Analogrechner Versuche 1: Versuchsbeschreibungen für Platte 38500

Complete English translation of the original German-language document (20 pages).

Experiment Description

Generating a Proportional Voltage

General

When working with an analog computer it is frequently necessary to set a voltage that is proportional to a given value — most importantly, to set the initial condition of an integrator. This presupposes, among other things, knowing how a proportional voltage can be adjusted in the simplest possible way and how to verify the adjustment.

1. Task

A circuit located on board 38500 is to be used for this purpose. Its function is to tap off the supply voltage by means of the slide-contact of a potentiometer and to apply a proportional voltage — as negative as possible (i.e. as close to zero as possible) — to the output of the operational amplifier as initial condition. The time constant of the integrator is set to correspond to a time step of 0.1 s per unit for each attempt.

2. Circuit Description

The circuit is fully contained in integrator 1 (I 4 units) of board 38509. The time constant is set by the potentiometer on board 38503, the fine adjustment is set at 0.

[page 1: figure on page 2 — schematic with relay contacts drawn in the working position]

Figure 1: Circuit — relay contacts drawn in working position

3. Equipment

1 Power supply, Type 36013
1 Integrator, Type 38503
1 Programming field, Type 38506
1 Operating and control unit, Type 38509

4. Experiment Procedure

4.1 Measure the voltage at the output of integrator I 4 in the switch position shown in Figure 1:

U_a = … V

4.2 Set the timer on board 38503 at I 4 to T = 40 s and set the fine adjustment to 0.

Set the switch on the control board 38509 to “Compute.”

At this timer setting, measure the output voltage U_a at f at intervals of 10 seconds and enter these measured voltages in a row of Table 1. The last value can be read when the switch on the control board is placed in “Pause” (only relay G is energized).

4.3 Set the switch again to “Set” (initial condition). Repeat the measurement with the various times T = 20, 10, 5 and 4 s of the timer, and enter the values in the table.

4.4 Plot the values from the table for U(t)/U_ref and U_a against t in a diagram (Figure 2) so that one can recognize the function of the timer from the curve shape. (Choose a suitable scale.)

Figure 2: Diagram of the time-dependent voltages

[page 3: diagram/figure — grid for plotting time-dependent voltages]

4.5 Measurement with the cathode-ray oscilloscope:

The oscilloscope is connected to the output of the integrator; the circuit is to be operated in the “automatic” mode. The oscilloscope trigger is to be set to –10 V so that the triggering of the oscilloscope occurs at the beginning of the integration step. The timer is set to correspond to a time step of 0.1 s per unit for each experiment.

Figure 3: Description of time measurement with the cathode-ray oscilloscope

[page 4: figure only — diagram showing integrator Pl 38503, oscilloscope connections, trigger connections labeled “Aufstellung” (setup) and “Trigger”]

In fact there is another possibility of carrying out the time measurement with the oscilloscope, namely to use the sawtooth signal of the oscilloscope as the time base and to display on the sawtooth the variation of the integrator output voltage U_a simultaneously. For this purpose one can use the “X-Y” mode, provided that the oscilloscope has this capability (“X-Y display”).

Experiment 3 shows a circuit that has been put together to demonstrate this on the oscilloscope.

Experiment Description

Generation of a Z-Voltage (Step Function)

General

The Z-voltage (step function) serves, as its name implies, to feed a step function into a control-engineering circuit so as to be able to specify the step height (amplitude) and the starting point (zero crossing) of the step as required.

Two conditions must be fulfilled:

A definable positive height (amplitude)
Knowledge of the starting point (zero crossing) of the step

To carry out the experiment, the circuit is assembled on the programming field, as shown in Figure 1 (below).

The circuit is designated with Z. It is also referred to as the “programming field.”

Caution: This circuit not with the correspondingly labeled terminals on the programming field.

Generation of the Initial Value Voltage (Z-Voltage)

For the generation of a proportional initial-value voltage, the front panel potentiometer from a single Gegenkopplungsstufe (feedback stage) is used. One applies a constant +15 V supply voltage to the input and switches on at the same moment. The output of the operational amplifier is thus the negative feedback to the output. In this way one forms the “step” at time t = 0: as long as the input potential is negative (i.e. below the negative reference voltage, which always equals –U_ref), the output of the operational amplifier is at the positive rail potential (≈ +15 V). As the input potential passes through Null (zero crossing), the output flips downward (“the spring jumps”). It is clearly impossible to have a negative output step at the same time as the beginning of the step, because only positive voltages are passed (with positive values of the upper input), and a step function can produce a positive as well as negative output (–I_ref).

