Elektronischer Langzeitrechner endim 2000: Programmier- und Bedienungsanleitung

Complete English translation of the original German-language document (88 pages).

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Electronic Long-Duration Computer — endim 2000 — Programming and Operating Instructions — VEB Rechenelektronik Glashütte

Foreword

The present document is intended for those who deal with programming the analog computer endim 2000 and with the execution of programs on it, and who either are already familiar with the operation of analog computers or wish to have a concise summary of the material they need. Those already working with the endim 2000 will find sufficient information here.

For the preparation and optimal programming of more complex problems, reference is made to earlier publications on the analog computer (e.g., B. Adler: Electronic Analog Computers, Gilo/Lauber: Analog Computers, the sections “Optimal Programming,” “Programming Aids,” and “Test Routines” of the companion volume). The genuinely demanding mathematical work done there should, however, be sufficient for the engineer who has a simpler problem to program, so that it is easily possible to prepare and carry out a program without outside assistance. For more technical details and for a summary of the analog computer endim 2000, more technical pages have been written, naturally grouped together, namely under “Technical Notes and Fault Elimination in Connection with the Circuit Diagrams — endim 2000.”

Chapter 1 provides, for those new to an analog computer of the endim 2000 type, information about the computing elements present, the input and output capabilities, and the operating and control equipment. The numbering of the central module Z_i (i = 1, 2, …, 16), each of which contains 4 amplifier boards, is done in meander fashion from the top left (see Fig. 1.1). A nonlinearity (multiplier or function generator) requires 2 slots, where both slots must be on the same central module, but the slots can be an even number apart. For populating this computer, the following combinations are possible:

4 amplifiers
2 nonlinearities
2 amplifiers and 1 nonlinearity

It should be noted that the nonlinearities may only be inserted in the central module positions 9 to 16 (counting from the left), and that positions 32, 36, 40, and 44 are reserved for multipliers. The multipliers operate according to the quarter-square method, such that the results can be approximated with the help of 10 diode segments. The input quantities must therefore be entered with both signs, as the sign must not change during the computation.

The function generators serve to represent a given function y = f(x) of any desired variable x. The function is approximated by straight-line segments, which are generated by superimposing 10 diode potentials, a constant voltage, and an x-proportional voltage. Break points, slopes in the four quadrants, and intercepts are set by switches or potentiometers.

For populating the computer, V = number of amplifier slots, M = number of multiplier slots, and F = number of function generator slots apply, with the relationships:

V, 2(M+F) ≤ 64
F ≤ 16
M ≤ 12
F, M ≤ 16

32 of the maximum 64 amplifiers can be freely selected from the programming panel as integrators or summers; the remainder are always used as summers.

Page

Structure of the Computing System … 6
Controls 2.1 Overview … 12 2.2 “Bia” … 13 2.3 “Ant” … 13
Operation 3.1 Control of the operating mode … 13 3.2 The control elements of the programming panel … 13 3.3 Operating mode control … 15 3.3.1 Overload display, switch and control options at the electronic programming panel … 16 3.3.2 Integrator … 19 3.3.3 Control of the coefficient potentiometers … 20 3.3.4 Control of the coefficient potentiometers of the electronic programming panel endim 2000 … 20 3.3.5 Comparators … 21 3.3.6 Selector for solution types … 22 3.4.1 Static check … 22 3.4.2 Summary of the operating-mode factors of the timer (electronic timer) … 23 3.4.3 Timer (timer part) … 23 3.4.4 Integrator (electronic timer) … 24 3.4.5 Operating mode overview … 25
Readout 4.1 Readout using the stylus pen (function receiver) … 27 4.1.1 EDAC … 27 4.2 Measurement readout … 27 4.3 Readout using the coordinate drawing instrument … 28
Programming 5.1 Programming panel … 29 5.1.1 Layout of the programming panel … 29 5.1.2 Structure of the program … 30 5.1.3 Building up the program field … 30 5.1.4 Setting the coefficient potentiometers … 31 5.1.5 Setting of general nonlinearities … 32 5.1.6 Form of the solution types … 32 5.1.7 Selection of the time scale … 33 5.1.8 Programming of switching and control operations, reference and overrun quantities … 33 5.1.11 Optimal programming … 35 5.2 Setting of the function-generator operating mode … 36 5.3 Optimal programming … 40 5.4 Sample programming … 42
Technical Supplement 6.1 Material data: BJTs, Op. BJTs 100 … 43 6.2 Component tables … 43 6.3 Potentiometer table … 44 6.4 Comparator … 44 6.5 Electronic clock … 44

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Contents continued (page 5):

5. Programming (continued) 5.1 Programming panel … 29 5.1.1 Layout of the programming panel … 29 — and so on through section 6.5 Electronic Clock … 44

1. Structure of the Computing System

The system to which the analog computer endim 2000 and the oscillograph endim 2100 with cart belong includes digital voltmeters (with printer output) and an xy-plotter as additional peripheral equipment. The coupling of two systems is possible. The computing elements are zero-point-stabilized tube amplifiers that operate with computing voltages between −100 V and +100 V.

All individual units of the computer are housed in easily exchangeable drawer-type modules under the chassis. When a module is exchanged, no intervention in the interior of the system is necessary. In the left part of the computer are the active computing elements. The numbering of the central module Z_i (i = 1, 2, …, 16), each of which contains 4 amplifier boards, runs in meander fashion from top left (see Fig. 1.1). A nonlinearity (multiplier or function generator) requires 2 slots, whereby the slots may only be placed on an even position. For populating the central module, one of the following combinations is therefore possible:

4 amplifiers
2 nonlinearities
2 amplifiers and 1 nonlinearity

It should be noted that the nonlinearities may only be inserted in the central module positions 9 to 16, and that positions 32, 36, 40, and 44 are reserved for multipliers. The multipliers operate according to the quarter-square method, such that the result is approximated with the help of 10 diode segments. The input quantities must therefore be entered with both signs, since the sign must not change during the computation.

The function generators serve to represent a given function y = f(x) of any desired variable x. The function is approximated by straight-line segments that are generated by superimposing 10 diode potentials, a constant voltage, and an x-proportional voltage. Break points, slopes in the four quadrants, and intercepts are adjustable by switches and potentiometers.

For populating the computer, V = number of amplifier slots, M = number of multiplier slots, and F = number of function generator slots apply, with the relationships:

V, 2(M+F) ≤ 64
F ≤ 16
M ≤ 12
F, M ≤ 16

32 of the maximum 64 amplifiers can be freely selected from the programming panel as integrators or summers; the remainder are always used as summers.

[page 7: figure only — Fig. 1.1 endim 2000 programming panel layout diagram, with labels Abb. 1.2, Abb. 1.3, Abb. 1.4, Abb. 1.5, Abb. 1.6, Abb. 1.7]

In each central module there is on the left a slot with the short designation SpN for the supply voltage of the amplifiers and nonlinearities. It must be ensured that each central module contains a functional SpN slot. The right part of the computer contains the power supply (2 slots SpU- and 3 slots SpU+) and the reference voltage supply (slot SpR ±100 V).

Above this is the control section (Felder; see Fig. 1.2 … 1.7). The individual fields have the following functions:

1. Measurement field: Readout, potentiometer control, central time-base transformation (X-tests).

2. Control field I: Control of the operating mode of the computer (see section 3).

3. Overload display field: Overload and fault display (1 pilot lamp per amplifier).

4. Control field III: Amplifier selection (selector) (see sections 5.1.1.3) and display of the amplifier inputs and outputs on the test jacks (tests TA) (see section 3.0).

5. Control field IV: Switching options (4 switches and a 10-position rotary switch, see section 5.1.11.4); selection of the voltages ±100 V, ±10 V (see sections 5.1.6 and 5.2); decoupling of the reference voltages from the locked programming panel (tests TA).

6. Control field II: Input and output changeover.

Above the control section is the central programming field with the exchangeable programming panel; above that the time-base field and to the right of it 3 potentiometer fields with 30 coefficient potentiometers each. The potentiometer numbering occurs again in meander fashion from the top left, for each field and potentiometer 3 fields are numbered; the two uppermost potentiometers, however, excluding the potentiometer numbering, are 2 exchangeable single-slot holders built into the back of the computer from the computer side (see Fig. 1.3). The left single-slot holder

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[page 10: figure only — shows Fig. 1.7 (control section front view) and Fig. 1.4, Fig. 1.5, Fig. 1.6 panel diagrams from above]

[page 11: figure only — Fig. 1.6: endim 2000 front view layout diagram with caption “Abb. 1.6 endim 2000 Frontansicht”]

2. Control of the Operating Mode

Before the computer endim 2000 can be operated, it must be checked and calibrated. All its internal elements must have been operating for some time, and all pre-programmed starting values must have been properly loaded. The program execution presupposes these conditions as given.

2.1 Overview

The electronic control of the endim 2000 is organized by means of the green pushbutton (key) in the upper part of the timer field. Each of these keys is designated with a label. The keys must be pressed in a specific order: pressing a wrong key (in the wrong sequence) causes the computer to proceed to the position “static check” (Statikprüfung). The individual keys are organized as follows:

2.1.1 Readiness condition

The LED glows when the computer is ready to operate.

2.1.2 Operating-mode control StP II (static check)

At control field I, the overloaded amplifiers are identified by the following criterion: ~ 2700 Ω/V, ÷ 100 (±2), indicating overload. At control field II, the Statikprüfung is then performed: amplifiers are reset, the Statikprüfung held, and on the x-tests on contacts BU and BN, it is also controlled whether the potentiometer voltages correspond to the programmed potentiometer settings. In this state, the integration phase is paused and the static (DC) circuit behavior checked.

2.1.3 Operating-mode control StP III

After a fault is identified, the cause can be localized via the test jacks BU and BN. A small fault leads to slight deviation, a larger fault leads to a larger deviation in the potentiometer reading. This can be checked by means of the toggle switch (Kippschalter).

2.1.4 Control of the operating-mode indicator

After the fault has been corrected (approximately 30 ms after the start of the following “Power-on” / “Heating” phase), the Programmierfeld (programming field) begins to operate; amplifiers, voltmeters, and nonlinearities begin functioning.

After switching the fault indicator 4, the control action proceeds to the static output checking: a transient fault display activates ‘Operating mode indicator’ in the “static check” position (“StP”); simultaneously LED “200” extinguishes.

3. Operation

3.1 Control of the Operating Mode

The operating modes of the integrators are controlled by control fields A and B. The integrators are switched to the operating modes as the inputs are set up, so that the voltage from which the integration starts is equal to the initial condition set on the potentiometer.

There are the following operating modes:

I	II	III	IV
Objectives	Resetting	Underbrace	Programming panel

The following conditions apply:

Operating mode	Initialized (Anfangswerteinstellung)	Hold	Initial condition selected
StP I	+	+	cf. Remarks Table 5.1

Transition over see Fig. 3.1.1.

3.2 Control Elements of the Programming Panel

The programming panel contains the elements with short designations M, Mu, Iu (see sections 5.1.10, 5.4.6, 5.4) for the central control of the operating mode (of the Integrators) and the reference voltage control by keys (see section 4.1 Nr. 5) coarsely and with a potentiometer (see Fig. 4.1 Nr. 6) finely. The entries on the keys correspond to the computing times in seconds; the potentiometer magnifies the time corresponding to the marked scale factor of 1 (accuracy ≥ 10%).

By raising the computing-time factor 5.3, the respective computing time can be multiplied up to a factor of 1 … 500x so that a range of 1 … 500x loose loops can be covered. The magnification factor is not linear. The relationship is:

Scale marking: 1, 2, 3, 4, 5, 6, 7, 7.5, 8, 8.5, 9, 10 Magnification factor: 1, 1, 1.15, 1.35, 1.5, 1.8, 2.2, 3.4, 2.95, 3.6, 5.0, 5.3

3.3 Operating Mode Control

The control equipment for Integrators is controlled by control fields A and B. The integrators are placed in the prescribed operating modes as the initial conditions are set up, so that the voltage from which the integration starts equals the initial value set on the potentiometer.

3.3.1 Overload Display, Switch and Control Options at the Electronic Programming Panel

Referring to Fig. 3.1, 5.1, 5.2, 5.4.8 at the programming panel, the following operating modes are programmable:

Operating mode I: Resetting (Anfangswerteinstellung) On line I, the fault indicator must be pressed, which is generated by the fault display using the following criterion: ~ 2700 Ω/V, ÷ 100 (±2) indicates an overload. On control field II (Fig. 3.1.2 right side) the first fuse lamp, position “BU” and “BN”, must both be controlled. In this state the integration phase is paused and the computer is in the initial-condition setting state.

