Analog Computers

English translation

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Faszination Analogrechnen

Complete English translation of the original German-language document (132 pages).

Overview — Terms — Mechanics — Early… — Computing Elements — System Examples — Sample Problems — Applications — Outlook

The Fascination of Analog Computing

Bernd Ulmann — [email protected]

Colloquium on the History of Natural Science, Mathematics, and Technology

2008

Overview

The following lecture is intended as an introduction to the history and technology of electronic analog computers. This topic is being treated as part of a doctoral dissertation by the author at the SPGN, under the supervision of Professor Wolfschmidt. Given the limited time frame of this lecture, most points can only be presented at a very basic level — for more detailed treatment the reader is referred, already at this point, to the doctoral thesis itself or to the author’s website:

http://www.vaxman.de/analog_computing/analog_computing.html

All figures without a source citation are by the author and represent items from the author’s own collection.

Overview

The present lecture is divided into the following parts:

  • Clarification of fundamental concepts
  • Mechanical analog computers
  • Early electronic analog computers
  • Typical computing elements
  • System examples
  • Programming
  • Sample problems
  • Typical fields of application
  • Outlook and future

Distinction between Analog and Digital Computers

Two essentially different perspectives exist for distinguishing analog computing from digital methods:

  1. Distinction by the differing value ranges of variables involved in computation: Analog computers use continuous value ranges, while digital computers can only operate with discrete values. Analog computers were sometimes also called measuring machines, and digital computers counting machines.

  2. Analogy formation as the distinguishing criterion: Analog computing installation vs. digital computing installation — the latter is controlled algorithmically by a program, while the former forms an analogue to the problem to be solved by interconnecting computing elements in a suitable manner.

Value Ranges as a Criterion

The following points argue against using differing value ranges as a distinguishing criterion:

  • With sufficiently precise measurement, even the value ranges assumed to be continuous — those accessible to an analog computer — turn out to be discrete.
  • A distinction based on the representation of values would not classify so-called digital differential analyzers within the class of analog computers, which is contrary to the intention of the following sections.

Analogy Formation as a Criterion

More suitable is the use of the concept of analogy to distinguish between analog and digital computers. Accordingly, an analogy computer — which in what follows will simply and habitually be referred to as an analog computer — is a system of computing elements that are interconnected in a suitable form to solve a problem, so as to form an analogue of the question under investigation.

Direct and Indirect Analog Computers

In forming analogies, a distinction must be made between so-called direct analogies — in which scale models are generally employed that are based on the same physical principles as the problem to be solved — and indirect analogies — in which the analogy formation is based on the intermediate step of an exact mathematical description, with implementation that usually departs from the actual problem. Direct analogy formation is generally used when no mathematical problem description is possible, or when it is only possible with great difficulty.

Direct Analogy Formation

The following figure (after [DRES72][p. 52]) shows a direct analogy formation in the context of a preliminary study for the roof of the Munich Olympic Stadium.

[page 8: figure only]

Implementation Techniques

Analog computers — or, more precisely, analogy computers — can be implemented using very different techniques. Techniques that come into use include, for example:

  • mechanical computing elements,
  • hydraulic elements,
  • analog-electronic computing components, and
  • purely digital techniques.

Strictly speaking, the computers treated as the primary focus of this work ought therefore to be called analog-electronic analogy computers; for simplicity’s sake, however, the term electronic analog computer will always be preferred.

Mechanical Analog Computers

In what follows, a number of simple mechanical analog computers are treated as representative examples, because their concreteness makes a series of essential properties of analog computers apparent:

  • Slide rules
  • Planimeters
  • Mechanical integrators
  • Further computing elements
  • Mechanical differential analyzers
  • Electromechanical developments

Slide Rules

Among the earliest mechanical analog computers are the so-called slide rules (for their history see, for example, [JEZI00]):

  • The fundamental operation consists in the addition of lengths (in the case of slide rules) or of angles (in the case of circular slide rules).
  • Through appropriate scales, more complex operations such as multiplication, logarithms, etc. can also be represented.

Slide Rules

The following figure shows a typical slide rule, consisting of the body, the slide, and the cursor, by means of which scales on the front and back can be linked with one another.

[page 12: figure only]

Special-Purpose Slide Rules

In certain fields — for example, aeronautical engineering — slide rules and circular calculators are still in use today in some cases, as the following example of the E6-B Flight Computer demonstrates:

[page 13: figure only]

Special-Purpose Slide Rules

Through appropriate scale design, problems for which only experimental data are available can also be addressed, as the following example of a rather rare Nuclear Weapon Effects Computer shows.

[page 14: figure only]

Planimeters

By means of so-called planimeters, the area of a region can be determined simply by tracing around its boundary — the following figure shows a typical polar planimeter.

[page 15: figure only]

More Complex Mechanical Analog Computers

In addition to such simple analog computers as slide rules and planimeters, complex installations also emerged over time; these are briefly described below. First, however, a few (select) fundamental mechanical analog computing elements are presented, which serve as the basic building blocks of such installations.

Differential Gear

The following figure shows a typical example of a differential gear as used for summation and differencing (after [BORD46][p. 174]).

