chapter four combinational logic 1. c ombinational c ircuits it consists of input variables, logic...
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Chapter FourCombinational Logic
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COMBINATIONAL CIRCUITS
It consists of input variables, logic gates and output variables.
Output is function of input onlyi.e. no feedback
When input changes, output changes (after a delay)
•••
•••
n inputs m outputsCombinational
Circuits
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COMBINATIONAL CIRCUITS
Analysis Given a circuit, find out its function Function may be expressed as:
Boolean function Truth table
Design Given a desired function, determine its circuit Function may be expressed as:
Boolean function Truth table
CBA
CBA
BA
CA
CB
F1
F2
?
?
?3
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ANALYSIS PROCEDURE Boolean Expression Approach
F1=T2+T3= AB'C'+A'BC'+A'B'C+ABCF2=AB+AC+BC
CBA
CBA
BA
CA
CB
F1
F2
T2=ABCT1=A+B+C
F2=AB+AC+BC
F’2=(A’+B’)(A’+C’)(B’+C’)
T3=AB'C'+A'BC'+A'B'C
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CBA
CBA
BA
CA
CB
F1
F2
ANALYSIS PROCEDURE
We can obtain the truth table directly from the logic diagram
Truth Table Approach
A B C F1 F2
0 0 0= 0 = 0= 0= 0= 0= 0
= 0= 0
= 0= 0
= 0= 0
T2 = 0
T1 = 0
0
0
0
F2 = 0
F’2 = 1
T3 = 0
0 0 0
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Full Adder Circuit
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CBA
CBA
BA
CA
CB
F1
F2
ANALYSIS PROCEDURE
Truth Table Approach= 0 = 0= 1= 0= 0= 1
= 0= 0
= 0= 1
= 0= 1
0
1
0
0
0
0
1
1
1
A B C F1 F2
0 0 0 0 00 0 1 1 0
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CBA
CBA
BA
CA
CB
F1
F2
ANALYSIS PROCEDURE
Truth Table Approach= 0 = 1= 0= 0= 1= 0
= 0= 1
= 0= 0
= 1= 0
0
1
0
0
0
0
1
1
1
A B C F1 F2
0 0 0 0 00 0 1 1 00 1 0 1 0
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CBA
CBA
BA
CA
CB
F1
F2
ANALYSIS PROCEDURE
Truth Table Approach= 0 = 1= 1= 0= 1= 1
= 0= 1
= 0= 1
= 1= 1
0
1
0
0
1
1
0
0
0
A B C F1 F2
0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 1
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CBA
CBA
BA
CA
CB
F1
F2
ANALYSIS PROCEDURE
Truth Table Approach= 1 = 1= 1= 1= 1= 1
= 1= 1
= 1= 1
= 1= 1
1
1
1
1
1
1
0
0
1
A B C F1 F2
0 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
B
0 1 0 1
A 1 0 1 0C
B
0 0 1 0
A 0 1 1 1C
F1=AB'C'+A'BC'+A'B'C+ABC F2=AB+AC+BC10
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DESIGN PROCEDURE Given a problem statement:
Determine the number of inputs and outputs Derive the truth table Simplify the Boolean expression for each output Produce the required circuit
Example: Design a circuit to convert a “BCD” code to
“Excess -3” code
4-bits 0-9 values
4-bits Value+3
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DESIGN PROCEDURE BCD-to-Excess 3 Converter
A B C D w x y z
0 0 0 0 0 0 1 10 0 0 1 0 1 0 00 0 1 0 0 1 0 10 0 1 1 0 1 1 00 1 0 0 0 1 1 10 1 0 1 1 0 0 00 1 1 0 1 0 0 10 1 1 1 1 0 1 01 0 0 0 1 0 1 11 0 0 1 1 1 0 01 0 1 0 x x x x
1 0 1 1 x x x x1 1 0 0 x x x x1 1 0 1 x x x x1 1 1 0 x x x x1 1 1 1 x x x x
C
1 1 1B
Ax x x x
1 1 x x
D
C
1 1 11
BA
x x x x
1 x x
D
C
1 11 1
BA
x x x x
1 x x
D
C
1 11 1
BA
x x x x
1 x x
D
w = A+BC+BD x = B’C+B’D+BC’D’
y = C’D’+CD z = D’ 12
It needs 7 AND gates and 3 OR gates
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DESIGN PROCEDURE
w = A + B(C+D)
x = B’(C+D) + B(C+D)’
y = (C+D)’ + CD
z = D’14
A B C D w x y z
0 0 0 0 0 0 1 10 0 0 1 0 1 0 00 0 1 0 0 1 0 10 0 1 1 0 1 1 00 1 0 0 0 1 1 10 1 0 1 1 0 0 00 1 1 0 1 0 0 10 1 1 1 1 0 1 01 0 0 0 1 0 1 11 0 0 1 1 1 0 01 0 1 0 x x x x
1 0 1 1 x x x x1 1 0 0 x x x x1 1 0 1 x x x x1 1 1 0 x x x x1 1 1 1 x x x x
It needs 4 AND gates and 4 OR gates
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SEVEN-SEGMENT DECODER a
b
c
g
e
d
f?
