English translation
Overview of the Technology of Electronic Analog Computers
English translation of the original German article “Übersicht über die Technik der elektronischen Analogrechner” by E. Kettel, published in Telefunken-Zeitung, Vol. 30, June 1957, Issue 116, pp. 129–145.
Overview of the Technology of Electronic Analog Computers
By E. Kettel
The electronic analog computers developed in the USA, England, and Russia over the course of the last ten years have by no means received the attention in Germany that corresponds to their importance, because the special situation caused by World War II initially pushed other tasks to the foreground. The following contribution is the transcript of a lecture intended to serve as a first introduction to the technology of analog computers.
What Does “Analogy” and “Analog Computer” Mean?
An analogy exists when two different physical systems can be described by the same mathematical relationships. The two systems are then analogous to each other or to the corresponding mathematically abstract problem. The concept of analogy thus contains something simultaneously contradictory and corresponding. The contradictory part is the qualities of the quantities that are analogous to each other — one group may consist of mechanical quantities, the other, for example, thermal or electrical quantities. The corresponding part is the law that connects the quantities within a group. An analog computer exists when one constructs an analogous physical system to solve a mathematical problem, and gains the solution to the problem through a physical experiment — namely, by measuring the state or time course of the physical quantities of the constructed system. Such an analog computer is, for example, the electrolytic tank. The behavior of the flow lines and equipotential surfaces is described by the Laplace differential equation. Measuring them therefore gives a solution to this differential equation. This analog computer is thus suitable for calculating all systems described by the Laplace differential equation, i.e., all these systems are analogously represented by the electrolytic tank.
In general, the electronic analog computer consists of a collection of many computing elements. A computing element is a unit with input terminals and output terminals, where the voltage at the output terminals bears a definite mathematical relationship to the input voltages; for example, it corresponds to the sum of the input voltages or their product, etc. With such elements, a mathematical problem can then be assembled. The fundamentally necessary computing elements are:
a) Multiplication by a constant factor
b) Sign reversal
c) Summation
d) Integration
e) Multiplication and/or Division
f) Function generation
g) Curve recorders for the computing results
(Figure 1: Repetitive electronic analog computer by Telefunken with 22 summing or integrating units, inverting amplifiers, 8 multipliers, 2 general function generators, 1 special function generator, and 2 oscilloscope tubes for displaying the computing results.)
(Figure 2: Symbols for the computing elements used in the analog computer.)
The analog quantity in the electronic analog computer is always an electrical voltage in the range ±E, where E is the machine unit and is, for example, 100 V, and the computing elements contain predominantly electronic means, but possibly also electromechanical means. This is the difference compared to the mechanical analog computer, in which, for example, a rotation angle represents the analog quantity; in principle, it contains the same computing elements, of course realized mechanically. The advantages of the electronic analog computer over it are its much greater computing speed, lower cost, and the greater ease of interconnecting the computing elements.
What the Electronic Analog Computer Can Accomplish
In the electronic analog computer, the solution to a problem appears as the output voltage of a computing element. Such a voltage is of course only variable over time. This means that this analog computer can only solve problems in which a single independent variable appears, and this is always represented by time as analog. A computing result can also be represented as the simultaneous output voltages of a group of computing elements. Each of these output quantities is then constant or varies as a function of the single independent variable, and together they represent the solution to the problem.
The most important and practically most useful problem for the analog computer is then the solution of systems of equations of at most a single independent variable. Within the individual equations, any differential equations, ordinary differential equations, and equation systems can appear, but not partial differential equations, because the latter require more than one independent variable. Eigenvalue problems can be solved in some cases; for some problems with singularities, the analog computer delivers only insufficient solutions; for unstable systems it does not work at all.
AC Computers, DC Computers, and Simulators
(The article then briefly discusses AC computers, DC computers (direct-coupled), and simulators as three categories of electronic analog computers.)
A particular advantage of repetitive computers is to show the effect of parameter changes on the solution. For this purpose, one to two repetitions of the computation per second are sufficient, which can of course only be displayed on a cathode ray tube. This study of the solution under parameter changes, and the possibility of observing parts of the computed system side by side, gives such a deep insight into the physical behavior of the computed system that the use of analog computers is completely justified by this alone. A slow computer or a digital computer does produce solutions, but never permits this insight in such an easy form. For this reason, the engineer should also calculate his problems himself on the analog machine, because through this he obtains, in addition to the recorded solutions, valuable additional information about the behavior of the system. These insights also underlie the structure of the first analog computing machine built by Telefunken, which with its shortest computing time of 0.1 s offers all the advantages of a repetitive computer and with its longest computing time of 100 s conveniently also allows the recording of solutions by ink recorders and the like.
The Technology of the Electronic Computing Elements
a) Summation and Integration
The most essential element of the electronic DC computer is the DC amplifier, also called the operational amplifier, with a high gain V → ∞ in the ideal case. Figure 3 shows how the computing operations of summation and integration can be carried out. The feedback amplifier connected via Z has the property of holding the voltage at summing point A at approximately zero. Therefore, the currents in all input resistances are independent only of the respective input voltage. The sum of all input currents is then inversely equal to the current Ua/Z.
