Analogrechner für den Lehrer (Analog Computer for the Teacher)

Complete English translation of the original German-language document (17 pages).

For the Teacher — Generation of a Sawtooth Voltage

(Document reference: A 1350.1 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) The voltage is −10 V.

5.2 (Re 4.2) The measured values are entered in the following table. This measurement is very scattered owing to the difficulty of reading.

T	s		40
t	s	0	10	20	30	40
Ua	V	−10.1	−6.1	−1.2	2.4	6.5

5.3 (Re 4.3) The measured values are entered in the following tables.

T	s		20
t	s	0	5	10	15	20
Ua	V	−10.1	−5.8	−1.2	3.6	8.2

T	s		10
t	s	0	2	4	6	8	10
Ua	V	−10.1	−6.7	3.5	0.6	4.8	8.4

T	s		5
t	s	0	1	2	3	4	5
Ua	V	−10.1	8.6	−4.1	0.6	4.6	9.5

(Document reference: A 1350.1 — L.2, January 1972)

5.4 (Re 4.4) Below, the curve corresponding to the table for a time constant of 40 s is plotted. For the other curves, the instructor should first gain a broader overview, since the determination of the values can be very scattered.

[Figure: Graph showing output voltage Ua (V) on the vertical axis from −10 V to +10 V, plotted against time t (s) on the horizontal axis from 0 to 45 s. The curve for T = 40 s shows a linear ramp rising from approximately −10 V at t = 0 to approximately +10 V at t = 40 s.]

5.5 (Re 4.5) The display on the oscilloscope is naturally particularly demonstrative and at the same time the most accurate. The notes on sheet S.3 regarding triggering must be observed.

For the Teacher — Generation of a Step Function (Z-Jump)

(Document reference: A 1350.2 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) Since the disturbance function is of great importance for understanding control engineering behavior, two matters must be treated in detail here.

Disturbance functions in control engineering.
Effect of the circuit on the computer. For this explanation, the circuit diagram (Fig. 4) can largely be consulted.

5.2 (Re 4.2) The screen image on the oscilloscope is to be observed. The step can be shifted in time across the entire width of the rising edge of the voltage, with the time-proportional progression.

The action of the “comparator” = summing amplifier with open input is to be repeated here, since this is also of importance for other applications, e.g. A/D converters.

For the Teacher — Formation of Factors

(Document reference: A 1352 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) Measurement of the voltages Ua (against ground) gives the values according to the following Table 1:

α	1	0.8	0.6	0.5	0.4	0.2	0	−
Ua	+10	+8	−2	0	−2	−6	−10	Volt

5.2 (Re 4.2) Measurement gives the values according to the following Table 2:

α	1	0.8	0.6	0.5	0.4	0.2	0	−
Ua	+10	+8	+6	+5	+4	+2	0	Volt

5.3 (Re 4.3) Measurement gives the values according to the following Table 3:

α	1	0.8	0.6	0.5	0.4	0.2	0	−
Ua	+10	+6	+2	0	−2	−6	−10	Volt

For the Teacher — Summation with the Analog Computer (I)

(Document reference: A 1353.1 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) The circuit can be plugged into the programming field by individual groups of trainees and then inserted into the computer. The circuit must be checked beforehand to ensure that no voltages are applied to the potentiometer taps that could destroy the potentiometers.

5.2 (Re 4.2) The sum is calculated according to the equation:

U₀ = −8 V

5.3 (Re 4.3) The voltages are given on the student sheet, namely:

U₁ = +2 V
U₂ = +1 V, 2 U₂ = +2 V
U₃ = +0.2 V, 20 U₃ = +4 V

5.4 (Re 4.4) The values for U₁ to U₃ are proposed in the following table and yield the stated output values (all values in volts):

Nr	U₁	Ua	Ub	Ua
1	−2	+2	+0.3	−8
2	+2	−2	+0.2	−2
3	−6	+3	−0.5	+10
4	−2	−2	−0.1	+8
5	+5	−5	−0.2	+9

For the Teacher — Summation with the Analog Computer (II)

(Document reference: A 1353.2 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) The circuit is to be checked in each individual group for errors; in particular, care must be taken to ensure that the potentiometers are not damaged.

