Elektronische Analogrechner

Complete English translation of the original German-language document (4 pages).

Title Page

MATHEMATICS FOR NATURAL SCIENCE AND TECHNOLOGY Edited by H. Heinrich and H. Schubert Volume 8

ELECTRONIC ANALOG COMPUTERS

Prof. Dr. rer. nat. habil. Helmut Adler Director of the Institute for Machine Computing Technology at the Technische Hochschule Ilmenau

With 222 figures and 7 tables Second, revised and expanded edition

VEB Deutscher Verlag der Wissenschaften Berlin 1968

[page 2 — Chapter IV: Operational Amplifiers, continuation of §12]

[IV] Operational Amplifiers 96

Because U₁ = U₂ sin ωt, one obtains:

$$\frac{U_2}{p} = \int U_2 ,\mathrm{d}t = -U_1 \cdot \frac{\sin!\left(\omega - \frac{\pi}{2}\right)}{\omega} = -\frac{1}{\omega} U_2$$

(4.12.6)

and therefore:

$$U_0 = \left(-|v| + \frac{2\omega t}{\omega RC} + \ldots\right) U_1$$

(4.12.7)

If one chooses ωRC = 1, equation (4.12.2) follows at last. Regarding the details of the remaining computing units for this special-purpose computing device, the reader is referred to the cited literature.

Fig. 69. Circuit for forming U₀ = (−|v| + i|u|) · ξₜ

§ 13. Reversible Operational Amplifiers

Various papers on so-called reversible operational amplifiers have been published in recent years by Puchow and his collaborators (cf. [336]). By this term one understands operational-amplifier circuits in which, in contrast to the circuits treated so far, a unique assignment of inputs and outputs does not exist. Fig. 70 shows the basic circuit of such a system, together with the currents, voltages, computing impedances Nᵢ, and the auxiliary impedances Nᵢ’. The currents Iᵢ that are fed into the circuit from outside are denoted by Iᵢ. The node equation yields the following relationships:

$$\sum_{j=1}^{n} \frac{U_K - U_j}{N_j} - I_n = 0$$ (4.13.1)

$$\frac{U_j - U_K - U_0}{N_j’} + I_j = 0 \quad (j = 1, 2, \ldots, n)$$ (4.13.2)

$$\sum_{j=1}^{n} \frac{U_0 - U_j}{N_j} = I_n$$ (4.13.3)

Between U₀ and U_K the following relationship again holds:

$$U_0 = A \cdot U_K$$

For U_K one obtains:

$$U_K = \frac{\displaystyle\sum_{j=1}^{n}\left[v_j!\left(\frac{U_j}{N_j} + \frac{U_j}{N_j’}\right) + \frac{U_j}{N_j}\right]}{n + A \cdot \displaystyle\sum_{j=1}^{n} \frac{N_j}{N_j’}}$$

[page 3 — §13 continued]

[§13] Reversible Operational Amplifiers 97

and because |A| → ∞, U_K → 0. Thus from (4.13.1):

$$\sum_{j=1}^{n} \frac{U_j}{N_j} = 0$$ (4.13.4)

This equation constitutes the fundamental relationship of the reversible operational amplifier. The remarkable property of this circuit is that any of the n terminals can be chosen arbitrarily.

Fig. 70. Basic principle of the reversible operational amplifier

(Fig. 71. The reversible constant multiplier [Rᵢ = 20 kΩ, Rᵢ’ = 200 kΩ])

(Fig. 72. The reversible integrator-differentiator [(C = 5 μF, Z’ = 20 kΩ) / (Z = 200 kΩ)])

If n − 1 terminal voltages Uⱼ are applied and one of the n terminal voltages is to be taken from equation (4.13.4) as the output voltage, this can be done. Fig. 71 shows an invertible inverter with the basic equation x₁ + x₂ = 0. If x₁ = −x₂ is given, x₂ = −x₁ is obtained, and vice versa; that is, either x₁ or x₂ can be set equal to zero (self-evidently at Rᵢ = Rᵢ’).

The use of this circuit as a differentiator element (x₁ input, x₂ output) principally eliminates at differentiation the problematic property of amplifying disturbances.

Equations (4.13.1), (4.13.2), and (4.13.3) establish further important properties of the reversible operational amplifier. The amplifier

[page 4 — §13 continued, end of section]

98 Operational Amplifiers [IV]

output voltage U₀ must, as follows from equations 2 and 1, be larger than the computing voltages Uᵢ; specifically, when Uᵢ is the result voltage:

$$U_0 = U_i!\left(1 + \frac{N_j}{N_j’}\right)$$

A reduction of Nⱼ’ is associated with an increase of the currents Iᵢ and thus with a heavier load on the amplifier. Conversely, the impedances Nⱼ cannot be made arbitrarily large either, since they are determining for the operations to be performed. The amplifier output current therefore fixes the dimensioning of the impedances and the loading conditions. However, a derivation of the relationships and a detailed discussion of the results will be dispensed with here.

Following this basic principle it is possible to construct reversible summers, reversible multiplication-division circuits, reversible function generators (whereby either the function set in the function generator or its inverse function can be formed), and others. By interconnection, computing circuits for systems of differential equations result, as well as circuits for linear systems of equations and linear optimization problems that are remarkable in their properties.

Since, however, precisely for the last-mentioned problems treatment by means of a digital computer is more expedient, it remains to be seen whether these reversible operational amplifiers will find broader application.