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4. Karnaugh Maps and Circuits

- Objective: To know how to simplify switching functions by Karnaugh maps,
- To understand what are the combinative and sequential circuits,
- To know the characteristics of the integrated circuits.

4.1 Simplification of Switching Functions

- Why simplify and optimize?
- Constraints
- Cost ($$$)!
- How?
- Algebraic method (still…)
- Karnaugh maps (wow!)

Algebraic Handling *

Canonical form:

L = A’B’C’+A’BC’+AB’C’+AB’C+ABC’

9 NOT (* 1) + 5 AND (* 3) + 1 OR (* 5) = 29

Simplified Form:

L = AB’ + C’

2 NOT (* 1) + 1 AND (* 2) + 1 OR (* 2) = 6

Karnaugh Maps (I)

- Simplification by algebraic method is DIFFICULT!
- Method of simplification graphically suggested: Karnaugh maps
- Usable with functions up to 6 variables

Karnaugh Maps (II)

- Can be conceived from:
- Truth tables
- Canonical CSOP or SOP form
- Canonical CPOS or POS form
- Can give result like:
- Minimal Sum of Products (SOP) form
- Minimal Products of Sums (POS) form

C

0

1

1

1

0

1

1

1

B

0

0

1

1

A

0

0

1

1

D

Simplification *- Simplify starting from the SOP form:f (A, B, C, D) = CD’+A’D+ACD

C

0

1

1

1

0

1

1

1

B

0

0

1

1

A

0

0

1

1

D

Simplification *- Simplify starting from the SOP form:f (A, B, C, D) = CD’+A’D+ACD

= C + A’D

Karnaugh Maps (III)

- Don’t-Care values (X)
- Certain switching functions are known as incompletely defined: certain combinations of their variables of inputs are never supposed to occur or not to have an effect on the result. One calls these combinations don’t-care values and one indicates them as ' X' in the truth tables.
- In the Karnaugh maps, one considers them like 1 (SOP) or of the 0 (POS) only to make larger groupings, but it is not necessary to gather them.

C

X

1

1

1

0

X

1

0

B

0

0

1

0

A

0

0

1

0

D

Don’t-Care Values *- Simplify f (A, B, C, D) = m (1, 2, 3, 7, 11, 15) X (0, 5)

- f SOP =A’B’ + CD

- The minterm 5 should not be included; it would not be minimal!

S

1

S

m

S

1

S

m

4.2 Circuits- Combinational:
- Sequential:

E

1

Output Variables

input Variables

combinational

E

n

circuit

E

1

Output Variables

Input Variables

combinational

E

n

circuit

States

memory

Integrated Circuits (I)

- The integrated circuits, material manufacture of logic gates and more complex functions, are characterized in several ways.
- Why they used are?
- Level of integretion? Quantity of transistors in a circuit.

Integrated circuits (II)

- Manufacturing Technologies
- Other characteristics

Complementary readings

- In Mano and Kime:
- Sections 2.4 and 2.5
- Simplification and Karnaugh maps
- Section 2.8 (Optional)
- Integrated circuits

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