Transcript Logic Gates - Physics & Astronomy | SFASU

Number Systems
Decimal (base 10)
{0 1 2 3 4 5 6 7 8 9}
o Place value gives a logarithmic representation
of the number
o 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
100
104
101
103
102
Powers of ten:
100 = 1
102 = 100
104 = 10000
101 = 10
103 = 1000
105 = 100000
Binary (base 2)
Binary
0
1
Decimal
0
1
10
11
100
101
2
3
4
5
110
111
6
7
1000
8
1001
1010
9
10
{0 1}
Example
1 1 0 1 1 0 0 1
27
20
26
21
25
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. 575
o Find the largest power of two less than the number
o
29 = 512
o Subtract that power of two from the number
o
575 – 512 = 63
o Repeat steps 1 and 2 for the new result until you reach zero.
25 = 32
o 24 = 16
o 23 = 8
o 22 = 4
o 21 = 2
o 20 = 1
o
63 – 32 = 31
31 – 16 = 15
15 – 8 = 7
7–4=3
3–2=1
1–1=0
o Construct the number
o
1000111111
Another Example
144
o 27 = 128
o 24 = 16
Result
144 – 128 = 16
16 – 16 = 0
10010000
Hexadecimal (base 16)
 {0 1 2 3 4 5 6 7 8 9 A B C D E F}
 Assignments Dec Hex Dec
0
1
0
1
8
9
Hex
8
9
2
3
4
5
2
3
4
5
10
11
12
13
A
B
C
D
6
7
6
7
14
15
E
F
Example
3B6E
163
160
162
161
3 X 163 = 12288
11 X 162 = 2816
6X
161
=
96
14 X 160 =
14
 15214
Hexadecimal is Convenient for
Binary Conversion
Binary
Hex
Binary
Hex
0
1
10
0
1
2
1001
1010
1011
9
A
B
11
100
3
4
1100
1101
C
D
101
110
111
5
6
7
1110
1111
1 0000
E
F
10
1000
8
 Nibble
Binary to Hex Conversion
 Group binary number by fours (nibbles)
o 1101 1001 0110
 Convert each nibble into hex equivalent
o 1101 1001 0110
D
9
6
Decimal to Hex Conversion
 Ex. 284
o 162 = 256
o 161 = 16
o Result 1 1 C
284 – 256 = 28
28 - 16 = 12 (Hex C)
Another Example with an Extension
 1054
o 162 = 256

But we have several multiples of 256 in 1054
o 1054/256 = 4.12 take integer part
o This eliminates 4*256 = 1024

1054 – 1024 = 30
o 161 = 16
o Result 4 1 E
30 – 16 = 14 (Hex E)
Truth Table
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Decimal
Hexadecimal
Truth Table
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Sexagesimal
(Base 60)
Practice
Convert 212 decimal to binary
o 212 – 27 = 84
o 84 – 26 = 20
o 20 – 24 = 4
o 4 – 22 = 0
o Result: 1101 0100
More Practice
Convert 1101 0010 binary to hex
o 0010 = 2
o 1101 = 13 = D
o Result 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 AB = C
(Sometimes AB = C)
A
0
0
B
0
1
C
0
0
1
1
0
1
0
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
Commutation
oA + B = B + A
oA  B = B  A
Same as
Same as
Commutation Circuit
AB
A+B
BA
B+A
Properties of AND and OR
Associative Property
A + (B + C) = (A + B) + C
=
A  (B  C) = (A  B)  C
Properties of AND and OR
Distributive Property
A + B  C = (A + B)  (A + C)
A+B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
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)
Logic:
QA
A
Q
0
1
1
0
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
0
1
0
1
B
0
0
1
1
S
0
1
1
0
Getting the XOR
Two ways of getting S = 1
A  B or A  B
A
B
S
0
0
0
1
0
1
0
1
1
1
1
0
Circuit for XOR
AB  AB  A B
Accumulating our results: Binary addition is the
result of XOR plus AND
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
3 (plus a carry)
4
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 Counting
Use 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)
Q  A B
A
B
Q
0
0
1
0
1
1
1
0
1
1
1
0
NOR (NOT OR)
Q  AB
A
0
0
1
1
B
0
1
0
1
Q
1
0
0
0
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
A
0
0
1
1
B
0
1
0
1
1
1
1
0
NAND
1
1
1
0
1
0
0
0
NOR
1
0
0
0
Exclusive NOR
Q  AB
Equality Detector
A
B
Q
0
0
1
0
1
0
1
0
0
1
1
1
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
OR
NOR
NOT
And More
XOR
NXOR
Multi-input Gates
Three input OR
Logic Gate ICs
Example 7400
More ICs
And More