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Chapter 4. Combinational Logic. Outline: . 4.1 Introduction. 4.2 Combinational Circuits. 4.3 Analysis Procedure. 4.4 Design Procedure. 4.5 Binary Adder- subtractor . 4.6 Decimal Adder. 4.7 Binary Multiplier. 4.9 Decoders. 4.10 Encoders. 4.11 Multiplexers. . 4.1 Introduction.

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combinational logic
Chapter 4

Combinational Logic

outline
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide3
4.1 Introduction

Logic circuits for digit systems maybe

combinational or sequential.

A combinational circuit consists of logic gates whose outputs at any time are determend from only the presence combinations of inputs

A Sequential circuits contain memory elements with the logic gates the outputs are a function of the current inputs and the state of the memory elements the outputs also depend on past inputs. (chapter 5)

2

outline1
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide5
4.2 Combinational Circuits

A combinational circuits

n

2 possible combinations of input values

Combinational circuits

n input m output

Combinatixnal

variables variables

xoxic Circuit

Specific functions

Adders, subtractors, comparators, decoders, encoders,

and multiplexers

4

outline2
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide7
4.3 Analysis Procedure

A combinational circuit

make sure that it is combinational not sequential

No feedback path or memory elements.

derive its Boolean functions (truth table)

design verification

slide9
F2= AB+AC+BC
  • T1= A+B+C
  • T2= ABC
  • T3= F2’. T1
  • F1= T3+T2
  • F1= T3+T2
  • = F2’. T1+ABC
  • = (AB+AC+BC)’.(A+B+C) +ABC
  • = (A’+B’)(A’+C’)(B’+C’).(A+B+C) +ABC
  • = (A’B’+A’B’C’+B’C’+A’C’). (A+B+C) +ABC
  • = AB’C’+A’B’C+A’B’C+ABC
outline3
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide12
4.4 Design Procedure

The design procedure of combinational circuits

From the scpecification of the circuit determine the required number of inputs and outputs.

For each input and output variables assign a symbol

Derive the truth table

Derive the simplified Boolan functions for each output as a function of the input variables

Draw the logic diagram and verify the correctness of the design

9

slide13
Example: code conversion

BCD to excess-3 code

11

slide15
The simplified functions

z = D'

y = CD +C'D‘

x = B'C + B‘D+BC'D'

w = A+BC+BD

Another i

mplementation

z = D'

y = CD +C'D' = CD + (C+D)'

x = B'C + B'D+BC'D‘ = B'(C+D) +B(C+D)'

w = A+B(C+D)

13

outline4
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide18
4-5 Binary Adder-Subtractor

Half adder

0 + 0 = 0 ; 0 + 1 = 1 ; 1 + 0 = 1 ; 1 + 1 = 10

two input variables: x, y

two output variables: C (carry), S (sum)

truth table

S = x'y+xy‚ S=xÅy

C = xy

15

full adder
Z

0

0

0

0

X

0

0

1

1

+ Y

+ 0

+ 1

+ 0

+ 1

C S

0

0

0 1

0 1

1 0

Z

1

1

1

1

X

0

0

1

1

+ Y

+ 0

+ 1

+ 0

+ 1

C S

0 1

1 0

1 0

1

1

Full-Adder
  • A full adder is similar to a half adder, but includes a carry-in bit from lower stages. Like the half-adder, it computes a sum bit, S and a carry bit, C.
    • For a carry-in (Z) of 0, it is the same as the half-adder:
    • For a carry- in(Z) of 1:
slide21
the arithmetic sum of three input bits

Full-Adder :

three input bits

x, y: two significant bits

z: the carry bit from the previous lower significant bit

Two output bits: C, S

18

slide23
S = x'y'z+x'yz'+ xy'z'+xyz

C = xy + xz + yz

S = zÅ (xÅy)

= z'(xy'+x‘y)+z(xy'+x'y)'

= z‘xy'+z'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

20

slide24
Binary adder

Note: n bit adder requires n full adders

21

slide25
Binary subtractor

A-B = A+(2’s complement of B)

4-bit Adder-subtractor using M as mode of operation

M=0, A+B; M=1, A+B’+1

26

slide26
Overflow

The storage is limited

Overfow cases :

1.Add two positive numbers and obtain a negative

number

2. Add two negative numbers and obtain a positive

number

V = 0, no overflow; V = 1, overflow

Example:

Note: XOR is used to detect overflow.

27

outline5
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide28
4-6 Decimal Adder

Add two BCD's

9 inputs: two BCD's and one carry-in

5 outputs: one BCD and one carry-out

A truth table with 2^9 entries

the sum <= 9 + 9 + 1 = 19

binary to BCD

slide30
In BCD modifications are needed if the sum > 9

Must add 6 (0110) in case:

C = 1

K = 1

Z8z4=1

Z

= 1

Z

8 2

d

Ification when C=1 we add 6:

mo

mo

C = K +Z Z + Z Z

8 4 8 2

outline6
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide33
4.7 Binary Multiplier

Partial products

–use AND operations with half adder.