Operating mode II — Abort: Fault: If there is a fault, caused by an error condition, the timer generates: ~ 2700 Ω/V, ÷ 100 (±2). In the timer readout period BU–BN, the counter counts position: a small fault produces a minor deviation, a large fault a larger deviation in the potentiometer reading.

Operating mode StP III — Control of the operating mode: Wait approximately 30 ms, then start the integrators (“Power-on”/“Heating” phase). After this, the programming panel starts; amplifiers, voltmeters, and nonlinearities begin to function.

Operating mode StP IV — Fault elimination of the operating mode: After the fault indicator 4 is activated, the control proceeds: a transient fault display activates “Operating mode indicator” in the “Static check” position “StP”, simultaneously LED “200” extinguishes.

3.3.2 Integrator

After the signal (field 24 “P”) on control field II is activated, the further control of the computer is triggered. After the control signal “static check” (StP) is given upward right, the computer proceeds into the “static check” position (“StP”).

3.3.2 Operating-mode control StP II — static check After the fault is identified, the cause can be identified via test jacks BU and BN: a small fault causes slight deviation, a larger fault a larger deviation in the potentiometer reading. The control is performed using a toggle switch (Kippschalter).

3.3.3 Operating-mode control StP III — Resetting and Null: Wait approximately 30 ms, then start the process from the following “Power-on”/“Heating” opening. After this, the programming panel starts working; amplifiers, voltmeters and nonlinearities begin to function.

3.3.4 Control of the coefficient potentiometers of the electronic programming panel endim 2000: After the fault indicator 4 is triggered, control proceeds: a transient fault display activates “Operating mode indicator” in the “StP” position, simultaneously LED “200” extinguishes.

3.3.5 Comparators: See section 5.1.9, Nr. 2.

3.4 Timer (Steuerzeit)

The timer can operate in the following modes, controlled by control field StF I:

3.4.1 Static check (StP): After the signal (field 24 “P”) at control field II is triggered, a further control action follows. The static check signal is activated at the top right; the computer then changes to the “StP” position.

3.4.2 Summary of the operating-mode factors of the timer (electronic timer):

The basic operation time for one Laufdurchlauf (one loop) is given at the timer field (BU and BN), see Fig. 4.1 Nr. 5.

Measurement range	Duration	Magnification
a two-stage	1.0	0.3 ms
off 4 times	1.0	0.5 ms
	1.0	1.0

The central readout equipment gives feedback that the time-base counter can be compared with the entered values (Turnus).

3.4.3 Timer (timer section) — “Bru”: The green key (Taste “Rückstellen”) at control field I activates the following further operation. After activation of the “Rückstellen” key a half key is pressed: the computer proceeds to the next right key 42 (according to Fig. 4.1 Nr. 5) precisely. The key “ZOf” then simultaneously activates the fault display “Statikprüfung” and “Null” at the display.

3.4.4 Integrator (electronic timer): For times t ≥ 30 ms, on account of the influence after “Power-on” / “Heating”: after this the programming field begins operating. Amplifiers, voltmeters, and nonlinearities begin working. After switching the fault indicator 4, the control proceeds: a transient fault display activates “Operating mode indicator” at the “StP” position; simultaneously LED “200” extinguishes.

3.4.5 Operating mode summary:

See Fig. 3.1 … 5.1.

3. Operation (continued)

3.1 Control of the Operating Mode (detailed)

The operating modes of the integrators are controlled by control fields A and B. The integrators are brought into the required operating modes as the inputs are set up, so that the voltage from which integration starts equals the initial value set on the potentiometer.

Fig. 3.1.1 — Integrator operating modes:

→ I (Objectives) → II (Resetting) → III (Underbrace) → IV (Programming panel) → return

The following conditions apply:

Operating mode	Initialized (Anfangswerteinstellung)	Hold	Initial cond. selected
I — Objectives	+	+	cf. Remarks Table 5.1
StP I	+
Übersteuer.	+

Transitions: see Fig. 3.1.1.

The above mentioned nonlinearities must be used as summators. In the series B, D, M, Mu, Iu (see section 5.1.10 etc.) the control of those elements is set for them; the transitions described are set through the key “Rückstellen” (Resetting); by pressing this key (see section 5.1.2), by “Null” at the display field and by the programming field via the key “Rechnen” (Compute), the computer is set in operation again.

3.5 Readout (Anzeigeteil) / Control of the Operating Mode

The readout sections for Integrators are controlled via control fields A and B (Abb. 5.1.1.5, 5.1.2, 5.1.4, 5.1). The integrators are placed in the prescribed operating modes as the initial conditions are set, so that the voltage from which integration starts equals the programmed initial value.

Fig. 3.1.2 — Operating mode table:

Commanded state	Hold bit	Overload check
		Static check
Steuerein	precised
Reported state

At repetitive (iterative) operation, the computing loop at static-check re-enters computing mode and is repeated. Within the loop the control field contains a pilot lamp. The pilot lamp at the top indicates the repetition mode “Overload” and “Null” in both fields. After this the repetition mode of the integration values and the overload-checking function are activated. In the repetitive mode, the time between “ÜbStellen” and “Rechnen” activates the comparator in the “Rückstellen” loop, and with repetitive integration a period is activated corresponding to the overflow factor 100 of the following output quantities:

Commanded state	Transfer rate	Hold bit during loop	Programmed state
a single effort	1 s	0.3 ms	cf. Sal. erect
off 4 times	1 s	0.5 ms
	1 s	1.0 ms

The central readout gives feedback that the time-base counter has reached the value (Turnus).

3.3 Selection of the Operating Mode of the Integrators

The selection equipment for the integrators is controlled by control fields A and B (see Figs. 5.1.1.5, 5.1.2, 5.4.8, 5.1). By switching the programming panel (Bedienpanel Nr. Mu, Iu) (see Fig. 5.1.1) through the key “Rückstellen” (see Fig. 5.1.2), by the key “Null” at the display field and through the programming field using “Rechnen,” the previously described states are set by the key “Rückstellen”:

Operating mode	Summator	Übersteuerung
I	+
II	+
III		+

Tab. 3.5

The following operating mode combinations for each Turnus value are also possible:

Turnus	Turnus value	Description
1.1	2
2.1	2.5	”Rückstellen”
1.1	3, 1.9, 5
6.1
8.1	None	”Übersteuerung” with the operating mode “Rückstellen”

Tab. 3.5

The illumination of the key “Rückstellen” is the sign that the computation sequence has stopped, that is, when a programming panel has been inserted and the machine is locked according to procedures.

3.4 Computing Time Setting at the Function Receiver endim 2160

In cases 3 and 4 from Tab. 5.3, if a function receiver is selected for readout, the computing time is set using keys (see Fig. 4.1 Nr. 5) coarsely and with a potentiometer (see Fig. 4.1 Nr. 6) finely. The numbers entered with the keys correspond to computing times in seconds; the potentiometer multiplies the time by the corresponding scale factor labelled 1 (accuracy ≥ 10%).

By the timer factor 5.3, the respective computing time can be maximally multiplied up to a factor of 1 … 500x so that a range of 1 … 500 loose loops can be covered. The magnification factor is not linear. The relationship is:

Scale marking: 1, 2, 3, 4, 5, 6, 7, 7.5, 8, 8.5, 9, 10 Magnification factor: 1, 1, 1.15, 1.35, 1.5, 1.8, 2.2, 3.4, 2.95, 3.6, 5.0, 5.3

3.5 Computing Time Setting, Switching and Control Options at the Electronic Clock

3.5.1 Time selection

At the display unit of the electronic clock, times in increments of 0.1 s can be selected using 4 separate switches (up to 4 switches per decade). The minimum selectable time is therefore t = 0.1 s; the maximum time t = 1000 s. For times t = 100 s, a relaying of the decade sequence according to the selected time is carried out by putting the decade selector of the next-higher decade to the “R” position. The time preselection is then carried out by both decade switches.

3.5.2 Selection of operating modes

By pressing the readout key on the display unit, a toggle between “continuous computing” and “repetitive computing” (repeating mode) is carried out; simultaneously the corresponding key on control field StF I must be pressed.

In cases 1 and 2 from Table 3.5, the computer is stopped after elapse of time t_1 or t_2, respectively, in the “interruption (Überbrechen) of computing” state; it can only be restarted after the keys “Rückstellen” and “Null” on the display unit and key “Rechnen” (Compute) on the control section have been pressed.

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In the “Repetitive Compute” mode, the selected time runs forward continuously; in this mode the fan runs at approximately 2 to 3 rpm, while in static operation it stands still.

3.5.3 Selection of the Data Printout

After pressing the “Null” key (first pressing and releasing it) and setting the computing mode, all data for the printout in column “0” are stored, so that additionally existing potentiometer data are stored in their Ausgangslage (initial position). An overview of the available switch and control options can be found in Table 3.5.

[Pages 20–21: Table 3.5 — Switch and Control Options for the Printout (Continuation)]

Tab. 3.5 — Switch and Control Options for Computing Mode Settings

No.	Computing Mode / Switch Setting	Additional Action	Setting of U_R
1	U_R / setting: initially on U_R	—	—
2	y_1, y_2	Initially set both potentiometers on U_R or to Null; set y_1 and y_2 via the Zeitkonstanten and values	—
3	y_1, y_2, y_3	Initially set potentiometers on U_R; set values individually	—
4	y_1 + y_2 + y_3	Used together with the Bewertungsfaktors; all potentiometer settings as with function sources; see printout for y_1,2,3	Potentiometer set accordingly, switch on U_R

(The table continues across pp. 20–21; the right columns indicate “initially set on U_R” or “initially set on U_R; then switch on U_S” for each row, and the right-most column gives notes such as “Potentiometereinstellungen wie für Fig. C, column 5.1.1.4” or similar cross-references.)

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3.6 Selection of Test Options

3.6.1 Stability Test

Pressing the “TA” key on the STF IV causes the reference voltage of ≈ 100 V to be switched off from the programming panel. The computing process can, however, continue to run so that the stability of circuits can easily be checked. Circuits without function generators are allowed to exhibit minor errors during the run. (This is caused, for example, by the circuit for generating y(t) = e^t despite y(0) = 0 after a short time of actuation.)

3.6.2 Setting and Measuring the Transfer Factor of the Summing Amplifiers (Static Test)

The “TA” key also provides the possibility of precisely setting the transfer factor of the summing amplifiers in the stopped program — specifically including consideration of the error of the weighting potentiometer (i.e. the ratio of feedback resistance to the corresponding input resistance). Pressing the key connects the potentiometer belonging to this measuring node exclusively to its input; a reference voltage is applied at the output of the summing amplifier via the measuring instrument and the correct transfer factor is then set at the potentiometer (see Section 5.1.6).

3.6.3 Setting and Measuring the Time Constants of the Integrators (Dynamic Test)

Setting the time constants of the integrators requires additionally the disconnection of the integrator outputs from the circuit and the use of a time-measuring device in conjunction with a comparator. This disconnection of the outputs of all amplifier positions is effected by the “TA” key on STF III. The time-measuring device including its associated comparator is activated by the electronic-timing mechanism of the upper relay row: bring the time-allocation switch for t_o (the initial value) to position “0” (the lower relay row for t_u is arbitrary); the push-button “continuous compute” (‘Dauerrechnen’) activates the time-measuring device. In the given sequence the following keys on the STF I are then to be pressed: “Dauerrechnen” (‘Continuous Compute’), then “Blockstarten” (‘Block Start’), then “NULL”. To measure or set the time constants of individual integrators (including the setting of the prescribed potentiometer), one integration over a reference voltage ≈ 100 V is performed, giving an output, so that at the comparator input of the electronic clock (socket CK on the programming panel) the output voltage of the

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integrator runs from 0 up to −100 V. One integrates until a positive reference voltage (≤ 10 V, ≥ 100 V) is reached. It must be ensured that the reference voltage CK does not leave the range 0 … −100 V (and must not go below −100 V!). When “Dauerrechnen” (‘Continuous Compute’) is pressed, the integration begins and the clock starts counting. When the comparator is reached at −100 V, the clock stops and the computer enters the “Unterbrechen” (‘Interrupt’) mode. The time constant at that moment is exactly the elapsed time in seconds if ≥ 100 V was integrated; dividing the integration of −100 V by the reference voltage gives the elapsed time divided by 10.

