WEEK 5

Watch this video before class!

Lesson Learning Outcomes

At the end of this lesson, students will be able to:

Define the minimization of circuits.
Use the Karnaugh Map method in two and three variables.
Apply the Karnaugh Map in minimization of the circuits.

Minimization of Circuits

Once the expression for a logic circuit has been obtained, we may be able to reduce it to a simpler form containing fewer terms or fewer variables in one or more terms. The new expression can then be used to implement a circuit that is equivalent to the original circuit but that contains fewer gates and connections.

Minimization of circuits using Karnaugh Map

Karnaugh maps provides and alternative way of simplifying logic circuits. Instead of using Boolean algebra simplification techniques, we can transfer logic values from a Boolean statement or a truth table into a Karnaugh map. The arrangements of 0's and 1's within the map helps us to visualize the logic relationships between the variables and leads directly to a simplified Boolean statement.

Introduction to Karnaugh Maps (K Map)

Karnaugh map is a technique used to minimize circuits
It is a graphical method to reduce the number of terms used in minimization of circuits of up to 4 variables
However, in this lesson, we will focus on 2 and 3 variable K-Maps

Karnaugh Map for 2 variables

Consider a 2-variable Boolean expression:

Step 1: Count the number of variable n, then 2n , that's the number of your cell on the table.

Step 2: Draw a table of four cells.

Step 3: Label rows and columns with your variables. The complement of the variables are first written followed by the uncomplemented variables.

Step 4: Fill in 1's in the cells representing the terms in the expression.

Step 5: Fill in 0’s in the empty cells.

Step 6: Group the 1's together according to the K-Map grouping rules.

Step 7: Simplify by:

i) Eliminate the 1s that map on A and A' (orange loop).

ii) Eliminate the 1s that map on B and B' (blue loop).

iii) Write down the variable that does not get eliminated, that's the simplest form.

NOTE:

If IN the same loop, then it's PRODUCT.

If NOT in the same loop, then it's SUM.

Karnaugh Map for 3 variables

Consider a 3-variable Boolean expression:

Step 1: Count the number of variable n, then 2n , that's the number of your cell on the table.

Step 2: Draw a table of four cells.

Step 3: Label rows and columns with your variables. The complement of the variables are first written followed by the uncomplemented variables.

Step 4: Fill in 1's in the cells representing the terms in the expression.

Step 5: Fill in 0’s in the empty cells.

Step 6: Group the 1's together according to the K-Map grouping rules.

Step 7: Simplify by:

i) Eliminate the 1s that map on A and A' then C and C'.

ii) Write down the variable that does not get eliminated, that's the simplest form.

NOTE:

If IN the same loop, then it's PRODUCT.

If NOT in the same loop, then it's SUM.

SIMPLIFY TIPS

K Map Grouping Rules

A group must contain only 1's, NO 0's.
A group can only be horizontal or vertical, NOT diagonal.
A group must contain the square of 2 (1, 2, 4, 8 min-terms).
Each group should be as large as possible ("THE BIGGER THE BETTER")
Groups may overlap.
Groups may wrap around a table("KMAP IN 3-D").
Every 1's must be in at least one group.

Quick Exercise.pdf

How to solve a 3-variable expression using K-Map

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