Practically however, the “spring” only occurs at the moment when the input potential passes through 0 V. Only in the instant in which the Eingangspotential (input potential) passes through 0 V does it fall to a value equal to the negative reference voltage. It is, as set out above, always negative, meaning that from the given connection scheme one obtains the negative Eingangsstufe (input stage).

At this point it is also worth mentioning that the “spring” only appears for the moment in which the Eingangspotential passes through the zero potential. This is solely due to the fact that the particular wiring at the Eingangspotential is always exactly –I_ref, no matter what one adjusts the potentiometer to.

Even if the input signals come close to the reference potential, there is always a very small proportional voltage here. The diagram (Figure 2) shows the Z-voltage together with the sawtooth voltage.

Figure 2: Waveforms — sawtooth and step voltage

[page 6: figure only — waveform diagram showing sawtooth and step voltage (Z-voltage)]

Figure 3 (accompanying text):

For a better overview of the voltage ratios, the potential relationships in Figure 3 are presented together. The accompanying figure (b) shows the voltage curve at the output of the Summiererstufe (summing stage). One can see in the axis (a) that the 10 V step begins at the negative supply potential (Sprunghöhe). The step (b) shows the voltage curve at the output of the computing amplifier.

At (c) the voltage curve at the output of the computing amplifier is shown.

In (a) the voltage-step sequence has already been cut off from the sawtooth voltage by the switching operation.

Figure 3: Impulse responses — accompanying potential relations

[page 7: figure only — diagram with labeled potential traces (a), (b), (c) showing sawtooth input and step responses]

1. Task

The voltage step occurring at the output of the switching stage is to be displayed on the cathode-ray oscilloscope so that one can also observe the time-base connection (Zeitablenkung) from the oscilloscope.

The time constant of the integrator is to be set to 0.1 s.

The experiment is to show, in automatic operation, one step function on the oscilloscope display at a time, in order to be able to see the variation of the voltage step also with the cathode-ray oscilloscope.

2. Circuit

The circuit, insofar as it is important for understanding, is shown in Figure 4 nochmals (once more). The connection is to be made from the front panel potentiometer P1 on board 38501 via the Summierfeld (summing field), the integrator (I 4), to the cathode-ray oscilloscope.

Figure 4: Test circuit

[page 8: figure only — block diagram showing Potentiometer P1 38501, Summierfeld Pl 38504, Integrator I 4 38503 (with Aufstellung/Trigger labels), and Oscilloscope connection]

3. Equipment

1 Power supply, Type 36013
1 Potentiometer, Type 38501
1 Integrator, Type 38503
1 Programming field, Type 38506
1 Operating and control unit, Type 38509
1 Oscilloscope, Type 2002

4. Experiment Procedure

4.1 Setting up the measurement circuit:

The circuit is assembled according to Figure 4. The selector switch on the programming field (board 38503) for I 4 is to be set to the “Automatic” position; the fine adjustment is set to linear. On board 38504 the output 4 is to be connected to the oscilloscope input.

The “Automatisch” (automatic) mode switch alternately energizes both relays (G and H) at defined intervals. This produces the step function.

The connection is made first without the oscilloscope trigger so as to produce a running oscilloscope image of the output voltage U_a.

For observing the oscilloscope image, two further aspects are important:

Y I on the oscilloscope at a sweep voltage from board 38504 side
Y II on the oscilloscope is at the proportional input voltage from the programming field

The corresponding ground connections between the computer and oscilloscope must not be forgotten. For clarity they are made from one of the black ground terminals of the computer.

4.2 Once the preparations according to 4.1 have been completed, the switch on board 38509 is set to “Automatic.” After making the corresponding settings on the oscilloscope, the dual-trace oscillogram must be visible.

One now varies the potential at potentiometer P 1 on board 38501 and observes the displacement of the “step” relative to the start of the sawtooth voltage.

Experiment Description

Formation of Coefficients

General

The circuits of the operational amplifiers in analog computers are designed so that only numbers smaller than a given magnitude can be used — in this case, smaller than ±10 V. The number 10 corresponds to the nominal machine voltage.

In such a computer there is the possibility of using potentiometers as multipliers to set arbitrary coefficients between 0 and ±1 as desired. One then finds the coefficient, as a rule, somewhere between 0 and an end. In the computer there are indeed three different Schaltungsmöglichkeiten (circuit options) for potentiometers, which are entered in Figure 1. They are also symbolized in Figure 2.

