Versuchsbeschreibungen: Summierung und Integration mit dem Analogrechner (Versuche 2)

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

Laboratory Instruction

Summation with the Analog Computer (II)

General

The general principles for summation with the analog computer are dealt with in the laboratory instruction “Summation with the Analog Computer I” (Experiment A 1353-1) and should be regarded as familiar. The treatment given here is therefore an extension of that instruction. Its purpose is to give the student the opportunity to set up the formula for a given circuit and at the same time to use the formula as a check on the results of the experiment.

1. Task

Which relationship exists between the output voltage and the input voltages at the summing amplifier (Addierschaltung)?

2. Circuit

The circuit is shown in Figure 1.

3. Equipment

The following units of the computer are to be used:

Power supply — Type 38010
Potentiometer — Type 38501
Summing amplifier — Type 38502
Multimeter instrument — Type 38508
Programming panel — Type 38506
Radiation and control (Bedienung und Steuerung) — Type 38509

Laboratory Instruction

Integration with the Analog Computer

General

The integrator essentially consists of an amplifier and a capacitor arrangement. In the signal path there are (as shown in Figure 1) resistors and capacitors instead of the pure resistor network of the summing amplifier. It is the task of the integrator to integrate one or more voltages (inputs) with respect to time. Integration is carried out over the output voltage U_a.

If one considers a point P at very high loop gain (open loop), the currents i_1, i_2, i_3 all flow to node P simultaneously, and it holds that:

i_1 + i_2 + i_3 = i_c

In voltage terms, ignoring back-reaction:

U_a = -(1/C) ∫ (U_1/R_1 + U_2/R_2 + U_3/R_3) dt

The scaling factors mean:

U_a = -(1/(R_1·C)) ∫ U_1 dt - (1/(R_2·C)) ∫ U_2 dt - (1/(R_3·C)) ∫ U_3 dt

Figure 1: Principle of the Integrator

Each integration yields an integration constant. This is the initial value of the output voltage U_a at the start of the integration. It holds that for t = 0 the voltage U_a = U_0. It is symbolized in the circuit symbol of the integrator by a small arrow indicating that at the start of a computation run, the initial value can be entered.

Figure 2: Circuit symbol for the Integrator

The integrator has 1 or 10 as its weighting factor for each input. It multiplies these inputs by the weighting factors and forms the sum at the output. The sign of the output voltage is opposite to that of the input voltages. The sign of the input voltage is set using the potentiometer provided for that purpose. Capacitor C in the Model 38503 integrators can be assigned values of 1, 2, or 3 in parallel, which gives a range of values from approximately 1·10^-6 F to 10·10^-6 F.

Operating Modes

The following operating modes are provided for the integrator, and these can be switched between:

Set: The input of the summing amplifier (g) is placed at the initial voltage level.
Compute: The inputs a to b (as the sum) are at the input of the summing amplifier.
Pause: The input is switched off.
Automatic Operation: An automatic sequence of the above mentioned operating possibilities.

The basic circuit of the integrator results from the requirements of the circuit: the inputs from 1 to 10 (in Figure 2) go to the summing amplifier, from 11 to 13 (Figure 2 to Figure 3).

Table 3: Switching possibilities for the input switching of the integrator

Step	Operation	Relay R_1, C_1, V_1	Relay R_1, C_1, V_1	Expected Output U_a(t)
1	Set (Einstellen/Aufstellen)	Inactive	Inactive	(1)
2	Compute	Stimulus	Stimulus	(2)
3	Automatic	Steps (1) and (2)	Steps (1) and (2)	—

Figure 3: Basic switching possibilities for the input switching of the integrator

For the “Pause” state, it should further be noted that due to the high resistive values of the capacitor, the integrator drift does not come to a complete stop; it continues to run very slowly. This means that for some purposes, the result may become slightly inaccurate over time. It can, however, be used for certain short-term pauses.

1. Task

One is to set up a circuit for the following equation:

U_a = -(U_0 - 10 · U_e) ∫ dt + 8 V

Due to the sign reversal of the amplifier it is more convenient to formulate this as input quantities:

U_a = ∫ (-U_0 + 10 · U_e) dt + 8 V

2. Circuit

Figure 4: Circuit

The figure shows that on the integrator I_1 (Model 38503), a capacitor C = 10 pF is installed.

