ON THE USE OF SPECIAL OPERATORS ON BOOLEAN VARIABLES IN LOGICAL METHODS FOR THE SYNTHESIS AND ANALYSIS OF POTENTIAL-PULSE RELAY DEVICES

E. L. SOBAKIN, V. M. NOVITSKII

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

Existing logical methods for the synthesis of relay devices in principle make it possible to find the structure of any relay device described by an ordinary, temporal, or recurrent Boolean function. The structural circuits obtained in this way can be constructed from any functionally complete set of logical elements, a delay element, and a switching device. The input and output signals of these elements are usually represented either in the form of voltage or current levels (we shall call them static signals), or in the form of voltage or current pulses (pulse signals).

We shall call an element a pulse element if, when a static signal is applied to its input, the output signal is pulsed. In fact, the output signal of a pulse element (a pulse signal) depends not on the state of the input signal, but on its change from 0 to 1, or from 1 to 0.

Since most relay devices use both static and pulse signals simultaneously, describing the operation of such devices by means of the notation used in Boolean algebra (and, still more, transforming structural formulas) presents considerable difficulties. This is explained by the fact that the designations of static and pulse variables (signals) are the same, whereas there exists a definite logical relation between these variables that does not express identity.

Attempts have already been made in the literature to denote pulse variables through static variables by means of special logical operators (transition operators) [1 and 2], or by means of special indices [3 and 4]. However, for solving engineering problems it is important to know not only which signals (variables) will be static and which pulse, but also to take into account the duration of pulse signals (pulses), which is determined both by the internal parameters of the pulse element and by the duration of the input signal.

We shall assume that the duration of a static signal \(T\) is always greater than the duration of a pulse \(\tau\) (Fig. 1, b), i.e., \(T > \tau\). Let us introduce the operator \(\tau\) to denote a pulse signal. Assigning it at the lower right (as a subscript) to Boolean variables, we shall consider that the expression obtained denotes a pulse signal formed from the signal to which this operator is applied. In this case the duration of the pulse signal is equal to \(\tau\). The quantity \(\tau\), expressed in some units—

in each particular case, quite definite, only at the last stage of synthesis—the creation of the basic circuit diagram.

When drawing up structural formulas and circuits, however, only the order of the values of \(\tau\) is important. The limits of variation of \(\tau\) may be taken as follows:

\[ 0 \leq \tau \leq T . \]

The limiting condition \(\tau = 0\) essentially means that the output signal does not depend on the input signal, and in this case the pulse element will implement, depending on its type, either the zero or the unit function.

For \(\tau = T\), the pulse signal is equivalent to a static one.

According to the functional dependence of the output signal on the input signal, all pulse elements may be divided into two types: a pulse repeater (abbreviated \(\tau\)-repeater) and a pulse inverter (abbreviated \(\tau\)—NOT).

The conventional designations of these elements and the timing diagrams explaining the functional dependence of the output signal on the input signal are shown in Figs. 1 and 2.

At the instant \(t_0\), when the input signal of the pulse repeater (Fig. 1, a) changes from 0 to 1 (Fig. 1, b) or from 1 to 0 (Fig. 1, c), the output signal repeats the change of the input signal. However, the output signal

Fig. 1

has the unit (zero) value only for the time \(\tau\), and then assumes its initial value, despite the fact that the input signal still has the unit (zero) value. The reverse change of the input signal from 1 to 0 for Fig. 1, b and from 0 to 1 for Fig. 1, c has no effect on the value of the pulse variable.

The output signal of the element \(\tau\)—NOT (Fig. 2, a) makes transitions opposite to the transitions of the input signal (Fig. 2, b, c). It should be noted that when a pulse signal of duration \(\tau_{\text{in}}\) is applied to the input of a pulse element, the output signal may have a duration \(\tau_{\text{out}}\) equal either to the duration of the input signal or to the duration \(\tau_{\text{own}}\), determined by the internal parameters of the pulse element.

If \(\tau_{\text{own}} > \tau_{\text{in}}\), then \(\tau_{\text{out}} = \tau_{\text{in}}\),

but if \(\tau_{\text{own}} < \tau_{\text{in}}\), then \(\tau_{\text{out}} = \tau_{\text{own}}\).

All the properties of pulse elements formulated above are satisfied by ordinary repeaters and inverters implemented with tubes, transistors, or other elements with differentiating circuits connected at the input.

