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CMPEN 270 Laboratory 4

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CMPEN 270 Laboratory 4

In this lab you will learn:
1. Given a truth table, how to make a K-Map
2. Given a K-Map how to solve and build a circuit
Karnaugh Map
The Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice
Karnaugh introduced it in 1953. The Karnaugh map reduces the need for extensive calculations by taking
advantage of humans’ pattern-recognition capability. It also permits the rapid identification and
elimination of potential race conditions. The required Boolean results are transferred from a truth table
onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code, and each cell
position represents one combination of input conditions, while each cell value represents the
corresponding output value. Optimal groups of 1s or 0s are identified, which represent the terms of a
canonical form of the logic in the original truth table. These terms can be used to write a minimal Boolean
expression representing the required logic.
Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using
a minimum number of physical logic gates. A sum-of-products expression can always be implemented
using AND gates feeding into an OR gate, and a product-of-sums expression leads to OR gates feeding an
AND gate. Karnaugh maps can also be used to simplify logic expressions in software design. Boolean
conditions, as used for example in conditional statements, can get very complicated, which makes the
code difficult to read and to maintain. Once minimized, canonical sum-of-products and product-of-sums
expressions can be implemented directly using AND and OR logic operators. Diagrammatic and
mechanical methods for minimizing simple logic expressions have existed since at least the medieval
times. More systematic methods for minimizing complex expressions began to be developed in the early
1950s, but until the mid to late 1980’s the Karnaugh map was the most common used in practice.
Activities
Activity 1: Making a K-Map from Truth Table
From your lectures, now you already know what Min-terms and Max-terms are. Here, we will use the
same techniques to make a K-Map from a given truth table with Min-terms. We will start with a small 3
variable table and make our way through to more complex problems.
Min-terms A B C Out (F)
M0 0 0 0 0
M1 0 0 1 0
M2 0 1 0 1
M3 0 1 1 0
M4 1 0 0 1
M5 1 0 1 1
M6 1 1 0 1
M7 1 1 1 1
Given the above table, construct the K-Map for the same and simplify this as minimal SOP. If you
are stuck somewhere, feel free to ask the instructors for directions.
CMPEN 270 Laboratory 4 2020
2
Activity 2: Validating your K-Map
Once you are done with the previous activity, using only NOR, NOT, and NAND gates and build the
equivalent circuit on bread board. Give all valid combinations as input and validate the output against
the truth table.
Fill in the following table with your observations from the circuit.
A B C F (expect) F (actual)
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
Activity 3: Solving a 4 variable K-Map
Using only NOR and NOT gates, build a circuit for the following function using the minimal SOP expression:
given a function f (a1, a2, a3, a4):
a1a2\a3a4 00 01 11 10
00 0 1 1 0
01 1 0 0 1
11 1 0 0 1
10 0 1 1 0
Finish the truth table below:
a1 a2 a3 a4 f a1 a2 a3 a4 f
0 0 0 0 1 0 0 0
0 0 0 1 1 0 0 1
0 0 1 0 1 0 1 0
0 0 1 1 1 0 1 1
0 1 0 0 1 1 0 0
0 1 0 1 1 1 0 1
0 1 1 0 1 1 1 0
0 1 1 1 1 1 1 1
0
0
1
0
1
1
1
1
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
CMPEN 270 Laboratory 4 2020
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Submission Requirements
Activity 1 – One filled out k-map
Activity 2 – Screenshot of the circuit and filled out truth table
Activity 3 – Screenshot of the circuit and filled out truth table
Make sure to attach all of these things to 1 file and submit that 1 file with everything on it.

 

 

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