4. Karnaugh Maps and Circuits

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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.

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4.1 Simplification of Switching Functions

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

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Algebraic Handling

• Canonical form
• L ABCABCABCABCABC
• 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

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Karnaugh Maps (I)

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

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Example

• Diagram - 2 variables
• f(A, B) ?m(0, 1) A

B
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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

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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
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Simplification

• Simplify starting from the SOP formf (A, B, C,
  D) CDADACD

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Simplification

• Simplify starting from the SOP formf (A, B, C,
  D) CDADACD

C AD
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Karnaugh Maps (III)

• Dont-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 dont-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.

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Dont-Care Values

• Simplify f (A, B, C, D) ?m (1, 2, 3, 7, 11,
  15) X (0, 5)

• f SOP AB CD

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

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4.2 Circuits

• Combinational
• Sequential

E
1
Output Variables
input Variables
combinational
E
n
circuit
E
1
Input Variables
Output Variables
combinational
E
n
circuit
States
memory
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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.

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Integrated circuits (II)

• Manufacturing Technologies
• Other characteristics

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Complementary readings

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