Calculation Examples: An Introduction to Analog Computer Methods

This document is an English translation of the original German text “Rechenbeispiele” (Vogel AR Einführung), a practical introduction to analog computing techniques.

Basic Operations with Summers
- 1.1 Multiplication by a constant
- 1.2 Sum of three variables
- 1.3 Solution of a linear system of equations with two unknowns
Basic Operations with Integrators
- 2.1 Integration of a constant input quantity y = ∫a dt
- 2.2 Integration of a linearly time-dependent variable y = ∫(a·t + b) dt
Computing Operations with the Multiplier Unit
- 3.1 Multiplication: y = x₁·x₂
- 3.2 Division: y = x₁/x₂
- 3.3 Squaring: y = x²
- 3.4 Square root: y = √x
- 3.5 Solution of a quadratic equation
Realization of Trigonometric Functions
- 4.1 Forming the function y = sin x
- 4.2 Forming the function y = cos x
- 4.3 Forming the function y = sin 2x
Realization of Discontinuous Functions
- 5.1 Comparison of two variables (comparator)
- 5.2 Forming the function y = sign x
- 5.3 Forming the function y = |x|
Solution of Linear First-Order Differential Equations
- 6.1 Solution of y’ = y
- 6.2 Solution of y’ = −k·y
Solution of Linear Second-Order Differential Equations
- 7.1 Solution of k₃·y” + k₂·y’ + k₁·y = 0
- 7.2 Solution of y” = −ω²·y

Introduction

The following calculation examples are intended to provide an introductory overview of the methods of analog computing. Programming details such as variable and time normalization, or the choice of initial conditions, are only addressed to the extent necessary for understanding. In the following examples, variables are given in dimensionless machine units. The machine unit +1 corresponds to a computing voltage of +10 Volts; the unit −1 corresponds to a voltage of −10 V.

1. Basic Operations with Summers

1.1 Multiplication by a Constant

The summers of the analog computer have the general transfer function:

y = −(x₁ + x₂ + 10·x₃) (1.1)

If only one input is used (circuit a), equation (1.1) reduces to:

y = −x

The summer then acts as a simple inverter. If the same input variable is applied to two summer inputs (circuit b), the output variable becomes:

y = −2x

The summer now acts like a multiplier that multiplies the input quantity by a constant factor of 2. If the output variable y is fed back to one input (circuit c), the summer has the same function as a coefficient potentiometer. Since y = −(x + y), the result is:

y = −0.5·x

For circuit variant d), from equation (1.1):

y = −(10·x + y + y) → 3y = −10·x → y = −3.33·x

1.2 Sum of Three Variables

y = x₁ − 2.5·x₂ + 0.5·x₃ + 1

The computing circuit uses three summers. Summer S1 outputs:

y₁ = −(−2.5·x₂ + x₁ + 1)

Summer S2 outputs: y₂ = −0.5·x₃. Summer S3 then gives: y = −(y₁ + y₂) = x₁ − 2.5·x₂ + 0.5·x₃ + 1.

Numerical example 1: x₁ = 0.4, x₂ = 0.2, x₃ = −1.0 → y = 0.4

Numerical example 2: x₁ = −0.7, x₂ = 0.7, x₃ = 0.9 → computed result would be y = −1.0. Although this result lies within the valid computing range −1.0 ≤ y ≤ +1.0, the actual measured result is wrong because summer S1 is already overdriven: y₁ = −(−1.75 − 0.7 + 1) = 1.45.

1.3 Solution of a Linear System of Two Equations

x₁ + 0.5·x₂ − 0.3 = 0 0.2·x₁ + x₂ + 0.3 = 0

Rearranging to separate variables:

x₁ = −(0.5·x₂ − 0.3) x₂ = −(0.2·x₁ + 0.3)

The computing circuit implements these two equations directly. The solutions are: x₁ = 0.5, x₂ = −0.4.

2. Basic Operations with Integrators

2.1 Integration of a Constant Input Quantity

The integrators of the analog computer have the general transfer function (2.1). If only the integrator input with weight factor 1 is used, the equation reduces to (2.2).

In the demonstration circuit, the input quantity x₁ is formed by a summer. When switch S1 is open, x₁ takes the constant value a = +0.4. When S1 is closed, x₁ changes from +0.4 to −0.6.