It is to be noted that with each time-constant measurement the elapsed time shown includes a small additional time resulting from the bounce period of the relay that arises when the clock is stopped. This amount may be derived from the intersection of Fig. No. 1 of Tab. 3.5 and the corresponding elapsed time (in general approximately 30 ms).

A setting of the electronic comparator belonging to the clock (see installation note Section 5.5) must be performed before time-constant measurements and at a new setting of the reference voltage ≈ 100 V.

3.7 Overload Indication

3.7.1 Overload Indication Display

The overload indication display assigns a signal lamp to each amplifier position via a secondary amplifier and reports through the illumination of the lamp an abnormally large error current at the summing point of the respective amplifier. This display does not indicate that the amplifier output has exceeded the machine unit range, but rather indicates in principle the occurrence of abnormal operation due to overloading (this can be caused by the output reaching more than ≈ 100 V, or by corresponding high loading due to small feedback resistance). The overload is announced centrally by illumination of the lamp “Uebersteuern loeschen” (‘Clear Overload’). The overload can be acknowledged by momentarily pressing the “Uebersteuern loeschen” key, provided it is no longer caused by the secondary amplifier output; otherwise it will be re-triggered. The function “Uebersteuern loeschen” can also be activated by pressing the “Unterbrechen” (‘Interrupt’) key from the STF I in the corresponding sequence.

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The overload indication and pressing of “Uebersteuern loeschen” work together with pressing “Registerrechnen” (‘Register Compute’): for these amplifiers the function “Uebersteuern loeschen” will be activated by pressing “Registerrechnen” simultaneously pressing “Unterbrechen” and “Dauerrechnen”.

3.7.2 “Hold on Overload”

When the “Halt bei Uebersteuern” (‘Hold on Overload’) key is engaged, together with “Registerrechnen” (‘Register Compute’), an overload causes the computer to enter the hold mode. For overload-free operation with the three keys “Schaltstarten” (‘Switch-Start’), “Unterbrechen” (‘Interrupt’), and “Dauerrechnen” (‘Continuous Compute’), the “Hold on Overload” function is enabled.

3.8 Connection of External Devices

To connect the control side, it is beneficial to have a small, easily adjustable Regelungsglied (control element) with a single loop. The circuit book (Abschn. B6 and B: Abb. 2.1) shows this. In such a book the reference voltage ≈ 100 V from the programming panel is used. A short-circuit protection is placed at the above-mentioned item, and the reference voltage is likewise switched off.

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4. Evaluation

4.1 Evaluation with the Oscilloscope (Function Receiver endim 2100)

The oscilloscope permits the display of 2 functions

y_I(t) and y_II(t) (i, k = 1, 2, …, 5)

(internal x-deflection) or the display of 2 functions

y_II(x_i) and y_III(x_j) (i, k = 1, 2, …, 5; l = 1, …, 4)

(external x-deflection) and their interconnection.

The voltages y_II_, y_IIk and x_l (i, k = 1, 2, …, 4) (l = 1, 2, 3) are set on the programming panel while computing and x_4 is applied directly to the oscilloscope sockets. For external x-deflection, 2 functions are to be displayed (y_II and y_IIk); pressing the key for the y-x-deflection switches a relay (approximately 50 ms) between y_II and y_IIk on the y-amplifier. If only one of the keys y_II and y_IIk is pressed, the relay switches only y_II to the y-amplifier. Pressing the left key “A” (14 in Fig. 4.1) disconnects the internal time-base voltage from the x-amplifier and connects the voltage from the signal-voltage divider instead; the right key “A” is used only in conjunction with external x-deflection and has no meaning without it.

In computing mode, time markers can be written between dark intervals. The spacings of the markers are selectable (0.1 s, 1 s, and 10 s) using one potentiometer. At the side panel there is a control for halving the spacing. In the upper part of the oscilloscope there are regulators for brightness, sharpness, position (x and y direction), and amplification (x and y direction). (In the specified sequence of the oscilloscope they are 1, 2, 15, 16, 4, 3 as in Fig. 4.1.) For dimming the oscilloscope screen, the brightness regulator should be turned to the far left when the oscilloscope is not in use.

The auxiliary voltages can be equalized with the Sieblassung ≈ 100 V (from the machine). At the same time the remaining input voltages are disconnected. For external x-deflection the following steps are necessary:

Set both amplification regulators to zero.
Adjust the shift regulators to set the right boundary of the raster.
Select y-amplification (1, 2, 10) and press “E” simultaneously; this controls the beam with the y-amplification regulator in the upper right corner of the raster.
Press “E” to make the beam jump from “E” to the upper right corner.
Set the x-amplification regulator so that the beam repetitively traverses from the raster against the raster.

[Page 26: Figure Only]

[page 26: figure only — Fig. 4.1, front panel diagram of the oscilloscope/function receiver with numbered controls 1–16, including controls labeled “Verschaltung” (left), “Rechnerfeld” (center top), “Zeitmarken” (center top right), “Weitrührung” (right), and control groups V through VIII numbered 4–8 below, and groups I–III plus items 9–13 at bottom)]

[Page 27]

When ending the computing process (stop computing), the beam must return to its starting position and upon interruption of the computing process the beam must remain stationary. The computing begins from the left hand side, and upon return to the start position an additional x-shift must be carried out. The scale begins with the potentiometer at the left hand side.

When operating with external x-deflection, the left “A” key is released. The internal time-base is thereby disconnected from the x-amplifier, and the x-amplifier now takes the voltage from the signal-voltage divider as the time-base input. The “Null” key (13 in Fig. 4.1) is then for both the y- and also for x-”Verstarkung”/“Einheiten” (amplification/scaling). For external x-deflection the following steps are needed:

Set amplification regulators to zero.
Set shift regulators on the raster center.
Select amplification (1, 2, 10) and press “E” to control the beam with the amplification regulators to the upper left corner of the raster.
At press of “E” the beam jumps to the upper right corner.

4.2 Measurement Readout

4.2.1 Direct Measurement

Connect the relevant amplifier output with socket “M1” [44;00] of the programming panel. After pressing the keys ”>” and ”↓”, the output voltage at the measuring instrument MF (see Fig. 1.2) reads up to approximately ±10% accurately. For voltages above 100 V the display is range-limited. For smaller voltages, the closest ranges can be selected via the keys “4.0”, “1.0”, “0.4” (see Fig. 1.2), whose label gives the measuring range limit in volts. (Note: with older equipment a different measuring range may apply!) The range extension may only be done at smaller magnitudes and after a corresponding reduction of it; a range extension of 4 V in the range means that a current as low as 0.4 V is allowed to be measured.

4.2.2 Measurement Evaluation using the Compensation Method

The procedure is similar to 4.2.1, but presses only the ”>” key. The compensating potentiometer then compensates the voltage by systematic comparison with the zero indicator, which the measuring instrument shows at 0 V. A range extension of the measuring instrument is not necessary, since the measuring instrument is switched to a sensitive offset with an electronic overload protection as soon as the ”↓” key is not pressed.

[Page 28]

The largest compensatable voltage is the machine unit of 100 V. If the voltage to be compensated is negative, the ”−” key must additionally be pressed.

4.2.3 Measurement Readout via Coefficient Potentiometers

When, for example, an optimization task requires setting particular values for one or more coefficient potentiometers, they can be compensated in the manner of Section 4.2.2 — however, the voltage is compensated via the potentiometer wiper as a gap voltage by systematic comparison with the zero indicator of the zero display as in 4.2.2.

4.3 Evaluation with Peripheral Devices

One connects the outputs to be measured to the corresponding input sockets of the peripheral device on the programming panel, following the operating instructions for those devices.

4.3.1 XY-Plotter

The xy-plotter permits the registration of computing results with high accuracy. It is connected to a 5-pole socket on the rear side of the computer (see Fig. 2.1). The plotter is controlled from the computer. Should the x-travel of the plotter be rectilinear, it is necessary to install integrators as a linearly time-connected signal source. The programming panel sockets for x and y lie at sockets [44;24] and [44;25].

When using the xy-plotter, the built-in normal voltage source of the computer serves as the reference source as for the normal machine.

4.3.2 Digital Voltmeter

The digital voltmeter is connected for accurate coefficient setting at a terminal strip on the rear of the computer (see Fig. 2.1). The voltage to be measured is applied to socket [44;01] of the programming panel. The connection for −100 V supply enables the voltmeter to be run with external reference voltage. At terminal strip 7 further external devices can be connected, in particular a multi-point recorder. For this purpose, 10 connections from the programming panel to terminal strip 7 are available, of which 7 are freely usable; the remaining 3 are connected to contacts of the electronic clock’s relay (see Section 5.1.10). In the following, all connections at terminal strip 7 are listed:

[Page 29]

Terminal Strip 7

Connection	Signal
+ 24 V
PrF-Buchse [44;01] “DV”	− 1a 1b − Mass
+ 100 V	− 2a 2b − 100 V
	− 3a
PrF-Buchse [44;20]	− 5a 5b − PrF-Buchse [44;27]
[44;28]	− 6a 6b −
[44;30]	− 7a 7b −
[44;32]	− 8a 8b −
[44;34]	− 9a 9b −

5. Programming

General Procedure

The scope of a program is limited to the working steps:

Plugging the program onto the programming panel and hanging it in the programming panel,
Setting the coefficient potentiometers,
Setting the eventual to-be-programmed function generators.

To avoid loading errors, the potentiometers are fundamentally first plugged into the finished-plugged programming panel in the computer.

The possibility of erroneous operation is largely excluded, provided that the following instructions are observed.

5.1 Programming Panel

5.1.1 Handling of the Programming Panel

In the ready-to-operate state, both the programming panel and the short-circuit plug are interlocked. This state is additionally secured externally by a lockable locking bar handle below the right locking bar.

Note: Any forced removal of programming patch cords or short-circuit plugs by unauthorized persons, or damage to the locking mechanism or contact elements of the panel, is to be avoided.

To unlock, after pressing the right “Entriegeln” (‘Unlock’) key, the locking bar can be pulled from right to left. This activates the following processes:

The reference and control voltages are disconnected from the programming panel.
The amplifiers used as integrators for summation are automatically switched to a particularly suitable drift compensation circuit.

[Page 30]

The programming panel, its patch cords, and short-circuit plugs are mechanically unlocked so that a program change or programming modifications can take place.
Any running computing process is interrupted, the computer enters the “Stop” mode.

To insert and lock, the reverse sequence is to be followed.

Note on programming panel replacement: It is fundamentally to be carried out with utmost care; when inserting, particular care must be taken that the programming panel is not distorted. During pressing from right to left while inserting, the panel slides into the guide rails approximately 2 cm further; the panel can then be slid into the panel from above. The panel is now at a lower level and can be raised further by pressing from its lower side. Upon inserting and locking, the reverse sequence is to be followed.

Should contact pins in the programming panel be damaged by careless handling, on no account should one attempt to self-repair these; contact a qualified repair service in all such cases.

Should a programming panel cause one of the reference voltages to be short-circuited, the affected computer will be placed in the “Stop” mode automatically after at least 3 seconds; the faulty correction is then to be carried out.

5.1.2 Patching the Program

The standard equipment for each programming panel consists of:

700 short-circuit plugs
50 patch cords, 20 cm long
80 patch cords, 40 cm long
5 patch cords, 75 cm long

For programming, the use of short-circuit plugs or short patch cords in combination with the long patch cords is recommended.

Points to note:

Patch cords and short-circuit plugs are to be plugged in fully to the programming panel sockets.
Patch cords are always to be inserted by both ends into the panel first (connection protection).
To avoid any feedback coupling of the computer amplifiers, at each summator/integrator amplifier input (see Section 5.2.1.5), the short-circuit plug is to be inserted when the affected amplifier is not used in the computing program.

[Page 31]

If the programming panel is plugged outside the machine, it must be placed on a sufficiently large, flat surface.

5.1.3 Layout of the Programming Panel

The programming panel contains connection elements for all computer elements:

Input/output elements of all computing amplifiers, input and output sockets, signal outputs, auxiliary sockets, aid junctions.

