Die Differentialgleichung und ihre Lösung mit den Operationsverstärkern; Grundlagen einfacher Regeleinrichtungen

Complete English translation of the original German-language document (18 pages).

The Differential Equation and Its Solution with the Operational Amplifier

(Document L 1112, January 1972 — pages 1–8; Document L 2310, January 1972 — pages 1–8)

Preface (Cover page — L 1112)

The following explanations, presented in leaflets 1 through 2.8, are useful for anyone who does not yet have much experience with analog computers. These explanations are essentially a handbook titled “Fundamentals of Circuit Technology for Operational Amplifier Applications” (order no. 38551), which should ideally be studied first.

These explanations are therefore written to be self-contained, with the intention that the user can work through them independently. The content has been tested internationally for its suitability in practice and found to be applicable. It is essentially a compact manual.

Important: these explanations also address the subject of time transformation, which is treated in depth in the full manual.

January 1972 — L 1112, 1 – S.1

[page 2: blank]

1. General

Physical processes can be described with the aid of differential equations. For example, it is of interest to determine the voltage of a resistance over the range 1 V to 5 V.

The solution of a differential equation with an analog computer is mathematically possible, but when one has an analog computer, it is not much more complicated. The analog computer is a versatile experimental device. It is a kind of “test setup” and is therefore not made available in the form of a specific type — it is assembled from components.

These explanations are meant to point out the most important techniques; they therefore do not claim to be a complete manual. Furthermore, the description of the time transformation, addressed in the full manual, is important internationally.

January 1972 — L 1112 – S.1

2. Integration in General (L 1112 – S.1, continued)

Mathematically, the integral of an input u_e is obtained from:

u_a = (1/C) ∫ i dt (a) since i = 0 (condition of an ideal op-amp)

Therefore u_a is:

u_a = (1/C) ∫ (u_e / R) dt (b) — since i = u_e/R

That is:

u_a = (1/RC) ∫ u_e dt

R_1 is a constant and can be set outside the integral. The integration constant k is defined as:

k = 1 / (R · C)

It is simultaneously the time constant and can be set at any desired value.

3. Realization of Integration with an Operational Amplifier

Figure 2 shows the circuit with op-amp and components.

Figure 2: Construction of an inverting integrator with an operational amplifier.

R₁ = 10 kΩ (input resistor)
C₁ = 10 µF (feedback capacitor)
R₂ = 10 kΩ (balance resistor at non-inverting input)

Integration constant:

k = 1 / (R₁ · C₁) ; R₁ = 10⁴ Ohm, C₁ = 10⁻⁵ F

k = 1 / (10⁴ Ohm · 10⁻⁵ F) = 1 / 10⁻¹ s⁻¹ = 10 s⁻¹

(Unit relationship: Ohm · F = s)

Figure 3: Symbol of the integrator from Figure 2.

The block symbol shows a triangle (amplifier) with the integration constant k inscribed (without units). Because Figure 2 is an inverting integrator, a minus sign is written before k (giving −10 in the symbol).

January 1972 — L 1112 – S.2

4. Multiple Integration with Operational Amplifiers (L 1112 – S.3)

It is shown here, using a single example, how to add up several integrals:

u_a = c · ∫ u_e dt = ∫ c · u_e dt

Figure 4: Schematic of a double integration using an inverting integrator with a summing input.

Figure 5: Construction of a double integration.

In Figure 4, the mathematical solution is converted into a circuit (O.-V. = op-amp). The first potentiometer provides a voltage source u_e/R₁. For the potentiometer setting of 0.5, an intermediate voltage 0.5 · u_e appears. Here u₁ and u₂ are the individual voltages; u₁ = u_e/1, u₂ = u_e/2. In the general case, the potentiometer settings can be between 0 and 1. For the capacitor values C₁, C₂ = 0.1 to 700 pF, the maximum intermediate voltage should be adjusted.

January 1972 — L 1112 – S.3

5. Simple Differentiation in General

The representation of differentiation with an op-amp follows Figure 6.

Figure 6: The differentiator in general.

