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4. Karnaugh Maps and Circuits

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- 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.

- Why simplify and optimize?
- Constraints
- Cost ($$$)!

- How?
- Algebraic method (still…)
- Karnaugh maps (wow!)

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

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

B

B’

A’

m0

m1

m2

m3

A

1

1

0

0

A

- Diagram - 2 variables
- f(A, B) = m(0, 1) = A’

B

- 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

1

1

0

1

0

1

0

0

0

0

0

0

1

1

0

1

- f (A, B, C, D) = m (0,1,2,5,8,9,10)
- f SOP=
- fPOS =

C

B'D' + B'C' + A'C'D

B

A

(A' + B') • (C' + D') • (B' + D)

D

C

0

1

1

1

0

1

1

1

B

0

0

1

1

A

0

0

1

1

D

- 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

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

= C + A’D

- 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

- 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

- Combinational:
- Sequential:

E

1

Output Variables

input Variables

combinational

E

n

circuit

E

1

Output Variables

Input Variables

combinational

E

n

circuit

States

memory

- 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.

- Manufacturing Technologies
- Other characteristics

- In Mano and Kime:
- Sections 2.4 and 2.5
- Simplification and Karnaugh maps

- Section 2.8 (Optional)
- Integrated circuits

- Sections 2.4 and 2.5