Combinational logic
circuits
Chapter 3
Analysis & design
Definition of combinational logic
Let X be the set of all input variables,X={X0,X1,……,Xn}
Let Y be the set of all output variables,Y={Y0,Y1,……,Ym}
Y=F(X) >> Yi=F(X0,X1,……,Xn)
The combinational function,F,operated on the input
variable set X,to produce the output variable set Y.
The output is related to the input as
Combinational
logic
function (F)
X0
Xn
Y0
Ym
Definition of combinational logic
Logic circuits without the feedback from output to
input
Logic circuits constructed from a functionally
completely gate set,contain no memory unit
Combinational
logic
function (F)
X0
Xn
Y0
Ym
Analyze a combinational logic circuit
Function statement
Construct truth table
Simplify equation
Switching equation
Logic diagram? Convert logic diagram to switching equation
Simplify the expression
derive the truth table from
the simplified switching
equation
Give the pertinent statement
of the circuit
(function\design)
Design a combinational logic circuit
Logic circuit built
Logic diagram drawn
Equations simplified
switching equations written
Truth table Construction
Problem statement
Develop a proper
statement of the problem
Based on the problem
statement,construct
truth table that clearly
established the
relationship between the
input and output variables
Design a combinational logic circuit
Determine the input variables and
output variables that are involved
Assign mnemonic or letter or
number symbols to each variable
Determine the size of the truth
table
Construct a truth table containing
all of the input variable combination
By careful reading of the problem
statement determine the
combinations of input that cause a
given output to be true
Truth table
Construction
Problem
statement
Elements
Two operator
Material
Interlock switch
Motor
put conveyor into action >> the motor is
turning on
Either of two operators is in position
The interlock switch is closed
Material must be present
Example,Conveyor system
4 input variables
a=1,operator 1 is in position
b=1,operator 2 is in position
m=1,material is present
s=1,interlock switch is closed.
1 output variable
M is the signal to turn the motor off or on
M=1,the motor is turning on
Example,Conveyor system
a b m s M
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
a b m s M
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
Example,Conveyor system
Elements
Two on-line computers
One redundant computer
Three self-checking diagnostics components
The control logic to connect or disconnect
the computers.
A warning,allow the third computer to come
on-line.
A warning,invoke the emergency procedures.
Example,NASN system
C1
C2
C3
connect
control
self-checking
diagnostics
W1
W2
O1 O2 O3
Switching
array
Oi=1,closed
Oi=0,open
Example,NASN system
Inputs/output variables
C1,C2,C3; Ci=1,computer-i is failed.
Represent the operation status of three
computers,generated by the self-checking
components,
O1,O2,O3; Oi=1,computer-i is connected.
Computer disconnect control signal output
Used to connect or disconnect computer
determined by the self-checking results.
W1,W2
Example,NASN system
C1 C2 C3 O1 O2 O3 W1 W2
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
Example,NASN system
Design a combinational logic circuit
Simplify the functions in
deriving the output
equation,the best solving
is the fewest gates and
gate input.
Arrange the simplified
equations to suit the logic
type which are requested
in realizing the circuit,
Draw the final logic
diagram.
Logic circuit built
Logic diagram drawn
Equations simplified
switching equations written
Truth table Construction
Problem statement
Design a combinational logic circuit
Design a combinational circuit that will convert a
8421 BCD code to 2421 BCD code.
Design a comparator that can compare two 3-bit
binary value,If the two number are equal,the
output is 1 else is 0
Design a full-adder that will add two 1-bit binary
values plus a carry-in from the less significant bit
and produce a sum and a carry-out.
Design a combinational circuit that will multiply two
2-bit binary values.
Example.1
Design a combinational circuit that will convert a
8421 BCD code to 2421 BCD code.
B3
B0
B2
B1
F3
F0
F2
F1
8421BCD 2421BCD
logic
circuit
Example.1
Input variables/ output variables
4 input variables,B3 B2 B1 B0
4 output variables,F3 F2 F1 F0
B3
B0
B2
B1
F3
F0
F2
F1
8421BCD 2421BCD
logic
circuit
B3 B2 B1 B0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
0
1
0
1
0
1
0
1
0
1
F3 F2 F1 F0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
1
0
1
1
1
1
0
0
1
1
0
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
Example.1
F3=B3B2’B1’+B3’B2B1+B3’B2B0
F2=B3B2’B1’+B3’B2B1+B3’B2B0’
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
11
1
1
F3 F2
Example.1
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
1
1
1
1
1
F1=B3B2’B1’+B3’B2’B1+B3’B2B1’B0
F0=B3’B0+B2’B1’B0
F1
F0
Example.1
Incompletely specified function (don’t care term)
When an output values is known for every possible
combination of input variables,the function is said to
be completely specified.
When an output values is unknown for every possible
combination of input variables,usually because all
combinations cannot occur,the function is said to be
incompletely specified.
The minterms or maxterms that are not used as part
of the output function are called don’t care terms.
It does not matter whether we assign them a value of
0 or 1,because the combinations of input variables
never occur.