The circuit consists of:

C₁ in series with the input (input capacitor)
R₁ as feedback resistor
R₂ as balance resistor at the non-inverting input

Derivation:

i_C = i_a (i = 0 condition)

i_C = C₁ · (d u_e / dt) (a)

U_a = i_a · R₁ = i_C · R₁ (b)

U_a = C₁ · (d u_e / dt) · R₁ [(a) substituted into (b)] (c)

U_a = C₁ · R₁ · (d u_e / dt) (d)

Because the capacitor C₁ is placed in series with the input, the circuit is very susceptible to interference at high frequencies. The solution of a differential equation is therefore mostly accomplished with the aid of an integrator circuit.

6. Circuit for Solving a Differential Equation with Operational Amplifiers

As already established, the differentiator circuit is not well suited for solving differential equations. It is shown here that the integrator can also be used for this purpose.

Figure 7 (capacitor symbol with label i at input, u across capacitor):

a) i = C · (du/dt) — the differential equation

b) u = (1/C) ∫ i dt — the integral

Figure 7 shows the behavior of current and voltage at a capacitor: a) the differential equation and b) the integral.

January 1972 — L 1112 – S.4

Integrator Circuit for Solving the Differential Equation (L 1112 – S.5)

Figure 8: Circuit of an integrator to be used for solving the differential equation.

R₁ (input resistor)
R₁’ (in parallel with C₁, to represent the loss resistance of the capacitor and prevent unwanted charging)
C₁ (feedback capacitor)
R₂ = protection resistor at the non-inverting input

Derivation:

u_e = (1 / C₁R₁) ∫ u_a dt (a)

u_e = (1 / C₁) ∫ (u_a / R₁) dt (b)

u_e = (1 / C₁) ∫ i dt Integral (c)

i = C₁ · (d u_a / dt) First-order differential equation (d)

where i = i₁ − i₂ (i₁ ≠ i₂ because capacitor C₁ must first be charged):

i₁ = u_e / R₁

i₂ = u_a / R₁

Substituting i₁, i₂ into i = i₁ − i₂:

C₁ · (d u_a / dt) = u_e/R₁ − u_a/R₁ (e)

Boxed result:

d u_a / dt = u_e/(R₁C₁) − u_a/(R₁C₁) (f)

This formula corresponds to the circuit of the operational amplifier according to the differential equation.

Figure 9: Block diagram of the differential equation

d u_a / dt = u_e/τ − u_a/τ (τ = R₁ · C₁)

The blocks 1/(R₁·C₁) are then simply emulated with potentiometers, i.e., the value 1/(R₁·C₁) is set as a resistance.

This is done — in order to obtain precise settings — with a multi-turn potentiometer.

January 1972 — L 1112 – S.6

7. Application of the Differential Equation for Emulation (L 1112 – S.6)

Example:

Figure 10: RL element

Series circuit of an inductance L and resistance R:

u_e = u_L + u_R (a)

u_e = L · (di/dt) + i · R (b) [L has dimension Ohm·s]

di/dt = u_e/L − i·R/L (c) : u_e/L − i·R/L

Figure 11: Block diagram for representing the mathematical relationships of an RL element.

Figure 12: Circuit construction for solving the differential equation of an RL element.

The output u_a corresponds to di/dt.

8. Time Transformation (L 1112 – S.7)

d u_a/dt is also designated mathematically as u̇_a; similarly:

di/dt = i̇ = ĩ

The differential equation reads:

u̇_a = u_e/CR − u_a/CR

u̇_a = u_e/τ − u_a/τ

Required computation time τ = 7 s for an assumed problem. If the writing width of the X-Y recorder or oscilloscope — due to the adjustable time scale — is insufficient for a desired recording duration, then a time transformation is necessary. This is carried out directly in the differential equation.

January 1972 — L 1112 – S.7

Time Transformation: Figures and Derivation (L 1112 – S.8)

Figure 13: Representation without time transformer (integration over 0 to 7 s).

Figure 14: Representation with time transformer (change of time scale; integration over 0 to 1 s).

The time τ = 7 s is to be reduced to τ’ = 1 s, i.e., the integration must run in 1 s instead of 7 s. The time saving is 1 : 7, from which follows the time-transformation factor:

λ = τ’/τ = 1/7

τ’ = λ · τ

The equation then reads:

u̇_a = u_e/(λτ) − u_a/(λτ)

u̇_a = u_e/τ’ − u_a/τ’

Figure 15a: Circuit without time transformation (e.g., τ = 7 s) — blocks labeled 1/τ.

