Digital Logic

Section 1: Intro, Notation and basic operations

Boolean Algebra is very similar to regular algebra with a couple key differences. The only possible outputs are 1 or 0. 1+1+1… = 1.

As with regular algebra 1+0=1

Addition is equivalent to a Or Gate. The mathematic representation of a OR Gate, with Inputs A,B and Output C is.

Multiplication is equivalent to Multiplication. The mathematic representation of a AND Gate, with Inputs A,B and Output C is.

Multiple Gates can be added and multiplied. Two And Gates with Inputs (A,B), (C,D) and Output E is.

A NOT Gate is another fundamental component. The simplest explanation is it inverts whatever it is give 1=0 and 0=1. It will give the complement of whatever it is given. The representation of NOT A with Boolean Algebra is.

It is distributive

This can be built into equations. A NOR Gate is on as long as Both inputs are 0.

A NAND Gate is similar.

Demorgan’s theorem is a way to convert between AND and OR Gates. This will be a useful feature later.

Converting a NAND Gate to a OR gate is as follows. A NAND Gate is equivalent to Two NOT Inputs and a OR Gate.

Converting a NOR Gate to a AND Gate is as follows. A NOR Gate is equivalent to Two NOT Inputs and a AND Gate.

Note: From this point further, i will be dropping the *, it would otherwise make the larger equations unnecessarily cumbersome.

XOR Gates

Section 2: Truth Tables

Truth tables provide every input and output, there are various useful things we can do with this, including create equations from truth tables.

A simple example for a AND Gate

A	B	C
0	0	0
0	1	0
1	0	0
1	1	1

This takes every possible input, and gives every possible output. Every Equation has a truth table and this can be used to convert equations and circuits into different gates. This is done occasionally for power efficiency, some types of gates are more efficient than others.

Section #: Minterm and Maxterms

You can derive various equations from a truth table. These equations are as a default in the MinTerm or Maxterm form. I will explain that in the next section

Section #: SR Latch

Section #: Karnaugh Maps

Karnaugh Maps are a type of Truth Table, which can be used to relatively easily simplify Equations into fewer Gates. In the given map, you draw various squares and rectangles in order to create various equations. With Karnaugh Maps, there are various rules and conventions for creating resulting equations/circuits.

Rules

• All selected outputs must be squares and rectangles.
• There must be a 2ⁿ number of ones, 2, 4, 8 are all valid while 3, 5, 6 and 7 are never valid.
• All cells must be horizontally or vertically, but not horizontally adjacent.
• There should be the fewest number of equations.

SAMPLE EQUATION

Truth Table

A	B	C	D	F

Karnaugh Map

AB/CD	00	01	11	10
00
01
11
10

Section #: Finite State Machines

Or Gates

And Gates

D Latches

Multiplexers

Section 3: Circuit Design

Truth Tables

Time Graphs

Minterms and max terms

SOP AND POS

Karnaugh Maps

Don’t care

Half Adders

Full adders

Half Subtractor

Full Subtractor

Finite State Machines