Logic and Computer Design Fundamentals
Chapter 3 – Combinational
Logic Design
Part 1- Implementation Technology and Logic
Design

 Part 1-Implementation Technology and Logic
Design
• Design Concepts
 Fundamental concepts of design Design Procedure
 The major design steps: specification, formulation, optimization, technology mapping, and verification
• Technology Mapping
 From AND, OR, and NOT to other gate types
• Verification
 Does the designed circuit meet the specifications?
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 Part 2 – Combinational Logic
• Rudimentary logic functions
• Decoding
• Encoding
• Selecting
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 A combinational logic circuit has:
• A set of n Boolean inputs,
• A set of m Boolean outputs, and
• Logic gates and interconnections
Chapter 3 - Part 1
Block Diagram of Combinational Circuit
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 The combinational circuit can be described by:
•
• Truth table that lists the output values for each of combination of the input variables m Boolean functions, one for each output variable. Each functioned is expressed as a function of n input variables
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 When a logic circuit is provided, we can analyse the circuit to obtain the logic expression.
 Example: What is the Boolean expression of
F4?
A'
A'B'
B'
A'B'+C (A'B'+C)'
F4
C
F4 = (A'B'+C)' = (A+B).C'
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Designing circuits
 The goal in circuit design is to build hardware that solves some problem.
 The basic approach is to express the solution as a Boolean function, which can then be converted to a circuit.
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1.
Specification
• make sure you completely understand all the design requirements
2.
Formulation
• Figure out how many inputs and outputs you need.
• Assign letter symbols to the input and output variables
• Derive a truth table or initial Boolean equations that define the required relationships between the inputs and outputs.
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3.
Optimization
• Find a simplified Boolean expression for the function.
• Draw a logic diagram using ANDs, ORs, and inverters
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4.
Technology Mapping
• Map the logic diagram to the implementation technology selected
5.
Verification
• Verify the correctness of the final design
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 PROBLEM : Design a combinational circuit with 3 inputs (A, B, C) and 1 output, f such that
• f is 1 when all 3 inputs are 1, or
• f is 1 when a is 1 and either b or c is 1, or
• f is 1 when all three inputs are 0
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Create a Truth Table
 List all possible inputs and outputs:
A B C F
0 0 0 1
F is one when: all three inputs are 1
A is 1 and either B or C is 1 all three inputs are 0
F is zero for all other inputs
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1 1 0 1
1 1 1 1
12

Chapter 3 - Part 1
 Use a Kmap to find the reduced SOP expression
A B C F
A
B
C
00 01 11 10
0 1
1 1 1 1
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1 1
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 The final circuit:
A
B
C
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ABC
AB
AC
14

Example: comparing 2-bit numbers
 Design a circuit that compares two 2-bit numbers, A and B. There are three possible results: A > B, A = B or A < B.
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Step 1: How many inputs and outputs?
 How many inputs and outputs will this circuit have?
• Two 2-bit numbers means a total of four inputs.
Let’s say the first number consists of bits called
A1and A0(from left to right), while second number has bits B1and B0.
• Inputs A and B will be 00, 01, 10, or 11 (0, 1, 2 or 3 in decimal).
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Step 1: How many inputs and outputs? (continued )
 We will represent the results using three separate outputs.
• G (“Greater”) should be 1 only when A > B.
• E (“Equal”) should be 1 only when A = B.
• L (“Lesser”) should be 1 only when A < B.
• For any inputs A and B, exactly one of the three outputs will be1.
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Step 1: How many inputs and outputs? (continued )
 Here is a block diagram that shows the inputs and outputs explicitly.
 Now the hard part is to design the circuitry that goes inside the box.
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Step 2: Functional specification
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Step 3: Simplified Boolean expressions
 use K-maps to simplify the circuit. There are three functions (each with the same inputs A1 A0 B1 B0), so we need three Kmaps.
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Step 3: Simplified Boolean expressions (continued)
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Step 4: Drawing the circuits
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 The mapping is accomplished by:
• Replacing AND and OR symbols,
• Pushing inverters through circuit fan-out points, and
• Canceling inverter pairs
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1.
Replace ANDs and ORs:
.
.
.
.
.
.
.
.
.
.
.
.
a+b= a . b
2.
Repeat the following pair of actions until there is at most one inverter between : a.
A circuit input or driving NAND gate output, and b.
The attached NAND gate inputs.
.
.
.
.
.
.
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 The mapping is accomplished by:
• Replacing AND and OR symbols,
• Pushing inverters through circuit fan-out points, and
• Canceling inverter pairs
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1.
Replace ANDs and ORs:
.
.
.
.
.
.
ab= a + b
.
.
.
.
.
.
2.
Repeat the following pair of actions until there is at most one inverter between : a.
A circuit input or driving NOR gate output, and b.
The attached NOR gate inputs.
.
.
.
.
.
.
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A
B
C
D
E
Chapter 3 - Part 1
(a)
C
D
E
A
B
F
C
D
E
A
B
(c)
1
X
3
2
(b)
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F
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F