On the Synthesis of Relay Circuits with Temporal Dependencies
Abstract Generated abstract
The paper addresses the synthesis of relay circuits whose behavior depends on signal duration, delay, and pulse formation, problems that ordinary Boolean algebra does not conveniently describe. It introduces static and pulse logical variables, including variables associated with leading and trailing signal transitions, and uses Boolean operations to express shortening, lengthening, and time shifting of signals as potential-pulse logical functions. The approach is illustrated through the synthesis of a pulse-duration discriminator, using state tables, transition matrices, and intermediate functions to derive structural and circuit realizations. The method is proposed as applicable to relay and pulse devices such as time-pulse encoders and decoders, pulse generators, and automatic switching circuits.
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ON THE SYNTHESIS OF RELAY CIRCUITS WITH TIME DEPENDENCIES
E. L. Sobakin
(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)
Relay devices are devices constructed from elements with relay characteristics. For the synthesis and analysis of such devices, Boolean algebra—the algebra of logic—is currently widely used [1].
In most practical cases, real devices include elements that are delayed upon “operation” or upon “release,” or both upon “operation” and “release” simultaneously. In addition, many devices provide for signals of different durations, and these devices themselves must transform the durations of signals.
Boolean algebra, in its usual form, is unsuitable for these purposes; therefore it is constantly being updated and supplemented—temporal and recurrent Boolean functions [2], etc., are appearing, making it possible to describe the operation of relay devices in time.
In the present work it is proposed to regard relay devices with time dependencies as potential-pulse devices. Then, having a mathematical apparatus that takes into account the difference between pulse and potential signals and their durations, one can, by using the operations of disjunction and conjunction, algebraically express the operations of decreasing and increasing the durations of signals and the operation of time shifting (delaying) signals. However, such a mathematical apparatus has not yet been encountered in the literature.
Let us introduce, to denote potential and pulse signals, static and, respectively, pulse variables. We shall give their mathematical definitions:
a) static variables
\[ a = \begin{cases} 0 & \text{for } t < t_0 \text{ and } t \geqslant t_1;\\ 1 & \text{for } t_0 \leqslant t < t_1; \end{cases} \qquad \overline{a} = \begin{cases} 1 & \text{for } t < t_0 \text{ and } t \geqslant t_1;\\ 0 & \text{for } t_0 \leqslant t < t_1; \end{cases} \]
b) pulse variables of the first kind (pulse variables of the transition \(0—1\))
\[ a_{\tau} = \begin{cases} 0 & \text{for } t < t_0 \text{ and } t \geqslant t_0 + |\tau|;\\ 1 & \text{for } t_0 \leqslant t < t_0 + |\tau|; \end{cases} \]
\[ \overline{a}_{\tau} = \begin{cases} 1 & \text{for } t < t_0 \text{ and } t \geqslant t_0 + |\tau|;\\ 0 & \text{for } t_0 \leqslant t < t_0 + |\tau|; \end{cases} \]
c) pulse variables of the second kind (pulse variables for transitions \(1\)—\(0\))
\[ \overline{a_{\tau'}}= \begin{cases} 0 \text{ for } t<t_1 \text{ and } t\geq t_1+|\tau'|,\\ 1 \text{ for } t_1\leq t<t_1+|\tau'|, \end{cases} \]
\[ a_{\tau'}= \begin{cases} 1 \text{ for } t<t_1 \text{ and } t\geq t_1+|\tau'|,\\ 0 \text{ for } t_1\leq t<t_1+|\tau'|. \end{cases} \]
In these formulas:
\(t\)—the current instant of time;
\(t_0\)—the instant at which the signal \(a\) changes from 0 to 1;
\(t_1\)—the instant at which the signal \(a\) changes from 1 to 0;
\(\tau\) and \(\tau'\)—special operators used to denote pulse signals;
\(|\tau|\) and \(|\tau'|\)—the moduli of the operators \(\tau\) and \(\tau'\), numerically equal to the durations of the pulse signals.
If \(T\) is the duration of the potential signal \(a\), then \(t_1=t_0+T\), and it is necessary that \(0<|\tau|<T\) and \(0<|\tau'|<T\). In principle, one may include the limiting conditions \(|\tau|=|\tau'|=0\) and \(|\tau|=|\tau'|=T\). Then the first case will correspond to the absence of pulse signals, while the second will mean that the durations of the pulse and potential signals are the same.