[page 17: figure only]

Mechanical Integrators

The heart of every analog computer, whether mechanical or electronic in nature, is an integration device. The following figure shows a typical friction-wheel integrator (after [SVOB48][p. 24]), which, by displacing the friction wheel across the rotating integrator disk, permits the computation of expressions of the form

$$x_3 = \int x_1 , dx_2$$

[page 18: figure only]

Double-Ball Integrator

The following presents a practical example of a mechanical integrator of the kind used in American fire-control computers of the early twentieth century (after [BORD46][p. 287]):

[page 19: figure only — double-ball integrator diagram]

Function Generators

The two following illustrations show, by way of example, methods for producing a square-law function and a function dependent on two parameters (after [SVOB48][p. 22] and [SVOB48][p. 23], respectively).

[page 20: figure only — function generator diagrams]

Example of a Function Generator

The function generator shown below was used, for instance, in fire-control computers (see [BORD40][p. 54]):

[page 21: figure only — function generator example]

Examples

The following slides show, by way of example, several instances of mechanical analog computers in order of increasing complexity. The examples presented are:

  • Tide predictors
  • Fire-control computers
  • Mechanical differential analyzers

Tide Predictor

Lord Kelvin was already concerned in the 1870s with the problem of tide prediction — a question of growing importance above all for the rapidly expanding steamship trade. The treatment of this problem can be reduced to a Fourier synthesis, that is, the formation of sums of harmonic functions of different periods, phases, and amplitudes. The individual functions in this context are called partial tides and represent effects such as:

  • the rotation of the Earth about its axis,
  • the rotation of the Moon around the Earth,
  • the rotation of the Earth around the Moon,
  • the precession of the lunar perigee,
  • and many others.

Kelvin’s Tide Predictor

The following illustration shows the mechanical tide predictor completed by Lord Kelvin in 1876, which already took into account 10 partial tides.

[page 24: figure only — Kelvin tide predictor photograph/diagram]

Fire-Control Computers

A wide field of application for mechanical analog computers emerged from the late nineteenth century onward in the form of so-called fire-control computers, which made it possible to improve the accuracy of long-range artillery.

The following presents, by way of example, the problem as it arises in a torpedo fire-control computer of the kind used during the Second World War — the fundamental problems involved in the deployment of long-range weapons are, however, essentially the same for ground-to-ground or ground-to-air artillery.

Fire-Control Computers

A simplified representation of the torpedo fire-control problem is shown in the following illustration, after [BORD44][p. 12]:

[page 26: figure only — torpedo fire-control problem diagram]

Fire-Control Computers

The following illustration shows the Torpedo Data Computer Mark 3 with the front panel removed (after [BORD44][p. 150]):

[page 27: figure only — Torpedo Data Computer Mark 3 photograph]

Mechanical Differential Analyzers

All of the mechanisms presented thus far, which cover only a small part of the spectrum of mechanical analog computers, were each developed for the treatment of a single problem, or at most a very limited class of problems. The idea of mechanically interconnecting these and other basic computing elements in arbitrary configurations in order to solve more general problems led to the development of so-called mechanical differential analyzers.

One of the first practically usable mechanical differential analyzers was developed by Vannevar Bush et al. at MIT from the late 1920s onward.

A Simple Differential Analyzer

The structure of a simple mechanical differential analyzer is shown in the following illustration (after [KARP58][p. 190]).

[page 29: figure only — simple differential analyzer diagram]

Mechanical Differential Analyzers

The following photograph shows, by way of example, two integrators from a mechanical differential analyzer built by Tim Robinson (reproduced with kind permission of Tom Robinson):

[page 30: figure only — photograph of two integrators]

Electromechanical Differential Analyzers

A major disadvantage in the use of such mechanical differential analyzers is, not least, the considerable effort required when changing the interconnection of the computing elements participating in a calculation.

The development of electromechanical differential analyzers — in which the actual computing operations are carried out mechanically, but the interconnection of the computing elements, whose input and output values are generally in the form of shaft rotations, is accomplished electrically with the aid of servo systems — was intended to remedy this disadvantage.

The Minden Integration System

The following illustration shows, by way of example, an integration system developed by Schoppe and Faeser in Minden, built on developments by the IPM from approximately 1948 onward:

[page 32: figure only — Minden integration system photograph]

Advantages and Disadvantages of (Electro-)Mechanical Analog Computers

Advantages:

  • Integration with respect to arbitrary integration variables is possible.

Disadvantages:

  • Low computing speed.
  • Mechanical components are expensive, maintenance-intensive, and failure-prone, and bring with them large weights and volumes.
  • Changing the computing circuit in purely mechanical computers is very time-consuming, meaning rapid program changes are nearly impossible.

The First Electronic Analog Computers

During the course of the Second World War, two developments in particular drove work on electronic analog computers, since for these applications mechanical or even electromechanical analog computers were in no way suitable.

The two stimulating applications in question were:

  • Fire-control computers (a development carried out principally at Bell Laboratories), and
  • the development of the A4 rocket and its guidance system (Helmut Holzer).