Figure 1: Circuit options for potentiometers

[page 11: figure — four circuit diagrams (a), (b), (c), (d) showing potentiometer configurations]

Figure 2: Symbol representations of potentiometer circuit options

[page 11: figure — symbol/schematic equivalents for the potentiometer circuit options (a), (b), (c), (d)]

In the following table the three circuit configurations with their Kennzeichnungen (designations) are given, as well as the relationship between the Abgriff (wiper position) α and the output voltage U_a relative to the input voltage U_e, and the ground terminal:

Pot. No.	Start	End	Reference
P 1 … P 3	—	—	to ground (at 0 V)
P 4 … P 9	+ U_ref	to ground	at 0 V
P 10 … P 15	+ U_ref	– U_ref	—

All potentiometers have the following specifications (manufacturer’s data):

Resistance: 2 kΩ ± 1 %
Linearity: ± 0.25 %

To find the setting for the task, a positive and negative Wert (value) is always present in the switch position of the circuit configuration.

P 1 … P 3:

U_a = α · U_e + (1 – α) · (–U_ref) = (2α – 1) · U_ref

P 4 … P 9:

U_a = α · U_e

P 10 … P 15:

U_a = α · (U_e – (–U_ref)) + (–U_ref) = α · (U_e + U_ref) – U_ref = (2α – 1) · U_ref

The coefficient k is at most equal to ±1 with the given potentiometers (P 1 to P 15).

Remark: When a coefficient must be set with great precision, it is not possible to read off the exact value from the potentiometer dial. Instead, the voltage output is measured directly with a digital voltmeter, or a fixed proportional portion of the voltage is set with the potentiometer. For certain purposes it has also proven useful not to use a freely settable potentiometer, but rather to use a fixed resistor divider for a fixed portion of the voltage in the circuit.

α) Applied to ground (chassis): voltages from +U_ref to –U_ref are possible:

1. Task

For the three types of potentiometers the relationship between output voltage and input voltage is to be recorded in succession.

2. Measurement Circuit

The measurement circuit for the three types of potentiometer is shown in Figure 3.

Figure 3: Measurement circuit

[page 13: figure only — schematic with U_e input, potentiometer symbol, U_a output, and ground; notation: “Digital-Spannungsmesser (Achtung jeweils umpolen)” (digital voltmeter — note: reverse polarity as required)]

3. Equipment

1 Power supply, Type 36013
1 Potentiometer, Type 38501
1 Programming field, Type 38506
1 Measuring instrument, Type 38508
1 Operating and control unit, Type 38509 (for the reference voltages)

4. Experiment Procedure

4.1 Connect potentiometer P 1 with the wiper s to the programming field and to the digital voltmeter.

Measure the voltage profile against ground as a function of the potentiometer setting according to the following table:

α	1	0.8	0.6	0.5	0.4	0.2	0	–
U_a								Volt

4.2 Connect potentiometer P 4 with point a to the reference voltage +U_ref, and the wiper s to the digital voltmeter.

Measure the voltage profile against ground as a function of the potentiometer setting according to the following table:

α	1	0.8	0.6	0.5	0.4	0.2	0	–
U_a								Volt

4.3 Connect potentiometer P 10 with point a to the reference voltage +U_ref, and point e to the reference voltage –U_ref, and the wiper s to the voltmeter.

Measure the voltage profile against ground as a function of the potentiometer setting according to the following table:

α	1	0.8	0.6	0.5	0.4	0.2	0	–
U_a								Volt

Experiment Description

Formation of Coefficients with the Analog Computer (I)

General

For the summation of, say, building blocks in analog computers there are strong operational amplifiers that, together with feedback resistors, can accept various inputs with a ratio of 1 : 10 in the Gegenkopplungszweig (feedback branch). These 1 : 10 resistors are used as gleichwertige (equivalent) resistors for equal weighting of all inputs.

In such a computer there is the possibility, as already described above, of using multiplying potentiometers to set arbitrary factors from 0 to ±1. One then multiplies by a factor found somewhere between 0 and end. In this computer there are indeed three different circuit options for the potentiometers as entered in Figure 1. They are symbolized in Figure 2.

The inputs E_0 to E_5 are labeled, as shown in the schematic. For the summation with “complete weighting” (full scale), both the analog computer stage and the amplifier provide a factor of 1 : 12 equally, so the factor is to some extent between 1/12 and 1. This ensures that the maximum machine voltage ±10 V between the input and output is never overstepped.

The inputs are labeled by the letters E_0 through E_5 according to Figure 1.

For the summation the so-called “Maschinen-Teiler” (machine dividers) are used, which reduce the input voltages by a factor as required. Hence one writes, with the designation E_i/factor:

F_i = F_0 · (1/12), F_1 = F_1 · 1, …

(Taken from Figure 1 on the right-hand side.)