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
F3=B3+B2B1+B2B0
d
d
d
d
d
d
Example.1
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
11
1
1
F3
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
F2=B3+B2B1+B2B0’
d
d
d
d
d
d
Example.1
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
F2
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
F1=B3+B2’B1+B2B1’B0
d
d
d
d
d
d
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1F1
Example.1
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1
F0=B0
d
d
d
d
d
d
Example.1
00 01 11 10
B3B2
B1B0
00
01
11
10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1
1
1
1
1F0
Design a comparator that can compare
two 3-bit binary value,If the two
number are equal,the output is 1 else
is 0
Example.2
6 input variables / 1 output variables
Solution 1
Construct Truth table
Simplify equation use K-map
Solution 2
F=1? A=B
A=a2a1a0 ; B=b2b1b0
Proposition:
A=B? (a2=b2)&(a1=b1)&(a0=b0)
f2=1? a2=b2;
f1=1? a1=b1;
f0=1? a0=b0;
F=1? A=B? f2?f1?f0=1
Example.2
Solution 2
f2=1? a2=b2;
f1=1? a1=b1;
f0=1? a0=b0
Example.2
F=f1·f2·f3
=(a1⊕ b1)’ (a2⊕ b2)’ (a3⊕ b3)’
=((a1⊕ b1)+(a2⊕ b2)+(a3⊕ b3))’
Example.2
a1
b1
f1
f2
f3
Fa2
b2
a3
b3
F=((a1⊕ b1)+(a2⊕ b2)+(a3⊕ b3))’
Design a full-adder that will add two 1-bit
binary values plus a carry-in from the less
significant bit and produce a sum and a
carry-out.
Example.3
Augend+ addend+ carry-in
= sum……carry-out
Ai+ Bi+ Ci-1= Si…… Ci
Example.3
Si Ci
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
0
1
0
1
1
1
0
1
1
0
1
0
0
1
Ai Bi Ci-1
Si(Ai,Bi,Ci-1)
=∑m(1,2,4,7)
Ci(Ai,Bi,Ci-1)
=∑m(3,5,6,7)
Example.3
Si(Ai,Bi,Ci-1)=∑m(1,2,4,7)
00 01 11 10
0
1
0
1
2
3
6
7
4
5
AB
C
MSB
LSB
1 1
1 1
Si= A’BC’+AB’C’+ A’B’C+ABC
= (A’B+AB’)C’+ (A’B’+AB)C
= (A⊕ B)C’+(A⊕ B)’C
= Ai⊕ Bi⊕ Ci-1
= (A⊕ B)C’+(A⊙ B)C
Example.3
Ci(Ai,Bi,Ci-1)=∑m(3,5,6,7)
00 01 11 10
0
1
0
1
2
3
6
7
4
5
AB
C
MSB
LSB
1
11 1
Ci= AiBi +BiCi-1 +AiCi-1
Si= Ai⊕ Bi⊕ Ci-1
Example.3
Ci= AiBi +BiCi-1 +AiCi-1
Si= Ai⊕ Bi⊕ Ci-1Ai
Bi
Ci-1
Si
AiBi
AiCi-1
BiCi-1
Ci
Example.3
Ci(Ai,Bi,Ci-1)=∑m(3,5,6,7)
00 01 11 10
0
1
0
1
2
3
6
7
4
5
AB
C
MSB
LSB
1
11 1
Ci= AiBi + Ai’BiCi-1 +AiBi’Ci-1
Si= Ai⊕ Bi⊕ Ci-1
00 01 11 10
0
1
0
1
2
3
6
7
4
5
AB
C
MSB
LSB
1 1
1 1
= AiBi + (Ai’Bi +AiBi’)Ci-1
= AiBi + Ci-1 (Ai⊕ Bi)
Example.3
Ci= AiBi + Ci-1 (Ai⊕ Bi)
Si= Ai⊕ Bi⊕ Ci-1
Ai
Bi
Ci-1
Si
AiBi
Ci
A=A1A0; B=B1B0;
AMAX=11 ; BMAX=11
Example.4
Design a combinational circuit that will multiply two
2-bit binary values.
A
B
CMultiplier
C=A*B
CMAX=11*11=1001
C3
Input variables/ output variables
4 input variables,A1 A0 B1 B0
4 output variables,C3 C2 C1 C0
Example.4
Design a combinational circuit that will multiply two
2-bit binary values.
A1
B0
A0
B1
C0
C2
C1
Multiplier
Solution 2
Binary multiplication is Achieved by
addition and shift.
A1A0
* B1B0
A1B0 A0B0
+ A1B1 A0B1
C2 A1B1+C1 A1B0+A0B1 A0B0
Example.4
Solution 2
Full adder
A1A0
* B1B0
A1B0 A0B0
+ A1B1 A0B1
C2 A1B1+C1 A1B0+A0B1 A0B0
Example.4
Ai Bi
Ci-1
CiSi
A1A0
* B1B0
A1B0 A0B0
+ A1B1 A0B1
C2 A1B1+C1 A1B0+A0B1 A0B0
Example.4
Ai Bi
Ci-1
CiSi“0”
Ai Bi
Ci-1
CiSi
A1B0 A0B1
A1B1,0”
C1
C2 C3
C0