The computing amplifiers V 00 to V 31 are divided into up to four functional groups according to their function. The amplifiers V 00 to V 31 may serve as summators or integrators and each amplifier can be addressed via its own trimmer; the coefficient potentiometers permit setting of the evaluation factors; the connection of the programming elements facilitates summation and/or weighting. This amplifier group contains connections of four non-switchable input lines at each side. In each functional group, the connections are numbered A, B, …, N, making the use of short patch cords particularly convenient. The connections for the integrators are identical in their row arrangement to the connections for the summing amplifiers.

The computing amplifiers V 32 to V 47 are switchable as summators or integrators via the address assignment; the addressing is possible via the serial identification; the connection can be used as a summator or integrator. The relay activation ensures that the inputs of the particular integrators are directly connectable, each with 4 connections; their relay switch connections are the same in every case; the connection of the computing amplifiers is done in the upper part of the programming panel via the yellow-marked computing amplifier connections.

The computing amplifiers V 48 to V 53 have the connections as summators; the external connections are connected directly to the corresponding amplifier outputs (for these amplifiers, the function generators G5 can be connected, for example connections for multipliers, single-variable functions, or otherwise be interconnected with the machine).

The weighting-factor inputs of the outputs are connected directly via the yellow-marked amplifier sockets in the lower part of the programming panel; these are connected with the yellow-marked computing amplifier sockets.

In the center part of the programming panel are routing sockets, distribution lines, diodes, compensators, free computing potentiometers, relay sockets, and other components.

Across the programming panel the connections are grouped into five groups of nearly adjacent sockets.

[Page 32]

5.1.4 Colors of the Programming Panel

Color	Meaning
red	outputs of active computing amplifiers
green	switchable inputs of active computing amplifiers
gray-striped	non-switchable inputs of active computing amplifiers, which also cannot be connected to the computing amplifier
blue-striped	initial value sockets
yellow	passive computing amplifiers (Zeitkonstantenwähler, diodes, single-variable functions)
black	amplified computing amplifiers
dark blue	reference voltage − 100 Volt
purple	reference voltage + 100 Volt
brown	sockets for switching and timing functions
beige-striped	equalization sockets
rosa	connections for measuring instruments
red-striped	sockets for continuous use
white	single-connections, distribution sockets, not yet belonging to specific groups

[Page 33: Figure Only]

[page 33: figure only — Fig. 5.1.5, schematic diagram titled “Steuerung Summator – Integrator” (Control: Summator – Integrator), showing the relay and signal routing circuit for switching an amplifier between summing and integrating modes, plus a truth-table matrix below indicating which relay contacts close for each mode; relay-contact keys are labeled KOO, KOI, VA, VMP, VO(EX)_VPT, etc.]

[Page 34: Table Only]

[page 34: figure/table only — Tab. 5.1.1, a table of amplifier circuit types with columns:

Circuit type (Schaltungsart)
Symbol
Patching/connection diagram on the programming panel for each mode: Vorzeichenbildung (sign inversion), Summation, Integration C = 1 μF, Integration C = 10 μF, Verwendung als Anfangswertschalter (use as initial-value switch) Each row shows the required programming panel sockets and cross-connections via a small schematic diagram.]

[Page 35]

5.1.3 Switching of the Computing Amplifiers

Four different circuit types are to be distinguished.

The switching of these amplifiers is shown in Fig. 5.1.5.

The various types of actuation are shown in Table 5.1.5.1.

All amplifiers not required in the program are fundamentally to be set according to a) to avoid overloading on summation.

When changing from circuit type a), a stopper is inserted into the connection sockets of the integrating-capacitors — so that the capacitor 1 μF is switched in, the corresponding integrator retains no central switching (Rückkopplung) on 0.1 μF possible.

To suppress initial value errors (see Fig. a)) it is necessary to use the potentiometer present at the initial value junction. Depending on whether the initial value of the potentiometer is positive or negative, it is to be connected to −100 V or +100 V.

The d) type circuit is only useful as an open amplifier if a newer feedback coupling occurs, for example through a particular H0-network, a multiplier (implicit function technique), etc.

The cross-coupling property also provides the possibility of connecting additional computing amplifiers to these amplifiers, where more than the immediately available number of inputs is required. The thereby additionally connected amplifier retains its usability as a sign inverter or simple integrator.

5.1.5.2 Amplifiers 01, 03, …, 29, 31

These amplifiers are also usable as summators. Since the cross-coupling of the input resistances is possible, the connection is possible. The connection is shown in Fig. 5.1.5.2.

At amplifiers 19, 23, 27, 31 each has 2 inputs available with weighting factors of 0.1.

[Figure 5.1.5.2 — schematic showing the connection layout for the summing amplifier group 01, 03, …, 29, 31 with input resistors and feedback network on the programming panel]

[Page 36: Table Only]

[page 36: figure/table only — Tab. 5.1.3, a table of amplifier types for amplifiers 00, 02, …, 28, 30, with columns:

Circuit type (Schaltungsart)
Symbol
Programming panel connection diagram for each of the following modes:
- Vorzeichenbildung (sign inversion)
- Summation
- Integration
- Summation C = 1 μF
- Anfangswertschalter (initial value switch), with note “Schaltung auf alle Rec-Verstärker V 00 bis V 31 anwendbar” (circuit applicable to all computing amplifiers V 00 to V 31) Each row shows the required socket connections via a small schematic.]

5.1.5.3 Potentiometers 30, 31, …, 46, 47

The diagram (see Fig. 5.1.5) shows these close to the integrators.

Along with the connection possibilities of these integrators, these are the variables generated from the programming panel that are processed.

Note: The output of the integrator programmable element is connected from 1 pF to 0.1 pF (at full range of the amplifier).

The precise assignment can be seen from the layout of the programming panel (Section 5).

5.1.5.4 Potentiometers 40, 41, …, 61, 43

The characteristics of potentiometers 40–61, 43 show these potentiometers as the summing node (within the Programming field) of the respective amplifiers.

Adjustment addresses the question of the interoperability of the input/output ranges of the ENDIM 2000. In this way one can select from either a standard range of ±10 V (as most analog computers possess) or from ±100 V, so that the ENDIM can be more easily connected to other devices (see Section 5).

5.1.6 Description of the Coefficient Entry Panel

The diagram for the general multiplier and function-generator modules is shown in Fig. 5.1.7.

The programming panel displays each module with a switchable variable. The display variables are each associated with switchable variables. The connection is established via the S-bus line (see also Fig. 1.2), an S-bus assembled out of lines.

Interplay of the function panels with the function generator (see Section 5.1.1):

[page 38: figure only — Fig. 5.1.7, schematic of general multiplier/function-generator elements with S-bus connections]

The interaction also applies at the function-generator panels to the proportion of function-panel connections to the function generator. The following connections are provided in this context.

5.1.6.1 All-purpose Multiplier Panel

[Page 39]

[page 39: figure only — Fig. 5.1.8.1, showing a multiplier module with interconnect diagram]

The use of 7 recommended special circuit configurations taken from the book “Analog Computer Techniques”:

Source: “Analog Computer Techniques”
Publisher: “Analogrechner”

This is applied wherever an integrating circuit includes a signal bus available to it, and also whenever input entries are present. Vertical arrangements are used in this context.

Source: “Elektronische Analogrechner” (“Electronic Analog Computers”), published in various references.

From this in the Busswork [−25; 30] and [−20; 25] and [−20; 25] one can see, and the Aufauf-section is described.

All listed Busswork positions (see Section 5.1.6.1), where the maximum input number is described, can have the following range:

[Page 40]

Beim Vertical adjustment a span of Multiplikant is recommended, to also the sub-programmable inputs to investigate.

The total vertical position of the device can be given a clear overview:

Designation: Spanning at lower input-bus: U₁; Spanning at upper input-bus: U₂

Range values:

U₁ > U₂: Reversed polarity at the summing point
U₁ < U₂: Normal polarity at the summing point

The assignment of the adjustment contacts is, for the given range of Page 84 (cf. contacts labeled P), as follows:

Contact range	Function
[40; 42]	Digital interface (as History Master digital assigner)
[42; 43]	Function-generator switch pages 2000
[42; 43]	Adjustment of the sub-potentiometers with 2000
[42; 43]	Contacts for sub-button use with the back contacts
[42; 43–N]	Multiplier units are subject, unless the contacts
[43; 43–N]	to History Master
[43; 53]	Contacts are blocked

(Assignment as in Section 7, cf. Section 5.1.7)

5.1.11 Programming switch and control contacts, Reset and Hold functions

5.1.11.1 Operator

For autonomous generation-programs, the Drauzählereinschub has available contacts. The Anschlüsse und Kontaktanordnungen give outputs immediately upon demand.

[Page 41]

This Drauzähler [counter/dial unit] advances the operating mode from “Repeating” to “Step-by-step” one step further, as soon as the contacts [C: 27–29] are interconnected with each other. Two sliding contacts of the counter are placed so that the two upper buswork contacts cover the upper two buswork rail areas, the two lower buswork rails the second areas. The slider is labelled with 0; the counting sequence of the panel proceeds accordingly.

5.1.11.2 Free Contacts

Three free, in the Drauzählereinschub accommodated contacts are connected to contacts [17–21; 45–49]; the terminals and contact assignments give outputs immediately upon demand.

5.1.11.3 Selector Contacts

The contact labelled “M” in contacts [19; 44] is the output of the selector switch, which selects a specific output V 00 … V 03 when the selector “M” connects the contact assigned to V 00, the corresponding field of the dialer III is activated.

5.1.11.4 Function Panel

In the contacts marked S 00, S 01, S 02, S 03, the contacts [18–20; 68], [18–20; 69], [18–20; 70], [18–20; 69] are assigned with the same symbols. Four contact panels are assigned to the Steuertafeln I to IV (dialer IV), on which during the procedure the connection outputs arise immediately from demand; the contact assignment corresponds to the currently assigned state of the corresponding text in dialer IV, which is indicated by the contact panel, which is entered in dialer IV.

For the same purpose serve the contacts [44; 02–12], as the 10-position dialer of dialer IV is assigned to them.

5.1.11.5 Programmable “Hold”, Compute, Hold/Compute

Through connecting contacts [26; 45–47] with S-size it is possible to switch the device to the operating modes Hold (“Ho”), Compute (“Re”), or Hold/Compute (“Dr”) to select.

5.1.11.9 Time Constant Measurement and Overshoot Contacts

The contacts with DK 0 to U 0 labelled contacts [17–18; 44–48] are associated with the electronic clock from Instruction and are treated in Sections 3.3 and 3.6.3 separately.

5.1.11.7 S- and S-Relay Contacts

Two S-relay contacts are in each Drauzählereinschub available in the programming panel [23–25; 20–23]. The Anschlüsse und Kontaktanordnungen contacts (see diagram Abb. 5.1.5, Table 3.1, and Relay control arrangements) give outputs immediately upon demand.

[Page 42]

5.1.11.8 Ref and S-Relay Contacts

As in 17 contacts [0–10; 25–49], [M 1; 25–49], [26; 47–48] are the beforehand listed contacts for autonomous switchable arrangements: the Anschlüsse und Kontaktanordnungen give outputs immediately upon demand.

Switching table:

Contact	A	B	Switch position (20 V)
S-	•	—	a: Switch position (−20 V)
S+	—	•

The reference voltage (−100 V at 48), not −100 V on the programming panel is applied. As the Busswork “N” on a 45, both contacts are always applied on the programming panel from it. The Reference voltage is available.

At each position an external reference voltage is possible from contacts [27; 45] to be adopted.

5.1.11.9 External Voltage Connection

The Sub-voltage of the panel is up to 8 Einstellungsspannungen (8 S-lines): 0 Einstellungen per line result.

5.1.11.10 Programmtafelslots

The Sub-contactpoints of the Programmtafel contain a total of 11 Contacts with their own bus structure. The assignment of this buswork can be found and described in the demand sheet at Instruction 5.

5.1.11.11 Vertical Orientation

The sub-Vertical-Orientation of the programmtafeln includes a total of 11 Contacts with their associated bus contact. The assignment of the Busslines is found in and described in the Demand Sheet.

