English translation
Tisch-Analogrechner: Beschreibung und Bedienungsanleitung (Desktop Analog Computer: Description and Operating Manual)
Complete English translation of the original German-language document (82 pages).
[page 1: title page — BBC / Tisch-Analogrechner / Beschreibung und Bedienungsanleitung / MV 2 460568 / Brown, Boveri & Cie, Aktiengesellschaft, Mannheim]
1. Fundamental Operating Principles of Electronic Analog Computers
1.1 Application
Electronic analog computers are used for solving differential equations or systems of differential equations, which arise as mathematical descriptions of physical processes.
As the name “analog computer” suggests, the variables being investigated — which are continuous functions of time — are represented by electrical voltages. Electronic analog computers are therefore particularly well suited for solving differential equations; being electronic, they can process these functions at many different speeds.
The presentation of the solution, expressed through voltage-time graphs or recorder traces, gives the user direct access to the results. This property also makes the analog computer an excellent tool for studying complex relationships through systematic variation of the individual parameters (the “knees” of the curves). It is also possible to study different models and to compare these with one another using an oscillograph or recorder. In the analog computer case, both of the latter instruments show the user a direct picture of the physical process being sought.
The great advantage of the analog computer lies in the ease with which parameters can be changed, so that the variable can be studied as the result of those parameter variations interactively. Problems can be programmed and the behavior of the inter-related physical variables examined by varying individual parameters (trial-and-error method). One can, for example, simultaneously study many different trajectories with an oscillograph or recorder.
[page 3]
A further advantage of the analog computer is its slight and straightforward manageability. It is a basic principle that as many as desired parameters can be set on the analog computer, the intermediate results of various computing elements can be studied by voltmeter or oscillograph, and finally the result can be read directly.
The study of very rapidly occurring processes (transient processes, i.e., very rapidly running events) is made possible by appropriate time scaling. These very fast processes can be slowed down, displayed on an oscillograph or plotter, and thus can be measured.
Very slowly occurring events (e.g., ageing phenomena) can be represented for study at a much higher speed, so that the results become available much more quickly.
Through the use of suitable repetitive operation (progressive scanning) — realized by means of so-called repetitive-operation generators — display of results on an oscilloscope becomes possible.
Through the use of a suitably designed hold circuit, very rapidly occurring processes (for example, surge voltage processes) can be interrupted at any chosen instant by a triggering device and held in a frozen state so that they can be read on a meter.
[page 4: figure only — figure caption reads “Bild 1a: Ersatzbild und Steuerkanäle eines Rechenverstärkers” (Fig. 1a: Equivalent circuit and control channels of a computing amplifier)]
With these properties, the analog computer is an ideal aid for research, design, and training.
1.2 Basic Computing Elements
1.21 Computing Amplifier
The most important element of an electronic analog computer is the computing amplifier, also called an operational amplifier. It is a DC amplifier that continuously amplifies a negative and a positive input voltage with a very high and constant gain. It delivers, at its output, a ramp voltage, i.e., a voltage of continuously variable magnitude. The gain is negative in sense at the output; that is, the output voltage has the opposite polarity from the applied input voltage (see Fig. 1).
Fig. 1a shows the equivalent circuit and control channels of a computing amplifier.
[page 5]
The gain, expressed through the voltage ratio:
$$V = -\frac{U_{aus}}{U_e} \tag{1}$$
is for the computing amplifier very large, i.e., the output voltage from null to full scale can result from a very small input voltage. The input resistance R_e of the computing amplifier is very small, so that it draws virtually no current from the input circuit. Since in an analog computer various inputs are combined, the different inputs must not influence one another. With an ideal amplifier the input resistance would be zero. In a practical analog computer the input resistance is always finite but small enough that it can be neglected for all computations.
The output voltage of the computing amplifier is not arbitrarily large; it is limited in amplitude by the power supply voltage. This is called the output voltage limiting. The value reached during this limiting determines the maximum permissible computing amplitude, since beyond this limit the results become falsified. Since all computing amplifiers have a common reference voltage (the full-scale value), one must measure all voltages U_1, U_2 and U_max relative to this common reference level.
The output voltage of the computing amplifier should not reach the value of the supply voltage; it is limited to 100 V, i.e., it is set at ±10 V by means of a voltage divider by a factor of 10, or by ±15 V with a factor of 15.
[page 6]
As can be seen from the circuit in Fig. 1, the output voltage is opposite in polarity to the input voltage, since the amplifier reverses the signal. This reversal is fundamental; it is needed for the following calculations.
By adding resistors and capacitors, one can, with the aid of a computing amplifier, perform the following computing operations:
1.22 Summation
Fig. 2 shows a computing amplifier that, through the connection of multiple input resistors, is able to perform addition as well as subtraction. Der computing amplifier becomes a summation amplifier by this means. It is not able, due to the individual input resistors, to perform more than one computing function; one must also be able to choose a suitable reference direction for all voltages. This is called the reference direction convention and it causes the computing amplifier to accept positive as well as negative signals at its input, since the output voltage is always in the opposite direction. The computing requirement is in this case always that, with the necessary reference direction, the amplifier supplies directly an inverting summing output.
[page 7: figure only with caption]
Fig. 2: Addition and subtraction with a computing amplifier
If one feeds, for example, a positive voltage U_1 into the input resistor R_1, then a current flows through R_1 toward the amplifier in the negative direction. The amplifier produces a negative output voltage to balance this input current. The amplifier works with a summation node, i.e., all the currents flowing through the input resistors and the feedback resistor must sum to zero (Kirchhoff’s current law). The output voltage is therefore in the opposite direction and the amplifier thus fulfills directly formula (2):
$$U_{aus} = -\frac{R_f}{R_1} \cdot U_1 \tag{2}$$
If R_1 = R_f, then U_max = –U_1, i.e., the signal has been amplified by a summation amplifier with a gain of 1.
[page 8]
If one makes R_f several times larger than R_1, it is then possible to achieve a gain V greater than 1; one obtains for any n inputs the condition (from formula 2):
$$U_{aus,max} = -(V_1 + V_2 + \ldots + V_n) \tag{3}$$
For the case where all resistors are equal and all voltage gains are 1, it follows:
$$U_{aus} = U_1 + U_2 + U_3 + U_4 \tag{4}$$
The output voltage is also the sum of all (positive) input voltages with the summing amplifier used in this way. The sign must be noted carefully; the output voltage is negative.
In order to obtain a positive output from a summation, one connects the signal once through a second summation amplifier. To achieve subtraction, one must also reverse the sign; this is done by connecting one input via a sign-reversal amplifier.
If one chooses R_f and R_1 in Fig. 4 so that R_f = 10 R_1, one obtains amplification by a factor of 10. If one also connects the inputs U_1 and U_2 via input resistors such that the voltages are multiplied by a constant factor, the result is:
$$U_{aus} = -(10 U_1 + 10 U_2 + 10 U_3 + 10 U_4 + 1 \cdot U_5) \tag{5}$$
[page 9]
Fig. 3 shows the circuit diagram representation that is used in Fig. 2 above.
Fig. 3: Circuit symbol for a summation amplifier
The feedback resistance was left unchanged, while the individual input resistors were changed to suitable values. The numbers alongside the individual inputs denote the multiplication factor for the respective input voltage. The triangle symbol for the amplifier shown in Fig. 3 corresponds to the block-diagram symbol used by manufacturers in circuit diagrams.
1.23 Integration
Fig. 4 shows an integration amplifier, which, through the replacement of the feedback resistor by a capacitor and the addition of a long-period resistor at the input, is capable of performing integration of the input voltage over time.
[page 10: figure]
Fig. 4: Integration with a computing amplifier
If one feeds into the input R_1 a positive DC voltage U_1, a current flows through the resistor R_1 in the negative direction toward the amplifier. This current charges the capacitor C in the negative direction, causing the output voltage to rise very slowly but not reaching the balance condition, since it is not able to cancel the charging current. Since the input current is very small in comparison with the charging current of the capacitor, it is essentially equal to the sum of the input resistor current and the capacitor current being zero. This gives:
$$I_{ein} = I_C \tag{6}$$
$$I_{ein} = \frac{U_1}{R_1} \tag{7}$$
$$I_C = -C \cdot \frac{d U_{aus}}{dt} \tag{8}$$
[page 11]
therefore:
$$-\frac{d U_{aus}}{dt} = \frac{1}{C} \cdot \frac{U_1}{R_1} \tag{9}$$
By integrating both sides one obtains:
$$-U_{aus} = \frac{1}{C \cdot R_1} \int U_1 \cdot dt + K \tag{10}$$
The output voltage of the amplifier thus represents the time integral of the input voltage. The factor C · R_1 is the integration time constant. It specifies the time in which the output voltage reaches the value of the input voltage, after this voltage has been abruptly switched in (see Fig. 5).
