Schaltungen der Analogrechentechnik

Complete English translation of the original German-language document (146 pages).

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Circuits of Analog Computing Technology

Collection of 95 fully derived and practically tested computing circuits

With 172 illustrations

By Dipl.-Ing. W. Ammon

R. Oldenbourg · Munich · Vienna 1966

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Digitized with the permission of the publisher. Further distribution as well as scientific and private use are permitted. Commercial use, in particular the sale of digital copies and printouts, remains prohibited.

SYMBOLS USED — 8

A. NOTES ON THE USE OF THE ANALOG COMPUTER AND THE CIRCUIT SHEETS — 11

B. CIRCUIT SHEETS

Section 1: Generally important circuits, also partially used in the following — 20
Section 2: Realization of selected rational transfer functions — 22
Section 3: Simulation of dead times — 28
Section 4: Division, square roots, exponentiation, and computing operations linked to these — 37
Section 5: Fractional rational functions with several variables — 56
Section 6: Formation of trigonometric functions and operations with them — 65
Section 7: Coordinate transformations — 72
Section 8: Generation of sine and cosine functions with fixed and with variable frequency — 78
Section 9: Simulation of special characteristic curves, in particular those with hysteresis properties — 96
Section 10: Generation of triangular and rectangular functions and pulses — 104
Section 11: Samplers and switches — 108
Section 12: Generation of special characteristic curves with free diodes — 116
Section 13: Miscellaneous — 139

C. LITERATURE COLLECTION — 145

Preface (Vorwort)

[page 6 — Preface, first portion:]

When solving problems with the aid of electronic analog computers, one must as a rule not only set up the computing equations but also select the circuits and combine the individual operations. One must know which circuits are available for this and be able to estimate the quality of these operations. The previous literature on this subject is not easily accessible to the non-specialist. The present collection aims to provide easy access to the most important circuits and to make them available to the practitioner. It should also enable verification of the feasibility of intended problems. The derivations are presented in full so that the reader can follow them step by step and can themselves implement further circuits. The author has endeavored to include, above all, circuits that have been practically tested and are suitable for use on the frequently-used analog computers from Dornier Electronic and Servomultiplier companies. The errors arising from the type of amplifier used are indicated in the circuit sheets under “Remarks.”

[page 7 — Preface, second portion:]

…my colleague, Dr. Haberstock, who subjected the manuscript to a very careful and critical review and to whom I owe many valuable suggestions and discussions. Frau Kahl carried out in an exemplary fashion the not exactly easy task of typing the manuscript, which contained so many formulas.

Since this booklet is a collection of many mutually independent parts — namely the circuits — the clarity of the whole depends essentially on a sensible ordering and labeling of the sections and the circuits they contain. The publisher has taken great pains over this clarity and a fine typographical appearance, and is thanked for this no less warmly than for the prompt publication and the good collaboration.

Munich, July 1965

W. Ammon

Symbols Used

1. Open Amplifier Without Input Resistor

[Circuit symbol: triangle with ∞ inside, input x_E, output x_A = −V x_E]

V = gain of the open amplifier
∞ = denotes the so-called summing point

2. Computing Amplifier With Specified Circuit (example)

[Circuit symbol: amplifier with feedback resistor R_0 and input resistor R_1]

$$x_A = -\frac{R_0}{R_1} x_E$$

3. Computing Amplifier Without Feedback (example)

[Circuit symbol: amplifier with multiple weighted inputs n_1, n_2 (equivalent to input resistors R_1, R_2)]

$$x_A = -\frac{V}{n_1 + n_2}[n_1 x_{E1} + n_2 x_{E2}]$$

$$= -\frac{V}{\dfrac{1}{R_1} + \dfrac{1}{R_2}} \left[\frac{x_{E1}}{R_1} + \frac{x_{E2}}{R_2}\right]$$

4. Summing Amplifier With Preceding Potentiometers

[Circuit symbol: amplifier with potentiometers α_i on inputs with weights n_i]

$$x_A = -\sum_{i=1}^{\nu} \alpha_i , n_i , x_{Ei}$$

α_i = setting of the potentiometers
n_i = input weights of the summing amplifier

5. Integrator

[Circuit symbol: integrator with potentiometers α_i and α_v]

$$x_A = -\int_0^t \left(\sum_{i=1}^{\nu} \alpha_i , n_i , x_{Ei}\right) dt$$

6. Two-Parabola Multiplier

[Circuit symbol: multiplier block with inputs +x_E2, −x_E2, +x_E1, −x_E1, output x_A = x_E1 · x_E2]

7. Servo-Multiplier (example)

[Circuit symbol: servo-multiplier block (X) with load (F), potentiometers driven by x_E1, tapped by +1 and −1, with potentiometers at +x_E2 and −x_E2 and −x_E3, outputs x_E1·x_E2 and −x_E1·x_E3]

For information on the center tap, refer to the note in Part A.

8. Servo-Resolver (example)

[Circuit symbol: resolver block (x), (AGC), (F), driven by x_E1, with potentiometers at −x_E2, +x_E3, outputs x_3 sin x_E1 and x_3 cos x_E1, and −x_E1·x_E2]

AGC = Automatic Gain Control (the larger the AGC voltage, the lower the gain. AGC is used in division circuits, coordinate transformations, and others.)

9. Function Generator With Diode Segments

[Circuit symbol: function generator block with inputs +x_E and −x_E, output x_A = f(x_E). Shown in two equivalent notations.]

10. Special Function Generator With Diode Segments (example)

[Circuit symbol: limiter block with ±δ bounds followed by amplifier with gain −4, output x_A]

x_A = −4 x_E in the linear range; limiting at x_A = ±δ

11. Relay (Comparator)

a) Comparison between x_E1 and x_E2

[Circuit symbol: relay block α with inputs x_E1 and x_E2, two output contacts (o) and (u), output x_A]

The contact rests at (o) as long as x_E1 < α x_E2. (α is set at a built-in potentiometer and is less than one.)

b) Comparison between x_E1 and a fixed voltage

[Circuit symbol: relay block with input x_E, fixed reference ±α, contacts (o) and (u), output x_A]

The contact rests at (o) as long as x_E < (±α).

A. Notes on the Use of the Analog Computer and the Circuit Sheets

The construction and operating principles of the usual components are assumed to be known. Since open amplifiers and servo-multipliers and servo-resolvers are frequently used in the following circuit sheets, a short explanation of these is given here. In addition, the methods used here for normalization and time transformation are explained.

Open Amplifier

An open amplifier is a computing amplifier without feedback:

[Circuit symbol: open amplifier with inputs x_1, x_2, x_3 through weights n_1, n_2, n_3 (∞ = summing point, i.e., grid of the first tube)]

The equivalent circuit is the same as a standard summing amplifier with input resistors R_1, R_2, R_3.

The transfer function of the open amplifier is derived as follows:

[Circuit: amplifier with input currents i_1, i_2, … i_v through resistors R_1, R_2, … R_v to summing point u_g, amplifier gain V, output x_A]

$$i_1 = \frac{x_1 - u_g}{R_1} \tag{1}$$

$$i_2 = \frac{x_2 - u_g}{R_2} \tag{2}$$

$$i_v = \frac{x_v - u_g}{R_v} \tag{3}$$

$$i_1 + i_2 + \cdots + i_v = 0 \tag{4}$$

$$x_A = -V u_g \tag{5} \quad (V = \text{gain of the open amplifier})$$

Substituting equations (1) to (3) into (4):

$$\frac{x_1}{R_1} + \frac{x_2}{R_2} + \cdots + \frac{x_v}{R_v} = u_g \left[\frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_v}\right]$$

[page 12 — continuation:]

and with (5) follows:

$$x_A = -\frac{V}{\dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots + \dfrac{1}{R_v}} \cdot \left[\frac{x_1}{R_1} + \frac{x_2}{R_2} + \cdots + \frac{x_v}{R_v}\right] \tag{6}$$

or, denoting $n_i = \frac{1}{R_i}$ as the input weight and $n_0 = \sum_{i=1}^{v} n_i$:

$$x_A = -\frac{V}{n_0} \sum_{i=1}^{v} n_i x_i \tag{6a}$$

wobei ein konstanter Faktor ist. The constant factor V/n_0 can also be written as V multiplied by a factor smaller than one. Special cases:

If only a single input is present:

$$x_A = -V x_1 \tag{7}$$

The open amplifier accordingly operates as a straightforward (uninverted) amplifier, apart from the sign reversal. At normal computer frequencies the gain of the open loop is usually sufficient (approximately 30 to 42 kΩ for Eingang “1”).

Servo-Multipliers and Servo-Resolvers

Notes on the use of servo-multipliers and servo-resolvers, and important hints:

1. When the direction of the output varies (tracking direction), one should ensure that the slider of the potentiometer moves in the direction that reduces the error Δ = (x − w). However, one can also achieve a reduction rather than a reduction of Δ by connecting the follow-up potentiometer with the wrong polarity — for example, it is possible for the follow-up potentiometer to be connected with the correct polarity of the input variable (z), such that the sign change acts on the product formation:

[Circuit diagrams showing two configurations with ±y, ±z outputs and potentiometer connections, with products −y, +xy, +xz]

2. In servo-resolvers, the linear potentiometers have trimmer potentiometers in the following circuit:

[Circuit diagram: resolver with trimmer potentiometer (Trimmerpot.), slider (Schleifer), and follow potentiometer or multiplications potentiometer (Folgepot. bzw. Multiplikationsp.)]

Because of the trimmer potentiometers, the following circuit, which is possible for servo-multipliers, cannot be implemented for resolvers:

[Circuit: servo-multiplier block with +1 input]

The circuit must be modified for the resolver as follows:

[Circuit: resolver block with center tap (Mittelanzapfung) and +1 input]

[page 14 — continuation:]

As can be seen from the following figures in Part B, servo-multiplier circuit sheets (e.g., pp. 33–42 for the most common use) include the following note:

The potentiometer midpoint tap is connected to +1. However, in order to achieve the correct operating behavior for resolvers, a separate summing point must be inserted. The Sinus/Cosinus potentiometer of the resolver is then used only for loading (−x_A), not for the product.

Symbols:

[Symbol: servo-multiplier block with (x) and (F) labels, and separate summing amplifier]

The Sinus/Cosinus potentiometer for the resolver is used only for loading of an input “−x_A”; therefore connecting a resolver to a ”+” summing point is required.

3. When the slider of a multiplier potentiometer moves to an end, one notes the following: the gain of the following amplifier must be small enough so that the output of the resolver cannot become larger than about ±10–20% above the machine unit.

4. When the slider of a servo-multiplier potentiometer at its end is supplied by the preceding full-load voltage, the potentiometer Schleifer (slider) is held at the same load value as the follow-up potentiometer with identical load:

[Circuit: resolver block (x), (F), with summing amplifier and feedback]

Symbols:

[Symbol diagrams for the above circuit]

The Sinus/Cosinus potentiometer for the resolver is used only for loading of input “−x_A”; each load from a potentiometer from resolver output then requires a separate summing amplifier.

5. When the center tap or another potentiometer is used as a final element, the following applies. The tapped-off section behaves as a loaded potentiometer:

[Circuit: two loaded potentiometers connected in the circuit; equivalent circuit shown]

Because of the non-negligible difference between an open and a loaded potentiometer, the rule applies that one should connect either both or neither of the follow-up potentiometers.

6. With resolvers there is an Automatic Gain Control (AGC): the larger the AGC voltage, the smaller the gain. This is used as a control signal in some circuits to maintain a constant output magnitude. The AGC voltage is such that it is zero (i.e., +0) so that the correction is zero.

7. For use of the circuits, a normalization of the equations is in many cases required.

8. For Abbildung (figure) notes see the following: for individual circuits the normalization of the machine variables requires an electronic null point from the follow-up potentiometer.

[page 16 — Normalization and Time Transformation:]

Normalization and Time Transformation

Since from the literature various methods for normalization and time transformation of equations are known, those used here are briefly given.

Given the equation:

$$a_2 \ddot{x}_A + a_1 \dot{x}_A + a_0 x_A + c = b x_E \tag{9}$$

It is solved after the highest derivative, and integrated:

$$\dot{x}_A = \int_0^t \left[\frac{b}{a_2} x_E - \frac{a_1}{a_2} \dot{x}_A - \frac{a_0}{a_2} x_A - \frac{c}{a_2}\right] dt + \dot{x}_A(0) \tag{10}$$

$$x_A = \int_0^t \dot{x}_A , dt + x_A(0) \tag{11}$$

For each variable, normalization values are introduced that are chosen such that they are greater than or at least equal to the maximum value the variable can assume in the course of the computation; thus, for example:

$$\left|\frac{x_A}{\hat{x}{Am}}\right| \leq 1 \qquad (\hat{x}{Am} = \text{normalization value for } x_A)$$

[page 17 — continuation:]

The normalization values are generally chosen to be equal to the next round (×10) greater-than or equal-to value (Anlehnung an Maschineneinheit). Normalization simplifies the equations considerably. Each new (normalized) variable thereby acquires the symbol of the old variable with a hat (Hut): the new variables are accordingly denoted as x̂_A, ẋ_A, etc. The normalization values become factors in the equations. These factors and the Koeffizient (coefficient) from equation (9) can be combined to form new (dimensionless) coefficients. The time transformation uses new time:

$$\tau = \frac{t}{t_m} \quad \text{or} \quad t = \tau \cdot t_m$$

Führt man diese Beziehungen in die Gleichungen (10) und (11) ein, erhält man:

Inserting these relations into equations (10) and (11) gives:

$$\left(\frac{\dot{x}A}{\hat{x}{Am}}\right) = \int_0^{\tau} \left[\frac{b \cdot t_m \cdot \hat{x}{Em}}{a_2 \cdot \hat{x}{Am}} \cdot \frac{x_E}{\hat{x}{Em}} - \frac{a_1 \cdot t_m}{a_2} \cdot \frac{\dot{x}A}{\hat{x}{Am}} - \frac{a_0 \cdot t_m^2}{a_2} \cdot \frac{x_A}{\hat{x}{Am}} - \frac{c \cdot t_m^2}{a_2 \cdot \hat{x}_{Am}}\right] d\tau + \left(\frac{\dot{x}A(0)}{\hat{x}{Am}}\right) \tag{12}$$

where $\hat{x}_{Am}$ is the normalization for $\dot{x}_A$ (not listed separately here). Here t_m and $\hat{x}$ are the Koeffizient and time scaling, respectively.

$$\frac{x_A}{\hat{x}{Am}} = \int_0^{\tau} \frac{\dot{x}A}{\dot{\hat{x}}{Am}} d\tau + \frac{x_A(0)}{\hat{x}{Am}} \tag{13}$$

Here t_m and $\hat{x}$ are the new (renamed) variables; the Fehler (error) is reduced by the Maschinen-β (machine factor):

$$\left(\frac{\dot{x}A}{\hat{x}{Am}}\right) = \int \left[ \cdots \right] d\tau \quad \text{etc.}$$

Substituting these relations into equations (12) and (13), one obtains:

$$\left(\frac{\dot{x}A}{\dot{\hat{x}}{Am}}\right) = \int_0^{\tau} [\cdots] , d\tau + \frac{\dot{x}A(0)}{\dot{\hat{x}}{Am}} \tag{14}$$

and

$$\frac{x_A}{\hat{x}{Am}} = \int_0^{\tau} \frac{\dot{x}A}{\dot{\hat{x}}{Am}} , d\tau + \frac{x_A(0)}{\hat{x}{Am}} \tag{15}$$

For some circuits where investigation of normalization values of the machine is required, the following guidelines apply:

τ_max ≈ 20
τ_min ≈ 2 (scaled to keep within ±1 machine unit)

[page 18 — machine data / typical component specifications:]

When z is between −1 and +2 schmaler, it becomes:

τ_max = 20
τ_min = 2

Typical specifications for the computing equipment used:

Computing Amplifier (Rechenverstärker):

Gain: 2 · 10⁵
Linearity error: ≤ 0.1%
Input weights (Eingangswertigkeiten): “1”

Two-Parabola Multiplier (Zweiparabel-Multiplikator):

Null error (Nullfehler): ≤ 0.4%
Produktfehler: ≤ 0.4%
Input weights of summing amplifier for Autosummation: approx. 45 kΩ per input “1”
Follow potentiometer (Folgepotentiometer) resistance: 0.1%
Total resistance of the servo system (Totaler Widerstand der Servopotentiometerserie inkl. Plus- und Minusanschluss): approx. 50 kΩ

Servo-Resolver:

Null error of the servo system: 0.1%
Follow potentiometer of multiplier: 0.1%
Follow potentiometer of resolver: 0.1%
Total resistance of the servo-Potentiometerserie: 50 kΩ

Function Generator (Funktionsgeber):

Summation amplifier: required for at least 45 kΩ input “1”

Die Betriebsspannung der Rechner und Funktionsgeber (Operating voltage of the computers and function generators):

10 mA at 100 V DC supply voltage (Gleichspannung)

Reference voltage (Referenzspannung):

100 V

B. CIRCUIT SHEETS

The collection of circuits on the following pages is organized by sections in accordance with the table of contents.

A new section begins with the bold-printed designation »SECTION …« at the head of a page.

Each section in turn comprises several circuits, each beginning with the heading »CIRCUIT …«. Where a circuit spans several pages, these are identified by the small print in the page headers carrying the relevant section and circuit numbers.

The errors stated are always relative errors, namely referenced to the respective nominal value that the quantity under consideration is supposed to have at that moment.

SECTION 1

CIRCUIT 1. Amplifiers with High Weighting Factors

In some cases, the available weighting factors n_v are too small to set a required factor. For summers, the highest weighting factor n is normally 10. For integrators, the highest n-value is usually 100; here one can resort to using a different integration capacitor. When values larger than those mentioned must be realized with summers, the following arrangement is used:

[Figure: schematic showing input −x fed through potentiometer α₁ at weight n₁ into the summing junction of amplifier V, with feedback from the output y via potentiometer α₂ back to the input. A stabilizing capacitor (Beruhigungs-Kapazität) is connected in the feedback path.]

Caption: Feed the output of an open amplifier back via a potentiometer.

$$y \approx \frac{\alpha_1 n_1}{\alpha_2} x = \frac{\alpha_1 R_1}{\alpha_2 R_2} x$$

For values α₂ < 1, weighting factors higher than those normally available can be set. A small stabilizing capacitor (a few pF) may be necessary.

CIRCUIT 2. Absolute-Value Formation with a Servo Multiplier: z = |x| · y

[Figure: schematic showing input +(x/x_m) entering a servo multiplier block labeled (X) with load and (F) = n, driven by +1 and −1 reference voltages. The output of the diode switching network feeds a summer/amplifier whose output is (z/z_m) = |(x/x_m)| · (y/y_m), followed by a potentiometer of weight n.]

$$z_m = x_m \cdot y_m$$

Remarks

When x ≈ 0 but y ≠ 0, the adjustment error of the center tap amplifies. For (x/x_m) > 0.05, the error is normally less than 0.5 to 1%.

SECTION 2

CIRCUIT 1. Differentiation: y = dx/dt

[Figure: schematic of a differentiator. The input −(x/x_m) is connected through capacitor C to the inverting input of a high-gain amplifier (∞). A resistor R is in the feedback path. A small capacitor of approximately 20 pF is placed in parallel with the feedback resistor for stabilization. Output is +(y/y_m).]

Derivation with Normalization

It holds that:

$$\left(\frac{y}{y_m}\right) = \frac{x_m}{y_m} \cdot \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{x}{x_m}\right)$$

Time Transformation

The new time t̄ is t̄ = βt,

$$\mathrm{d}\bar{t} = \beta,\mathrm{d}t.$$

With this it becomes:

$$\left(\frac{y}{y_m}\right) = \frac{x_m}{y_m},\beta,\frac{\mathrm{d}}{\mathrm{d}\bar{t}}\left(\frac{x}{x_m}\right).$$

The circuit forms:

$$\left(\frac{y}{y_m}\right) = CR,\frac{\mathrm{d}}{\mathrm{d}\bar{t}}\left(\frac{x}{x_m}\right), \qquad \text{(for the computer, } \bar{t} \text{ is always the independent variable!)}$$

therefore:

$$CR = \frac{x_m}{y_m},\beta.$$

Remarks

If the noise level is too high, improvements can be achieved by connecting a small capacitor in parallel with R, or a small resistor in series with C.

Approximate differentiation is provided by the circuit of Section 2, Circuit 2: a VD-element (lead-lag element) with very small T.

(page 23 — continuation of Section 2, Circuit 1)

A differentiating element cannot be used when x originates from a multiplier, because in that case a very high noise level is to be expected. This noise level is caused by the following:

a) in two-parabola multipliers, by high-frequency brush noise, b) in mechanical multipliers, by commutator jumps and contact impurities, c) in time-division multipliers, by the incompletely suppressed harmonics.

CIRCUIT 2. Realization of Selected Rational Transfer Functions

[Figure: Table with three columns — “Structural symbol,” “Coupling diagram (Koppelplan),” and “Required normalization of the output quantity for the following input quantities.” Two rows are shown:]

Row 1:

Structural symbol: block with gain k, transfer function G(p) = k / (1 + pT), step input from x_E to x_A.
Coupling diagram: integrator fed from −(x_E/x_{Em}) through potentiometer α₁ n₁ summed with feedback from output (x_A/x_{Am}). Setting: α₁ n₁ = k · x_{Em} / (βT · x_{Am}); α₂ n₂ = 1/(βT).
Output normalization: x_{Am} = k · x_{Em}.

