14. DIGITAL CIRCUITS

8/7/2019 14. DIGITAL CIRCUITS

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COMPUTER

PLAY WORKORGANIZE THEIR LIVES


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DIGITAL CIRCUITS are the

brains of the technologicalworld


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PC have subdivided the circuit

board into four areas


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FOUR AREAS

1. BASIC LOGIC AREA. This consists of the

basic gates (glue chips).

2. MEMORY AREA. This consists of bothRAMs and ROMs,

3. MICROPROCESSOR. This is the heart of

the PC and contains several millions oftransistors and be able to execute over 100

millions instruction/second.


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4. PERIPHERAL ICs. These ICs

primarily involved withInput/Output (I/O)


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How this digital

circuits works in a

computer?

To understand the operation

and application digital

electronics mastered theMATHEMATICS AND

LOGIC DEVICES, then

BASIC GATES


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COMPUTERS are designed to handle numbers, not

letters

Use a more efficient code called theBINARY NUMBERS

To understand the basic operation ofcomputer it is a prerequisite to knowthe conversion of each number system


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Binary system- uses only two

distinctive numbers: LOGIC 1 ------------ON

LOGIC 0 -----------OFF

An action of a switch0 and 1 called BIT

Bit Quantities:

4 bits = One nibble or hexadecimal digit

8 bits = Two nibble or one byte

16 bits = Two bytes or one word

32 bits = Two words or one longword

(Motorolas Term) one Double word (Intels term)


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BINARY TO DECIMAL

CONVERSION

To find the equivalent decimal

numbers, we will use thepolynomials expansion:

N = +(B2R2) + (B1R

1) + (B0R0)


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CONVERSION OF BINARY

FRACTIONS TO DECIMALSN = (B-1R

-1) + (B-2R-2) +

MIXED NUMBERS

can be converted from binary to

decimal by working on the integersand fraction portion separately.


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EXAMPLES:1. 101112

2. 1101012

3. 1100010112

4. 11011.101112

5. 0.000100112


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DECIMAL TO BINARY

CONVERSION1. Obtain the N ( the decimal number to be

converted_

2. Determine if N is odd or even.3. (a) If N is odd, write 1 and subtract 1 from N.

Go to step 4. (b) If N is even, write 0. Go to step4.

4. Obtain a new value of N by dividing the N ofstep 3 by 2.

5. (a) If N >1, go back to step 2 and repeat theprocedure. (b) If N = 1 , write 1.


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NOTE: The number written is thebinary equivalent of the original

decimal number. The number

written first is the least

significant bit (LSB), and the

number written last is the mostsignificant bit (MSB)


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EXAMPLES:1. 1010

2. 2510

3. 25010

4. 7710

5. 8910


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CONVERTING DECIMAL

FRACTION TO BINARY

FRACTIONS1. Obtain N.

2. Double N.3. (a) If the new value of N is greater than 1,

write 1 as the next most significant bit,subtract 1 from N, and go back to step 2.(b) If the new value of N is less than 1,write 0 as the next most significant bit andgo back to step 2.


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MIXED NUMBERSMixed numbers can be converted from

decimal and fraction portions separately.


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EXAMPLES:1. 0.510

2. 0.2510

3. 3.72510

4. 0.69310

5. 0.35610


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TWO ALTERNATIVES

SYSTEMS USEDF

ORREPRESENTING BINARY:

1. OCTAL NUMBER SYSTEM-uses 0 to 7 and base 8

2. HEXADECIMAL NUMBERSYSTEM- uses 0 to 9, and Athrough F corresponding tonumbers 10 through 15 and base16


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BINARY TO OCTAL

CONVERSION1. Converting binary to octal is grouped into

three from the least significant bit (LSB)

to the most significant (MSB)Examples:

1. 111111112

2. 1000111023. 0000111124. 101010102


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OCTAL TO BINARY

CONVERSION Converting octal to binary just write the binary

equivalent of the numbers in octal form

EXAMPLES:

1. 778

2. 578

3. 6584. 468

5. 528


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DECIMAL TO OCTAL

CONVERSION1. Convert the decimal number to its binary

equivalent

2. Then grouped the binary equivalent into

three and convert each group separately to

octal equivalent

Examples:

1. 4410, 5410, 6610


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OCTAL TO DECIMAL

CONVERSION1. Convert the octal number to its binary equivalent

by assigning a 3-bit binary equivalent to each

octal digit2. Then convert the binary back to decimal using

the usual method.