Fig. 5: Waveform of input and output voltage of an integrator; integration time
[page 12]
Assuming that the constant K is zero, i.e., that the output voltage has a value of zero at the start of integration, equation (10) can also be written analogously to formula (5) for the case of multiple inputs:
$$-U_{aus} = \frac{1}{C \cdot R_1} \int U_1 \cdot dt + \frac{1}{C \cdot R_2} \int U_2 \cdot dt + K \tag{11}$$
The output voltage is thus also the weighted sum of the time integrals of the individual input voltages.
If one chooses as an example in Fig. 4: C = 5 µF, R_1 = 400 kΩ, R_2 = 20 kΩ, the integration time constants are:
- C · R_1 = 5 · 10⁻⁶ · 4 · 10⁵ = 2 s
- C · R_2 = 5 · 10⁻⁶ · 2 · 10⁴ = 0.1 s
One can set the integration input gains via the integration factor 10 obtained above. In this example, the integration time is 2 s, which means: the output voltage reaches the value of the input voltage 2 s after the integration has begun.
[page 13]
It was stated above that the integrator requires, at the beginning of the computing run, to be brought to a specified initial condition. As can be seen from equation (10) and (12), this initial condition is represented by the constant K. In Fig. 6, drawn schematically, a capacitor C is charged through a very high-resistance resistor to the initial value U_0. The computing amplifier output is connected to a feedback resistor. At the start of the computation the switch S is opened and the capacitor then supplies its charge as an initial voltage of the integrator, while the integration process begins. The initial condition is thus represented by the voltage U_0, and the integration proceeds from this initial value U_0.
Fig. 6: Circuit symbol for a single-input integrator
The triangle symbol in Fig. 6 with a circle in the middle is the symbol of the integration amplifier. The number or fraction at the top of the triangle gives the gain of the various input channels and the associated integration time constant. The circle in Fig. 6 symbolizes the manufacturer’s identification of the integrator, to differentiate it from the summation amplifier.
[page 14]
A constant current source and a potentiometer at ±100 V supply the voltage for the circuit. The voltage divider ratio is given by 100 / (100 + R), i.e., 100 Ω. There is thus a voltage of 5.25 V at the output; in this case the output voltage serves as the initial value for the integrator.
1.24 Coefficient Setting
With the aid of a coefficient potentiometer it is possible to set the following two basic operations:
a) Multiplication of an input voltage by a constant between zero and 100 and multiplication by a fraction (variable multiplication).
b) Fixing of a constant potential between zero and 100 (or any fractional value). This second function is also used for operations such as a) and b), and is also used for the initial condition of the integrator.
In the first case the potentiometer is used as a voltage divider, i.e., its sliding contact moves from zero to full scale. In the second case, the slider sets a fixed DC voltage which can be read by the voltmeter scale at 1 or 100 divisions.
[page 15]
The output voltage U_aus of the slider of the potentiometer is set by the ratio of the two resistances (the wiper position), and can, when the wiper can be varied continuously:
$$U_{aus} = k \cdot U_{ein} \tag{13}$$
where the factor k can be set from zero to full scale; the slider is graduated proportionally. Since k is proportional to the angle of the slider, the setting of the scale can be made with high reproducibility. The measurement is sharp and the scale can be calibrated. The potentiometer provides a strongly defined “number of turns” output, which makes a very precise adjustment possible.
For multiplication by one of the constant coefficients, a potentiometer with a factor set to exactly 1 or 100 is used. The settings between 1 and 100 are directly proportional to the divisor ratio.
For division, the same instrument is used as for multiplication; only the assignment of input and output is reversed. As a result, the potentiometer also serves as a divider. When this is the case one uses the same circuit as in Fig. 7 and reads the quotient directly.
Treating the potentiometer as a coefficient-setting element, it is possible to multiply by a constant factor:
A DC computational value is fed to the input of the potentiometer, and the potentiometer slider is adjusted to the desired coefficient value.
[page 16]
In Fig. 8, the actuation of the potentiometer is set equal to zero by the position of the slider, so that the fine adjustment can also read zero and 100. The setting is performed in Fig. 8 with a coefficient potentiometer. The circle gives the divisor ratio k.
Fig. 8: Circuit symbol for a coefficient-potentiometer multiplier
1.5 Multiplication and Division of Two Variable Quantities
The multiplication of one variable by another can be performed with an element known as a multiplier. One can, with this element, also multiply by any number of constant intermediate values. The multiplication of two variable quantities requires a special method, known as the quarter-square method, which is implemented in the BBC desktop analog computer.
[page 17]
The method is based on the relationship:
$$(a + b)^2 - (a - b)^2 = 4ab \tag{14}$$
The computation of a · b requires the following steps:
- Formation of the sum a + b
- Squaring of the sum a + b
- Formation of the difference a – b
- Squaring of the difference a – b
- Multiplication of the result (a + b)² – (a – b)² by the constant factor 0.25
The sub-computations of addition, subtraction, and multiplication by a constant are performed as described under 1.2 to 1.4. For steps 2 and 4, a squaring circuit whose input-output characteristic is described by a parabola must be built in. For this purpose, two so-called function generators (diode function generators, DFG) — implemented as break-point approximators — are used; they are described under 1.4.
Fig. 9: Principle circuit of a multiplier using the quarter-square method
[page 18]
At this point it is noted that the individual computing voltages that are presented through the various computing elements are limited to a maximum of ±10 V (or in some cases ±15 V), which can be standardized in the computation.
Fig. 10 shows the circuit symbol for a multiplier.
Fig. 10: Circuit symbol for a multiplier
The multiplier used here accepts two input voltages U_a and U_b, each lying in the range ±10 V. The product ab is delivered at the output; the result is also in the range ±10 V.
The divider is implemented using the multiplier in a feedback loop configuration. In the feedback loop configuration (Fig. 10 b), the output of the multiplier is connected to the feedback resistor of the computing amplifier; this enables division to be performed with a high degree of precision (see Fig. 10 b). The circle in the symbol for the multiplier represents the manufacturer’s identification, by which it can be distinguished from other computing elements.
Division of Two Variable Quantities
Basic Principle of Division
Fig. 11 shows the basic circuit for dividing two variable quantities.
Applying the quantities a and b to the inputs of the multiplier causes the output voltage to change as a function of the variable b. The sign of this change shall be very large. The output voltage of the multiplier Z is connected through a resistor to the input of a summing amplifier. To prevent this summing amplifier from becoming saturated, one needs to arrange for the input voltage to always be zero, i.e., so that an output voltage is always achieved, which leads to:
d = b · c
and since from (15) and (16):
c = 1/b
it follows that:
d = a / b (1)
Of note is the question of the operating range, in which the circuit operates without error. In order to avoid overdriving the amplifier, it is necessary that:
e ≤ 1 Max. 100%
or
b ≥ a (1)
That is:
b ≥ a (1)
1.4 Non-Linear Function Generators
For the simulation of non-linear Relationships, so-called non-linear function generators are used. These consist essentially of stabilized diode networks, which produce a piecewise-linear approximation of the desired function. These are passive elements; diodes (transistors, etc.) on their own exhibit the same characteristics. The output of the function generator is then connected to a subsequent summing amplifier, which amplifies as a lossless Tailor [relay], acting as an inverter, and as its output delivers the desired function.
Page 20
From there:
d = a · c (2)
yields from (15) and (16):
c = 1/b
and thus:
d = a / b (1)
To be noted is the question of the operating range, in which the circuit operates without error. In order to avoid overdriving the amplifier, it is necessary that:
e ≤ 1 Max. 100%
or
b ≥ a (1)
That is.
b ≥ a (1)
1.4 Non-Linear Function Generators
For the simulation of non-linear relationships, the so-called non-linear function generators are used.
Page 21
A switching amplifier (sign-changing amplifier, inverter) converts the output voltage into a rectangular wave U_out = −1 · U_in, which depends on the sign of the input voltage. The relationship is:
U_out = −K_max · f (U_in) (20)
Fig. 12 shows the basic circuit of such a function generator.
Fig. 12: Basic circuit of a function generator; non-linear relationship.
Fig. 13 shows the realization of a non-linear characteristic curve using back-biased diodes.
Fig. 13: Circuit and characteristic curve of a function generator.
Page 22
If the previously described circuit is held at a fixed reference voltage U_ref and the diode current is blocked step by step: the potential at point P is initially zero. The voltage U_ref … U_P passes through the resistor network and is blocked at the diodes. The fixed reference voltage U_ref is applied to the resistors.
Below this, the diodes conduct depending on their threshold voltages. An increase of U_b from zero: the input current flows through the diode chain. At a certain current level the operating point on the diode characteristic rises above the knee, and the diode turns on. Then a larger current flows through the resistor, and U_P rises.
