lobed spiral converging inward]

Figure 6.6.6.8 — [Lissajous figure: very dense ring converging to a tight spiral center]

[page 132]

Figure 6.6.7 — [Circuit diagram]

x-component. The frequency of the second sine oscillation is set correspondingly to section 6.6.6 via the potentiometer in the feedback coupling. The sum of both x-components and the sum of both y-components respectively give the coordinates of the superimposed motion.

The registration begins by opening switches S₁ and S₂. Figures 6.6.7.1 and 6.6.7.2 show two solutions with slight damping of the slow circular motion. The radius c of the slow circular motion is larger than the radius r of the second circular motion. The two figures differ in the frequency ratio of the two oscillations. In figure 6.6.7.1, b < a applies in particular; in figure 6.6.7.2, b > a.

Figure 6.6.7.1 — [Lissajous figure: daisy/flower with many narrow petals arranged radially]

Figure 6.6.7.2 — [Lissajous figure: dense ring of rounded lobes, more open center]

The following figures show three solutions for a superposition of a slow spiral (damping not equal to zero) and one circular motion. In figure 6.6.7.4, c < r; in figure 6.6.7.5, c > r. In figure 6.6.7.3, c = r; during the course of the registration the amplitude of the sine generator is reduced by hand at the potentiometer, and thereby the radius of the second circular motion decreases.

The figures in sections 6.6.1 to 6.6.7 are only a small selection of the “computer graphics” producible with the analog computer. The imagination is barely limited.

[page 133]

Figure 6.6.6.3 — [Lissajous figure: spiral pattern with many petals converging inward] (figure only)

Figure 6.6.6.6 — [Lissajous figure: ring of rounded lobes, uniform spacing] (figure only)

Figure 6.6.6.4 — [Lissajous figure: large outward ring with 5-lobed outline] (figure only)

Figure 6.6.6.7 — [Lissajous figure: dense multi-petal inward spiral ring] (figure only)

Figure 6.6.6.5 — [Lissajous figure: 4- to 5-lobed spiral converging inward] (figure only)

Figure 6.6.6.8 — [Lissajous figure: very dense ring converging to a tight center] (figure only)

[page 134]

Figure 6.6.7 — [Circuit diagram, continuation]

x-component. The frequency of the second sine oscillation is set correspondingly to section 6.6.6 via the potentiometer in the feedback coupling. The sum of the two x-components and the sum of the two y-components respectively give the coordinates of the superimposed motion.

The registration begins by opening switches S₁ and S₂. Figures 6.6.7.1 and 6.6.7.2 show two solutions with slight damping of the slow circular motion. The radius c of the slow circular motion is greater than the radius r of the second circular motion. The two figures differ in the frequency ratio of the two oscillations.

Figure 6.6.7.1 — [Lissajous figure: daisy with many narrow petals]

Figure 6.6.7.2 — [Lissajous figure: ring of rounded lobes]

The following images show three solutions for a superposition of a slow spiral (damping not equal to zero) and one circular motion. In figure 6.6.7.4, c < r; in figure 6.6.7.5, c > r. In figure 6.6.7.3, c = r; during the course of registration the amplitude of the sine generator is reduced by hand at the potentiometer and thereby the radius of the second circular motion is decreased.

The images in sections 6.6.1 to 6.6.7 are only a small selection of the “computer graphics” that can be produced with the analog computer. The imagination is barely limited.

Figure 6.6.7.3 — [Lissajous figure: spiral pattern with many lobes, converging inward]

Figure 6.6.7.4 — [Lissajous figure: large outward ring with open center]

Figure 6.6.7.5 — [Lissajous figure: dense multi-petal flower with slight inward spiral]

[page 135]

6.7 Games

A further non-scientific application of the analog computer is to create programs for interesting two-player games. Two examples are described below.

6.7.1 Game 1

The program is reproduced again in circuit diagram 6.7.1. Player A generates a signal a with a potentiometer on the left (each in the range −1 ≤ a ≤ 1). Player B multiplies the signal a by a factor b, for which the same condition holds: −1 ≤ b ≤ 1.

The product a · b is fed to an integrator (with time constant RC) as input; one obtains:

s(t) = −(1/RC) ∫ a · b dt

The game begins with s(0) = 0, and both potentiometers set to zero (a = 0 and b = 0). Then an integrator is opened. The goal for Player A, as the output signal s starts from zero, is to drive s as close as possible to +1 or −1 (in the range −1 ≤ s ≤ 1). The chances are equal. Whoever turns their potentiometer (thereby changing s) more quickly wins. Going too fast, since the time constant (resistance) can be doubled, makes the game somewhat slower.

In the event of a dispute as to whether the output signal s is +1 (= +10 V) or −1 (= −10 V), the double voltmeter (442.84 together with 442.74) is well-suited as an indicating instrument; in a competition involving a larger group, this game can be conducted using a measuring display board.

For the referee, the most suitable instrument is a voltmeter with its zero point in the middle, with the integrator and one further operational amplifier (on the right below) forming:

z(t) = (1/2)(s(t) + 1)

Thus the output signal lies between −0.1 and +1.1. A voltmeter with a 10 V scale can be used. Player A attempts to reach +1 (= 10 V) and Player B attempts to reach 0 (= 0 V).

6.8 Essays, Reports, and Lectures from the Operational Amplifier / Analog Computer Area

[1] J. Sammet: Der Operationsverstärker und seine Einschränkungsgrenzen. Vortr. MRG Berlin 30 (1971), S. 191.