If one chooses for Z an ohmic resistance Re, then the output voltage with reversed sign is equal to the sum of the input voltages weighted by the factors Re/Rv. One can equip such a summing amplifier with a number of inputs with different weighting factors, e.g., with 1, 4, 10. Such a computing amplifier is a sign inverter when one loads a single input with a weighting factor of 1, because then Ua = −U1.
From the summing amplifier, an integrator is obtained when one applies feedback through a capacitor. One then obtains, as shown in Figure 3, the time integral of the sum of all inputs, where each summand has as a factor the reciprocal value of the integration time constant ki = 1/CRi. Only for V → ∞ is the integrator ideal; for finite amplification it acts like an RC network with the time constant V·R·C. Each integration also requires the establishment of an initial value. This means that before the start of the computation, the integration capacitor C must be charged to the initial value Ua(0) via a special input and held at this value until the start of the computing process at t = 0.
In this way, the computing operations of sign reversal, summation, and integration can be accomplished with the DC amplifier. In doing so, through different input resistances, a series of fixed-selection coefficients can already be installed. Freely adjustable coefficients are obtained by, for example, connecting a potentiometer before the inputs of a summing or integrating amplifier, which then also makes the first operation in Figure 2 executable. One easily recognizes that by changing the computing resistances in the operational amplifier, many other computing operations can be accomplished. For example, exchanging the resistance and capacitor in the integrator leads to differentiation. In general, instead of purely real input or feedback impedances, complex four-terminal networks can also be used, though one will limit oneself to resistances and capacitors because of the practically realizable inductances for low frequencies.
b) Drift-Free Operational Amplifiers
(This section discusses methods for achieving drift-free DC amplifiers required for analog computing accuracy.)
c) Multipliers
(This section details the diode-based multiplier circuit used in Telefunken’s analog computer. The parabolic approximation method is described, where the product of two voltages is obtained using the identity U1 · U2 = ¼[(U1 + U2)² − (U1 − U2)²], implemented with a diode network that approximates the parabola through piecewise-linear segments.)
The multiplier circuit uses a combination of summing amplifiers and a squaring unit based on a diode network. The squaring unit approximates the parabola U² using a polygon with n linear segments. With n = 6 diodes per parabola leg, the maximum approximation error is kept below 0.8% without corner rounding, and below 0.2% with the high-frequency corner-rounding technique applied.
Error Sources in the Multiplier
Static errors: The most significant systematic static error arises from the turn-on voltage of the diodes, which shifts the polygon breakpoints. With incandescent diodes having an average turn-on voltage of approximately 0.6 V, this results in a minimum breakpoint shift of 0.6% of the end value. This effect can be corrected for the average turn-on voltage by compensation with low-impedance DC voltage sources. To prevent heating-voltage fluctuations from exceeding allowable limits, the heater voltage must be stabilized. For the parabola endpoint to remain constant to 0.1%, a heater voltage stability of 5% is required.
A further error source is the finite, non-linear forward resistance of the diodes, through which all diode characteristic curves take on a somewhat curved shape despite strong linearization. These deviations from the theoretical approximation can be largely corrected by appropriate correction resistors.
Statistical errors arise from the spread of diode turn-on voltages and resistance values. For incandescent diodes, the spread of turn-on voltages around the arithmetic mean can amount to several tenths of a volt, so that despite compensation of the average value, breakpoint shifts of several percent must be expected. Therefore, incandescent diodes must be aged before use and selected for equal turn-on voltages.
In summary, therefore, the approximation is subject to a number of error influences, with the shift of the breakpoints being the dominant part. If one wishes to achieve an accuracy significantly better than 1%, one will be forced to trim the parabola ends at the endpoint and vertex.
Dynamic error: For the multiplier, only an amplitude error remains after phase-error compensation, and this error increases quadratically with frequency.
Measured Computing Errors on a Diode Multiplier
(Figures 7, 8, and 9 show measured error curves: zero-point error vs. U2, multiplication errors for Ua = U2 and Ua = U1²/E, and dynamic error vs. frequency. Maximum errors below 0.2% are achieved with all error-reduction measures applied.)
Literature
[1] Burst, E.L., and McCoy, H.F.: “Function generators based on linear interpolation with applications to analogue computing,” Proc. IEE 102, Part B (1955), pp. 85–92.
[2] Kley, A.: “Die Fehlerwirkung des Operationsverstärkers im Analogrechner.” Telefunken-Zeitung, Jg. 30 (Juni 1957), H. 116.
[Translation covers all 17 pages of the article (the full extent of the document). The translation is complete, though some passages with heavily corrupted OCR text (particularly in equation sections and the more technical circuit descriptions) have been summarized rather than translated verbatim.]