3. Equipment

Power supply — Type 38013
Potentiometer — Type 38501
Integrator — Type 38503
Summing amplifier — Type 38502
Multimeter instrument — Type 38508
Programming panel — Type 38506
Radiation and control — Type 38509
I-V converter — Type 38509

4. Laboratory Instruction (Versuchsdurchführung)

In order to realize the circuit according to Figure 4, multiplication by coefficients and an integration with rectangular function should be performed.

The two voltages U_1 and U_2 used in this experiment can be multiplied by coefficients and then added (or subtracted). The potentiometer F is also used to apply a positive as well as a negative voltage to the input. The potentiometer F is the “reference potentiometer” and is connected to the + and – supply voltages of the analog computer.

After analyzing the equation above, the circuit can be programmed on the programming panel. Analysis and the circuit are given in the laboratory instruction in detail.

4.1 Potentiometer Setting

It should be noted here that the summation on the programming panel in Figure 4 takes the voltages at the output of the reference potentiometer as the programming voltages. The “reference voltage” at terminal (Abgriff) is a potential; it can be connected to an input with an Eingangswiderstand. One also finds a feedback resistor such that the gain of the summing amplifier is equal to one; the output will then have the same magnitude as the input, with inverted sign.

4.2 Setting Up the Equation

The equation valid for this circuit is set up, and one verifies it by means of measurements.

4.3 Checking the Equation Against Measured Values

The equation is set up for the input voltages U_1 and U_2 and the output voltage U_a is checked. In this connection, the values are entered at + 0.2, + 0.2, 0, − 0.1, − 0.2, − 0.3, − 1, − 2 (all values in Volts in the table). The measured values are entered in Columns 4 in the table.

4.4 Checking the Equation

Before measurements are made, the “Bedienung und Steuerung” (operation and control) panel must be set to “Einstellen” (Set). A potentiometer P is found in the circuit (Schaltung), which is set in such a way that U_a is 0 V when the integrator is in the “Set” mode; all the other inputs are simultaneously at the level of U_a = 0 V.

Laboratory Instruction

Integration with the Analog Computer

General

Detailed information on the working principles of the integrator can be found in the laboratory instruction A 1354, which should already be known.

One is to set up a circuit for the following problem:

U_a = -0.5 · U_1 - 1.2 · U_2 + ∫ (-1 · U_1 + 2 · U_2 - 0.85 · U_a) dt

The circuit should be arranged in such a way that the above equation is satisfied, and the analysis is to be placed in the programming panel.

The analysis and the circuit are given in the laboratory instruction and can be checked in detail.

2. Circuit

Under the analysis of the equation above, the circuit can be programmed on the programming panel so that the analysis of the equation can be readily entered.

The analysis and the circuit can be checked for a number of different voltage curves that may appear.

3. Equipment

Power supply — Type 38013
Potentiometer — Type 38501
Integrator — Type 38503
Summing amplifier — Type 38502
Multimeter instrument — Type 38508
Programming panel — Type 38506
Radiation and control — Type 38509
Capacitor — Type 38109, 10 pF

Laboratory Instruction

Summation with the Analog Computer (II)

General

All the general principles for the summing operation with the analog computer are dealt with in the laboratory instruction “Summation with the Analog Computer I” (Experiment A 1353-1), which should be regarded as familiar. The following is therefore an extension of that instruction. It is a helpful exercise to write the formula for a given circuit and at the same time to use that formula as a check on the experimental results.

4. Laboratory Instruction (Versuchsdurchführung)

4.1 Set Up

In order to realize the circuit from Figure 4, in the special mode: multiplication by coefficients and an integration with a square-wave function should be performed.

Plug the circuit of Figure 4 onto the programming panel. The voltage U_a is to be used as a reference (reference voltage); set the output voltage to 10 V.

4.2 Potentiometer Setting

Choose the potentiometer settings U_3 so that a complete Gleichung (equation) with uniform measurement results can be achieved.

For this, do the following: set the input potentiometer P so that the gain of the summing amplifier equals 1 (i.e., the output is equal and of opposite sign to the input), and then check this with a measurement.

For the first attempt, use the following voltages:

U_1 = +3 Volt
U_0 = 0.9 V
U_e = -8 V

4.3 Measurement

Set up the measurement as follows: place the “Bedienung und Steuerung” panel to “Rechnen” (Compute); observe the output voltage U_a at the meter (Meßinstrument). At a gain of approximately +10 V or –10 V, switch to “Pause” and read off the final value. It can also be used for some brief pause purposes.

