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Chapter 6 Function of Combination Logic

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- 6-1 Bisic Adders

- 6-2 Parallel Binary Adders

- 6-3 Comparators

- 6-4 Decoders

- 6-5 Encoders

- 6-6 Code Converters

- 6-7 Multiplesers

- 6-8 Demultiplesers

- 6-9 Parity Generators

- 6-10 Troubleshooting

- 6-11 Programmable Logic Devices

- 6-12 Digital System Application

6-1 Bisic Adders

Logic Circuit

Give the express gradually

predigest

Truth Table

Cive the logic function

6-1 Bisic Adders

Half Adder

S= xy +xy

C= xy

6-1 Bisic Adders

S=x⊕y

C=xy

6-1 Bisic Adders

Full Adder

Truth Table

∑

CO

CI

The full adder can be also implemented with two half adders and one OR gate.

F3

F2

F1

F0

CO

CO

∑

∑

CI

CI

CO

CO

∑

∑

CI

CI

B3

A3

B2

A2

B1

A1

B0

A0

6-2 Parallel Binary Adders

Binary Adder

C0 must be 0

4-Bit Adder

Only after the carry propagates and ripples through all stages will the output S4 and carry C4 settle to their final correct value.

6-2 Parallel Binary Adders

Carry Propagation

Pi = Ai⊕ Bi

Si = Pi⊕ Ci

Ci+1 = Gi + Pi Ci

Gi = Ai Bi

6-2 Parallel Binary Adders

Carry lookahead

C0 = input carry

C1 = G0+ P0 C0

C2 = G1+ P1 C1 = G1+ P1(G0+ P0 C0 )

= G1+ P1 G0+ P1 P0 C0

C3 = G2+ P2 C2 = G2+ P2 G1+ P2 P1 G0 + P2 P1 P0 C0

6-2 Parallel Binary Adders

Carry lookahead

Ci+1 = Gi + Pi Ci

F4⑽

F1⑷

F3⒀

F2⑴

CO4⑼

=1

=1

=1

=1

Y4

Y3

Y2

X4

X3

X2

X1

Y1

≥1

≥1

≥1

≥1

&

&

1

&

&

&

&

&

&

&

&

&

&

&

1

&

1

1

1

&

≥1

&

≥1

&

≥1

&

≥1

1

A4⑿

A3⒁

B3⒂

A2⑶

B2⑵

A1⑸

B1⑹

CI1⑺

B4⑾

4-Bit Adder with Carry Lookahead

A0

0

3

COMP

A1

P

A2

A3

FP＜Q

FP＝Q

FP＞Q

P＜Q

P＝Q

P＞Q

A＜B

A＝B

A＞B

＜

＝

＞

B0

0

3

B1

Q

B2

B3

6-3 Comparators

Truth Table of 1 bit comparators

Truth Table of 4 bit comparators

A0

0

3

COMP

A1

P

A2

A3

FP＜Q

FP＝Q

FP＞Q

P＜Q

P＝Q

P＞Q

A＜B

A＝B

A＞B

＜

＝

＞

B0

0

3

B1

Q

B2

B3

(A > B) = A3 B3 + x3 A2 B2 + x3 x2 A1 B1 + x3 x2 x1 A0 B0

(A < B) = A3 B3 + x3 A2 B2 + x3 x2 A1 B1 + x3 x2 x1 A0 B0

6-3 Comparators

if A ≠ B,

6-3 Comparators

4-Bit Magnitude Comparator

&

Y0

1

1

1

1

1

ST

&

Y1

&

Y2

A0

&

A1

Y3

6-4 Decoders

Y0

Y2

Y1

ST

Y3

BIN/OCT

0

1

2

3

1

2

A0

A1

EN

2-to-4-Line Decoder

Y3

Y1

Y2

Y0

ST

ST

Y1

Y3

Y0

Y2

3 2 1 0

3 2 1 0

BIN/OCT

BIN/OCT

Ⅰ

Ⅱ

1 2

EN

1 2

EN

1

A0

A1

A2

Decoders with enable inputs can be connected together to form a larger decoder circuit.

3-to-8-Line Decoder

Y3

Y0

Y2

Y1

STB

STC

Y7

Y6

Y4

Y5

BIN/OCT

0

1

2

3

1

2

4

A0

A1

A2

4

5

6

7

&

STA

EN

3-to-8-Line Decoder

A

B

C

D

VCC

&

&

&

&

1

1

1

1

9

8

7

6

5

4

3

2

1

0

1kW×9

6-5 Encoders

An encoder is a digital circuit that performs the inverse operation of a decoder.