Row 2:

Structural symbol: block with gain k and lead zero, transfer function G(p) = pkT / (1 + pT), step input from x_E to x_A.
Coupling diagram: integrator with additional direct path; three potentiometers. Settings: α₁ n₁ = k · x_{Em} / (βT · x_{Am}); α₂ n₂ = 1/(βT).
Output normalization:
- x_{Am} = k · x_{Em} (when T_1 = T_2)
- x_{Am} = k(T_1/T_2 − 1) x_{Em} (when T_1 > T_2)

(page 25 — continuation of Section 2, Circuit 2)

Row 3 (continued):

Structural symbol: second-order block G(p) = k / (1 + pDT + (pT)²), with input x_E and output x_A.
Coupling diagram: two integrators in series with feedback potentiometers. Settings: α₁ n₁ = k · x_{Em} / (βT · x_{Am}²); α₂ n₂ = 2D / (βT · x_{Am}); α₃ n₃ = 1/(βT).
Output normalization: x_{Am} = 2k · x_{Em} (when D = 0); x_{Am} = k/β · x_{Em} (general).

CIRCUIT 3. General Simulation of a Linear Differential Equation

Given, for example, the equation:

$$a_3 \ddot{x}_A + a_2 \dot{x}_A + a_1 \dot{x}_A + a_0 x_A + c = b_2 \ddot{x}_E + b_1 \dot{x}_E + b_0 x_E$$

with arbitrary x_E(t).

It is assumed that x_A(t = −0), ẋ_A(t = −0), …, ẍ_A(t = −0) = 0 so that the system has stored no values of the disturbance function x_E(t) before it appears. The actual initial conditions x_A(t = +0); ẋ_A(t = +0) and ẍ_A(t = +0) follow automatically from the function x_E(t) at t = +0 and need not be set separately.

To clarify the case that x_A(t = −0); ẋ_A²(t = −0) ≠ 0, reference is made to the literature [13].

Transformation:

$$x_A = \frac{b_2}{a_3} x_E + \int\limits_0^t \left[\frac{b_2}{a_3} x_E - \frac{a_2}{a_3} x_A\right]\mathrm{d}t + \int\limits_0^t \int\limits_0^t \left[\frac{b_1}{a_3} x_E - \frac{a_1}{a_3} x_A\right]\mathrm{d}t,\mathrm{d}t +$$

$$+ \int\limits_0^t \int\limits_0^t \int\limits_0^t \left[\frac{b_0}{a_3} x_E - \frac{a_0}{a_3} x_A - \frac{c}{a_3}\right]\mathrm{d}t,\mathrm{d}t,\mathrm{d}t$$

or:

$$x_A = \frac{b_2}{a_3} x_E + \int\limits_0^t \left[\frac{b_2}{a_3} x_E - \frac{a_2}{a_3} x_A + \int\limits_0^t \left(\frac{b_1}{a_3} x_E - \frac{a_1}{a_3} x_A + \int\limits_0^t \left(\frac{b_0}{a_3} x_E - \frac{a_0}{a_3} x_A - \frac{c}{a_3}\right)\mathrm{d}t\right)\mathrm{d}t\right]\mathrm{d}t.$$

Normalization and Time Transformation:

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{b_2}{a_3} \frac{x_{Em}}{x_{Am}} \left(\frac{x_E}{x_{Em}}\right) + \frac{u_m}{x_{Am}} \left(\frac{u}{u_m}\right),$$

with:

$$\left(\frac{u}{u_m}\right) = \int\limits_0^{\bar{t}} \left[\frac{b_2 x_{Em}}{a_3 \beta \cdot u_m} \left(\frac{x_E}{x_{Em}}\right) - \frac{a_2 x_{Am}}{a_3 \beta u_m} \left(\frac{x_A}{x_{Am}}\right) + \frac{v_m}{\beta u_m} \left(\frac{v}{v_m}\right)\right]\mathrm{d}\bar{t},$$

(page 27 — continuation of Section 2, Circuit 3)

with:

$$\left(\frac{v}{v_m}\right) = \int\limits_0^{\bar{t}} \left[\frac{b_1 x_{Em}}{a_3 \beta \cdot v_m} \left(\frac{x_E}{x_{Em}}\right) - \frac{a_1 x_{Am}}{a_3 \beta w_m} \left(\frac{x_A}{x_{Am}}\right) + \frac{w_m}{\beta v_m} \left(\frac{w}{w_m}\right)\right]\mathrm{d}\bar{t},$$

with:

$$\left(\frac{w}{w_m}\right) = \int\limits_0^{\bar{t}} \left[\frac{b_0 x_{Em}}{a_3 \beta \cdot w_m} \left(\frac{x_E}{x_{Em}}\right) - \frac{a_0 x_{Am}}{a_3 \beta w_m} \left(\frac{x_A}{x_{Am}}\right) - \frac{c}{a_3 \beta w_m}\right]\mathrm{d}\bar{t}.$$

[Figure: full coupling diagram (Koppelplan) for the third-order differential equation simulation. The circuit consists of a chain of four integrators/summers. Input −(x_E/x_{Em}) enters the leftmost summer, and the output +(x_A/x_{Am}) is fed back through the chain. Potentiometers α₁ through α₁₂ with settings n₁ through n₁₂ implement the equation coefficients.]

The potentiometer settings are:

$$\alpha_1 n_1 = \frac{b_2 \cdot x_{Em}}{a_3 \cdot x_{Am}}; \qquad \alpha_5 n_5 = \frac{v_m}{\beta u_m}; \qquad \alpha_9 n_9 = \frac{a_1 x_{Am}}{a_3 \beta w_m};$$

$$\alpha_2 n_2 = \frac{x_{Em}}{x_{Am}}; \qquad \alpha_6 n_6 = \frac{a_2 x_{Am}}{a_3 \beta u_m}; \qquad \alpha_{10} n_{10} = \frac{b_0 x_{Em}}{a_3 \beta w_m};$$

$$\alpha_3 n_7 = \frac{b_1 x_{Em}}{a_3 \beta v_m}; \qquad \alpha_{11} n_{11} = \frac{c}{a_3 \beta w_m};$$

$$\alpha_4 n_4 = \frac{b_2 x_{Em}}{a_3 \beta u_m}; \qquad \alpha_8 n_8 = \frac{w_m}{\beta v_m}; \qquad \alpha_{12} n_{12} = \frac{a_0 x_{Am}}{a_3 \beta w_m}.$$

Using the same schema, any other linear differential equations can be built and normalized.

SECTION 3. Dead-Time Approximation, General

The following scaling refers to the maximum total amplitude (Vollaussteuerung):

$$y_{\max} = x_E(t - T_t) \tag{1}$$

The scaling of following amplifiers depends on the type of subsequent circuit and on the type of input function. The input function x_E is applied at the entry of the circuit; the actual output (dead-time approximation) must not drive subsequent stages into overload.

[Figure (p. 28): time-response graph showing how the approximation output follows the input x_E delayed by the dead time T_t, with annotations for maximum amplitude and overshoot during the interval from the switching instant to T_t.]

The ratio v (= Normierungsverhältnis, normalization ratio) and the time transformation β determine the normalization range in this case. The amplitude of the maximum and minimum during the time interval of the approximation circuit is critical for machine scaling.

The time from maximum to minimum (and vice versa) in the approximation circuit’s transient corresponds to the dead time T_t on the machine (in machine-time units).

CIRCUIT 1. Dead-Time Approximation. Padé Approximation, 1st Order

Transfer function:

$$\frac{x_A}{x_E} = \frac{1 - 0{,}5 \cdot T_t p}{1 + 0{,}5 \cdot T_t p}$$

[Figure: coupling diagram showing input +(x_E/x_{Em}) entering a summer (with potentiometers α₂ and α₃) that feeds into an integrator, whose output +(x₁/x_{1m}) goes through potentiometer α₁ n₁ to a second summer/amplifier (with weight n₁) that produces +(x_A/x_{Am}). The output is also fed back negatively through a potentiometer of gain 1.]

When x_{Am} = x_{Em}:

$$\alpha_1 n_1 = x_{1m},$$

$$\alpha_2 n_2 = \alpha_3 n_3 = \frac{2}{x_{1m} \beta T_l}.$$

Minimum normalization of the intermediate quantity and maximum overshoot:

v	x_{1m}	Max. overshoot during t = 0 to T_t
0.1	2	−0.015 ME *)
0.2	2	−0.03
0.5	2	−0.07
1	2	−0.12
2	2	−0.21
5	2	−0.38
10	2	−0.5

*) 1 ME = full-scale deflection (Vollaussteuerung)

CIRCUIT 2. Dead-Time Approximation. Padé Approximation, 2nd Order

Transfer function:

$$\frac{x_A}{x_E} = \frac{1 - 0{,}5 T_t p + 0{,}0833(T_t p)^2}{1 + 0{,}5 T_t p + 0{,}0833(T_t p)^2}$$

[Figure: coupling diagram with more amplifier stages than Circuit 1, showing input −(x_E/x_{Em}) feeding a cascade of integrators and summers with potentiometers α₁ through α₆ and weights n₁ through n₆, output +(x_A/x_{Am}) via a summing amplifier n₁ at the top right.]

When x_{Am} = x_{Em}:

$$\alpha_1 n_1 = x_{1m};$$

$$\alpha_2 n_2 = \alpha_3 n_3 = \frac{6}{x_{1m} \cdot \beta T_l};$$

$$\alpha_4 n_4 = \frac{x_{2m}}{x_{1m} \beta T_l};$$

$$\alpha_5 n_5 = \alpha_6 n_6 = \frac{12}{x_{1m} \cdot \beta T_l}.$$

Minimum normalization of the intermediate quantities and maximum overshoot:

v	x_{1m}	x_{2m}	Max. overshoot during t = 0 to T_t
0.1	0.1	12.5	±0.006 ME
0.5	0.4	12.5	±0.03
1	0.6	12.5	±0.06
2	1	12.5	±0.1
5	1.2	12.5	±0.2
10	1.2	12.5	±0.3

CIRCUIT 3. Dead-Time Approximation. Padé Approximation, 3rd Order

Transfer function:

$$\frac{x_A}{x_E} = \frac{1 - 0{,}5 T_t p + 0{,}1(T_t p)^2 - 0{,}0083(T_t p)^3}{1 + 0{,}5 T_t p + 0{,}1(T_t p)^2 + 0{,}0083(T_t p)^3}$$

[Figure: coupling diagram extending the 2nd-order circuit with an additional integrator stage and potentiometers α₇ through α₉, n₇ through n₉ for the third-order terms. Input −(x_E/x_{Em}), output +(x_A/x_{Am}).]

When x_{Am} = x_{Em}:

$$\alpha_1 n_1 = x_{1m};$$

$$\alpha_2 n_2 = \alpha_3 n_3 = \frac{12}{x_{1m} \cdot \beta T_l};$$

$$\alpha_4 n_4 = \frac{x_{2m}}{x_{1m} \cdot \beta T_l};$$

$$\alpha_5 n_5 = \alpha_6 n_6 = \frac{60}{x_{2m} \cdot \beta T_l};$$

$$\alpha_7 n_7 = \frac{x_{3m}}{x_{2m} \cdot \beta T_l};$$

$$\alpha_8 n_8 = \alpha_9 n_9 = \frac{120}{x_{3m} \cdot \beta T_l}.$$

Minimum normalization of the intermediate quantities and maximum overshoot:

v	x_{1m}	x_{2m}	x_{3m}	Max. overshoot during t = 0 to T_t
0.1	2	1	125	±0.004 ME
0.5	2	5	125	±0.02
1	2	6	125	±0.04
2	2	10	125	±0.08
5	2	15	125	±0.15
10	2	15	125	±0.2

CIRCUIT 4. Dead-Time Approximation. Padé Approximation, 4th Order

Transfer function:

$$\frac{x_A}{x_E} = \frac{1 - 0{,}5 T_t p + 0{,}107(T_t p)^2 - 0{,}0119(T_t p)^3 + 5{,}95 \cdot 10^{-4}(T_t p)^4}{1 + 0{,}5 T_t p + 0{,}107(T_t p)^2 + 0{,}0119(T_t p)^3 + 5{,}95 \cdot 10^{-4}(T_t p)^4}$$

When x_{Am} = x_{Em}:

$$\alpha_1 n_1 = x_{1m}; \qquad \alpha_5 n_5 = \alpha_6 n_6 = \frac{180}{x_{2m} \cdot \beta T_l}; \qquad \alpha_{10} n_{10} = \frac{x_{4m}}{x_{3m} \cdot \beta T_l};$$

$$\alpha_2 n_2 = \alpha_3 n_3 = \frac{20}{x_{1m} \cdot \beta T_l}; \qquad \alpha_7 n_7 = \frac{x_{3m}}{x_{2m} \cdot \beta T_l}; \qquad \alpha_{11} \alpha_{11} = \alpha_{12} \alpha_{12} = \frac{1680}{x_{4m} \cdot \beta T_l};$$

$$\alpha_4 n_4 = \frac{x_{2m}}{x_{1m} \cdot \beta T_l}; \qquad \alpha_8 n_8 = \alpha_9 n_9 = \frac{840}{x_{3m} \cdot \beta T_l}.$$

Minimum normalization of the intermediate quantities and maximum overshoot:

v	x_{1m}	x_{2m}	x_{3m}	x_{4m}	Max. overshoot during t = 0 to T_t
0.1	0.1	40	20	2000	±0.003 ME
0.5	0.4	40	60	2000	±0.012
1	0.6	40	100	2000	±0.03
2	1	40	150	2000	±0.05
5	1.2	40	200	2000	±0.12
10	1.2	40	200	2000	±0.17

[Page 33: figure only — coupling diagram for the 4th-order Padé dead-time approximation circuit (Section 3, Circuit 4). The diagram shows five cascaded stages, each consisting of a summer amplifier with three potentiometer inputs (labeled n₁ through n₁₂ and α₁ through α₁₂). The input is −(x_E/x_{Em}) at the bottom, output is +(x_A/x_{Am}) at the top, with intermediate normalized signals +(x₁/x_{1m}), −(x₂/x_{2m}), +(x₃/x_{3m}), −(x₄/x_{4m}) appearing at successive stages. An inverter (gain −1) is shown in a secondary feedback path.]

CIRCUIT 5. Dead-Time Approximation. Padé Approximation, 5th Order

Transfer function:

$$\frac{x_A}{x_E} = \frac{1 - 0{,}5 p T_t + 0{,}111(p T_t)^2 - 0{,}0139(p T_t)^3 + 9{,}92 \cdot 10^{-4}(p T_t)^4 - 3{,}306 \cdot 10^{-5}(p T_t)^5}{1 + 0{,}5 p T_t + 0{,}111(p T_t)^2 + 0{,}0139(p T_t)^3 + 9{,}92 \cdot 10^{-4}(p T_t)^4 + 3{,}306 \cdot 10^{-5}(p T_t)^5}$$

When x_{Am} = x_E:

$$\alpha_1 n_1 = x_{1m}; \qquad \alpha_3 n_3 = \frac{420}{x_{2m} \cdot \beta T_l}; \qquad \alpha_{10} n_{10} = \frac{x_{5m}}{x_{3m} \cdot \beta T_l};$$

$$\alpha_2 n_2 = \alpha_3 n_3 = \frac{30}{x_{1m} \cdot \beta T_l}; \qquad \alpha_7 n_7 = \frac{x_{3m}}{x_{2m} \cdot \beta T_l}; \qquad \alpha_{11} n_{11} = \alpha_{12} n_{12} = \frac{60 \cdot 252}{x_{4m} \cdot \beta T_l};$$

$$\alpha_4 n_4 = \frac{x_{2m}}{x_{1m} \cdot \beta T_l}; \qquad \alpha_8 n_8 = \alpha_9 n_9 = \frac{3360}{x_{3m} \cdot \beta T_l}; \qquad \alpha_{13} n_{13} = \frac{x_{5m}}{x_{4m} \cdot \beta T_l};$$

$$\alpha_{14} n_{14} = \alpha_{15} n_{15} = \frac{120 \cdot 252}{x_{5m} \cdot \beta T_l}.$$

Minimum normalization of the intermediate quantities and maximum overshoot:

v	x_{1m}	x_{2m}	x_{3m}	x_{4m}	x_{5m}	Max. overshoot during t = 0 to T_t
0.1	2	3	800	300	30 000	±0.003 ME
0.5	2	10	800	1000	30 000	±0.01
1	2	20	800	2000	30 000	±0.02
2	2	25	800	2500	30 000	±0.04
5	2	35	800	3500	30 000	±0.1
10	2	40	800	3500	30 000	±0.13

[Page 35: figure only — coupling diagram for the 5th-order Padé dead-time approximation circuit (Section 3, Circuit 5). The diagram shows six cascaded stages with potentiometers α₁ through α₁₅ / n₁ through n₁₅. Intermediate signals +(x₁/x_{1m}), −(x₂/x_{2m}), +(x₃/x_{3m}), −(x₄/x_{4m}), +(x₅/x_{5m}) appear at successive integrator outputs. Input is +(x_E/x_{Em}) at the bottom, output +(x_A/x_{Am}) at the top right. Two inverters (gain −1) are shown in the feedback paths.]

CIRCUIT 6. Dead-Time Approximation. Padé Approximation, 2nd Order using Computing Amplifiers and RC Networks

Transfer function:

$$\frac{x_A}{x_E} = \frac{1 - 0{,}5 p T_t + 0{,}0833(p T_t)^2}{1 + 0{,}5 p T_t + 0{,}0833(p T_t)^2}.$$

[Figure: circuit schematic using two operational amplifiers and passive RC networks. Input −(x_E/x_{Em}) connects through resistor R₀ (in series) to the summing junction of the main amplifier (∞), which also receives inputs through resistors R and R₁. Capacitor C₂ is connected between the junction of R and R₁ and the summing junction. A second amplifier (∞) with capacitor C₁ in its input and resistor R in its feedback forms an auxiliary network. The main amplifier has feedback resistor R₀. Output is +(x_A/x_{Am}).]

When x_{Am} = x_{Em}:

$$R = k_0 \cdot T_t,$$

$$C_1 = \frac{0{,}333}{k_0},$$

$$C_2 = \frac{0{,}25}{k_0},$$

$$R_1 = R_0 \cdot 0{,}333.$$

T_t is to be substituted in seconds. k₀ and R₀ are freely selectable. When R₀ is substituted in MΩ and k₀ as 10⁺⁶, R and R₁ appear in MΩ and C₁ and C₂ in μF.

SECTION 4

Circuit 1. Division x/y, sign of y restricted

[page 37: figure — Circuit diagram (a): The input −(x/x_m) is fed through attenuator α_1 into an open amplifier (gain n_1, feedback gain n_2) with approximately 50 pF stabilization capacitor. The output drives one input of a servo multiplier. The servo wiper position drives a unity-gain sign follower. Outputs: +(z/z_m) when y is positive, −(z/z_m) when y is negative. The second servo input receives +(y/y_m) when y is positive only, or −(y/y_m) when y is negative only.]

or b)

[page 37: figure — Circuit diagram (b): Same topology as (a) but with approximately 100 pF capacitor and a 1 MΩ servo multiplier block labeled (x)/(F). Outputs: +(z/z_m) for positive y only, −(z/z_m) for negative y only; +(y/y_m) for positive y only, −(y/y_m) for negative y only.]

or c)

[page 38: figure — Circuit diagram (c): The input −(z/z_m) and the signals +1/−1 (when y is positive only) and −1 (when y is negative only) are fed to a block containing elements AGC/(x)/(F) and a “Last&n” stage. Output passes through a unity-gain stage marked “0 −1 bzw. +1”. Outputs: +(y/y_m) and −(y/y_m).]

Required normalization: z_m = x_max / y_max

In all circuits the variables may also be introduced with the opposite sign from that shown, provided the circuits are adapted accordingly.

Derivation and Normalization

(for the circuits with the exception of those in parentheses)

Because of the open amplifier:

$$-!\left(\frac{z}{z_{\max}}\right) = \frac{V}{n_1+n_2}\left[-\alpha_1 n_1\left(\frac{x}{x_{\max}}\right) + n_2\left(\frac{y}{y_{\max}}\right)\left(\frac{z}{z_{\max}}\right)\right],$$

$$\left(\frac{z}{z_{\max}}\right) = \frac{\alpha_1 n_1!\left(\dfrac{x}{x_{\max}}\right)}{\dfrac{n_1+n_2}{V}+n_2!\left(\dfrac{y}{y_{\max}}\right)}.$$

For V >> 1:

$$\left(\frac{z}{z_{\max}}\right) \approx \frac{\alpha_1 n_1}{n_2} \cdot \frac{!\left(\dfrac{x}{x_{\max}}\right)}{!\left(\dfrac{y}{y_{\max}}\right)}.$$

Since z = x/y must hold, the condition

$$\frac{\alpha_1 n_1}{n_2} = \frac{x_{\max}}{y_{\max} \cdot z_{\max}}$$

must be satisfied.

