CS1104: Computer Organisation

http://www.comp.nus.edu.sg/~cs1104

Lecture 4: Logic Gates and Circuits

CS1104-4 Lecture 4: Introduction to Logic Gates

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Lecture 4:Logic Gates and Circuits

Logic Gates The Inverter The AND Gate The OR Gate The NAND Gate The NOR Gate The XOR Gate The XNOR Gate

Drawing Logic Circuit Analysing Logic Circuit Propagation Delay

CS1104-4 Lecture 4: Introduction to Logic Gates

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Lecture 4: Logic Gates and Circuits

Universal Gates: NAND and NOR NAND Gate NOR Gate

Implementation using NAND Gates

Implementation using NOR Gates

Implementation of SOP Expressions

Implementation of POS Expressions

Positive and Negative Logic

Integrated Circuit Logic Families

CS1104-4 Logic Gates 4

Logic Gates Gate Symbols

EXCLUSIVE OR

a

ba.b

a

ba+b

a a'

a

b(a+b)'

a

b(a.b)'

a

ba b

a

ba.b&

a

ba+b1

AND

a a'1

a

b(a.b)'&

a

b(a+b)'1

a

ba b=1

OR

NOT

NAND

NOR

Symbol set 1 Symbol set 2

(ANSI/IEEE Standard 91-1984)

CS1104-4 Logic Gates: The Inverter 5

Logic Gates: The Inverter

The InverterA A'

0 11 0

A A' A A'

Application of the inverter: complement.

1

0

0

1

0

1

0

1

1

0

0

1

1

0

1

0

Binary number

1’s Complement

CS1104-4 Logic Gates: The AND Gate 6

Logic Gates: The AND Gate (1/2)

The AND Gate

A B A . B0 0 00 1 01 0 01 1 1

A

BA.B

&A

BA.B

CS1104-4 Logic Gates: The AND Gate 7

Logic Gates: The AND Gate (2/2)

Application of the AND Gate

1 sec

A

1 secEnable

A

EnableCounter

Reset to zero between Enable pulses

Register, decode and frequency display

CS1104-4 Logic Gates: The OR Gate 8

Logic Gates: The OR Gate

The OR Gate

1

A

BA+B

A

BA+B

A B A + B0 0 00 1 11 0 11 1 1

CS1104-4 Logic Gates: The NAND Gate 9

Logic Gates: The NAND Gate

The NAND Gate

&A

B(A.B)'

A

B(A.B)'

A

B(A.B)'

NAND Negative-OR

A B (A.B)'0 0 10 1 11 0 11 1 0

CS1104-4 Logic Gates: The NOR Gate 10

Logic Gates: The NOR Gate

The NOR Gate

NOR Negative-AND

1

A

B(A+B)'A

B(A+B)'

A

B(A+B)'

A B (A+B)'0 0 10 1 01 0 01 1 0

CS1104-4 Logic Gates: The XOR Gate 11

Logic Gates: The XOR Gate

The XOR Gate

=1A

BA B

A

BA B

A B A B0 0 00 1 11 0 11 1 0

CS1104-4 Logic Gates: The XNOR Gate 12

Logic Gates: The XNOR Gate

The XNOR Gate

A

B(A B)'

=1A

B(A B)'

A B (A B) '0 0 10 1 01 0 01 1 1

CS1104-4 Drawing Logic Circuit 13

Drawing Logic Circuit (1/2)

When a Boolean expression is provided, we can easily draw the logic circuit.

Examples:

(i) F1 = x.y.z' (note the use of a 3-input AND gate)

xy

z

F1

z'

CS1104-4 Drawing Logic Circuit 14

Drawing Logic Circuit (2/2)

(ii) F2 = x + y'.z (if we assume that variables and their complements are available)

(iii) F3 = x.y' + x'.z

x

y'z

F2

y'.z

x'z

F3

x'.z

x.y'xy'

CS1104-4 Quick Review Questions (1) 15

Quick Review Questions (1)

Textbook page 77.

Questions 4-1, 4-2.

CS1104-4 Analysing Logic Circuit 16

Analysing Logic Circuit

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'.B'

A'.B'+C (A'.B'+C)'

A'

B'

CF4

F4 = (A'.B'+C)' = (A+B).C'

CS1104-4 Propagation Delay 17

Propagation Delay (1/3) Every logic gate experiences some delay (though

very small) in propagating signals forward.

This delay is called Gate (Propagation) Delay.

Formally, it is the average transition time taken for the output signal of the gate to change in response to changes in the input signals.