The shape of the curve depends on the magnitude and polarity of the reference voltages U_1, U_2, U_3, U_4, which may be set by the potentiometers. At the breakpoints the slope of the curve changes. The available number of breakpoints is 1, 2, 3, 4, which are arranged as standard configurations. The curves with increasing slope are described. The shape of the Diodenkennlinie [diode characteristic] is adjusted by appropriate setting of the Potentiometers (R_1 and R_2 B_1) and B_2.
Page 23
By appropriate combination of linear and non-linear segments, arbitrary curves can be produced. Between 0 and 7 segments are possible, and these can be freely selected. At each segment, an arbitrary slope can be set. Fig. 14 shows a circuit arrangement for the representation of a curve with decreasing slope.
Fig. 14: Circuit for representing a curve with decreasing slope.
The desired curve is approximated as the difference of two curves. The function generator represents the linear component. The resistor R is the amplifier in the background, which amplifies the positive portion. The circuit portion is shown dashed in the diagram.
Page 24
The part contributed by the function generator is shown dashed. The output voltage of the subsequent amplifier corresponds to the difference of both portions and thereby the desired curve with decreasing slope.
2. Description of the BBC Desktop Analog Computer
2.1 General
The BBC desktop analog computer is housed in a compact unit. On the front side there is a clearly arranged panel with all the controls and operating elements needed for control and monitoring of the computing operations, as well as the Bedienungstafel [operator panel] and the Steckerbrett [patch board] on the upper side of the device.
Fig. 15: Front view of the BBC Desktop Analog Computer.
Page 26
Fig. 16: Rear side of the BBC Desktop Analog Computer.
The computing elements and stabilized power supply units are mounted on the rear side and can be removed as required. The Hilfsgeräte [auxiliary devices] and stabilized circuit boards, on which the passive components (transistors, resistors, diodes, etc.) are mounted, can be seen on the front of the computing elements. The patch board is equipped with a metal frame. To protect against mechanical and atmospheric influences, the finished and tested patch board is hermetically sealed on both sides with a synthetic cover plate.
Page 27
An example of such a printed circuit board is shown in Fig. 17, which depicts a sign-changing amplifier.
On the left side of the printed circuit board there are several pre-connected contacts (integrated circuits), in a simply connected circuit layout with pre-wired contacts. On the printed circuit board there are also fixed resistors and appropriately trimmed potentiometers. The contacts can be fitted as sockets or as plugs depending on use, and can be exchanged.
The integrated circuits are the so-called “Pasta” type, which are built into the chassis in groups of 12 units.
The housing accommodates a total of three integrated circuit boards, which are built in a staggered arrangement: in the upper range (row 1) an amplifier board of 154 mm width (about 3) and one of 69 mm width (row 2).
Page 28
2.2 Electrical Data
2.21 Connection Values
- Mains voltage: 220 V ± 10%
- Mains frequency: 50 Hz (45 … 65 Hz)
- Power consumption: approx. 80 W
- Dimensions: Screw-terminal pitch 0.8 mm, 5 × 20 mm
2.22 Basic Equipment
12 Coefficient potentiometers
Helipot single-turn type; for setting the proportionality coefficients. The potentiometers are suitable as summing elements with a setting range of 0.1 to 1 and with a setting accuracy of ±0.01%.
12 Inverting amplifiers
Absolute value of the voltage gain: > 10⁵
Input resistance: > 1400 kΩ
Working range: operating as amplifiers (summing amplifiers) or as inverters. Each amplifier is used for one operation, and the full-range output voltage is ±100 V.
5 Integrators
A fractional-capacitor [integrating capacitor] is used as the feedback element of the amplifier. Each integrator contains a stable capacitor component and is equipped with a “hold” function and suitable initial conditions.
1 Multiplier
For performing accurate multiplication, two sign-correct multiplication operations can be performed simultaneously; with a multiplication accuracy of the order of ±0.3%.
1 Function generator
For positive and negative input ranges. The unit contains up to 10 logarithm-like elements per range. The desired curve is approximated from them. Each approximation can be constructed so that it can be read off continuously, while the Abtastsystem [scanning system] remains at the Ankopplungspunkt [coupling point].
Page 29
8 Integrators
With a fractional-capacitor condenser as the integrating-capacitor of the amplifier. Each integrator contains a stable capacitor component and is equipped with a “hold” function as well as suitable initial-condition facilities.
1 Diode unit
With four germanium diodes for constructing a simple non-linear circuit with some particular characteristics.
1 Operating instrument
1 measuring instrument, Class 1
Measuring range: −15 … 0 … +15 V
Scale: 100 … 0 … 100
1 Multi-pole switch
For switching the measuring instrument, type Drehschalter [rotary switch].
3 Mode switches
For selecting the operating modes:
- “Problem setup” (P)
- “Initial conditions” (A)
- “Compute” (B)
- “Hold” (S)
- “Regulate” (Reg.)
1 Potentiometer
For setting the repetition rate with two contacts serving as a common synchronous pulse generator for the 12 Synchronisationspotentiometer [synchronization potentiometers].
Page 30 (continued specifications)
1 Mains switch
For switching the mains voltage on and off.
1 Mains indicator lamp
1 Equipment fuse lamp
With a fuse element of 0.8 A, medium-acting type, 5 × 20 mm. When the fuse blows, the built-in lamp lights up.
1 Multi-socket
For optionally connecting a second computer.
2 Power supply units
Electronically stabilized, for powering the computing elements.
1 Power supply unit
Non-stabilized, for powering the relays, indicator lamps, and voltage indicator tubes.
The basic equipment also includes:
1 Mains cord
2 m long, with Schuko plug and cold-equipment socket.
100 Connecting cables
of which:
- 50 pieces, 0.2 m
- 25 pieces, 0.4 m
- 25 pieces, 0.8 m
1 Operating manual
Page 31 (continued specifications)
1.25 Signal level
The signal level of the BBC desktop analog computer is ±15 V.
2.3 Technical Data
-
Dimensions:
Height: 740 mm
Width: 575 mm
Depth: 500 mm -
Weight: 40 kg
Transport/Packaging:
The device is transported in a stable wooden crate fitted with foam padding (Schaumstoff). It is fitted with screwed metal mounts for transport.
When transporting via air freight, attention must be paid to the instrument, so that transport is carried out with adequate cushioning.
2.4 Individual Amplifiers
2.41 DC Amplifier
The DC amplifiers are fully stabilized (DC-stabilized) amplifiers, which meet the different requirements with respect to characteristic data. The characteristic data are:
- Voltage gain: > 10⁵
- Input resistance: > 1400 kΩ
- Übergangsfrequenz [transition (unity-gain) frequency]: > 1400 kHz
- Operates as summing amplifier with full-range output at ±100 V
Each amplifier is designed for integration, so that an integration time of 1 ms (at full range) is achieved. At working frequency of 10 kHz, five Integratorzeitkonstanten [integrator time constants] of 0.1 ms are available.
Page 32
5 Integrators
With a fractional-capacitor condenser as the integrating-capacitor amplifier. Each integrator contains a stable capacitor component and is equipped with a “hold” function and suitable initial-condition equipment.
1 Diode unit
With four germanium diodes to build a simple non-linear circuit with certain special characteristics.
1 Operating instrument
1 measuring instrument, Class 1
Measuring range: −15 … 0 … +15 V
Scale: 100 … 0 … 100
1 Multi-pole switch
For switching the measuring instrument, type Drehschalter [rotary switch], with 12 contacts.
3 Mode switches
For selecting the operating modes:
- “Problem setup” (P)
- “Initial conditions” (A)
- “Compute” (B)
- “Hold” (S)
- “Regulate” (Reg.)
1 Potentiometer
For setting the repetition rate with two contacts as a common synchronous pulse generator for the 12 Synchronisationspotentiometer.
12 Spannungsindikatorröhren [voltage indicator tubes]
As trigger-pulse indicator for the 12 Spannungsindikatorpotentiometer, i.e., as display device for the voltage indication.
Page 33
Fig. 18 shows the circuit and panel cross-section of a DC amplifier.
On the left side of the panel there is one amplifier per printed circuit board (Printplatte), indicated with the two Eingangsstiften [input pins] of the amplifier. On the right side of the input there is a simple Eingangsnetz [input network], which is connected to the amplifier.
Fig. 18: Circuit and panel cross-section of a DC amplifier.
From the input of the amplifier, from the upper row (row 2), the Widerstandsnetz [resistor network] is connected. The amplifier is the Gegenkopplungswiderstand [feedback resistor] (R), through which the Verstärkerausgang [amplifier output] is connected to an inverting unit. The Eingangsabschlußwiderstand [input termination resistor] is set accordingly. The three are connected so that the Eingangsspannung [input voltage] and the Ausgangsspannung [output voltage] are opposite in sign, and the Rückkopplungswiderstand [feedback resistor] is matched accordingly.
