Operating Manual: Educational Analog Computer Model EC-1-E

This document translates the original German Bedienungsanleitung: Schul-Analogrechner Modell EC-1-E, published by Heathkit (Heath Gerätebau GmbH, Buxheim/Allgäu). The original is a 32-page scanned image PDF with no text layer.

Operating Manual

EDUCATIONAL ANALOG COMPUTER

MODEL EC-1-E

Technical Specifications

9 DC operational amplifiers, each assembled from discrete components such as integrators, adders, and inverters
Power supply: approx. 3 W from a 220 V / 50–60 Hz mains; frequency: 1 kHz to 60 Hz, adjustable by the user
1 variable, adjustable reference voltage of 100 V (DC)
3 variable, stabilized DC voltages for the initial conditions of the integrators (initial condition voltages)
Time constant: 10–0.01 s; selectable and fixed during operation
Solution time: 0.01–0.02 s, selectable and variable through the choice of time scale factor (TSF)
Meter: 0–100 mA full scale
Dimensions: 465 × 285 mm; Weight: approx. 4.7 kg; Power consumption: 1.6 A at 220 V (AC)
Can be connected directly to the analog oscillograph Model OA-1.

Supplementary Sheet for Resistors and Capacitors

Resistors

Resistor color codes identify the resistance value and tolerance. The color bands are printed on the body of the resistor in a defined order. From one end, the first three bands encode the numeric value (first digit, second digit, multiplier), and the fourth band encodes tolerance.

Capacitors

Capacitor types include tubular ceramics and electrolytic (polarized) capacitors. Electrolytic capacitors must be installed with correct polarity. Capacitor values are indicated on the body directly or by a color code conforming to DIN standard IEC 62 and DIN 41426.

Part 1. The Educational Analog Computer EC-1-E

Analog computers are used to solve mathematical-physical equations and to simulate processes. The circuits shown here use discrete electronic components such as integrators, adders, and inverters, thus demonstrating the principle of analog computation. The operational amplifiers within each computer element have sufficiently high open-loop gain.

The computer can operate from 220 V, 50–60 Hz mains at 3 W. Operating frequency: 1 Hz to 60 Hz. The computer incorporates 4 continuously adjustable initial conditions, 4 coefficient potentiometers for setting up program coefficients, and a time-base oscillator for repetitive operation so that CRT display devices such as oscilloscopes can be used. The solution time is 0.01 to 0.02 s. This makes the EC-1-E compatible with the analog oscillograph Model OA-1.

2. Theory of Operation (Brief)

2.1 The Operational Amplifier as a Summing Amplifier and Inverter

An amplifier with one input resistor R₁ and one feedback resistor R_f satisfies:

u_a = − (R_f / R₁) · u_e

For equal resistances R_f = R₁ = R the output is:

u_a = − u_e

This is the inverter. When multiple input resistors R₁, R₂, … R_n feed the summing node, the output is:

u_a = − (R_f/R₁ · u₁ + R_f/R₂ · u₂ + … + R_f/R_n · u_n)

If all resistors are equal, this becomes a sign-inverting adder. The factor can be varied by choosing resistors of different values.

Practical note on gain: When only one input is used with the same resistor value, the gain is −1. Using a 100 kΩ feedback with a 10 kΩ input gives gain of −10.

2.2 The Operational Amplifier as an Integrator

When the feedback resistor is replaced by a capacitor C_f, the circuit integrates the input:

u_a = − (1 / R₁C_f) ∫ u_e dt

The time constant τ = R₁ · C_f determines the integration speed. For τ = 1 s, a constant input of 1 V yields an output ramp of 1 V/s.

The integration constant (initial condition) is the voltage to which the capacitor is pre-charged before the computation run begins. During the “Initial Condition” (IC) phase, the feedback capacitor is pre-charged to the desired starting value. When the “Operate” (OP) phase begins, integration proceeds from that point.

2.3 The Operational Amplifier as Integrator with Summation

An integrator may also receive multiple input signals through separate input resistors. The output is then:

u_a = − (1/C_f) ∫ (u₁/R₁ + u₂/R₂ + …) dt

2.4 Nonlinear Functions: The Diode

Diodes pass current primarily in one direction only. The forward voltage drop across a small-signal diode is very small, and the output of a diode circuit provides a piecewise-linear approximation. Limiter and clipping circuits can be built in conjunction with operational amplifiers. The output waveform of such a clipper is shown: when the input voltage exceeds a threshold, the output voltage is clamped. Depending on the direction the diode is inserted, the clamping acts on positive or negative peaks.

One significant application is the squarer (triangle-wave to parabola conversion), realized by a diode-function-generator arrangement connected to the amplifier feedback path.

