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Digital System Design

Combinational Circuits

Submitted To : Fernaz Naurin Nur

Lecturer

Dept. Of CSE.

Submitted By : MD. Arafat HossenID: UG02-22-09-012

Dept. Of CSE.

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AdderDesign an Adder for 1-bit numbers

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AdderDesign an Adder for 1-bit numbers

1. Specification:2 inputs (X,Y)

2 outputs (C,S)

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AdderDesign an Adder for 1-bit numbers

1. Specification:2 inputs (X,Y)

2 outputs (C,S)

2. Formulation:

X Y C S

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

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AdderDesign an Adder for 1-bit numbers

1. Specification: 3. Optimization/Circuit2 inputs (X,Y)

2 outputs (C,S)

2. Formulation:

X Y C S

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

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Half AdderThis adder is called a Half Adder

X Y C S

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

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Full AdderA combinational circuit that adds 3 input bits to generate

a Sum bit and a Carry bit

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Full AdderA combinational circuit that adds 3 input bits to generate

a Sum bit and a Carry bit

X Y Z C 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 0

1 1 1 1 1

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Full AdderA combinational circuit that adds 3 input bits to generate

a Sum bit and a Carry bit

X Y Z C 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 0

1 1 1 1 1

X

YZ

0

1

00 01 11 100 1 0 1

1 0 1 0

X

YZ

0

1

00 01 11 100 0 1 0

0 1 1 1

Sum

Carry

S = XYZ + XYZ

+ XYZ +XYZ

= X Y Z

C = XY + YZ + XZ

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Full Adder = 2 Half AddersManipulating the Equations:

S = X Y Z

C = XY + XZ + YZ

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Full Adder = 2 Half AddersManipulating the Equations:

S = ( X Y ) Z

C = XY + XZ + YZ

= XY + XYZ + XYZ + XYZ + XYZ

= XY( 1 + Z) + Z(XY + XY)

= XY + Z(X Y )

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Full Adder = 2 Half AddersManipulating the Equations:

S = ( X Y ) Z

C = XY + XZ + YZ = XY + Z(X Y )

Src: Manos Book

Think of

Z as a

carry in

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Half Adder

The half adder is an example of a simple, functional digital

circuit built from two logic gates. The half adder adds to one-bit

binary numbers (AB). The output is the sum of the two bits (S)and the carry (C). Note how the same two inputs are directed to

two different gates. The inputs to the XOR gate are also the

inputs to the AND gate. The input "wires" to the XOR gate are

tied to the input wires of the AND gate; thus, when voltage is

applied to the A input of the XOR gate, the A input to the ANDgate receives the same voltage.

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Full Adder:The full-adder circuit adds three one-bit binary numbers (C A B)

and outputs two one-bit binary numbers, a sum (S) and a carry

(C1). The full-adder is usually a component in a cascade ofadders, which add 8, 16, 32, etc. binary numbers. The carry

input for the full-adder circuit is from the carry output from the

circuit "above" itself in the cascade. The carry output from the

full adder is fed to another full adder "below" itself in the

cascade. If you look closely, you'll see the full adder is simplytwo half adders joined by an OR.

Ahmad Almulhem, KFUPM 2010

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