  Digital Teaching Aid (DED Philippinen, 86 p.)  Circuit Analysis and Design - Lesson 3  Lesson Plan Handout No. 2 Worksheet No. 3

(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

Handout No. 2

Boolean Algebra Theorems

 No Theorem Name 1a1b A + B = B + AAB = BA Commutative law 2a2b (A + B) + C = A + (B + C)(AB) C = A (B + C) Associative law 3a3b A (B + C) = AB + ACA + (BC) = (A+ B) (A+ C) Distributive law 4a4b A + A = AAA = A Identity law 5a5b  Negation 6a6b A + AB = AA (A + B) = A Redundancy 7a7b7c7d 0 + A = A1 A = A1 + A = 10A = 0 8A8b  9a9c  10a10b  De Morgan's laws

Worksheet No. 3

No. 1 Simplify the Boolean equation and discribe the logic circuit: No. 2 Simplify the following Boolean expressions: No. 3 A digital system has a 4-bit input from 0000 to 1111. Design a logic circuit that produces a high output whenever the equivalent decimal input is greater than 13.

No. 4 In a heating plant the burner X has to be switched on, when the circulating pump A is actuated and the temperature probe B for the warm water supply or the room temperature probe C respond.

a) Develop the truth table
b) Write down the sum of products equation
c) Draw the logic circuit
d) Use Boolean algebra to simplify the equation
e) Draw the corresponding logic circuit.