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Outline:

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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

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

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.

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

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.

4.3 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

6

- 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

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.

4.4 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

12

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

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.

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

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:

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

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

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

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

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.

4-6 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

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

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.

4.7 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.

4-bit by 3-bit binary multiplier

Fig. 4.16

Four-bit by three-bit binary

multiplier.

Digital Circuits

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.

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

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

- 3-to-8-Line Decoder: example: Binary-to-octal conversion.

D0 = m0 = A2’A1’A0’

D1= m1 = A2’A1’A0

…etc

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?

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.

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

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

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.

4.10 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

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

■ 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

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.

4.11 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

n

n-to- 2 decoder

add the 2 input lines to each AND gate

n

OR(all AND gates)

an enable input (an option)

Quadruple two-to-one-line multiplexer

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

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

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

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

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