5.1.11.11 Free Contacts

In the contacts [26–27] and [24–25] are available constant contacts:

[Page 43]

Contact range	Value	Unit
[26; 20–21]	0.1	pF
[26; 22–23]	0.1	”
[26; 24–25]	1	”
[24–25; ?]	1	”

In the contacts [15–17; 31] additional calculation capacitors are accommodated:

Value	Unit
C 00	1
C 01	1
C 02	0.1
C 03	0.1
C 04	0.1

5.2 Adjustment of Coefficient Potentiometers

5.2.1 Decimal Potentiometer

With the aid of the compensation facility, coefficient potentiometers can be set with a load-independent error of less than 0.1 %. The procedure is as follows:

Make entries independent of the previously-set programming panel; press the Dauersprechend (“continuous compute”) mode and on the dialer panel IV, press the key “100 V”.
Make sure that the measuring field shows none of the keys ”?”, “1” or “L”, that the left Tastenstufenfeld key is pressed (see Fig. 1.2).
Set the measuring field to the three-digit decimal keypad (see Abb. 1.2) to the desired single-variable coefficient to be set (from among the three adjacent potentiometer positions, 1–2 assigned to the corresponding key).
Press the potentiometer fieldwork under the one-number mode button (hold-down switch). The measuring field instrument shows 0 V as long as the pot is in the condition that it was when it left the potentiometer.
Turn the potentiometer as long until the measuring instrument shows 0 Volt.
Return the hold-down switch again under the potentiometer to the rest position.

A built-in inhibit circuit prevents the circumstance that simultaneously two buslines being switched to the same measuring lead can be. When the measuring instrument beats at the same time, when one rotates the potentiometer, then the cause is that the previous potentiometer was not returned correctly to its rest position.

For 1 S operation, corrected decimal potentiometers:

Uncorrected single pot: P₀, …, P₃
P₁₀, …, P₁₄
M₀₁
M₀₄, M₀₅, M₀₆

For 1 S operation, corrected decimal potentiometers:

P₀₁, P₀₂, P₀₃
M₀₁, M₀₄, M₀₅, M₀₆
P₁₁, P₁₂, P₁₃
P₇₁, P₇₂, P₇₃, P₇₄

5.2.2 Decimal Potentiometer with Correction

The correction function of the (Fine-) compensation-device provides the possibility, that when the set coefficient indicates in the measuring field the correct, i.e., the true value, when the potentiometer is under load.

Procedure:

The correction is carried out on contact corrections from P 0 … P n once and with corrections from K 1 N M Correction is made available then. Contact P is set to [41; N], corrected contact K to [41; N Korrektur].

[Page 44]

5.2.2 Decimal Potentiometer (continued)

The installation of the (Fine-) correction device provides the programmer with the possibility, that the actually generated coefficient can be used as a measuring device even with a certain load ratio greater than 1, so that the potentiometer will then signal its loading by a correction term. The correction term is connected when the potentiometer is set up, so that on completion of the potentiometer its correction is always present upon the measuring field.

The loaded (corrected) individual potentiometers are: P₀, …, P₆₃

For 1 S operation, corrected decimal potentiometers:

P₀₀, …, P₀₄
P₁₀, …, P₁₄
M₀₁
M₀₄, M₀₅, M₀₆
P₇₁, P₇₂, P₇₃, P₇₄

5.3 Description of the Function Generator

5.3.1 Setting Up an Arbitrary Function y(x) through Polygonal Approximation

An arbitrary function y(x) can be approximated with the aid of the Diodenstrecken (diode segments), using a piecewise-linear representation (Polygonal trace). Each piecewise-linear segment is represented by the aid of a fixed diode-constant (active diode constant a). A representation using this approach can be calculated.

[Page 45: figure only — Fig. 5.3.1, Principle schematic of the function generator]

[Page 46: figure only — Fig. 5.3.2, Function generator assembly plates with switch arrangement, showing left and right rear halves and front and rear main portions with: “Relay generator”, “add. Konstante” (additive constant), “1 V”, “1 250V”, and associated diode breakpoint switches]

[Page 47: figure only — Fig. 5.3.3, Front view of the function generator, showing “add. Konstante” (additive constant), “Nullpunktgerade” (zero-point line), “FS 900 / 16”, and “Ersatzpuf I…II” / “Steigung I…II” (replacement curve I…II / slope I…II) potentiometer knobs]

[Page 48]

Two separate variables, one with no variables (Nullvariable) and two potentiometers (Potentiometer 1 to 5) are located on the right side of the diode breakpoints. The 20 Potentiometer, die 1 to 5 on the front view from the right rear half (see Fig. 5.3.3, Fig. 5.3.2), from the Diodenstrecken (diode segments) 1 are shown, with one potentiometer one for each variable and one additional constant variable. The 20 Potentiometer, die on the right side from the Diodenstrecken (diode segment) to be related, give the adjustment values in context.

The four quadrants, each at ±270 V at ±Strom, are:

Quadrant	±270 V	±Current	Direction
I.	+	+	Right
II.	−	+	Left
III.	−	−	Left
IV.	+	−	Right

[page 48: figure — Fig. 5.3.4, “Diodenstrecken in den einzelnen Quadranten” (Diode segments in the individual quadrants) — showing four quadrant diagram with diode-segment line segments in each quadrant]

[Page 49]

K or A designates that the Kathode (cathode) or Anode respectively is applied to the input of the system.

The breakpoints are determined so that the function is approximated as closely as possible by a straight line (piecewise linear). Graphically, the best possible approximation for a given number of ξ-intervals can be achieved by drawing the function (f:ξ) and the zero-crossing line with the fewest kinks in the ξ-intervals; the approximation error is thereby always smaller than ξ. For setting the breakpoints, extend the baseline further and bring it with the straight line x = −1 (for x₀ > 0) or x = −1 (for x₀ < 0) to the intersection. The ordinates of the intersection points can be read off. One arranges the straight-line segments beginning with positive x-values from the zero point and arranges the straight-line segments with negative x-values (reversed from the zero point). These are divided in (s. Abb. 5.3.5). After the breakpoint settings are made, the values to be set in each Einstellungsregler follow one after another:

y₀(1)  = a            add. Konstante (additive constant)
y₁(1)  = a+b          Nullpunktgerade (zero-point straight line)
y₂(1)  = a+b+d₁(1)    }
  ⋮                     }  Anstiege (slopes) 1 ... 10
yₙ(1)  = a+b+ Σ dₖ(1)  }
         k=1

at x = 1 and

yᵢ₋₁(−1) = a+b+dᵢ₋₁(−1)
  ⋮
y₁₁(−1)  = a+b+ Σ dₖ(−1)
                k=1

at x = −1.

For the convenient setting of a function generator, a table is prepared with the required data (s. Abb. 5.3.6). After this table, the values at each successive Einstellungsregler of the function generator module are entered. The switches of all the un-needed diode segments remain in an arbitrary position. The breakpoint settings proceed in the following manner: All slope controls are turned against the three-grip stops to the Anschlag (stop), likewise the controls for “add. Konstante” and “Nullpunktgerade”. Using a voltmeter and two amplifiers (to avoid the loading of the control-resistance) the dial is switched to the function generator. The dial associated with the diode assigned slope control is turned in the clockwise direction to its stop. The output bus of the function generator…

[Page 50: figure only — Fig. 5.3.5, “Zur Einstellung einer vorgegebenen Funktion auf dem Funktionsgenerator” (For setting a given function on the function generator) — showing a curve plotted on a ξ-axis with unit-input-voltage label, multiple labeled breakpoints (A1–A4), and corresponding piecewise-linear approximation segments in both positive and negative quadrants]

[Page 51: figure only — Table 5.3.6, “Tabelle zur Kurve der Abb. 5.3.5” (Table for the curve of Fig. 5.3.5) — a data table with columns for: Konturer (curve number), ±250 V / ±range, Diodenstrecke-entries (diode segment values) numbered 1–10, and Drehregler (rotary control), Nullpunkt (zero point), Anstiege (slopes) 1–10, annotated with symbols k (clockwise), l (anti-clockwise), and numeric values]

[Page 52]

…is connected on the programming panel with the Bus M instrument. The instrument is used in the measuring field in particularly sensitive range. Now the intercept-point control of the diode segment is set to zero at the null point. The slope control is then turned counter-clockwise until the pointer stops. After completing the measurement the slope controls give the following values: y₀(−1) (n = 0, 1, …, 1) and y₁₁(−1) (n = −1, −1, …, 1). For the precise setting, the measuring instrument is used as a compensation instrument. A check of the set function z(t)-(t−1) (0 ≤ t ≤ 2) is recommended.

For the various operating steps in setting the function generator there are self-evidently various methods and paths with different advantages and disadvantages. One is the above described method of setting the breakpoints greater than, e.g., at x = y 1. In the above described method one can also set more simply and more easily than the above-described method. The approximation of the given curve by piecewise-linear segments can in the same way be applied, although here it is also possible, without previous working, to set the curve by reading values graphically or from a tabulated function directly, although then one accepts not having the optimum diode segment accuracy.

For the setting one proceeds as follows:

For a convenient setting with the compensation device, contacts [“A”] are connected with the switching contact of one of the contacts S00, …, S03, e.g., S 300 contact, the working- or base-contact, the input, the output of the function generator. (This option offers itself particularly for the freely switchable S- and S-relay contacts, where contacts S00 press the keys “Hold”, “Compute”, or “STFIT” to be set.)
All switch controls of the function generator are set as described above.
All intercept-point controls and the controls “Nullpunktgerade” are set against the clockwise stop; all slope controls to the clockwise stop.
Setting and recording of the breakpoint voltages xₙ as described above.

[Page 53]

With the control “additive Konstante”, turn x₂(t) to a specific value directed at the Nullpunkt; then turn the key S00 and on output z₂(xₙ) to be set.
The intercept-point control for the breakpoint is in a specific direction from the start-of-integration (Nullpunkt). Set breakpoint xₙ and with the des start-of-integration (Nullpunkt) set f(xₙ) connected. In the same way it is advanced in a specific successive direction; adjust this breakpoint and use it as the successive-given direction for the next breakpoint.

One proceeds in the same direction for the other breakpoints.

[Page 54]

5.4 Optimum Programming

The mathematical operations are carried out by the computing elements only within a defined voltage range. Therefore, all the differential equations or, in the Differentialgleichungssystem (differential equation system) occurring variables must be so transformed that a transformation from the given system into machine equations in which the machine variables do not exceed the defined voltage range and moreover are utilized as well as possible (to avoid Genauigkeitsverluste — accuracy losses). In the programming and in the machine equations, voltage ranges appear as maximally ±100 V at ±U₀ = 100 V. In the programming ranges the Spanningsgrenzen (voltage limits) are ±100 V, when the maximum working span is 1 (machine unit). The working span is thus always between −1 and +1.

For optimum utilization of all amplifiers, the solution functions and their derivatives must be as close as possible to their maximum values. The highest order derivative of the Maschinengrössen can be reached by means of amplitude transformations, such that the working range of all amplifiers is fully utilized at a Single-Ein-Austeuerung (maximum excursion). An excitation of the multiplier (see Beispiel I — Example I) is in general not possible.

For optimum utilization of all amplifiers, the solution functions and their derivatives must pass as close as possible to their maximum values. During this time, without exceeding them, it serves the amplitude transformation for this purpose, so that the Coefficients of the machine equations are as close as possible to their values between 0.1 and 10 so that they can generally be assumed.

For simplicity of approach, it is proposed that a given arbitrary task is described as a so-called Problem-variable with small letter names, and it is assumed that the independent variable is denoted as the variable t occurring in the variable. These Problem-variables will therefore be connected by linear relationships of the form:

x = mₓ · x̃ ,   y = mᵧ · ỹ , ...

where the scale-factors mₓ, mᵧ, … are referred to as Maßstabsfaktoren (scale factors). The maximum values are determined from:

mₓ = |x|max ,   mᵧ = |y|max , ...

from estimates or from the analog-computer program-derived results, so that for the machine variable X applies:

X = x / |x|max    such that |X| ≤ 1 .

In the same way all derivatives are to be transformed. In general for the i-th (except the highest) derivative:

x⁽ⁱ⁾ = mᵢ · x̃⁽ⁱ⁾   with  |x̃⁽ⁱ⁾|   (i = 1, 2, ..., m−1)

[Page 55]

For the simplest case, the following approximation is used instead of a general nonlinear transformation:

$$y^{(n)} = \frac{1}{T_1} \left( y^{(n-1)} \right)$$

which is achieved by ensuring that the dimensionless coefficients are taken directly from the approximating polynomial and then replacing the T transformation by a scaling step. This method of program preparation is described in the sections on the computing elements (Section 2 of the operating manual); it is carried out as an addition to the computational circuit.

If M_k = 1, so that the Laplace-transformation of the scale-change variable V^(n) is performed, the Bézier coefficient T in the Laplace-transformation corresponds to a dimensionless variable, and must be evaluated accordingly.