Using the definitions of disjunction and conjunction for Boolean variables and the definitions of the variables \(a\), \(a_\tau\), \(a_{\tau'}\) and their inversions, it is not difficult to establish that the expressions \(\overline{a_\tau}\cdot a=a^\tau\), \(a_{\tau'}+a=a^{\tau'}\), and \(a+\overline{a_{\tau'}}=a^{\tau\tau'}\) denote, respectively, the operations of shortening, lengthening the signal \(a\), and the operation of time shift. On the right-hand sides of these equalities stand the operator expressions \(a^\tau\), \(a^{\tau'}\), and \(a^{\tau\tau'}\), which show a second way of using the operators \(\tau\) and \(\tau'\). The moduli of the operators in this case indicate the magnitude of the decrease (increase) in the duration of the signal, as well as the magnitude of the time shift.
Algebraic expressions containing the indicated static and pulse logical variables and composed according to the laws of Boolean algebra will be called potential-pulse logical functions.
Table 1 gives the normal disjunctive forms of potential-pulse logical functions of one Boolean variable, their names, and their conventional designations. This table shows that a large class of functions taking account of time dependences and variable durations is added to the arsenal of ordinary Boolean functions.
As an illustration of the proposed mathematical apparatus in the synthesis of relay circuits with time dependences, let us consider an example of the synthesis of a pulse-duration discriminator.
It is required to design a device having one input and one output. At the output a pulse signal must appear every time the duration of the input unit signal is equal to \(\tau_1\). If the duration of the input signal is \(\tau_{\text{in}}<\tau_1\) or \(\tau_{\text{in}}>\tau_1\), then the output must have the signal 0. The duration of the output signal may be arbitrary, but significantly smaller. Obviously, the device must compare the duration of the input signal with the duration of another, intermediate signal, having duration \(\tau_1\) independently of the duration of the input signal, and, if these durations are equal, issue a pulse signal at the output.
Let us denote the input signal by \(a\), the intermediate signal by \(X\), and the output signal by \(Y\). Then the condition of the problem can be written in the following form.
If the leading edge of the signal \(X\) coincides with the leading edge of the signal \(a\), then the output signal \(Y\) assumes the value 1 only in the case when the trailing edges of these signals also coincide, i.e.,
\[ Y=\bar a_{\tau'}\cdot \bar x_{\tau'} . \]
Table 1
| No. | Function name | \(a\) | \(a_\tau\) | \(a_{\tau'}\) | Value | Normal disjunctive form |
|---|---|---|---|---|---|---|
| \(v_0\) | Zero | 0 | 0 | 0 | \(0\) | \(\bar a \bar a_\tau \bar a_{\tau'}+\bar a\bar a_\tau a_{\tau'}+a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_1\) | Repetition of \(a_\tau\) | 0 | 0 | 1 | \(a_\tau\) | \(aa_\tau a_{\tau'}\) |
| \(v_2\) | Delay on switching off \(a\) | 0 | 1 | 0 | \(a^\tau\) | \(a\bar a_\tau a_{\tau'}\) |
| \(v_3\) | Repetition of \(a\) | 0 | 1 | 1 | \(a\) | \(a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_4\) | Delay on switching off \(\bar a\) | 1 | 0 | 0 | \(\bar a^\tau\) | \(\bar a\bar a_\tau a_{\tau'}\) |
| \(v_5\) | Delay on switching on and off \(\bar a\) | 1 | 0 | 1 | \(\bar a^{\tau\tau'}\) | \(\bar a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_6\) | Differentiation of \(\bar a\) | 1 | 1 | 0 | \(\delta\bar a\) | \(\bar a\bar a_\tau a_{\tau'}+a\bar a_\tau a_{\tau'}\) |
| \(v_7\) | Repetition of \(a_{\tau'}\) | 1 | 1 | 1 | \(a_{\tau'}\) | \(\bar a\bar a_\tau a_{\tau'}+a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_8\) | Inversion of \(a_{\tau'}\) | 1 | 0 | 0 | \(\bar a_{\tau'}\) | \(\bar a\bar a_\tau \bar a_{\tau'}\) |
| \(v_9\) | Differentiation of \(a\) | 1 | 0 | 1 | \(\delta a\) | \(\bar a\bar a_\tau \bar a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_{10}\) | Delay on switching on and off \(a\) | 1 | 1 | 0 | \(a^{\tau\tau'}\) | \(\bar a\bar a_\tau \bar a_{\tau'}+a\bar a_\tau a_{\tau'}\) |
