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

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