Three different propagation delay times associated with a logic gate: tPHL: output changing from the High level to Low level

tPLH: output changing from the Low level to High level

tPD=(tPLH + tPHL)/2 (average propagation delay)

CS1104-4 Propagation Delay 18

Propagation Delay (2/3)

Input Output

Output

InputH

L

L

H

tPHL tPLH

CS1104-4 Propagation Delay 19

Propagation Delay (3/3)

A B C

Ideally, no delay:

1

0

1

0

0

1

time

Signal for C

Signal for B

Signal for A

In reality, output signals normally lag behind input signals:

1

0

1

0

0

1

time

Signal for C

Signal for B

Signal for A

CS1104-4 Calculation of Circuit Delays 20

Calculation of Circuit Delays (1/3) Amount of propagation delay per gate depends on:

(i) gate type (AND, OR, NOT, etc) (ii) transistor technology used (TTL,ECL,CMOS etc), (iii) miniaturisation (SSI, MSI, LSI, VLSI)

To simplify matters, one can assume (i) an average delay time per gate, or (ii) an average delay time per gate-type.

Propagation delay of logic circuit= longest time it takes for the input signal(s) to propagate to the

output(s).

= earliest time for output signal(s) to stabilise, given that input signals are stable at time 0.

CS1104-4 Calculation of Circuit Delays 21

Calculation of Circuit Delays (2/3) In general, given a logic gate with delay, t.

If inputs are stable at times t1,t2,..,tn, respectively; then the earliest time in which the output will be stable is:

max(t1, t2, .., tn) + t

LogicGate

t1

t2

tn

: :

max (t1, t2, ..., tn ) + t

To calculate the delays of all outputs of a combinational circuit, repeat above rule for all gates.

CS1104-4 Calculation of Circuit Delays 22

Calculation of Circuit Delays (3/3) As a simple example, consider the full adder circuit

where all inputs are available at time 0. (Assume each gate has delay t.)

where outputs S and C, experience delays of 2t and 3t, respectively.

XY S

C

Z

max(0,0)+t = t

t

0

0

0

max(t,0)+t = 2t

max(t,2t)+t = 3t2t

CS1104-4 Quick Review Questions (2) 23

Quick Review Questions (2)

Textbook page 77.

Questions 4-3 to 4-5.

CS1104-4 Universal Gates: NAND and NOR 24

Universal Gates: NAND and NOR AND/OR/NOT gates are sufficient for building any

Boolean functions. We call the set {AND, OR, NOT} a complete set of

logic.

However, other gates are also used because:(i) usefulness(ii) economical on transistors(iii) self-sufficient

NAND/NOR: economical, self-sufficientXOR: useful (e.g. parity bit generation)

CS1104-4 NAND Gate 25

NAND Gate (1/2)

NAND gate is self-sufficient (can build any logic circuit with it).

Therefore, {NAND} is also a complete set of logic.

Can be used to implement AND/OR/NOT.

Implementing an inverter using NAND gate:

(x.x)' = x' (T1: idempotency)

x x'

CS1104-4 NAND Gate 26

NAND Gate (2/2)

((x.y)'(x.y)')' = ((x.y)')' idempotency = (xy) involution

((x.x)'(y.y)')' = (x'.y')' idempotency = x''+y'' DeMorgan = x+y involution

Implementing AND using NAND gates:

Implementing OR using NAND gates:

xx.y

y

(x.y)'

x

x+y

y

x'

y'

CS1104-4 NOR Gate 27

NOR Gate (1/2)

NOR gate is also self-sufficient. Therefore, {NOR} is also a complete set of logic

Can be used to implement AND/OR/NOT.

Implementing an inverter using NOR gate:

(x+x)' = x' (T1: idempotency)

x x'

CS1104-4 NOR Gate 28

NOR Gate (2/2)

((x+x)'+(y+y)')'=(x'+y')' idempotency = x''.y'' DeMorgan = x.y involution

((x+y)'+(x+y)')' = ((x+y)')' idempotency = (x+y) involution

Implementing AND using NOR gates:

Implementing OR using NOR gates:

xx+y

y

(x+y)'

x

x.y

y

x'

y'

CS1104-4 Implementation using NAND gates

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Implementation using NAND gates (1/2)

Possible to implement any Boolean expression using NAND gates.

Procedure:

(i) Obtain sum-of-products Boolean expression:

e.g. F3 = x.y'+x'.z

(ii) Use DeMorgan theorem to obtain expression using 2-level NAND gates

e.g. F3 = x.y'+x'.z

= (x.y'+x'.z)' ' involution

= ((x.y')' . (x'.z)')' DeMorgan

CS1104-4 Implementation using NAND gates

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Implementation using NAND gates (2/2)

F3 = ((x.y')'.(x'.z)') ' = x.y' + x'.z

x'z

F3

(x'.z)'

(x.y')'xy'

CS1104-4 Implementation using NOR gates 31

Implementation using NOR gates (1/2)

Possible to implement any Boolean expression using NOR gates.