For the first Bézier approximation:

$$\frac{dg}{dt} = \frac{M_k}{T_1}$$

In general, for the k-th approximation:

$$\frac{d^k g}{dt^k} = \frac{1}{T_1^k} \quad (k = 1, 2, \ldots, n)$$

It should be noted that the pre-programmed scale-change transformation is to be converted, since a Laplace-transformation can give an incorrect result if a scale transformation has not been applied beforehand.

For the explanation of the general Bézier ratio, the differential equation

$$F\left(y, \dot{y}, \ddot{y}, \ldots, y^{(n)}\right) = f(t)$$

with the substitution transformations serves the series

$$f(t) = f_0 \quad \phi$$

In order to make the variables y and v_y comparable, we obtain the solution of the differential equation as a function of the independent variable:

$$y = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \cdots$$

and correspondingly

$$v_y = \left[ y \right]_{\min} \cdots$$

[Page 56]

In most cases, the solution of the problem of computing the optimal value is not directly available to the user. In many cases, however, the user can derive the denominator from the problem data with the aid of some knowledge about the trajectory of the solution. This method of preparing the denominator is described better in the section on mapping (Section 2 of the description); it is carried out first as an approximation in the upper domain of the denominator.

With the substitution:

$$-y_0 = f = 2 Y \quad \text{and} \quad -v_{y_0} = T = \dot{Y}$$

one obtains the mapping equation as:

$$f \cdot v_y - 2.5 Y$$

or, using the division rule:

$$F \cdot y = F \cdot v_y - 225 \cdot 1.1$$

In addition, the following substitution algebraic equation is given:

$$(1)$$

and the resulting input conditions are:

$$y_0 = \frac{1}{L_1} \cdot y_j \cdot \frac{1}{L_1} \cdot \frac{1}{L_1} \cdot \frac{1}{k} \quad \frac{1}{k} = 0.4 \quad (2)$$

$$(3)$$

From relations (1), (2) and (3), a simple circuit diagram with a Laplace-transformation can be assembled (see Section 5.4.1).

The input conditions are:

$$f(t) - f_0 = 0$$

In order to make the variables y and v_y comparable, we obtain the solution of the differential equation using the Laplace-transformation of the adjoint equation, giving:

$$y = \sqrt{\frac{\pi}{2}} \sin kt + \sqrt{\frac{1}{2}} \cos kt \quad \cdots \quad \left[ \text{or} \quad -\arctan \frac{\pi}{k_y} \right]$$

$$v_y = y \left| \cdots \right|_{\min}$$

and correspondingly

$$v_y = \left[ y \right]_{\min} \cdots$$

[figure: Abb. 5.4.1 — block diagram with summation, integration, and inversion elements]

[Page 57]

While the multiplying coefficients from the first integrator are passed directly as the denominator coefficients, which are used as a better approximation, there are two equivalent methods within the integration intervals (Section 2) for passing the denominator computation. The better solution for each case receives additional attention, so that an appropriately detailed description is given in that section. For the particular program, the additional factor 1/T_1 is introduced before each integrator.

For the formation of the denominator coefficients, the time-transformation is applied. With:

$$t \to \tau \quad \frac{d}{dt} \to \frac{1}{T_1} \frac{d}{d\tau}$$

the mapping equation (1) gives:

$$\frac{d^2\tau}{d\tau^2} \cdot \frac{1}{T_1^2} = -\tau \cdot 35 \cdot T_1$$

The corresponding circuit diagram (circuit diagram Abb. 5.4.2) shows that, for the actual program, the factor 1/T_1 is introduced before each integrator.

[figure: Abb. 5.4.2 — block diagram showing coefficient scaling and time-transformation with integrators]

In the interval t = 0 … 9 K (i.e., τ = 0 … 9), the initial values for the mapping with the time-transformation are computed as follows; accordingly:

$$k_0(\tau) = k_0 \left( \frac{\tau}{T_1} \right)$$

If one needs the Bézier ratio Y and the abbreviation y^(n), which is not in the denominator, then for one variable one should:

$$\sum$$

The initial values for the input conditions remain unchanged.

[Page 58]

The choice of the Bézier ratio M_k is to be carried out such that all coefficients can be set as accurately as possible to within the requirements of the denominator computation (coefficient M_k at most 1, in the range 1 to the denominator coefficient at the computing element). At the same time, the computational accuracy is best preserved. The circuit diagram in Abb. 5.4.3 shows the optimal arrangement for which an additional circuit modification is required.

[figure: Abb. 5.4.3 — block diagram with modified coefficient arrangement]

It should also be pointed out that for linear differential equations with constant coefficients, the Bézier approximation through normalization of the problem can be used to achieve the best programming. By this normalization, all values within the computation can be reduced to the same order of magnitude. This is an important matter for any differential equations that require a higher-order solution.

For the case of the differential equation of the k-th order, a general statement applies: the optimal programming for the solution is obtained by normalizing the problem through a suitable choice of the Bézier coefficients. The normalized differential equation is:

$$y^{(n)} = -a_{n-1} y^{(n-1)} - \ldots - a_1 \dot{y} - a_0 y + f(t)$$

$$y^{(k)}(0) = y^{(k)}_0 \quad (k = 0, 1, \ldots, n-1)$$

Selecting a time-scale M (n = 0, 1, …, n-1) yields:

$$y^{(k)}\left(t_0\right), \quad y^{(k)}_0 = M, \quad (k = 0, 1, \ldots, n-1) \quad \text{satisfied.}$$

[Page 59]

$$y^{(k)} = \left{ -a_{n-1} y^{(n-1)} - \ldots - a_1 \dot{y} - a_0 y + f \right} \Big|_{t_0}^{\mathstrut}$$

$$y^{(k)}\Big|_{t_0} \quad (k = 0, 1, \ldots, n-1)$$

$$= \quad a_i \cdot y^{(i)}\Big|_{\max} \quad (i = 0, 1, \ldots, n)$$

This form of the differential equation has the advantage that it can be used to determine the optimal Bézier coefficients for the given problem; the derivative values appear normalized on the right-hand side and on the initial conditions side. The Bézier denominator ratios are thus determined directly.

It should be noted, however, that the method described above assumes a general programming in which all operational elements can be given their full output range. The denominator coefficients (for example: 43 integrations for the Bézier ratio) are set according to the mapping; they must be determined through a sufficiently accurate computation. For this reason the program is checked first after each setting of a parameter, and then the denominator coefficients are adjusted accordingly until the solution is within the allowable range.

For the analog computer (in the sense described here), each has its own procedure, and each system has only a small number of elements, so that the requirement for small denominator coefficients is not an obstacle.

[Page 60]

5.5 Application Examples

Example 1: Computation of the step function f(t) = e^{−at}

The commonly required step function f(t) = e^{−at} can be computed by multiplication of the functions of f(t) = e^{−at} and y(t) = e^{−at} where the initial conditions are f(0) = 0 and f(1) = 1.

In the first step, one must examine the Bézier differential equation to find a suitable solution:

$$\dot{y} = -a \quad \text{with} \quad y(0) = 0 \quad \text{and} \quad f(1) = 1$$

or equivalently:

$$\dot{y} - a \cdot y = 0$$

With [y]_max = 1 and [y_0]_max = 1, one obtains the circuit diagram of Abb. 5.5.1 and the mapping equation:

$$F = 2F \cdot \frac{1}{T_r} \cdot \tau$$

and the corresponding initial conditions are found from Abb. 5.5.1.

[figure: Abb. 5.5.1 — two-part block diagram: a) exponential decay generation using integrator with feedback; b) combined circuit for generating e^{−at}]

[Page 61]

The circuit (a) consists in its first part of Bézier elements and requires a further application of the multiplier (the multiplication of the functions of y(t) with cos(y(t)) is then brought to completion). The circuit (b) requires that the problem be set up with 2 comparators; a simple configuration with only 2 comparators is sufficient, as shown in Abb. 5.5.1.

Example 2: A function generator (sawtooth/parabolic) is to be programmed. At most two computing elements (simple comparators, see Abb. 5.5.1) are used for the circuit configuration with the sampler and for the memory elements.

The approach for the function generator is to perform the programming with all computing elements (comparator, memory, sampler and all associated test elements). In the simplest form of the memory elements, it is sufficient to use two integrators for each function period.

Given preassigned values t_p and t_q, these are to be used as the function generator’s initial values:

$$t_p = \frac{1}{f_{\text{max}}} \quad t_q = \frac{1}{f_{\text{max}}}$$

[figure: Abb. 5.5.2 — block diagram for parabolic/sawtooth function generator with comparators and integrators]

[Page 62]

Example 3: Simulation of an electrical RC-network

The following RC-network is to be simulated on the analog computer:

[figure: Abb. 5.5.3 — electrical RC-circuit schematic with two resistors and two capacitors]

$$e_1 = \begin{cases} -1 \text{ for } 2nT_1 \leq t < (2n+1)T_1 \ +1 \text{ for } (2n+1)T_1 \leq t < 2(n+1)T_1 \end{cases}$$

$$e_2 = \begin{cases} -1 \text{ for } 0 \leq t < (2n+1)T_1 \ +1 \text{ for } (2n-1)T_1 \leq t < 2n T_1 \end{cases}$$

and, i = 1, …, n = 1, 2, …

To determine the positive initial values of e_2, an exact treatment is required. The Laplace-transform of the input function at node 2 is assumed to be completely positive; because at node 2, at the initial time point of a half-period, the output is fully charged; as a result, an output is obtained that is characterized by an alternating negative current:

$$(V_0)_1 \cdot (V_0)_2 \quad \text{gives the mapping equation}$$

$$(V_0)_1 \quad (V_0)_2$$

From the outputs e_2 and e_2, the denominator factor gives the differential equations from the mapping equation:

$$\frac{d}{dt}\left(V_0\right) = \ldots$$

and the corresponding programming is shown in Abb. 5.5.4.

[Page 63]

[figure: Abb. 5.5.4 — block diagram for electrical RC-network simulation with summation and integration elements, showing signal flow from e_2 through comparators to outputs C_1 and C_2]

The initial conditions are for both integrators: zero. An amplitude-transformation is not necessary because the output does not go beyond the normal domain. The comparators (for the sawtooth circuit, see Abb. 5.5.1) are shared; the function generator unit is divided into two parts (see Abb. 5.5.3, two time-scale elements). For qualitative purposes, this two-integrator arrangement is sufficient; it can be extended to a four-integrator circuit.

[figure: Abb. 5.5.5 — simplified block diagram with two integrators and two comparators for qualitative simulation]

For qualitative evaluation, the integrators can be assigned to the forward and the feedback path, as shown in the simplified arrangement (see Abb. 5.5.5); the two function-generator units each contain one integrator. In the inner loop of the comparator for this problem, the corresponding negative-feedback region of the two comparators must be observed.

[Page 64]

Example 4: Pendulum oscillation

The equation of motion is:

$$\ddot{y} + \omega^2 \sin y = 0 \qquad (*)$$

for the initial conditions y_{max} = k_{\pi} / n \quad (k = 1, 2, \ldots, N)

The initial conditions for the given problem are found from a single application of the denominator:

$$y = \sqrt{\text{const}} \cdot \cos y \cdot \frac{1}{\sqrt{k}}$$

where y(0) = y_{\text{max}} and f(0) = y_{\text{max}} / k

The circuit for y and y_t is to be determined separately by the differential equation. For the two cases:

$$y(0) = \cos kx / 2 \quad \text{and} \quad y(0) = \sqrt{\text{const}} / k$$

The Bézier ratio for y and y_t is computed as a function of the parameter k. It is known:

$$y_{\max} \quad \omega_y = \sqrt{\text{const}} / \omega_k$$

For the solution, the differential equation y = −ω²·sin(y) is solved as f(y) = sin(y), y(0) = y_0, and y(1) = y_0/k. For both trajectories, the following programming is obtained (see Abb. 5.5.6).

[figure: Abb. 5.5.6 — block diagram for pendulum oscillation simulation showing sin-function generation, multiplier, and integration stages]

[Page 65]

[figure: Abb. 5.5.7 — block diagram for pendulum simulation with nonlinear function generation, multiplier, and two integrators for full solution including parameter variation]

The circuit requires fewer function generators and has in itself four initial conditions to set. The function-generator units are set with y via +k√2/k·√2 of the Bézier denominator, so that this allows all trajectories from the Bézier approximation described in Section 5.3.8 to be determined and subsequently plotted by the function generator.