From integral calculus: ∫a dt = a·t (2.3). The integrator output variable is therefore:

y = −x₁·t + IC (2.4)

At time t₀, the integrator mode is switched from IC (initial condition) to RUN, starting integration with initial value +0.8. The variable y decreases linearly over time. At t₁, x₁ is switched to a = −0.6, causing the direction of y to reverse and the slope to change (since |x₁| also increases from 0.4 to 0.6).

2.2 Integration of a Linearly Time-Dependent Variable

The circuit from the previous example is extended with a second integrator. Following the integral calculus rule:

∫(a·t + b) dt = a·t²/2 + b·t

The output variable of the second integrator takes a quadratic time profile.

At t₀ both integrators are started. At t₁ the input x₁ of the first integrator is again switched to a negative value. At each zero crossing of x₂, y has an inflection point. At t₂ both integrator outputs exceed their linear computing range of ±10 V and shortly afterwards enter the saturation region, limited by the supply voltage (±15 V).

3. Computing Operations with the Multiplier Unit

The general transfer function of the multiplier unit is:

U = A·(X₁·(Y₁ − Y₂) + Z₂ − Z₁) (3.1)

where X₁, Y₁, Y₂, Z₁, Z₂ are the inputs, U is the output, and A is the internal gain factor (A > 10,000).

The permitted value ranges for the multiplier input variables depend on the operating mode:

Mode	Input range
Multiplication	−1 ≤ x₁,x₂ ≤ +1
Division	−1 ≤ x₁ ≤ +1, 0 ≤ x₂ ≤ +1
Squaring	−1 ≤ x ≤ +1
Square root	0 ≤ x ≤ +1

3.1 Multiplication

For multiplication, the circuit yields from (3.1):

y = A·(x₁·x₂ − y) → y·(A+1) = A·x₁·x₂

Since A >> 1: y = x₁·x₂

3.2 Division

For division: y·(1 + A·x₂) = A·x₁. Since A >> 1: y·x₂ = x₁, therefore y = x₁/x₂

3.3 Squaring

If x₁ = x₂ = x, the multiplier unit acts as a squarer: y = x²

3.4 Square Root

A diode inserted in the feedback loop ensures that no negative voltages appear at inputs X₁ and Y₂. From (3.1): y = A·(−y² + x). Since A >> 1: y² = x, therefore y = √x

Since the square root can only be computed for positive values of x, a pre-circuit must convert the bipolar signal from the triangle-wave generator to a unipolar signal.

3.5 Solution of the Quadratic Equation

y = x² − 0.7·x − 0.2

The first summer outputs: y₁ = −(0.7·x + 0.2). The second summer outputs: y₂ = −(y₁ + x²) = −(x² − 0.7·x − 0.2). An inverter then gives: y = x² − 0.7·x − 0.2. The equation has two roots at x₁ = −0.22 and x₂ = +0.92. For x ≤ −0.8, the computing circuit overdrives.

4. Realization of Trigonometric Functions

4.1 Forming the Function y = sin x

The diode function generator of the analog computer produces y = sin x for −1 ≤ x ≤ +1. Via the scaling factor x = φ/90°, the input variable x is proportional to an angle φ in the range −90° ≤ φ ≤ +90°.

4.2 Forming the Function y = cos x

The cosine function is realized using: cos φ = sin(90° − φ) (4.1). With the same scaling x = φ/90°:

cos x = sin(1 − x) (4.2)

Since cos x takes only positive values in −1 ≤ x ≤ +1, the absolute value |x| is fed to the summer rather than x itself. The result is y = cos x.

4.3 Forming the Function y = sin 2x

From two harmonic variables x₁ = cos x and x₂ = sin x, a variable with double frequency is formed using:

sin 2x = 2·sin x·cos x (4.3)

Two integrators generate x₁ and x₂, which are fed to a multiplier. The product y* = x₁·x₂ = ½·sin 2x (4.4) has double the frequency of the inputs but only half the amplitude. A subsequent summer and inverter form the final result: y = 2·y* = sin 2x (4.5).