Figure 15b: Circuit with time transformation (e.g., τ’ = 1 s) — blocks labeled 1/(τ·λ).

January 1972 — L 1112 – S.8

Fundamentals of Simple Control Systems

(Document L 2310, January 1972)

1. General (L 2310 – S.1)

The nature of the controller best demonstrates the operating principle through the output variables, i.e., by means of the step response.

Physical control processes can be described with differential equations. The electronics of operational amplifiers has been gaining ground increasingly in the area of control.

2. Proportional Controller (P-Controller)

Figure 1: Simple liquid-level control system.

If the float is exactly over the center, then the valve is half open (Figure 1). If the float level is too low (valve position fully open), more water flows in. The available length of the lever arm is L₁₂₀; with this, the ratio of water level to valve position is determined. The difference in the lever arm length L₁ determines the feedback to the controller and establishes the proportional behavior of the system.

The linear operational amplifier characteristic of the P-controller is shown in Figure 2.

v = U_a / U_e = R₂ / R₁

Figure 3: Step response (U_a) of a P-controller.

The upper graph shows the step input (U_e) and below it the corresponding output (U_a) in inverted form — a step response with immediate proportional reaction.

Note: P-controllers produce an adjustment of the manipulated variable that is proportional to the control deviation. In a P-controller, due to the rigid coupling, a permanent control deviation (offset) arises.

The transfer function for the P-controller is:

F = v (transfer function is the function of the controller)

U_a = F · U_e = v · U_e

January 1972 — L 2310 – S.1 / S.2

3. Integrating Controller (I-Controller) (L 2310 – S.2)

To reduce the mostly undesirable control deviation at the output, the so-called I-controller is used. By incorporating an integrator into the controller, the output variable is changed in a time-independent manner. By pre-integration or incomplete discharge of the capacitor, initial conditions can be entered.

Figure 4: I-controller with initial condition u_anfang (U_initial).

Circuit components:

R₃ (series input resistor)
C₁ (feedback capacitor)
R₁ (summing input resistor)
R₂ (balance resistor)
U_anfang input (initial condition voltage)

Figure 5a: Test with step (U_e — step input signal).

Figure 5b: Output voltage (U_a) — shows linear ramp rising from −U_a, starting at U_anfang up to U_Grenze (limit voltage).

Note: The I-controller has a linear rise over time. I-controllers have a “long” control time but achieve zero control deviation at the end of the control process.

Pure I-controllers are very rare; usually a proportional component is also present. This comes about through the internal resistance of the op-amp and the ohmic portion of the capacitor.

Transfer function:

F = 1/Tp = 1/(R₁C₁p) (p = Laplace operator)

January 1972 — L 2310 – S.2 / S.3

4. Proportional-Integral Controller (PI-Controller) (L 2310 – S.3)

The PI-controller first performs a proportional step and then continues to control by integration. Initial values can again be entered here.

Figure 6: PI-controller with initial condition U_anfang.

Circuit components:

R₄ (initial-value input resistor)
R₂, C₁ (feedback network — integrating branch)
R₁ (summing input resistor)
R₃ (balance resistor)
U_anfang (initial condition input)

Figure 7: Step response of a PI-controller.

The output variable U_a changes abruptly at first (proportional jump), then rises linearly (integral action). T_N = reset time (Nachstellzeit).

Note: The magnitude of the output voltage U_a changes proportionally at first; then the temporal change of U_a acts, i.e., the derivative of U_a is also relevant. This means the set value changes more quickly. In a pure P-controller, this effect is not achievable solely by changing the proportional gain.

The transfer function is:

F = v + 1/(T_N p) = (R₁C₁p + 1) / (R₁C₁p)

The PI-controller is applied in temperature control so that the temperature of the liquid can be regulated. The Proportional component alone cannot do this.

January 1972 — L 2310 – S.3 / S.4

5. Differentiating Controller (D-Controller) (L 2310 – S.4)

The D-controller has an infinitely high rate of change and therefore reacts to every change of the input variable, even if it is infinitesimally small. This makes it very difficult in practice to implement a pure D-controller. It produces a step response (spike/pulse) in response to a step, and it is better examined with a ramp input.

