Ee 101_logic Gates_implementation With Dtl.pdf 5y6g2o

Boolean Logic and Boolean Algebra

Boolean algebra: • Boolean algebra is the mathematics of logic circuits. • Devised for dealing mathematically with philosophical propositions which have ONLY TWO possible values: TRUE or FALSE, Light ON or OFF. • Boolean algebra deals with the rules which govern various operations between the binary variables.

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LOGIC SYSTEMS

LOGIC SYSTEM

INPUTS

OUTPUTS

COMBINATORIAL LOGIC Output = Logical combination of the Inputs

Zt= f (It) • For any given set of Inputs (I) you will always get the same

output (Z) and the output will not vary with time. Examples: Gates: AND, OR …, ADDER, MULTIPLEXER, PARITY GENERATOR, BINARY to BCD converter 2

TRUTH TABLES Shows the relationship between all combinations of inputs and resulting outputs. (for n inputs there are 2n combinations - so truth tables can get pretty big). Truth tables contain : Inputs (>1) Intermediate (only for very long functions) Outputs (>1) Note - multiple outputs (i.e. >1) are listed on the same table but treated separately. Example: AND truth table (shown for two inputs and 22 outputs)

Inputs

Output

A

B

X

A1

B1

C1

A2

B2

C2

A3

B3

C3

A4

B4

C4

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Concept of GATE

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Truth Table

Timing Diagram

Inputs

Output

A

B

X

0

0

0

0

1

0

1 1

0 1

0 1

0

A

1

0

1

1

B

1 1

1

0 0

1

0

1

0

0

t1

t2

t3

t4

t5

C

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AND gate implementation with DTL

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Truth Table Inputs

Timing Diagram Output

A

B

X

0

0

0

0

1

1

1

0

1

1

1

1

A

1

0 0

1

B

1

1

0

0

1

1

0

1

t1

t2

t3

t4

C

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OR gate implementation with DTL

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Timing Diagram

Truth Table Input

Output

0

1

1

0

A C

1

0

1

1

0

0

1

0

0

1

t1

t2

t3

t4

t5

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NOT Gate implementation with BJT

NAND Gate

NAND gate implementation with DTL

NOR Gate

NOR gate implementation with DTL

Boolean Algebra • These are the mathematical rules which describe how logic gates combine. • The three fundamental rules are:

AND

Z = A.B

OR

Z=A+B

NOT

Z= A

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Rules of Boolean algebra 1. COMUTATIVE LAW: A+B=B+A A.B=B.A 2. ASSOCIATIVE LAW: A + (B + C) = (A + B) + C A(BC) = (AB)C 3. DISTRIBUTIVE LAW: A(B + C) = AB + AC

THESE LAWS CAN BE EXTENDED TO INCLUDE ANY NUMBER OF VARIABLES. 19

Circuit presentation of DISTRIBUTIVE LAW (B+C) A(B + C) = AB + AC = A(B + C)

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RULES OF BOOLEAN ALGEBRA (Continuing…)

1.

A+0=A

2.

A+1=1

3.

A.0=0

4.

A.1=A

5.

A+A=A

6.

A+A=1

7.

A.A=A

8.

A.A=0

9.

A=A

10.

A + AB = A

11.

A + AB = A + B

12.

(A+B)(A+C) = A+BC 21

PROOF OF RULES 10, 11, 12:

Rule 10: A+AB = A(1+B) =A.1 =A Rule 11: A + AB = = (A + AB) + AB = (AA + AB) + AB = AA +AB + AA + AB i.e. adding AA =0 = (A+A)(A + B) = 1 . (A + B) =A+B

(Rule 2) (Rule 4)

(Rule 10) (Rule 7) (Rule 8) (Factoring) (Rule 6) 22

Rule 12: (A + B)(A + C) = = AA + AC + AB + BC = A +AC +AB +BC = A(1+C) + AB + BC = A . 1 + AB + BC = A + AB + BC = A(1 + B) + BC = A .1 + BC = A + BC

(Distributive) (Rule 7) (Distributive) (Rule 4) (Rule 2) (Distributive) (Rule 2) (Rule 4)

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