  Digital Teaching Aid (DED Philippinen, 86 p.)  Circuit Analysis and Design - Lesson 3  Lesson Plan  (introduction...) Introduction Boolean laws and theorems Sum of product method Design example

### (introduction...)

Titel: Circuit Analysis and Design

Objectives:

- Know how to apply the basic Boolean laws
- Able to create digital circuits with the help of the sum of products method

 Time Method Topic Way Remark * Review Lesson 2 * Introduction * Boolean laws and theorems HO Handout No. 2 (Boolean Algebra Theorems) - Basic laws - Boolean relation about OR operations - Boolean relations about AND operations * Sum of products method - Fundamental products - Sum of products equations * Review Exercise WS Worksheet No. 3 S: SpeechD: DiscussionQA: Question/AnswerE: Exercise B: BoardscriptP: PictureEx: ExampleHO: Hands-OnWS: WorksheetHT: Hand-Out

### Introduction

Circuit Analysis and Design

### Boolean laws and theorems

Basic laws

Commutative law:

A + B = B + A

A B = B A Fig. 3-1: Commutative law

Associative law:

A + (B + C) = (A + B) + C

A (B + C) = (A B) C Fig. 3-2: Associative law

Distributive law:

A (B + C) = AB + AC Fig. 3-3: Distributive law

A + 0 = A

Proof:

when A is 0

0 + 0 = 0

when A is 1

1 + 0 = 1

A + A = A

Proof:

when A is 0

0 + 0 = 0

when A is 1

1 + 1 = 1

A + 1 = 1

Proof:

when A is 0

0 + 1 = 1

when A is 1

1 + 1 = 1

A + A = 1

If one input is high, the output is high no matter what the other input is.

A · 1 = A
A · A = A
A · 0 = 0

HO: Check the equations above in the same way as we did it before.

A · A = 0

If one input is low, the output is low no matter what the other input is.

Double inversion De Morgan's theorems see also Lesson 1 Duality theorem

1. Change each OR sign to an AND sign

2. Change ech AND sign to an OR sign

3. Complement any 0 or 1 appearing in the expression

 Ex: A + 0 = A ® A 1 = A Ex: A (B + C) = AB + AC ® A + B C = (A + B) (A + C)

TIP: Proof it with a truth table

Ex: Simplify the following Boolean equation  Handout 2 No. 3a X = A (1) No. 8a X = A No. 7b

HO: Simplify the following Boolean equation Solution:   X = B + B
X = B

### Sum of product method

Fig. 3-4: Example truth table with fundamental products

 A B X Fundamental Products 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ### Design example

Sum of products equation

Given is the following truth table:

Fig. 3-5: Example truth table

 A B C X 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 ® 1 0 0 0 1 0 1 1 ® 1 1 0 1 ® 1 1 1 1 ® We have to locate each output 1 in the truth table and write down the fundamental product.

The next step is to OR the fundamental products: Now we can derive the corresponding logic circuit: Fig. 3-6: Logic circuit

HO: What is the sum of product circuit for the given truth table?

Fig. 3-7: Truth table

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