5.2 (Re 4.2) The equation for the given circuit is derived as follows:

U₀ = −(10 U₁ + U₀)
Pay attention to the signs!
2U₀ = −10 U₁
U₀ = −5 U₁

5.3 (Re 4.3) The table yields the following computed and measured values:

U₁	+2	+1	+0.5	+0.2	+0.1	0	−0.1	−0.2	−0.5	−1	−2
U₀	−10	−5	−2.5	−1	−0.5	0	+0.5	+1	+2.5	+5	+10	All in Volt

5.4 (Re 4.4 and 4.5) The following relationship emerges corresponding to the general equation:

y = a · x, with a = −5.

[Figure: Graph of U₀ (V) on the vertical axis from approximately −10 to +10, plotted against U₁ (V) on the horizontal axis from 0 to 2.5. The curve shows a straight line with negative slope passing through the origin, consistent with U₀ = −5 U₁.]

For the Teacher — Integration with the Analog Computer

(Document reference: A 1354 — L.1, January 1972)

5. Findings

5.1 (Re 4.3) The total input voltage is U_ges = −2 V.

5.2 (Re 4.4.1) The output voltage at the initial position is:

U₀ = +6 V

5.3 (Re 4.4.2) The voltage changes continuously.

5.4 (Re 4.4.3) Table 1 yields the values entered below:

U₀	V	+6	4.4	3.5	2.8	1.6	0.2
t	s	0	1	2	3	4	5

5.5 (Re 4.4.4) The following Fig. 6 shows the curve obtained:

[Figure: Graph of U₀ on the vertical axis, plotted against t (time). The curve shows a decreasing approximately linear ramp from +6 V at t = 0 to near 0 V at approximately t = 5 s.]

5.6 (Re 4.4.5) An increase of the input voltage produces a greater slope, i.e. the voltage rises more steeply in a shorter time.

A reduction of the input voltage flattens the curve.

Note: When working with the X–Y recorder, if the input voltage is changed in equal steps over equal time intervals — i.e. a linear change is performed — a polygonal line results that corresponds to a parabola.

For the Teacher — Computing (Arithmetic Operations)

(Document reference: A 1355 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) Circuit analysis

It is the task of the instructor to enable the trainees to perform the circuit analysis. For this reason the equation for the computing circuit is not fully given on the student sheet.

The equation should be developed step by step. The procedure is described in detail in the following; this is not however presented in its entirety on the student sheet, since no special initial condition U₀ is specified.

U₀ = −(0.9 U₁ + 1.2 U₂ − (1.3 U₁ − 0.85 U₁) dt)

(Note: the exact form of the equation as legible from the scan; under the integral sign the input combination varies.)

It should be noted that the equation for integration is not explicitly given. Among the inputs to the integrator, no particular initial condition U₀ is stated.

[Figure (Bild 1) — Computing circuit: A block diagram showing several operational amplifier stages (labeled S 1 … D 3 on flat 38502; F 1 … on flat 38502; etc.) with interconnections representing summer, integrator, and sign-inversion stages producing output Ua. The diagram includes potentiometers and feedback paths.]

Caption — Fig. 1: Computing circuit

S 1 … D 3 on flat 38502
F 1 … on flat 38502
(further labels partially illegible due to scan quality)

Note: The output voltage U₀ may not exceed ±10 V at the inputs of integrator S 3 (more exactly ±1); in this case the voltage at the potentiometer tap must be reduced by the ratio n or alternatively other potentiometer settings must be selected.

(Document reference: A 1355 — L.2, January 1972)

5.2 (Re 4.2) Patching plan for the programming field

It is the instructor’s task to leave the patching of the circuit to the trainees. In doing so, the instructor should make concrete suggestions for the choice of the circuit components.