[page 39: text — partially legible]

When z_max = x_max / y_min is chosen, it follows that α_1 n_1 = y_min / y_max.

One can choose z_max = x_max / y_max, so that α_1 n_1 = 1 can be set (which simplifies adjustment). To obtain less amplifier overdrive and better resolution for large signals, however, z_max = x_max / y_min is preferable, at the cost of a smaller setting α_1 n_1 = y_min / y_max.

Remarks

Circuit a with a two-parabola multiplier has an error down to (x/x_m) α_1 n_1 ≥ 0.025, approximately 2%; below this the error rises to about 10–20%.

The inaccuracy of circuit b is down to (x/x_m) α_1 n_1 ≥ 0.005, approximately 1%; below this the error rises. At x = 0 the value (z/z_m) is influenced by the drift voltage of the open amplifier.

It is not recommended to use the circuit for (z/z_m) — only the servo output. As soon as this signal is no longer required, the AGC can be omitted, leaving only the servo output available.

The inaccuracy of (z/z_m) is then determined purely by the servo characteristics. If AGC is present, the circuit for generating u works only with greater errors (see also Section 4, Circuit 1); therefore w is also heavily affected by errors (approximately 5%, for small values of (u/u_m) < 0.2, possibly up to 10%).

Furthermore (x/x_m) is restricted to (x/x_m) < (y/y_m).

This circuit is not recommended.

Circuit 2. Division x/y, signs of x and y arbitrary

[page 40: figure — Circuit diagram (a): Input +(z/z_m) feeds a servo multiplier block (x)/(1 MΩ) with ±1 switching. The output passes through a 20 pF-stabilized amplifier (gain Ω_a) to a sign-selecting stage. Further summer networks V and W produce the output +(z/z_m). Input −(z/z_m) enters the loop. Bottom input −(z/z_m) connects back. The connection point labeled −(x/x_m) at lower left.]

[page 40: figure — Circuit diagram (b): Similar to (a) but simplified; input +(z/z_m) feeds block (x)/(1 MΩ) with +1/0/−1 switching; 20 pF capacitor on amplifier Ω_a. Output +(z/z_m). Bottom of loop labeled −(z/z_m) and entry −(x/x_m).]

Derivation and Normalization for Circuit a

Because of the open amplifier:

$$\alpha_1 n_1\left(\frac{x}{x_m}\right) \approx |\alpha|\left(\frac{u}{u_m}\right); \quad (\alpha = \text{wiper position})$$

Since α = (y/y_m), it follows that:

$$\left(\frac{u}{u_m}\right) = \alpha_1 n_1\frac{!\left(\dfrac{x}{x_m}\right)}{|\alpha|} = \alpha_1 n_1 \cdot \frac{!\left(\dfrac{x}{x_m}\right)}{!\left|\left(\dfrac{y}{y_m}\right)\right|}.$$

[page 41]

Furthermore −v ≈ w, where

$$v = -\left(\frac{u}{u_m}\right)\alpha, \quad w = |\alpha|\left(\frac{z}{z_m}\right).$$

Therefore:

$$\left(\frac{z}{z_m}\right) = \frac{\alpha}{|\alpha|}\left(\frac{u}{u_m}\right) = \alpha_1 n_1 \cdot \frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)} \cdot \frac{!\left(\dfrac{y}{y_m}\right)}{!\left|\left(\dfrac{y}{y_m}\right)\right|}.$$

This, however, is equal to

$$\left(\frac{z}{z_m}\right) = \alpha_1 n_1 \cdot \frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)}.$$

Required setting for α_1 n_1:

$$\alpha_1 n_1 = \frac{x_{\max}}{y_{\max} \cdot z_{\max}}.$$

When z_m = x_max / y_min, it follows that α_1 n_1 = y_min / y_max.

Remarks

Circuit a has errors of up to 20% when (y/y_m) < 0.05 (setting and tracking errors); above this, only approximately 0.5–2% error, largely independent of (x/x_m).

Circuit b is to be preferred over a because fewer multiplication potentiometers are needed and setting errors are therefore less noticeable. Down to (y/y_m) ≥ 0.1 the result can be computed with errors smaller than 0.5%; for smaller (y/y_m) the error increases to approximately 5%. The quantity (x/x_m) does not enter into the accuracy, apart from very small values (α_1 n_1 (x/x_m) < 0.005).

A brief overdrive of the circuit at a sign change of (y/y_m) is naturally unavoidable. If chopper-stabilized computing amplifiers are used, these must be artificially limited to approximately ±1 ME (machine unit), because otherwise, due to the recovery time of the amplifier after an overdrive, transient computing errors arise.

Circuit 3. Division x/y, signs of x and y arbitrary

[page 42: figure — Circuit diagram (top): Input +(z/z_m) feeds servo multiplier block (x)/(1 MΩ) with +1/0/−1 switching and a +(x/x_max) reference; the servo output u goes to amplifier A. Output is +(z/z_m). Input −(z/z_m) also present.]

[page 42: figure — Circuit diagram (bottom, “or”): Alternative using two cross-multipliers. Inputs +(z/z_max) and +(x/x_max) feed respective multiplier blocks and a further two-multiplier block; output u to amplifier A; output +(z/z_max). Reference −(z/z_max).]

A warning is given regarding the use of two-parabola multipliers due to the large inaccuracy of the result (see Remarks).

Derivation and Normalization

Because of the open amplifier:

$$-u = \frac{V}{3}\left[\gamma\left(\frac{x}{x_m}\right) + ku - \left(\frac{y}{y_m}\right)\left(\frac{z}{z_m}\right)\right]. \tag{1}$$

[page 43]

$$u = \frac{!\left(\dfrac{z}{z_m}\right)!\left(\dfrac{y}{y_m}\right)-\gamma!\left(\dfrac{x}{x_m}\right)}{\dfrac{3}{V}+k}. \tag{2}$$

Furthermore, neglecting the servo settling time:

$$-\left(\frac{z}{z_m}\right) = \int u\left(\frac{y}{y_m}\right) dt = \int n\left(\frac{y}{y_m}\right) \cdot \frac{!\left(\dfrac{z}{z_m}\right)!\left(\dfrac{y}{y_m}\right)-\gamma!\left(\dfrac{x}{x_m}\right)}{\dfrac{3}{V}-k} , dt, \tag{3}$$

or rewritten:

$$\left[\frac{3}{V}+k\right]\frac{d}{dt}\left(\frac{z}{z_m}\right) + n\left(\frac{y}{y_m}\right)^2\left(\frac{z}{z_m}\right) = \gamma n\left(\frac{x}{x_m}\right)\left(\frac{y}{y_m}\right). \tag{4}$$

In the steady state:

$$\left(\frac{z}{z_m}\right) = \gamma \cdot \frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)}.$$

Since z = x/y is to be formed, the setting

$$\gamma = \frac{x_{\max}}{y_{\max} \cdot z_{\max}}$$

must be made.

When z_max = x_max / y_min is chosen, it follows that γ = y_min / y_max.

Remarks

A capacitor cannot be used in place of the potentiometer k; otherwise the circuit oscillates.

The settling time is influenced by k, n, y and the settling time of the servo, and for V → ∞ from equation (4) with a slight rearrangement:

$$\frac{k}{n\left(\dfrac{y}{y_m}\right)^2} \cdot \frac{d}{dt}\left(\frac{z}{z_m}\right) + \left(\frac{z}{z_m}\right) = \gamma\left(\frac{x}{x_m}\right)\left(\frac{y}{y_m}\right).$$

[page 44 — partially legible]

The settling time depends on k, n, y, and the servo settling time exponentially on y. This means that for small y, the settling becomes very slow. However, the open amplifier’s voltage is not excessively overdriven even for small (y/y_m), which is a positive feature.

It is nevertheless possible to make the circuit work for small (y/y_m) as well.

The inaccuracy of the results for (x/x_m) ≥ 0.05 is approximately 2%, below which the error rises, but is practically independent of (x/x_m).

Using two-parabola multipliers (generally for (y/y_m) ≥ 0.005, approximately 1%), below which the error rises strongly. At x = 0, (z/z_m) is influenced by the drift voltage of the open amplifier.

Circuit 4. Multiplication/Division and/or Reciprocal Formation with a Servo Multiplier: u = z · x/y and/or w = y/x · v

[page 45: figure — Circuit diagram: Input +(z/z_m) feeds a servo multiplier block (AGC)/(x)/(F)/(1 MΩ) with −(z/z_m) also present. A +(y/y_m) input and −(y/y_m) reference drive the servo. The servo wiper position feeds an amplifier (gain n_1) with approximately 50 pF capacitor and attenuator α_1 forming a feedback loop to −(y/y_m). The output −(w/w_m) is taken from the amplifier. The output labeled +(u/u_m) is taken from the servo wiper via AGC.]

Signs of x, z, and v: arbitrary. Signs of y: restricted to positive only or negative only (swap connections at the follow-up potentiometer and at (z/z_m) for sign reversal).

Derivation and Normalization

The servo multiplier automatically sets:

$$\left(\frac{x}{x_m}\right) = \alpha\left(\frac{y}{y_m}\right), \quad (\alpha = \text{wiper position})$$

Also:

$$\left(\frac{u}{u_m}\right) = \left(\frac{z}{z_m}\right)\cdot\frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)},$$

with the required normalization:

$$u_m = z_m \cdot \frac{x_m}{y_m}.$$

Furthermore:

$$-\left(\frac{w}{w_m}\right) = \frac{V}{1+n_1}\left[-\alpha_1 n_1\left(\frac{v}{v_m}\right)+|\alpha|\left(\frac{w}{w_m}\right)\right].$$

[page 46]

From this, for V → ∞:

$$\left(\frac{w}{w_m}\right) = \alpha_1 n_1 \cdot \frac{!\left(\dfrac{v}{v_m}\right)}{|\alpha|},$$

$$\alpha = \frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)}.$$

Therefore:

$$\left(\frac{w}{w_m}\right) = \alpha_1 n_1 \left|\frac{!\left(\dfrac{y}{y_m}\right)}{!\left(\dfrac{x}{x_m}\right)}\right| \left(\frac{v}{v_m}\right).$$

α_1 n_1 is to be set as follows:

$$\alpha_1 n_1 = \frac{y_m \cdot v_m}{x_m \cdot w_m}.$$

Remarks

The circuit for generating u works only with greater errors (see also Section 4, Circuit 1); therefore w is also heavily affected by errors (approximately 5%, for small values of (u/u_m) < 0.2, possibly up to 10%).

Furthermore, (x/x_m) is restricted to (x/x_m) < (y/y_m).

This circuit is not recommended.

Circuit 5. Square Root z = √x

(Cube root and others via further multiplications in the feedback path)

[page 47: figure — Circuit diagram (a): Inputs −(x/x_m) and +(x/x_m) (in square brackets) feed attenuator α_1 and open amplifier with gains n_1, n_2 and approximately 50 pF capacitor. Amplifier output drives a cross multiplier block labeled [−][+]. Feedback through unity-gain stage back to amplifier. Outputs: +(z/z_m) and [−(z/z_m)]. The signs in square brackets indicate alternatively usable sign connections.]

The signs in square brackets may optionally be used as alternatives.

[page 47: figure — Circuit diagram (b): Inputs −(x/x_m) and +(x/x_m) feed attenuator α_1 and open amplifier (n_1, n_2) with 20 pF capacitor. Output drives a servo multiplier block (x)/(F)/(Last & n_2). The servo output passes through sign-selecting stages [−]/[+] and [−]/[+] in cascade, with +1/[−1] switching. Outputs +(z/z_m) and [−(z/z_m)].]

[page 47: figure — Circuit diagram (c): Input +(x/x_m) feeds a block (AGC)/(x)/(F) via 1 MΩ resistor. Output (z/z_m) passes through a unity-gain stage then a summing junction. Feedback to the multiplier input. Result: (z/z_m) = √(x/x_m); required normalization z_m = √x_m.]

$$\left(\frac{z}{z_m}\right) = \sqrt{\left(\frac{x}{x_m}\right)}; \quad \text{required normalization } z_m = \sqrt{x_m}$$

Derivation and Normalization

For circuits a and b:

$$-\left(\frac{z}{z_m}\right) = \frac{V}{n_1+n_2}\left[-\alpha_1 n_1\left(\frac{x}{x_m}\right)+n_2\left(\frac{z}{z_m}\right)^2\right],$$

$$\left(\frac{z}{z_m}\right) = \frac{\alpha_1 n_1!\left(\dfrac{x}{x_m}\right)}{\dfrac{n_1+n_2}{V}+n_2!\left(\dfrac{z}{z_m}\right)}.$$

For V >> 1 this gives:

$$\left(\frac{z}{z_m}\right) \approx \sqrt{\frac{\alpha_1 n_1}{n_2}}\sqrt{\left(\frac{x}{x_m}\right)}.$$

Since (z/z_m) = (√x_m / z_m) · √(x/x_m) must hold, the condition

$$\frac{\alpha_1 n_1}{n_2} = \frac{x_m}{z_m^2}$$

must be satisfied.

For circuit c:

The servo multiplier automatically sets:

$$\left(\frac{x}{x_m}\right) = \alpha\left(\frac{z}{z_m}\right),$$

and additionally (z/z_m) = α.

Therefore:

$$\left(\frac{z}{z_m}\right) = \sqrt{\left(\frac{x}{x_m}\right)}.$$

Required normalization:

$$z_m = \sqrt{x_m}.$$

Remarks

For circuits a and b: It is preferable to set n_2 = 1, because the larger n_2 is, the more poorly the multiplier is driven; furthermore, for n_2 > 1 the settling capacitance must be increased.

[page 49]

When a resolver is used instead of a servo multiplier, circuit b must be modified accordingly (see under Section A).

The inaccuracy of circuit a is down to (x/x_m) α_1 n_1 ≥ 0.025, approximately 2%; below this the error rises to about 10–20%.

The inaccuracy of circuit b is down to (x/x_m) α_1 n_1 ≥ 0.005, approximately 1%; below this the error rises. At x = 0, (z/z_m) is influenced by the drift voltage of the open amplifier.

For circuit c:

The inaccuracy of the circuit is down to (x/x_m) ≥ 0.05, approximately 2%; below this the error rises. AGC is necessary only for values (x/x_m) ≥ 0.3; it can be omitted entirely but for smaller values of (x/x_m) this greatly increases the error at small values. When AGC is present, it can happen during the computing pause at x = 0 that the servo rotates through zero. In such cases, AGC is permissible and the servo gain may be slightly reduced.

Circuit 6. Square Roots w = ±√|x| and z = sign x · √|x|

[page 50: figure — Circuit diagram: Input −(z/z_m) feeds attenuator α_1, amplifier n_1, then into a servo multiplier block (x)/(F)/(1 MΩ) with approximately 20 pF capacitor. The servo output passes through a summing stage with +1 and −1 inputs. Outputs: +1 (top right) and +(w/w_m) (far right). Input +(y/y_m) via AGC shown at upper right.]

Derivation and Normalization

Because of the open amplifier:

$$-\left(\frac{z}{z_m}\right) = \frac{V}{1+n_1}\left[-\alpha_1 n_1\left(\frac{z}{z_m}\right)+\left(\frac{u}{u_m}\right)\right],$$

$$\left(\frac{u}{u_m}\right) = \text{sign}, z\left(\frac{z}{z_m}\right)^2.$$

Therefore:

$$\left(\frac{z}{z_m}\right)\left[1+\frac{V}{1+n_1}\left(\frac{z}{z_m}\right)\text{sign}, z\right] = \frac{V}{1+n_1}\alpha_1 n_1,$$

and for V >> 1:

$$\left(\frac{z}{z_m}\right)^2 = \frac{1}{\text{sign}, z},\alpha_1 n_1,$$

and since the circuit dictates sign z = sign x, and x is also real for negative x:

$$\left(\frac{z}{z_m}\right) = \text{sign}, x\cdot\sqrt{\alpha_1 n_1}\sqrt{\left|\left(\frac{x}{x_m}\right)\right|},$$

where

$$\sqrt{\alpha_1 n_1} = \frac{\sqrt{x_m}}{z_m}, \quad \text{i.e.,} \quad \alpha_1 n_1 = \frac{x_m}{z_m^2}.$$

[page 51]

The quantity (w/w_m) is, according to the circuit:

$$\left(\frac{w}{w_m}\right) = \left|\left(\frac{z}{z_m}\right)\right|,$$

so that it follows:

$$\left(\frac{w}{w_m}\right) = +\frac{\sqrt{x_m}}{z_m}\sqrt{\left|\left(\frac{x}{x_m}\right)\right|}. \quad \text{Required normalization: } w_m = z_m.$$

Remarks

The inaccuracy of the circuit is, for (x/x_m) · α_1 n_1 > 0.005, approximately 1%; below this the error increases.

At x = 0, (z/z_m) is influenced by the drift voltage of the open amplifier.

Circuit 7. Square Root with Various Operations under the Radical

$$z = \sqrt{x \cdot y}; \quad u = \sqrt{\frac{x}{y}}; \quad z = \sqrt{v^2 - w^2}.$$

x and y must have the same sign. The circuit is given for x and y positive.

[page 52: figure — Circuit diagram: Input +(x/x_m) feeds servo multiplier block (AGC)/(x)/(F)/(1 MΩ), labeled w. Output feeds a summing stage receiving +(z/z_m) through a unity-gain stage. Further summing stages receive −(y/y_m) and +1. Output +(u/u_m) at far right.]

Derivation and Normalization

The multiplier automatically sets:

$$\left(\frac{x}{x_m}\right) = w$$

Furthermore:

$$w = \alpha\left(\frac{z}{z_m}\right); \quad \alpha = \text{wiper position}$$

$$\left(\frac{z}{z_m}\right) = \alpha\left(\frac{y}{y_m}\right).$$

From this it follows:

$$\alpha = \sqrt{\frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)}} = \left(\frac{u}{u_m}\right)$$

and

$$\left(\frac{z}{z_m}\right) = \sqrt{\left(\frac{x}{x_m}\right)\cdot\left(\frac{y}{y_m}\right)}.$$

The required normalization follows from the following consideration. It is desired that:

a) u = √(x/y)

[page 53]

$$\left(\frac{u}{u_m}\right) = \frac{1}{u_m}\sqrt{\frac{x_m}{y_m}}\sqrt{\frac{!\left(\dfrac{x}{x_m}\right)}{!\left(\dfrac{y}{y_m}\right)}},$$

therefore:

$$u_m = \sqrt{\frac{x_m}{y_m}}.$$

b) z = √(x · y)

$$\left(\frac{z}{z_m}\right) = \frac{\sqrt{x_m \cdot y_m}}{z_m}\sqrt{\left(\frac{x}{x_m}\right)\cdot\left(\frac{y}{y_m}\right)},$$

therefore:

$$z_m = \sqrt{x_m \cdot y_m}.$$

When (x/x_m) = (v + w)/x_m and (y/y_m) = (v − w)/y_m are entered, the result is:

$$\left(\frac{z}{z_m}\right) = \sqrt{\frac{v^2 - w^2}{x_m \cdot y_m}}$$

Required normalization:

$$z_m = \sqrt{x_m \cdot y_m}.$$

Remarks

The inaccuracy of the circuit is, for (x/x_m) > 0.05, approximately 2%; below this the error increases, but is practically independent of y. The same applies to AGC as stated in Section 4, Circuit 5. Furthermore, (x/x_m) is restricted to (x/x_m) < (y/y_m).

Circuit 8. Formation of Roots of Higher Degree and Fractional Powers

of the form u = ⁿ√(x^m), (m < n), and z = ⁿ√x

[page 54: figure — Circuit diagram: Input −(z/z_m) feeds attenuator α_1 and amplifier (n_1, n_2) with approximately 50 pF capacitor. Amplifier output drives a servo multiplier block (x)/(F)/(1 MΩ). The output is labeled u_a. The servo output passes through a chain of n unity-gain or inverting stages labeled 1, 2, 3, …, n (with four diamond-shaped inverter symbols shown). Each stage feeds the next through summing junctions. Output: (z/z_m).]

Derivation and Normalization

The servo multiplier automatically sets:

$$\alpha_1 n_1\left(\frac{x}{x_m}\right) = u_a.$$

It is further the case that:

$$u_n = \alpha^{n-1}\left(\frac{z}{z_m}\right), \quad (\alpha = \text{potentiometer setting})$$

therefore:

$$\left(\frac{z}{z_m}\right) = \alpha_1 n_1 \cdot \frac{1}{\alpha^{n-1}};$$

and since α = (z/z_m), it follows:

$$\left(\frac{z}{z_m}\right)^n = \sqrt[n]{\alpha_1 n_1}\sqrt[n]{\left(\frac{x}{x_m}\right)}.$$

Required normalization:

$$\alpha_1 n_1 = \frac{x_m}{z_m^n}.$$

Section 4, Circuit 8 (continued)

z_m can be chosen freely within the constraint z_m >= n-th root of x_m. For the normalization of an intermediate quantity, the following holds:

(u / u_m) = [ n-th root of ( alpha_1 * n_1 * (x / x_m) ) ]^m

w_m = n-th root of ( (x_m / (alpha_1 * n_1))^m ).

If alpha_1 * n_1 = x_m / z_m^n is taken into account, it follows that

w_m = z_m^m.