Fire-Control Computers

The underlying idea of a fire-control computer is to track one or more artillery pieces as automatically as possible in response to a moving target.

The input data may consist, for example, of range and angle information from radar equipment, from which the fire-control computer calculates the necessary orientation of the gun, taking into account a number of additional parameters.

The first developments in this direction took place from the late 1930s onward, principally in the United States.

Fire-Control Computers

The following illustration shows the first sketch of an electronic fire-control computer (by D. B. Parkinson, after [FAGE78][pp. 135 f.]).

[page 36: figure only — first sketch of electronic fire-control computer]

Fire-Control Computers

The following illustration shows a modified fire-control computer of the T-15 type, which was also capable of accounting for curved trajectories (after [FAGE78][p. 154]).

[page 37: figure only — modified T-15 fire-control computer with curved-trajectory capability]

Helmut Hölzer’s Work

In the early 1940s, Helmut Hölzer developed the world’s first electronic analog computer, which was successfully employed in the investigation of several control-engineering questions that arose during the development of the A-4 rocket at Peenemünde. The following illustration shows a photograph of this analog computer (source: NASA, Marshall Space Flight Center).

[page 38: figure only — photograph of Hölzer’s electronic analog computer, NASA/MSFC]

Helmut Hölzer’s Electronic Analog Computer

[page 39: figure only — detailed view of Hölzer’s electronic analog computer]

Helmut Hölzer’s Electronic Analog Computer

This first electronic analog computer already possessed nearly all the characteristics of later commercial developments:

  • Purely electronic construction; integration and differentiation achieved through the use of suitable RC networks.
  • Use of operational amplifiers (though this term was not coined until later).
  • Programming by interconnecting the computing elements via simple plug-in connections.

The Mischgerät (“Mixing Device”)

While this first electronic analog computer was successfully used in the laboratory investigation of questions relating to missile guidance, the actual control of the A-4 rocket demanded a development that went beyond this computer. Controlling a vehicle as complex as the A-4 rocket required the development of a flight-worthy electronic analog computer. This had to be small, robust, and as maintenance-free as possible. The guidance computer developed by Helmut Hölzer for this purpose was designated the Mischgerät (“mixing device”) for reasons of camouflage.

The Mischgerät

The following illustration (after [A4 45][Fig. 82]) shows the problem of attitude control for the A-4 rocket: a purely proportional control law leads — due to the control-loop time constants — to an oscillatory divergence and ultimately to the destruction of the missile. For successful guidance, at least differential terms must additionally be taken into account.

[page 42: figure only — diagram illustrating the attitude-control problem of the A-4 rocket; purely proportional control causes oscillatory divergence]

The Mischgerät

[page 43: figure only — figure illustrating the A-4 attitude-control problem (after [A4 45][Fig. 82])]

The Mischgerät

The following illustration gives an impression of the guidance system of the A-4 rocket (after [A4 45][Fig. 80]):

[page 44: figure only — overview diagram of the A-4 rocket guidance system]

The Mischgerät

The following illustration shows the Mischgerät as installed in the nose of the rocket (after [TREN82][p. 134]):

[page 45: figure only — photograph/diagram of the Mischgerät installed in the A-4 rocket nose]

The Mischgerät

The following illustration shows the complete Mischgerät (photo: Adri de Keijzer, reproduced with kind permission):

[page 46: figure only — photograph of the complete Mischgerät unit; photo credit: Adri de Keijzer]

Later Developments

Proceeding primarily from these two lines of development, the first commercially available electronic analog computers emerged in the early post-war years, marking the beginning of a line of development that extended into the late 1980s.

All of these electronic analog computers — which, incidentally, operated almost exclusively with direct-current voltages — share a set of typical computing elements that are briefly described on the following slides, before the programming of such systems is examined in greater detail.

Later Developments

All electronic analog computers considered in the following share the following characteristics:

  • Problem variables are represented by voltages.
  • These voltages cannot exceed or fall below certain upper and lower limits. This limit is called the machine unit and is typically ±100 V or ±10 V.
  • The interconnection of computing elements is accomplished by establishing appropriate electrical connections between the outputs and inputs of computing elements.

Fundamental Computing Elements

The computing elements briefly described in the following are:

  • Coefficient potentiometers
  • Operational amplifiers
  • Summers (summing amplifiers)
  • Integrators
  • Function generators
  • Multipliers
  • Comparators
  • Typical output devices

Coefficient Potentiometer

The simplest (passive) computing element of an electronic analog computer is the so-called coefficient potentiometer, whose circuit and computing-diagram symbol are shown below:

[circuit diagram: potentiometer with input and output]
[symbol: block labeled 'a' with 0 ≤ a ≤ 1]

With its help, an input signal can be multiplied by a freely selectable multiplier 0 ≤ a ≤ 1.

Operational Amplifier

The most important active element of an electronic analog computer is the so-called operational amplifier:

[symbol: op-amp with inverting (−) and non-inverting (+) inputs]

Such an (idealized) operational amplifier has two inputs, one of which is inverting, and provides at its output the sum of the two input signals multiplied by a gain factor V = ∞.

