Combinational Logic - Design and Analysis - Lecture Notes, Study notes for Digital System Design. Jaypee University of Engineering & Technology

Digital System Design

Description: Combinational Logic, Combinational Circuits, Design Procedure, Binary Adder Subtractor, Decimal Adder, Binary Multiplier, Magnitude Comparator, Decoders are the key points in this lecture handout.
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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):
F2
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 FT 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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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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