English translation
Analog Computer Experiments 3: Servo-Multiplier and PID Controller Experiments
This is an English translation of the original German document “Versuchsbeschreibung: Anwendung des Servo-Multiplizierers im Analogrechner / Versuchsanleitung: Aufbau und Wirkungsweise eines PID-Reglers” published by PEK Electronic, January 1972. The document covers two separate application-note sections (A 1357 and A 1360) bound together as experiment set 3.
Part 1: Application of the Servo-Multiplier in the Analog Computer
Experiment Description
In addition to purely electronic means for multiplication using diode networks or integrated circuits, there are also electromechanical means for multiplication. One such device is the servo-multiplier. The operating principle of a servo-multiplier is as follows: the motor is controlled in two directions via a reference voltage. The motor positions a wiper on two potentiometers simultaneously, so that the angle corresponds to one input voltage — in this case, the reference voltage. The voltage picked off at the potentiometer is then proportional to the product of the reference voltage and the potentiometer supply voltage. This can be viewed as a division operation: one measures the ratio of the output voltage to the supply voltage, which then gives the quotient of the reference voltage divided by the supply voltage. For use in an analog computer the servo-multiplier type SM 1 is manufactured by PEK; in this case, both potentiometers are replaced by a single potentiometer with center tap, whose position corresponds to a voltage. The formula for the servo-multiplier is:
v_out = (u_1 × u_2) / 10 V
Image 1 shows the relationship between the two input voltages u_1 and u_2 and the potentiometer settings.
Tasks
For the program module Servo-Multiplier SM 1, the following tests are to be carried out:
Task 1: Both voltages u_1 and u_2 are to be multiplied by one another. They are each half the reference voltage. A voltage proportional to the product is to be obtained. To avoid sign errors, one should also measure both input voltages and the output voltage.
Task 2: At the same input u (second potentiometer), multiply u_1 by itself several times in succession. Measure the resulting output voltage and also note the final value. This yields a simple squaring circuit.
Task 3: This task deals with division. A circuit for performing a division is to be designed:
v_out = u_2 / u_1, with reference voltage > 10 V
The task can be solved by recalling that the servo-multiplier represents a multiplication, and one can easily derive a circuit that performs the given multiplication. A multiplier forms part of this circuit (Image 2 — shown below).
Task 4: Parts listing
- 1× Power supply Type 36073
- 2× Summing amplifier Type 38502
- 1× Frequency generator Type 38506
- 1× Servo-multiplier Type 38507
- 1× Measuring instrument Type 38508
- 1× A-D display Type 38509/A3
Task 4.1: Experiment execution
The two voltages u_1 and u_2, corresponding to the values from Table 1 (below), are to be set. Both voltages are to be normalized to half the reference voltage. The voltage from the servo-multiplier is then to be measured and entered into the table. One should note that the servo-multiplier introduces no sign reversal; one can also set negative values for the potentiometer (5, 10, 20, 40 V versions are used).
Table 1 — all voltages in Volts:
| u_1 | +10 | +5 | 0 | -5 | -10 |
|---|---|---|---|---|---|
| u_2 | -10 | -5 | 0 | +5 | +10 |
Task 4.2: Measure nine subsequent values as the circuit is driven step by step as shown in Table 1, and check whether the output of the servo-multiplier corresponds to the product (divided by 10).
Table 2 — all voltages in Volts:
| u_1 | +10 | +5 | +2.5 | 0 |
|---|
Task 4.3: The circuit now needs a divider: design a circuit. Note that the divider requires a sign reversal.
Task 4.4: Measure the values obtained in the division circuit. Enter them in Table 2. Note the found circuit in Table 2.
Part 2: Construction and Operation of a PID Controller
General
For analog computer operation, an X-Y recorder is provided; this is driven by the PEK type 38507 units installed on the type 38009 rack. To use the X-Y recorder, two Jackboxes I and II are required. Box I has the inputs on the front panel of the analog computer; Box II is connected to the oscilloscope. For “automatic” computation mode, the recorder is set to “Compute.” On the return stroke (“Operate”), the computation is stopped.
Task 1: PID Controller — Second-Order System
The following second-order ordinary differential equation is to be solved by a PID controller:
y” + 0.5 y’ + y = f(t)
The solution for a step input leads to a damped second-order response. For the simulation, one uses two integrators I1 and I2, three amplifiers acting as proportional, differential, and integral gain stages in parallel, and a summing amplifier II that combines them. The values can be set on potentiometers P1 through P5.
Task 2: Divider Circuit
The circuit is shown in Image 2. Since a multiplier has no input for the tracking signal, a summing amplifier is used as an adder. If there is no input at the multiplier, the output corresponds to the reference voltage. The wiper is driven from one output through a control amplifier toward the reference voltage.
Circuit Description
The circuit also includes a summing amplifier I, which adds the reference quantity x with the command variable w. Three “amplifiers” follow in parallel — acting as proportional, differential, and integral stages — each connected to summing amplifier II as output. The values can be adjusted with potentiometers P1 through P5.
Section 4: Experiment Execution
4.1 Building the control path
Image 2 shows the structure of the control path. By varying the damping values P4 and P5, the oscillation can be influenced; with P6, the oscillation amplitude is set.
The connections are made on the programming field. The resulting damped oscillations should be recorded using an oscilloscope or X-Y recorder.
4.6 Plotting the functions
The functions recorded by the X-Y recorder should be transcribed onto paper. Note the extreme values in each case.
4.7 Circuit for a growing oscillation
By a small change one can represent a variant of the construction. By inverting the feedback from I1, the circuit for a growing (self-excited) oscillation is obtained. The disturbance for this case must be made very small so that the circuit is not overdriven.
The circuit to be programmed is shown in Image 7.
4.8 Plotting the function
The function recorded with the X-Y recorder should be transcribed onto millimeter paper in DIN A4 format.
[Translation covers the first 12 pages (all pages); the document is complete.]