ON THE USE OF PULSE BOOLEAN VARIABLES IN THE SYNTHESIS OF MULTI-CYCLE POTENTIAL-PULSE RELAY DEVICES

E. L. Sobakin

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

The introduction of pulse variables, serving to denote pulse signals [1], naturally increases the number of possible combinations of logical variables. Since each Boolean variable in this case is represented in the form of two variables (static and pulse), the number of possible combinations will increase by \(2^n\) times. Therefore

\[ N = 2^{2n} = 4^n, \]

where \(N\) is the total number of combinations of logical variables (pulse and static);

\(n\) is the number of static variables.

However, of the four combinations of the pulse and static variables corresponding to one Boolean variable, the combination \(\overline{a}a_\tau\) is identically equal to zero (see Corollary 2 [1]) and, consequently,

\[ N = (4 - 1)^n = 3^n. \tag{1} \]

The total number of logical functions describing a potential-pulse relay device will then be expressed by the formula

\[ M = 2^{3^n}. \tag{2} \]

Table 1 gives the names, designations, and normal forms of the potential-pulse logical functions of one Boolean variable.

In fact, these functions are functions of two variables; however, the number of these functions is half the number of ordinary Boolean functions of two variables [2].

Comparing the obtained potential-pulse functions with ordinary logical functions of two variables, the following conclusions may be drawn: function \(v_0\) is analogous to the function “inhibition of \(a\),” \(v_1\) to logical product, \(v_2\) to the function of nonequivalence and the function “inhibition of \(a_\tau\),” \(v_3\) to logical sum, \(v_4\) to Peirce’s function (the NOR function), \(v_5\) to the function of equivalence and the function “implication of \(a_\tau\),” \(v_6\) to Sheffer’s function, and \(v_7\) to the function “implication of \(a\).”

The number of potential-pulse logical functions of two Boolean variables, each of which corresponds to a separate relay device, can be determined from formula (2).

The most convenient of the existing ways (methods) of specifying logical functions is the matrix method, in which the functions

are specified in the form of Karnaugh matrices (maps) [2], [3], and Veitch diagrams [4]. The number of squares of the matrix depends on the number of Boolean (static) variables and is determined by the formula \(N=2^n\) for ordinary functions and by formula (1) for potential-pulse functions. Consequently, matrices of poten-

Table 1

tial-pulse logical functions, unlike the matrices of ordinary Boolean functions, will have a number of squares that is a multiple of \(3^n\).

As examples, Fig. 1, \(a\), \(b\), and \(c\), shows matrices of potential-pulse functions of one, two, and three Boolean variables. The matrix in Fig. 1, \(a\), specifies the logical function “delay in switching off \(a\)”; the matrices in Fig. 1, \(b\), and Fig. 1, \(c\), specify, respectively, the logical sum \(a_\tau+b_\tau\) and the logical product \(a_\tau b_\tau c_\tau\).

From Fig. 1 it follows that the matrix for potential-pulse functions of two variables (Fig. 1, \(b\)) consists of three matrices of potential-pulse logical functions of one variable; the matrix of potential-pulse functions of three variables (Fig. 1, \(c\)) consists of three matrices of potential-pulse functions of two variables, etc. The rules for distribu-

Fig. 1

\(a)\quad f=\bar a+a_\tau\)

\(b)\quad f=a_\tau+b_\tau\)

\(c)\quad f=a_\tau b_\tau c_\tau\)

of the unit and zero regions of the independent, i.e., input and intermediate, variables remain the same as in matrices for ordinary Boolean functions [2] and [7].

The matrix method of specifying logical functions possesses, as is known, a number of advantages, among which one should first of all mention the possibility of using matrices for the minimization of functions and the synthesis of relay devices under incompletely specified initial conditions. At the same time, as in the case of ordinary logical functions, the specification of multicycle potential-pulse logical functions by means of matrices is complicated by the necessity and difficulty of determining the number of intermediate variables from the formulated design conditions for the device.

It is quite obvious that the use of pulse variables will lead to a reduction in the required number of intermediate variables, since each pulse variable may be regarded as an elementary multicycle function that passes from one definite state to another automatically, without changing the value of the independent variable.

At present the two most widespread methods for synthesizing multicycle relay devices are the following: one uses, for describing the operating conditions of devices, the language of switching tables [5], [6]; the other uses the language of state tables, transition tables, and function matrices [2], [3].

Using the second method, let us consider an example of the synthesis of a multicycle relay device whose operating conditions may be formulated as follows: “In the initial state (in the absence of an input signal) element \(X\) is on, and element \(Y\) is off. When an input signal appears, element \(X\) is switched off and element \(Y\) is switched on. After a time \(\tau_1\), \(Y\) is switched off and \(X\) is switched on; after a time \(\tau_2\), \(X\) is switched off and \(Y\) is switched on, and so on. The cycles repeat as long as there is a signal equal to 1 at the input of the device. If the input signal assumes the value 0, then the device returns to the initial state. It is required that the elements \(X\) and \(Y\) at no instant of time be in the same state.”

It is known from electronics that such conditions are satisfied by an electronic relaxation oscillator—a multivibrator—which is normally inhibited and is switched into a self-oscillating mode by an input signal.

It is not difficult to establish that the multicycle logical function describing the operation of such a multivibrator will have one input variable (let us denote it by \(a\)) and two intermediate variables \(x\) and \(y\), which are simultaneously output variables.

