# Basic Combinational Circuits: Types & Examples Video

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• 0:04 What Are Combinational…
• 0:24 Combinational Circuit Types
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Instructor: Lyna Griffin

Lyna has tutored undergraduate Information Management Systems and Database Development. She has a Bachelor's degree in Electrical Engineering and a Masters degree in Information Technology.

In this lesson, we will examine the different types of combinational circuits, like the adder, subtractor, multiplexer, demultiplexer, etc. We will examine their functions, characteristics, logic diagrams, and truth tables.

## 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.

## Combinational Circuit Types

Let's look at some of the most common combinational circuits:

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 adders compute single digit numbers, while full adders compute larger numbers.

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:

In addition, the following truth table 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

The next figure represents the logic circuit of the half adder:

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

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 this figure:

The next truth table shown here 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. This is where A and B are all the possible binary inputs and C is the carry in. For 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 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 in this image as well as the half subtractor table:

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

The Boolean expressions are as follows:

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 three 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

For 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're 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 2N, 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.

The next table 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's also referred to as a data distributor. It converts a single serial input into parallel data outputs on the output line. This next figure is a block diagram representation of the de-multiplexer:

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