What Are Combinational Circuits?

What Are Combinational Circuits?

Combinational Circuits (CC) are circuits made up of different types of logic gates. A logic gate is a basic building block of any electronic circuit.

The output of the combinational circuit depends on the values at the input at any given time. The circuits do not make use of any memory or storage device.

Lets look at some of the most common combinational circuits:

The Adder

An Adder is a digital circuit that is used to perform the addition of numeric values. It is one of the most basic circuits and is found in Arithmetic Logic Units of computing devices. There are two types of adders. Half Adderscompute single digit numbers while Full Adders compute larger numbers.

Half Adder

The Half Adder adds two single digit binary numbers and forms the foundation for all addition operations in computing. If we have two single binary digits A and B then the Half Adder adds them with the circuit carrying two outputs; the sum and the carry. The carry represents any overflow from the addition of the two numbers. This is represented in the following block diagram figure 1.

The following truth table shown in Table 1, demonstrates all the possible outputs for various input combinations of the half adder.

Table 1: Truth Table - Half Adder

A	B	S	C
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Figure 2 - Represents the logic circuit of the Half Adder

The sum S is represented by the Boolean Expression S = A'B + AB' and C = AB

Full Adder

The Full Adder overcomes the disadvantages of the half adder in that it can add two single bit numbers in addition to the carry digit at its input as seen in Figure 3.

The following truth table shown in Table 2 demonstrates all the possible outputs for various input combinations with the carry input digit.

Table 2: Truth Table - Full Adder

A	B	Cin	Co	S
0	0	0	0	0
0	0	1	0	1
0	1	0	0	1
0	1	1	1	0
1	0	0	0	1
1	0	1	1	0
1	1	0	1	1
1	1	1	1	1

Boolean expression for the full adder is S= A'B'Cin + A'BCin' + AB'Cin' + ABCin

and C = A'BCin + AB'Cin + ABCin' + ABCin

where A and B are all the possible binary inputs and C is the Carry in

Example if A is 0 and B is 0 and the Cin is 1 then

S = (0'0'1)+(0'01')+(00'1')+(001) = (111)+(100)+(010)+(001) = (1)+(0)+(0)+(0) = 1

C = (0'01)+(00'1)+(001')+(001) = (101)+(011)+(000)+(001) = (0)+(0)+(0)+(0) = 0

S = 1 and C = 0

Subtractors

A Subtractor is used to subtract one number from the other another. Because we are dealing with binary digits, the 1s complement and 2s complement of the numbers are used to achieve this. Three bits are involved in performing the basic subtraction: the minuend (X), the subtrahend (Y) and the borrow (Bi) which is input from the previous bit. The outputs are the difference (D) and the borrow bit (Bout).

Half Subtractor

When a subtraction is done between just two bits a Half Subtractor is used, similar to the Half Adder. The Half Subtractor's combinational circuit is represented below.

Table 3: Truth Table - Half Subtractor

X	Y	D=(X-Y)	Bout
0	0	0	0
0	1	1	1
1	0	1	0
1	1	0	0

Boolean expressions:

D = X'Y + XY'

Bout = X'Y

Example, if our inputs X and Y are 0 and 1 then Compliment of 0 is 1 and vice versa

D = (0'1)+(01') = (11)+(00) = 1 and Bout = (0'1) = (11) = 1

Full Subtractor

The combinational circuit of the Full Subtractor performs a subtraction operation on 3 bits, the minuend, the subtrahend and the borrow-in bits. The circuit generates two outputs comprising of the calculated difference D and the borrow-out.

Table 4: Truth Table - Full Subtractor

X	Y	Bin	D=X-Y-Bin	Bout
0	0	0	0	0
0	0	1	1	1
0	1	0	1	1
0	1	1	0	1
1	0	0	1	0
1	0	1	0	0
1	1	0	0	0
1	1	1	1	1

Boolean expressions:

D= X'Y'Bin + X'YBin' + XY'Bin' + XYBin

Bout = X'Y'Bin + X'YBin' + X'YBin + XYBin

Example: When X = 1 Y = 0 and Bin = 1

D = (1'0'1)(1'01')(10'1')(101) = (011)+(000)+(110)+(101) = (0)+(0)+(0)+(0) = 0

Bout = (1'0'1)(1'01')(1'01)(101) = (011)+(000)+(001)+(101) = (0)+(0)+(0)+(0) = 0

Multiplexers

Multiplexers are combinational circuits designed to select one of multiple data inputs and produce a single output. They are commonly used in communication transmissions.