Figure 8: D-controller circuit.

Figure 9: Step response of a D-controller — shows an ideal impulse (spike) at the moment of the step.

Figure 10: Ramp response of a D-controller — a ramp input (linearly increasing U_e) produces a step-like constant output (−U_a).

The transfer function is:

F = T · p

F = R₁ · C₁ · p

January 1972 — L 2310 – S.4 / S.5

6. Proportional-Differential Controller (PD-Controller) (L 2310 – S.5)

PD is the combination of P- and D-controller; the distinctive characteristics need not always be clearly visible in the circuit behavior. This controller is also advantageously examined with the ramp.

Figure 11: PD-controller circuit.

Components:

R₁ (input resistor)
R₂ (feedback resistor)
C₁ (capacitor in parallel with R₁, forming the differentiating branch)
R₃ (balance resistor)

Figure 12: Step response of a PD-controller — shows both ideal (solid line) and actual (dashed line) behavior; initial spike decays toward a proportional steady-state level.

Figure 13: Ramp response (rise response) of a PD-controller — the output −U_a shows a ramp rising with the input, but shifted forward in time by T_V (lead time/advance time, Vorhaltezeit).

Note: The PD-controller controls faster than the P-controller, but results in a permanent control deviation (offset).

The transfer function is:

F = v (1 + T_V p)

F = (R₂/R₁) (1 + R₂ · C₁p)

Figure 14: Alternative circuit of a PD-controller with improved frequency response — op-amp with input E and output A, with combined RC feedback network.

January 1972 — L 2310 – S.5 / S.6

7. Proportional-Integral-Differential Controller (PID-Controller) (L 2310 – S.6 / S.7)

The step response of a PID-controller is composed of all 3 components. It has eliminated almost all the disadvantages of the other controllers through this combination.

Figure 15: PID-controller circuit.

Components:

C₁ in parallel with R₁ (differentiating input branch)
R₂, C₂ in series (integrating feedback branch)
R₃ (initial-condition summing input via U_anfang)
R₄ (balance resistor)

Figure 16: Step response of a PID-controller.

The output U_a shows:

An initial spike (D-action)
Then a ramp (I-action)
T_N = reset time (Nachstellzeit)
T_V = lead/advance time (Vorhaltezeit)
Ideal behavior shown as solid line; actual behavior as dashed line.

The transfer function is:

F = v · [(1 + T₁p)(1 + T₂p)] / (T_N p)

F = (R₂/R₁) · [(1 + R₁C₂p) · (1 + C₁R₂p)] / (R₁ · C₂p)

Note: The PID-controller corrects a control deviation quickly and without permanent offset.

Figure 17: Alternative circuit possibility of a PID-controller with improved frequency response — op-amp with combined RC networks in input and feedback paths.

January 1972 — L 2310 – S.7

Summary Table: Controllers and Time Transformation (L 2310 – S.8)

The following table summarizes controller types, their step/ramp responses, transfer functions, and circuit configurations. Controllers represented:

Controller type	Step response	Ramp response	Transfer function F	Circuit
P-controller	Step (immediate, no offset elimination)	Ramp (proportional)	F = v = R₂/R₁	Inverting amplifier
I-controller	Linear ramp rise	Parabolic	F = 1/(T_p · p) = 1/(R₁C₁p)	Integrator
PI-controller	Proportional jump + ramp	Ramp with initial proportional offset	F = v(1 + 1/(T_N p))	Amplifier + integrating feedback
PD-controller	Spike + proportional step (with offset)	Step + ramp (advanced by T_V)	F = v(1 + T_V p) = (R₂/R₁)(1 + R₂C₁p)	Amplifier with differentiating input
PID-controller	Spike + ramp, no permanent offset	Advanced ramp	F = v · (1 + T₁p)(1 + T₂p) / (T_N p)	Combined

Note on time transformation: The time transformation factor λ = τ’/τ compresses or expands the time axis of the computation. It is applied directly in the differential equation by replacing τ with τ’ = λ·τ (or equivalently multiplying all integration constants by λ).

January 1972 — L 2310 – S.8