Combinational Logic
Chapter 4
1
Combinational Circuits
Combinational Circuits
• Adders
• Subtractors
• Comparators
• Decoders
• Encoders
• Multiplexers
Available as MSI Circuits and as Standard Cells in
VLSI (Bonus Assignment: Get one example of each
type of combinational circuits in the CMOS family, 5
points for the second exam)
Analysis Procedure
• Given a logical diagram, determine one or
more of the following:
– Boolean functions;
– Truth table;
– Explanation of circuit operation
• Make sure the circuit is combinational, not
sequential (No feedback loops)
Analysis Procedure
1. Label all gate outputs that are a function of
input variables. Determine the Boolean function
for each gate output
2. Label the gates that are a function of input
variables and previously labeled gates. Find the
Boolean functions for these gates
3. Repeat step 2 until output of circuits are
obtained
4. By repeated substitution of previously defined
functions, obtain the output Boolean functions
in terms of input variables
Analysis Procedure
Analysis Procedure
Step 1
𝑇2 = 𝐴𝐵𝐶
𝑇1 = 𝐴 = 𝐵 + 𝐶
𝐹2 = 𝐴𝐵 + 𝐴𝐶 + 𝐵𝐶
Step 2
𝑇3 = 𝐹2′ 𝑇1
𝐹1 = 𝑇3 + 𝑇2
Steps 3 and 4
𝐹1 = 𝑇3 + 𝑇2 = 𝐹2′ + 𝐴𝐵𝐶
= 𝐴𝐵 + 𝐴𝐶 + 𝐵𝐶 ′ 𝐴 + 𝐵 + 𝐶 + 𝐴𝐵𝐶
= 𝐴′ + 𝐵′ 𝐴′ + 𝐶 ′ 𝐵′ + 𝐶 ′ 𝐴 + 𝐵 + 𝐶 = 𝐴𝐵𝐶
= 𝐴′ + 𝐵′ 𝐶 ′ 𝐴𝐵′ + 𝐴𝐶 ′ + 𝐵𝐶 ′ + 𝐵′ 𝐶 + 𝐴𝐵𝐶
= 𝐴′ 𝐵𝐶 ′ + 𝐴′ 𝐵′ 𝐶 + 𝐴𝐵′ 𝐶 ′ + 𝐴𝐵𝐶
Analysis Procedure
1. Determine the number of input variables in the
circuit. For 𝑛 inputs, form the 2𝑛 possible input
combinations and list the binary numbers from 0 to
2𝑛 − 1in a table
2. Label the outputs of selected gates with arbitrary
symbols
3. Obtain the truth table for the outputs of those gates
which are a function of the input variables only
4. Proceed to obtain the truth table for the outputs of
those gates which are a function of previously defined
values until the columns for all outputs are
determined
Analysis Procedure
Design Procedure
1. From the specifications of the circuit, determine
the required number of inputs and outputs and
assign a symbol to each
2. Derive the truth table that defines the required
relationship between inputs and outputs
3. Obtain the simplified Boolean functions for each
output as a function of the input variables
4. Draw the logical diagram and verify correctness
of the design
Code conversion example
Code conversion example
Code conversion example
𝑧 = 𝐷′
𝑦 = 𝐶𝐷 + 𝐶 ′ 𝐷′ = 𝐶𝐷 + 𝐶 + 𝐷)′
𝑥 = 𝐵′ 𝐶 + 𝐵′ 𝐷 + 𝐵𝐶 ′ 𝐷′ = 𝐵′ 𝐶 + 𝐷 + 𝐵𝐶 ′ 𝐷′
= 𝐵′ 𝐶 + 𝐷 + 𝐵(𝐶 + 𝐷)′
𝑤 = 𝐴 + 𝐵𝐶 + 𝐵𝐷 = 𝐴 + 𝐵(𝐶 + 𝐷)
Code conversion example
Binary Adder-Subtractor
Half adder
𝑆 = 𝑥 ′ 𝑦 + 𝑥𝑦′
𝐶 = 𝑥𝑦
Half adder
Full adder
Full adder
Implementation of full adder in sum-of-products
𝑆 = 𝑥 ′ 𝑦 ′ 𝑧 + 𝑥 ′ 𝑦𝑧 ′ + 𝑥𝑦 ′ 𝑧 ′ + 𝑥𝑦𝑧
𝐶 = 𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧
Full Adder
Full adder
Implementation of full adder using two half adders and one or gate
𝑆 = 𝑧 ⊕ (𝑥 ⊕ 𝑦)
= 𝑧 ′ 𝑥𝑦 ′ + 𝑥 ′ 𝑦 + 𝑧(𝑥𝑦 ′ + 𝑥 ′ 𝑦)′
= 𝑧 ′ 𝑥𝑦 ′ + 𝑥 ′ 𝑦 + 𝑧(𝑥𝑦 + 𝑥 ′ 𝑦 ′ )
= 𝑥𝑦 ′ 𝑧 ′ + 𝑥 ′ 𝑦𝑧 ′ + 𝑥𝑦𝑧 + 𝑥 ′ 𝑦 ′ 𝑧
𝐶 = 𝑧 𝑥𝑦 ′ + 𝑥 ′ 𝑦 + 𝑥𝑦 = 𝑥𝑦 ′ 𝑧 + 𝑥 ′ 𝑦𝑧 + 𝑥𝑦
Binary adder
Carry propagation
Full Adder with 𝑃 and 𝐺 shown
Carry Generate
Carry propagation
𝑃𝑖 = 𝐴𝑖 ⊕ 𝐵𝑖
𝐺𝑖 = 𝐴𝑖 𝐵𝑖
𝑆𝑖 = 𝑃𝑖 ⨁𝐶𝑖
𝐶𝑖+1 = 𝐺𝑖 + 𝑃𝑖 𝐶𝑖
𝐶0 = input carry
𝐶1 = 𝐺0 + 𝑃0 𝐶0
𝐶2 = 𝐺1 + 𝑃1 𝐶1 = 𝐺1 + 𝑃1 𝐺0 + 𝑃𝑜 𝐶0 = 𝐺1 + 𝑃1 𝐺0 + 𝑃1 𝑃0 𝐶0
𝐶3 = 𝐺2 + 𝑃2 𝐶2 = 𝐺2 + 𝑃2 𝐺1 + 𝑃2 𝑃1 𝐺0 + 𝑃2 𝑃1 𝑃0 𝐶0
Carry lookahead generator
Four-bit adder with carry lookahead
Carry lookahead