Page 34
Additional input circuits carry their own 22 kΩ input resistors. Ten of the inputs are therefore equipped with 22 kΩ input resistors, while the two remaining inputs carry 10 kΩ resistors.
Should the amplifier operate as a subordinate (lower) amplifier, its inputs are connected to the 10-kΩ inputs of that amplifier. This makes it possible, without additional devices, to achieve a second-level amplifier circuit through interconnection, while the upper amplifier is used with an integration accuracy of 0.1%.
The self-contained feedback resistors are assembled in this way: the feedback network of the amplifier consists of a chain of amplifiers connected with a low coupling, in a single circuit, through which the amplifier can be built up in a chain. Each of this circuit constitutes a sign-changing amplifier (Vorzeichen), which ensures that at the input the test function “P” (Positive) or “Neg” (Negative) can be applied, and with this and hence as ±1 switches continuously re-energized.
If the three upper left buttons of the amplifiers are pressed without operating the amplifier (i.e., in the “normal” mode), then this constitutes a Leerlaufverstärker [open-circuit/no-load amplifier] — and from that point on the DC amplifier can also be used as a sign-changing amplifier (cf. Fig. 11).
The DC feedback amplifier is constructed so that the feedback chain (absolute value of the feedback ratio) achieves a high Leerlaufverstärkung [open-loop gain] and has low output resistance — and thus achieves the oscillation stability criterion through a Kettenschaltung [cascade circuit] of amplifiers with small Stabilisierungskondensatoren [stabilizing capacitors] (stray-type capacitors) to achieve high frequency performance.
Page 35
Fig. 19 shows the Schwingungserzeugung [oscillation generation] situation in a closed chain of Summierverstärkern [summing amplifiers].
If such a closed chain cannot be avoided, then either one or more Verstärker [amplifiers] with a sufficiently high open-loop gain must be inserted; or one or more of the computing amplifiers must each be decoupled by means of a small Integrierkondensator [integrating capacitor]. Either of these measures will ensure that the chain is stable with respect to oscillation.
The same Stabilitätsanforderung [stability requirement] also applies in the Bedienungsanleitung [operating manual] when a computing operation is not carried out directly by a “sign-changing amplifier” but rather with an RC element — in this case, it is also necessary to install a Gegenkopplungskondensator [feedback capacitor].
Fig. 19: Elimination of oscillation tendency in a closed chain of summing amplifiers.
Page 36
[page 36: figure only — Fig. 19 continued; diagram showing a closed feedback chain of summing amplifiers (three triangular amplifier symbols in series) connected back to the first input, with small Integrierkondensatoren (integrating capacitors) shown at the feedback path to suppress oscillation tendency. No additional text beyond the caption on this page.]
[Page 37]
On the patch panel there are also a number of potentiometers of 10 kΩ available (g, a₁, g₁).
The potentiometers are equipped with 2 Beckman resistors, i.e., with:
- 2 coefficient potentiometers (A 10 kΩ)
- or 4 10Ω resistors from Beckman/variable (A 20 kΩ)
- or 40 Ω resistors × Beckman/variable (A 332 Ω)
- or 1 multiplier × 10Ω resistors from Beckman instruments.
2.42 Coefficient Potentiometers
The coefficient potentiometers are linear ten-turn potentiometers of 10 kΩ each (identified as W) and serve for the adjustment (Einstellung) of the instantaneous coefficient values. They are equipped with a dial (Skalierung) of 0 to 1 in increments of 100. In each revolution (Drehung) there are 100 subdivisions, so the finest (kleinste) readable division is 1/1000 of the full scale. The right border of the drum (Trommel) shows the digit of one full revolution. When the coefficient is to be set at 15 V, den den Potentiometer D is set to the decimal equivalent of the reference voltage, approximately at the decimal value assigned.
[Page 38 — figure only]
Figure 20: Circuit and front panel of three coefficient potentiometers.
The figure shows the front panel with three large rotary potentiometer knobs (two rows of three) and their corresponding miniature trim adjustments, together with the schematic circuit diagram below. In the circuit, the three 10 kΩ potentiometers are connected between +U_R and ground; the wiper outputs go via a push-button switch (Taste P) to the measuring instrument (Messgerät) on the operator panel (Bedienungstafel). The output of each potentiometer is also taken to the patch panel.
[Page 39]
The balancing (Abgleichen) procedure proceeds as follows:
-
Press push-button “P” (Potentiometervergleich — potentiometer comparison). Thereby the voltage from the reference source (Bezugsspannungsquelle) is switched through to the measuring instrument on the panel (Anzeigeinstrument), and the summing line (Summenleitung) of the potentiometer network is connected to this line.
-
Press the balancing switch of the respective potentiometer.
The measuring instrument now measures the free output of the wiper (Schleifer) of the potentiometer between the set coefficient value and ground. The output wiper of the potentiometer lies in series with a 100 Ω resistor; this causes the potentiometer to be loaded slightly at the balance point. The wiper can also be driven (betätigt) so that the measuring instrument shows approximately zero. During the reference span (Referenzspannung) of 15 V, the output voltage of the coefficient network at each balance point is approximately zero. Thus the balancing (Abgleich) is independent of the reference voltage. The balance point is indicated by a null reading (Nullanzeige) — see Figure 22 in section 1.3.
-
To trim the potentiometer P 10 … P 13 according to the correct coefficient value, one must refer to the corresponding volume in the applicable Beckman manual.
[Page 40]
To display the initial value (Anfangsbedingung), the push-button “IB” (Anfangsbedingungsanzeige — initial-condition display) is pressed for this purpose. Thus the outputs from the free resistor (Freiwiderstands) of the integrator are switched through and displayed on the measuring instrument on the panel. A potentiometer is provided at the output of each integrator for this purpose, so that the initial conditions (Anfangsbedingungen) are adjustable from 0 to ±100 V in small steps. This voltage can be switched to the measuring instrument via a key switch. The potentiometer is equipped with a 10-turn dial.
In the reference span (Bezugsspannung) of 15 V the full-deflection of the initial-condition potentiometer is controlled by the output of the summing amplifier. The initial-condition is therefore adjustable from 0 to the output maximum. The polarity of the Anfangsbedingungsspannung (initial-condition voltage) at the integrator output depends on the setting of the wiring and the input connections. The initial-condition voltage is set in relation to the reference voltage through the dial position (see Figure 21, Bild 21).
[Page 41]
[Continued text from page 40, regarding the integrator initial-condition display:]
To display the initial condition, push-button “IB” (Anfangsbedingungsanzeige) is pressed. The relays of the integrators are then activated and the initial-condition voltage is switched from the initial-condition source to the operational amplifier summing input (Summierungspunkt). The output of the integrator can then be measured at a free output connection (freie Ausgangsbuchse) of the patch panel (Schalttafel) with a suitable external instrument (Außengerät — oscillograph, voltmeter, etc.).
The initial-condition voltage source is connected to the reference voltage source (Bezugsspannungsquelle) through the initial-condition potentiometer. When the right-hand scale (rechte Skala) of the dial indicates a negative value, the initial condition is negative; when it indicates a positive value, the initial condition is positive. (Cf. Bild 21, Abschnitt 1.3.)
[Page 42 — figure only]
Figure 22: Connections of the integrator units to the patch panel.
The figure shows the front-panel layout of two integrator modules (labeled 11 and 12) with their input and output connection sockets. Annotations indicate:
- Left side: negative Ausgangsspannung (negative output voltage)
- Right side: positive Ausgangsspannung (positive output voltage)
Caption (Bild 22): When push-button “IB” is pressed (Drücken), the relay is operated and the initial-condition voltages are switched to the integrator inputs. The output voltage of the integrator is now free (frei) and corresponds to the previously set initial-condition voltage (Anfangsbedingungsspannung). This is equal to the negative (definite) integral of the input quantities (Eingangsgrößen).
[Page 43]
2.43 Integrator Units
Each integrator unit comprises two precision (hochstabilen) broadband operational amplifiers, one of which serves as the integrating amplifier proper and the other as the detector (Detektor) / damping device (Dämpfungsglied). Above the patch panel, the two stable precision amplifiers are always identified by a sign. The integrator is equipped with a 1 kΩ precision potentiometer driven from 1 volt; this serves as the command input (Stelleingang) and at the same time as the output of the function-generator potentiometer. The potentiometer is approximately 10 V in full deflection and is fitted with a Federrückstellgriff (spring-return grip).
Figure 21: Circuit and front panel of an integrator unit.
[Page 44]
Figure 23: Circuit and front panel of the reference voltage unit.
Figure 25: Circuit and front panel of the reference voltage unit.
