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close this bookDigital Teaching Aid (DED Philippinen, 86 p.)
close this folderCircuit Analysis and Design - Lesson 3
close this folderLesson Plan
View the document(introduction...)
View the documentIntroduction
View the documentBoolean laws and theorems
View the documentSum of product method
View the documentDesign 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: Speech
D: Discussion
QA: Question/Answer
E: Exercise


B: Boardscript
P: Picture
Ex: Example
HO: Hands-On
WS: Worksheet
HT: Hand-Out


Introduction

Circuit Analysis and Design

Boolean laws and theorems

Basic laws

(see also Handout No. 2)

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

Boolean relations about OR operations

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.

Boolean relations about AND operations

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

Solution: (see also Handout No. 2)

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