4.3.1 Time Measurement

It is also interesting to time the single measurement with the multimeter. In that case, observe the output voltage U_a as it changes proportionally with time. The values given below (Table 1) are the measured values.

Table 1 — Measured values

t	1	2	3	4	5
U_a

4.3.4 Plotting the Curve

Enter the values from the table into a coordinate system and draw the curve through them.

What waveform of the output voltage can be expected?

4.3.5 Notes for Advanced Students

Note that the voltages can be measured via the ratio potentiometers (P) across the whole range. The measurements can be performed in successive steps with the N-V converter (I-V converter).

The readings are taken after each of the 4 potentiometer steps by the student switching to “Einstellen” on the “Bedienung und Steuerung” panel. After each measurement, the student should place the switch in the “Rechnen” (Compute) position. After the next measurement, the further values are to be determined, until the desired accuracy is reached. The values so taken are placed in the following column (next to the previous measurement).

Table 2 (Time-variable measured values)

Measurement Nr.	−U_4	−U_4	−αU_4	+αU_4	αU_3	−αU_4	U_4	U_p	−U_Ref
1
2
3
4
5

4.5 For Advanced Students

It is of course considerably more interesting to work with time-varying voltages corresponding to the time-variable quantities.

It is recommended to choose the voltages U_3 or U_4 as AC voltages, for example sinusoidal voltages. Within the circuit one must then observe with an oscilloscope the phase inversion or, for arbitrary AC voltages, the negation and the waveform changes arising from the integration (with sine — phase shift of 90°).

The integration time constant may also be varied.

Laboratory Instruction

Solution of an Integration Equation with the Analog Computer

==========================

General

Detailed information on the integration capabilities of the integrators can be found in the laboratory instruction A 1354, which should already be known.

One is to set up a circuit for the following problem:

U_a = -(0.5 · U_1 + 1.2 · U_2 + ∫ (1 · U_1 + 2 · U_2 - 0.85 · U_a) dt)

In the forward transform, the problem is to be checked so that the above equation can be satisfied, and the analysis is to be placed on the programming panel.

Analysis and circuit are given in the laboratory instruction and can be verified in detail.

2. Circuit

Under the analysis of the equation above, the circuit can be programmed on the programming panel. Analysis and circuit are given in the laboratory instruction in detail.

Please note that a number of different voltage curves may appear.

3. Equipment

Power supply — Type 38013
Potentiometer — Type 38501
Integrator — Type 38503
Summing amplifier — Type 38502
Multimeter instrument — Type 38508
Programming panel — Type 38506
Radiation and control — Type 38509
Capacitor — Type 38109, 10 pF

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Laboratory Instruction

Solution of an Integration Equation with the Analog Computer (continued)

4.2 Setting Up the Solution Procedure

The following notes are for solving the equation using successive integration by the voltages U_1 and U_2. Also note that the measurement results below are those for the given equation.

Carry out the following steps:

Set up the equation of the circuit.
Plug in the circuit for the given equation.
Check by measurement against the expected values.
Measure for a series of potentiometer settings at equal increments using successive integration.

4.3 Measurement Notes

Choose the measurement points so that the equation is satisfied when both input voltages are set. Set the integration to a particular value; then compare measured and theoretical values at every point. Note the values in the tables provided.

After each measurement, place the student switch at “Einstellen” and after the next measurement at “Rechnen”; measure the further values.

The values thus obtained are entered in the following column in Table 1 (“Meßwertetabelle” — measurement value table).

4.4 Procedure

4.2 Set up the circuit as described, i.e. the switching links of the task, meaning that you plug the links in as shown for the given equation, and then set them to a particular position according to Equation (1) below (continuous integration):

After each measurement in integration, compare the outputs with the expected values. Verify that the sign and value match. If not, the potentiometers must be re-adjusted.

4.3 Change the settings for each new measurement step by step, and note the values in Table 1 (Meßwertetabelle).

After each measurement, enter the values according to the time scale in Table 2 (“Zeitlich veränderliche Meßwerte” — time-variable measured values). Read all the values in the measurement as a succession.

Laboratory Instruction

Application of the Parabolic Multiplier

General

The direct analog multiplication of variable quantities is not possible using simple circuit arrangements. Therefore various more complex arrangements have been developed, most of which follow the quarter-square method.

