Chapter 4 part I Simplification of Switching Function

1

Objective

s s s

s

Simplifying logic circuit Minimization using Karnaugh map Using Karnaugh map to obtain simplified SOP and POS expression Five-variable Karnaugh map

2

Simplifying Logic Circuits

• • •

A

A Boolean expression for a logic circuit may be reduced to a simpler form The simplified expression can then be used to implement a circuit equivalent to the original circuit Consider the following example: B C

A B C + A BC

Y

AB C + AB C

Y = A B C + A BC + AB C + AB C

3

Continue ...

Checking for common factor: Y = A B C + A BC + AB C + AB C

= A C ( B + B ) + AB (C + C )

Reduce the complement pairs to ‘1’

Y = A C ( B + B ) + AB (C + C ) = A C + AB

Draw the circuit based on the simplified expression

A B C

Y

4

Continue ...

•

A

Consider another logic circuit:

B C

Y

Y = C( A + B + C ) + A + C

Convert to SOP expression:

Y = C( A + B + C ) + A + C = AC + B C + AC

Checking for common factor:

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

5

Continue ...

• • Simplification of logic circuit algebraically is not always an easy task The following two steps might be useful: i. The original expression is convert into the SOP form by repeated application of DeMorgan’s theorems and multiplication of terms ii. The product terms are then checked for common factors, and factoring is performed wherever possible

6

Continue ...

• Consider the truth table below:

A 0 0 0 0 1 B 0 0 1 1 0 C 0 1 0 1 0 Y 0 0 1 0 0

Minterm Boolean expression: Simplify to yield:

Y = A BC + ABC + AB C

Y = BC ( A + A) + AB C = BC + AB C

1 0 1 1 1 1 0 1 1 1 1 0

•

If minterms are only differed by one bit, they can be simplified, e.g. A BC & ABC

7

Continue ...

• More example:

A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Y 0 1 1 0 0 1 1 0

Minterm Boolean expression:

Y = A B C + A BC + AB C + ABC

Minterms 1 and 5, 2 and 6 are only differ by one bit:

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

A B C Y 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0

Minterm Boolean expression:

Y = A B C + A BC + AB C + ABC

Checking and factoring minterms differed by only by one bit:

Y = A C ( B + B ) + AC ( B + B ) = A C + AC = C ( A + A) =C

8

Continue ...

• • • • Though truth table can help us to detect minterms which are only differed by one bit, it is not arranged in a proper way A Karnaugh map (K-map) is a tool, which help us to detect and simplify minterms graphically It is a rearrangement of the truth table where each adjacent cell is only differed by one bit By looping adjacent minterms, it is similar to grouping the minterms with a single bit difference on the truth table

9

Karnaugh Map

• • A K-map is just a rearrangement of truth table, so that minterms with a single-bit difference can be detected easily Figure below shows 4 possible arrangement of 3-variable K-map A BC

00

0

01

1

11

3

10

2

C

AB

00

0

01

2

11

6

10

4

0 1

4 5 7 6

0 1

1 3 7 5

AB

C

0

0

1

1

BC

A

0

0

1

4

00 01

2 3

00 01

1 5

11

6 7

11

3 7

10

4 5

10

2 6

10

Continue ...

• Figure below show two possible arrangement of 4variable K-map CD AB

00

0

01

1

11

3

10

2

AB 00 CD

01

4

11

12

10

8

00 01

4 5 7 6

00

0

01

1 5 13 9

11

12 13 15 14

11

3 7 15 11

10

8 9 11 10

10

2 6 14 10

•

Notice that the K-map is labeled so that horizontally and vertically adjacent cells differ only by one bit. 11

Continue ...

• The K-map for both SOP and POS form are shown below:

C D C D CD C D AB AB

AB

0 1 3 2

C+D C+ D C + D C +D

A +B

0 1 3 2

4

5

7

6

A+B A+B A +B

4

5

7

6

12

13

15

14

12

13

15

14

AB

8 9 11 10

8

9

11

10

SOP form (minterm)

POS form (maxterm)

•...