The input lines are selected depending on the selection inputs called control lines. The binary state of these inputs can either be low '0' or high '1'. Multiplexers have an even number of data input lines D as 2^N with a corresponding number of control lines S.

Multiplexers are designed at different levels. There are 2:1, 4:1, 16:1 and 32:1 multiplexers.

Table 5 shows the Truth Table for a 2:1 Multiplexer

From the truth table we can see that whenever E is low at logic 0 input, data D0 is blocked while D1 passes data through the multiplexer to the output Y. When E is high at logic 1, D1 is blocked as D0 passes input through the multiplexer to the output.

Table 5: Truth Table - 2:1 Multiplexer

E	D1	D0	Y
0	0	0	0
0	0	1	0
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

Boolean expression: Y = (D1E') + (D0E)

Example: When D1 = 0 and D0 =1 and E = 1

Y = (01')+(11) = (0)+(1) = 1

De-multiplexer

The De-multiplexer does the reverse operation of the multiplexer. This means that it receives a single data input and depending on the selection of its control lines it produces multiple outputs. It is also referred to as a data distributor. It converts a single serial input into parallel data outputs on the output line. Figure 8 is a block diagram representation of the De-multiplexer.

Figure 9 is the logic circuit of a 1:4 De-multiplexer where Din is the input data, S0 and S1 are select lines and A, B, C, and D are the outputs.

Table 6: Truth table - De-multiplexer

S1	S2	A	B	C	D
0	0	Din	0	0	0
0	1	0	Din	0	0
1	0	0	0	Din	0
1	1	0	0	0	Din

This 1:4 De-Multiplexer carries the Boolean expression for the various outputs A, B, C or D which are selected depending on the logic state of line S1 and S2 as follows:

A = S1'S2'Din

B = S1'S2Din

C = S1S2'Din

D = S1S2Din

Encoder

An Encoder is a combinational circuit that has a maximum of 2ⁿ input lines producing n outputs, n being the number of bits. Figure 10 shows the block diagram representation of a 4:2 encoder. It has 4 input bits (2²) and 2 output bits (n=2).

Table 7: Truth Table - Encoder

X3	X2	X1	X0	Y1	Y2
0	0	0	1	0	0
0	0	1	0	0	1
0	1	0	0	1	0
1	0	0	0	1	1

There are 4 inputs X0, X1 X2 X3. A successful output is indicated by a high denoted by an output value = 1. Looking at the truth table Y1 gives an output only when X2 or X3 are high (1).

The Boolean expression therefore Y1 is X2+X3. Again Y2 only gives an output when X1 or X3 are high(1). The Boolean expression for Y2 therefore is Y2 = X1+X3.

Circuit Boolean expression: Y1 = X2+X3 and Y2 = X1+X3

Input X0 when high (=1) does not produce any output which is why it does not play a part in the output expressions for Y1 and Y2. Its redundancy therefore results in its absence in the logic circuit.

The logic circuit derived from this Boolean expression is as follows:

Decoder

A Decoder converts n input lines into a maximum of 2ⁿ output lines. One output line will be an active high based on the input combinations.

Figure 13 shows the logic circuit of a 2:4 Decoder. E is the enable bit, X are inputs and Y are outputs.

Table 8: Truth Table - 2:4 Decoder

E	X1	X2	Y1	Y2	Y3	Y4
0	-	-	0	0	0	0
1	0	0	0	0	0	1
1	0	1	0	0	1	0
1	1	0	0	1	0	0
1	1	1	1	0	0	0

Boolean Expressions:

Y1 = EX1X2

Y2 = EX1X2'

Y3 = EX1'X2

Y4 = EX1'X2'

Lesson Summary

Combinational circuits are made up of different building blocks of electronic circuits called logic gates. There are different types of combinational circuits serving different purposes. Some perform arithmetic operations like the Half Adder, Full Adder, Half Subtractors and Full Subtractors. Others, such as the Multiplexers and De-Multiplexers, Encoders and Decoders, serve as data distributors in communication and telecommunication networks. These circuits are volatile signal carriers and do not use any form or memory or storage device.