exam for Electronic Engineering

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EE 1301: MODERN ELECTRONIC TECHNOLOGY

SESSION #22: BINARY MATH AND YOUR COMPUTER 3/19/2018

Instructor: Joseph Cleveland, Ph.D. Email: [email protected]

Thought for the Day

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A large portion of the world’s working population are poorly

educated but work as entrepreneurs.

With all of this education could we do that?

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Topics & Comments

• Binary math (cont) – Addition

– Subtraction

• Logic gates (collection of transistors)

• Binary adder (collection of gates)

• Lab: use this week for make ups

• Exam #2: Friday, April 6

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Binary Addition

• Let’s work the following binary addition:

1 1 0 0 1 2510 + 1 1 0 1 1310

3810

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0

1 carry

1

0

1

0

0

1

0

1

1

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Subtraction

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4 7 - 0 6

4 1 1

- 0 0 0 1 1 0

0011

1 0 1 1 1 1

0

4 6 - 9 3 7

4 6 - 9

163

How to we perform borrowing in binary math?

Borrow 10

3 7

Subtraction …

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1 0 1 1 1 0

0 0 1 0 0 1

4 6

-9

3 7

= 1 + 1

1

1 0 1 1 0 0

1 0 0 1 0 0 1

Borrowing

01001 = 3710

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Subtraction …

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1 0 1 1 1 0

0 0 1 0 1 1

4 6

-1 1

3 5

= 1 + 1

1

1 0 1 1 0 0

= 1 + 1

0 0 1 0 1 1 1

Borrowing

1

0

Notice that you have a lot to keep track of!

2’s Complement Subtraction

• There is an easier way: 2s complement

• Example: Using the 2’s Complement method in the following example:

1 0 1 1 1 0 4 6 – 1 0 0 1 -1 1

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2’s Complement Subtraction

• Step 1: Make both numbers have the same number of digits

1 0 1 1 1 0 46 – 0 0 1 0 1 1 -11

• Step 2: Take 2’s complement of the number to be subtracted:

001011  110100 + 1 = 110101.

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Complement of 001011

Add 1

2’s Complement Subtraction …

• Step 3: Add the result

1 0 1 1 1 0 46 + 1 1 0 1 0 1 -11

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01111 0

1100011

· · · · · ·

Note: Final carry

If the final carry is “1”, the answer is +. If it is “0” the answer is “-”.

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Binary Subtraction …

• We need to have – Registers

– Compliment operation

1  NOT 1 or  0 0  NOT 0 or  1

– Adder with carry

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Binary Multiplication

• Recall decimal number multiplication

4 5 6

7 8 9

4 1 0 4

3 6 4 8

3 1 9 2

3 5 9 7 8 4

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55 Carry4 444 Carry 553 Carry

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Binary Multiplication

• Binary multiplication follows the same format

1 0 1 1 1110 x 1 0 1 510

1 0 1 1

0 0 0 0

1 0 1 1

1 1 0 1 1 1 5510

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Shift left Shift left

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Logic Gates

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• Digital computers contain a large number of circuit blocks called logic gates – NOT

– AND

– NAND

– OR

– NOR

– XOR

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Compliment or NOT Function

• The complement of X is denoted as which is defined by a truth-table:

• A switch circuit that performs a complement

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X

0 1

1 0



R

D

G

S

5 V

ID

Vin

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Complement or “NOT” …

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X



R

D

G

S

5 V

5V A NOT gate

Notice that the electronic circuit is hidden.

Only the input/output is shown – standardized.

This is synthesis.

We use the NOT gate in 2’s complement.

NAND Gate

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Both transistors must be ON for current to flow in R3: to drop VOUT to LOW (~0.5V)

+5V

VOUT X

Y

X Y VOUT 0 0 1

0 1 1

1 0 1

1 1 0

X

Y

NAND Logic Gate Symbol

VOUT

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AND Logic Gate

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X

Y

VOUT

5V

NAND Gate NOT Gate

AND Logic Gate …

The AND of X and Y is denoted as XY and has the following truth-table:

Can be viewed as “multiplication”.

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X Y XY

0 0 0

0 1 0

1 0 0

1 1 1

X

Y XY

AND Logic Gate Symbol

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OR Logic

The OR of X and Y is denoted as X+Y and has the following truth-table:

Can be viewed as “addition”.

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X Y X+Y

0 0 0

0 1 1

1 0 1

1 1 1

X

Y X+Y

OR Logic Gate Symbol

NOR Logic

The NOR of X and Y is denoted as and has the following truth-table:

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X Y X+Y

0 0 1

0 1 0

1 0 0

1 1 0

X Y

NOR Logic Gate Symbol

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The XOR or Adder

The is denoted by XY and has the following truth-table:

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X Y XY

0 0 0

0 1 1

1 0 1

1 1 0

This looks like binary addition with no carry – a partial sum

X

Y XY

XOR Logic Gate Symbol

FET circuit combination is “hidden”. Only the input-output is shown. This is Synthesis

The Carry

• Truth table for the Carry

• This is the same as an AND gate

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X Y C

0 0 0

0 1 0

1 0 0

1 1 1

X

Y C=XY

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The Adder

• Let’s combine AND, OR, XOR gates to make an adder with carry

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XY (XY)C

(XY)C

Cin XY

S

XY

Cout

Y X

S

Cin Cout FA

Transistor Count XOR: 6 AND: 6 OR: 6

~30 MOSFETs

The Adder

• Adding two, 4-bit numbers

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Y0 X0

S0

C0

Y1 X1

S1

C1

Y2 X2

S2

C2

Y3 X3

S3

C3

C4

FA FA FA FA

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How The Adder works

• Check it out

• Adder truth table: 8 X,Y,C input possibilities

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1 1 0 0 1

0 1 1 0 1

X Y C S C

0 0 0 0 0

0 0 1 1 0

1 0 0 1 0

1 0 1 0 1

X Y C S C

0 1 0 1 0

0 1 1 0 1

1 1 0 0 1

1 1 1 1 1

0

11=0 00

S=0

0 1

011=1

0 C=1

0

End of Session

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