CSI-2111 Structure of Computers I page 4-1

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004. 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.

CSI-2111 Structure of Computers I page 4-2

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004.1 Simplification of Switching Functions

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

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

CSI-2111 Structure of Computers I page 4-3

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00 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

CSI-2111 Structure of Computers I page 4-4

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00 Karnaugh Maps (I)

Simplification by algebraic method is DIFFICULT!

Method of simplification graphically suggested: Karnaugh maps

Usable with functions up to 6 variables

CSI-2111 Structure of Computers I page 4-5

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00 Example *

Diagram - 2 variables

f(A, B) = m(0, 1) = A’

m0

m1

m3

m2

BB’

A

A’

1 1

00A

B

CSI-2111 Structure of Computers I page 4-6

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00 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

CSI-2111 Structure of Computers I page 4-7

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00 Example *

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

fPOS =

D

C

A

B

1

0

1

1

0

0

1

0

0

1

0

1

0

0

0

1(A' + B') • (C' + D') • (B' + D)

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

CSI-2111 Structure of Computers I page 4-8

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00 Simplification * Simplify starting from the SOP

form:f (A, B, C, D) = CD’+A’D+ACD

D

C

A

B

0

0

1

1

1

1

1

1

0

0

0

0

1

1

1

1

CSI-2111 Structure of Computers I page 4-9

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00 Simplification * Simplify starting from the SOP form:

f (A, B, C, D) = CD’+A’D+ACD

D

C

A

B

0

0

1

1

1

1

1

1

0

0

0

0

1

1

1

1

= C + A’D

CSI-2111 Structure of Computers I page 4-10

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00 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.

CSI-2111 Structure of Computers I page 4-11

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00 Don’t-Care Values * Simplify f (A, B, C, D)

= m (1, 2, 3, 7, 11, 15) X (0, 5)

D

C

A

B

X

0

1

X

1

1

1

0

0

0

0

0

1

1

0

0

f SOP =A’B’ + CD

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

CSI-2111 Structure of Computers I page 4-12

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00 4.2 Circuits

Combinational:

Sequential:E1En

S 1Sm

combinationalcircuit

memory

Input Variables Output Variables

States

E1En

S 1Sm

combinationalcircuit

input Variables Output Variables

CSI-2111 Structure of Computers I page 4-13

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00 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.

CSI-2111 Structure of Computers I page 4-14

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00 Integrated circuits (II)

Manufacturing Technologies Other characteristics

CSI-2111 Structure of Computers I page 4-15

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00 Complementary readings

In Mano and Kime:– Sections 2.4 and 2.5

Simplification and Karnaugh maps

– Section 2.8 (Optional) Integrated circuits