Operational Amplifier

Of course, an infinite gain factor is not realizable — however, in order to keep the following discussion simple, ideal amplifiers are assumed throughout: amplifiers that exhibit neither drift nor input current. With the aid of operational amplifiers, summers and integrators can be constructed in a straightforward manner; these rank among the most important computing elements of all.

Summer (Summing Amplifier)

Such an (inverting) summer has the following basic circuit:

U₁ ──R₁──┐
U₂ ──R₂──┤
          ├──[−∞ op-amp]── Uout
Uₙ ──Rₙ──┘
              RF

For an ideal operational amplifier, the following relation holds:

$$\sum_{i=1}^{n} \frac{U_i}{R_i} = -\frac{U_{\text{out}}}{R_F}$$

Summer (Summing Amplifier)

Overall, such a summer computes:

$$-U_{\text{out}} = \sum_{i=1}^{n} a_i U_i$$

with

$$a_i = \frac{R_F}{R_i}$$

It is typically represented by the following symbol:

U₁ ──┐
U₂ ──┤ [summer block: gains 1, 5, 10] ── −(U₁ + 5U₂ + 10U₃)
U₃ ──┘

The example shown computes −(U₁ + 5U₂ + 10U₃), with individual input gains of 1, 5, and 10 respectively.

Integrator

The following figure shows the basic circuit of an integrator, which likewise relies on an (idealized) operational amplifier as its active element:

U1
R1
U2  R2
         -+C
Un
Rn        Uout

In contrast to a summer, a capacitor is placed in the feedback branch of the amplifier.

Integrator (continued)

Using the current flowing through the capacitor, I_C = C * U_out_dot, it follows analogously that:

sum_{i=1}^{n} (U_i / R_i) = -C * U_out_dot

With a_i as before, the operation of the integrator is thus ultimately:

-U_out = sum_{i=1}^{n} a_i * integral_0^t U_i(t) dt + U(0)

Note that integration can only be performed with respect to time t.

Integrator (continued)

Such an integrator is typically represented by the following symbol:

[integrator block symbol with multiple inputs and one output]

It has several operating modes (simplified):

  • Initial condition (IC): The capacitor is charged to an initial value U(0).
  • Operate (OP): Integration runs.
  • Hold (HALT): Integration is suspended; the output of the integrator remains constant at the last output value.

Function Generator

Many problem formulations require the representation of prescribed functions. For this purpose, so-called function generators are available, for the implementation of which a large number of different variants have been developed. The general symbol of such a function generator has the following form:

[block with input x and output f(x)]

Multiplier

The multiplication of two voltages is a considerably more complex operation than, for example, integration. Over time, a large number of different techniques for performing multiplications have been proposed and implemented. However, the general symbol of a multiplier has in almost all cases the following form:

[multiplier block symbol with two inputs and one output]

By means of a trick, a multiplier can also be used — with the aid of a so-called open amplifier — to compute divisions and square roots, so that as a rule no separate computing elements are provided for these operations.

Comparators

Some problem formulations require, for their treatment, step functions or a means of selecting between two values, etc. For this purpose, most electronic analog computers offer so-called comparators, which are controlled switches. The symbol of such a comparator is shown in the following figure:

[comparator symbol with inputs a and b, and a switch output]

If the sum of the two input voltages a and b is greater than 0, the switch takes the upper position; if it is less than or equal to 0, the lower position is active.

Output Devices

Since an electronic analog computer computes with voltages, the natural output devices are oscilloscopes and x,t- or x,y-plotters.

More advanced systems also offer the capability to digitize measured values and either display them, print them out, or forward them to a connected digital computer. (If this digital computer additionally has the capability to intervene in the analog computing program in a controlling manner, the combination is referred to as a hybrid computer.)

Output Devices (continued)

An example of an early digital voltmeter (Telefunken, 1960) is shown in the following figure (after [BALL60], p. 215):

[page 62: figure only]

Output Devices (continued)

A typical, simple single-channel x,y-plotter is shown in the following figure:

[page 63: figure only]

Output Devices (continued)

A typical oscilloscope with camera attachment, as it was frequently used with electronic analog computers, is shown in the following figure (after [STEIN]):

[page 64: figure only]

System Examples

The following slides present, as representative examples, some typical electronic analog computers spanning a period of approximately 1950 through the 1980s. Given the character of the underlying work, the focus here will be on systems from the manufacturer Telefunken. The following picture (after [KETT57], p. 129) shows the first electronic analog computer from Telefunken, the RA 1 system in its expansion stage as of 1957. Already clearly visible are the modular construction without a central patch panel, as well as the dual oscilloscope for displaying computation results.

RA 1

[page 66: figure only]

RA 1 (continued)

The RA 1, which was a prototype on which different circuit techniques in the area of multipliers were already being tested, was still equipped exclusively with tubes.

The right cabinet contains the four anode-voltage power supplies, the precision power supply for the computing voltages of ±100 V, and a voltage stabilizer.

The center cabinet contains exclusively computing elements (function generators, multipliers, summers/integrators, coefficient potentiometers, etc.), while the left cabinet contains, in addition to some general computing elements, the dual oscilloscope as the central output device.