The Eingänge (inputs) have the following relative weighting:

F_0 = F_0 / 12 up to F_5 = F_5 / 12

In the following table the formulation is given in detail.

Theory of the Summation Amplifier

The summation amplifier is built from a feedback amplifier according to the following principles:

It has high voltage gain, so that as a consequence the output can always be driven to a full output voltage.
There is always a Phasendrehung (phase inversion) by 180° between the input and the output.
It has an extremely high input impedance so that the individual source impedances of each input have no loading effect.

From a circuit standpoint, the following simplified considerations apply:

Its high gain means that, for full output voltage, the differential input voltage between the input and the feedback input (the “Summenpunkt” or summing point) is always set equal to Null (zero). This may be called “virtual ground” (the summing point is always at zero volts).
The phase inversion by 180° between the input and the output.
The extremely high input impedance means that the individual input resistors R_1 through R_n flow their currents directly into the summing point, practically without loading each other.

Figure 2: Current and voltage paths at the summation amplifier

[page 16: figure only — schematic showing summing junction with input resistors R_1, R_2 … R_n, feedback resistor R_f, and operational amplifier; input voltages U_1, U_2 … U_n and output voltage U_a labeled]

The sum of all currents equals zero, so:

i_1 + i_2 + … + i_n = 0

From this, for a single input:

U_a = –(R_f / R_1) · U_1 – (R_f / R_2) · U_2 – … – (R_f / R_n) · U_n

The output voltage is thus independent of the individual input voltages and the external loading, which only depends on the ratio of the feedback resistor to the input resistors.

1. Task

The following equation is to be solved using the analog computer:

U_a = –(U_1 + 2 · U_2 + 20 · U_3)

In this task it is to be shown that when using two factors, the factor must be approached by two building blocks in the Verstärker (amplifier stage). This means that in this case two potentiometers are always placed in series in the signal path (Cases b and c, below). One should note at this point that the summation amplifiers can accept only identical summing inputs at equal weighting, i.e. the voltages are always proportional to each other before proceeding. One also notes that in that case a full Vorzeichenwechsel (sign change) can be prevented by connecting each factor inversely through the appropriate circuit for that potentiometer (as has been described before).

It is worth noting at this stage that the summation may include both positive and negative addends simultaneously, and that the sign inversion inherent in the summation amplifier can be used to advantage.

In particular, when connected via an intermediate amplifier stage, one must also take into account the additional sign reversal through the routing.

For the preparation one should consider the following simplified aspects: that the summation amplifiers can process only proportional inputs at the same time — individual units are the rule for the task at hand. It is possible to cover all values with one stage or with several stages; it depends on the circuit configuration selected.

2. Circuit

The circuit corresponding to the task is shown in Figure 3.

In this figure all selectable potentiometers and the summation amplifier are indicated. The interconnections listed in Table 1 are to be made from the programming field.

Figure 3: Circuit according to the task

[page 18: figure only — block diagram with two potentiometers and a summation amplifier, output labeled –U_a]

3. Equipment

From Figure 3 the following boards are required:

1 Power supply, Type 36013
2 Potentiometers, Type 38501 (×2)
1 Summation amplifier, Type 38502
1 Programming field, Type 38506
1 Measuring instrument, Type 38508
1 Operating and control unit, Type 38509

4. Experiment Procedure

4.1 First set up the interconnection on the programming field.

From the reference voltage the partial voltages to the summing junction of the summation amplifier are tapped in parallel; these form the part corresponding to the task from the equation. The output of the summation amplifier is the negative sum (inverted).

4 Analog Computer Measurement

The scale of the analog computer is to be maintained throughout. The analog computer first normalizes the partial voltages to be processed. (Small voltages [below 1 volt] are not handled reliably with the digital voltmeter.)

4.2 Approach for Voltages

To convert voltages, the following voltages must be assumed:

U₁ = 2 V
U₂ = 1 V
U₃ = 0.2 V

4.3 Measurement

The required voltages are set at the inputs and checked individually at the outputs of the summing amplifier. This is done at the inputs of the digital voltmeter.

4.4 Additional Setpoints

Additional setpoints, also negative, for the voltage control of the summing amplifier are listed in the following table. These are then to be connected accordingly.

Nr.	U₁	U₂	U₃	U₄	Sum voltages in Volt
2
3
4
5

Note: In the wiring of the programming panel, inputs to the summing amplifier that are unused must be connected to the negative reference voltage.

One can use both potentiometers to set positive and negative reference voltages, so that the task of summing positive and negative values at the inputs does not present a problem.

[page 20: blank]