Remarks

Both z and the intermediate quantities ( alpha_1 * n_1 * (x / x_m) )^((n-1)/n) are computed with good accuracy using a carefully adjusted servo multiplier; specifically (z / z_m) with an inaccuracy <= 0.5%, and all other quantities with an inaccuracy <= 1%.

Section 5

Circuit 1. Broken rational function z = ax / (1 + by), for by > (-1) and sign of x arbitrary, (b > 0)

[page 56: two circuit diagrams shown]

First circuit variant: An operational amplifier with potentiometers alpha_1 (gain n_1) and alpha_2 (gain n_2) drives a multiplier. The output +(z/z_m) is fed back through the multiplier. The input is -(x/x_m). A summation point feeds a reference +(y/y_m).

Second (alternative) circuit variant: Uses -(x/x_m) as input, potentiometers alpha_1 (n_1) and alpha_2 (n_2), a unity-gain inverter, a summing junction, a servo element with a 1 MΩ load denoted (X) and (F), with reference voltages +1 and -1, giving output +(y/y_m).

Derivation and Normalization

The following holds:

-(z / z_m) = -alpha_1 * n_1 * (x / x_m) + alpha_2 * n_2 * (y / y_m) * (z / z_m),

(z / z_m) = [ alpha_1 * n_1 * (x / x_m) ] / [ 1 + alpha_2 * n_2 * (y / y_m) ].

Since z = ax / (1 + by) is required, it follows that

alpha_1 * n_1 = a * (x_m / z_m),

alpha_2 * n_2 = b * y_max.

z_max can be chosen freely within the following limits:

z_max >= a * x_max / (1 + b * y_min) for by > 0,

z_max >= a * x_m / (1 - b * |y_N|) for (-1) < by < 0,

where y_N is to be the most negative value that y can assume during the computation.

Remarks

For by < (-1) the circuit is overdriven or unstable.

When alpha_2 * n_2 * (y / y_m) becomes larger than approximately 1, a damping (settling) capacitor must be used.

Accuracy of the circuit better than 1% when using servo multipliers or two-parabola multipliers.

Section 5, Circuit 2. Broken rational function z = ax / (1 - by), for by < 1 and sign of x arbitrary, (b > 0)

[page 58: two circuit diagrams shown]

First circuit variant: Input -(x/x_m) feeds an amplifier with potentiometers alpha_1 (n_1) and alpha_2 (n_2). A multiplier is connected in the feedback path. Output is +(z/z_m). The multiplier lower input connects to -(y/y_m).

Second (alternative) circuit variant: Input -(x/x_m), potentiometers alpha_1 (n_1) and alpha_2 (n_2), a unity-gain inverter, a summing junction with a servo multiplier (1 MΩ load, (X) and (F)), reference voltages +1 and -1, giving output +(y/y_m).

Derivation and Normalization

The following holds:

-(z / z_m) = -alpha_1 * n_1 * (x / x_m) - alpha_2 * n_2 * (y / y_m) * (z / z_m),

(z / z_m) = [ alpha_1 * n_1 * (x / x_m) ] / [ 1 - alpha_2 * n_2 * (y / y_m) ].

Since z = ax / (1 - by) is required, it follows that

alpha_1 * n_1 = a * (x_m / z_m),

alpha_2 * n_2 = b * y_max.

z_max can be chosen freely within the following limits:

z_max >= a * x_max / (1 - b * y_max) for 0 < by < 1,

z_max >= a * x_max / (1 + b * |y_min|) for by < 0.

Remarks

For by > 1 the circuit is overdriven. When alpha_2 * n_2 * (y / y_m) becomes larger than approximately 1, a damping capacitor must be used. Accuracy of the circuit better than 1% with servo or two-parabola multipliers.

Section 5, Circuit 3. Formation of broken rational functions

The circuit generates functions of the following type:

s = x / ( ±k_1y ± k_2u ± … ),

v = (x * u) / ( ±k_1y ± k_2u ± … ) = s * u,

w = (+x * y) / ( ±k_1y ± k_2u ± … ) = s * y,

etc.

The sign of the overall denominator is restricted.

General Circuit

[page 60: block diagram shown]

The general circuit uses input +(x/x_m) fed into a block labeled (X) / (F) (representing a servo multiplier with load), whose output drives a chain of summing junctions. The summing junctions receive ±(y/y_m), ±(u/u_m), etc., and also ±1. The output of the chain is +(s/s_m). The intermediate outputs of the servo multiplier (proportional to the denominator terms) are labeled ±(w/w_m), ±(v/y_m), fed through n_1, n_2, …, n_m to the main amplifier (Hauptverstärker).

Specific Circuit Variants for Different Cases

a. s = x / (k_1y + k_2u + …) Denominator > 0

[circuit diagram: input +x, servo block (X)/(F), summing junctions with -y, -u, -1 at top and +y, +u, +1 at bottom, output +s; feedback amplifier z with inputs -w, -v]

b. s = x / (k_1y + k_2u + …) Denominator < 0

[circuit diagram: input +x, servo block (X)/(F), summing junctions with +y, +u, -1 at top and -y, -u, +1 at bottom, output +s; feedback amplifier z with inputs -w, -v]

c. s = x / (-k_1y - k_2u - …) Denominator > 0

[circuit diagram: input +x, servo block (X)/(F), summing junctions with +y, +u, +1 at top and -y, -u, -1 at bottom, output +s; feedback amplifier z with inputs +w, +v]

d. s = x / (-k_1y - k_2u - …) Denominator < 0

[circuit diagram: input +x, servo block (X)/(F), summing junctions with -y, -u, -1 at top and +y, +u, +1 at bottom, output +s; feedback amplifier z with inputs +w, +v]

e. s = x / (k_1y - k_2u) Denominator > 0

[circuit diagram: input +x, servo block (X)/(F), summing junctions with -y and +u at top, +y and -u at bottom, output +s; separate summing circle with +1 and -1 for s; feedback amplifier z with -w, +y, -u]

Derivation and Normalization (for Circuit a)

The multiplier automatically sets:

(x / x_m) = (z / z_m).

Furthermore:

-(z / z_m) = -n_1 * alpha * (y / y_m) - n_2 * alpha * (u / u_m) - …

From both equations, the potentiometer position alpha follows:

alpha = (x / x_m) / [ n_1 * (y / y_m) + n_2 * (u / u_m) + … ].

Furthermore:

(v / v_m) = alpha * (u / u_m) = [ (x / x_m) * (u / u_m) ] / [ n_1 * (y / y_m) + n_2 * (u / u_m) + … ],

etc.

Normalization Conditions:

n_1 = k_1 * y_m; n_2 = k_2 * u_m; etc.

s_m = x_m = z_m;

v_m = x_m * u_m; w_m = x_m * y_m; etc.

Remarks

As indicated in the circuit, n_1 = n_2 = n_m must hold so that the servo potentiometers all carry the same load; otherwise additional amplifiers and potentiometers must be switched between the servo outputs and the inputs of the main amplifier.

Accuracy is better than 1% for (y / y_m), (u / u_m), … > 0.1. AGC cannot be used.

Section 5, Circuit 4. Interpolation

The following is computed:

u = w + (v - w) * (x - z) / (y - z)

[page 63: graph showing linear interpolation between points (z, w) and (y, v), with u the interpolated value at abscissa x]

[circuit diagram: input +(x/x_m) feeds a servo multiplier block (X)/(F) with slider output s; two summing junctions combine +(y/y_m) with +(z/z_m) and +(v/v_m) with +(w/w_m) respectively; a subtraction yields the interpolated output +(u/u_m) via an amplifier]

Derivation and Normalization

The servo multiplier automatically sets:

(x / x_m) = s.

Furthermore:

s = (z / z_m) + [ (y / y_m) - (z / z_m) ] / 2 * (1 + alpha) = (x / x_m)

(alpha = slider position measured from center).

From this it follows:

1 + alpha = 2 * [ (x / x_m) - (z / z_m) ] / [ (y / y_m) - (z / z_m) ].

Furthermore:

(u / u_m) = (w / w_m) + [ (v / v_m) - (w / w_m) ] / 2 * (1 + alpha)

and substituting the above expression for alpha:

(u / u_m) = (w / w_m) + [ (v / v_m) - (w / w_m) ] * [ (x / x_m) - (z / z_m) ] / [ (y / y_m) - (z / z_m) ].

This corresponds, for example, to the computation of an interpolated value:

[graph: axes show (z/z_m), (x/x_m), (y/y_m) on horizontal axis and (w/w_m), (u/u_m), (v/v_m) on vertical axis, with the interpolated point marked]

Remarks

The obvious conditions are:

(y / y_m) > (z / z_m),

(z / z_m) < (x / x_m) < (y / y_m).

The error depends on (y / y_m) and (z / z_m). The smaller (y / y_m) and (z / z_m) are, the larger the error. For (y / y_m) and (z / z_m) > 0.1 the error is approximately 1 to 3%. AGC cannot be used.

Section 6

Circuit 1. Formation of the functions u = y · tan x and z = y / cos x

a) for (-90°) < x < 90°

[circuit diagram: input +(x/x_m) feeds a servo block (X)/(F) with a Last (load) connection; the servo output connects to a summing junction that adds +1 and -1 reference to drive a sin/cos resolver; the cos output drives an amplifier with gain n_1 and potentiometer alpha_1 (approx. 20 pF stabilization capacitor shown); output is -(y/y_m); the sin output gives +(u/u_m)]

b) for 90° < x < 180°

[circuit diagram: same arrangement as (a) but with an additional unity-gain inverter (gain factor 1) inserted in the cos feedback path; output remains -(y/y_m) with +(u/u_m) from sin tap]

Derivation and Normalization

Because of the open amplifier, the following holds:

alpha_1 * n_1 * (y / y_m) = (z / z_m) * cos x.

From this it follows:

(z / z_m) = alpha_1 * n_1 * (y / y_m) / cos x.

It is required that:

z = y / cos x.

The normalization conditions are therefore:

x_m = 1.11 pi or 200°,

alpha_1 * n_1 = y_m / z_m.

If z_m = y_m / (cos x)_min is chosen, it follows that:

alpha_1 * n_1 = (cos x)_min.

At the sin tap, the following is then available:

(u / u_m) = (z / z_m) * sin x = alpha_1 * n_1 * (y / y_m) * tan x = (y_m / z_m) * (y / y_m) * tan x.

So that u = y * tan x, the necessary normalization is:

u_m = z_m.

Remarks

For x = 70° accuracy is better than 2%; for x > 70° the error increases. At x = 80°, for example, accuracy is only approximately 8%.

Section 6, Circuit 2. Formation of the functions u = y · cot x and z = y / sin x

a) for 0 < x < 180°

[circuit diagram: input +(x/x_m) feeds a servo block (X)/(F) with Last; summing junction adds +1 and -1; sin/cos resolver outputs drive an amplifier with n_1 and potentiometer alpha_1 (approx. 20 pF stabilization capacitor shown); cos output gives +(u/u_m); output of amplifier is -(y/y_m)]

b) for 0 > x > (-180°)

[circuit diagram: same as (a) but with an additional unity-gain inverter (factor 1) inserted in the path before the amplifier; cos output gives +(u/u_m); output is -(y/y_m)]

Derivation and Normalization

Because of the open amplifier, the following holds:

alpha_1 * n_1 * (y / y_m) = (z / z_m) * sin x.

From this it follows:

(z / z_m) = alpha_1 * n_1 * (y / y_m) / sin x.

The necessary normalization is:

x_m = 1.11 pi or 200°,

alpha_1 * n_1 = y_m / z_m.

If z_max = y_m / (sin x)_min is chosen, it follows that:

alpha_1 * n_1 = (sin x)_min.

At the cos tap the following is then available:

(u / u_m) = (z / z_m) * cos x = alpha_1 * n_1 * (y / y_m) * cot x.

Required normalization:

u_m = z_m.

Remarks

Accuracy better than 2% for |x| > 20°. Below |x| <= 20° the error rises sharply and can reach, for example, approximately 8% at x = 10°.

Section 6, Circuit 3. Formation of the functions z = u · arcsin(x/y) and w = u · arccos(x/y)

a) u arcsin(x/y)

[circuit diagram: input +(x/x_m) feeds an arcsine resolver block (arcsin, (F)); the sin output of the resolver drives a sin/cos element; output is -(z/z_m)]

b) u arccos(x/y)

[circuit diagram: input +(x/x_m) feeds an arccosine resolver block (arccos, (F)); the cos output of the resolver drives a sin/cos element; output is +(w/w_m)]

Derivation and Normalization

The servo multiplier automatically sets:

(x / x_m) = v,

v = (y / y_m) * sin alpha (alpha = slider position).

From this it follows:

alpha = arcsin [ (x / x_m) / (y / y_m) ];

and:

(z / z_m) = (u / u_m) * alpha = (u / u_m) * arcsin [ (x / x_m) / (y / y_m) ].

Required normalization:

x_m = y_m = u_m * 1.11 pi or z_m = u_m * 200°,

depending on whether z is to be expressed in radians or degrees.

If instead of the sin tap the cos tap of the resolver F is used, one obtains:

(w / w_m) = (u / u_m) * arccos [ (x / x_m) / (y / y_m) ]

with the same normalization conditions as above.

Remarks

The functions arcsin and arccos are multivalued.

[page 70: graph showing the multivalued nature of arcsin and arccos, indicating the usable ranges of ±90° for arcsin and 0° to -180° for arccos]

When forming arcsin, only the range ±90° is used; at all times y must be positive:

for positive x: 0 <= q <= 90°,
for negative x: 0 >= q >= (-90°)

(q = angular position of the sin/cos potentiometer).

When forming arccos, only the range 0° to -180° is used (in the range 0 to +180° the servo cannot stably balance; see Remarks in Part A), and y must always be positive:

for positive x: 0 <= q <= -90°,
for negative x: (-90°) >= q >= (-180°).

(At the terminals for u, the output of arccos is displayed as positive, however.)

The following applies to both circuits:

The use of AGC in the manner described is expedient. The AGC supply voltage must always be positive. When using AGC, amplification and damping of the servo must be re-adjusted.

Accuracy of both circuits lies between 1 and 3%, with the exception of the ranges around:

±90° for arcsin,
0° and -180° for arccos,

where the error rises sharply.

For (y / y_m) < 0.1 the error rises sharply even with AGC.

Section 6, Circuit 4. Formation of the functions z = u · arctg(x/y) and w = u · arcctg(x/y)

a) z = u arctg(x/y)

[circuit diagram: a servo resolver block (X)/(AGC)/Last/(F) with sin and cos outputs; the sin output connects to a summing junction receiving -(y/y_m) and -(x/x_m); the cos output connects to a summing junction receiving +(y/y_m) and +(x/x_m); additional +(u/u_m) input; the resolver is driven by a unity-gain inverter (factor 1); output is +(z/z_m)]

b) w = u arcctg(x/y)

[circuit diagram: same as (a) but with inputs -(x/x_m) and -(y/y_m) swapped relative to case a; output is +(w/w_m)]

A detailed description for both circuits is given under Coordinate Transformation: Section 7, Circuit 1.

Section 7

Circuit 1. Coordinate Transformation, Rectangular Coordinates to Polar Coordinates

Given: x; y

To compute: r; phi.

[graph: Cartesian coordinate system showing a vector of length r at angle phi from the x-axis to the point (x, y)]

r = sqrt(x^2 + y^2); phi = arctg(y/x)

[circuit diagram: input -(x/x_m) feeds a resolver block (X)/(F)/(AGC) denoted Last; the resolver sin and cos outputs are connected to two-resolver arrangement; the first resolver drives summing circles receiving -(x/x_m) and providing the feedback signal -(x/x_m)·sin(phi); a second resolver receives -(y/y_m); its cos output provides -(y/y_m)·cos(phi) = -(r/r_m)·cos^2(phi); this feeds into a unity-gain amplifier giving +(r/r_m); simultaneously a summing circle with +1 and -1 reference provides phi as (phi/200°)]

Derivation and Normalization

The resolver automatically sets:

(x / x_m) * sin z = (y / y_m) * cos z,

where z is the position of the sliders.

Therefore:

z = arctg [ (y / y_m) / (x / x_m) ], … (1)

Section 7, Circuit 1 (continued)

and

$$z = \varphi \quad \text{provided that} \quad x_m = y_m. \tag{2}$$

φ is to be normalized to 200° or 1.11π.

Furthermore, according to the circuit:

$$\left(\frac{r}{r_m}\right) = \frac{1}{2}\left[\left(\frac{x}{x_m}\right)\cos z + \left(\frac{y}{y_m}\right)\sin z\right]. \tag{3}$$

Since

$$\sin\varphi = \frac{y}{r} \quad \text{and} \quad \cos\varphi = \frac{x}{r},$$

it follows from (2) and with the preliminary assumption from (4):

$$\sin z = \sin\varphi = \frac{\left(\dfrac{y}{y_m}\right)}{2\left(\dfrac{r}{r_m}\right)}; \qquad \cos z = \cos\varphi = \frac{\left(\dfrac{x}{x_m}\right)}{2\left(\dfrac{r}{r_m}\right)}.$$

Substituting into (3) gives:

$$\left(\frac{r}{r_m}\right)^2 = \frac{\left(\dfrac{x}{x_m}\right)^2}{4\left(\dfrac{r}{r_m}\right)} + \frac{\left(\dfrac{y}{y_m}\right)^2}{4\left(\dfrac{r}{r_m}\right)},$$

$$\left(\frac{r}{r_m}\right) = \frac{1}{2}\sqrt{\left(\frac{x}{x_m}\right)^2 + \left(\frac{y}{y_m}\right)^2}.$$

Hence, since $r = \sqrt{x^2 + y^2}$, it follows that

$$x_m = y_m = \frac{r_m}{2} \quad \text{must hold.} \tag{4}$$

If one were to choose $r_m = x_m = y_m$, the direct feedback of the summing amplifier would be eliminated, but this amplifier would then be driven with $\left(\dfrac{y}{x_m}\right) = 1$ and simultaneously overdriven.

The following quantities are available at the various brushes of potentiometers 1 and 2:

Pot. 1 sin-brush:

$$-\left(\frac{x}{x_m}\right)\sin\varphi = -\left(\frac{r}{r_m}\right)\sin\varphi\cos\varphi,$$

Pot. 1 cos-brush:

$$-\left(\frac{x}{x_m}\right)\cos\varphi = -\left(\frac{r}{r_m}\right)\cos^2\varphi,$$

Pot. 2 sin-brush:

$$-\left(\frac{y}{y_m}\right)\sin\varphi = -\left(\frac{r}{r_m}\right)\sin^2\varphi,$$

since

$$\left(\frac{y}{y_m}\right) = \left(\frac{r}{r_m}\right)\sin\varphi.$$

Pot. 2 cos-brush:

$$-\left(\frac{y}{y_m}\right)\cos\varphi = -\left(\frac{r}{r_m}\right)\sin\varphi\cos\varphi.$$

Remarks

When AGC is used, the gain and damping of the resolver must be readjusted in comparison to operation without AGC. A relative error of approximately 1% in the computation of r and φ is to be expected.

For small $\left(\dfrac{r}{r_m}\right)$, errors are large because of the AGC — the amplification is set very high for large $\left(\dfrac{r}{r_m}\right)$ — the gain becomes too weak, and therefore larger errors occur when $\left(\dfrac{r}{r_m}\right) < 0.05$ (error approximately 3%). It is even less favorable if, in addition to a small $\left(\dfrac{r}{r_m}\right)$, the angle φ lies in the range 0° to 180° or 180° to 360°, since errors in φ and r of up to 20% are then possible.

The coordinate conversion is only possible for $(-180°) \leq \varphi \leq 180°$ due to the nature of the AGC. An angle of +270° is interpreted as, for example, −90°. If one were to use an n-turn potentiometer, one could also tap off $\varphi = \pm n \cdot 180°$.

Section 7, Circuit 2 — Coordinate Transformation: Polar to Rectangular Coordinates

Given:

$$r_1; \quad \varphi,$$ $$r_2; \quad \varphi,$$

To compute:

$$x_1; \quad y_1,$$ $$x_2; \quad y_2.$$

[page 75: figure showing block diagram with resolver units]

Normalization

$$r_{1m} = x_{1m} = y_{1m},$$ $$r_{2m} = x_{2m} = y_{2m},$$ $$\varphi_m = 200° \quad \text{or} \quad 1.11\pi.$$

Remarks

For arbitrary φ and r, the accuracy is better than 1%.

Section 7, Circuit 3 — Coordinate Transformation: Rotation of the Coordinate System

Given:

$$u; \quad v; \quad \varphi,$$

To compute:

$$x; \quad y.$$

[page 76: figure showing rotated coordinate axes and block diagram]

Derivation and Normalization

The relationships between two mutually rotated coordinate systems are given by:

$$x = u\cos\varphi + v\sin\varphi,$$ $$y = -u\sin\varphi + v\cos\varphi;$$

normalized:

$$\left(\frac{x}{x_m}\right) = \left(\frac{u}{u_m}\right)\cos\varphi + \left(\frac{v}{v_m}\right)\sin\varphi,$$

$$\left(\frac{y}{y_m}\right) = -\left(\frac{u}{u_m}\right)\sin\varphi + \left(\frac{v}{v_m}\right)\cos\varphi.$$

It is necessarily the case that $x_m = y_m = u_m = v_m$. Amplifier input gains other than 1 are not possible, since the sin/cos potentiometers are wound for a load of 1 MΩ.