RA 463 and RA 463/2

From the RA 1, two production systems quickly emerged:

  • RA 463: A small system consisting of only two racks.
  • RA 463/2: An expanded system that, like the RA 1, consists of a total of three cabinets.

Both systems were marketed with some success, but remained the only tube-equipped analog computers from Telefunken. The following figure shows an RA 463/2 system (after [RA463 58]).

RA 463/2

[page 69: figure only]

The First Transistorized Analog Computer

As early as the mid-1950s, Dr.-Ing. Gunter Meyer-Brotz at Telefunken began investigations into the application of transistors in precision operational amplifiers, such as those that would also be suitable for use in electronic analog computers. No other manufacturer of electronic analog computers ventured so early to move away from proven tube technology toward fully transistorized circuits. From these investigations emerged the laboratory prototype of a transistorized analog computer shown in the following picture (after [ERNS60], p. 255).

The First Transistorized Analog Computer (continued)

[page 71: figure only]

RA 800

From this small prototype — which cannot conceal its structural affinity to the RA 1 and RA 463 / RA 463/2 systems — a large precision analog computing system was developed in a short time, which was already presented at the Hannover Industrial Trade Fair in 1960.

This was the world’s first commercially available, fully transistorized precision analog computer, which placed other manufacturers — above all EAI — under pressure to also develop transistorized systems.

The following figure shows such an RA 800 system (after [KLEY64], p. 133) — note the central, easily exchangeable programming patch panel.

RA 800

Based on the transistor technology of the RA 800 system, a series of small transistorized desktop computer systems were developed by Telefunken over the following years (up to the early 1970s), which achieved wide market penetration. These systems were as follows:

RAT 700: The very first model — a direct by-product of the RA 800 development; it was introduced before the RA 800.

RA 740: An improved model with a larger feature set and increased accuracy.

RA 741: Successor system to the RA 740 — no significant new features.

RA 742: The last desktop computer model — new patch-panel design, electronic switches in the integrators.

RA 710: Low-cost model with reduced accuracy.

[page 75: figure only — photographs of the RAT 700 and RA 742 desktop analog computers]

RA 770

Later electronic analog computers possessed, in addition to purely analog computing elements, digital elements as well, by means of which complex computational sequences could be controlled — such as those arising, for example, in optimization tasks. The following image shows the last analog computer system developed by Telefunken, the RA 770. On the left side is the digital control unit; in the center, the various operator and readout systems are combined; while the analog programming panel is arranged on the right.

[page 77: figure only — photograph of the RA 770 analog computer system]

Dornier 960

The Dornier 960 — one of the last analog computing systems ever developed — rounds out the systems presented here as representative examples.

The Dornier 960 is a complete hybrid system, meaning that a digital computer is associated with the analog computer and can work together with it on a problem via DA and AD converters as well as corresponding control lines.

Hybrid computing systems of this kind were built by Dornier well into the late 1980s and are still in use today in isolated areas.

The following illustration is taken from [DO960].

[page 79: figure only — illustration of the Dornier 960 hybrid computer system, from reference DO960]

Sample Problems

In the following, two sample problems are treated by way of example with the aid of an analog computer, in order to illustrate the fundamentals of programming such systems. These are:

  • a mass-spring-damper system, and
  • a simple predator-prey system.

In general, all problems that can be described by differential equations can be treated with the aid of an (electronic) analog computer, which applies to the majority of technically relevant problems.

Basic Elements of the Mass-Spring-Damper System

[figure: mass-spring-damper schematic]

The forces acting in the system are:

  • Inertial force: F_m = ma = m*y”
  • Spring force: F_s = s*y
  • Damping force: F_d = d*y’

The Mass-Spring-Damper System

The equilibrium condition requires:

F_m + F_d + F_s = 0

which gives:

my” + dy’ + s*y = 0

Therefore:

y” = -(dy’ + sy) / m

Setting Up the Computing Plan

A first integration yields:

[diagram: integrator block producing -y’ from y”, with initial condition y’_0]

After a further integration:

[diagram: second integrator block producing y from -y’, with initial condition -y_0; the term s/m feeds back from y]

Setting Up the Computing Plan (continued)

Forming the expression -(dy’ + sy):

[diagram: summer block combining d/m * (-y’) and s/m * y to produce -(dy’ + sy), then divided by m; the full signal-flow path is shown with initial conditions y’_0 and -y_0]

The Complete Computing Circuit

[diagram: complete analog computing circuit for the mass-spring-damper system, showing:

  • Initial condition inputs: y’_0 and -y_0
  • First integrator producing -y’
  • Second integrator producing y
  • Coefficient potentiometers s/m and d/m
  • Summer forming -(dy’ + sy)
  • Coefficient amplifier 1/m producing y”]

The closed-loop circuit implements: y” = -(1/m)(dy’ + sy)

[page 86: figure only — photograph of the physical patch-panel realization of the mass-spring-damper circuit on an analog computer]

Results

The two following images show simulation results for different values of damping:

[page 87: figure only — oscilloscope / recorder output plots showing the response of the mass-spring-damper system for varying damping coefficients (underdamped, critically damped, overdamped cases)]

Predator-Prey System

The predator-prey system presented as the second example is somewhat more complex than the mass-spring-damper system, since for the first time two coupled differential equations appear. A closed ecosystem consisting of a lynx population (l) and a hare population (r) is considered, for which:

r’ = alpha_1 * r - alpha_2 * r * l

l’ = -beta_1 * l + beta_2 * r * l

Here, alpha_1 is the birth rate of the hares, alpha_2 is the rate at which hares are killed by lynxes, beta_1 is the mortality rate of the lynxes, and beta_2 is the increase in the lynx population produced by the food supply.