3. Setup and Operation of the Educational Analog Computer EC-1-E

3.1 Amplifiers and Balancing Elements

The computer is approximately 465 × 285 × 130 mm and weighs about 4.7 kg. It contains nine (9) tube-type amplifiers connected through a 100 kΩ resistor network and four (4) coefficient potentiometers (100 kΩ, linear). The balancing elements include 1 Si-Zenerdiode chain, 3 Stabivolt tubes, and 2 resistors. The machine also contains an external Stabivolt BA 205A for the IC voltage supply.

Amplifier Frequency Response:

The amplifier’s open-loop gain is very high at low frequencies and rolls off at approximately 12 dB/octave above the break frequency. The gain-bandwidth product is a key specification. The Bode plot (Fig. 22) shows the gain (in dB) versus frequency, with the break point near 1 kHz and unity gain at about 100 kHz for the type used.

The amplifier’s phase response (Fig. 23) shows that at the nominal operating frequency range (1 Hz – 60 Hz), the phase shift is negligible. As frequency increases toward and beyond the break point, phase lag increases significantly, which limits the useful integration bandwidth.

3.2 Coefficient Potentiometers and Switches

The panel contains four (4) 10-turn coefficient potentiometers, each calibrated from 0.00 to 1.00. These set the coefficients for the problem being solved. Switch selections allow each amplifier’s feedback element to be either a resistor (summing/inverting mode) or a capacitor (integration mode).

3.3 The Balancing Procedure

The balance (offset null) adjustment is performed as follows:

Switch the function selector to “METER RANGE.”
Set “AMPLIFIER BALANCE” potentiometers to null the meter reading.
Select “METER FUNCTION” and balance each amplifier in turn.

The balance adjustment corrects for DC offset in each operational amplifier. For the 9 amplifiers in the EC-1-E, this is done individually.

3.4 Reference Voltage and Initial Conditions

The reference voltage of 100 V DC is stabilized by the Zener/Stabivolt arrangement. The initial-condition voltages (IC voltages) for the three integrators are set through their respective 10-turn potentiometers on the panel. These voltages pre-charge the integrating capacitors at the beginning of each problem solution.

The IC voltages are adjustable from −100 V to +100 V. The panel markings show the three IC controls and a fourth “OPERATE” switch that transitions the computer from the IC phase to the compute phase.

3.5 The Servo (Repetitive Operation)

The time-base oscillator generates a square wave at the operating frequency (1 Hz to 60 Hz). During the IC half-cycle, all integrators are reset to their initial conditions. During the Operate half-cycle, integration proceeds. By setting the frequency to 50 Hz, the problem is solved 50 times per second, allowing a CRT oscilloscope to display a steady trace of the solution.

The time-scale factor (TSF) is set by the choice of integration resistors and capacitors. The RC product (time constant τ) determines how fast the machine time corresponds to real (problem) time. For τ = 0.01 s, the machine operates 100 times faster than real time if the problem is formulated in seconds.

4. Programming and Solving Problems

4.1 Solving a Linear Differential Equation

The simplest differential equations solvable on this machine involve constant coefficients.

Example: Solve the equation
2x + 3y = 5
and
2x − 3y = 1

The Lösungsschaltung (solution circuit) in Fig. 33 shows two amplifiers (A1 and A2) connected through cross-coupled resistors. The coefficient potentiometers set the ratio values. The result is read on the voltmeter.

Example: Solving a First-Order Differential Equation

dy/dt + ay = b

This is solved by programming:

dy/dt = −ay + b

An integrator integrates the right-hand side, and its output is fed back with gain −a to the same integrator’s input. The initial condition y(0) is set on the IC potentiometer.

4.2 The Linear Word Problem

The programming steps for a linear simultaneous problem are:

Identify the unknowns and write the equations.
Rearrange each equation to express the highest derivative (or an unknown) in terms of the others.
Assign an amplifier to each variable.
Set coefficient potentiometers to match equation coefficients.
Apply initial conditions.
Switch to Operate and read or record results.

Example given in the manual:

4x + 5y = −8
6x − 7y = 2
Start with x = 40, y = −20

The Schaltungsplan (patch diagram) in Fig. 34 shows the interconnections.

4.3 Solving a First-Order Differential Equation

Example:

dx/dt = −ax + b, x(0) = x₀

where a = 2, b = 5, x₀ = 3.

Programming:

Assign integrator V2 to integrate −ax + b.
Set coefficient pot P1 to value a/scale.
Feed x back to the integrator input with the correct sign.
Set IC = x₀.

The solution is x(t) = (x₀ − b/a)·e^(−at) + b/a.

4.4 A Linear Differential Equation of Second Order

The standard form is:

d²y/dt² + 2ζω_n dy/dt + ω_n² y = 0

This is programmed using two integrators in cascade with feedback, implementing the equation as:

d²y/dt² = −2ζω_n dy/dt − ω_n² y

Fig. 44 shows the patch diagram. The first integrator output is dy/dt; the second integrator output is y. These are fed back through coefficient potentiometers to the input summing amplifier.

