The Electronic Analog Computer and Its Use in Industry

This is an English translation of the original German-language document (“Der elektronische Analogrechner und seine Verwendung in der Industrie”).

Part I

History of the analog computer
Structure and operating principle of the analog computer
Programming, explained with an example

Part II

Application of the analog computer, illustrated using the example of electric braking of a field-controlled motor
Industrial deployment of the analog computer, explained through various technical problems
Comparison of the essential characteristics of digital and analog computers

Telefunken Analog Computer, built 1957 (tube-equipped)

Part I

1. History of the Analog Computer

The analog computer is an aid for solving ordinary differential equations. The differential equation is modeled by an electrical circuit, i.e., the model is described by the same differential equation that is submitted for solution. Smaller analog computers have existed since 1880; however, these operated purely mechanically and were called “integrating systems.” With them, only integrals could be determined or differential equations of low order solved, because the mechanical implementation was very elaborate and bulky. Only during and after World War II did the development of electronic computers begin — computers that can solve extensive problems with far less effort. On a medium-sized computer, for example, a nonlinear differential equation of 30th order can easily be represented; hence, electronic computers are now very widespread.

2. Structure and Operating Principle of an Electronic Analog Computer

The modern electronic analog computer is a so-called “DC voltage computer.” The problem variable corresponds at the associated computing circuit to a voltage as the “model variable,” which behaves exactly as the problem variable.

What electronic components are needed to model an ordinary differential equation? A simple example may help clarify this question.

Given is the equation:

L·(di/dt) + i·R = u

(Current and voltage behavior at an iron choke coil connected to a voltage source.)

The computing scheme for solving this equation is:

di/dt = (u − i·R)/L
i = ∫(di/dt)dt
i = f(V_F)

It is easy to state that in a solution scheme for any ordinary differential equation, only the following computing operations ever occur:

a) Summation: y = x₁ + x₂ b) Integration: y = ∫x dt c) Multiplication by a constant: y = k·x d) Multiplication of two variables: y = x₁·x₂ e) Formation of a function: y = f(x) f) Sign reversal: y = −x

For each of these operations there is an electronic building block. Every analog computer contains a large number of such “computing elements.” How they must be interconnected for a given differential equation or an entire system is determined by “programming.” Programming therefore means adapting a task to the capabilities of the analog computer. In contrast to the digital computer, however, no decomposition into a sum of individual steps is necessary — only a “rewriting” and “rescaling” of the given differential equation. The term “rewriting” refers to the reformulation of the differential equation as shown above; by “rescaling,” normalization and time transformation are meant (see Section 3).

In the analog computer, all interconnected building blocks work simultaneously on the solution. The computation is therefore not divided into temporally successive steps as in a digital computer. Since integration in the electronic analog computer is modeled by the charging of an ideal capacitor, only one independent variable can appear, namely time. The system of equations may then also have only one independent variable, which is represented at the computer by time. Therefore, for example, partial differential equations can only be handled — if at all — by indirect methods. However, technical problems are very often described by ordinary differential equations with initial or boundary values, for which the analog computer is particularly suitable. The equations can be arbitrarily nonlinear. This is precisely where the great advantage of the analog computer lies, because there are no general solution methods for such equations.

Since the “electrical model” is isomorphic to the system of equations under investigation, the solution initially runs in real time when time is the independent variable. By suitable substitution of the independent variable, however, a time compression or expansion can be achieved. This is an extremely effective aid in the investigation of very slowly or very rapidly progressing processes. One can always ensure that the solution runs in a time span of seconds to a few minutes, regardless of the scope and complexity of the given system of equations.

The “solution” of the represented system appears in curve form, namely as an oscilloscope trace of the corresponding voltages, which are proportional to the sought system quantities.

3. Programming, Explained with an Example

Already in Section 2, the scheme for solving a differential equation was given. Now consider the following equations (a sprung mass with damping and an exciting force):

ds/dt = v
dv/dt = [P(t) − c·s − d·v] / m
s = ∫v dt + s(0)

At the analog computer, the quantities P = force, s = displacement, v = velocity correspond to variable voltages, which naturally cannot assume arbitrary values but are limited by the construction of the computer’s building blocks. The computer has a certain voltage defined as the “unit voltage” or “machine unit (m.u.),” which the amplifiers can still process correctly — usually 100 V or 10 V equals 1 m.u. This unit is dimensionless. The given system of equations must be reformulated so that the variables also appear as dimensionless quantities whose absolute value remains ≤ 1. If this succeeds, it is guaranteed that the permissible computing voltage at the computer will not be exceeded.

The new variables are introduced as follows: the previous variables — in the example P, s, and v — are referenced to “normalization values” that are greater than or at least equal to the maximum values these variables can take; these normalization values are denoted for the example as P_m, s_m, and v_m.

After substitution with normalized variables P/P_m, s/s_m, v/v_m, the equations are rearranged. The coefficients such as c·s_m / (m·v_m), P_m / (m·v_m), etc., are fixed values that can be set on potentiometers.

The previously mentioned “time transformation” — i.e., the replacement of the independent variable (in the example, time t) by a new “machine variable” — is accomplished as follows: t is substituted by T, with both linked by the relation T = λ·t (0 < λ < ∞). The factor λ indicates whether a time compression or expansion is being performed:

0 < λ < 1 means time compression, i.e., the solution runs at the computer correspondingly faster than in nature.
1 < λ < ∞ means time expansion, i.e., the solution runs correspondingly slower.