Computing -r’

[diagram: signal-flow for computing -r’:

  • Initial condition +r_0 feeds an integrator producing r
  • r feeds back through coefficient alpha_1 (producing alpha_1 * r)
  • A multiplier receives r and l, producing r*l; this is scaled by alpha_2
  • A summer computes: -(−alpha_1 * r + alpha_2 * rl) = alpha_1 * r − alpha_2 * rl, which equals r’
  • The integrator output is -r]

Computing -l’

[diagram: signal-flow for computing -l’:

  • The product r*l (from the multiplier shared with the r’ computation) is scaled by beta_2
  • Initial condition +l_0 feeds an integrator producing l (or -l)
  • l feeds back through coefficient beta_1 (producing -beta_1 * l)
  • The sum beta_2 * r*l − beta_1 * l gives l’]

The Complete Circuit

[page 91: figure only — schematic of the complete circuit with labeled nodes: +1 lr0, +1 ll0, l1, l2, l, Er, El, and integrator connections]

Implementation

[page 92: figure only — photograph or diagram of the physical implementation]

Results

[page 93: figure only — results plots or output traces]

Fields of Application

During the period from approximately 1950 to 1980, electronic analog computers were indispensable in virtually every area of engineering, technology, and research. They played an essential role in space travel, aeronautical engineering, the development and optimization of industrial processes, mechanical engineering, nuclear engineering, and more. The most important fields of application for electronic analog computing are presented below by way of example; no claim to completeness is made — given the breadth of the subject areas, this would not be possible in any case.

Mathematics

  • Differential equations (ordinary ODEs, boundary-value problems, nonlinear ODEs, partial differential equations)
  • Integral equations
  • Conformal mappings
  • Fourier synthesis and analysis
  • Stochastics and statistics
  • Optimization problems
  • Monte Carlo methods

Mathematics (continued)

The following example shows the application of a conformal mapping to generate streamlines around a Joukowski airfoil profile using an analog computer.1

[page 96: figure only — streamline plot around a Joukowski airfoil generated by the analog computer]

Physics

  • Planetary orbits (many-body problems)
  • Optics
  • Magnetic lenses
  • Magnetic bearings
  • Heat conduction
  • Ferromagnetic thin films
  • Semiconductor physics

Physics (continued)

The following figure shows the Electronic Analog Frost Computer for investigations of frost effects in soil layers (after [ALDR55], p. 259):

[page 98: figure only — photograph of the Electronic Analog Frost Computer]

Mechanics and Mechanical Engineering

  • Vibration and oscillation studies
  • Shock absorber development (automotive engineering)
  • Rotating systems (bearings, compressors, etc.)
  • Materials science (non-destructive testing, plasto- and elastomechanics)
  • Pneumatics and hydraulics
  • Servo systems

Nuclear Engineering

  • Fundamental research (sodium vapor bubble simulation, xenon poisoning, breeding processes, etc.)
  • Simulation
  • Control and regulation of reactors

Nuclear Engineering (continued)

The following figure shows an AEG reactor simulator with an RA 463 analog computer (after [GERW58]):

[page 101: figure only — photograph of the AEG reactor simulator with RA 463 analog computer]

Biology and Medicine

  • Ecosystems and population dynamics
  • Metabolic studies
  • Pharmacokinetics
  • Cardiovascular systems
  • Neurophysiology

Biology and Medicine (continued)

The following figure shows Richard FitzHugh solving the Hodgkin–Huxley equations on a Beckman analog computer (reproduced with kind permission of Dr. Izhikevich):

[page 103: figure only — photograph of Richard FitzHugh at a Beckman analog computer]

Geology and Oceanography

  • Mineral resources
  • Earthquake simulation
  • Propagation of acoustic waves

Geology and Oceanography (continued)

The following figure shows the Russian ZIS analog computing system for the investigation of hydraulic questions in the context of reservoir research (after [USHA58], p. 1812):

[page 105: figure only — photograph or diagram of the Russian ZIS analog computing system]

Geology and Oceanography (continued)

The following figure shows the analog ray tracer of Light, Badger, and Barnes (after [LIGH66], p. 724):

[page 106: figure only — diagram or photograph of the analog ray tracer]

Energy Technology

  • Transmission lines
  • Electrical supply networks
  • Network simulation
  • Frequency control and network synchronization, interconnected-grid regulation
  • Switching transients
  • Generators
  • Scheduling of power plants
  • Transformers, alternating-current and rectifier systems
  • Direct-current transmission

