A general n-terminal circuit was defined in Section (Non-series-parallel circuits). In this section, we will consider the problem of designing certain special types of n-terminal circuits, where the design problem can be treated as the problem of combining a set of specified 2-terminal circuits into an n-terminal circuit in such a way as to share switches or parts of circuits so far as possible. To illustrate the possibilities, consider the two functions f = a(x + y) and g = axy. The separate circuits and a combined circuit are shown in Fig. 1-1. It is apparent that the combined circuit can be used to obtain either function f or function g and with one less switch than the two circuits require if drawn separately. In addition, a new function, f₂₄, has been introduced  (representing the circuit joining terminals T₂ and T₄). Provided that the possibility of a closed circuit between T₂ and T₄ does not create other problems in connection with the apparatus being designed, the 3-terminal circuit may be used in place of the two 2-terminal circuits with a saving in the number of switches needed. This illustration leads to the following definition of the problem we will consider.

We assume given a set f₁, f₂, . . . , f_nof functions (or a statement of properties leading to such functions), where some or all of the switches involved are common to more than one function. We will design an

FIG. 1-1. Two 2-terminal circuits and a combined circuit.

(n + 1)-terminal circuit with terminals T₀, T₁, ... , T_n . where the function f₂ represents the circuit connecting T₀ to T_i for each i = 1, 2, . . . , n.  We will assume that the possible closed paths between T_i and T_j, i ≠0,  j ≠0, will cause no complications in the application of the circuit, and we will ignore the functions representing all such circuits.

This problem is, of course, a severe simplification of the more general problem of design in which each of the possible circuits is to have specified properties. The problem is still sufficiently general to have considerable value, and has the advantage of being easy to solve. In fact, the problem consists solely of examining the functions for common factors. Any such factors indicate portions of the circuits which may be shared.

If the functions involved are given explicitly, one may factor by trial and error in the attempt to find common factors, or each function may be placed in conjunctive normal form, from which common factors may be determined by inspection.

EXAMPLE 1. Design a 4-terminal circuit to realize the following three functions, using common switches wherever possible:

                       f = x y'z + (xy' + x'y)zw,

                      g = xy'zw' + x'yz w',

                      h = x'y + (xy' + x 'y) (z +w').

FIGURE 1-2

Solution. Since g can be most easily factored, we begin by writing it in the form

                             g = (xy' + x'y)zw'.

Examining f and h for possible factors in common with g, we find

Similarly,

Hence the required circuit is that of Fig. 1-2, where f = f₀₂, g = f₀₁, and h = f₀₃.

If the functions are not given, but the properties are specified by a table as in Section (Design of circuits from given properties), the best procedure is to first write each function in conjunctive normal form. Common factors may be found either from the table or from the functions. The circuit is then drawn in such a way as to make the best possible use of common factors.

EXAMPLE 2. Construct a 4-terminal circuit to realize the functions f, g, and h, with properties given by Table 1-1.

TABLE 1-1

Solution. First, each function is written in conjunctive normal form:

FIGURE 1-3

The common factors are noted and are indicated below with vertical lines.  Simplifications are made where possible.

The final circuit is shown in Fig. 1-3, where f = ,f₀₁. g = f₀₂, and h = f₀₃-In addition to cases in which the functions have common factors, there are certain other cases in which sharing of switches is possible. These cases will be easier to discuss in connection with the use of special types of switches called transfer contacts. We will postpone this discussion until the next chapter.

مواضيع ذات صلة

Zero Polynomial

Kronecker,s Polynomial Theorem

THE ALGEBRA OF SETS-The number of elements in a set

THE ALGEBRA OF SETS-Solution of equations

THE ALGEBRA OF SETS-Conditional equations

THE ALGEBRA OF SETS-Properties of set inclusion

THE ALGEBRA OF SETS- Expanding, factoring, and simplifying

THE ALGEBRA OF SETS-Fundamental laws

THE ALGEBRA OF SETS-Venn diagrams

THE ALGEBRA OF SETS-The combination of sets

THE ALGEBRA OF SETS-Element and set

THE ALGEBRA OF SETS-Introduction