2.45 Multiplier
The multiplier operates according to the already described (section 1.3) operating principle and consists of two quarter-square multiplier elements, two inverting amplifiers, two front-panel jacks, and the two functional diode programs (Diodenprogramme). Bild 24 shows a schematic of the multiplier circuit, enabling the simultaneous formation (simultane Bildung) of the sum a + b and the difference a − b.
[Page 45 — figure only]
Figure 24: Circuit and front panel of the multiplier.
The figure shows the front-panel layout with four input sockets (a, −a, b, −b on the left) and four output sockets on the right, together with the symbol Π (multiplication). The lower schematic shows the full circuit: inputs a, b, −a, −b are each fed through resistors into two diode function networks (one with the curve U_aus vs. U_ein in the first quadrant, one with −U_ein vs. −U_aus). Their outputs are summed in an operational amplifier whose output is labeled −a·b.
[Page 46]
The multiplier forms (bildet) the product of two quantities, providing it at the output as −a·b. To obtain any one of the four products a·b, −a·b, a·(−b), or (−a)·(−b), corresponding input polarities must be applied. Each of these four products has a single value at the output jack. The summing amplifier forms the difference (Differenzbildung) of the result; the output is then −a·b. When the sign of the product a·b is negative, the output voltage is positive, and vice versa.
For use with a multiplier in multi-variable computation tasks (mehrveränderliche Rechenaufgaben), an input circuit (Eingangsverschaltung) with inverting amplifiers (Umkehrverstärker) is available for use. The achievable accuracy values of the multiplier are shown in Bild 25.
[Page 47 — figure only]
Figure 25: Circuit of a multiplier with inverting amplifiers at the inputs.
The top diagram shows inputs a and b, each passing through a separate inverting amplifier (triangular symbol) which also produces −a and −b. All four signals (a, −a, b, −b) are fed to the multiplier block Π, yielding output −a·b.
Figure 26: Circuit of a multiplier without sign inversion.
The bottom diagram shows a simplified connection: inputs a and b pass through inverting amplifiers providing −a and −b; the four signals enter the multiplier block Π, giving output −a·b at one terminal and +a·b at another (by connecting one input without inversion).
Caption (Bild 26): If the output voltage of the multiplier is required in opposite polarity (entgegengesetzter Polarität), it is sufficient to reverse the polarity of one input quantity (einer Eingangsgröße), as shown in Figure 26 for the input quantity a.
[Page 48]
The multiplier can in principle also be used for squaring (Quadrierung) a computational quantity. In that case the inputs a and b and −a and −b are connected together.
Figure 27: Squaring with a multiplier.
The diagram shows input a passing through an inverting amplifier to produce −a; then a, −a (and both again) are applied to the four inputs of multiplier Π, giving output −a².
The multiplier can, as described in section 1.3, also be used for the division of two variable computational quantities. Figure 28 shows the complete circuit.
Figure 28: Circuit for dividing two quantities.
The diagram shows U₁ applied to the a input (and through an inverter to −a), and U₃ (from the output of a feedback integrator) connected to the b and −b inputs, with a 10 nF capacitor in the feedback of the integrator. The combined output of the multiplier and integrator yields U₃ such that:
$$U_3 = \frac{U_2}{U_1}$$
[Page 49]
It should be noted that in the division circuit shown in Figure 28 the input quantity U₁ must always be positive (positiv).
As stated in section 1.3, the feedback integrator (Rückkopplungsintegrierer) operates in such a manner that when the voltage U₃ is negative, the multiplier’s output drives the feedback amplifier, which then adjusts U₃ back toward zero equilibrium. The feedback loop becomes stable (stabil) only when the input quantity U₁ has a positive value, because the overall feedback polarity depends on the sign of U₁. The circuit breaks down (wird instabil) for negative values of U₁. The circuit is therefore only valid for positive values of U₁.
[Page 50]
2.46 Function Generator
The function generator operates according to the principle already described in section 1.4 with specially adapted diode programs (angepassten Diodenprogrammen). The function block of Figure 29 shows the diode programs with positive input voltages, while the corresponding block for negative input voltages provides a negative output. Figure 29 shows the front panel of the function generator.
Figure 29: Front panel of the function generator.
The front panel shows multiple dial knobs (Skalenscheiben) arranged in rows for adjusting the diode breakpoints (Knickpunkte), together with associated output jacks.
The function generator basically constitutes a piecewise linear approximation (stückweise lineare Annäherung) device. In Bild 30 the principal circuit of the left diode program (linken Diodenprogramms) is shown.
[Page 51 — figure only]
Figure 30: Principal circuit of the left diode program.
The figure shows the detailed schematic of the diode function-generator network. The circuit comprises a series of diodes and potentiometers connected between the reference supply rails. Each diode–potentiometer pair defines a breakpoint (Knickpunkt) at which the slope of the piecewise linear characteristic changes. The potentiometers, labeled P 1 … P n, allow individual adjustment of the breakpoint voltages. A summing amplifier at the output (rechts, with the triangular op-amp symbol) combines the contributions from all segments to produce the desired nonlinear transfer characteristic (gewünschte nichtlineare Übertragungskennlinie).
Caption: The desired curve (gewünschte Kurve) must be approximated by adjusting each of the individual segments. The individual breakpoints must lie at specific voltage levels (Spannungspegeln); the potentiometers allow these to be set precisely. The diodes conduct (leiten) when the input voltage exceeds a given breakpoint voltage, thereby adding (addieren) a new slope segment to the composite output. At the summing amplifier output the full piecewise linear function is assembled. The more segments (Dioden) used, the closer the approximation to the desired function.
[Page 52]
The curve approximated (angenäherte Kurve) by the function generator can be obtained completely satisfactorily by the following means:
1. Setting from the Abscissa Axis (x-axis):
This is possible for input (Abszissengröße) values between −1 and +1 (as shown in the example in Figure 35 of the corresponding section). The input quantity is connected directly to the summing point (Summierungspunkt — summing input) of the coefficient amplifier (Koeffizientenverstärkers). The output of the amplifier corresponds to the desired function. The coefficient potentiometer brings the desired curve (gewünschte Kurve) from the input to the output.
If the output voltage of the function generator matches (entspricht) the input voltage, the curve is said to be set correctly (richtig eingestellt). The curve is adjusted (eingestellt) on the function generator via the coefficient potentiometers so that the output matches the desired function as closely as possible.
2. Setting using an External Instrument (Stelleingang):
For simpler problems, it is possible, as shown in Bild 30, to feed a sawtooth voltage (Sägezahnspannung) from an external instrument (Außengerät) simultaneously with a ramp input (Rampenspannung) through a coefficient potentiometer (as described in section 1.3), and the amplifier output then indicates (zeigt an) the approximated (angenäherte) function on an oscilloscope. An approximate setting is then achieved by multiplying the input voltage by the coefficient and adjusting (stellen) the potentiometers until the desired curve shape is displayed.
[Page 53 — figure only]
Figure 31 a, b: In Figure 31, the circuits of the function generator for a number of different curves are given.
a) Upper graph: Transfer characteristic U_aus vs. U_ein showing a nonlinear saturation curve (S-shape), with the dashed portion indicating the limiting behavior. The corresponding block diagram on the right shows input e1 going into the function-generator block, with output a1 going to an inverting amplifier.
b) Lower graph: Transfer characteristic U_aus vs. U_ein showing a monotonically decreasing hyperbolic curve, with a dashed extension. The corresponding block diagram on the right shows input e4 going into the function-generator block with output a2 going to an inverting amplifier.
[Page 54 — figure only]
Figure 31 c, d, e: Further circuit arrangements of the function generator for different curve shapes.
c) Graph: U_aus vs. U_ein showing a parabolic (squared, U²) characteristic, symmetric about the vertical axis. The corresponding block diagram shows two inputs e1·z4 and e2·z5 entering the function-generator block with output a1 going to an inverting amplifier. (Two diode programs feed a summing amplifier to produce the absolute-value or squared function.)
d) Graph: U_aus vs. U_ein showing a combined S-curve and hyperbolic characteristic, crossing the origin, with dashed extensions in both quadrants. The corresponding block diagram shows inputs e1 and e4 into two function-generator blocks (a1 and a2) going into an inverting amplifier, allowing the construction of antisymmetric functions.
e) Graph: U_aus vs. U_ein showing a steeply rising curve in the positive quadrant only (one-sided hyperbola or exponential), with the dashed portion indicating the asymptotic behavior. The corresponding block diagram shows a single input e6 into function-generator block a2 going to an inverting amplifier.
[page 55: figure only — Fig. 31 f, g, h. Characteristic curves (U_out vs. U_in) and block diagrams for nonlinear function generation using the diode unit. Panel f shows a smoothed S-curve (single-input configuration with inverter and summing junction). Panel g shows an S-curve with steeper slope (two-input configuration). Panel h shows a V-shaped or absolute-value-like characteristic (two-input configuration with diodes at the output stage).]