EXAMPLES:

1. 7782. 5783. 6584. 468


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BINARY TO HEX

CONVERSION To convert binary to hex, grouped the

binary digits into four bits (from LSB to the

MSB) and get the equivalent in hex.

EXAMPLES:

1. 1010101111002

2. 1100000101111111012


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HEX TO BINARY

CONVERSION Get the binary equivalent of each digit in

the proper order.

EXAMPLES:

1. 1CB0916


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DECIMAL TO HEX

CONVERSION1. Convert the decimal number to binary

using the usual approach.

2. Then, grouped the binary digits into 4 bits

and take the hex equivalent of each group

separately.

EXAMPLES:

1. 76410


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HEX TO DECIMAL

CONVERSION1. Using the direct approach, the hex digit

are multiplied with the weighed bit

equivalent of the number (Polynomialexpansion)

2. Convert the hex number to binary by

assigning 4-bit equivalent for each hexdigit and then converting the binary to

decimal number.


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EXAMPLE 1. 2FC16

Since C = 12, F = 15

H2 + H1 + H0

= 162 x 2 + 161 x 15 + 160 x 12

= 512 + 240 + 12

= 764


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BCD Code The BCD is one popular form of coding used to

simplify the conversion of binary into its decimalequivalent.

This is generally used for coding binary in 7-segment displays.

Note: this code is weighed code, not a numbersystem.

Similar to the hexadecimal number system, thebinary bits are grouped into four to facilitateconversion; but unlike hex, the acceptable binarycodes are obviously for digits 0 to 9 only.


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BCD 1. 321 = 0011 0010 0001

2. 10 = 0001 0000

3. 11 = 0001 0001

4. 22 = 0010 0010

5. 100 = 0001 0000 0000


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BASIC BINARY ARITHMETIC ADDTION

Rules of Binary Addition:

0+0 = 00+1 = 1

1+0 = 1

1+1 = 0, and carry 1 to the next most significantbit

Examples: 1. 00011010 + 00001100

2] 11+11 , 3] 11 + 101 , 4] 1001 + 1101


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SUBTRACTION Rules of binary subtraction:

0 0 = 0

1 0 = 1

1 1 = 0

0 1 = 1, and borrow 1 from the next most significant

bit.

EXAMPLES:

A] 00100101 00010001

B] 00110011 00010110


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MULTIPLICATIONRules of binary multiplication:

0 * 0 = 0

0 * 1 = 0

1 * 0 = 0

1 * 1 = 1, and no carry or borrow bits

EXAMPLES:A] 00101001 * 00000110

B] 00010111 * 00000011


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DIVISIONBinary division is the repeated process of

subtraction, just as in decimal division.

EXAMPLES:

A] 00101010


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2s Complementing Numbers The 2s complement number system is used in all

modern computers to express numbers.

It is similar to the binary number system, but bothpositive and negative numbers can be represented.

MSB of a 2s complement number denotes:

0 means the number is positive

1 means the number is negativeNOTE: 2s complement notation, positive numbers are

represented as simple binary numbers with therestriction that the MSB is 0.


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To obtain the representation of a

negative number, use the following

algorithm:1. Represent the number as a positive binary

number.

2. Complement it. (Write 0s where there are

1s and 1s where there are 0s in the

positive number)

3. Add 1.

4. Ignore any carries out of the MSB.


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Example:1. Given 8-bit word, find the 2s complement

representation of:

a. 25 = 000110012

b. 25

c. 1


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Adding 2s Complement Numbers

C = A + B

If A and B are both positive, addition is

required.

But if one of the operands is negative and

the other is positive, a subtraction operation

must be performed.


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Example:1. Express the numbers 19 and 11 as 8-bit, 2s

complement numbers, and add them.

+ 19 = 00010011

Get the 2s complement of - 11

= 11110101Then add:

00001000


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Example:2. Add 11 and 19.

- 11 = 11110101

- 19 = 11101101

Then add:

11100010


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Subtraction of Binary Numbers The 2s complement of the subtrahend is taken

and added to the minuend.

Example:

Subtract 30 from 53. Use 8-bit numbers.

53 = 00110101 (Minuend)

2s complement of 30 = 11100010 (Subtrahend)

Then add:

00010111


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Subtract 30 from 19.- 19 = 11101101

- 30 = 11100010 = 2s complement =

00011110

Then add:

00001011

14. DIGITAL CIRCUITS

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