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Lecture 16 Overview

Logic gatesBoolean AlgebraDeMorgan's TheoremKarnaugh maps

Simple gate summary

INPUTOUTPUTABA AND B000100010111

INPUTOUTPUTABA NAND B001101011110

INPUTOUTPUTABA OR B000101011111

INPUTOUTPUTABA NOR B001100010110

The XOR gateCC=A XOR BC=ABThe XNOR gateCC=A NOR BC=AB

ABC(in)(in)(out)000011101110

ABC(in)(in)(out)001010100111

How to build a digital gate We can build an inverter with one switch: A NAND gate takes two switches in series:INPUT=0INPUT=1INPUT AINPUT B

CMOS gates Gates are very easy to build using MOSFET transistors (recall; transistors can be considered as a voltage controlled switch) p-type conduct when the input=0 n-type conduct when the input=1

CMOS NAND gate NAND gates are built using 4 MOSFETs p-type conduct when the input=0 n-type conduct when the input=1

INPUTOUTPUTABA NAND B001101011110

CMOS NAND gate The NAND gate is by far the most important It is cheapest to construct It can be used to produce all other logic operations

CMOS NAND gate The NAND gate is by far the most important It is cheapest to construct It can be used to produce all other logic operations

XOR

In general, how do we figure out how to build a complex logic circuit?

Method I: Boolean AlgebraAND:OR:NOT:Associative Law:Commutative Law:Distributive Law:Precedence:DeMorgan's Theorem:

Complete Rules of Boolean AlgebraAND:OR:NOT:Associative Law:Commutative Law:Distributive Law:Precedence:DeMorgan's Theorem:

DeMorgan's Theorem Proof"The contradictory opposite of a disjunctive proposition is a conjunctive proposition composed of the contradictories of the parts of the disjunctive proposition (William of Ockham, Summa Logicae)." Duality between AND and OR means that any logic function can be implemented by using just OR and NOT gates , or by just AND and NOT gatesOR"break the line, change the sign"

Using the Rules of Boolean AlgebraExample: Simplify the following function:=1expand this14 gates3 gates

Method II: Karnaugh Maps There are often many solutions available to implement a given logic expression How do we find the most efficient (least number of gates)? Use a Karnaugh map. Set up the Karnaugh map like this:

Two inputs; A and BThree inputs; A,B and CThree inputs; A, B and C4 cells8 cells8 cells Each map consists of 2n cells, where n is the number of inputs (logic variables) Row and column assignments arranged such that adjacent terms change by only one bit so: use 00,01,11,10 instead of 00,01,10,11 Makes it easier to identify subcubes

Karnaugh Maps: Setting up the MapsFour inputs; A,B,C and D16 cells

Karnaugh Maps: Setting up the Maps Here's an example: Truth Table:Karnaugh Map The Karnaugh map "wraps around itself" - i.e. the top and bottom, right and left edges are touching. Adjacent cells contain terms which vary by only one input variable.A subcube is defined as a set of 2m adjacent cells with the same value. m is an integer, so the subcube can be 1,2,4,8... cells This is how we want the circuit to behave.

Karnaugh Maps: Example 1 "Box the ones" This subcube is represented by A'BThis subcube is represented by AC' So, the output is true if (A'ANDB) OR (AANDC') are true "sum of products": A'B + AC' This requires 5 gates: note clean schematic layout The method only works if ALL of the 1's are considered The minimal expression uses the smallest number of maximal subcubes

Karnaugh Maps: Example 2:ABCD Draw the table Find any isolated cells Find any 2-cell subcubes which are not adjacent to other 2-cell subcubes. Remember wrappingABCD

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

00011110001101010100110110101110

Karnaugh Maps: Example 2:ABCD Draw the table Find any isolated cells Find 2-cell subcubes. Find 4-cell subcubes which are not adjacent to other four cell subcubesFind 8-cell subcubes etc etc. Minimal expression is formed by the smallest number of maximal subcubes

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

Karnaugh Maps: Example 2:ABCD A'B'D'

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

Karnaugh Maps: Example 2: B'C'ABCD

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

Karnaugh Maps: Example 2: C'DABCD

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

Karnaugh Maps: Example 2: ADABCD

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

Karnaugh Maps: Example 2: So sum-of products realization is: O=A'B'D'+B'C'+C'D+AD Requires 9 gatesABCD

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111

00011110001101010100110110101110

Karnaugh Maps: Example 2: So sum-of products realization is: O=A'B'D'+B'C'+C'D+AD Requires 9 gatesA'B'D'B'C'C'DAD

ABCDO00001000110010100110010000101101100011101000110011101001011111000110111110011111