Procedure:

(i) Obtain product-of-sums Boolean expression:

e.g. F6 = (x+y').(x'+z)

(ii) Use DeMorgan theorem to obtain expression using 2-level NOR gates.

e.g. F6 = (x+y').(x'+z)

= ((x+y').(x'+z))' ' involution

= ((x+y')'+(x'+z)')' DeMorgan

CS1104-4 Implementation using NOR gates 32

Implementation using NOR gates (2/2)

F6 = ((x+y')'+(x'+z)')' = (x+y').(x'+z)

x'z

F6

(x'+z)'

(x+y')'xy'

CS1104-4 Implementation of SOP Expressions

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Implementation of SOP Expressions (1/2)

Sum-of-Products expressions can be implemented using: 2-level AND-OR logic circuits 2-level NAND logic circuits

AND-OR logic circuit

F = A.B + C.D + E

F

A

B

D

C

E

CS1104-4 Implementation of SOP Expressions

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Implementation of SOP Expressions (2/2)

NAND-NAND circuit (by circuit transformation)

a) add double bubbles

b) change OR-with- inverted-inputs to NAND & bubbles at inputs to their complements

F

A

B

D

C

E

A

B

D

C

E'

F

CS1104-4 Implementation of POS Expressions

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Implementation of POS Expressions (1/2)

Product-of-Sums expressions can be implemented using: 2-level OR-AND logic circuits 2-level NOR logic circuits

OR-AND logic circuit

G = (A+B).(C+D).E

G

A

B

D

C

E

CS1104-4 Implementation of POS Expressions

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Implementation of POS Expressions (2/2)

NOR-NOR circuit (by circuit transformation):

a) add double bubbles

b) changed AND-with- inverted-inputs to NOR & bubbles at inputs to their complements

G

A

B

D

C

E

A

B

D

C

E'

G

CS1104-4 Quick Review Questions (3) 37

Quick Review Questions (3)

Textbook page 77.

Questions 4-6 to 4-8.

CS1104-4 Positive & Negative Logic 38

Positive & Negative Logic (1/3)

In logic gates, usually: H (high voltage, 5V) = 1 L (low voltage, 0V) = 0

This convention is known as positive logic.

However, the reverse convention, negative logic possible: H (high voltage) = 0 L (low voltage) = 1

Depending on convention, same gate may denote different Boolean function.

CS1104-4 Positive & Negative Logic 39

Positive & Negative Logic (2/3)

A signal that is set to logic 1 is said to be asserted, or active, or true.

A signal that is set to logic 0 is said to be deasserted, or negated, or false.

Active-high signal names are usually written in uncomplemented form.

Active-low signal names are usually written in complemented form.

CS1104-4 Positive & Negative Logic 40

Positive & Negative Logic (3/3)

Positive logic:

Negative logic:

EnableActive High: 0: Disabled 1: Enabled

Enable

Active Low: 0: Enabled 1: Disabled

CS1104-4 Integrated Circuit Logic Families 41

Integrated Circuit Logic Families (1/2)

Some digital integrated circuit families: TTL, CMOS, ECL. TTL: Transistor-Transistor Logic.

Uses bipolar junction transistors Consists of a series of logic circuits: standard TTL, low-power

TTL, Schottky TTL, low-power Schottky TTL, advanced Schottky TTL, etc.

TTL Series Prefix Designation Example of Device

Standard TTL 54 or 74 7400 (quad NAND gates)

Low-power TTL 54L or 74L 74L00 (quad NAND gates)

Schottky TTL 54S or 74S 74S00 (quad NAND gates)

Low-powerSchottky TTL

54LS or 74LS 74LS00 (quad NAND gates)

CS1104-4 Integrated Circuit Logic Families 42

Integrated Circuit Logic Families (2/2)

CMOS: Complementary Metal-Oxide Semiconductor. Uses field-effect transistors

ECL: Emitter Coupled Logic. Uses bipolar circuit technology. Has fastest switching speed but high power consumption.

Performance characteristics Propagation delay time. Power dissipation. Fan-out: Fan-out of a gate is the maximum number of inputs

that the gate can drive. Speed-power product (SPP): product of the propagation

delay time and the power dissipation.

CS1104-4 Summary 43

SummaryLogic Gates

AND, OR, NOT

NAND

NOR

Drawing Logic Circuit

Analysing Logic Circuit

Given a Boolean expression, draw the circuit.

Given a circuit, find the function.

Implementation of a Boolean expression using these Universal gates.

Implementation of SOP and POS Expressions

Positive and Negative Logic

Concept of Minterm and Maxterm

CS1103 Chapter 1: Introduction

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