For a clear representation of the required sequence of operation of the device being designed, its operating conditions may be represented in the form of the following conditional notation:

\[ A_{01}\to X_{10}\to \overline{Y}_{01}\xrightarrow{\tau_1}\overline{Y}_{10}\to \overline{X}_{01}\xrightarrow{\tau_2}\overline{X}_{10}\to \overline{Y}_{01}\xrightarrow{\tau_1}\overline{Y}_{10}\to \ldots A_{10}\to X_{01}\to Y_{10}. \]

Here the subscript 01 denotes switching on, and the subscript 10 denotes switching off of the corresponding element. Analyzing the conditional notation of the transitions, one can establish that the input signal \(a\) will be static, while the connection between elements \(X\) and \(Y\) is effected by the pulse signals \(x_{\tau_2}\) and \(y_{\tau_1}\). Thus, the multicycle function describing the operation of the multivibrator will be potential-pulse.

According to the conditions of the problem, we compile the state table of this multicycle potential-pulse logical function (Table 2).

This table is constructed by a known method (see [2]), but, unlike similar tables for ordinary multicycle Boolean functions, each

of an intermediate variable there correspond two columns, “s” and “p,” serving to describe the static and pulse variables.

To find the matrix of the general multicycle function, it is necessary to assign to each state of the function a definite combination of static and pulse variables and to construct the transition matrix.

Table 2

In Fig. 2, a combined correspondence and transition matrix is given for the present example. The transitions are indicated by arrows. Replacing the states in the matrix of Fig. 2a by the corresponding combinations of static variables taken from the table of states, we obtain the matrix of the multicycle function $XY$ in Fig. 2b, from which it is not difficult to obtain the matrices of the intermediate functions $X$ and $Y$ (Fig. 2c and d).

Fig. 2

Using the matrices of these functions, and forming the corresponding contours (Fig. 2c and d), one can find the algebraic expressions for the intermediate functions $X$ and $Y$:

\[ X=\bar{x}\bar{y}_{\tau_1}+x_{\tau_2}\bar{y}_{\tau_1}+\bar{a} =\bar{a}+\bar{y}_{\tau_1}(\bar{x}+x_{\tau_2}), \tag{3} \]

\[ Y=(x+y_{\tau_1})(\bar{x}_{\tau_2}+y_{\tau_1})a =a(y_{\tau_1}+x\bar{x}_{\tau_2}). \tag{4} \]

It follows from these expressions, and also directly from the matrices of the functions \(X\) and \(Y\), that the intermediate variables are related to one another by the following relations:

\[ x=\bar y \quad \text{and} \quad y=\bar x . \]

Replacing in expression (3) the variables \(\bar x\) and \(x_{\tau_2}\) by the corresponding values of the variable \(y\), and in expression (4) the variable \(y_{\tau_1}\) by the corresponding value \(x_{\tau_1}\), we obtain, for \(\tau_1<\tau_2\):

\[ X=\bar a+\bar y_{\tau_1}(y+y_{\tau_2})=\bar a+\bar y_{\tau_1}=\overline{a y_{\tau_1}}, \tag{5} \]

\[ Y=a(\bar x_{\tau_1}+x\bar x_{\tau_2})=a(\bar x_{\tau_1}+x)(\bar x_{\tau_1}+\bar x_{\tau_2})=a\bar x_{\tau_2}. \tag{6} \]

In principle, the structural formulas obtained for a multicycle potential-pulse function may correspond to a relay device both on contact and on contactless elements.

Of definite interest is the implementation of the obtained intermediate functions on electronic or semiconductor devices (in order to compare the resulting and existing schematic diagrams of a multivibrator). Therefore, as an example, let us consider the implementation of functions (5) and (6) on semiconductor devices. For a multivibrator on contactless elements, formulas (5) and (6) correspond to the functional (structural) diagram shown in Fig. 3, a.

Using the circuit of the \(\tau\)—NOT element given in [1] and an inverter on triodes that are in the normally closed state, one can obtain the schematic diagram of a multivibrator shown in Fig. 3, b.

Fig. 3

Here, on triode \(\mathrm{PT}_1\), the function \(X=\overline{a y_{\tau_1}}\) is implemented; triode \(\mathrm{PT}_2\) and capacitor \(C_2\) form the \(\tau\)—NOT element; the coincidence circuit II for negative potentials is formed by resistance \(R\) and diodes \(D_1\) and \(D_2\), through which the signals \(\bar x_{\tau_2}\) and \(a\) are applied. From the output of coincidence circuit II the function \(Y=a\bar x_{\tau_2}\) is formed.

It is not difficult to see that the circuit in Fig. 3, b uses the operating principle of a multivibrator with “positive bases.”

If transistors in the normally open state are used as inverters in elements \(I\) and \(\tau\)—NOT (Fig. 3, a), then one can obtain a prin-

the principal multivibrator circuit in a simpler form (Fig. 3, c), since the coincidence circuit II can be omitted in this case.

It is not difficult to verify that this circuit too is completely described by the structural diagram of Fig. 3, a.

REFERENCES

1. E. L. Sobakin, V. M. Novitskii. On the question of the use of special operators over Boolean variables in logical methods for the synthesis and analysis of potential-pulse relay devices. (In the present collection.)

2. N. P. Vasil’eva, I. Gashkovets. Logical elements in industrial automation, GEI, 1962.

3. S. Caldwell. Logical synthesis of relay devices, IL Publishing House, 1962.

4. E. N. Vavilov, G. P. Portnoi. Synthesis of circuits of electronic digital machines, Sovetskoe Radio Publishing House, 1963.

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

6. A. N. Yurasov. Theory of the construction of relay circuits, GEI, 1962.

7. A. D. Zakrevskii. A visual-matrix method for minimizing Boolean functions, Avtomatika i telemekhanika, vol. 21, no. 3, 1960.