Through the time transformation, one can always ensure that the solution of a system of equations is carried out on the computer in seconds to minutes, regardless of how complex the relevant system of equations is. The voltages analog to the problem variables are either displayed on an oscilloscope or recorded with a plotter. Through normalization and the setting of λ, one knows the scale relationship between actual quantities and the analog voltages.

The accuracy of the solution amounts to 1–5% depending on the type and scope of the problem. For technical purposes, this is usually sufficient, since when setting up equations one already makes simplifications, and moreover the coefficients can usually only be determined to within a few percent — consider, for example, modeling an electrical machine. If greater accuracy is required, a digital computer must be used.

Parameter variations — e.g., variation of c, d, or m in the example — can be performed very quickly at the analog computer; they are simply carried out by adjusting potentiometers. Introduction of nonlinearities is equally easy. It suffices if the nonlinearity is given in the form of a curve. This curve is immediately approximated by a piecewise linear function in a function generator. Discontinuities, such as the operation of contactors, etc., can also easily be modeled, either by function generators or by relays.

Part II

1. Application of the Analog Computer: Electric Braking of a Field-Controlled Motor

As an example of the representation and investigation of technical systems on the analog computer, the following problem is used:

A DC motor with separate excitation is at a certain initial speed n₀ and is to be electrically braked. The voltage at the load resistance is to be kept constant by field control.

The control is described by the following system of equations:

(1) u_A = k₁·Φ·n
(2) Δu = u_soll − u_ist = u_soll − k₂·u_A = u_soll − k₂·k₁·Φ·n
(3) i_F = f(Δu)  [given as a measured curve]
(4) V_F = ∫[u_F − i_F·R_F(V_F)]dt
(5) Φ = f(V_F)  [given as a measured curve]
(6) E_a = k₃·Φ·n
(7) L_A·(di_A/dt) + R_K·i_A = E_a − u_A
(8) M_A = i_A·Φ
(9) M_D = k₃·Φ·i_A
(10) n = n₀ − k₄·∫M_D dt

Disregarding the normalization step (which only changes the coefficients), one can already indicate the computing circuit in principle. The dashed-line boxes indicate the equation realized at each point. The amplifiers drawn as dashed lines are required only for technical reasons and are not essential to the principle of the circuit.

Building blocks used:

Summing amplifiers
Integrators
Multipliers
Function generators

2. Industrial Deployment of the Analog Computer

Above all in control engineering, the analog computer is indispensable for investigating the dynamic behavior of a system with changes in setpoint values or with the occurrence of disturbances. One example of such an investigation is the voltage control of a synchronous generator for a power plant. Modeled on the analog computer were the dynamic characteristics of the generator, the exciter machine, an Amplidyne as amplifying machine, as well as several magnetic amplifiers and electronic amplifiers — i.e., the entire control loop.

In controls generally, the controller must first be designed so that the system is stable (does not tend to oscillate) and achieves the most aperiodic possible response to setpoint changes or disturbance rejection. For this purpose, the controller must also have a time behavior that can be determined on the analog computer to meet the above requirements. After the controller for voltage regulation was designed with the aid of the computer, load switching operations were modeled on the computer and the resulting voltage dips and overshoots were investigated, as well as the times in which these disturbances were brought back under control.

As an example from drive control, the investigation of the Leonard drive for reversing rolling mills is cited. Besides determining the controller based on the mentioned criteria, it was necessary to investigate, among other things, how the enormous load changes during rolling — from no-load to instantaneous full load and back — would affect the speed of the drive machine and the generator voltage, and what measures should be taken to achieve the smallest possible and brief speed drop. Further questions concerned the design of a current limiter and the dimensioning of various auxiliary machines.

The use of the analog computer is, however, not limited only to the investigation of electrical systems. For example, the dynamic behavior of nuclear reactors and their control has been modeled and studied. This involved modeling the so-called neutron kinetics — i.e., the actual nuclear fission process — as well as the generation of steam, steam pressure, and other quantities as a result of the released fission energy. As examples from mechanics and thermodynamics, the modeling of a turbine with intermediate superheating is cited. A well-known mechanical problem is the design of a car’s suspension, which can also be done with the aid of the analog computer. Another interesting problem is the determination of trajectories of elementary particles under the influence of magnetic fields in a cyclotron. This list of already completed investigations could be extended at will.

3. Comparison of Essential Characteristics of Digital and Analog Computers

Feature	Digital Computer	Analog Computer
Architecture	A computing unit that handles all operations sequentially	Separate building block for each operation (modular), all elements operate simultaneously
Computing time	Depends on problem scope and required accuracy	Independent of problem scope and accuracy
Accuracy	Can be increased at the cost of computing time	Depends on construction of components; cannot be changed
Parameter variation range	Practically unlimited	Limited
Program storage	Possible	Only conditionally possible when a patch board is present
Number storage capacity	Large	Very limited in number and quality of storage

Conclusion:

The digital computer is a universal computing device for solving tasks of any kind; especially suited for processing large amounts of data (business calculations), solving algebraic and transcendental equations, computing function values, etc. Since computing steps are executed sequentially and there is no dedicated element for integration, it is poorly suited for solving differential equations.
The analog computer is primarily suited for solving ordinary differential equations, because it possesses integrators and can solve equations directly — without decomposing them into individual steps. Processing of data material is not possible due to limited storage capacity and the architecture of the computer.

[Translation covers all 13 pages of the original document.]