[page 56: figure only — Fig. 31 i, k, l. Additional nonlinear transfer characteristics using the diode unit. Panel i shows a sigmoid (saturation) curve (two-input configuration with inverter). Panel k shows an anti-symmetric saturation curve. Panel l shows a two-level clipping (relay-like) characteristic implemented with a simple amplifier connected to +15 V and −15 V via diodes.]
[page 57: figure only — Fig. 31 m, n, o. Representation of nonlinear relationships using the diode unit. Panel m shows a multi-level step function (staircase characteristic) realized with a summing amplifier, diodes, and ±15 V reference voltages. Panel n shows a piecewise-linear ramp function using an integrator-like arrangement with diode clamps to ±15 V. Panel o shows a multi-slope piecewise-linear characteristic using several diode breakpoints and summing stages. Caption reads: “Fig. 31 m, n, o: Representation of nonlinear relationships with the diode unit.”]
2.47 Diode Unit
The diode unit contains 4 multiplier diodes together with which curvilinear (nonlinear) transfer characteristics can be set up. At one end it holds up to 4 condensers of C = 10 nF for stabilization; in both positions these range from 0 to 100 numbered, whereby the digit corresponds to 15 V. The control knob “2” is equipped with the corresponding operational amplifier and, with the aid of styli, position “1” can be connected to the left front panel. Beyond this the diode unit can also be used as a coefficient potentiometer (cf. section 2.47) and as an output potentiometer (cf. Fig. 2.47, Fig. 13).
In Fig. 32 there is also shown a nonlinear relationship that can be realized entirely with diodes in closed-loop circuit without function generators.
[page 58: figure only — Fig. 32. Realization of nonlinear relationships with diodes. The left diagram shows a transfer characteristic with two breakpoints forming a piecewise-linear approximation. The right diagram shows the corresponding circuit: a summing/integrating amplifier with feedback diodes creating the breakpoint nonlinearity in a closed-loop configuration.]
The use of diodes for limiting the input voltage of an amplifier is shown in Fig. 31 l, m and n.
2.48 Control Panel
The control panel mounted on the front of the desk contains all elements required for the operation and monitoring of the analog computer. (Cf. Fig. 33)
[page 59: figure only — Fig. 33: Control panel. The panel photograph shows labeled positions: 1 (upper center), 2 (lower center), 3 (lower right of center), 4 (upper left), 5 (left side), 6 (right upper), 7 (right), 8 (lower right), 9 (upper right), 10 (lower left), 11 (lower left center), 12 (lower left of center). These correspond to the control elements described in sections 2.481 through 2.486.]
2.481 Voltmeter (Control Element 1)
The voltmeter “1” is used for checking the reference voltages, e.g. to check the input voltages of multipliers. It is equipped with an internal resistance of 500 kΩ and is. The decimal places are divided from 0 to 10 in both directions, where the digit 10 corresponds to 15 V. The control knob “2” is assigned to the respective operational amplifier and, with the aid of styli, position “1” can also be connected to the left front panel.
Beyond this, the voltmeter can also be used to check the coefficient potentiometer (cf. section 2.42) and the output potentiometer (cf. Fig. 13).
2.482 Overrange Indicator
On the control panel there is also an overrange indicator that illuminates when any amplifier output voltage exceeds 15 V × 100 in magnitude, in order to indicate overloading of the respective operational amplifier. At this point the output voltage is clamped to a supply voltage of +15 V and the appropriate correction can be made.
2.483 Repetition Operation (Control Element “Rep”)
The analog computer is capable of performing a repetitive calculation, allowing the results to be continuously observed. When using the key “Rep” (repetition), the electronic differential analyzer is activated, and the output of integrators is switched. The “Rep” position switches the automatic “Betrieb” (operation) position simultaneously; the potentiometer “6” on the control panel is the repetition-rate potentiometer. With this the repetition rate can be set from approximately 1 Hz to 100 Hz (approximately 1 s to 10 ms).
[page 62 continued — Voltmeter function (Pos. 1), continued:]
2.481 Voltmeter Function (continued)
As described in section 2.48, the key “T” is used to connect individual amplifier outputs to the voltmeter. When in the “T” position, the voltmeter “1” on the control panel, the voltmeter “4” and the switch “0” are combined into one unit; the switch “T” connects the output of the selected operational amplifier to a supply voltage of +15 V and the appropriate correction can be made.
2.482 Overrange Indicator (continued)
As described in section 2.48, the key is used during the initial checking of input voltages of a multiplier; in this position, the voltmeter “4” is connected by the switch “0” and the switch “T” connects the output of the selected operational amplifier.
Interlock function (Pos. 2)
For monitoring the machine, the key “2” activates an electronic interlock that controls the machine state via the selector switch “4” on the control panel and the switch “T” connected to the associated interlock switch “1”.
Phase 1 (Pos. 1)
At the beginning of Phase 1 the machine starts the integration process and releases the initial conditions. It lasts as long as desired, until calculation of the new beginning conditions is needed. The shorter Phase 1 is, the more quickly Phase 2 can begin.
Calculation (Pos. Cal)
In this operating mode the machine runs continuously, performing the calculation. After a certain time with determined initial conditions, the calculation process returns to the starting conditions again.
2.485 Repetition Rate
The repetition rate “P” (also written on the front panel as “Rep”) allows the repetition rate of the calculation process to be set. It can vary from 1 Hz to 100 Hz.
2.486 Power Supply
On the left side of the control panel near the power-on switch is an indicator lamp “L” and a fuse with control knob “12”. At the output of the power supply, there is a fuse indicator; if a defect occurs in this fuse, a fuse indication lamp can indicate the cause of the fault.
3. Programming the Analog Computer
3.1 General
As has already been mentioned, the analog computer allows one to set up an analogy between the mathematical/physical relationships of a given problem and the corresponding electrical quantities generated by the machine. In the process, the variables of the problem are represented as electrical voltages. This is particularly the case when the variables of the problem can be represented as differential equations, as differential quotients or as systems of differential-equation solutions.
The analog solution process consists of the following steps:
-
Setting up the circuit diagram corresponding to the mathematical problem, in which the individual computing elements are used in a corresponding arrangement.
-
Scaling the circuit diagram (i.e. the determination of the machine variables) by specifying appropriate amplitude and time scale factors.
-
Setting up the programming panel according to the circuit diagram.
3.2 Setting Up the Circuit Diagram
Under the term “block diagram” is understood a diagram that represents the interconnection of the individual computing elements. The coupling between the elements is depicted as a signal flow. The block diagram shows the analogy between the variables of the problem and the electrical voltages between the elements; it should be set up so that all required functions of the variables can be provided. A simple example will now be used to illustrate this. It should be possible to set up a block diagram for a simple Feder-Masse-System (spring-mass system) as shown in Fig. 34.
- P = Patent-elasticity (spring stiffness)
- d = Damping
- m = Mass
- F = Äussere Kraft der “Störfunktion” (external force, the “disturbance function”)
- x = Displacement of the “u”
Fig. 34: Schematic diagram of a Feder-Masse-System (spring-mass system)
The differential equation of this system reads:
$$m\ddot{x} + d\dot{x} + c \cdot x = F \tag{21}$$
[page 66: figure with caption]
The setup of the circuit diagram is done by the Gleichungsstrom (equation flow), where the Gleichung (equation) is brought step by step — from the left — one order lower. All the circuits are thus established so that the highest derivative stands on the left side; as many integrators as there are integration steps are lined up in sequence, all with the sign changed (inverted output). The output of the last integrator is to be found on the right side of the equation at point “1” in Fig. 35.
Equation (21) can be rearranged as follows:
$$\frac{1}{m_u}\left[-d\dot{x} - c \cdot x + F\right] = \ddot{x} \tag{22}$$
For this it is necessary that the Diskriminante (left-hand expression) at each step must be evaluated before the next integration. The output of the first integrator gives x-dot, and so on. If the signal is led from point “1” to “2”, then it passes through the Gleichungsschreibung (equation path) along the feedback path.
Fig. 35: Coupling diagram for the described Feder-Masse-System (spring-mass system)
[page 67 — continuation of the circuit-diagram discussion]
If the system is to be connected, the signal path must come from integrator “3”, the output of which gives the Gleichungsfaktor X. The integrator “4” follows it. The task is such that the output of integrator X at the Faktor “4” is a simple inverter. The feedback from point “1” and point “2” is connected; when these points are connected, the signal flows through the Gleichungslösung along the feedback path; the output follows in sequence from the Diskriminante.
