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- Chapter 4Combinational Logic

Combinational Logic Circuit

- Combinational circuit logic circuit whose

outputs at any time are determined directly and

only from the present input combination. - A combinational circuit performs a specific

information-processing operation fully specified

logically by a set of Boolean functions. - Sequential circuit one that employ memory

elements in addition to (combinational) logic

gatestheir outputs are determined from the

present input combination as well as the state of

the memory cells.

Introduction

- The state of the memory elements, in turn, is a

function of the previous inputs (and the previous

state). - Its behavior therefore is specified by a time

sequence of inputs and internal states. - In many applications, the source and the

destination are storage registers. - A combinational circuit also can be described by

m Boolean functions, one for each output

variable.

Block Diagram of a Combinational Circuit

Combinational Circuit

n inputs

m outputs

Fig. 4-1 Block Diagram of Combinational Circuit

Analysis Procedure

ABC

(A B C)(AB AC BC) ABC

A B C

(A B C)(AB AC BC)

AB

(AB AC BC)

AB AC BC

AC

BC

Design Procedure

- 1. Define the problem.
- 2. Determine the number of input/output
- variables.
- 3. Assign letter symbols to input/output

variables. - 4. Derive the that defines the required
- relationships between inputs outputs.
- 5. Obtain the simplified Boolean function for

each - output.
- 6. Draw the logic diagram.

Code Conversion Example

Maps for BCD to Excess-3 Code Converter

Maps for BCD to Excess-3 Code Converter

Code Conversion

- The expressions obtained may be manipulated

algebraically for the purpose of using common

gates for 2 or more outputs. - w A BC BD A B(C D)
- x BC BD BCD B(C D) B(C D)
- y CD CD CD (C D)
- z D

Logic Diagram for BCD to Excess-3 Code Converter

Adder

- The most basic arithmetic operation is the

addition of 2 bits. A combinational circuit that - performs this operation is called a

half-adder. - A combinational circuit that performs the

addition of 3 bits is called a full-adder, which - can be implemented by 2 half-adders.

Half Adder

Full Adder

Full Adder

Implementation of FA

Implementation of FA

S z ?(x ? y) z(xy xy) z (xy xy)

z(xy xy) z(xy xy) xyz

xyz xyz xyz The carry output is C z(xy

xy) xy xyz xyz xy

Binary Adder

- Iterative Logic Array (ILA)

Carry Propagation

- The total propagation time is equal to the

propagation delay of a typical gate times the

number of gate levels in the circuit. - The longest propagation delay time in an adder is

the time it takes the carry to propagate through

the full adders. - The number of gate levels for the carry

propagation can be found from the circuit of the

full adder.

Carry Propagation

Ai

Si

Bi

Pi

Gi

Ci1

Ci

- pi carry propagate
- ci carry generate
- These two signals are common to all full adders

and depend only on the input augend and addend

bits.

Carry Propagation

- Pi Ai ? Bi
- Gi AiBi
- The output sum and carry can be expressed as
- Si Pi ? Ci
- Ci1 Gi PiCi
- C0 input carry
- C1 G0 P0C0
- C2 G1 P1C1 G1 P1(G0 P0C0) G1

P1G0 P1P0C0 - C3 G2 P2C2 G2 P2G1 P2P1G0

P2P1P0C0

Carry Lookahead Generator

4-Bit Adder with Carry Lookahead

Binary Subtractor

- Subtraction addition of minuend and

2s-complemented subtrahend. - Also can implement subtraction directlywith

half-subtractors and full-subtractors.

Subtractor

4-Bit Adder Subtractor

BCD Adder

Binary Multiplier

4-Bit by 3-Bit Binary Multiplier

Magnitude Comparator

A A3A2A1A0 B B3B2B1B0 xi AiBi

AiBi (AB) x3x2x1x0

Decoder

2-to-4-Line Decoder with Enable Input

4 ?16 Decoder Constructed with Two 3 ? 8 Decoders

Combinational Logic Implementation

Priority Encoder

- A priority encoder is an encoder that includes

the priority function.

Priority Encoder

Multiplexers

- A multiplexer is a combinational circuit that

selects binary information from one of many input

lines and directs it to a single output line.

A 4-to-1 Multiplexer

Quadruple 2-to-1-Line Multiplexer

Boolean Function Implementation

Implementing a 4-Input Function with a Multiplexer

Three-State Gates

Multiplexers with Three-State Gates

Three-State Gates

2-to-1-Line Multiplexer

Stimulus and Design Modules Interacion

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