[2] H. Becker, G. Becker: Rechenverfahren. Vortr. MRG Berlin 30 (1971), S. 263.

[3] J. Becker, R. Becker: Rechentechnik der analogen Rechner, Praxis 31 (1972), S. 253.

[4] A. Kafka: Der Einfluß des Analogrechners in Schulen. Vortr. MRG Berlin 37 (1973).

[5] R. Burkert: Ein Analogrechner zur Lösung linearer Differentialgleichungen. Praxis 22 (1974), FAK 256.

[6] H. Haas: Simulation und Lösung physikalischer Probleme mit dem Analogrechner. Vortr. MRG Berlin 35 (1974), FAK 235.

[7] Kettner: Beschreibungstechnik des Analogrechners. MRG FAK 40 (1973).

[8] M. Hund: Simulation und Lösung physikalischer Probleme mit dem Analogrechner über elektronischen Tischrechner. Vortr. MRG FAK 40 (1973).

[9] R. Burkert: Ein Analogrechner zur Lösung linearer Differentialgleichungen. Praxis 37 (1973), S. 325.

[10] R. Burkert: Ein Analogrechner zur Lösung linearer Differentialgleichungen. Praxis 22 (1974).

[11] G. Limper: Kopplung von Analogrechner und TV-Kamera. Praxis 22 (1974).

[12] G. Limper: Analog- und Hybridrechner-Tischrechner. MRG FAK 37 (1973), S. 3.

[13] D. Sammet: Vortr. MRG Berlin 30 (1973), S. 285.

[14] G. Limper: Kopplung von Analogrechner und TV-Kamera. Praxis 25 (1974).

[15] D. Sammet: Operationsverstärker-Kopplung. MRG FAK 40 (1973).

[16] G. Limper: Kopplung von Analogrechner und TV-Kamera. Praxis 25 (1975).

[17] J. Sammet: Vortr. MRG Berlin 30 (1973), S. 285.

[18] G. Limper: Kopplung von Analogrechnern und TV-Kamera. Praxis 25 (1975).

Der Analogrechner

The Analog Computer

Table of Contents

Introduction

1.1.3 Programmable Switch

1.2. Description

1.2.1 Analog Computer

1.2.2 Programmable Switch

1.3. Operation

1.3.1 General Notes on the Circuit Diagrams

1.3.2 Amplitude Scaling

1.3.3 Time Scaling at an Integrator

(page 14 continued)

1.3.4 Time Scaling with a Programmable Switch

1.4. Calibration of the Analog Computer

1.4.1 Drift Compensation of an Operational Amplifier

1.4.2 Offset Compensation of Multipliers

1.5. Conversion to Mains Supply Voltage 110 V

2. Basic Circuits

2.1. Basic Circuits of an Operational Amplifier

2.1.1 The Open-Loop Amplifier

2.1.2 Multiplication of a Variable by a Constant

2.1.3 Error Additions

2.1.4 Determination of the Cut-Off Frequency

2.1.5 Summation of Two Variables

2.1.6 Integration of a Variable

2.1.8 Integration of a Variable

[Page 19]

2.1.7 Integration with a Defined Initial Condition

2.1.8 Integration with a Defined Initial Condition (Practical Procedure)

2.1.8 Differentiation of a Variable

[Page 20]

2.1.9 Differentiation with Lag

[Page 21]

2.2.1 Multiplication of Two Variables

2.2.2 Squaring of a Variable, y = x²

2.2.3 Squaring of a Variable, y = x³

2.2.4 Squaring of a Variable, y = x⁴

[Page 22]

2.2.5 Formation of the Square Root, y = −√x, x > 0

2.2.6 Formation of the Quotient, y = Z/N, N < 0

2.2.7 Construction of a Multiplier from Two Squarers

[Page 23]

2.3. Circuits with Multiplier and Open-Loop Amplifier

2.3.1 Formation of the Square Root, y = −√−x, x < 0

2.3.2 Formation of the Cube Root, y = −∛x

[Page 24]

2.3.4 Formation of the 4th Root, y = −∜x, x < 0

[Page 25]

2.3.5 Formation of the Quotient, y = −Z/N, N > 0

2.4. Basic Circuits with Diodes in the Feedback Path

2.4.1 Limiter Function

2.4.2 Limiter Function with Improved Blocking Behavior

2.4.3 Formation of the Absolute Value

[Page 26]

2.4.4 Step Function

2.4.5 Step Function with Variable Step Height

2.4.6 Kinked Characteristic with Limitation

2.4.7 Kinked Characteristic without Limitation

[Page 27]

2.4.8 Comparator

2.4.9 Two-Point Switch

2.5. Basic Circuits with Diodes in the Input Path

2.5.1 Characteristic Curve of a Single Diode

[Page 28]

2.5.2 Characteristic Curve of a Diode with Limited Slope

2.5.3 Generation of a Quadratic Characteristic Through Piecewise-Spanned Diodes

2.5.4 Dead Zone

[Page 29]

2.5.5 Dead Zone with Variable Width

2.5.6 Gear Function

[Page 30]

2.5.7 Gear Function with Variable Play

2.5.8 Gear Function with Limitation (Hysteresis)

[Page 31]

2.6. Logical Operations, SIMULOG-Compatible

2.6.1 Negation

2.6.2 NAND Gate

2.6.3 NOR Gate

2.6.4 Full Adder