| \(v_{11}\) | Delay on switching off \(a\) | 1 | 1 | 1 | \(a^{\tau'}\) | \(\bar a\bar a_\tau \bar a_{\tau'}+\bar a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_{12}\) | Inversion of \(a\) | 1 | 1 | 0 | \(\bar a\) | \(\bar a\bar a_\tau \bar a_{\tau'}+\bar a\bar a_\tau a_{\tau'}\) |
| \(v_{13}\) | Delay on switching on \(a\) | 1 | 1 | 0 | \(\bar a^\tau\) | \(\bar a\bar a_\tau \bar a_{\tau'}+\bar a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
| \(v_{14}\) | Inversion of \(a_\tau\) | 1 | 1 | 1 | \(\bar a_\tau\) | \(\bar a\bar a_\tau \bar a_{\tau'}+\bar a\bar a_\tau a_{\tau'}+a\bar a_\tau a_{\tau'}\) |
| \(v_{15}\) | Unit | 1 | 1 | 1 | \(1\) | \(\bar a\bar a_\tau \bar a_{\tau'}+\bar a\bar a_\tau a_{\tau'}+a\bar a_\tau a_{\tau'}+aa_\tau a_{\tau'}\) |
Let us synthesize an intermediate device which, in the absence of an input signal \(a\), would have at its output the signal \(x=0\), and would issue a unit signal of duration \(\tau_1\) when an input unit signal is applied, independently of its duration. Under these conditions we compile the state table of the intermediate device (Table 2) [3]. Minimizi-
form the state table is impossible; therefore we encode the states by combinations of the intermediate variables \(z\) and \(z_{\tau_1}\) (Table 3). We construct the transition matrix (Fig. 1a) and the matrix of the intermediate function \(Z\) (Fig. 1b).
Table 2
| \(s\) | \(a=0\) | \(a=1\) | \(X\) | State duration |
|---|---|---|---|---|
| 1 | \(\langle 1\rangle\) | 2 | 0 | — |
| 2 | 3 | 3 | 1 | \(\tau_1\) |
| 3 | 1 | \(\langle 3\rangle\) | 0 | — |
Table 3
| \(s\) | \(z\) | \(z_{\tau_1}\) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 1 | 1 |
| 3 | 1 | 0 |
From the matrix of the intermediate function \(Z\) we find its algebraic expression
\[ Z=a+z_{\tau_1}. \]
Fig. 1
a) transition matrix; b) matrix of the intermediate function \(Z\).
For the intermediate output \(X\) we shall have
\[ X=z_{\tau_1}\quad \text{or}\quad X=(a+x)_{\tau_1}. \]
According to the rules derived by the author, the last formula can be transformed into
\[ X=\overline{(a+x)}_{\tau_1}^{\prime}. \]
The general formulas describing the operation of a pulse-duration discriminator will have the form:
\[ X=\overline{(a+x)}_{\tau_1}^{\prime}\quad \text{and}\quad Y=\bar a_{\tau}+x_{\tau}. \]
Using these formulas and the conventional designations of logical and pulse elements, it is not difficult to construct the structural (functional) circuit of the device (Fig. 2a). As an example, Fig. 2b gives a schematic circuit of a pulse-duration discriminator using semiconductor diodes and triodes.
It should be noted that in the structural diagram, pulse elements of the second kind, introduced by the author, are shown by conventional designations in the form of a rectangle. In addition, the author has proposed structural designations for pulse elements of the first kind, as well as concrete
variants of realizing pulse elements on contact and contactless elements.
In conclusion, we note that the scope of the article does not allow us to present all the consequences and conclusions used in the synthesis example considered. However, as can be seen, the synthesis of potential-pulse relay devices, when the introduced mathematical variables are used, can already be carried out by existing methods, in particular by a method using the language of tables of states, transitions, and matrices of intermediate and output functions.
Fig. 2
The proposed method for describing relay circuits with time dependencies can be used in the synthesis of encoders and decoders of time-pulse-interval signals, pulse generators with a specified pulse duty factor, various automatic switching devices, etc.
REFERENCES
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S. Caldwell. Logical Methods of Relay-Circuit Synthesis. IL Publishing House, 1962.
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D. A. Pospelov. Logical Methods of Analysis and Synthesis of Circuits. Energiya Publishing House, Moscow–Leningrad, 1964.
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N. P. Vasil’eva and I. Gashkovets. Logical Elements in Industrial Automation. GEI Publishing House, Moscow–Leningrad, 1962.