One can however assign the task to trainees without stipulating a circuit layout, leaving them to find a suitable patching plan on their own.

Patching plan:

Inputs:

u₁ … F 1 a
u₂ … P 5 a
u₃ … P 5 a

Connections:

7 f … F 4 a, F 4 a … 53 b
P 5 a … 53 d, F 4 a … P 7 e
3 2 f … P 7 e, P 7 e … I 1 e
I 1 f … 53 b

Output:

53 f

5.3 (Re 4.3) Selection of the measured voltages

If no satisfactory voltages are found in the following 4 columns of Table 1, the following can be inserted:

m
1	0	0	0
2
3

(Table 1: Suggested voltages for the selection of the settings. Remaining entries partially illegible.)

(Document reference: A 1355 — L.3, January 1972)

5.4 (Re 4.4) Measurement of the voltages

Table 2 (Supplementary)

Nr	−U₁	−U₂	−U₃	−U₄	U₁/u₁	U₂/u₂	−U_out
1	−0.22	−0.22	—	—	—	6.1	+4.5
2	+1.04	+1.05	—	—	—	2.53	+1.5
3	1.25	1.26	0.62	—	2.55	—	−6.4
4	(further values partially illegible)

(Note: The table columns include measured voltages at various nodes and the corresponding computed output; scan resolution limits precise reading of some entries.)

5.5 (Re 4.5) This experiment is particularly well suited for demonstration purposes. It is however necessary, when dealing with differential equations, that a functional generator be available. The experiment is easier to carry out since all functions that one would wish to demonstrate are available. A function generator (e.g. type 3592/A2) should be on hand for conducting this experiment.

For the Teacher — Application of a Parabolic Multiplier

(Document reference: A 1356 — E.1, January 1972)

5. Findings

5.1 (Re 4.1) Squaring gives the values in the following Table 1:

Table 1 (Teacher)

Ue	+10	+9	+8	+7	+6	+5	+4	+3	+2	+1	0
Ua	+10	+8.1	+6.4	+4.9	+3.6	+2.5	+1.6	+0.9	+0.4	+0.1	0

Ue	−1	−2	−3	−4	−5	−6	−7	−8	−9	−10
Ua	+0.1	+0.4	+0.9	+1.6	+2.5	+3.6	+4.9	+6.4	+8.1	+10	Volt

5.2 (Re 4.2) Fig. 6 shows the resulting parabola:

[Figure: Graph of Ua (V) on the vertical axis from 0 to 10, plotted against Ue (V) on the horizontal axis from −10 to +10. The curve is a parabola (U-shape) with its minimum at the origin, symmetric about the vertical axis, consistent with Ua = (Ue)²/10.]

Bild 6 — Parabola

(Document reference: A 1356 — L.2, January 1972)

5.3 (Re 4.3) Multiplication yields the values in the following Table 2:

U₁	+10	+8	+6	+4	+2	0	−2	−4	−6	−8	−10
U₂	−10	−8	−6	−4	−2	0	+2	+4	+6	+8	+10	Volt
Ua	−10	−6.4	−3.6	−1.6	−0.4	0	−0.4	−1.6	−3.6	−6.4	−10	Volt

5.4 (Re 4.4) The values in the table yield the function represented on the following sheet (negative parabola):

[Figure: Graph of Ua (V) on the vertical axis from −10 to 0, plotted against Ue (V) on the horizontal axis from −10 to +10. The curve is an inverted parabola with its maximum at the origin, consistent with Ua = −(Ue)²/10 or Ua = −U₁ · U₂/10.]

Bild 6 — “Negative” Parabola

For the Teacher — Application of the Servo-Multiplier in the Analog Computer

(Document reference: A 1357 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) What must be particularly noted in this circuit is that division occurs through the machine voltage at the output of the multiplier. The instructor should explain precisely how this affects the division.