The forming from points “1” and “2” directly gives the Gleichungsfaktor x; the size to be matched, which is summed at integrator “3” by multiplication with the factor d/m, makes up the required Rückkopplungs-Leitung (feedback path). The output of the Integrators “3” through multiplication with the factor c/m gives the required size x. This c/m value is then supplied to the summing at the output side in the Rückkopplungs-Leitung (feedback loop).
[page 68 — continued discussion of scaling]
Factor “4” by multiplication with the factor d/m in Potentiometer “4”.
In the same way also for equations of higher order one proceeds, whereby more or less steps must be performed. These steps are:
-
Isolation of the highest derivative — the highest derivative is placed on the left side of the equation.
-
Assumption — the highest derivative is assumed to be available as the input (i.e., the right-hand side of the equation).
-
Formation on the right side — the terms on the right side of the equation are formed from lower derivatives using successive integration steps through intermediate stages.
-
Formation on the right side — the output is connected back from the right side of the equation via the feedback path, closing the loop.
7.1 Determination of the Scaling Factors
The scaling factors arise from the requirement that the machine variables must lie within defined limits. The computation variable U must be a dimensionless Bezugsgroesse (reference quantity), so that it neither exceeds the maximum voltage of the machine nor drops too low in absolute value.
$$U = a \cdot x \tag{23}$$
In this it should not be in Volt, but rather in the Vollausschlag (full-scale) quantities; the Bezugsgroesse (reference quantity) is therefore also not in Volt, where the Hüllkurve (envelope) of the computation also is not in Volt. For the die allgemeine Gleichung (general equation):
$$a = \frac{150}{x_{\max}} = 2.667 \frac{1}{m} \tag{24}$$
For the general equations:
$$\begin{aligned} \delta \cdot \dot{x} &= a \cdot \dot{x} \ \delta \cdot \ddot{x} &= a \cdot \ddot{x} \end{aligned}$$
or in general:
$$\delta \cdot x^{(n)} = a \cdot x^{(n)} \tag{25}$$
When determining the Amplitudenskalierung (amplitude scaling) several points must be taken into account so that the greatest precision is achieved:
- The voltage that enters an amplifier should be as large as possible, i.e. the Bezugsgroesse (reference quantity) should reach as close to 10 V (= 100%) as possible.
- In the intermediate stages, the output of the Maschineneinheiten (machine units) should not exceed ±10 V; then a Fehler (error) of 0.5% can arise. When the voltage exceeds 10 V = 100%, a Fehler of approximately 1% can arise. On the other hand, if the value falls to about 100 mV, an error of 5% can readily arise.
[page 70 — continued amplitude-scaling discussion]
In order to arrive at a as precise as possible Berechnung (calculation) of the operational amplifiers, it is necessary that the amplifiers are as fully driven as possible; it is therefore required that the amplified value and the integration initial conditions are as precisely pre-set as possible.
3.4 Determination of the Time Scale
The time scale (also called Zeitmaßstab or Zeitbezugsgroesse) relates the real time T of the problem to the machine time t. Let the Verhältnis (ratio) be:
$$U = a \cdot x \tag{29}$$
The Zeitmaßstab should not be in Volt, but rather in Vollausschlag units; the corresponding Bezugsgroesse is also not in Volt, because the Hüllkurve is also not in Volt. For the general case, to be taken as a Vollausschlag:
$$a = \frac{150}{x_{\max}} = 2.667 \frac{1}{m} \tag{24}$$
The general equations (26), (27), and (28) give:
$$\begin{aligned} \delta T &= a \cdot t \ \delta T^2 &= a^2 \cdot t^2 \end{aligned} \tag{25}$$
or in general:
$$\delta \cdot x^{(n)} = a \cdot x^{(n)} \tag{26}$$
When two very extreme boundary conditions exist, they must both be satisfied simultaneously:
- The time is such that the Rechenzeit does not take too long — one does not want to wait too long.
- The time is such that the Rechenzeit is not too short — so that it can be reliably measured with accuracy. Consequently, the Rechenzeit should lie between approximately 1 s and 100 s.
[page 71 — figure and text on open and closed integrator configurations]
Fig. 36: Example of an open and a coupled (closed-loop) integrator circuit
In Fig. 36 a test circuit is shown consisting of two integrators and a sign-inverting amplifier. In the open configuration (a), the Rechenzeit starts at zero with both integrators. If the computation begins, it should start at the “B” position. Both integrators begin integrating at that point: The output voltage U_2 rises in a relatively short time. Due to the inevitable Eingangsspannung (input voltage) offset errors of the amplifiers, the output voltage U_2 at integrator output 2 will eventually deviate and may drift; this means the output voltage will in relatively short time reach a limiting value.
Illustrated here is the “open” configuration: even a small offset input voltage offset leads to significant drift. One of the reasons is that in practice the input error of integrators is magnified by the integrating action.
[page 72 — continued discussion of time scaling and dimensional analysis]
This leads back to a state and indeed in the way that the voltage U_2 also at any arbitrary time can be brought back to a fixed value, so that the Ausgangsspannung (output voltage) of integrator 2 drifts freely. Among these extreme Abschaltungen (disconnections) there is the case in which both integrators are coupled (closed-loop, configuration b). The Dimensionsanalyse (dimensional analysis) dictates that the Zeitmaßstab can be set as follows:
$$\tilde{T} = t \tag{29}$$
where T means:
- T = Real time (Rechenzeit, problem time)
- t = Machine time (Maschinenzeit)
If τ is also to be given as a dimensionless quantity, then it is a dimensionless time:
$$\tau \left(\dot{x} = \dot{x}, , t\right) \tag{30}$$
$$\left(\dot{\tau}^2 = x^2, , \dot{t}^2\right) \tag{31}$$
or in general:
$$\dot{\tau}^n = x^n \cdot \dot{t}^n \tag{32}$$
With the help of equations (26), (27), (28) and (30):
Page 73
From (51), (52) one obtains the differential equation. To facilitate integration, the substitution U, which abbreviates the acceleration variable, is introduced as an auxiliary variable. The notation U = ẍ/(ω² · x_max) is then contained in the computation diagram.
$$\frac{d^2x}{dt^2} = -\frac{d}{dt}\dot{x} - \frac{d}{dt_5}\dot{x} - \frac{d^2}{dt_4}\dot{x} - \frac{1}{4}\omega^2 \cdot x - U \tag{37}$$
and
$$\frac{d^2\dot{x}}{dt^2} = -\frac{d}{dt}\ddot{x} - \frac{d}{dt_5}\ddot{x} - \frac{d^2}{dt_4}\ddot{x} - \frac{1}{4}\omega^2 \cdot \dot{x} - \dot{U} \tag{38}$$
3.5 Example of a Complete Programming
The example in Section 3.5 illustrates a complete Runge–Kutta computation process, and then Sections 3.4 and 3.5 describe a step-by-step programming procedure.
The following numerical values are to be programmed:
$$m = 4600 ; \frac{\text{kg} \cdot \text{s}^2}{\text{m}} \tag{5}$$
$$d = 300 ; \frac{\text{kg}}{\text{cm}} \tag{6}$$
$$c = 150 ; \frac{\text{kg}}{\text{cm}} \tag{7}$$
The system is excited at 45 cm on the reference level and then released freely, i.e., the following initial conditions apply:
$$x_0 = 45 \text{ cm} \tag{30}$$
$$\left(\frac{dx}{dt}\right)_0 = 0 \tag{39}$$
Page 74
A higher-order step function is not to be applied:
$$f = 0 \tag{40}$$
In order to determine the amplitude ratios, which determine which variable governs the scaling (the Amplitudenmaßstab), one must proceed as follows. The amplitudes that occur in computation should remain, as far as possible, below the machine unit (Recheneinheit), yet they should also be as large as possible — i.e., the amplitudes of all variables should lie in the range between 1 and the amplitude limit. Thus one must look at the calculation unit, and then fix the amplitude scale so that the variables use as much of the available range as possible without exceeding it. For integrator “5” the scale factor is:
$$a = \frac{100 %}{45 \text{ cm}} = 2.22 ; \frac{%}{\text{cm}} \tag{41}$$
The scale factor for the integrator “5” is:
$$V_0 = 100 ; \text{S} \tag{42}$$
The derivatives of the computation variable are computed from equations (26) and (30) to (31) as follows:
$$\frac{dx}{dt} = \frac{d , x}{1 \cdot t} \tag{43}$$
$$\frac{d\dot{x}}{dt} = \frac{d , \dot{x}}{1 \cdot t} \tag{44}$$
Under the assumption that a sine-like oscillation is carried out, the frequency condition gives:
$$\left(\frac{d\dot{x}}{dt}\right)_{\max} = \omega \cdot s \tag{45}$$
Page 75
The maximum values of the first and second derivatives of the computation variable are:
$$\left(\frac{dx}{dt}\right){\max} = \omega^2 \cdot x{\max} \tag{46}$$
$$\left(\frac{d^2x}{dt^2}\right){\max} = \omega^2 \cdot \omega \cdot x{\max} \tag{47}$$
The actual occurring maximum values are somewhat lower.
