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COEN 312 DIGITAL SYSTEMS DESIGN - LECTURE NOTES

Chapter 4: Combinational Logic

NOTE: For more examples and detailed description of the material in the lecture

notes, please refer to the main textbook:

Digital Design 3rd Edition, By Morris Mano, Publisher Prentice Hall, 3rd Edition

All examples used in the lecture notes are from the above reference.

Combinational Circuits

- A combinational circuit consists of input variables, output variables, and logic gates

that transform binary information from the input data to the output.

Combinational

circuit

n inputs m outputs

- A combinational circuit cannot have any storage elements (registers) or any feedback

paths (connections from the output of one gate to the input of a second gate that

directly or indirectly affects the input to the first gate).

- As an example, consider the following combinational circuit with 3 inputs and 2

outputs (n = 3, m = 2):

F’2

A

B

C

A

B

C

A

B

A

C

B

C

T1

T2

T3

F2

F1

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2

- If there are more than two levels of gates to generate a function, label the output of

the corresponding gates and determine the Boolean expression for each of them (here,

we have used the labels T1, T, T, and F’

2 3 2 to find the Boolean expression for F1).

231

123

2

1

2

.'

..

...

TTF

TFT

CBAT

CBAT

CBCABAF

+=

=

=

++=

++=

- By using the properties of Boolean algebra, the expression for will be equal to:

1

F

CBACBACBACBAF ..''..'.'.'.'.

1+++=

- The truth table for the outputs of this combinational circuit can be obtained by using

the above expressions obtained for and , or by using the labeled gate outputs in

the truth table and obtaining

1

F2

F

CBCABAF .

..

2

+

+

=

231 TTF

+

=

and for different

combinations of the input variables as follows:

A B C F F’T T T F

2 2 1 2 3 1

0 0 0 0 1 0 0 0 0

0 0 1 0 1 1 0 1 1

0 1 0 0 1 1 0 1 1

0 1 1 1 0 1 0 0 0

1 0 0 0 1 1 0 1 1

1 0 1 1 0 1 0 0 0

1 1 0 1 0 1 0 0 0

1 1 1 1 0 1 1 0 1

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3

[

]

[

]

12 FFCBA

=

+

+

- This is in fact the logic circuit for a full adder ( ).

Design Procedure

- Important design constraints include the number of gates, number of inputs to a gate,

propagation time of the signal through the gates, number of interconnections, etc.

Example

- Find a combinational circuit, which converts the binary coded decimal (BCD) to the

excess-3 code for the decimal digits.

Input BCD Output Excess-3 Code

A B C D w x y z

0 0 0 0 0 0 1 1

0 0 0 1 0 1 0 0

0 0 1 0 0 1 0 1

0 0 1 1 0 1 1 0

0 1 0 0 0 1 1 1

0 1 0 1 1 0 0 0

0 1 1 0 1 0 0 1

0 1 1 1 1 0 1 0

1 0 0 0 1 0 1 1

1 0 0 1 1 1 0 0

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4

X

0

X

0

0

1

1

0 1

1

CD

AB

00

01

00 01 11 10

D’ D’

D

C’ C

X

1 0 X

X

X

11

10

B’

B’

B

A’

A

- Simplified expression:

'Dz =

X

1

X

0

1

1

1

0 0

0

CD

AB

00

01

00 01 11 10

D’ D’

D

C’ C

X

1 0 X

X

X

11

10

B’

B’

B

A’

A

- Simplified expression: ''.. DCDCy

+

=

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University:
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Digital System Design

Upload date:
03/11/2012