w
x
y
z
abcdefg
w x y z a b c d e f g
0 0 0 0 1 1 1 1 1 1 00 0 0 1 0 1 1 0 0 0 00 0 1 0 1 1 0 1 1 0 10 0 1 1 1 1 1 1 0 0 10 1 0 0 0 1 1 0 0 1 10 1 0 1 1 0 1 1 0 1 10 1 1 0 1 0 1 1 1 1 10 1 1 1 1 1 1 0 0 0 01 0 0 0 1 1 1 1 1 1 11 0 0 1 1 1 1 1 0 1 11 0 1 0 x x x x x x x
1 0 1 1 x x x x x x x1 1 0 0 x x x x x x x1 1 0 1 x x x x x x x1 1 1 0 x x x x x x x1 1 1 1 x x x x x x x
y
1 1 1
1 1 1x
wx x x x
1 1 x x
z
BCD code
a = w + y + xz + x’z’ b = . . .c = . . .d = . . .
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BINARY ADDER – SUBTRACTOR Digital computers perform a variety of information-processing tasks. Among the functions encountered are the various arithmetic operations. The most basic arithmetic operation is the addition of two binary digits.
Half Adder Adds 1-bit plus 1-bit Produces Sum and Carry
HAx
yS
C
x y C S
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
x+ y───C S
x
y
S
C 16
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Full Adder Adds 1-bit plus 1-bit plus 1-bit Two significant bits and a previous carry Produces Sum and Carry
x y z C S0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 0 11 0 1 1 01 1 0 1 01 1 1 1 1
x+ y+ z───C S
FAxyz
S
C
y
0 1 0 1
x 1 0 1 0z
y
0 0 1 0
x 0 1 1 1z
S = xy'z'+x'yz'+x'y'z+xyz = x y z
C = xy + xz + yz 17
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BINARY ADDER Full Adder
xyzx
y
z
xy
xz
yz
S
C
S = xy'z'+x'yz'+x'y'z+xyz = x y z
C = xy + xz + yz
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Implementation of full adder in sum of products form
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BINARY ADDER Implementation of Full Adder with two half adder
and an OR gate.
x
y
z
S
C
HAxy
z
HAS
C
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BINARY ADDER c3 c2 c1 .+ x3 x2 x1 x0
+ y3 y2 y1 y0
────────Cy S3 S2 S1 S0
FA
x3 x2 x1 x0
FAFAFA
y3 y2 y1 y0
S3 S2 S1 S0
C4 C3 C2 C1
0
Binary Adder
x3x2x1x0 y3y2y1y0
S3S2S1S0
C0C4
20
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Subscript i 3 2 1 0
Input carry 0 1 1 0 Ci
1 0 1 1 xi
0 0 1 1 yi
Sum 1 1 1 0 Si
Output carry
0 0 1 1 Ci+1
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BINARY SUBTRACTOR
Use 2’s complement with binary adder x – y = x + (-y) = x + y’ + 1
Binary Adder A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
CiCy 1
x3 x2 x1 x0 y3 y2 y1 y0
F3 F2 F1 F022
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BINARY ADDER/SUBTRACTOR
Binary Adder A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
CiCy
Mx3 x2 x1 x0 y3 y2 y1 y0
F3 F2 F1 F0
M: Control Signal (Mode) M = 0 F = x + y M = 1 F = x – y
F = x + y’+ 1
y Å 0 = yy Å1 = y'
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OVERFLOW: When two numbers with n digits each are added and the sum is a number occupying n + 1 digits,we say that an overflow occurred.
Unsigned Binary Numbers
2’s Complement Numbers (signed numbers)
FA
x3 x2 x1 x0
FAFAFA
y3 y2 y1 y0
S3 S2 S1 S0
C4 C3 C2 C1
0
Carry
Overflow
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0 1+70 0 1000110+80 0 1010000
+150 1 0010110 1 0-70 1 0111010- 80 1 0110000
-150 0 1101010
25
An overflow cannot occur after an addition if one number is positive and the other is negative. Since adding a positive number to a negative number produces a result whose magnitude is smaller than the larger of the two original numbers.
An overflow may occur when the two numbers added are both (+) or both (-). Consider 8 bit register (127 --- -128).
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10 1101 1010-------- 0111
11 1110 1101-------- 1011
01 0011 0110-------- 1001
00 0010 0011-------- 0101
00 0010 1100-------- 1110
11 1110 0100-------- 0010
Addition cases and overflow
OFL OFL
+2+3+5
+3+6-7
-2-3-5
-3-6+7
+2-4-2
-2+4+2
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MAGNITUDE COMPARATOR
Compare 4-bit number to 4-bit number 3 Outputs: < , = , > Expandable to more number of bits 0010, 1000 (Using XNOR Gates for equality)
Magnitude Comparator
A3A2A1A0 B3B2B1B0
A<B A=B A>B
33333 BABAx
22222 BABAx
11111 BABAx
00000 BABAx
0123)( xxxxBA
00123112322333)( BAxxxBAxxBAxBABA
00123112322333)( BAxxxBAxxBAxBABA
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MAGNITUDE COMPARATOR