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presentation on logic gates
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Number Systems
Decimal (base 10) {0 1 2 3 4 5 6 7 8 9}o Place value gives a logarithmic representation
of the numbero Ex. 4378 means
4 X 103 = 4000 3 X 102 = 300 7 X 101 = 70 8 X 100 = 8
o The place also gives the exponent of the base
Example
• 432,600
4 3 2 6 0 0
105
104
103
100
101
102
Powers of ten:
100 = 1 102 = 100 104 = 10000
101 = 10 103 = 1000 105 = 100000
Binary (base 2) {0 1}Binary Decimal
0 0
1 1
10 2
11 3
100 4
101 5
110 6
111 7
1000 8
1001 9
1010 10
Example
1 1 0 1 1 0 0 1
27
26
25
20
21
22
24 23
Decimal Equivalent
1101 1001 1 X 27 = 128 + 1 X 26 = 64 + 0 X 25 = 0 + 1 X 24 = 16 + 1 X 23 = 8 + 0 X 22 = 0 + 0 X 21 = 0 + 1 X 20 = 1 217
Notice how powers of two stand out:
20 = 1
21 = 10
22 = 100
23 = 1000
Decimal to Binary Conversion
Ex. 575o Find the largest power of two less than the number
o 29 = 512o Subtract that power of two from the number
o 575 – 512 = 63o Repeat steps 1 and 2 for the new result until you reach zero.
o 25 = 32 63 – 32 = 31o 24 = 16 31 – 16 = 15o 23 = 8 15 – 8 = 7o 22 = 4 7 – 4 = 3o 21 = 2 3 – 2 = 1o 20 = 1 1 – 1 = 0
o Construct the numbero 1000111111
Another Example
144o 27 = 128 144 – 128 = 16o 24 = 16 16 – 16 = 0
Result 10010000
Hexadecimal (base 16)
{0 1 2 3 4 5 6 7 8 9 A B C D E F} Assignments Dec Hex Dec Hex
0 0 8 8
1 1 9 9
2 2 10 A
3 3 11 B
4 4 12 C
5 5 13 D
6 6 14 E
7 7 15 F
Example
163
162
160
161
3 B 6 E
3 X 163 = 12288
11 X 162 = 2816
6 X 161 = 96
14 X 160 = 14
15214
Hexadecimal is Convenient for Binary Conversion
Binary Hex Binary Hex
0 0 1001 9
1 1 1010 A
10 2 1011 B
11 3 1100 C
100 4 1101 D
101 5 1110 E
110 6 1111 F
111 7 1 0000 10
1000 8 Nibble
Binary to Hex Conversion
Group binary number by fours (nibbles)o 1101 1001 0110
Convert each nibble into hex equivalento 1101 1001 0110
D 9 6
Decimal to Hex Conversion
Ex. 284o 162 = 256 284 – 256 = 28o 161 = 16 28 - 16 = 12 (Hex C)
oResult 1 1 C
Another Example with an Extension
1054o 162 = 256
But we have several multiples of 256 in 1054o 1054/256 = 4.12 take integer parto This eliminates 4*256 = 1024
1054 – 1024 = 30
o 161 = 16 30 – 16 = 14 (Hex E)
oResult 4 1 E
Truth TableBinary Decimal Hexadecimal
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
Truth TableBinary Decimal Hexadecimal0000 0 00001 1 10010 2 20011 3 30100 4 40101 5 50110 6 60111 7 71000 8 81001 9 91010 10 A1011 11 B1100 12 C1101 13 D1110 14 E1111 15 F
Sexagesimal(Base 60)
Practice
Convert 212 decimal to binaryo 212 – 27 = 84o 84 – 26 = 20o 20 – 24 = 4o 4 – 22 = 0oResult: 1101 0100
More Practice
Convert 1101 0010 binary to hexo 0010 = 2o 1101 = 13 = DoResult D2
Notation
Some books use a subscript to denote the base.o Ex: 1210 = 12 decimal
o 1216 = 12 hex = 18 decimal
Logic Gates
Transistors as Switches• VBB voltage controls whether the transistor
conducts in a common base configuration.
• Logic circuits can be built
Boolean Algebra
AND
In order for current to flow, both switches must be closed¤ Logic notation AB = C
(Sometimes AB = C)
A B C
0 0 0
0 1 0
1 0 0
1 1 1
OR
Current flows if either switch is closed¤ Logic notation A + B = C
A B C
0 0 0
0 1 1
1 0 1
1 1 1
Properties of AND and OR
Commutationo A + B = B + Ao A B = B A
Same as
Same as
Commutation Circuit
A + B B + A
A B B A
Properties of AND and OR
Associative PropertyA + (B + C) = (A + B) + C
A (B C) = (A B) C
=
Properties of AND and OR
Distributive PropertyA + B C = (A + B) (A + C)
A + B CA B C Q
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
Distributive Property
(A + B) (A + C)
A B C Q
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
Binary Addition
A B S C(arry)
0 0 0 0
1 0 1 0
0 1 1 0
1 1 0 1
Notice that the carry results are the same as AND
C = A B
Inversion (NOT)
A Q
0 1
1 0Logic:
AQ
Exclusive OR (XOR)
Either A or B, but not both
This is sometimes called the inequality detector, because the result will be 0 when the inputs are the same and 1 when they are different.
The truth table is the same as for S on Binary Addition. S = A B
A B S
0 0 0
1 0 1
0 1 1
1 1 0
Getting the XOR
A B S
0 0 0
1 0 1
0 1 1
1 1 0
Two ways of getting S = 1
BAor BA
Circuit for XOR
Accumulating our results: Binary addition is the result of XOR plus AND
BA BABA
Half Adder
Called a half adder because we haven’t allowed for any carry bit on input. In elementary addition of numbers, we always need to allow for a carry from one column to the next.
18
25
4
3 (plus a carry)
Half Adder
Full Adder
INPUTS OUTPUTS
A B CIN COUT S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
Full Adder Circuit
Chaining the Full Adder
Possible to use the same scheme for subtraction by
noting that
A – B = A + (-B)
Binary CountingUse 1 for ON
Use 0 for OFF
= 00101011
So our example has 25 + 23 + 21 + 20 = 32 + 8 + 2 + 1 = 43
Counting in Binary
1 1 11 1011 21 10101
2 10 12 1100 22 10110
3 11 13 1101 23 10111
4 100 14 1110 24 11000
5 101 15 1111 25 11001
6 110 16 10000 26 11010
7 111 17 10001 27 11011
8 1000 18 10010 28 11100
9 1001 19 10011 29 11101
10 1010 20 10100 30 11110
NAND (NOT AND)
A B Q
0 0 1
0 1 1
1 0 1
1 1 0
BAQ
NOR (NOT OR)
A B Q
0 0 1
0 1 0
1 0 0
1 1 0
BAQ
DeMorgan’s Theorem
A NAND gate is equivalent to an inversion followed by an OR
A NOR gate is equivalent to an inversion followed by and AND
DeMorgan Truth Table
NAND NOR
Exclusive NOR
A B Q
0 0 1
0 1 0
1 0 0
1 1 1
Equality Detector
BAQ
Summary
Summary for all 2-input gates
Inputs Output of each gate
A B AND NAND OR NOR XOR XNOR
0 0 0 1 0 1 0 1
0 1 0 1 1 0 1 0
1 0 0 1 1 0 1 0
1 1 1 0 1 0 0 1
Logic Gates and Symbols
AND
NAND
More Gates and Symbols
NOR
NOT
OR
And More
XOR
NXOR
Multi-input Gates
Three input OR
Logic Gate ICs
Example 7400
More ICs
And More
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