Gates and Circuits

Monday/Wednesday

Week 7

Electronic Circuits

Two types of circuit diagrams See pp. 238 in The Analytical Engine by Decker

and Hirshfield for the "mechanical switch" type of diagram.

Second style (which we will call "symbolic diagrams") uses pictures for the boolean logic operators. These pictures are called gates.

Three Main Gates

AND OR NOT

Gate Diagrams

Example 1: (MR) + S

Gate Diagrams

Example: What does it represent?

Gate Diagrams

Example: ((MR) + S) (RS)’

Truth Table to Gates

First, build the Boolean algebra expression that gives Z Z = AB + A’B’ Z = (A AND B) OR

(NOT A AND NOT B)

A B Z

T T T

T F F

F T F

F F T

Truth Table to Gates

Z = AB + A’B’ Next, build the

circuit that goes with the Boolean algebra expression Z

A B Z

T T T

T F F

F T F

F F T

Z = AB + A’B’

Binary Arithmetic

We can add binary numbers just like decimal numbers only using base two arithmetic.

For example:

5 101

1110

101

+ 7 + 111 + 111

12 1100 1100

Binary Addition

Notice in addition:

0 0 1 1

+ 0 + 1 + 0 + 1

0 1 1 10

False True True False

False False False True

Sum

Carry

A B Sum

(1) T (1) T (0) F

(1) T (0) F (1) T

(0) F (1) T (1) T

(0) F (0) F (0) F

Sum and Carry

A B Carry

1 1 1

1 0 0

0 1 0

0 0 0

A B Sum

1 1 0

1 0 1

0 1 1

0 0 0

Sum Circuit

A B Sum

1 1 0

1 0 1

0 1 1

0 0 0

Sum = AB’ + A’B

Carry Circuit

A B Carry

1 1 1

1 0 0

0 1 0

0 0 0

Carry = AB

Half Adder - Sum and Carry

Half Adder

Handles the case where we add two binary digits with no inward carry.

Full Adder

Takes a carry in and produces the result and carry out.

So, we have 3 inputs and two outputs. Combine two half-adders together with an or

gate to get a full adder for each binary digit. How many half adders would we need to add

two 8-digit binary numbers? How many gates?

Full Adder

Binary Subtraction

We do binary subtraction like decimal subtraction only the borrowing is done in 2’s instead of 10’s.

122 1111010

- 7 - 111

115 1110011

Subtraction

A B Sub

1 1 0

1 0 1

0 1 1

0 0 0

A B Borrow

1 1 0

1 0 0

0 1 1

0 0 0

Binary Multiplication

Again, just like decimal except we add and multiply in binary.

* 0 1

0 0 0

1 0 1

5 101

x 7 x 111

35 100011

Exercises

1 - From the book, p 266, Construct circuits with the following properties (using AND, OR or NOT gates):

Exercises

A) B)A B Z

1 1 0

1 0 1

0 1 1

0 0 1

A B C Z

1 1 1 1

1 1 0 0

1 0 1 1

1 0 0 0

0 1 1 0

0 1 0 1

0 0 1 0

0 0 0 1

NAND Gates and NOT

This gate represents (A NAND NOT B).

NAND Truth Table

A B A NAND B

T T F

T F T

F T T

F F T

Exercises

2 - Fill in a truth table and give a Boolean expression for the following circuits.

Exercises

3 - How would you create a one binary digit multiplier? A two-digit by one-digit multiplier? A two-digit by two-digit multiplier? * 0 1

0 0 0

1 0 1