‫طراحی مدارهای منطقی‬
‫دانشگاه آزاد اسالمی واحد پرند‬
‫نیمسال دوم ‪93-92‬‬
‫طراحی مدارهای منطقی‬
‫دانشگاه آزاد اسالمی واحد پرند‬
‫بهینه سازی مدارهای منطقی‪:‬‬
‫جدول کارنو‬
Optimization
 Algebraic procedures problems
 The procedures are difficult to apply in a
systematic way
 It is difficult to tell when you have arrived at a
minimum solution
 Karnaugh map
(For 3 and 4 variables)
 Quine-McCluskey
Optimization
 Cost directly related to 
The number of gates
&
Gate inputs used
Optimization
 Karnaugh 
 Minimum cost two-level circuits composed of AND and OR
gates
 SOP: a group of AND gates feeding a single OR gate
 POS:
a group of OR gates feeding a single AND gate
 Minimum SOP/POS 
• has minimum number of terms (Minimum number of gates)
• has minimum number of literals (minimum number of gate inputs)
Karnaugh Map
Karnaugh map of a function
 Specifies the value of the function for every
combination of values of the independent variables
• Like Truth table
 Terms near each other with one bit difference
• Maximum terms in one group (2,4,8,… terms)
• Minimum groups
Karnaugh Map (2-variable)
Karnaugh Map (3-variable)
Karnaugh Map (3-variable)
Karnaugh Map (3-variable)
Karnaugh Map (3-variable)
 Proving boolean algebra equations  Consensus theorem
Karnaugh Map (3-variable)
 Minimized SOP  Not unique
Karnaugh Map (4-variable)
Karnaugh Map (4-variable)
Karnaugh Map (4-variable)
 With Don’t cares
 Group X’s if they simplify the function
Karnaugh Map (4-variable)
 POS extraction
Karnaugh Map (EPI)
 Using PIs and EPIs
 Implicant
 Prime Implicant (PI)
 Essential Prime Implicant (EPI)
Karnaugh Map (EPI)
 In the process of finding PIs:
 don’t-cares are treated just like 1’s
Note  PI composed entirely of X’s can never be part
of the minimum solution
Karnaugh Map (EPI)
Karnaugh Map (EPI)
Karnaugh Map (5-variable)