[Page 66]

[figure: Abb. 5.5.8 — time-domain waveform (oscillatory, decaying) for the pendulum trajectory, labeled with amplitude and time axes]

[figure: Abb. 5.5.9 — overall function-generator block diagram showing complete chain for pendulum computation as described in Abb. 5.5.7]

In circuit 2, the saturation condition is satisfied and the Bézier ratio y is as a result only slightly positive (set to ε). The integrator T_1 in the circuit acts as a saturation limiter for the result; the integrator is kept at the same point without being driven by the sampling element. In certain cases, due to the saturation condition, all variables must be checked, so that the optimal programming of the saturation is achieved. For this reason, the denominator (variable y^{(n)}) is required from 0 all the way to the Bézier maximum.

[Page 67]

Example 5: Solution of the van der Pol differential equation — function generator

For the stationary solution of the van der Pol differential equation:

$$\ddot{Y} - \varepsilon(1 - Y^2)\dot{Y} + Y = 0 \qquad (**)$$

functions can be provided.

Since for linear differential equations with constant coefficients the initial conditions result in an approximation that is acceptable only to a limited extent, and since the initial conditions of the approximation are only approximately satisfied, in the stationary approximation the computation is carried out as follows, so that the simplest Bézier approximation yields the result directly from the mapping equation.

With T = 1 and Y = 1, the mapping equation is:

$$\ddot{Y} - Y = T \cdot \varepsilon \cdot \left(1 - Y^2\right) \dot{Y} = 0$$

with which the circuit diagram (see Abb. 5.5.10) directly results.

[figure: Abb. 5.5.10 — block diagram for van der Pol oscillator simulation with multipliers, integrators, and nonlinear feedback representing ε(1−Y²)Ẏ]

[Page 68]

[page 68: figure only]

Upper portion: time-domain waveform sketch showing a periodic oscillation with labeled axes (Y, t) and annotations indicating the amplitude region.

[figure: Abb. 5.5.11 — block diagram for van der Pol equation showing comparator-based circuit with two comparator stages (labeled 1K1 and 1K2), sawtooth generator, and two integration/output stages; the block diagram illustrates how the stationary limit-cycle solution is maintained by the comparator logic]

[Page 69]

From the equation it is evident that the maximum is not obtained by direct substitution into linear differential equations with constant coefficients. When sampling frequency ω_0 (t = 2) is equal to t_1, the substitution of ω_0 gives rise in the frequency domain to a positive shift, such that the transformation is used.

The Fourier coefficients are obtained by solving the differential equation:

$$\frac{d^2 e}{dt^2} + \left(\frac{\pi k}{T_0}\right)^2 e = x \qquad \text{with} \quad f(0) = f(0) = 0$$

Over a period T_0 \leq t \leq T_1, for the stationary solution.

The values at t_0 (T_0 \leq t \leq T_1) are the Fourier coefficient values proportional to the denominator; these are extracted from the representation over a single period of the stationary solution. For the denominator computation, 4 comparators (see Abb. 5.5.11) are required. The comparator K 01 computes the product ε·Y and by using the comparator K 01 the multiplier register is activated, so that the time interval from t = t_0 to t_1, the computation is underway. Comparator K 03 is driven by the sampling register, and it verifies that the integrators are reset at the beginning of the interval t_0. The starting values are provided to the register from the comparator K 01. The values are read from one sample-and-hold element; this can be read out during the subsequent interval and printed with a digital voltmeter or via a connected recording printer.

5.6 Test Circuits

In the following, a number of test circuits are assembled that are suited to providing the most important statements about the accuracy and operational behavior of analog computing elements.

Test circuit 1: Determination of the relative mean accuracy of the computing coefficients of linear computing elements

Literature: A. Kley, Test circuits for the evaluation of operational amplifiers, Elektronische Rundschau 14 (1960) 10, pp. 403–404.

Theoretical basis:

The adjacent circuit has the characteristic equation:

$$0 + s^2 U = 0 \qquad \text{with}$$

where s = \sqrt{k_{10} k_{20}}, with the limit values k_{10} and k_{20} being the computing coefficients of the computing elements; further, e_D is the error; the deviations are:

$$k_i = k_{i0}(1 - \xi_i), \quad i = 1, 2$$

[figure: Abb. 1 — block diagram for test circuit with two computing elements and feedback, illustrating the oscillation test for coefficient accuracy]

[Page 70]

With k_0 = k_1 · k_2 · x = 1 and t_0 ≥ 0, one obtains the frequency error:

$$\left| \Delta \omega \right| = \frac{1}{2} \left| \xi_1 \right| + \frac{1}{2} \left| \xi_2 \right| + \frac{1}{4} \left| \xi_1 \xi_2 \right| \approx \frac{1}{2}\left(\left|\xi_1\right| + \left|\xi_2\right|\right)$$

with

$$\xi = \text{(see previous equations)} \left| \xi_i \right|$$

Viewed with the simplest quasi-linear approximation, one obtains from the two functions f_1(t) and f_2(t) on a sine wave with phase and amplitude, if each:

$$f_1(t) = 100 \cdot \cos(\omega t + \varphi) \quad \text{and} \quad f_2(t) = 100 \cdot \cos(\omega t + \varphi), \quad \text{respectively,}$$

Then the frequency error is:

$$\left|\Delta \omega\right| = \ldots \quad \omega_0 = \ldots \quad \omega_0 = \ldots \quad \text{(refer to equations)}$$

From this one obtains additionally the denominator of the two computing coefficient deviations for all values and other variables. For the frequency–coefficient combination, the following is obtained:

$$v_k^2 \cdot 1 = a \cdot N$$

For any arbitrary sine with value:

$$v^* = \sqrt{k_{10} k_{20}} \quad a = 1, 2, \ldots$$

The time t* can be set:

$$t^* = \frac{2\pi}{\omega} \cdot a \quad \text{and}$$

Time prescription:

The relative mean-squared denominator error t is at most 1 with:

$$t = \frac{2\pi}{\omega} \left[ 1 + \frac{1}{\omega_i} \left| \xi_1 \right|^2 \left| \xi_2 \right|^2 \cdots \right]$$

For the denominator k’, a mean is obtained from diagram 1 (interpolated), where:

$$k’ = k_0 \left(1 - \xi_k\right)$$

For parameter k’: an independent computation is carried out in diagram 1, where the maximum value of the deviation from the true value V_i is read.

[Page 71]

[page 71: figure only — Diagram 1]

Diagram 1 is a nomogram (alignment chart) with three diagonal parallel lines slanting from lower left to upper right on a grid. The axes are labeled and the lines appear to correspond to values of relative frequency deviation as a function of the accuracy parameters t_0, t_1, and t_2 of the computing coefficients. The chart is used to read off the overall relative mean accuracy from the individual coefficient deviations.

[Page 72]

Test circuit 2: Determination of the phase-frequency behavior of computing elements (amplifiers)

Literature: A. Kley, Considerations on the evaluation of operational amplifiers, Elektronische Rundschau 14 (1960) 19, pp. ___

Theoretical basis:

As the solution of the circuit shown in Abb. 2, the characteristic equation gives instead of the desired ideal solution:

$$U(t) = \left(\tan \Psi_a - \frac{Y_a(s)}{Y(s)}\right) \cdot e^{-at} \qquad \text{with} \quad a = -b \text{ und}$$

Further, b = −1.

$$\psi = \left(\tan \Psi_a\right) \cdot \frac{Y_a}{Y(s)} \cdot \frac{1}{\left|V\right|} \quad \text{bzw.} \quad \frac{Y_a}{s}$$

where: Y_a(s) — Laplace-transform of the output signal Y(s) — Laplace-transform of the input signal V(s) — transfer function of the computing element Ψ_a(s) — phase angle of the computing element

Computing-coefficient bias:

The function is (see circuit diagram Abb. 2) computed with the initial condition:

$$\dot{y}(0) = -b$$

It is experimentally necessary to ensure that the time t_0 is measured accurately, for which the experiment is carried out.

Test procedure:

The circuit (see Abb. 2) is programmed approximately according to Diagram 2. One then interpolates between lines b and the value V_i, where:

The function f_0(t) with the initial condition according to Abb. 2 is integrated; the result is computed in a loop with 1 integrator, where the summation element takes the result.

[figure: Abb. 2 — block diagram for phase-frequency test circuit with computing elements, showing feedback path for integration and phase-measurement arrangement]

Exercise 5a: Direct Measurement of the Null-Position Error

Literature: A. Klug, The Calibration of Servo-Amplifier Systems in Analog Computers, Akademie-Verlag, Berlin (June 1957), p. 30.

Theoretical Foundations:

[Circuit diagram: An integrator with feedback capacitor C₁₀ and input resistor R₁₀, driven by input voltage u_E, producing output voltage u_A. See Fig. 3a.]

Due to the drift, even with no externally applied input the output undergoes a continuous motion. If a constant k is selected, which is determined in the analog computer by the integrator gain, then:

u_A = u_E · (1 + k)

Test Procedure:

According to Fig. 3a, a DC test signal of 2–50 mV (measured with the DC millivoltmeter 11) is applied to the input of the verifier. The filter of the verifier is set so that it passes signals from DC to 10 kHz (bandwidth approximately 100 mV) as the null voltmeter.

The null voltage u_N can thus be calculated as:

u_N = u_E · (k − 1) / k

Exercise 5b: Measurement of Null-Position Error by Integration

Theoretical Foundations:

Since it can be assumed that the null-position error of the comparator remains approximately constant during a short measurement time, the output voltage u_A(t) behaves at the output according to:

u_A(t) · ∫u_E · dy = 4 + k · t

This describes, with one error voltage E_0 and one time constant T_e, an approximately linear output.

Test Procedure:

The comparator whose drift should be measured is connected as an integrator (Fig. 3b). The output voltage u_A(t) is then observed over time t₀ = 10–50 [s]. During this time the measured input voltage U_E is recorded, from which the drift (characteristic 100 mV) can be read.

[page 75: figure only]

Diagram 3b — Graph showing output voltage |u_A| as a function of time t (in seconds), with parameter k = 10 and u(t) = 100 mV. The curve shows a steeply falling exponential decay from an initial high value toward zero as time increases.

Exercise 6: Simulation of the Multiplier Characteristic through Addition of u₁ = u₂, with u₁ = 0.5083 and u₂ = 0.5

Theoretical Foundations:

[Circuit diagram Fig. 4: A multiplier circuit comprising two summers and a multiplier element, with inputs x₁, x₂ and output ξ(t).]

Test Procedure:

Exercise 6, Fig. 4, is to be programmed with the following parameter values:

x = −0.1, k = 20 s, x = −0.1, k = 20 s u_D = 1, u_D = +k

The output variable is measured for u_E: u₂ = u_D, u_G = u_D, over a runtime of 20 s.

To check the symmetry of the multiplier, at one point u₁ and u₂ are to be interchanged.

Exercise 7: Testing of the Multiplier Characteristic for Generating x^n = sin^n·x + 1 for Various Processors

Theoretical Foundations:

[Circuit diagram Fig. 5: A more complex computing circuit using two multiplier elements, summers, and an integrator, with inputs and output ξ(t).]

With the aid of the input–output relationship:

y’ = x(n+1) · 1; x(k) = 0

both polynomials from k to n are generated, for squares x^n and on to an arbitrary number of multiplier elements.

Test Procedure:

For programming Fig. 5, used for various processors (k = 1, 2, 3, 30, 20 s), the error function E at one unit Bezier point k = 0 is

E = u_E − u_E · u_E(k)

Exercise 8: Division Using Interconnected Squares

Literature: Richter, Electronic Measuring Devices, Akademie-Verlag, Berlin.

Theoretical Foundations:

[Circuit diagram Fig. 6: A division circuit consisting of a multiplier X and an operational amplifier feedback loop, with inputs u₁, u₂ and output u_G(t).]

The quotient u_G = u₁/u₂ is not directly achievable with the analog computer, since from u₂ = 0 the multiplier cannot be made to invert. By closing the division circuit in the manner shown (Fig. 6), however, the division is realized. The relationship has a stable equilibrium point.

For programming (Fig. 6), U_E = 0 and stability shall be verified, with the following parameters:

U_E = −500 V, U_G = 0.3, u_D = −0.1

For u_G results from the interval 500 V to u_G = 10 V.