Example: Spring-Mass System (Abb. 43)

For a spring-mass-dashpot system:

m·ẍ + c·ẋ + k·x = 0

with m = 20 kg, c = 0.24 N·s/m (Dämpfungskonstante), k = 0.48 N/m (Federkonstante), and initial displacement x₀, initial velocity ẋ₀.

The time-scale factor τ_s = 7 ms is used. With this scaling, the machine solves the problem 1 / τ_s times faster than real time. The result displayed on the oscilloscope (or voltmeter) represents the position x(t) as a function of machine time.

A Simulation of a free spring-mass system: the damped oscillation appears as a decaying sinusoid. By varying the potentiometer for c, the effect of varying the damping constant on the response can be explored in real time.

4.5 Solving Linear Differential Equations: Polynomial Simulation

Higher-order polynomials can be evaluated by a cascaded integrator/summation chain.

Example: Simulating a Third-Order Polynomial

p(t) = a₀ + a₁t + a₂t² + a₃t³

This is programmed by successive integration:

dp/dt = a₁ + 2a₂t + 3a₃t²
d²p/dt² = 2a₂ + 6a₃t

Starting from the highest derivative and integrating successively, the variable t increases linearly and drives all terms.

5. Analog Examples (Analogiebeispiele)

The manual includes a table of physical systems and their analog-computer equivalents (Fig. 45):

Physical System	Symbol	Equation	Analog Element	Symbol
Velocity	v	i = ∫F dt / m	Integrator	(amp with cap)
Mass	m	—	—	—
Force	F	—	—	—
Spring	k	F = kx	Summing amp	(amp with R)
Dashpot	c	F = cx’	Coefficient pot	P
Inductance	L	u = L·di/dt	Integrator	—
Capacitance	C	i = C·du/dt	Integrator	—
Resistance	R	u = Ri	Multiplier (pot)	—

The table lists velocity analogous to current, force analogous to voltage, spring stiffness analogous to resistor, dashpot to resistor, mass to inductance/capacitance, etc. These correspondences allow mechanical, hydraulic, thermal, and electrical systems to be simulated interchangeably.

5.1 Fully and Partially Coupled Programs

A partially coupled program uses only some of the available amplifiers, leaving others free for additional subproblems. A fully coupled program employs all nine amplifiers in an interconnected network.

Fig. 46 shows a network of all amplifiers interconnected for a complex simulation (e.g., a multi-degree-of-freedom mechanical or electrical system).

Fig. 47 shows a more elaborate example: the PID controller simulation, with proportional (P), integral (I), and derivative (D) elements. The patch diagram (Fig. 48) shows how the three terms are generated and summed:

P term: direct connection through a coefficient pot
I term: integrator with gain
D term: differentiator (R–C on input to amplifier)

The output Y = K_P·e + K_I ∫e dt + K_D·de/dt represents the controller output, where e is the error signal.

5.2 The Bouncing Ball

The bouncing ball problem is a classic nonlinear differential-equation demonstration. The ball (mass m) falls under gravity g and bounces elastically from the floor. The equations are:

ẍ = −g (in free fall)
ẋ → −e·ẋ (at bounce, e = restitution coefficient)

The computer implements this with a diode circuit that detects when the position x = 0 (ball hits floor) and reverses the velocity with a given coefficient of restitution.

Components required (from the text):

1 Phototransistor 10 kΩ
3 Widerstände 100 kΩ
1 Kondensator 1 µF
2 Si-Diodes

The patch diagram (Fig. 50) shows the complete circuit with the floor-detection nonlinearity.

Schaltungssymbole (Circuit Symbols)

The back inside cover (page 31) contains a reference chart of commonly used circuit symbols (Schaltungssymbole für häufig gebrauchte Einzel-teile), including:

Resistor (fixed and variable/potentiometer)
Capacitor (fixed, electrolytic, variable)
Diode (signal and rectifier)
Transistor (NPN, PNP)
Vacuum tube (triode, pentode)
Battery
Zener diode
Transformer
Loudspeaker
Microphone
Antenna
Earth/ground
Toggle switch, push-button
Lamp / pilot light
Fuse

Back Cover

HEATHKIT
Das Gütezeichen für Elektronische Bausätze von Weltrang
(“The mark of quality for world-class electronic kits”)

HEATHKIT-GERÄTE
Heath Gerätebau GmbH
Buxheim/Allgäu
Postfach 60, Robert-Bosch-Straße 32–35
Tel. 08331 / 6484–6488

[End of translation. Original German document: Bedienungsanleitung Schul-Analogrechner Modell EC-1-E, Heathkit / Heath Gerätebau GmbH, Buxheim/Allgäu, Germany. 32 pages, scanned image PDF, no text layer.]