Remarks

Accuracy for arbitrary values of φ, u, v is better than 1%, with the exception of value combinations where x or y is supposed to be zero (e.g., φ = ±45°; u = v = 1 should yield y or x = 0, but this is erroneous due to unavoidable alignment errors of the servo potentiometers, etc.).

Section 8, Circuit 1 — Generation of y = a·sin(ωt + φ) and z = a·cos(ωt + φ) with Amplitude Stabilization

[page 78: figure showing block diagram of the circuit]

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\frac{\omega}{\beta},t + \varphi\right); \qquad \left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\frac{\omega}{\beta},t + \varphi\right)$$

Derivation and Normalization with Time Transformation

First, the circuit is considered without branches 1 and 2. The functions

$$y = a\sin(\omega t + \varphi) \tag{1}$$

$$z = a\cos(\omega t + \varphi) \tag{2}$$

are to be generated.

By differentiation one obtains:

$$\dot{y} = \omega a\cos(\omega t + \varphi) = \omega z,$$ $$\dot{z} = -\omega a\sin(\omega t + \varphi) = -\omega y,$$

or:

$$y = \int_0^t \omega z,dt + y(0); \qquad y(0) = a\sin\varphi, \tag{3}$$

$$z = \int_0^t -\omega y,dt + z(0); \qquad z(0) = a\cos\varphi. \tag{4}$$

In normalized form and with a time transformation $\bar{t} = \beta\cdot t$, equations (3) and (4) become:

$$\left(\frac{y}{y_m}\right) = \int_0^{\bar{t}} \frac{\omega}{\beta},\frac{z_m}{y_m}\left(\frac{z}{z_m}\right)d\bar{t} + \left(\frac{a\sin\varphi}{y_m}\right), \tag{5}$$

$$\left(\frac{z}{z_m}\right) = \int_0^{\bar{t}} -\frac{\omega}{\beta},\frac{y_m}{z_m}\left(\frac{y}{y_m}\right)d\bar{t} + \left(\frac{a\cos\varphi}{z_m}\right). \tag{6}$$

From these two relations the given circuit arises without branches 1 and 2. Since the circuit delivers:

$$\left(\frac{y}{y_m}\right) = \int_0^{\bar{t}} \alpha_1 n_1 \left(\frac{z}{z_m}\right)d\bar{t} + z_4,$$

$$-\left(\frac{z}{z_m}\right) = \int_0^{\bar{t}} \alpha_2 n_2 \left(\frac{y}{y_m}\right)d\bar{t} + z_3,$$

the required settings of α_ν n_ν follow as:

$$\alpha_1 n_1 = \frac{\omega}{\beta}\frac{z_m}{y_m}, \qquad \alpha_2 n_2 = \frac{\omega}{\beta}\frac{y_m}{z_m},$$

$$\alpha_3 = \frac{a\cos\varphi}{z_m}, \qquad \alpha_4 = \frac{a\sin\varphi}{y_m}, \qquad \alpha = \frac{a}{y_m}.$$

Normally one makes $y_m = z_m$.

Amplitude Stabilization — Branches 1 and 2

Through the positive feedback via branch 1, the naturally present damping of the amplitude of $\left(\dfrac{y}{y_m}\right)$ and thereby also of $\left(\dfrac{z}{z_m}\right)$ can be eliminated. The setting δ is made so that the amplitudes of $\left(\dfrac{y}{y_m}\right)$ and $\left(\dfrac{z}{z_m}\right)$ tend to grow slightly with time (δ approx. 0.002).

The dead zone — branch 2 — is set so that $z = \dfrac{a}{y_m}$. As soon as the amplitude of $\left(\dfrac{y}{y_m}\right)$ exceeds its set value, branch 2 introduces damping via small cross-coupling δ of y, so that overall the amplitude is held at the value a or $\dfrac{a}{y_m}$ and remains constant. Too large a δ produces nonlinear distortions of the sine waveform.

Remarks

The circuit holds the amplitude constant to better than 1% even for smallest values of α_ν n_ν, provided δ was carefully set by experiment.

Section 8, Circuit 2 — Generation of y = a·sin(∫ω(t) dt + φ) and z = a·cos(∫ω(t) dt + φ) with Amplitude Stabilization

[page 80: figure showing block diagram]

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$\left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$\alpha_1 n_1 = \frac{\omega_m}{\beta}\frac{z_m}{y_m}; \qquad \alpha_2 n_2 = \frac{\omega_m}{\beta}\frac{y_m}{z_m}$$

$$\alpha_3 = \frac{a}{z_m}\cos\varphi; \qquad \alpha_4 = \frac{a}{y_m}\sin\varphi$$

$$\alpha = \frac{a}{y_m}$$

δ approx. 0.002 ÷ 0.005 to be set by experiment; see Remarks.

Circuit with Servo Multipliers for Rapid Changes in ω(t)

[page 81: figure showing block diagram with servo multipliers]

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right); \qquad \frac{z}{z_m} = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$n_3 n_4 = n_1 = \frac{\omega_m}{\beta}; \qquad y_m = z_m$$

$$\alpha_3 = \frac{a\cdot\cos\varphi}{z_m} = \frac{a}{y_m}\cos\varphi$$

$$\alpha_4 = \frac{a\cdot\sin\varphi}{y_m} = \frac{a}{z_m}\sin\varphi$$

$$\alpha = \frac{a}{y_m}$$

δ approx. 0.002 ÷ 0.005 to be set by experiment.

Circuit with Servo Multiplier for ω(t) ≤ ω, (ω = Corner Frequency of the Servo)

[page 82: figure showing block diagram]

$$\left(\frac{y}{y_m}\right) = \left(\frac{a}{y_m}\right)\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right); \qquad \left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$n_3 n_1 = n_1 = \frac{\omega_m}{\beta}; \qquad y_m = z_m \quad \text{(condition)}$$

$$\alpha_3 = \frac{a\cos\varphi}{z_m}$$

$$\alpha_4 = \frac{a\sin\varphi}{y_m}$$

$$\alpha = \frac{a}{y_m}$$

δ approx. 0.002 ÷ 0.005 to be set by experiment. The input resistances of n_1 and n_3 must be equal.

Derivation and Normalization with Time Transformation

The circuit using electronic multipliers is first considered without branches 1 and 2.

The functions

$$y = a\sin\left(\int_0^t \omega(t),dt + \varphi\right) = a\sin(z(t) + \varphi), \tag{1}$$

$$z = a\cos\left(\int_0^t \omega(t),dt + \varphi\right) = a\cos(z(t) + \varphi) \tag{2}$$

are to be generated.

By differentiation one obtains:

$$\dot{y} = \omega(t)\cos\left(\int_0^t \omega(t),dt + \varphi\right) = \omega z,$$

$$\dot{z} = -\omega(t)\sin\left(\int_0^t \omega(t),dt + \varphi\right) = -\omega y,$$

or:

$$y = \int_0^t \omega(t)z,dt + y(0); \qquad y(0) = a\sin\varphi, \tag{3}$$

$$z = \int_0^t -\omega(t)y,dt + z(0); \qquad z(0) = a\cos\varphi. \tag{4}$$

In normalized form and with a time transformation $\bar{t} = \beta\cdot t$, equations (3) and (4) become:

$$\left(\frac{y}{y_m}\right) = \int_0^{\bar{t}} \frac{\omega_m^2\omega}{\omega_m\beta}\left(\frac{\omega}{\omega_m}\right)\left(\frac{z}{z_m}\right)d\bar{t} + \frac{a\sin\varphi}{y_m}, \tag{5}$$

$$\left(\frac{z}{z_m}\right) = -\int_0^{\bar{t}} \frac{\omega_m z_m}{\omega_{m0}\beta}\left(\frac{\omega}{\omega_m}\right)\left(\frac{y}{y_m}\right)d\bar{t} + \frac{a\cos\varphi}{z_m}. \tag{6}$$

From these two relations the given circuit arises without branches 1 and 2.

An analysis of the circuit (without these branches) yields:

$$\left(\frac{y}{y_m}\right) = \int_0^{\bar{t}} \alpha_1 n_1 \left(\frac{\omega}{\omega_m}\right)\left(\frac{z}{z_m}\right)d\bar{t} + \alpha_4, \tag{7}$$

$$-\left(\frac{z}{z_m}\right) = \int_0^{\bar{t}} \alpha_2 n_2 \left(\frac{\omega}{\omega_m}\right)\left(\frac{y}{y_m}\right)d\bar{t} + \alpha_3. \tag{8}$$

From equations (5) to (8) it follows:

$$\alpha_1 n_1 = \frac{\omega_m}{\beta}; \qquad \alpha_2 n_2 = \frac{\omega_m}{\beta}.$$

Normally one makes $y_m = z_m$.

The potentiometer settings for the amplitude stabilization are given analogously to those in Section 8, Circuit 1.

If $\left(\dfrac{a}{y_m}\right) \approx 0.1$ and α_ν n_ν are not too small, the amplitude control holds the amplitude constant to better than 1%. The error of the amplitude stabilization depends on the magnitude of the electronic multipliers and the potentiometer settings. With electronic multipliers $\left(\dfrac{a}{y_m}\right) \approx 0.5$ can be achieved without degrading the waveform.

Section 8, Circuit 3 — Generation of y = a·sin(∫ω dt + φ) and z = a·cos(∫ω dt + φ) with Amplitude Stabilization

[page 85: figure showing block diagram]

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},dt + \varphi\right)$$

$$\left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},dt + \varphi\right)$$

$$y_m = z_m \quad \text{(condition)}$$

$$z_1 n_1 = \frac{\omega_m}{\beta}$$

$$\alpha_3 = \frac{a}{z_m}\cos\varphi; \qquad \alpha_4 = \frac{a}{y_m}\sin\varphi$$

$$\delta \approx 0.010$$

For this relay it should be noted that $\pm\left(\dfrac{a}{y_m}\right)$ u.U. can be ±100 V, and it must be checked whether the contacts can cause a short circuit when ±100 V is applied (contact spacing and contact arcing!).

Circuit with Servo Multipliers

[page 86: figure showing block diagram with servo multipliers]

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right);$$

$$\left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right);$$

$$y_m = z_m \quad \text{(condition)}$$

$$n_1 = \frac{\omega_m}{\beta}$$

$$\alpha_3 = \frac{a}{z_m}\cos\varphi; \qquad \alpha_4 = \frac{a}{y_m}\sin\varphi; \qquad \delta \approx 0.010.$$

For slow changes in $\dfrac{\omega}{\beta}$, the circuit can be built with only one servo multiplier; see also Section 8, Circuit 2.

Mode of Operation of the Circuit without Amplitude Stabilization

With proper alignment of the cos function, the amplitude of $\left(\dfrac{y}{y_m}\right)$ starts at a value $\left(\dfrac{a}{y_m}\right)$; this value does not remain constant but decreases slowly. The reason for this is the minute unavoidable deviation of the cos function from the ideal. A small deviation already causes the amplitude to decrease appreciably. This effect is corrected by branches 1 and 2, which accomplish the amplitude stabilization.

At proper alignment of α_ν n_ν, the amplitude stabilization holds the amplitude to better than 1% even for small values of α_ν n_ν, provided δ was carefully set by experiment. With electronic multipliers, the amplitude stabilization keeps the value constant to better than approximately 1% even for small potentiometer settings.

Section 8, Circuit 4 — Generation of y = a·sin(∫ω dt + φ) and z = a·cos(∫ω dt + φ) with Amplitude Stabilization (Amplitude a variable)

[page 88: figure showing block diagram]

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$\left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$y_m = z_m \quad \text{(condition)}$$

$$\alpha_1 n_1 = \frac{\omega_m}{\beta}; \qquad \alpha_2 = \frac{a}{z_m}\cos\varphi; \qquad \alpha_4 = \frac{a}{y_m}\sin\varphi$$

$$\alpha_5 = \frac{a^2}{y_m^2} = \frac{a^2}{z_m^2}$$

(For ω = const, the multipliers marked with * can be omitted.)

Circuit with Servo Multipliers

[page 89: figure showing block diagram with servo multipliers]

Note: Upper corner frequency of the servos must be observed!

$$\left(\frac{y}{y_m}\right) = \frac{a}{y_m}\sin\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right); \qquad \left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos\left(\int_0^{\bar{t}}\frac{\omega}{\beta},d\bar{t} + \varphi\right)$$

$$y_m = z_m \quad \text{(condition)}$$

$$n_1 = \frac{\omega_m}{\beta}$$

$$\alpha_3 = \frac{a}{z_m}\cos\varphi; \qquad \alpha_4 = \frac{a}{y_m}\sin\varphi; \qquad \alpha_5 = \frac{a^2}{y_m^2} = \frac{a^2}{z_m^2}$$

Derivation and Normalization

For the analysis of the circuit it is expedient to proceed as described in the source [9]. The output quantities of the individual components are linked by the following relations:

$$-\frac{d}{d\bar{t}}\left(\frac{z}{z_m}\right) = \alpha_1 n_1 u + n_2 r_2,$$

$$-\frac{d}{d\bar{t}}\left(\frac{y}{y_m}\right) = \alpha_1 n_1 x + n_2 r_1,$$

$$x = -\left(\frac{\omega}{\omega_m}\right)\left(\frac{z}{z_m}\right),$$

$$u = \left(\frac{\omega}{\omega_m}\right)\left(\frac{y}{y_m}\right),$$

$$r_1 = -q\left(\frac{y}{y_m}\right); \qquad -q = \left(\frac{z}{z_m}\right)^2 + \left(\frac{y}{y_m}\right)^2 - \alpha_5;$$

$$r_2 = -q\left(\frac{z}{z_m}\right).$$

Combining these equations gives:

$$-\frac{d}{d\bar{t}}\left(\frac{z}{z_m}\right) = \alpha_1 n_1 \left(\frac{\omega}{\omega_m}\right)\left(\frac{y}{y_m}\right) + n_2 \left(\frac{z}{z_m}\right)\left[\left(\frac{z}{z_m}\right)^2 + \left(\frac{y}{y_m}\right)^2 - \alpha_5\right] \tag{1}$$

$$-\frac{d}{d\bar{t}}\left(\frac{y}{y_m}\right) = -\alpha_1 n_1 \left(\frac{\omega}{\omega_m}\right)\left(\frac{z}{z_m}\right) + n_2 \left(\frac{y}{y_m}\right)\left[\left(\frac{z}{z_m}\right)^2 + \left(\frac{y}{y_m}\right)^2 - \alpha_5\right] \tag{2}$$

Since the circuit is to yield $y = a\sin\left(\int_0^{\bar{t}}\dfrac{\omega}{\beta},d\bar{t} + \varphi\right)$ and $z = a\cos\left(\int_0^{\bar{t}}\dfrac{\omega}{\beta},d\bar{t} + \varphi\right)$, the following trial solution is substituted for the solution of the above equations:

$$\left(\frac{z}{z_m}\right) = R(t)\cos\left(\Phi(t) + \varphi\right), \tag{3}$$

$$\left(\frac{y}{y_m}\right) = R(t)\sin\left(\Phi(t) + \varphi\right). \tag{4}$$

Since

$$R(t) = \frac{a}{y_m} = \frac{a}{z_m}, \qquad (y_m = z_m) \tag{5}$$

Section 8, Circuit 4 (continued)

and

$$\Phi(t) = \int_{0}^{t} \frac{\omega}{\beta}, dt \tag{6}$$

one then obtains the desired functions

$$z = a \cos!\left(\int_{0}^{t} \frac{\omega}{\beta}, dt + \varphi\right) \quad \text{and} \quad y = a \sin!\left(\int_{0}^{t} \frac{\omega}{\beta}, dt + \varphi\right).$$

Substituting (3) and (4) into (1) and (2) yields

$$-R’\cos(\Phi+\varphi) + R\dot\Phi’\sin(\Phi+\varphi) = \alpha_1 n_1!\left(\frac{\omega}{\omega_m}\right) R\sin(\Phi+\varphi) + n_z R\cos(\Phi+\varphi),[R^2 - \alpha_5], \tag{7}$$

$$-R’\sin(\Phi+\varphi) - R\dot\Phi’\cos(\Phi+\varphi) = -\alpha_1 n_1!\left(\frac{\omega}{\omega_m}\right) R\cos(\Phi+\varphi) + n_z R\sin(\Phi+\varphi),[R^2 - \alpha_5]. \tag{8}$$

Multiplying (7) by $\sin(\Phi+\varphi)$ and (8) by $\cos(\Phi+\varphi)$ and subtracting gives

$$R\dot\Phi’ = \alpha_1 n_1!\left(\frac{\omega}{\omega_m}\right) R$$

$$\dot\Phi’ = \int_{0}^{t} \alpha_1 n_1!\left(\frac{\omega}{\omega_m}\right) dt. \tag{9}$$

Multiplying (7) by $\cos(\Phi+\varphi)$ and (8) by $\sin(\Phi+\varphi)$ and adding gives

$$-R’ = n_2 R(R^2 - \alpha_5). \tag{10}$$

Equation (9) is identical to (6) when

$$\alpha_1 n_1 = \frac{\omega_m}{\beta} \quad \text{is made to hold.}$$

This requirement is thereby fulfilled.

Now (10) is further examined: Instead of $R$ write $\varepsilon = R^2 - \alpha_5$ in (10):

$$\frac{d}{dt}\sqrt{(\varepsilon + \alpha_5)} = \frac{1}{2}(\varepsilon + \alpha_5)^{-\frac{1}{2}}\frac{d\varepsilon}{dt} = -n_2 \varepsilon \sqrt{(\varepsilon + \alpha_5)}.$$

Section 8, Circuit 4 (continued)

Since $\varepsilon \ll \alpha_5$ will hold approximately,

$$\frac{d\varepsilon}{d\bar{t}} = -n_2 \cdot 2\varepsilon\alpha_5.$$

The solution of this equation yields

$$\varepsilon = k, e^{-2n_2\alpha_5 \bar{t}}.$$

For $\bar{t} = 0$, $\varepsilon$ equals the initial error $\varepsilon_0$:

$$\varepsilon = \varepsilon_0, e^{-2n_2\alpha_5 \bar{t}},$$ $$R^2 = \varepsilon_0, e^{-2n_2\alpha_5 \bar{t} + \alpha_5}. \tag{11}$$

One sees that with large $n_2$ the error $\varepsilon$ is quickly corrected, and then

$$R^2 = \alpha_5.$$

Setting

$$\alpha_5 = \left(\frac{a}{y_m}\right)^2 = \left(\frac{a}{z_m}\right)^2,$$

so at vanishing $\varepsilon$ one actually obtains

$$\left(\frac{y}{y_m}\right) = \left(\frac{a}{y_m}\right)\sin!\left(\int_{0}^{t} \frac{\omega}{\beta}, d\bar{t} + \varphi\right),$$

$$\left(\frac{z}{z_m}\right) = \frac{a}{z_m}\cos!\left(\int_{0}^{t} \frac{\omega}{\beta}, d\bar{t} + \varphi\right),$$

$$y_m = z_m \quad \text{is a prerequisite.}$$

$\alpha_5 = \dfrac{a^2}{y_m^2}$ thus determines the amplitude of the oscillation.

Furthermore,

$$\alpha_1 n_1 = \frac{\omega_m}{\beta}.$$

$n_2$ determines the “settling time” and is therefore to be made as large as possible.

$$\alpha_3 = \frac{a}{z_m}\cos\varphi,$$

$$\alpha_4 = \frac{a}{y_m}\sin\varphi.$$

Section 8, Circuit 4 (continued)

For the circuit with servo-multipliers this derivation also applies, but $\alpha_1 = 1$ must be chosen.

Remarks

The amplitude is — even when electronic multipliers are used — stable for all values $\alpha_1 n_1$ and $\left(\dfrac{\omega}{\omega_m}\right)$ down to approximately 2 per mille. When using electronic multipliers it must however be noted that at $\left(\dfrac{\omega}{\omega_m}\right) \approx 0$ considerable errors in the frequency occur, whereas the circuit with servo-multipliers works exactly up to $\left(\dfrac{\omega}{\omega_m}\right) = 0$.

When a variable voltage is applied to $\alpha_5$, the amplitude can also be varied at will. However, the settling time from one amplitude to another depends very strongly on $n_2$; for $n_2 = 10$ the settling time is approximately 1 s for a step change in $\alpha_5$, for $n_2 = 100$ approximately 0.1 s. For small values of $n_2$ the settling time increases somewhat.

Regarding loading of the follow-up potentiometer (i.e., $n_1$ or $n_3$): it must be set to $n_1$ (i.e., $n_3$). It can happen that $\sin^2 + \cos^2 = 1$ is no longer formed exactly, but the error in the worst case is 4%; instead 2% is normal. A run-out error of the servo-potentiometer before $n_2$ is negligible.

Even for step changes of $\left(\dfrac{\omega}{\omega_m}\right)$, the amplitude is held stable within the stated limits.

Section 8, Circuit 5

Generation of Three-Phase Voltage with Constant Frequency and Amplitude Stabilization

A three-phase voltage $y_1 = a\sin(\omega t + \varphi)$, $y_2 = a\sin(\omega t + \varphi - 120°)$, $y_3 = a\sin(\omega t + \varphi + 120°)$ is to be generated. The following circuit is to be used.