Energy Technology (continued)

The following figure shows the GEDA Power Dispatch Computer (after [GEDA57]):

[page 108: figure only — photograph or diagram of the GEDA Power Dispatch Computer]

Electronics and Communications Engineering

  • Circuit simulation
  • Frequency response and resonance studies
  • Spectral analysis
  • Filter design
  • Radio networks
  • Correlation analysis

Instrumentation, Control, and Regulation Engineering

  • Data acquisition and processing
  • Control loops
  • Servo systems
  • Sampled-data systems
  • Dedicated computers in control applications

Instrumentation, Control, and Regulation Engineering (continued)

The following illustration shows several typical analog computing modules from General Dynamics that were used in the control of a nuclear reactor:

[page 111: figure only]

Process Engineering

  • Heat exchangers, evaporators, columns
  • Solvent recovery
  • Reaction kinetics
  • Process control
  • Process simulation
  • Adaptive control systems
  • Parameter determination and optimization

Transportation Systems

  • Suspension and shock absorber systems
  • Steering systems
  • Transmission development
  • Traffic flow simulation
  • Dynamic behavior of rail vehicles
  • Magnetic levitation trains
  • Hovercraft
  • Marine engineering

Transportation Systems (continued)

The following illustration shows a simple steering system simulator (after [MCLE58][p. 1995]):

[page 114: figure only]

Aerospace Engineering

  • Flight tables
  • Autopilot development
  • Landing gear vibrations
  • Flight simulation
  • In-flight simulations
  • Engine development
  • Helicopter rotors
  • Flight guidance systems
  • Arresting cable systems
  • Human-machine interface development
  • Rocket engines, multi-stage launch vehicles
  • Flight behavior and control of rockets
  • Launch window determination
  • Rendezvous maneuvers

Aerospace Engineering (continued)

The following illustration shows the EASE 2133 installation at MBB (520 computing amplifiers, 72 multipliers, 64 function generators, 240 servo potentiometers, etc.) after [MBB]:

[page 116: figure only]

Aerospace Engineering (continued)

The development of the control system for the Mercury space capsules also required the use of electronic analog computers, as the following illustration shows:

[page 117: figure only]

Military Applications

  • Combat simulations
  • Projectile trajectories
  • Anti-aircraft simulators
  • Missile guidance (Nike, Atlas, Polaris, etc.)

Military Applications (continued)

The following image shows the Tactical Avionics System Simulator (reproduced with the kind permission of Bruce Baker):

[page 119: figure only]

Military Applications (continued)

Several early missile systems were guided with the aid of ground-based analog computers — one example of this is the Nike Research and Development computer (ca. 1951) shown below, after [FAGE78][p. 383] (computer shown open):

[page 120: figure only]

Art, Music, and Entertainment

Analog computers have been used — and in some cases continue to be used — in the fields of art, music, and entertainment, as the following examples show. In the realm of music, the composer Hans Kulk (Netherlands) works with analog computers, which he employs to control analog synthesizers.

Art, Music, and Entertainment (continued)

The following illustration shows a graphic produced with the aid of an electronic analog computer by the artist Prof. Herbert W. Franke (reproduced with the kind permission of the artist):

[page 122: figure only]

Art, Music, and Entertainment (continued)

Heinrich Heidersberger also created a large number of well-known artworks using mechanical analog computers — one example is the picture “Prelude” shown below (reproduced with the kind permission of Benjamin Heidersberger):

[page 123: figure only]

Decline of Analog Computing

As early as the beginning of the 1970s, the decline of electronic analog computing began to make itself apparent. The principal cause was the enormous gain in performance of digital computing systems, together with their falling prices and rapidly growing proliferation. Today, electronic analog computers have almost completely disappeared from the market as well as from research and education, and the knowledge of analog computing technology is in danger of being forgotten. Nevertheless, analog computers still retain certain advantages over digital computers — advantages that, through the use of digital implementation techniques while avoiding the drawbacks of analog electronics, might perhaps lead to a renaissance of the analog computing concept.

Advantages of Analog Computing

  • Extremely close proximity to the problem — the question being studied is not obscured by its algorithmic description.
  • High interactivity — running computations can be intervened in at any time, either manually or automatically.
  • Extreme parallelism of computing elements, which is in principle unattainable by stored-program digital computers.

Future of Analog Computing

Using modern technologies such as FPGAs, purely digital implementations of typical analog computing elements in the style of earlier digital differential analyzers would be conceivable.

With the aid of such an approach, the advantages of analog computing — above all its inherent parallelism — could be combined with the advantages of modern digital computers — arbitrary computational precision, program-controlled configuration. Such an implementation could, for example, take the form of a coprocessor with which dynamic simulations could be carried out that still pose difficulties for conventional digital computers even today.

A Personal Note

Electronic analog computing is not only the subject of the author’s doctoral research but above all a great personal passion — for this reason, a request:

Should anyone be aware of an analog computer that is in need of a good new home, please make this known. Every effort is made to rescue such machines from impending scrapping and oblivion; even large systems, such as an RA 800(H), are not a deterrent.