Exercise 7 (continued): Computation of the Legendre Functions P_n for 0 ≤ n ≤ 5

Literature: Abramowitz–Stegun, Tables of Special Functions (Tafeln besonderen Funktionen), Leipzig 1956.

Theoretical Foundations:

The Legendre differential equation is:

(1 − x²) y” − 2x y’ + n(n + 1) y = 0

It is possible to solve it with the analog computer. One obtains the following recurrence relation:

d/dx [(1 − x²) dy/dx] = −n(n + 1) ∫y dx + L(0)

Then one obtains the recurrence relation:

[u_max / (∫y dx)_max] · [u_max / (∫y dx)_max] · [a(n+1) · ∫y_n dx + u_max / (∫y dx)_max] · u_A(0) =

n	a_n	B_n	a(n+1)	b(n+1)	b_n(0)
1	1	2	1	2	−1
2	1	3.180	6.380	−0.004	−0.230
3	1	−0.402	−0.148	−0.315	−0.423
4	10	32.533	−6.205	−0.375	0
5	15	0.266	0.260	−0.375	−0.125
6	21	20.520	0.4062	−0.513	−0.241

Table 7

[page 80: figure only — circuit diagram]

The following scaling relationships are defined:

B₁ = −u_max / 1
B₂ = 1 / (∫y_n dx)_max
B₃ = a(n+1) · (∫y_n dx)_max
B₄ = u_A(0)

[Fig. 7: Block diagram of the analog computer circuit for the Legendre functions. It comprises two integrator chains, a multiplier, and several summing amplifiers, with inputs labeled 1) and −1) and the output denoted as e² · u_max.]

[page 81]

Test Procedure:

The circuit (Fig. 7) is to be programmed from the parameterized Table 7.

For the Legendre polynomials (Input x):

n = 2: P₂ = 1
n = 3: P₃ = ½ (3x² − x)
n = 4: P₄ = ⅛ (5x³ − 3x)
n = 5: P₅ = ⅛ (35x⁴ − 30x² + 3)
n = −5: P₋₅ = ⅛ (63x⁵ − 70x³ + 15x²)
n = −6: (additional expression involving powers up to x^6 with coefficients 63, 70, 15, −1)

Also to be verified alongside the tabulated forms are the Legendre forms. Note that a time transformation τ = t/8 was employed.

Value Table for Legendre Polynomials:

t	P₁(t)	P₂(t)	P₃(t)	P₄(t)	P₅(t)	P₆(t)
0	0.000	1.000	0.000	1.000	0.000	1.000
0.1	−0.100	−0.485	−0.148	0.336	0.375	−0.813
0.2	−0.200	−0.940	−0.280	0.538	0.466	−0.410
0.3	−0.448	−1.200	−0.148	0.412	−0.90	−0.325
0.4	0.214	0.280	−0.153	−0.313	0.271	−0.293
0.5	0.564	−0.266	−0.148	−0.115	−0.449	0.323
0.6	0.448	1.064	−0.153	−0.812	−0.271	−0.352
0.7	0.236	0.280	−0.475	−0.812	0.271	−0.241
0.8	0.716	0.721	0.473	0.216	−0.441	−0.241
1.0	1.000	1.000	1.000	1.000	1.000	1.000

6. Commissioning and Operation

6.1 Controls

Operation of Both Kreis (Type II and Type III):

Before commissioning, both Kreis are to be set into operation: for type II, the Bus No. 2 and type III must not previously be in operation, the Bus No. 2 and the internal Bus (type III, e.g. from the VEB Gerätewerk Karl-Marx-Stadt) are activated. To set the scaling, the potentiometers are set at the front panel of the VB; the Digit input voltage is set at the front panel of the ZRK and the Display is observed. These controls are numbered. The Display is calibrated at the front of the ZRK instrument.

6.1.1 Kreis Type II (II. Kreis)

Switch 1 is set as the reference voltage amplifier: from u_S, the output spans the reference voltage range and is connected, so that the Display amplifier cannot be bypassed. By this step the division is realized. The relationship shows stable point.

For programming (Fig. 6), for U_E = 0, stability shall be tested, and the parameter settings are:

U_E = −500 V, U_G = 0.3, u_D = −0.1

6.1.2 Kreis Type III (III. Kreis)

Switch 1 is set as a reference voltage amplifier; from u_S the output spans the reference voltage range. This control is directly visible at the front panel of the ZRK. The Display amplifier is monitored by this control and by the calibration of the machine from 0 to 0.

6.2 Operational Amplifier Test

During operation of the machine a regular check of the zero-point and drift of the operational amplifiers should be performed. In a single inspection, information about the state over time of the operational amplifier can be obtained. The inspection is carried out at least once per day.

To do this, the following test procedure uses the Program Field (Bus No. 2 included) and a voltmeter with about 100 scale divisions; a switch is provided so the button can be used to test each amplifier in sequence — pressing “Check” (Russian: “Контроль”) for the desired amplifier. Each test is done in succession.

6.2.1 Control of the Supply Voltage

With the aid of Potentiometer 20 the supply voltage is monitored. The machine is normally supplied with voltages of −— and +—.

6.2.2 Control of the Drift Voltage

For the drift voltage test, the Potentiometer 42 is set to 0; this is the integrator. After this, the integrator gain is connected so that it integrates (with a gain of 1 × RF₁ · 1 = 1 × 1 = 1) with a time constant. The text reads “Check” (Russian), where “P” stands for the potentiometer position.

6.3 Functional Test of the Multiplier

a) Alignment (u_A)²

With a single bus and Regner 11 (Counting Regner 2 from the program field), the output is set to −100. The input of the Program-Operating-Bus “MZ” (Multiplier Zero) is connected with a voltage of −100 V.

The output is connected using Switch “Eingang 1” and “Eingang 2” so that the Supply Voltage is set to −400 and the Switching Voltage is switched to +40 V supply. One should observe whether the “Eingang 1” and “Eingang 2” controls are both set to left, and also check Regner 2 “Eingang 2” — they should each yield an effectively equal value.

b) Alignment (u_A)² (second arrangement)

Now Regner 11 is connected to the bus; the alignment proceeds as follows: in each case for Regner 2 (above), each set Regner 2, Eingang 1 and 2, each Regner 12, Eingang 1 and 2 is tried and the inputs to the Multiplier are compared.

[page 85: figure only — panel diagram]

Fig. 6.3 — Front-panel layout diagram of the multiplier module, showing the following labeled controls:

W2 IV, W78, W73 (top row, potentiometers)
Einstr. 1 (Input trimmer 1)
Einstr. 2 (Input trimmer 2) with associated Ru1, Ru2, Ru3 feedback resistors
Einstr. 1, Potenz 2 (lower section)
FM 380/13 (module type designation)
W79, W32 (bottom row, potentiometers)

6.4 Comparator

The device also has three comparators housed. As can be seen from Fig. 2 of the description of the ZRK (see Sec. 4.1 for layout), on the right side of the instrument the Regulator K 13 comparator at position 60 and K1 comparator at position 17 are found, and the Program Field controls the comparators.

The comparator should be checked at least once during the operation period. Over the entire operation time, the comparator supplies an output voltage of about 100 V (positive and negative). The text reads: “Check” in Russian, the key labeled “Контроль” is used for the test — upon each press the comparator’s state is verified.

6.4.1 Control of the Supply Voltage

During the comparator test, the supply voltage is checked. The comparator is normally supplied with voltages of −29 V. These controls are visible at the front of the ZRK display and the calibration of the machine takes place from 0.

[Description continues:] The comparator check is carried out as follows: by an additional comparison of the input voltage, a positive and a negative voltage from the Regner 2 bus, K 2, is applied. Through the additional operation, the null-signal is checked, and the Regner 2 bus is then brought to the zero point — so that the device ensures that the Multiplier arrives at the null point.

[page 87: figure only — comparator diagrams]

Fig. 6.4 — Three diagrams illustrating the comparator:

a) Programmer circuit (Reference: 5 sec.) Schematic showing a summer with +50 V and −50 V inputs driving a comparator element, with output labeled 001.

b) Unbalanced comparator (b: unausgeglichener Komparator) Graph with axes showing the switching characteristic of an unbalanced comparator — a stepped curve with hysteresis, with the switching point offset from zero on the input voltage (Eingangspegel) axis.

c) Balanced comparator (c: ausgeglichener Komparator) Graph showing the ideal comparator switching characteristic — the transfer curve (Übertragungskennlinie) and working curve (Arbeitskennlinie) centered symmetrically about zero, with the switching point precisely at zero on the input voltage axis.

6.5 Calibration Voltage

The device takes its reference directly from the Calibration of the Comparators, which are self-calibrating at 0 V nominal reference voltage. From the description, this can be automatically set so that the reference voltage of 100 V corresponds to full scale.

Procedure:

At the upper right-hand side of the upper Potentiometer block (see Fig. 2 for layout, see Sec. 4.1) the “Synchronize” / “Überbrücken” button provides the possibility to press it in sequence to bring in turn each of the following:

The Regner is brought to a value; the “Überbrücken” means the device (for the Regner being tested) is bridged. The Regner for this state is calibrated, i.e. the display control amplifier is switched to “Check” (Russian/label), and from this the Regner is again set to “Check” as required.

Leaving the program side: one presses in sequence Regner 20, then 2 to set the output voltage comparison at Regner 20 for the Machine calibration function — this is labeled Overbridge “Überbrücken” on the instrument.

This procedure is to be repeated in alternation.

Elektronischer Langzeitrechner endim 2000: Programmier- und Bedienungsanleitung

Foreword

Table of Contents

1. Structure of the Computing System

2. Control of the Operating Mode

2.1 Overview

2.1.1 Readiness condition

2.1.2 Operating-mode control StP II (static check)

2.1.3 Operating-mode control StP III

2.1.4 Control of the operating-mode indicator

3. Operation

3.1 Control of the Operating Mode

3.2 Control Elements of the Programming Panel

3.3 Operating Mode Control

3.3.1 Overload Display, Switch and Control Options at the Electronic Programming Panel

3.3.2 Integrator

3.4 Timer (Steuerzeit)

3. Operation (continued)

3.1 Control of the Operating Mode (detailed)

3.5 Readout (Anzeigeteil) / Control of the Operating Mode

3.3 Selection of the Operating Mode of the Integrators

3.4 Computing Time Setting at the Function Receiver endim 2160

3.5 Computing Time Setting, Switching and Control Options at the Electronic Clock

3.5.1 Time selection

3.5.2 Selection of operating modes

[Page 19]

3.5.3 Selection of the Data Printout

[Pages 20–21: Table 3.5 — Switch and Control Options for the Printout (Continuation)]

[Page 22]

3.6 Selection of Test Options

3.6.1 Stability Test

3.6.2 Setting and Measuring the Transfer Factor of the Summing Amplifiers (Static Test)

3.6.3 Setting and Measuring the Time Constants of the Integrators (Dynamic Test)

[Page 23]

3.7 Overload Indication

3.7.1 Overload Indication Display

[Page 24]

3.7.2 “Hold on Overload”

3.8 Connection of External Devices

[Page 25]

4. Evaluation

4.1 Evaluation with the Oscilloscope (Function Receiver endim 2100)

[Page 26: Figure Only]

[Page 27]

4.2 Measurement Readout

4.2.1 Direct Measurement

4.2.2 Measurement Evaluation using the Compensation Method

[Page 28]

4.2.3 Measurement Readout via Coefficient Potentiometers

4.3 Evaluation with Peripheral Devices

4.3.1 XY-Plotter

4.3.2 Digital Voltmeter

[Page 29]

5. Programming

General Procedure

5.1 Programming Panel

5.1.1 Handling of the Programming Panel

[Page 30]

5.1.2 Patching the Program

[Page 31]

5.1.3 Layout of the Programming Panel

[Page 32]

5.1.4 Colors of the Programming Panel

[Page 33: Figure Only]

[Page 34: Table Only]

[Page 35]

5.1.3 Switching of the Computing Amplifiers

5.1.5.2 Amplifiers 01, 03, …, 29, 31

[Page 36: Table Only]

5.1.5.3 Potentiometers 30, 31, …, 46, 47

5.1.5.4 Potentiometers 40, 41, …, 61, 43

5.1.6 Description of the Coefficient Entry Panel

5.1.6.1 All-purpose Multiplier Panel

[Page 39]

[Page 40]

5.1.11 Programming switch and control contacts, Reset and Hold functions

5.1.11.1 Operator

[Page 41]

5.1.11.2 Free Contacts

5.1.11.3 Selector Contacts