[page 94: figure — block diagram of the three-phase oscillator circuit]

Derivation and Normalization

Derivation Using Phase Lead with Frequency

A single-stage lag element with gain $k = 2$ and time constant

$$T = \frac{\sqrt{3}}{\omega}$$

is used. The condition

$$\frac{k}{\sqrt{1+(\omega T)^2}} = 1, \quad \text{gives with } \omega T = \sqrt{3} \text{ the value } k = 2.$$

Every lag element with gain $k = 2$ and time constant $T = \dfrac{\sqrt{3}}{\omega}$ therefore produces from a given sinusoidal oscillation

$$y = a\sin(\omega t + \varphi)$$

an oscillation

$$y_1 = a\sin(\omega t + \varphi - 60°) = -a\sin(\omega t + \varphi + 120°).$$

Taking into account the sign inversion of the computing amplifier, the circuit thus generates three mutually phase-shifted sinusoidal oscillations of equal frequency and amplitude. In normalized form and with the time transformation $\bar{t} = \beta \cdot t$, one obtains

$$\left(\frac{y_1}{y_m}\right) = \frac{a}{y_m}\sin!\left(\frac{\omega}{\beta},\bar{t} + \varphi\right),$$

$$\left(\frac{y_2}{y_m}\right) = \frac{a}{y_m}\sin!\left(\frac{\omega}{\beta},\bar{t} + \varphi + 120°\right),$$

$$\left(\frac{y_3}{y_m}\right) = \frac{a}{y_m}\sin!\left(\frac{\omega}{\beta},\bar{t} + \varphi + 240°\right).$$

The realization conditions for this are:

$$\alpha_2 \mu_1 = \frac{2\omega}{\beta}, \quad \alpha_2 \mu_2 = \frac{\omega}{\beta\sqrt{3}}, \quad \alpha_3 = \frac{a}{y_m}\sin\varphi,$$

$$\alpha_1 = \frac{a}{y_m}\sin(\varphi + 120°), \quad \alpha_5 = \frac{a}{y_m}\sin(\varphi + 240°), \quad \alpha = \frac{a}{y_m}.$$

Amplitude Stabilization

The amplitude stabilization works as described in Section 8, Circuit 1. The potentiometer marked with * is set slightly higher, so that the amplitudes of the oscillations can grow slightly. The necessary damping is introduced via the dead zone.

The amplitudes of all three oscillations are in every case large and stable over any computation time.

Section 9

Circuit 1. Backlash-Free (Getriebelose)

[page 96: figure — characteristic curve (upper): output $x_A$ vs. input $x_E$ showing the backlash-free (deadband-free) parallelogram-shaped characteristic; (lower): block diagram with two inputs $-\left(\dfrac{x_E}{x_{Em}}\right)$ via coefficients $\alpha_1, n_1$ and $\alpha_2, n_2$, feeding a dead-zone element with thresholds $a^$, $b^$, followed by a high-gain amplifier ($\infty$) and a gain-100 stage ($n_3$), output $+\left(\dfrac{x_A}{x_{Am}}\right)$]

Derivation and Normalization

The relation is

$$-x = -\alpha_2 n_2!\left(\frac{x_E}{x_{Em}}\right) + \alpha_1 n_1!\left(\frac{x_A}{x_{Am}}\right).$$

As an example, for $\left(\dfrac{x_E}{x_{Em}}\right) > b^$ and $\dot{x}_E \geq 0$, and with high gain in the integrator: $x = b^$, i.e.

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{\alpha_2 n_2}{\alpha_1 n_1}\left(\frac{x_E}{x_{Em}}\right) - \frac{b^*}{\alpha_1 n_1}. \tag{1}$$

Similarly,

for $\left(\dfrac{x_E}{x_{Em}}\right) < (-|a^|)$ and $\dot{x}_E \leq 0$: $x = -|a^|$, i.e.

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{\alpha_2 n_2}{\alpha_1 n_1}\left(\frac{x_E}{x_{Em}}\right) + \frac{|a^*|}{\alpha_1 n_1}. \tag{2}$$

Section 9, Circuit 1 (continued)

From a coefficient comparison with the desired normalized relations, e.g.

$$\left(\frac{x_A}{x_{Am}}\right) = \tan\alpha \cdot \frac{x_{Em}}{x_{Am}}\left(\frac{x_E}{x_{Em}}\right) - \frac{b \cdot \tan\alpha}{x_{Am}} \tag{1a}$$

it follows that

$$\tan\alpha = \frac{x_{Em}}{x_{Am}}\cdot\frac{\alpha_2 n_2}{\alpha_1 n_1},$$

$$a^* = \frac{\alpha_2 n_2}{x_{Em}}, a,$$

$$b^* = \frac{\alpha_2 n_2}{x_{Em}}, b.$$

Remarks

The integrator must receive a high input value, because the settling time of the circuit is determined by

$$T = \frac{1}{\alpha_1 n_1 n_3}.$$

However, too high a gain $n_3$ leads to considerable disturbances ($x_A$ oscillates when $x_E = \text{const}$). A value of $n_3 = 100$ is recommended. A high value of $\alpha_1 n_1$ is advantageous.

For values of $a^$ and $b^$ smaller than 0.1, the corners of the characteristic curve become strongly rounded, which leads to greater inaccuracies. The accuracy of the circuit is only approximately 5%, because the operating point of the characteristic curve is always located in the uncertain initial curvature region. It must be ensured that for the chosen values $\alpha_1 n_1$; $\alpha_2 n_2$; $n_3$, $x_A = \text{const}$ holds for $x_E = \text{const}$, above all within the dead zone. If this is not the case, an attempt must be made to improve matters by increasing $\alpha_1 n_1$ or $n_3$.

Section 9, Circuit 2. Special Step Functions

[page 98: figure — (upper): characteristic curve showing $x_A$ vs. $x_E$ with a jump discontinuity of height $b$ at $x_E = 0$, slope $\alpha$ on both sides; (lower): block diagram with input $+\left(\dfrac{x_E}{x_{Em}}\right)$ via $\alpha_1, n_1$ and $\alpha_2, n_2$, a signal $x^$ fed into a dead-zone / 45° element with thresholds $a^$, $b^*$, a high-gain stage, and gain-100 output stage $(n_3)$, giving $+\left(\dfrac{x_A}{x_{Am}}\right)$]

Derivation and Normalization

The steady-state condition requires $x = 0$. From

$$-x = \alpha_1 n_1 x^* + \alpha_2 n_2!\left(\frac{x_E}{x_{Em}}\right) = 0$$

it follows that

$$x^* = -\frac{\alpha_2 n_2}{\alpha_1 n_1}\left(\frac{x_E}{x_{Em}}\right). \tag{1}$$

It is $\left(\dfrac{x_A}{x_{Am}}\right) = f(x^*)$.

For $\left(\dfrac{x_A}{x_{Am}}\right) > b^$: $-x^ = \left(\dfrac{x_A}{x_{Am}}\right) - b^*$,

Section 9, Circuit 2 (continued)

for $\left(\dfrac{x_A}{x_{Am}}\right) < -|a^|$: $-x^ = \left(\dfrac{x_A}{x_{Am}}\right) + |a^*|$.

From equation (1) the circuit yields

for $\left(\dfrac{x_E}{x_{Em}}\right) > 0$: $\left(\dfrac{x_A}{x_{Am}}\right) = b^* + \dfrac{\alpha_2 n_2}{\alpha_1 n_1}\left(\dfrac{x_E}{x_{Em}}\right)$, \tag{2}

for $\left(\dfrac{x_E}{x_{Em}}\right) < 0$: $\left(\dfrac{x_A}{x_{Am}}\right) = -|a^*| + \dfrac{\alpha_2 n_2}{\alpha_1 n_1}\left(\dfrac{x_E}{x_{Em}}\right)$. \tag{3}

From a comparison with the normalized relations derived from the characteristic curve

$$x_A = b + x_E \tan\alpha \quad \text{for } x_E > 0, \tag{2a}$$

$$x_A = -a + x_E \tan\alpha \quad \text{for } x_E < 0, \tag{3a}$$

the setting conditions are obtained by coefficient comparison with equations (2) and (3):

$$a^* = \frac{a}{x_{Am}}; \quad b^* = \frac{b}{x_{Am}}; \quad \frac{\alpha_2 n_2}{\alpha_1 n_1} = \frac{x_{Em}}{x_{Am}}\tan\alpha.$$

Remarks

The slope at $x_E = 0$ is in practice not infinite but approximately 50 to 100. The circuit has a time constant $T = \dfrac{1}{\alpha_2 n_1 n_3}$. A high value of $\alpha_1 n_1$ (or $n_3$) is therefore advantageous.

Settings of $a^$ and $b^$ below 0.1 lead to rounded corners of the characteristic curve.

Section 9, Circuit 3. Characteristic Curves with Hysteresis Properties

[page 100: figure — (upper left): characteristic $x_{A1}/x_{A1m}$ vs. $x_E$ showing rectangular hysteresis loop with inner levels $-b^$, $+a^$ and switching at $g$, $f$; (upper right): $x_A$ vs. $x_E$ showing the parallelogram-shaped hysteresis characteristic with slopes $\alpha$ and offsets $a$, $b$, $d$, $p$; (lower): block diagram with input $+\left(\dfrac{x_E}{x_{Em}}\right)$ and $\left(\dfrac{x_E}{x_{Em}}\right)$ through $\alpha_1$ and $\alpha_2$ branches, a dead-zone element $a^$, $b^$, a high-gain amplifier, outputs $+\left(\dfrac{x_{A2}}{x_{A2m}}\right)$ and $+\left(\dfrac{x_{A1}}{x_{A1m}}\right)$, a decoupling capacitor of approximately 100 pF, and a $b \pm 1$ switch]

At $\left(\dfrac{x_E}{x_{Em}}\right) = 0$ the sign of the small value $\varepsilon$ determines the rest position of $\left(\dfrac{x_{A1}}{x_{A1m}}\right)$ and $\left(\dfrac{x_{A2}}{x_{A2m}}\right)$. Only one of the two characteristic curves can be set freely at a time.

Derivation and Normalization

When

$$\alpha_2 n_2!\left(\frac{x_E}{x_{Em}}\right) + \alpha_1 n_1 |b^*| > 0, \quad \text{i.e. however}$$

$$\left(\frac{x_E}{x_{Em}}\right) > \left(\frac{\alpha_1 n_1}{\alpha_2 n_2}|b^*|\right) \text{ holds, then} \tag{1}$$

$$\left(\frac{x_{A2}}{x_{A2m}}\right) = +\alpha_1 n_1 |b^*| + \alpha_2 n_2!\left(\frac{x_E}{x_{Em}}\right).$$

Section 9, Circuit 3 (continued)

When

$$\left(\frac{x_E}{x_{Em}}\right) < \left(\frac{\alpha_1 n_1}{\alpha_2 n_2}a^*\right) \text{ holds, then} \tag{2}$$

$$\left(\frac{x_{A2}}{x_{A2m}}\right) = -\alpha_1 n_1 a^* + \alpha_2 n_2!\left(\frac{x_E}{x_{Em}}\right).$$

[page 101: figure — characteristic curve $x_{A2} = f(x_E)$ showing the hysteresis loop with labeled dimensions: slope $\tan\gamma^* = \alpha_2 n_2$, offsets $b^\alpha_1 n_1$ and $a^\alpha_1 n_1$ (= $\alpha_1 n_1(a^, b^)$), switching levels $b^\alpha_1 n_1/\alpha_2 n_2$ and $a^\alpha_1 n_1/\alpha_2 n_2$ on the $x_E/x_{Em}$ axis]

From the characteristic $x_{A2} = f(x_E)$ one directly reads off

$$x_{A2} = \tan\gamma \cdot x_E + a \quad \text{for } x_E > c, \tag{3}$$

$$x_{A2} = \tan\gamma \cdot x_E - b \quad \text{for } x_E < d. \tag{4}$$

From a comparison of equations (3) and (4), still to be normalized, with equations (1) and (2), one obtains

$$\left.\begin{aligned} \alpha_2 n_2 &= \tan\gamma,\frac{x_{Em}}{x_{A2m}},\ a &= \alpha_1 n_1 b^* x_{A2m},\ b &= \alpha_1 n_1 a^* x_{A2m},\ c &= a^\frac{\alpha_1 n_1}{\alpha_2 n_2}x_{Em},\ d &= b^\frac{\alpha_1 n_1}{\alpha_2 n_2}x_{Em}. \end{aligned}\right} \quad \text{freely selectable e.g.:} \quad x_{Em};; x_{A2m};; \alpha_1 n_1.$$

For setting the characteristic $x_{A1} = f(x_E)$:

$$f = \frac{\alpha_1 n_1}{\alpha_2 n_2}a^* \cdot x_{Em},$$

$$g = \frac{\alpha_1 n_1}{\alpha_2 n_2}b^* \cdot x_{Em}.$$

Section 9, Circuit 3 (continued)

Remarks

Settings of $\alpha_1 n_1 > 1$ should be avoided if possible, because at higher loop gain a larger settling capacitor must be switched in.

The circuit also works well for the smallest values of $a^$ and $b^$.

Section 9, Circuit 4. Function with Hysteresis Properties

[page 103: figure — (upper): three-dimensional characteristic surface showing a hysteresis-type function; (lower): block diagram with input $\left(\dfrac{x_E}{x_{Em}}\right)$ through a comparator, dead-zone/relay element with thresholds, high-gain stages, a function generator block $F$, and output $\left(\dfrac{x_A}{x_{Am}}\right)$; below: the required characteristic of $F$: a smooth S-shaped (sigmoid-like) curve]

The circuit for “backlash-free” (getriebelose) operation: see the corresponding circuit sheet.

The function generator $F$ must implement the following approximation:

[figure: smooth S-shaped characteristic that $F$ must reproduce]

Section 10

Circuit 1. Generation of Symmetric Triangle and Square Waveforms

[page 104: figure a — block diagram: a relay/comparator block (with value $\pm 1$) receives a feedback ramp signal; its output is integrated to produce a ramp (triangular wave). An auxiliary potentiometer set to $+k_1$ and a level comparator at $+\varepsilon$ complete the loop. Waveforms shown: upper trace — triangular wave reaching $k_1$ with period $2T$; lower trace — square wave $K$ with pulse width $\Delta t_\varepsilon$]

$$\tan\gamma = \alpha n, \quad T = \frac{1}{\alpha n}.$$

[page 104: figure b — block diagram: integrator with approximately 100 pF feedback capacitor; $\pm 1$ input; integrator output feeds a comparator chain and relay; output is taken after a further amplifier stage. Waveforms shown: triangular wave (period $2T$), and square wave output]

$$\tan\gamma = \alpha n, \quad T = \frac{1}{\alpha n}.$$

Remarks on Circuit b

If the square wave is to make its first jump at time $2T$, the integrator must receive $\pm 1$ as the initial condition.

When $\varepsilon = 0$ the starting direction of the square wave is indeterminate. If one wishes to start with a defined starting direction of the square wave, a value of approximately 0.3 to 0.5 must be switched in for $\varepsilon$ and immediately switched off again after the start; otherwise the triangular wave is shifted in the ordinate by $\pm\varepsilon$, but the period remains unchanged.

Section 10, Circuit 2. Generation of Asymmetric Triangle and Square Waveforms

[page 105: figure a — block diagram with two coefficient potentiometers $\alpha_1$ and $\alpha_2$ controlling the integrator rate in each direction; the relay provides feedback; waveforms show: asymmetric triangular wave with positive slope $\tan\gamma_1 = \alpha_2 n$ and negative slope $\tan\gamma_2 = \alpha_1 n$; the initial position is controlled by the sign and value of $\alpha_3$]

$$\tan\gamma_1 = \alpha_2 n, \quad \tan\gamma_2 = \alpha_1 n.$$

The sign and value of $\alpha_3$ determine the starting position.

[page 105: figure b — block diagram: approximately 100 pF capacitor; integrator; two coefficient inputs $\alpha_1$ and $\alpha_2$; comparator; output stage. Waveforms show asymmetric triangle and square with slopes $\tan\gamma_1 = \alpha_2 \alpha n$ and $\tan\gamma_2 = \alpha_1 \alpha n$]

$$\tan\gamma_1 = \alpha_2 \alpha n, \quad \tan\gamma_2 = \alpha_1 \alpha n.$$

Remarks on Circuit b

Through an initial value at the integrator the initial conditions for the oscillations can be changed. When the oscillations are to be symmetric in their amplitude, an offset voltage must additionally be added; for the triangular wave, the switch-on offset voltage alone is sufficient for the shift.

In general, the sign of $\varepsilon$ (approximately 0.3 to 0.5) determines the starting direction; however, $\varepsilon$ must be switched off immediately after the start if an ordinate shift of the triangle is not desired.

[page 106: figure c — block diagram: approximately 100 pF capacitor; $\pm 1$ input with coefficient $\varepsilon$; first integrator with thresholds $+1$/$-1$ and high gain; second stage with potentiometers $\alpha_1$/$\alpha_2$ and high gain; feedback path with gain 1, potentiometer $n$, and phase-inverter $\alpha$. Waveforms: square wave switching between $\alpha_2$ and $\alpha_1$; asymmetric triangle with slopes $\tan\gamma_1 = \alpha_2 \alpha n$ and $\tan\gamma_2 = \alpha_1 \alpha n$]

$$\tan\gamma_1 = \alpha_2 \alpha n, \quad \tan\gamma_2 = \alpha_1 \alpha n.$$

Regarding the choice of $\varepsilon$: see the remarks for Circuit b.

Circuit c offers the possibility of obtaining the asymmetric triangle and square oscillations with amplitudes $\pm 1$.

Section 10, Circuit 3. Generation of Pulse Sequences

[page 107: figure a — block diagram: input $-1$ via coefficient $\alpha$ and protective resistor $R$; integrator with high gain produces a ramp (slope $\tan\gamma = \alpha n$). When the ramp reaches $+k_1$, a latching relay (potentiometer $+k_1$, switch $u$) closes briefly and resets the integrator; simultaneously a comparator at level $+\varepsilon$ produces a pulse output $K$. Waveforms: ramp (sawtooth) reaching $k_1$ with reset interval $\Delta t_\varepsilon$; square pulse $K$ of width $\Delta t_\varepsilon$]

$$\tan\gamma = \alpha n.$$

Where possible, a small protective resistor $R$ of approximately 10 kΩ should be provided so that relay contacts do not burn.

[page 107: figure b — block diagram: triangular wave (amplitude $\pm k$, with inner levels $\pm\delta$) input to two level-triggered relay/comparator stages with thresholds $+(k-\delta)$ (upper path, output $\alpha_2$) and $+(k-\delta)$ (lower path, output $\alpha_1$); both feed a unity-gain summing amplifier. Waveforms: triangular wave input with dashed levels $\pm\delta$; output: narrow positive pulse at $+\alpha_1$ and narrow negative pulse at $-\alpha_2$ occurring at successive peaks]

Section 11

Circuit 1. Sample-and-Hold Circuit (Sampler)

Block Diagram (Strukturbild):

[page 108: figure — block diagram symbol: input $x_E$, a summing junction, switch $S$, a diode/hold element, output $x_S$ (or $-x_S$); labeled “Schaltfunktion” (switching function)]

The sample-and-hold circuit stores a value $x_{E\nu}$ when the switch $S$ is briefly closed at time $t_\nu$. When $S$ is again briefly closed at time $t_{\nu+1}$, the value $x_{E\nu+1}$ is stored.

Circuits:

[page 108: figure a — circuit: input $x_E$; two series multiplier/amplifier stages; switching function applied to the multiplier; output $-x_S$. Waveforms: pulsed “Schaltfunktion” signal; sampled output]

The sign of the multiplier must be observed; the loop must always be a negative feedback with respect to the position of the switching function.

[page 108: figure b — circuit: input $x_E$; coefficient potentiometer $\pm\delta$; integrator with high gain $(\infty)$; switching function controls the integrator; output $x_S$. Waveforms: either impulse train or staircase approximation output shown]

Section 11, Circuit 1 (continued)

Circuit variant c)

[page 109: figure — schematic showing a switching-function block connected to a potentiometer integrator with two equal resistors R of approximately 10–50 kΩ each at the input]

The time constant with which x_A approaches the value x_g is T = CR, where C is the integration capacitance.

Circuit variant d)

[page 109: figure — schematic showing a switching-function block with a multiplier and potentiometer integrator, with an equivalent simplified form at right using two potentiometers set to α = 0.500]

Both potentiometers are to be balanced so that for x_E = a, x_A = a as well. The time constant of the circuit is T = CR(1 − α).

Remarks

For all circuits, the hold time must be set long enough that the integrator can charge fully to the respective value of x_g.

On Circuit a

n must not be chosen too large. Although the time constant T = 1/n decreases with larger n, the unavoidable null-point oscillations of the multipliers then become strongly effective during the hold phase; that is, x_A fluctuates considerably depending on the hold duration. A value of n ≈ 100 is likely to be the optimum. Overall, this circuit is not recommended.

Section 11, Circuit 1 (continued) — Circuits b through d

On Circuits b through d

The value n should be chosen as large as possible, because T = 1/n. To reduce time constants one can connect smaller capacitors externally in place of the built-in integrating capacitors; care must be taken, however, that this capacitance has a high isolation resistance, otherwise the value x_g will not be maintained.