If anyone possesses literature, articles, notes, etc. relating to analog computing and/or its applications, it would be greatly appreciated if these could be made available on loan for scanning, in order to make them accessible to a wider audience.

Contact is always possible at [email protected] or by telephone at 0177/5633531. Many thanks for your understanding and support.

References

[ALDR55] St. P. Aldrich, H. M. Paynter, “First Interim Report on Analytic Studies of Freezing and Thawing of Soils (for the Arctic Construction and Frost Effects Laboratory New England Division, Corps of Engineers)”, in [PAYN55], pp. 247–260

[BALL60] B. Ball, “Ein Digitalvoltmeter höherer Genauigkeit mit Transistoren” [A high-accuracy digital voltmeter using transistors], in Telefunken Zeitung, Vol. 33 (September 1960), Issue 129, pp. 211–216

[BORD40] Bureau of Ordnance (ed.), Basic Fire Control Mechanisms, OP 1140, September 1940

[BORD44] Bureau of Ordnance Publication (ed.), Torpedo Data Computer, Mark 3, Mods. 5 to 12 inclusive, June 1944

[BORD46] Bureau of Ordnance (ed.), Basic Fire Control Mechanisms: Maintenance, OP 1140 A, 1946

[DRES72] Fritz Dreslер, “Das Dach” [The Roof], in hobby — Das Magazin der Technik, Nr. 8/72, pp. 50 ff.

[ERNS60] Dietrich Ernst, Elektronische Analogrechner — Wirkungsweise und Anwendung [Electronic Analog Computers — Operating Principles and Applications], R. Oldenbourg Verlag München, 1960

[FAGE78] M. D. Fagen (ed.), A History of Engineering and Science in the Bell System — National Service in War and Peace (1925–1975), Bell Telephone Laboratories, Inc., First Printing, 1978

[GERW58] Robert Gerwin, “Atom-Strom für deutsche Städte” [Atomic power for German cities], in hobby — Das Magazin der Technik, Nr. 9, September 1958

[JEZI00] Dieter von Jezierski, Slide Rules — A Journey Through Three Centuries, Astragal Press, Mendham, New Jersey, 2000

[KARP58] Walter J. Karplus, Walter W. Soroka, Analog Methods — Computation and Simulation, McGraw-Hill Book Company, Inc., 1958

[KETT57] E. Kettl, “Übersicht über die Technik der elektronischen Analogrechner” [Survey of the technology of electronic analog computers], in Telefunken Zeitung, Vol. 30 (June 1957), Issue 116, pp. 129–135

[KLEY64] Adolf Kley, “Analogrechner” [Analog computers], in Kybernetische Maschinen, edited by Helmut Frank, S. Fischer Verlag, 1964, pp. 174–183

[LIGH66] L. Light, J. Badger, D. Barnies, “An Automatic Acoustic Ray Tracing Computer”, in IEEE Transactions on Electronic Computers, Vol. EC-15, No. 5, October 1966, pp. 719–725

[MCLE58] John H. McLoed, “Instruments and Automation”, in Simulation Council Newsletter, Vol. 31, December 1958, pp. 1991–1997

[A4 45] N. N., Das Gerät A4 Baureihe B, Teil III, Gerätebeschreibung V2 [The A4 Device, Series B, Part III, Equipment Description V2], OKH/Wa A/Wa Prüf, Annex to Bb.Nr 19/45 gK, 1.2.1945

[RA463 58] Telefunken, N. H., “Elektronischer Analogrechner RA 463/2” [Electronic Analog Computer RA 463/2], 5. Apr. 58

[MBB] N. N., MBB Simulation, company publication, Messerschmitt Bölkow Blohm GmbH, Unternehmensbereich Flugzeuge [Aircraft Division]

[GEDA57] N. N., “New GE DA Power Dispatch Computer”, in Instruments and Automation, Vol. 30, February 1957, p. 179

[DO960] N. N., “Simulationssystem Dornier 960” [Dornier 960 Simulation System], Dornier System GmbH

[PAYN55] Henry M. Paynter (ed.), A Palimpsest on the Electronic Analog Art, printed by Geo. A. Philbrick Researches Inc., AD 1955

[STEI] N. N., “Steinheils optische Werkstätten, Schirm C. A., Monographien” [Steinheil Optical Works, Screen C. A., Monographs], Steinheil Söhne GmbH (Optik), München

[SVOB48] Antonin Svoboda, Computing Mechanisms and Linkages, McGraw-Hill Book Company, Inc., 1948

[TREN82] Fritz Trenkle, Die deutschen Funklenkverfahren bis 1945 [German Radio Control Methods up to 1945], AEG-TELEFUNKEN AKTIENGESELLSCHAFT, 1982, Anlagentechnik, Geschäftsbereich Hochfrequenztechnik [Systems Technology, High-Frequency Technology Division]

[USHA58] V. B. Ushakov, “Soviet Trends in Computers for Control of Manufacturing Processes”, in Instruments and Automation, December 1958, pp. 1810–1813

Footnotes

  1. See http://www.vaxman.de/analog_computing/joukowski/joukowski.html ↩