Post on 10-Feb-2016
description
Digital Logic Systems
Combinational Circuits
Basic Gates&
Truth Tables
Basic Gates
AND Gate OR Gate NOT Gate
More Gates
NAND Gate NOR Gate BUF Gate
More Gates
XNOR GateXOR Gate
n-Input Gates
3-Input XOR Gate
5-Input NOR Gate 5-Input AND Gate
4-Input OR Gate
Definitions
AND It gives a logical output true only if all the inputs are true
OR It gives a logical output true if any of the inputs is true
XOR It gives a logical output true only if an odd-number of inputs is true
NOT It gives a logical output true if the input is false and vice versa
Truth Table
A truth table is a tabular procedure to express the relationship of the outputs to the inputs of a Logical System
Truth Tables for Gates
a b fAND
0 0 00 1 01 0 01 1 1
a b fOR
0 0 00 1 11 0 11 1 1
a fNOT
0 11 0
AND Operation OR Operation
NOT Operation
AND Gate OR Gate NOT Gate
Truth Tables for Gates
a b fNAND
0 0 10 1 11 0 11 1 0
a b fNOR
0 0 10 1 01 0 01 1 0
a fBUF
0 01 1
NAND Operation NOR Operation
BUF Operation
NAND Gate NOR Gate BUF Gate
Truth Tables for Gates
a b fXOR
0 0 00 1 11 0 11 1 0
a b fXNOR
0 0 10 1 01 0 01 1 1
XOR Operation XNOR Operation
XNOR GateXOR Gate
A Bubble Implies a Logical Inversion
Bubbles can be replaced by NOT Gates to get logically equivalent
circuits
Bubbles
Generate tables for all combinations of bubbles and a XOR gate
Gate Equivalence
===
Gate Equivalence
== ?
Gate Equivalence
= =
Switching Expressions
Basic Switching Expressions
AND f = a . b
OR f = a + b
NOT f = a’f = ā
Is there an expression for XOR operation?
Switching Expressions
Switching Expressions
Switching Expressions
f1 = a . b’f2 = (a + b)’
Switching Expressions
Switching Expressions
Switching Expressions
f = ?
Switching Expressions
f = m + n
n = a’ . bm = a . b’
Switching Expressions
f = (a . b’) + (a’ . b)This is the equivalent circuit and equivalent
expression for a XOR operation
From Digital Design, 5th Edition by M. Morris Mano and Michael Ciletti
Switching Expressions
Switching Expressions
Switching Expressions
f1 = a . bf2 = a ^ bf2 = (a . b’) + (a’ . b)
Switching Expressions
x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1
x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0
x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 0 0 00 0 1 0 0 0 1 00 1 0 1 0 0 1 00 1 1 1 0 1 0 11 0 0 1 0 0 1 01 0 1 1 0 1 0 11 1 0 0 1 0 0 11 1 1 0 1 0 1 1
x y z s c0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1
s = sc = m + g
s = sc = m + g m = p . z
g = g
s = p ^ z
s = sc = m + g m = p . z
g = g
p = x ^ y g = x . y
s = p ^ z
s = sc = m + g
p = x ^ y g = x . y m = (x ^ y) . z
g = g
s = (x ^ y) ^ z
s = (x ^ y) ^ zc = ((x ^ y) . z) + (x . y)
p = x ^ y g = x . y m = (x ^ y) . z
g = g
s = (x ^ y) ^ z
s = (x ^ y) ^ zc = ((x ^ y) . z) + (x . y)
s = ((x . y’) + (x’ . y)) ^ zc = (((x . y’) + (x’ . y)) . z) + (x . y)
s = (((x . y’) + (x’ . y))’ . z) + (((x . y’) + (x’ . y)) . z’)c = (((x . y’) + (x’ . y)) . z) + (x . y)
Procedure
To obtain the output functions from a logic diagram, proceed as follows:
1. Label with arbitrary symbols all gate outputs that are a function of the input variables. Obtain the Boolean Functions for each gate.
2. Label with other arbitrary symbols those gates that are a function of input variables and/or preciously labeled gates. Find the Boolean functions of these gates.
3. Repeat the process in step 2 until all the outputs of the circuit are obtained.4. By repeated substitution of previously defined functions, obtain the output
Boolean functions in terms of input variables only.