Instead of a relay, the diode bridge (see Section 11, Circuit 2) may also be used.

On Circuit d

In Circuit d, standard potentiometers are used. The potentiometer setting must be chosen so that the potentiometers are not overloaded. Consideration of the load on the potentiometers:

[page 110: figure — schematic showing two potentiometer sections with taps at α·R and (1−α)·R, connected to a switch S and an amplifier, illustrating the load analysis when S is closed]

When switch S is closed (hold), point a lies practically at earth potential. The potentiometer circuit can then be simplified as shown:

[page 110: figure — equivalent circuit with two sections (1−α)·R each, showing the simplified load model]

The current flowing through the potentiometer sections (1 − α) is:

$$i = \frac{-x_E - x_A}{2R(1-\alpha)},$$

$$|i| = \frac{|x_E|}{R(1-\alpha)},$$

since −x_A = −x_E must hold.

When R = 50 kΩ; −x_E = +100 [V] and therefore x_A = −100 [V], the result is:

$$i = \frac{2}{(1-\alpha)} \quad [\text{mA}].$$

It can be seen that an excessively large α can lead to overloading of the potentiometers.

Section 11, Circuit 2. Switch

The switch is intended to temporarily interrupt a function or to reverse its sign.

Circuit variant a)

[page 112: figure — schematic with a multiplier symbol (×) having one input at 1 and the other at x_E, driven by a switching function (e.g. a square wave between +1 and −1), followed by a unity-gain amplifier, output labelled x_A = x_E switched]

Circuit variant b)

[page 112: figure — schematic with a potentiometer-integrator block (±α, outputs o and u) receiving x_E and a switched inverted feedback (−1 block, dashed), output x_A = x_E switched]

Circuit variant c)

[page 112: figure — diode-bridge circuit with resistors of approximately 100 kΩ, driven by two square-wave switching voltages of opposite polarity, with resistance R in series with the bridge output going to a high-gain amplifier, output x_A; the switching voltage E is also applied through ca. 100 kΩ]

This circuit is suitable only as a switch, not as a changeover switch.

Remark on Circuit c

The switching voltage E of the switching function must be greater than the maximum value of x_E:

$$|E| > |x_{Em}|.$$

The resistance R must be set so that, in series with the internal resistance of the diode section, it yields the desired transmission factor between x_E and x_A (e.g. 1 when R + R_i = 1 MΩ).

Section 11, Circuit 3. Sampling and Storage of Extreme Values

a) The absolute maximum value of a function x_E = f(t) is to be detected and stored.

[page 114: figure — time-varying signal x_E(t) showing a maximum (Max) and minimum (Min); below it a schematic with an integrator, relay contact, and two potentiometer-integrator blocks (with α settings), using dashed connections for quantities in square brackets, output labelled (x_Am) (for the maximum) and (−x_Em)]

(dashed connections for quantities in square brackets)

b) The absolute minimum value of a function x_E = f(t) is to be detected and stored.

[page 114: figure — analogous schematic for minimum tracking, with corresponding dashed connections]

(dashed connections for quantities in square brackets)

Derivation and Normalization for Circuit a

Since the function x_E is to be sampled, it is expedient to set x_Am = x_Em.

If (x_A / x_Am) < (x_E / x_Em), then z < (−ε) and the relay contact lies at 0; the circuit then corresponds to a delay element with time constant T = CR(2 − α), where C is the integration capacitance and R is the resistance of the potentiometer.

Section 11, Circuit 3 (continued)

As long as (x_E / x_Em) > 0, (x_A / x_Am) ≈ (x_E / x_Em) holds. As soon as ẋ_E < 0, z > 0 and the relay contact opens, so that the last value of x_E remains stored.

Circuit b operates in a corresponding manner. If instead of +(x_E / x_Em) the quantity −(x_E / x_Em) is fed into the circuits, the signs in the square brackets and in the dashed connections apply.

Remarks on the Circuits

ε must be adjusted by experiment so that for small (x_E / x_Am) and small ẋ_E the circuit still operates cleanly. For this purpose ε ≈ 0.005 is required. The amount ε is (x_A / x_Am) of (x_E / x_Em) shifted to a maximum.

The dashed extreme values are detected and stored until overwritten by the next larger extreme value.

The initial conditions on the integrators ensure that maxima are detected even when the curve x_A(t) lies entirely in the negative region, and similarly minima when the curve lies entirely in the positive region.

Section 12, Circuit 1. Generation of Break-Lines (Piecewise-Linear Segments)

[page 116: graph — x_A vs. x_E showing a line with a single knee (Knick) at the origin, with three slopes corresponding to u < 0, u = 0, and u > 0]

Circuit variant a)

[page 116: figure — circuit with a potentiometer (setting α) biased by +u, a diode connected to a node z, followed by resistors R_1 and R_0 into a high-gain (∞) amplifier, output +(x_A / x_Am); input −(x_E / x_Em)]

Circuit variant b)

[page 116: figure — circuit using a multiplier-type element (n, ∞) with a diode in the feedback, potentiometer α, input −(x_E / x_Em), bias voltage u, output +(x_A / x_Am)]

Circuit variant c)

[page 116: figure — circuit with two diodes D_1 and D_2 and a stabilizing capacitor of ca. 100 pF in the feedback of a high-gain amplifier; inputs via R_2 (from bias u) and R_1 (from −(x_E / x_Em)), output +(x_A / x_Am)]

Derivation and Normalization for Circuit a

The diode is blocked as long as z > 0; this is the case for:

$$\left(\frac{x_E}{x_{Em}}\right) < \frac{\alpha}{1-\alpha}, u.$$

From (x_E / x_Em) ≥ α/(1−α) · u the diode conducts and:

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{R_0}{R_1}\left(\frac{x_E}{x_{Em}}\right); \quad \frac{R_0}{R_1} = \frac{x_{Em}}{x_{Am}},\tan\gamma.$$

Remarks on Circuit a

The knee is sufficiently sharp, though less pronounced than in Circuit c. If the knee is to lie on the negative abscissa, then instead of −u the quantity −α/(1−α) must be applied to the potentiometer. If the output voltage (x_A / x_Am) is not to extend into the positive but only into the negative range, the diode must be reversed. A positive knee voltage then arises when +u is applied to the potentiometer.

Derivation and Normalization for Circuit b

The diode is blocked for positive (x_A / x_Am). Positive (x_A / x_Am) is generated when:

$$\left[\frac{x_E}{x_{Em}}\cdot 2n - u\right] \geq 0 \quad \text{holds.}$$

The knee voltage therefore lies at:

$$\left(\frac{x_E}{x_{Em}}\right) = \frac{u}{2n},$$

$$2n = \tan\gamma \cdot \frac{x_{Em}}{x_{Am}}.$$

Remarks on Circuit b

The knee is sufficiently sharp, even for 2n ≥ 1, but less pronounced than in Circuit c. Due to the forward voltage of the diode, (x_A / x_Am) in the limiting region is not clamped to zero but instead a residual voltage of up to 0.05 machine units arises.

Section 12, Circuit 1 (continued)

Derivation and Normalization for Circuit c

For positive (x_A / x_Am), Diode 1 is conducting and Diode 2 is blocked, and therefore:

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{R_0}{R_1}\left(\frac{x_E}{x_{Em}}\right) - \frac{R_0}{R_2},u.$$

From this it follows that:

$$\frac{x_{Em}}{x_{Am}}\tan\gamma = \frac{R_0}{R_1}.$$

The knee point lies at (x_A / x_Am) = 0, i.e.:

$$\left(\frac{x_E}{x_{Em}}\right) = \frac{R_1}{R_2},u.$$

Remarks on Circuit c

This circuit delivers a very sharp knee. Moreover, in the limiting region (x_A / x_Am) is actually = 0. Note that the necessary capacitance for high computing frequencies introduces a dynamic behavior. Diode D2 prevents the amplifier from overdriving in the limiting region and contributes to the formation of a sharp knee.

Section 12, Circuit 2. Bent Characteristic Curves and Limiting

[page 119: two graphs — left: x_A vs. x_E showing a piecewise-linear curve with two slope changes at break-points γ_1 and γ_2; right: x_A vs. x_E showing a limiter (flat saturation) characteristic; solid curves for x_E2 = 0, dashed curves for x_E2 < 0]

Note: solid curves for x_E2 = 0; dashed curves for x_E2 < 0.

Circuit variant a)

[page 119: figure — circuit with two diode branches (labeled 1 and 2) each with resistors R_11, R_12 and bias voltages +u_1, −u_2 (with a stabilizing capacitor C of ca. 100 pF), input −(x_E / x_Em) via R_1, output +(x_A / x_Am); also input −x_E2 through a resistor]

Circuit variant b)

[page 119: figure — more complex circuit with two parallel amplifier chains, each using potentiometers α_1 through α_4, diodes D_1 through D_4 in feedback, capacitors C (≤ 100 pF), inputs −1 and +1 (for biasing), −(x_E / x_Em), and output −(x_A / x_Am)]

Circuit variant c)

[page 120: figure — diode bridge circuit with four diodes D_1 through D_4, resistors R, R_1, R_0, biased by +u (or u_i) and −u (or −u_i), input −(x_E / x_Em), output −(x_A / x_Am), with an infinite-gain amplifier]

Derivation and Normalization for Circuit a

It is assumed that x_E2 = 0 and R ≪ R_11; R_12 is used.

As long as Diodes 1 and 2 are blocked:

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{R_0}{R_1}\left(\frac{x_E}{x_{Em}}\right); \quad \frac{R_0}{R_1} = \frac{x_{Em}}{x_{Am}},\tan\gamma.$$

Diode 1 opens when:

$$\left(\frac{x_E}{x_{Em}}\right) < \left(-u_1,\frac{1-\alpha_1}{1-\alpha_1}\right)\frac{R_1}{R_0} \quad (u_1 \text{ expressed in machine units})$$

For such (x_E / x_Em):

$$\left(\frac{x_A}{x_{Am}}\right) \approx \frac{R_0 R_{11}}{R_1(R_0 + R_{11})}\left(\frac{x_E}{x_{Em}}\right); \quad \frac{R_0 R_{11}}{R_1(R_0 + R_{11})} = \frac{x_{Em}}{x_{Am}},\tan\gamma_1. \tag{1}$$

Diode 2 opens when:

$$\left(\frac{x_E}{x_{Em}}\right) > \left(u_2,\frac{\alpha_2}{1-\alpha_2}\right)\frac{R_1}{R_0} \quad (u_2 \text{ expressed in machine units}).$$

For such (x_E / x_Em):

$$\left(\frac{x_A}{x_{Am}}\right) \approx \frac{R_0 R_{12}}{R_1(R_0 + R_{12})}\left(\frac{x_E}{x_{Em}}\right); \quad \frac{R_0 R_{12}}{R_1(R_0 + R_{12})} = \frac{x_{Em}}{x_{Am}},\tan\gamma_2. \tag{2}$$

Remarks on Circuit a

The break-points in the limiting region do not lie exactly at the theoretically derived values because R and the diode forward resistance are not negligible compared to R_11 and R_12. Only for the ideal case R_{11} = R_{12} = 0 (i.e. pure limiting without slope change) does the circuit give an exact result. For small slopes in the limiting region and for steep approach slopes, it is advisable — as the formulae show — to keep α·R as small as possible and to use inputs ×1, so that R_1/R_0 becomes as large as possible.

If (x_E / x_Em) = 0 is also to be forced to x_A = 0, a small compensating voltage is sometimes required to nullify the small residual output that exists at null passage (< 10^−3). One must also ensure that the diode forward voltage traces the desired curve in the region around null; at x_E = 0, a small compensation voltage may be needed to force x_A = 0 as well.

Because of the possible need for a stabilizing capacitance, at high frequencies a dynamic behavior of the circuit may become apparent.

With this circuit, integrators can also be limited.

Section 12, Circuit 2 (continued)

Derivation and Normalization for Circuit b (continued)

$$\tan\gamma_1^* = \frac{x_{Em}}{x_{Am}},\tan\gamma_1.$$

For (x_E / x_Em) > 0 and (x_E / x_Em) · α_4 n_4 > α_2 n_2, Diodes 2 and 3 are blocked, D 1 and D 4 open, i.e. x_1 = 0:

$$x_2 = \alpha_1 n_1 \left(\frac{x_E}{x_{Em}}\right) - \alpha_2 n_2,$$

and therefore:

$$\left(\frac{x_A}{x_{Am}}\right) = (\alpha_2 n_2 - \alpha_1 n_1)\left(\frac{x_E}{x_{Em}}\right) + \alpha_2 n_2.$$

From this it follows:

$$\alpha_4 n_2 = \frac{x_{Em}}{x_{Am}},\tan\gamma_2.$$

For (x_E / x_Em) < 0 and [−(x_E / x_Em)] · α_3 n_3 > α_1 n_1, correspondingly:

$$\left(\frac{x_A}{x_{Am}}\right) = (\alpha_4 n_4 - \alpha_3 n_3)\left(\frac{x_E}{x_{Em}}\right) - \alpha_1 n_1.$$

Section 12, Circuit 2 (continued)

Therefore:

$$(\alpha_5 n_5 - \alpha_3 n_3) = \frac{x_{Em}}{x_{Am}},\tan\gamma_1.$$

[page 123: graph — x_A / x_Am vs. x_E / x_Em showing three-segment piecewise-linear characteristic with slopes tg γ_2* = α_5 n_5 − α_1 n_1, tg γ* = α_5 n_5, and tg γ_1* = α_5 n_5 − α_3 n_3]

Remarks on Circuit b

The circuit operates for all (x_E / x_Em) and arbitrary limiting in a cross-over-free manner. The limiting knees are very sharp. The slope in the limiting region is at “ideal limiting” practically zero (< 10^−3). For large slopes α_5 n_5 > 1, however, a small compensating voltage at α_5 is required to force x_A = 0 at x_E = 0.

The dashed-line plot of x_A(t) shows the dynamic behavior that can make itself felt at high frequencies due to the required stabilizing capacitance. [page 123: two side-by-side time-domain sketches showing x_E(t) with transitions x_E1, x_E2, x_E3 and x_A(t) with the corresponding dynamic overshoot]

It should also be noted that the open amplifiers overdrive for:

$$\left(\frac{x_E}{x_{Em}}\right) \geq \frac{1 + \alpha_2 n_2}{\alpha_4 n_1},$$

or:

$$\left(\frac{x_E}{x_{Em}}\right) \leq \frac{-1 - \alpha_1 n_1}{\alpha_3 n_2};$$

beyond these values x_A is no longer limited.

Derivation and Normalization for Circuit c

With the input open, a current i_1 = u_1/R flows from u_1 into the summing point, and from u_2 a current i_2 = −u_2/R. The output quantity (x_A / x_Am) is therefore with the input open:

$$\left(\frac{x_A}{x_{Am}}\right) = -(i_1 + i_2)R_0 = \frac{R_0}{R}(u_2 - u_1).$$

When (x_E / x_Em) ≠ 0, an additional current i_E = −(x_E / x_Em) / R_1 flows into the summing point, and the output becomes in the linear region:

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{R_0}{R}(u_2 - u_1) + \frac{R_0}{R_1}\left(\frac{x_E}{x_{Em}}\right).$$

When (x_E / x_Em) > u_1 · R_1/R, Diode 2 blocks because the partial current driven by x_E through D2 cancels the current from u_1; therefore for such values:

$$\left(\frac{x_A}{x_{Am}}\right) = \frac{R_0}{R},u_2.$$

For [+(x_E / x_Em)] < [−u_2 · R_1/R], Diode 4 blocks and:

$$\left(\frac{x_A}{x_{Am}}\right) = -\frac{R_0}{R},u_1.$$

[page 125: graph — x_A / x_Am vs. x_E / x_Em showing the characteristic with upper plateau R_0/R · u_2 and lower plateau −R_0/R · u_1, with linear central slope tg γ* = R_0/R_1; break-points at ±u_i/R and (u_2−u_1) · R_0/R offset; also dashed curve for x_E2 ≠ 0]

tg γ* = R_0/R_1

The horizontal shift of the characteristic with respect to the coordinate origin can, if necessary, be compensated by an offset voltage x_E2 ≠ 0 (dashed curve).

Remarks on Circuit c

The knees are not as sharp as in Circuit 2 but are adequate for normal requirements.

If instead of −(x_E / x_Em) the quantity +(x_E / x_Em) is to be fed into the circuit, all signs in the circuit must be reversed and the diodes reversed in polarity; the above relations then remain valid.

Section 12, Circuit 3. Special Step Functions and Relay Characteristics

[page 126: two graphs — left: x_A vs. x_E showing a relay characteristic with hysteresis: upper level a, lower level below zero, slopes γ_1 and γ_2, and horizontal offset b; right: the same for R_11 = R_12 = 0 (ideal relay); solid curves for x_E2 = 0, dashed curves for x_E2 < 0]

Solid curves for x_E2 = 0; dashed curves for x_E2 < 0.

Circuit variant a)

[page 126: figure — circuit with two diode branches labeled 1 (with R_11, diode, potentiometer α_1, bias +u_1) and 2 (with R_12, diode reversed, potentiometer α_2, bias −u_2), both connected through summing resistors to an infinite-gain amplifier with a stabilizing capacitor of ca. 100 pF; input −x_E2 through a resistor, input −(x_E / x_Em) through R_1, output (x_A / x_Am)]

Circuit variant b)

[page 126: figure — diode-bridge circuit with four diodes D_1 through D_4, two bias resistors R connected to +u_1 (top) and −u_2 (bottom), feedback resistor R_0 to an infinite-gain amplifier, input −(x_E / x_Em), output (x_A / x_Am)]

Section 12, Circuit 3 (continued)

Derivation and Normalization for Circuit a

It is assumed that x_E2 = 0 and R_11, R_12 ≫ R. As long as diodes 1 and 2 are blocked, the following holds:

(x_A / x_Am) = V · (x_E / x_Em) where V = gain of the computing amplifier.

As soon as

(x_A / x_Am) < −u_1 · α_1 / (1 − α_1) (u_1 in machine units),

diode 1 opens and the following holds:

(x_A / x_Am) ≈ (R_11 / R_1) · (x_E / x_Em). (1)

When

(x_A / x_Am) > u_2 · α_2 / (1 − α_2) (u_2 in machine units),

diode 2 opens and the following holds:

(x_A / x_Am) ≈ (R_12 / R_1) · (x_E / x_Em). (2)

It therefore follows that:

R_11 / R_1 = tan γ_1 · (x_Em / x_Am) and

R_12 / R_1 = tan γ_2 · (x_Em / x_Am) must hold.

Diagram annotation:

tan γ_2* = R_12 / R_1; a* = u_2 · α_2 / (1 − α_2)
tan γ_1* = R_11 / R_1; b* = u_1 · α_1 / (1 − α_1)

Remarks on Circuit a

The slopes in the limiting region do not correspond exactly to the theoretical values because the forward resistance R and the series resistance of the diodes have been neglected. R_11 and R_12 must therefore be fine-adjusted empirically.

The slopes at the breakpoints are not infinite because the values 50 to 100 — as given for +(x_E / x_Em) — are not infinite. Similarly the slopes at zero are approximately 0.01 to 0.02.

If, instead of −(x_E / x_Em), the quantity +(x_E / x_Em) is fed into the circuit, all signs of the circuit must be reversed and all diodes must be reversed in polarity; the relations given above remain unchanged.

The exact calculation taking R into account yields, in place of equation (1):

(x_A / x_Am) = (x_E / x_Em) · [R α_1 (1 − α_2) + R_11] / [R_1 (1 − α_2)] − u_2 α_1 / (1 − α_2) (1a)

and the corresponding expression for equation (2). It therefore follows that:

tan γ_2* = x_Em/x_Am · [R α_1 (1 − α_2) + R_12] / [R_1 (1 − α_2)]

tan γ_1* = x_Em/x_Am · [R α_2 (1 − α_1) + R_11] / [R_1 (1 − α_1)]

If sharp limiting is desired (R_11 = R_12 = 0), it is advisable — as the formulas above show — to make α / R_1 as small as possible, that is, to choose R small and R_1 large.

With x_E2 ≠ 0 the characteristic can also be given a horizontal offset with respect to the coordinate zero point.

Derivation and Normalization for Circuit b

The point ∞ lies approximately at earth potential. With the input open, current u_1 flows from u_1 via the summing point:

i_1 = u_1 / R

and from u_2 the current:

i_2 = −u_2 / R

into the summing point. Consequently:

(x_A / x_Am) = −(R_0 / R) · (a_1 − u_2)

For (x_E / x_Em) ≥ 0, diodes D 1 and D 4 are blocked, so only current from D 2 flows into the summing point:

i_ = −u_2 / R

Thus:

(x_A / x_Am) = −i_ · R_0 = u_2 · R_0 / R

For (x_E / x_Em) < 0, diodes D 1 and D 4 are open, D 2 is blocked, so only current from D 2 flows into the summing point:

i_+ = u_1 / R

Thus:

(x_A / x_Am) = −i_+ · R_0 = −u_1 · R_0 / R

[Page 129: figure shows the characteristic with a sharp corner at the origin — positive slope for positive x_E, negative for negative x_E.]

