Simplifying logic using Karnaugh maps
Equipment: Pen and paper!
PC with “logisim” installed
Introduction
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The purpose of this laboratory is to solidify the concepts of combinational logic design presented in the lectures. The learning outcomes of this laboratory session will be the ability to take a combinational logic design specification and transform it into a truth table, and transform the truth table into a “sum of products” and a “product of sums” expression. You will gain experience in simplifying the logic output expression using Karnaugh maps. You will also gain some familiarity with digital circuit simulation using the Logisim digital circuit simulator.
You will also build the circuit using discrete logic gates on a breadboard in the Lab and verify its operation.
Logisim familiarisation: Exclusive OR gate
You will have learnt in you lectures that the following identity holds:
To help you become familiar with the logisim software you are required to build and test the following circuit to verify the operation of the above relationship.
Use this circuit to verify the following truth table.
Your Numbers:
UCD Student Number: 1 9 5 3 1 2 8 1
A 4-bit Number Detector
You are required to design a simple logic circuit with takes a 4 bit input and a 1 bit output. The circuit will take in a 4 bit number, and output a logic-high when presented with an input number that corresponds to:
Your student number digits (2,4,6,8), eg. to detect the values 9,3,2, 1.
You may use any standard logic gates (AND, NAND, OR, NOR, NOT). The circuit should accept a 4-bit binary input () and a 1-bit binary output.
In addition, you are given instructions below detailing the information that you will report.
Design Procedure
The following steps indicate the design procedure you should following in order to construct the number detector. If you have any difficulty, you should ask a lab demonstrator for help.
Write out the truth table for the combinational logic circuit which will detect when the
input is a selected number from your student number . The circuit should output a logic 1 when the input is a is a selected number from your student number, and output a logic 0 otherwise.
Write the “sum of product” expression.
Write the “product of sums” expression.
Use the method of Karnaugh maps to simplify the “sum of product” SOP expression. (a) Convert to NOR/NOR and NAND/NAND logic
Sketch the circuit diagram for this.
Using the Karnaugh map, write down the simplified “product of sums” POS.
(a) Convert to NOR/NOR and NAND/NAND logic Sketch the circuit diagram for this.
Determine which of the 2 possible circuits uses the fewest transistors using the following rules:
An n-input OR and AND gates require 2n+2 transistors. An n-input NOR and NAND gates require 2n transistors.
An inverter uses 2 transistors.
Simulate the circuit which uses fewer resources (i.e. transistors) in Logisim. Ensure your design is working as required.
• After you complete the Lab, a short report will be written with the following:
• Screenshot of XOR Gate implementation.
• Show truth table for decoder design
• Number detector Designs with K-map – POS and SOP.
• Circuit Designs: OR/AND, AND/OR, NOR/NOR, NAND/NAND designs + transistor count for each design – (can be hand written if
image is very clear).
• Screenshot of chosen design implementation in simulation.
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