As an example, in Fig. 1, c and Fig. 2, c, pulse elements on transistors of the \(n—p—n\) type are presented, performing the functions of a \(\tau\)-repeater (Fig. 1, c) and a \(\tau\)-inverter (Fig. 2, c). The duration of the output

Fig. 2

pulse will be determined by the time constant of the capacitor charge and by the cutoff potential of the triode.

In view of the specificity of contact elements, the time parameters of the signal at the output of pulse elements implemented on electromagnetic relays vary considerably. However, the basic functional dependence of the output signal on the input signal can also be realized on contact elements.

In Fig. 3 a circuit with one electromagnetic relay \(P\) is shown, realizing both the function of a pulse repeater and the function of a pulse inverter.

If the output of the circuit is taken through the normally open contact of relay \(P\), then the function of a \(\tau\)-repeater will be realized; and if through its normally closed contact, then the function of a \(\tau\)-inverter. The duration of the output pulse can be varied by changing the capacitance of the capacitor and the release time of relay \(P\).

Fig. 3

Considering the operation of circuits corresponding to the right- and left-hand sides of identities expressing one or another basic law of Boolean algebra, one can verify the validity of these laws for pulse and static variables. However, because of the non-equivalence of pulse and static variables from a single signal, the former consequences of these laws change and new ones appear. Below are given the principal consequences following from the non-equivalence of pulse and static variables and from the definitions of logical product and logical sum.

Consequences of logical multiplication:

\[ a \cdot a_\tau = a_\tau, \tag{1} \]

\[ \bar a\, a_\tau = 0 \tag{2} \]

\[ \bar a\, \bar a_\tau = \bar a, \tag{3} \]

\[ a\, \bar a_\tau = a^\tau \text{— delay in switching on} \tag{4} \]

Expression (4), in which \(\tau\) is not an exponent but an operator, corresponds to the signal at the output of an AND circuit, to whose inputs the signals \(a\) and \(\bar a_\tau\) are applied (Fig. 4,a). From the timing diagrams corresponding to this case (Fig. 4,b), it is seen that the output signal changes its value from 0 to 1 with a delay of magnitude \(\tau\) relative to \(t_0\)—the instant at which the input signal \(a\) appears.

Thus, such a circuit produces a delay of the leading edge of the input signal by the amount \(\tau\), while the trailing edge of the output signal coincides with the trailing edge of the signal \(a\).

Consequences of logical addition:

\[ a + a_\tau = a, \tag{5} \]

\[ a + \bar a_\tau = 1, \tag{6} \]

\[ \bar a + \bar a_\tau = \bar a_\tau, \tag{7} \]

\[ \bar a + a_\tau = \bar a^\tau \text{— delay in switching off} \tag{8} \]

Expression (8) corresponds to the signal at the output of an OR circuit, to whose inputs the signals \(\bar a\) and \(a_\tau\) are applied (Fig. 4,c). The timing diagrams illustrating the operation of such a circuit are presented in Fig. 4,d. In this case as well, the output signal is switched off (assumes the value 0) with a delay of time \(\tau\) after the signal \(\bar a\) is switched off, i.e., the circuit produces a delay in switching off the signal \(\bar a\).

Fig. 4

Proofs of the remaining consequences can be carried out in an analogous manner with the aid of timing diagrams illustrating the operation of the corresponding circuits.

In addition to the consequences of logical multiplication and addition noted above, let us give a number of further consequences that may be used for transforming and simplifying structural formulas:

\[ \begin{aligned} a_{\tau}+a^{\tau}&=a &&(9)\\ a_{\tau}+\overline{a^{\tau}}&=\overline{a_{\tau}} &&(10)\\ a+a^{\tau}&=a &&(11)\\ \overline{a}+a^{\tau}&=\overline{a_{\tau}} &&(12)\\ a_{\tau}+\overline{a^{\tau}}&=\overline{a^{\tau}} &&(13)\\ \overline{a_{\tau}}+\overline{a^{\tau}}&=1 &&(14)\\ a+\overline{a^{\tau}}&=1 &&(15)\\ \overline{a}+\overline{a^{\tau}}&=\overline{a^{\tau}} &&(16) \end{aligned} \qquad \begin{aligned} a_{\tau}\cdot a^{\tau}&=0 &&(17)\\ \overline{a_{\tau}}\cdot a^{\tau}&=a^{\tau} &&(18)\\ a\cdot a^{\tau}&=a^{\tau} &&(19)\\ \overline{a}\cdot a^{\tau}&=0 &&(20)\\ a_{\tau}\cdot \overline{a^{\tau}}&=a_{\tau} &&(21)\\ \overline{a_{\tau}}\cdot \overline{a^{\tau}}&=\overline{a} &&(22)\\ \overline{a}\cdot \overline{a^{\tau}}&=\overline{a} &&(23)\\ a\cdot \overline{a^{\tau}}&=a_{\tau} &&(24) \end{aligned} \]

