Digital design chap 2

DIGITAL ELECTRONICSCHAPTER 2

DEE 204

DIGITAL ELECTRONICSDigital fundamentals

Difference between analog and digital systems.

Logic levels and families.Logic values, truth tables and logical

operations.Boolean algebra.De Morgan’s Theorem

DIGITAL FUNDAMENTALS

• Analog versus digital systemAnalog system: process information that varies continuously, time varying signals that take any value across continuous range


• Analog versus digital system

Digital system: use discrete quantities to represent information, distinct or separated quantities


• Advantages of digital system:– ease of design– reproducibility of result– flexibility– functionality– programmability– speed– economy


• Logic levelsBinary logic used in digital system assumes only TWO values: HIGH or LOWThese two levels or states can represent two numerals: 1 and 0 of the binary system ortwo logic states: TRUE and FALSE of the logic

operations


• Logic family- fundamental approach used to produce different types of digital integrated circuit- different logic functions belonging to the same logic family will have identical electrical characteristics: supply voltage range, speed of response, power dissipation, input and output logic levels, current sourcing and sinking capability, etc., making it compatible with each other


• Types of logic families:1. Bipolar – diode logic (DL), resistor

transistor logic (RTL), diode transistor logic (DTL), transistor transistor logic (TTL), emitter couple logic (ECL), current mode logic (CML), integrated injection logic (IIL or I2L)

2. MOS – PMOS, NMOS, CMOS3. Bi-MOS – using both bipolar and MOS


• Binary variables have either logic ‘0’ state or logic ‘1’ state which usually represents two different voltage or current levels

• It may be a more positive(1) or less positive (0) value referred as positive logic system

• Or may be the more positive (0) and less positive (1) referred as negative logic system


• Example:A positive logic system for values 0V and +5V0V = 0, +5V = 1

A negative logic system for values 0V and +5V0V = 1, +5V = 0


• Example:A positive logic system for 0V and -5V0V = 1, -5V = 0

A negative logic system for 0V and -5V0V = 0, -5V = 1


• Truth table- lists all possible combinations of input binary variables and the corresponding outputs of a logic system- depends on the number of binary input variables; one will have two possibilities, two will have four possibilities, while 3 will have 8 possibilities

DIGITAL FUNDAMENTALS• Thus, for n input variables, the possible inputs

combinations are given as 2n

Two input logic system

and truth table


• Truth table for a three input logic system


• Logic gates- most basic building block of any digital system

- a piece of hardware or an electronic circuit used to implement basic logic expression- the three basic logic gates are OR gate, AND gate and NOT gate


• OR gate- to perform OR operation for two or more logic variables with two or more inputs and one output- written as Y = A + B (Y equals to A OR B)- output of OR gate is LOW when all inputs are LOW and HIGH for any other input combinations


• OR gateFor a two input OR gate:


• Example:For an input waveform fed into a OR gate, sketch the output waveform.


• Solution:The output waveform produced by a OR gate


• AND gate- also with two or more inputs and one output- the output is HIGH when all inputs are HIGH and LOW for any other combinations- the output will become ‘1’ only when all inputs are ‘1’- written as Y = A.B (Y equals to A AND B)


• AND gateFor a two input AND gate:


• For a three input AND gate

For a four input AND gate


• NOT gate- a one input one output logic circuit which complements the input- the input is HIGH when the input is LOW and vice versa- a logic ‘0’ produces a logic ‘1’- known as ‘complementing’ or ‘inverting’ circuit


• NOT gate- written as and reads as Y equals to NOT A AY

A Y

0 1

1 0

DIGITAL FUNDAMENTALS• Summary of all logic gates

DIGITAL FUNDAMENTALS• Standard and alternative symbols


• Example:Draw an alternative circuit of the circuit shown


• Solution


• Boolean algebra- used to do manipulation of binary variables and simplify logic expressions- is basically the mathematics of logic- composed of a set of symbols and a set of rules to manipulate these symbols


• Rules of Boolean algebra

1. A + 0 = A 7. A.A = A

2. A + 1 = 1 8. 3. A.0 = 0 9. 4. A.1 = A 10. A + AB = A

5. A + A = A 11.

6. 12. (A + B)(A + C) = A + BC1 AA

0. AA

AA

BABAA


• Boolean algebra–There are cases when Boolean

algebra is used to simplify a Boolean expression

–Simplification means fewer gates for the same function


• Example:Simplify the given expression using Boolean algebra techniques

CBBCBAAB

DIGITAL FUNDAMENTALS• Solution:

ACB

BACAB

BCBACAB

BCBBACABAB

CBBCBAAB

B)B(AB applying

B)BC(B applying

AB)AB(AB and B)(BB applying

givesequation theexpanding

DIGITAL FUNDAMENTALS• DeMorgan’s theorem

1. The complement of a product of variables is equal to the sum of the complements of the variables (The complement of two or more ANDed variables is equivalent to the OR of the complements of each variables)

2. The complement of a sum of variables is equivalent to the product of the complements of the variables (The complement of two or more ORed variables is equivalent to the AND of the complements of each variables)


• DeMorgan’s theorem gives an expression

ZYXXYZ

YXYX

.


• Example:Apply DeMorgan’s theorem to the given expressions:

EFDCBA

DEFABC

DCBA


• Solution

FEDCBAEFDCBAEFDCBA

FEDCBADEFABCDEFABC

DCBADCBADCBA

..


• Example:Apply DeMorgan’s theorem to the given expressions:

)ZY()YX(F 1

ZYYXF

)ZY( )YX(F

)ZY( )YX(F

)ZY( )YX(F

)ZY()YX(F

1

1

1

1

1

DIGITAL FUNDAMENTALS• Example:

Apply DeMorgan’s theorem to the given expressions:

Y XZ XF

)XY()Z X(F

)XY()Z X(F

)XY()ZX(F

)XY()ZX(F

)XY)(ZX(F

2

2

2

2

2

2

Engineering

Digital design chap 2