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# 4. Karnaugh Maps and Circuits - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' 4. Karnaugh Maps and Circuits' - nora

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Presentation Transcript

• 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’

A’

m0

m1

m2

m3

A

1

1

0

0

A

Example *

• 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

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

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

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

• 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.

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!

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

• 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