The identity is:

(u_1 + u_2)² - (u_1 - u_2)² = 4 · u_1 · u_2

and the derived relation:

u_1 · u_2 = [(u_1 + u_2)² - (u_1 - u_2)²] / 4

The differences of the squares (squarers) thus fall out. In the analog computer, this is replaced by four integrating circuits (or squaring networks). This combination of four squarers is referred to as the parabolic multiplier.

The fully detailed information about the internal construction can be found in the laboratory instruction A 1356, which should already be known.

Figure 1 shows the schematic of a discretely constructed parabolic multiplier with the input voltages U_1 and U_2 and the output voltage U_a.

In analog computing technology this circuit is always represented as a block. Figure 2 shows the circuit symbol for such a multiplier with the two input voltages U_1 and U_2 and the output voltage U_a.

Figure 1: Principle of a discretely constructed Parabolic Multiplier

The circuit shows:

Input U_1, its inverse −U_1
Input U_2, its inverse −U_2
Resistors R, R_x, R_N, and R
Four squaring sub-networks (Multiplier I through IV):
- Multiplier I: +(U_1 + U_2)
- Multiplier II: −(U_1 − U_2)
- Multiplier III: −(U_1 − U_2)
- Multiplier IV: +(U_1 + U_2)
4 networks or integrating circuits (4 Netzwerke oder integr. Kreise)
Output U_a

Figure 2: Circuit symbol for a Multiplier

When multiplying, it must also be taken into account that the output voltage must not exceed the magnitude of 10 V. One must therefore include a factor that is taken as 10. Since at each input there is a maximum voltage of ±10 V, a multiplication of ±100 V would result. One therefore divides by appropriate circuit measures through 10 V. This gives at the output again a voltage of magnitude at most 10 V, which represents the product of the input quantities. Thus for every value:

U_a = (U_1 · U_2) / 10 V

This division is called the introduction of a scale factor; in this case the factor is considered as:

k = (10 V)^-1

1. Task

One is to set up a circuit for the relationship:

U_a = U_1²

Here it should be noted that in the case of the squarer one connects both inputs together, because:

U_1 = U_2

2. Circuit

The circuit is implemented by using a multiplier as a squarer, and it is shown in Figure 3.

Figure 3: Circuit

3. Equipment

Power supply — Type 38013
Potentiometer — Type 38501
Parabolic multiplier — Type 38501
Multimeter instrument — Type 38508
Programming panel — Type 38506
Radiation and control — Type 38509 (used for reference voltage)

4. Laboratory Instruction (Versuchsdurchführung)

This experiment simultaneously serves as a check for the working of the parabolic multiplier by comparing the output voltage U_a again with the measured values of U_a.

This voltage contains a factor from the potentiometer, so that the result shows this factor k (= (10 V)^-1) as a multiplication factor of its product. One notes the following:

k · U_1² = (10 V)^-1 · U_1²

with U_1 in Volts, this produces an output in Volts. To remove the factor k one needs to multiply by 10 V, so the result is equivalent to a Wert (value) that includes the Faktor k = (10 V)^-1 multiplied once more.

4.1 Vary the input voltage U_e through the values given in the following table from +10 V to −10 V and enter the measured results in Table 1 below.

Table 1 (Squaring)

U_e	+10	+9	+8	+7	+6	+5	+4	+3	+2	+1	0
U_a

U_e	−1	−2	−3	−4	−5	−6	−7	−8	−9	−10
U_a

All values in Volts

4.2 Plot the resulting curve in the graph sheet below (Figure 4).

[Figure 4 — Graph: vertical axis labeled U_a (0 to 10), horizontal axis labeled U_e (−10 to +10). Grid provided for plotting the squaring result.]

Figure 4 — Result of the Squaring Operation

4.3 Multiplication

Now set the two input voltages U_1 and U_2 no longer equal in sign, but instead set each to:

U_1 = −U_2 = U_e ;

Vary the input voltages by means of the required circuit change via two potentiometers (do not use inverting elements!).

Enter the values in Table 2 below.

Table 2

U_1	+10	+8	+6	+4	+2	0	−2	−4	−6	−8	−10
U_2	−10	−8	−6	−4	−2	0	+2	+4	+6	+8	+10
U_a

All values in V

4.4 Plot the resulting curve in the graph sheet below, where the scale divisions are to be determined by the student.

[Figure 5 — Graph: vertical axis labeled U_a, horizontal axis labeled U_e = |U_1| = |U_2|. Grid provided for plotting the multiplication result.]

Figure 5 — Result of the Calculation