Note: A*B=1 only if A=B=1

Oherwise 0.

fig. 4.15

Two-bit by two-bit binary multiplier.

slide34
4-bit by 3-bit binary multiplier

Fig. 4.16

Four-bit by three-bit binary

multiplier.

Digital Circuits

outline7
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide36
4-9 Decoder

A decoder is a combinational circute that converts n-input

lines to 2^n output lines.

We use here n-to-m decoder

n

a binary code of n bits = 2 distinct information

n

n input variables; up to 2 output lines

only one output can be active (high) at any time

slide37
An implementation

Fig. 4.18

Three-to-eight-line decoder.

38

Digital Circuits

slide38
Demultiplexers

a decoder with an enable input

receive information

in a single line and transmits

it in one of 2 possible output lines

n

Fig. 4.19

Two-to-four-line decoder with enable input

decoder examples
Decoder Examples
  • 3-to-8-Line Decoder: example: Binary-to-octal conversion.

D0 = m0 = A2’A1’A0’

D1= m1 = A2’A1’A0

…etc

slide40
Expansion

two 3-to-8 decoder: a 4-to-16 deocder

Fig. 4.20

4 16 decoder

constructed with two

3 x 8 decoders

a 5-to-32 decoder?

decoder expansion example 2
A0

A1

A2

3-8-line Decoder

D0 – D7

E

3-8-line Decoder

D8 – D15

A3

A4

E

2-4-line Decoder

3-8-line Decoder

D16 – D23

E

3-8-line Decoder

D24 – D31

E

Decoder Expansion - Example 2
  • Construct a 5-to-32-line decoder using four 3-8-line decoders with enable inputs and a 2-to-4-line decoder.
slide42
Combination Logic Implementation

each output = a minterm

use a decoder and an external OR gate to

implement any Boolean function of n input

variables

A full-adder

S(x,y,z)=S(1,2,4,7)

C(x,y,z)=

C(x,y,z)=

S

S

(3,5,6,7)

(3,5,6,7)

Fig. 4.21

Implementation of a full adder with 1 decoder

slide43
two possible approaches using decoder

OR(minterms of F): k inputs

NOR(minterms of F'): 2 - k inputs

n

In general, it is not a practical implementation

outline8
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide45
4.10 Encoders

The inverse function of decoder

a decoder

z

=

D

+

D

+

D

+

D

1 3 5 7

The encoder can be implemented

y

=

D

+

D

+

D

+

D

2 3 6 7

with three OR gates.

x

=

D

+

D

+

D

+

D

4 5 6 7

slide46
An implementation

limitations

illegal input: e.g. D

=D

x1

3 6

The output = 111 (¹3 and ¹6)

slide47
Priority Encoder

resolve the ambiguity of illegal inputs

only one of the input is encoded

D

has the highest priority

3

has the lowest priority

D

0

X: don't-care conditions

V: valid output indicator

slide48
■ The maps for simplifying outputs x and y

fig. 4.22

Maps for a priority encoder

slide49
■ Implementation of priority

x =

D

+

D

Fig. 4.23

2 3

Four-input priority encoder

y=

¢

D

+

D D

3 1 2

V

=

D

+

D

+

D

+

D

0 1 2 3

outline9
Outline:

4.1 Introduction.

4.2 Combinational Circuits.

4.3 Analysis Procedure.

4.4 Design Procedure.

4.5 Binary Adder-subtractor.

4.6 Decimal Adder.

4.7 Binary Multiplier.

4.9 Decoders.

4.10 Encoders.

4.11 Multiplexers.

slide51
4.11 Multiplexers

select binary information from one of many input

lines and direct it to a single output line

2 input lines, n selection lines and one output line

n

e.g.: 2-to-1-line multiplexer

Fig. 4.24

Two-to-one-line multiplexer

slide52
4-to-1-line multiplexer

Fig. 4.25

Four-to-one-line multiplexer

slide53
Note

n

n-to- 2 decoder

add the 2 input lines to each AND gate

n

OR(all AND gates)

an enable input (an option)

slide54
Fig. 4.26

Quadruple two-to-one-line multiplexer

slide55
Boolean function implementation

MUX: a decoders an OR gate

n

2 -to-1 MUX can implement any Boolean function

of n input variable

a better solution: implement any Boolean function

of n+1 input variable

n of these variables: the selection lines

the remaining variable: the inputs

slide56

an example: F(A,B,C) = S(1,2,6,7)

Fig. 4.27

Implementing a Boolean function with a multiplexer

slide57
Procedure:

Assign an ordering sequence of the input variable

the rightmost variable (D) will be used for the input

lines

assign the remaining n-1 variables to the selection

Lines with construct the truth table

lines w.r.t. their corresponding sequ

consider a pair of consecutive minterms starting

from m

0

determine the input lines

slide58
Example: F(A, B, C, D) = S(1, 3, 4, 11, 12, 13, 14, 15)

Fig. 4.28

Implementing a four-input function with a multiplexer

slide59
Three-state gates

A multiplexer can be constructed with three-state

gates

Output state: 0, 1, and high-impedance (open ckts)

Fig. 4.29

Graphic symbol for a three-state buffer

slide60
Example: Four-to-one-line multiplexer

Fig. 4.30

Multiplexer with three-state gates

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