The actual task here is the determination of the amplitude scale (Amplitudenmaßstab). The amplitude scale and its correction is to be established from the following partially known requirements:
-
The coefficients are to be chosen so that the amplitudes do not exceed the machine range and are not unnecessarily small.
-
The coefficients in the computation circuit must be consistent with the differential equation (match the differential equation).
Requirement 1, from the output of integrator “5”, places the initial condition equal to 1.0 S (Volt), that is:
$$\left(\frac{x}{x_{\max}}\right) \cdot 100 ; \text{S} = 1.0 ; \text{S} \tag{48}$$
Page 76
The output amplitude of integrator “4” is, like that of integrator “5”, also 100 S (from (26) and (55)):
$$\left|\frac{1}{T_1} \cdot \frac{d}{dt}\omega \cdot x_{\max} \cdot 100 ; \text{S}\right| \leq 100 ; \text{S} \tag{49}$$
The absolute value is also equal to 100 S.
The output amplitude of integrator “3”, likewise equal to that of summing amplifier “6” (from (27) and (30)):
$$\left|\frac{1}{T_1} \cdot \frac{d}{dt}\omega^2 \cdot x_{\max} \cdot 100 ; \text{S}\right| \leq 100 ; \text{S} \tag{50}$$
The absolute value is also equal to 100 S.
The output amplitude of integrator “3”, equal to the output amplitude of summing amplifier “6” (from (27) and (31)):
$$\left|\frac{1}{T_1 \cdot d_1} \cdot \omega \cdot x_{\max} \cdot 100 ; \text{S}\right| \leq 100 ; \text{S} \tag{51}$$
The absolute value is also equal to 100 S.
Requirement 2 provides two more determination equations for the variables. The amplitude of the summing amplifier “6” equals the product of the respective parts of the differential equation (the respective coefficients of the differential equation), and in each case the output of the corresponding amplifier of the differential equation is multiplied. The outputs of amplifiers “3” and “4” denote in the differential equation the partial expression:
$$\frac{d^2x}{dt^2} = -\frac{d}{dt}\dot{x} - \frac{d^2}{dt^2}x \tag{53}$$
In the variable “f” and “m”, one obtains the following condition equations from the amplifier outputs. The output of the summing amplifier “4” is equal to the product of the corresponding part of the differential equation, and the outputs of the amplifiers “3” and “4” are given in the differential equation as the partial expression:
$$T_1 \cdot x_4 = \frac{d}{dt} + \frac{d_1}{dy} \tag{54}$$
Page 77
For the variable “T” and “m”, one obtains from the inputs
$$x_4 \cdot x_5 = \frac{d}{t} \cdot \frac{1}{t} \tag{55}$$
Similarly, from the output of integrator “4” and the summing amplifier “5”, the last condition equation is obtained:
$$x_3 \cdot x_4 = x_5 \cdot \frac{d_1}{y} \tag{56}$$
From these 5 condition equations, one can determine the size of the amplitude scale factors.
Equation (41) yields the ratio of the amplitude scale factors a_m and a:
$$\frac{x_{\max}}{x_{\max}} = \frac{100 ; %}{45 ; \text{cm}}$$
The coefficients K₃ and K₄ (the gain factors) for the integrators are initially unknown, i.e., one is free to choose the gain, subject to fulfilling the conditions stated above. If K₃ is set equal to 1 s⁻¹, then from (50) it follows:
$$K_3 = 10^{-1} ; \text{s}^{-1}$$
This yields from condition equation (52):
$$x_g = \frac{2.22 ; \frac{%}{\text{cm}} \cdot 0.158 ; \text{s}^{-1} \cdot 45 ; \text{cm}}{1 ; \text{s}^{-1} \cdot 100 ; %} = 10 ; \text{s}^{-1}$$
Page 78
Using the known formula:
$$\omega = \sqrt{\frac{c}{m}} = \sqrt{\frac{150 ; \frac{\text{kg}}{\text{cm}}}{4600 ; \frac{\text{kg} \cdot \text{s}^2}{\text{m}}}} \approx 0.158 ; \text{s}^{-1} \tag{59}$$
is confirmed. The factor K₃ can then be determined from equation (52):
$$K_3 = \frac{a \cdot \omega^2 \cdot x_{\max}}{a^2 \cdot 100 ; % \cdot K_4}$$
$$= \frac{2.22 ; \frac{%}{\text{cm}} \cdot 0.158^2 ; \text{s}^{-2} \cdot 45 ; \text{cm}}{1 ; \text{s}^{-1} \cdot 100 ; % \cdot 10^{-1} ; \text{s}^{-1}} = 10^{-1} ; \text{s}^{-1} \tag{60}$$
Equation (54) gives the coefficient x₄:
$$x_4 = \frac{1}{x_3} \cdot \frac{1}{x_5} = \frac{300 ; \frac{\text{kg}}{\text{m}}}{6000 \cdot 0.0158 ; \text{s}^{-1} \cdot 0.0158 ; \text{s}^{-1}} \cdot 10^{-1} ; \text{s}^{-1}$$
$$= 0.317 \tag{61}$$
And from equation (55) the coefficient K₀:
$$K_0 = \frac{150 ; \frac{\text{kg}}{\text{m}}}{6000 \cdot 0.0158 ; \text{s}^{-1} \cdot 10^{-1} ; \text{s}^{-1} \cdot 10^{-1}}$$
$$= \frac{150 ; \frac{\text{kg}}{\text{cm}}}{6000 \cdot 0.0158 ; \text{s}^{-1} \cdot 10^{-1} \cdot 10 ; \text{s}^{-1} \cdot 1} \tag{62}$$
With that, all coefficients are determined and the programming is complete. The eigenfrequency in the computer is in this case zero (i.e., the undamped natural frequency on the computer corresponds to the excitation frequency).
Page 79
$$f_r = \frac{\omega}{2\pi} \cdot \frac{1}{z} = \frac{0.158 ; \text{s}^{-1}}{2\pi \cdot 0.0158} = 1.59 ; \text{Hz} \tag{63}$$
If a lower oscillation frequency is desired — for example when a mechanical pen recorder is used as a readout device — the gain factors K₃ and K₄ can both be set equal to 1 s⁻¹.
In this case, from (50):
$$z = \frac{2.22 ; \frac{%}{\text{cm}} \cdot 0.158 ; \text{s}^{-1} \cdot 45 ; \text{cm}}{1 ; \text{s}^{-1} \cdot 100 ; %} = 0.158 \tag{64}$$
and the oscillation frequency on the computer:
$$f_r = \frac{0.158 ; \text{s}^{-1}}{2\pi \cdot 0.158} = 0.159 ; \text{Hz} \tag{65}$$
Even slower oscillations can be achieved by connecting potentiometers upstream of integrators “3” and “4”.
Page 80
References
-
C. A. Korn and Th. M. Korn: Electronic Analog Computers McGraw-Hill Book Company, New York 1952
-
C. L. Johnson: Analog Computer Techniques McGraw-Hill Book Company, New York 1956
-
D. Ernst: Elektronische Analogrechner (Electronic Analog Computers) R. Oldenbourg-Verlag, Munich 1960
Page 81
Table of Contents
1. Fundamentals of Electronic Analog Computers
- 1.1 Scope of application — Page
- 1.2 Basic elements
- 1.21 Operational amplifier — ”
- 1.22 Integrator — ”
- 1.23 Summing amplifier — ”
- 1.24 Characteristics of operational amplifiers — ”
- 1.3 Initialization and operation of a simple analog computation — ”
- 1.4 Block diagram of a simple computing machine — ”
2. Description of the BBC Desktop Analog Computer
- 2.1 General construction
- 2.21 General design — ”
- 2.22 Panel layout — ”
- 2.23 Mains connection — ”
- 2.4 Technical data
- 2.41 Amplifier section — ”
- 2.42 Coefficient potentiometer — ”
- 2.43 Interconnection panel — ”
- 2.44 Reference voltage unit — ”
- 2.45 Multiplier/divider — ”
- 2.46 Stabilizing unit — ”
- 2.47 Voltmeter — ”
- 2.48 Diode circuit — “
Page 82
3. Programming the Analog Computer
| Section | Title | Page |
|---|---|---|
| 3.1 | General | 63 |
| 3.2 | Setting up the coupling diagram (patch diagram) | 64 |
| 3.3 | Determining the amplitude scale | 67 |
| 3.4 | Determining the time scale | 69 |
| 3.5 | Example of a complete programming | 72 |
| References | 79 |