5.2 (Re 4.2) Results of the measurements according to Table 1:

U₁	+10	+5	0	−5	−10	+10	+5	0	−5	−10
U₂	+10	+10	+10	+10	+10	−10	−10	−10	−10	−10
Ua	+10	+5	0	−5	−10	−10	−5	0	+5	+10

5.3 (Re 4.3)

[Figure (Bild 4): Circuit diagram for radication (square-root extraction). The diagram shows the servo-multiplier circuit configuration with an operational amplifier (labeled with α), feedback path including the multiplier block (labeled a, b, c, d), and an output Ua. The feedback arrangement forces the output to satisfy the square-root relationship.]

Bild 4 — The circuit for square-root extraction (radication)

5.4 (Re 4.4) Results of the measurements according to Table 2:

Ue	V	+10	+9	+6	+5	+2.5	0
Ua	V	3.16	3	2.45	2.24	1.58	0

For the Teacher — Construction and Operation of PID Controllers

(Document reference: A 1360 — L.1, January 1972)

5. Findings

5.1 (Re 4.1) Patching plan

For the given component groups the programming field must be patched as follows:

2 … P 6 a
2 1 f … P 4 a … 1 2 e
7 4 … 1 2 e
7 e …

Output on oscilloscope or recorder at: 1 2 / 1 2 … 9

At B it is described elsewhere, where a time-proportional progression is produced.

5.2 (Re 4.2) Proportional controller (P-controller) circuit

Additionally to B.1 the programming field is patched as follows:

1 2 f … 7 7 e
1 af … 7 7 e

The variation of the control behavior through adjustment of the potentiometer value is observed.

5.3 (Re 4.3) Proportional-integral controller (PI-controller) circuit

Additionally the following is patched to the programming field:

2 5 a … Kondensator 1 µF (Pl. 1)
1 5 a … Kondensator 1 µF (Pl. 2)
2 5 a … P 9 a

At that point, to obtain a time constant of 10, the output of the flat Summierer 38504 is used.

(Document reference: A 1360 — L.2, January 1972)

5.4 (Re 4.4) Patching plan — Proportional-integral control (PI controller)

The program from 5.3 is used again; additionally 4.1 to 4.4 is patched on top:

5 9 f … P 5 b
1 3 f … P 5 b
P 10 a …

5.5 (Re 4.5) Patching plan — PD controller

Numerically the program from 5.3 is used again; additionally the following is patched:

(connections partially illegible in scan)

5.6 (Re 4.6) Recording of the functions

The functions have been recorded on the plotter. The resulting curve is shown in Fig. 7 (see companion sheet A 1360 — L.3).

5.7 (Re 4.7) Damped oscillation

On the programming field:

P 4 a … P 5 b
(further connections partially illegible)
… 5 8 e

5.8 (Re 4.8) Recording of the functions

The function has been recorded. It yields the curve shown in Bild 8 (see companion sheet A 1360 — L.4).

(Document reference: A 1360 — L.3, January 1972)

[Page 16: Figure only]

Bild 5 — Diagrams for various dimensioned controllers

[Figure: X–Y recorder chart showing multiple overlapping curves on graph paper, representing the step responses of various controller types. The curves are labeled with controller type designations visible in the scan:

“keine Regel” (no control / open-loop)
“P = 3” (P controller, gain 3)
“P = 8” (P controller, gain 8)
“PD-Regler” (PD controller)
“PI-Regler” (PI controller)
“P-Regler” (P controller)
“PID und optimal dimensionierter Regler” (PID and optimally dimensioned controller) The horizontal axis represents time and the vertical axis represents the controlled variable. The curves show characteristic step responses including overshoot, oscillation, and settling behavior for the various controller configurations.]

(Document reference: A 1360 — L.4, January 1972)

[Page 17: Figure only]

Bild 8 — Aufklingende Schwingung (Growing / Divergent Oscillation)

[Figure: X–Y recorder chart on graph paper showing a single curve representing a divergent (growing-amplitude) oscillation. The curve begins near zero and exhibits sinusoidal oscillations whose amplitude increases progressively over time, characteristic of an unstable closed-loop system or underdamped resonance driven above the stability boundary.]