5. Realization of Discontinuous Functions

5.1 Comparison of Two Variables (Comparator)

Variables x₁ and inverted x₂ are applied to an open-loop amplifier. Due to its very high gain, the output x₃ satisfies:

x₃ = −U_S for x₁ > x₂ ; x₃ = +U_S for x₁ < x₂

(±U_S are the saturation voltages of the open-loop amplifier, approx. ±14 V). A diode suppresses positive portions of x₃; a coefficient potentiometer limits negative portions to −1; an inverter delivers the final result:

y = +1 for x₁ > x₂ ; y = 0 for x₁ < x₂ (5.1)

5.2 Forming the Function y = sign x

Setting x₂ = 0 in the previous example gives:

y = +1 for x > 0 ; y = 0 for x < 0 → y = sign x

(The inverter is no longer needed when x₂ = 0.)

5.3 Forming the Function y = |x|

The circuit computes the absolute value of a variable x:

For positive x: diode D1 is reverse-biased, D2 conducts, and the open-loop amplifier acts as a simple inverter: x₁ = −x.
For negative x: D2 is reverse-biased: x₁ = 0.

The summer then produces y = −(x + 2·x₁), giving:

y = x for x > 0 ; y = −x for x < 0 → y = |x|

6. Solution of Linear First-Order Differential Equations

6.1 Solution of y’(t) = y(t)

Assuming y’(t) is at the integrator input, −y(t) appears at its output. The output is fed back through an inverter to satisfy y’(t) = y(t). Using the integration rule ∫e^(a·t) = (1/a)·e^(a·t) with a = 1:

y(t) = y(0)·eᵗ (6.3)

An integrator time constant T = 1 s is used. Solutions for various initial conditions y(0) are shown.

6.2 Solution of y’(t) = −k·y(t)

The setup is identical to 6.1, but the inverter is replaced by a coefficient potentiometer. With a = −k:

y(t) = y(0)·e^(−k·t) (6.5)

Solutions for various coefficients k and initial conditions y(0) = ±1 are shown, with integrator time constant T = 1 s.

7. Solution of Linear Second-Order Differential Equations

7.1 Solution of k₃·y” + k₂·y’ + k₁·y = 0

The equation is rearranged to isolate the highest derivative:

y” = −(1/k₃)·(k₂·y’ + k₁·y) (7.2)

With y” at the first integrator input, −y’ appears at its output, and y at the next integrator output. A summer provides −(k₂·y’ + k₁·y), which is fed back via coefficient potentiometer k₃ to satisfy (7.2).

This homogeneous second-order ODE describes the motion of a spring-mass system with mass m, spring constant c, and damping constant d:

m·y” + d·y’ + c·y = 0 (7.3)

Correspondence: k₃ = m, k₂ = d, k₁ = c. (Unlike previous examples, these coefficients are not dimensionless.)

With k₃ = m = 0.8, k₂ = d = 0.01, y(0) = −1, y’(0) = 0, the influence of spring constant k₁ on oscillation frequency is demonstrated. The influence of damping constant k₂ is shown for k₃ = 0.8, k₁ = 0.8, y(0) = −1, y’(0) = 0, with k₂ varying from 0.02 to 1.0 (underdamped through overdamped behavior).

7.2 Solution of y” = −ω²·y

Setting the damping factor k₂ = 0 in equation (7.1) yields an undamped oscillation:

k₃·y” + k₁·y = 0 → y” = −(k₁/k₃)·y (7.4)

The expression √(k₁/k₃) is called the angular frequency ω, where ω = 2π·f (7.5). The equation becomes:

y” = −ω²·y (7.6)

With the ansatz y” = −ω²·sin ωt (7.7), integrators with time constant T = 1 s and weight factor C yield:

y’ = C·ω·cos ωt (7.8) y = C²·sin ωt (7.9)

Connecting the second integrator’s output through an inverter back to the first integrator’s input requires: −ω²·sin ωt = −C²·sin ωt, so ω = C. Therefore:

f = C/(2π)

For C = 1: f = 0.159 Hz. For C = 10: f = 1.59 Hz.

The two integrator outputs produce two sinusoidal signals 90° out of phase. Due to the non-ideal properties of real computing amplifiers and integration capacitors with losses, a slight oscillation damping occurs in practice, causing the amplitudes to decay slowly.

[Translation covers all 18 pages of the original document — complete translation.]