\[ \overline{a}+a^{\tau}+a_{\tau}=1. \tag{25} \]

It is not difficult to prove these identities, taking into account the consequences of logical multiplication and addition for pulse and static variables. In the case where, in the device being designed, it is necessary to provide for pulse signals (or delays) of different duration and to take this into account in the structural formulas, one must bear in mind that

\[ a_{\tau_1}\cdot a_{\tau_2}=a_{\tau\min}, \tag{26} \]

\[ a_{\tau_1}\cdot \overline{a_{\tau_2}} \begin{cases} 0, & \text{for } \tau_1 \leqslant \tau_2,\\ (a_{\tau_1})^{\tau_2}, & \text{for } \tau_1>\tau_2, \end{cases} \tag{27} \]

\[ a_{\tau_1}+a_{\tau_2}=a_{\tau\max}, \tag{28} \]

\[ a_{\tau_1}=\overline{a_{\tau_2}}= \begin{cases} 1, & \text{for } \tau_1>\tau_2,\\ \dfrac{1}{(\overline{a_{\tau_2}})^{\tau_1}}, & \text{for } \tau_1<\tau_2, \end{cases} \tag{29} \]

where \(\tau_{\min}\) is the duration of the shortest pulse,
\(\tau_{\max}\) is the duration of the longest pulse,
\((a_{\tau_1})^{\tau_2}\) is the pulse \(a_{\tau_1}\), whose leading edge is delayed by the time \(\tau_2\), i.e., the pulse \(a_{\tau}\) of duration

\[ \tau=\tau_1-\tau_2; \]

\[ (\overline{a_{\tau_2}})^{\tau_1} \]

is the pulse \(\overline{a_{\tau_2}}\), whose leading edge is delayed by the time \(\tau_1\), i.e., the pulse \(\overline{a_{\tau}}\) of duration

\[ \tau=\tau_2-\tau_1. \]

It should be noted that the operator \(\tau\) can be applied not only to individual Boolean variables, but also to entire algebraic expressions.

Conclusions

1. By applying the operator \(\tau\) to Boolean variables, one can, in algebraic form, simultaneously operate with static variables, pulse variables, and variables with delays on switching on and switching off.

2. With the aid of these variables one can describe the operation of any single-cycle and multicycle relay device. In this case, from the structural formula of the device one can immediately judge the character of the individual variables and the time dependences in the circuit.

1. A definite merit of such an operator-logical method for describing circuits may be considered to be the introduction of the time factor into Boolean algebra, without requiring a separate variable for this purpose, as is the case when time-dependent Boolean functions are used.

2. The possibility of taking into account and writing in algebraic form delays of variables only for switching off or switching on is of definite interest for the engineering application of the proposed method, since recurrent Boolean functions do not provide such a possibility.

3. One of the important advantages of the method may be considered to be the substantial simplification of the synthesis and analysis of multi-cycle potential-pulse relay devices, since the second transition of pulse variables occurs automatically. Consequently, to carry out this transition it will not be necessary to introduce new intermediate variables (new intermediate elements), and the structural diagrams of multi-cycle relay devices implemented both on contact and on contactless elements will be considerably simpler.

4. The method requires the development of a rational form for specifying potential-pulse logical functions.

REFERENCES

1. A. D. Talantsev. On the analysis and synthesis of certain electrical circuits by means of special logical operators, Avtomatika i telemekhanika, vol. 20, No. 7, 1959.

2. V. G. Lazarev and E. I. Piil. Synthesis of asynchronous finite automata, Nauka Publishing House, 1964.

3. V. N. Roginskii. Construction of relay control circuits, Energiya Publishing House, 1964.

4. M. A. Gavrilov. Theory of relay-contact circuits, Publishing House of the Academy of Sciences of the USSR, 1950.