generator
Four-bit adder with carry lookahead
Binary subtractor
Overflow
• Occurs only when adding two positive
numbers or two negative numbers;
• Overflow produces change in result sign
Example: eight-bit adder
Carry bits
0
1
1
0
+70
0 1000110
-70
1 0111010
+80
0 1010000
-80
1 0110000
+150
1 0010110
-150
0 1101010
Decimal Adder
• Consider adding two decimal digits in BCD
• Output sum cannot exceed 9+9+1=19 (the last
1 is the carry from previous digit)
Decimal Adder
Need correction
Carry
𝐶 = 𝐾 + 𝑍8 𝑍4 + 𝑍8 𝑍2
Condition for correcting result
Decimal Adder
Binary Multiplier
Exercise:
Multiply 10 × 11
Explain how you carried the
multiplication out. How many bits at
the output?
Binary Multiplier
Two-bit multiplier
Binary Multiplier
Exercise: With your
neighbor classmate
discuss its operation.
Magnitude Comparator
Exercise: Discuss with your neighbor classmate
and write down how you would compare two
four-bit binary numbers 𝐴 and 𝐵, where 𝐴 =
𝐴3 𝐴2 𝐴1 𝐴0 and 𝐵 = 𝐵3 𝐵2 𝐵1 𝐵0 . You should have
three outputs corresponding to 𝐴 = 𝐵, 𝐴 > 𝐵,
and 𝐴 < 𝐵. Explain how you determined each
condition.
Magnitude Comparator
Does this circuit
correspond to what you
wrote down in the
exercise? Discuss again
with your neighbor
classmate the
comparison of your result
with this circuit.
Decoders
Exercise:
Minimize the
functions for
two of the eight
output lines.
Decoders
Exercise: Compare
your function with
the circuit.
Three-to-eight-line
decoder
Decoders
0
Decoders
4 × 16 decoder
constructed with two 3 ×
8 decoders
Exercise: Explain how this
decoder works.
Combinational logic implementation
Exercise: For the maps of the full adder shown above, express the sum 𝑆
(left) and the carry bit 𝐶 (right) as a sum of minterms.
Combinational Logic Implementation
𝑆 𝑥, 𝑦, 𝑧 =
(1,2,4,7)
𝐶 𝑥, 𝑦, 𝑧 =
(3,5,6,7)
Compare in terms of propagation time and number of gates this Full Adder
with the previously studied implementation.
Encoders
𝑧 = 𝐷1 +𝐷3 +𝐷5 + 𝐷7
𝑦 = 𝐷2 +𝐷3 +𝐷6 + 𝐷7
𝑥 = 𝐷4 +𝐷5 +𝐷6 + 𝐷7
Priority encoder
Highest priority
Valid output
Priority encoder
Exercise: obtain the function for 𝑉
Priority encoder
Multiplexers
• Selects binary information from one of many
input lines
• Directs input line to output, controlled by a
set of selection lines
• 2𝑛 input lines and 𝑛 selection lines
Multiplexers
Two-to-one-line multiplexer
Multiplexers
Four-to-one-line multiplexer
Multiplexers
Quadruple two-to-one-line multiplexer
Boolean function implementation
• Multiplexer is essentially a decoder with OR gates
• Thus can implement Boolean functions, similar to
decoders
• Minterms generated by circuit associated with
selection inputs
• Individual minterms can be selected by data
inputs
• Boolean function of 𝑛 variables and 2𝑛 data
inputs
Boolean function implementation
• More efficient method for implementing a
Boolean function of 𝑛 variables with
multiplexer of 𝑛 − 1 selection inputs
• Remaining variable of function is used for the
data inputs
Boolean function implementation
• Example: 𝐹 𝑥, 𝑦, 𝑧 = (1,2,6,7)
Selected
input line
0
1
2
3
Implementation of a three-input Boolean function
Boolean function implementation
𝐹 𝐴, 𝐵, 𝐶, 𝐷 =
0
1
2
3
4
5
6
7
(1,3,4,11,12,13,14,15)
Three-state gates
Three-state gates
Multiplexers with three-state gates
Homework Assignment
•
•
•
•
•
•
4.3
4.9
4.17
4.27
4.33
Using LogicWorks, simulate a 4-bit full adder
with and without carry-lookahead.