Remarks on Circuit b

The corners are sharp, and the slope in the jumping region is approximately 50. Instead of −(x_E / x_Em), if the quantity +(x_E / x_Em) is fed into the circuit, all signs of the circuit must be reversed and all diodes must be reversed in polarity; the relations given above remain unchanged.

Section 12, Circuit 4. Dead Zone

[Page 130: figure shows the characteristic curve with a dead zone. The output x_A is zero for a band of input values between b and a, with slope γ for x_E > a and slope γ_1 for x_E < b. Circuit variant (a) uses a computing amplifier with bias supply voltages +u_1 and −u_2, resistors R_1, R_2, R_0, and diodes D_1 and D_2; or, when tan γ = tan γ_1, a simplified version without the individual attenuators. Circuit variant (b) uses two amplifier stages with feedback capacitor C ≤ 100 pF and diodes D_1 through D_4. Circuit variant (c) uses resistors R_1, R_2, R_0, and a diode bridge D_1–D_4 with supplies +u_1 and −u_2.]

Derivation and Normalization for Circuit a

It is assumed that R ≪ R_1.

Then:

z = α_1 u_1 + (1 − α_1)(x_E / x_Em), (z and u in machine units)

Diode 1 is open for z < 0, i.e.

(x_E / x_Em) < −u_1 · α_1 / (1 − α_1).

[Page 132 continued:]

Diode 2 is open for z > 0, i.e.

(x_E / x_Em) > u_2 · α_2 / (1 − α_2).

It therefore follows that:

b = u_1 · α_1 / (1 − α_1) · x_Em, respectively α_1 = 1 / (u_1 x_Em / b + 1)

and

a = |u_2| · α_2 / (1 − α_2) · x_Em, respectively α_2 = a / (|u_2| x_Em + a)

(u_1 and u_2 expressed in machine units).

Furthermore:

x_Em / x_Am · tan γ = R_0 / R_2; x_Em / x_Am · tan γ_1 = R_0 / R_1;

tan γ* = R_0 / R_2 more precisely tan γ* = R_0 / (1 − α_2) R_2 [approximately]

tan γ_1* = R_0 / R_1 [diagram shows characteristic with breakpoints at ±b/x_Em·|u_2|, slopes labeled.]

Remarks on Circuit a

Both α_1 and α_2 as well as R_0 / R_1 must be fine-adjusted empirically, because the computed values do not give exactly the desired curve owing to the approximation R ≪ R_1. The differences are however slight.

If instead of −(x_E / x_Em), the quantity +(x_E / x_Em) is fed into the circuit, all signs and all diodes of the circuit must be reversed; the relations given above remain unchanged.

With good diodes (e.g. OA 132) the quantity b (and likewise a) can be made as small as 0.01 ME (machine units). The corners are sharp.

Derivation and Normalization for Circuit b

For (x_E / x_Em) > 0 and (x_E / x_Em) · α_3 n_2 < α_1 n_1, diodes D 2 and D 3 are blocked, D 1 and D 4 are open, so x_1 and x_2 are both zero. Therefore:

(x_A / x_Am) = 0.

When (x_E / x_Em) > 0 and (x_E / x_Em) · α_3 n_2 > α_1 n_1, diodes D 1 and D 3 are blocked, D 2 and D 4 are open, so x_2 = 0 and:

−x_1 = α_3 n_2 · (x_E / x_Em) − α_1 n_1 = (x_A / x_Am).

From a coefficient comparison with the desired curve:

x_Em / x_Am · tan γ = α_3 n_2,

a = (α_1 n_1 / α_3 n_2) · x_Em.

For (x_E / x_Em) < 0 and [−(x_E / x_Em) · α_4 n_4] < α_2 n_2, the following also holds:

(x_A / x_Am) = 0.

For (x_E / x_Em) < 0 and [−(x_E / x_Em) · α_4 n_4] > α_2 n_2, x_1 = 0 and:

−x_2 = (x_A / x_Am) = α_1 n_4 · (x_E / x_Em) + α_2 n_2.

Consequently:

x_Em / x_Am · tan γ_1 = α_1 n_4,

b = (α_2 n_2 / α_4 n_4) · x_Em.

[Page 134 figure: characteristic with slopes tan γ* = α_3 n_2 and tan γ_1* = α_1 n_4, breakpoints at b/x_Em and a/x_Em.]

Remarks on Circuit b

The dead zone can be reduced to zero on both sides. The corners are very sharp even for α_2 n_2, α_3 n_1 > 1.

If instead of −(x_E / x_Em), the quantity +(x_E / x_Em) is fed into the circuit, all signs of the circuit must be reversed and all diodes must be reversed in polarity; the relations given above remain unchanged.

Derivation and Normalization for Circuit c

When x_E = 0 and x_E2 = 0, a current i_1 = u_1 / R flows from u_1 into the summing point, and from u_2 a current i_2 = −u_2 / R. The output (x_A / x_Am) must assume a small value ε sufficient to maintain a counter-current through the diode bridge to balance these two currents. Since ε is small, the current caused by ε is already large due to the very small forward resistance of the diodes.

As soon as (x_A / x_Am) > 0, diodes D 1 and D 4 are blocked, and then only currents from (x_E / x_Em) and from (x_A / x_Am) flow into the summing point. This is the case for:

(x_E / x_Em) > (R_1 / R) · u_2; it holds then:

(x_A / x_Am) = (R_0 / R_1) · (x_E / x_Em) − (R_0 / R) · u_2; (u_2 in ME).

From this:

x_Em / x_Am · tan γ = R_0 / R_1,

a / x_Em = (R_1 / R) · u_2.

For (x_E / x_Em) < −(R_1 / R) · u_1, it follows from the same reasoning:

(x_A / x_Am) = (R_0 / R_1) · (x_E / x_Em) + (R_0 / R) · u_1; (u_1 in ME).

x_Em / x_Am · tan γ_1 = R_0 / R_1,

b / x_Em = (R_1 / R) · u_1.

When x_E2 ≠ 0, the dead zone is made asymmetric.

Remarks on Circuit c

The corners are sufficiently sharp, though better in circuit b. The dead zone can be reduced to zero.

[Page 135 figure: characteristic with equal slopes tan γ* = R_0 / R_1 and breakpoints u_1 R_1 / R and u_2 R_1 / R.]

Section 12, Circuit 5. Absolute Value Function

[Page 136: figure shows the V-shaped absolute value characteristic x_A = |x_E|. Three circuit variants are shown:

(a) simple circuit using a unity-gain amplifier stage followed by diodes D_1 and D_2 and a second unity-gain stage;
(b) uses a high-gain amplifier with 100 pF feedback capacitor, diodes D_1 and D_2, and a unity-gain output stage with x_1 as intermediate node;
(c) uses two amplifier stages, diodes D_1 through D_4, and a summing output stage with intermediate nodes x_1 and x_2, capacitor C ≤ 100 pF.]

Derivation and Normalization for Circuit a

For (x_E / x_Em) > 0, diode 1 is open and D 2 is blocked, and the following holds:

(x_A / x_Am) = (x_E / x_Em).

For (x_E / x_Em) < 0, D 1 is blocked and D 2 is open, and the following holds:

(x_A / x_Am) = −(x_E / x_Em).

Therefore for all x_E:

(x_A / x_Am) = |(x_E / x_Em)| for all x_E.

The self-evident normalization condition is x_Am = x_Em.

Remarks on Circuit a

[Page 137 figure: deviation from ideal characteristic near x_E = 0 — a small rounded region about 0.05 ME wide due to non-ideal diode properties.]

Due to the non-ideal diode characteristics, there is a deviation from the desired behavior near x_E = 0: the curve departs from the ideal near zero, with the slope at γ* being unity.

Derivation and Normalization for Circuit b

For (x_E / x_Em) > 0, D 2 is blocked and D 1 is open, so x_1 is therefore zero. Therefore:

(x_A / x_Am) = (x_E / x_Em).

For (x_E / x_Em) < 0, D 2 is open and D 1 is blocked, x_1 = (x_E / x_Em), and:

(x_A / x_Am) = (x_E / x_Em) − 2 x_1 = −(x_E / x_Em).

Therefore for all x_E:

(x_A / x_Am) = |(x_E / x_Em)| for all x_E.

The normalization condition is x_Am = x_Em.

Remarks on Circuit b

The circuit delivers an ideal absolute value characteristic. Instead of −(x_E / x_Em), if +(x_E / x_Em) is fed into the circuit, all signs of the circuit must be reversed and all diodes must be reversed in polarity; the relations given above remain unchanged.

Derivation and Normalization for Circuit c

For (x_E / x_Em) > 0, diodes D 1 and D 3 are blocked, D 2 and D 4 are open, so x_2 = 0 and:

−x_1 = (x_A / x_Am) = (x_E / x_Em).

For (x_E / x_Em) < 0, diodes D 2 and D 4 are blocked, D 1 and D 3 are open, so x_1 = 0 and:

−x_2 = (x_A / x_Am) = −(x_E / x_Em).

Therefore for arbitrary (x_E / x_Em):

(x_A / x_Am) = |(x_E / x_Em)|.

The self-evident normalization condition is x_Am = x_Em.

Remarks on Circuit c

The circuit delivers an ideal absolute value characteristic. It offers no advantages over the simpler circuit b.

Section 13, Circuit 1. Sign-Dependent Coefficient Setting

It is desired to form: y = ax for x ≥ 0, y = bx for x < 0.

Circuit variants:

(a) A diode pair routes the input +(x/x_m) to two attenuators α_1 and α_2 that feed separate amplifier inputs n_1 giving output −(y/y_m).
(b) A servo potentiometer labeled (α) with a function generator (F) receives +(x/x_m), with +1 and −1 reference supplies and outputs +a* and +b* to a summing amplifier yielding +(y/y_m).
(c) A relay-switched arrangement with relay Q receives +(x/x_m) and routes it through attenuators α_1 and α_2 before two amplifier inputs to give −(y/y_m).

Implementation conditions:

α_1 n_1 = a · (x_m / y_m), α_2 n_2 = b · (x_m / y_m),

or equivalently:

a* = a · (x_m / y_m), b* = b · (x_m / y_m).

(Since y_m = ax_m or bx_m must hold, a* and b* are always ≤ 1.)

Remarks

For circuit (a), significant errors can occur for small values (x/x_m) due to the diode characteristic (e.g. f ≈ 10% for (x/x_m) < 0.02).

For circuit (b), the accuracy at small values of (x/x_m) depends on the precise adjustment of the servo potentiometer.

Section 13, Circuit 2. Reflection (Mirroring) of a Function

When setting up an even or odd function with a function generator, a potentially large number of segments may be required. This can sometimes be avoided by using the following circuits, which restrict the function generation to the positive abscissa:

a) When the function is even:

x_E is passed through an absolute value circuit (dead zone removed), and the result is fed into the function generator (shown schematically). A sign-controlled switching multiplier (relay block labeled 0/φ) receives x_E and applies it to a second function generator output x_A. The characteristic is symmetric about the vertical axis (V-shaped).

[Page 140: figure showing two block-diagram paths for the even case and the resulting symmetric x_A vs. x_E plot.]

b) When the function is odd:

x_E is passed through an absolute value circuit, the result is fed into a function generator, then into a sign-controlled multiplier (relay block labeled 0/φ) driven by a sign detector. A separate sign-inversion block (with ±1 reference) supplies the multiplier control. The output x_A has an S-shaped antisymmetric characteristic.

[Page 140: figure showing the block diagram for the odd case.]

[Page 141: figure showing additional detail of the circuit for odd-function mirroring using relay-switched multiplier blocks (0/φ), a unity-gain amplifier, and a function generator. The resulting x_A vs. x_E characteristic is an odd S-curve passing through the origin.]

Remarks

Attention must be paid to the zero crossings!

Section 13, Circuit 3. Inversion of a Function

Let the function y = f(z) be set in a function generator. The inverse function z = f⁻¹(y) can be formed as follows:

[Page 142: figure — the block diagram shows −x fed into a unity-gain input of a high-gain amplifier (gain ∞) whose output is +z. The output z also drives the input of a function generator block F, which produces output f(z). The amplifier compares f(z) with y via its summing junction.]

Because of the open amplifier (very high gain), the following holds approximately:

x ≈ y.

Since in the function generator

y = f(z)

is set, it holds:

x ≈ y ≈ f(z)

and therefore:

z ≈ f⁻¹(y).

Section 13, Circuit 4. Detection of Drift Voltage

a) Drift Voltage at the Summing Amplifier

[Page 143: figure — a summing amplifier circuit with feedback resistor R_0, input resistor R_1, input voltage x_E, internal drift voltage u_D (referred to the grid), actual grid potential u_g, and ideal grid potential u_g*. The output is x_A.]

The following holds:

(x_E − u_g*) / R_1 = −(x_A − u_g*) / R_0,

u_g = u_g* + u_D,

x_A = −V u_g.

From this:

x_E / R_1 + x_A / R_0 = (1/R_1 + 1/R_0) · (−x_A / V − u_D);

x_A = −x_E / [R_1 (1/R_0 · (1 + 1/V) + 1/(V R_1))] − u_D (1/R_0 + 1/R_1) / [1/R_0 · (1 + 1/V) + 1/(V R_1)]

For x_E = 0 (short-circuit of the input!) and V ≫ 1:

x_A = −u_D · (1 + R_0 / R_1).

If the input is not short-circuited but left open, one obtains only:

x_A = −u_D.

To detect drift voltage it is therefore advisable to short-circuit one input to ground.

b) Drift Voltage at the Integrator

[Page 144: figure — an integrator circuit with feedback capacitor C_0, input resistor R_1, input voltage x_E, internal drift voltage u_D (referred to the grid), ideal grid potential u_g*, and actual grid potential u_g. The output is x_A.]

The following holds:

(x_E − u_g*) / R_1 = −C_0 · d/dt (x_A − u_g*),

u_g = u_g* + u_D,

x_A = −V u_g.

From this:

x_E / R_1 + C_0 · dx_A/dt = u_g* / R_1 + C_0 · du_g*/dt = −x_A / (V R_1) − u_D / R_1 − (C_0 / V) · dx_A/dt − C_0 · du_D/dt.

For V ≫ 1:

x_A = −∫₀ᵗ [1/(C_0 R_1) · (x_E + u_D)] dt − u_D.

If the input is short-circuited, x_E = 0, one obtains:

x_A = −u_D − ∫₀ᵗ (1/(C_0 R_1)) · u_D dt.

With the input left open, one would obtain only:

x_A = −u_D.

To detect drift voltages it is therefore advisable to short-circuit one input to ground.

C. Literature

I. Cited Literature

[1] Korn and Korn: Electronic Analog Computers. McGraw-Hill Book C., 1956

[2] Johnson: Analog Computer Techniques. McGraw-Hill Book C., 1956

[3] Soroka: Analog Methods in Computation and Simulation. McGraw-Hill Book C., 1954

[4] Witsenhausen: Utilisation optimum des multiplicateurs électro-mécanique. Acte-proceedings of second international analogue computation meetings, 1958

[5] Oldenbourg: Elektrische Analogrechner. VDI-Z, Band 3, 1956

[6] Ammon: Über die Nachbildung von Vorstufen mit Analogrechenelementen. Elektronische Rechenmaschinen, Heft 3, 1961

[7] Ammon: Zur Frage der Nachbildung des Verhaltens von Analogrechnerelementen. Elektronische Rechenmaschinen, Heft 3, 1961

[8] Paul; McLennen: Measurement of Phase and Amplitude at Low Frequencies. Electronic Engineering, August 1959, p. 386

[9] Ammon; Westphal: Dämpfungsregelung. German Patent Application, 1961

[10] Fryer: Analogue Computing. McGraw-Hill Book C., 1961

[11] Flügener: Über die Anfangsbedingungen bei linearen Übertragungsgliedern. Elektronische Rechenmaschinen, Heft 3, 1961

[12] Lauer, Lesnick, Matson: Servomechanism Fundamentals. McGraw-Hill Book C., 1947

[13] Giloi, Lauber: Analogrechnen. Springer-Verlag, 1963

[14] Geometry: (see Viermannshausen); Wirksungsplan. (unpublished)

[15] Flügener: Über die Anfangsbedingungen bei Übertragungsgliedern. Elektronische Rechenmaschinen, Heft 3, 1961

[16] Watt: Electronic Analogue Computers. McGraw-Hill Book C., 1956

[17] Electronic: Elektronische Analogrechner. Oldenbourg-Verlag, 1960

[18] Watt: Electronic Analogue Computers. Iliffe & Sons Ltd., London, 1970

LITERATURE COLLECTION

Source Index:

Section 1

Circuit 1: [16]; Circuit 2: [4]

Section 2

Circuit 1: [1]

Section 3: [1]; [2]; [7]

Section 4

Circuit 1: [1]; [16]; Circuit 2: [4]; Circuit 3: [5]
Circuit 5: [2]; [4]; Circuit 6: [4]; Circuit 7: [4]
Circuit 8: [4]

Section 5

Circuit 1: [4]; Circuit 2: [4]; Circuit 4: [4]

Section 6

Circuit 1: [6]; Circuit 2: [6]

Section 7

Circuit 1: [1]; Circuit 2: [1]; Circuit 3: [1]

Section 8

Circuit 1: [10]; Circuit 2: [2]; [10]; Circuit 4: [9]

Section 9

Circuit 1: [1]; Circuit 3: [1]

Section 10

Circuit 1: [1]; [12]; [14]; Circuit 2: [1]
Circuit 3: [17];

Section 11

Circuit 1: [1]; Circuit 2: [1]; Circuit 3: [15]

Section 12

Circuit 1: [1]; Circuit 2: [1]; [2]; [3]; [14]
Circuit 3: [1]; [3]; Circuit 3: [1]; [2]; [14]
Circuit 5: [2]; [14]

Section 13

Circuit 1: [16]; Circuit 2: [8]; Circuit 3: [1]

Schaltungen der Analogrechentechnik

Circuits of Analog Computing Technology

Table of Contents

Preface (Vorwort)

Symbols Used

1. Open Amplifier Without Input Resistor

2. Computing Amplifier With Specified Circuit (example)

3. Computing Amplifier Without Feedback (example)

4. Summing Amplifier With Preceding Potentiometers

5. Integrator

6. Two-Parabola Multiplier

7. Servo-Multiplier (example)

8. Servo-Resolver (example)

9. Function Generator With Diode Segments

10. Special Function Generator With Diode Segments (example)

11. Relay (Comparator)

A. Notes on the Use of the Analog Computer and the Circuit Sheets

Open Amplifier

Servo-Multipliers and Servo-Resolvers

Normalization and Time Transformation

B. CIRCUIT SHEETS

SECTION 1

CIRCUIT 1. Amplifiers with High Weighting Factors

CIRCUIT 2. Absolute-Value Formation with a Servo Multiplier: z = |x| · y

Remarks

SECTION 2

CIRCUIT 1. Differentiation: y = dx/dt

Derivation with Normalization

Time Transformation

Remarks

CIRCUIT 2. Realization of Selected Rational Transfer Functions

CIRCUIT 3. General Simulation of a Linear Differential Equation

Normalization and Time Transformation:

SECTION 3. Dead-Time Approximation, General

CIRCUIT 1. Dead-Time Approximation. Padé Approximation, 1st Order

CIRCUIT 2. Dead-Time Approximation. Padé Approximation, 2nd Order

CIRCUIT 3. Dead-Time Approximation. Padé Approximation, 3rd Order

CIRCUIT 4. Dead-Time Approximation. Padé Approximation, 4th Order

CIRCUIT 5. Dead-Time Approximation. Padé Approximation, 5th Order

CIRCUIT 6. Dead-Time Approximation. Padé Approximation, 2nd Order using Computing Amplifiers and RC Networks

SECTION 4

Circuit 1. Division x/y, sign of y restricted

Derivation and Normalization

Remarks

Circuit 2. Division x/y, signs of x and y arbitrary

Derivation and Normalization for Circuit a

Remarks

Circuit 3. Division x/y, signs of x and y arbitrary

Derivation and Normalization

Remarks

Circuit 4. Multiplication/Division and/or Reciprocal Formation with a Servo Multiplier: u = z · x/y and/or w = y/x · v

Derivation and Normalization

Remarks

Circuit 5. Square Root z = √x

Derivation and Normalization

Remarks

Circuit 6. Square Roots w = ±√|x| and z = sign x · √|x|

Derivation and Normalization

Remarks

Circuit 7. Square Root with Various Operations under the Radical

Derivation and Normalization

Remarks

Circuit 8. Formation of Roots of Higher Degree and Fractional Powers

Derivation and Normalization

Section 4, Circuit 8 (continued)

Remarks

Section 5

Circuit 1. Broken rational function z = ax / (1 + by), for by > (-1) and sign of x arbitrary, (b > 0)

Derivation and Normalization

Remarks

Section 5, Circuit 2. Broken rational function z = ax / (1 - by), for by < 1 and sign of x arbitrary, (b > 0)

Derivation and Normalization

Remarks

Section 5, Circuit 3. Formation of broken rational functions

General Circuit

Specific Circuit Variants for Different Cases

Derivation and Normalization (for Circuit a)

Normalization Conditions:

Remarks

Section 5, Circuit 4. Interpolation

Circuit with Servo Multiplier for ω(t) ≤ ω, (ω = Corner Frequency of the Servo)