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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?

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4. Karnaugh Maps and Circuits

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4 karnaugh maps and circuits

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

4.1 Simplification of Switching Functions

  • Why simplify and optimize?

    • Constraints

    • Cost ($$$)!

  • How?

    • Algebraic method (still…)

    • Karnaugh maps (wow!)


Algebraic handling

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

Karnaugh Maps (I)

  • Simplification by algebraic method is DIFFICULT!

  • Method of simplification graphically suggested: Karnaugh maps

  • Usable with functions up to 6 variables


Example

B

B’

A’

m0

m1

m2

m3

A

1

1

0

0

A

Example *

  • Diagram - 2 variables

  • f(A, B) = m(0, 1) = A’

B


Karnaugh maps ii

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


Example1

1

1

0

1

0

1

0

0

0

0

0

0

1

1

0

1

Example *

  • 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


Simplification

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


Simplification1

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

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.


Don t care values

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!


4 2 circuits

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

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

Integrated circuits (II)

  • Manufacturing Technologies

  • Other characteristics


Complementary readings

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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