3-to-8-Line Decoder

YS(15)

≥1

≥1

≥1

≥1

&

&

&

&

&

&

&

&

&

&

&

&

&

1

1

1

1

1

1

1

1

1

1

1

1

1

IN7(4)

IN0(10)

IN1(11)

IN3(13)

IN4(1)

IN5(2)

YEX(14)

IN2(12)

ST(5)

Y0(9)

Y1(7)

Y2(6)

IN6(3)

CT54/74148

IN3

IN6

IN5

ST

IN2

IN1

IN4

IN7

Y0

YEX

Y2

Y1

IN0

HPRI/BIN

0/Z10

1/Z11

2/Z12

3/Z13

4/Z14

5/Z15

6/Z16

7/Z17

10

11

12

13

14

15

16

17

≥1

YS

18

a

V18

ENa

1a

2a

3a

ST

0 …… 7

8 …… 15

ST

0 1 2 3 4 5 6 7 EN

0 1 2 3 4 5 6 7 EN

HPRI/BIN

HPRI/BIN

YS

YS

Y2

Y0

Y1

Y2

YEX

Y0

Y0

YEX

YEX

Y1

Y2

Y1

Y3

&

&

&

&

Enlarge 8-3 line Encoder to 16-4 line Encoder

Output S1 through S3 have equal propagation delay times.

6-6 Code Converters

4-Bit Adder with Carry Lookahead

An－1 … A1 A0

D0

D0

D1

D1

D2

D2

MUX

DMUX

…

…

D2n－1

D2n－1

6-7 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.

Logic Diagram of 4-to-1-line Multiplexer

6-7 Multiplexers

Logic Diagram of Quadruple 2-to-1-Line Multiplexer

Function Table of 4 Quadruple 2-to-1-Line Multiplexer

Express of 2n to1 is

8 to1(CT54/74151) is

MUX

ST

EN

A0

0

2

0

7

A1

G

A2

Y

D0

0

1

2

3

4

5

6

7

W

D1

D2

D3

D4

D5

D6

D7

Logic symbol of 8to1 Multiplexer

Truth Table of 8to1 Multiplexer

Step 3: Each data input will be z , z , 1, or 0.

6-7 Multiplexers

Boolean Function Implementation

Method for implementing a Boolean function of n variables with a multiplexer that has n-1 selection inputs is as follows,

Step 1 : The first n-1 variables are connected to the selection inputs.

Step 2: The remaining single variable, denoted by z , is used for the data input.

A4

EN

2

1

A3

BIN/OCT

Ⅰ

A2

A1

0 1 2 3

A0

…

…

…

…

D0

D7

D8

D15

D16

D23

D24

D31

EN

0 7

EN

0 7

EN

0 7

EN

0 7

MUX

MUX

MUX

MUX

Ⅱ

0

2

0

7

0

2

0

7

Ⅲ

0

2

0

7

Ⅳ

0

2

0

7

Ⅴ

G

G

G

G

Y

Y

Y

Y

≥1

Y

Enlarge 8to1 to 32 to1 Multiplexer

…

…

…

…

D0

D7

D8

D15

D16

D23

D24

D31

EN

0 7

EN

0 7

EN

0 7

EN

0 7

MUX

MUX

MUX

MUX

Ⅰ

Ⅱ

0

2

0

7

0

2

0

7

0

2

0

7

Ⅲ

0

2

0

7

Ⅳ

G

G

G

G

Y

Y

Y

Y

A0

A1

EN

0 1 2 3

A2

MUX

A3

0

1

0

3

Ⅴ

G

A4

Y

Another method to Enlarge 8to1 to 32 to1 Multiplexer

A

S2

B

S1

C

S0

6-8 Demultiplexers

Consider the Implementation of the Boolean Function

F( A,B,C,D) = ∑( 1,3,4,11,12,13,14,15 )

6-9 Parity Generators

Modeling techniques:

Using procedural assignment statements with keyword always

Gate-level modeling

To design at the MOS transistor level

Using instantiation of primitive gates and user-defined modules

Dataflow modeling

Behavioral modeling

Using continuous assignment statements with keyword assign

Switch-level modeling

8 other gates are declared as

and, nand, or, nor, xor, xnor, not, buf

6-9 Parity Generators

Gate-Level Modeling:

System assigns a four-valued logic set to each gate

Verilog recognizes 12 basic gates as predefined primitive.

4 primitive gates are of the three-state type.

12 basic gates

An unknown value is considered during simulation for the case when an input or output is ambiguous.

A high-impedance condition occurs in the output of three-state gates or if a wire is inadvertently left unconnected

6-9 Parity Generators

Truth table for and gate

For example:

6-9 Parity Generators

Three State Gates:

Y4

Y6

Y5

STC

Y3

Y7

Y1

STB

Y0

Y2

CP

BIN/OCT

0

1

2

3

1

2

4

A0

Q1

A1

A2

4

5

6

7

Q2

&

STA

EN

Q3

Q4

Z

6-10 Troubleshootings

A3 A2 A1 A0

1

1

1

1

&

W0

W1

W2

W3

W4

W5

W6

W7

W8

W9

W10

W11

W12

W13

W14

W15

≥1

≥1

≥1

≥1

D3 D2 D1 D0

B3 B2 B1 B0

1

1

1

1

&

W0

W1

W2

W3

W4

W5

W6

W7

W8

W9

W10

W11

W12

W13

W14

W15

≥1

≥1

≥1

≥1

G3 G2 G1 G0

Pass

Stop

6-12 Digital System Application

Pass

Stop

6-12 Digital System Application

Pass

Stop

6-12 Digital System Application

6-12 Digital System Application

RY

G

00 01 11 10

0 1