Higher Engineering Mathematics

BOOLEAN ALGEBRA AND LOGIC CIRCUITS 95

A

Figure 11.3

Figure 11.4

To achieve a given output, it is often neces-
sary to use combinations of switches connected
both in series and in parallel. If the output from a
switching circuit is given by the Boolean expression
Z=A·B+A·B, the truth table is as shown in

Fig. 11.4(a). In this table, columns 1 and 2 give all

the possible combinations ofAandB. Column 3 cor-

responds toA·Band column 4 toA·B, i.e. a 1 output

is obtained whenA=0 and whenB=0. Column 5

is theor-function applied to columns 3 and 4 giv-

ing an output ofZ=A·B+A·B. The corresponding

switching circuit is shown in Fig. 11.4(b) in which

AandBare connected in series to giveA·B,Aand

Bare connected in series to giveA·B, andA·Band

A·Bare connected in parallel to giveA·B+A·B.

The circuit symbols used are such thatAmeans the

switch is on whenAis 1,Ameans the switch is on

whenAis 0, and so on.

Problem 1. Derive the Boolean expression and

construct a truth table for the switching circuit

shown in Fig. 11.5.

Figure 11.5

The switches between 1 and 2 in Fig. 11.5 are in

series and have a Boolean expression ofB·A. The

parallel circuit 1 to 2 and 3 to 4 have a Boolean

expression of (B·A+B). The parallel circuit can be

treated as a single switching unit, giving the equiv-

alent of switches 5 to 6, 6 to 7 and 7 to 8 in series.

Thus the output is given by:

Z=A·(B·A+B)·B

The truth table is as shown in Table 11.2. Columns 1

and 2 give all the possible combinations of switches

AandB. Column 3 is theand-function applied to

columns 1 and 2, givingB·A. Column 4 isB, i.e., the

opposite to column 2. Column 5 is theor-function

applied to columns 3 and 4. Column 6 isA, i.e. the

opposite to column 1. The output is column 7 and is

obtained by applying theand-function to columns

4, 5 and 6.

Table 11.2

1 2 3 4 5 6 7

A B B·A B B·A+B A Z=A·(B·A+B)·B

0 0 0 1 1 1 1

0 1 0 0 0 1 0

1 0 0 1 1 0 0

1 1 1 0 1 0 0