Digital Logic Design
-Switch Logic & Basic Gates-
A Simple Switch
Instr: Dr. Awais M. Kamboh.
2
Switches in Series
Instr: Dr. Awais M. Kamboh.
3
Switches in Parallel
Instr: Dr. Awais M. Kamboh.
4
Normal vs Inverting Switch
Instr: Dr. Awais M. Kamboh.
5
Boolean Logic
Boolean Logic is based on two states, e.g., True (1) or False (0)
Boolean Logic is a form of algebra which is based on three simple
Operators: “Not”, “Or,” and “And,”.
There are different ways to represent Boolean logic
• True / False,
• 1 / 0,
• ON / OFF,
• High / Low,
• Active / Inactive
Instr: Dr. Awais M. Kamboh.
6
The Inverter (NOT Gate)
The inverter performs the Boolean NOT operation. When the
input is LOW, the output is HIGH; when the input is HIGH,
the output is LOW.
Input
Output
A
X
0
1
1
0
A
X
The NOT operation (complement / invert) is shown with an
overbar or apostrophe. Thus, the Boolean expression for an
inverter is X = A or X = A’
Instr: Dr. Awais M. Kamboh.
7
The Inverter (Not Gate)
A
X
As time passes, we may change the input of the gate if we
want to, as a result the output changes. Such changes can be
represented as a waveform (also called a timing diagram).
The x-axis is the time, the y-axis the value of the input or the
output.
Example waveforms:
A
X
Instr: Dr. Awais M. Kamboh.
8
Decimal vs Binary Numbers
Decimal Numbers
Commonly used in our daily lives
They have 10 possible symbols (called digits) 0,1,2,3,4,5,6,7,8,9
Binary Numbers
Used in all digital systems
They have 2 possible symbols (called bits) 0,1
Decimal numbers larger than 9 can be represented by adding more digits
Example: 247635, 526, 83474
Binary numbers larger than 1 can be represented by adding more bits
Example: 10110, 110, 00110011
Instr: Dr. Awais M. Kamboh.
9
The Inverter (Not Gate)
A group of inverters can be used to get the 1’s complement
of a binary number:
1
0
Binary number
0
0
1
1
0
1
1
1
0
0
1’s complement
0
1
1
0
1’s complement of a binary number means that we invert each
bit of that number.
Instr: Dr. Awais M. Kamboh.
10
The AND Gate
The AND gate produces a HIGH output when all inputs are
HIGH; otherwise, the output is LOW. For a 2-input gate,
the truth table is
Inputs
Output
A
B
X
0
0
1
1
0
1
0
1
0
0
0
1
A
X
B
The AND operation is usually shown with a dot between the
variables, but it may be implied (no dot). Thus, the AND
operation is written as X = A .B or X = AB.
Instr: Dr. Awais M. Kamboh.
11
The AND Gate
A
X
B
Example waveforms:
A
B
X
If the binary number 1010 0011
Instr: Dr. Awais M. Kamboh.
is ANDed with
0000 1111
what is the result?
0000 0011
12
The OR Gate
The OR gate produces a HIGH output if any input is HIGH;
if all inputs are LOW, the output is LOW. For a 2-input gate,
the truth table is
Inputs
Output
A
B
X
0
0
1
1
0
1
0
1
0
1
1
1
A
B
X
The OR operation is shown with a plus sign (+) between the
variables. Thus, the OR operation is written as X = A + B.
Instr: Dr. Awais M. Kamboh.
13
The OR Gate
A
B
X
Example waveforms:
A
B
X
If the binary number 1010 0011
Instr: Dr. Awais M. Kamboh.
is ORed with
0000 1111
what is the result?
1010 1111
14
Timing Diagram / Waveform
Instr: Dr. Awais M. Kamboh.
15
Logical Operations
Assuming A=0101 and B=1001 as inputs and X as output
AND Logic Gate
X = A.B or AB
A= 0101
B= 1001
X= 0001
OR Logic Gate
X =A+B
A= 0101
B= 1001
X= 1101
NOT Logic Gate
X =Ā or A’ (A Complement)
A= 0101
X= 1010
Instr: Dr. Awais M. Kamboh.
16
The NAND Gate
A
X
B
The NAND gate produces a LOW(0) output when all inputs
are HIGH (1); otherwise, the output is HIGH (1).
Think of it as AND followed by NOT
Inputs
For a 2-input gate, the truth table is
Input
NAND
A
B
AND
AND-NOT
X
0
0
0
1
1
0
1
1
0
0
0
1
1
1
1
0
1
1
1
0
Output
A
B
X
0
0
1
1
0
1
0
1
1
1
1
0
The NAND operation is written as X = A .B (Also as, X = AB) = (AB)’
pronounced as AB whole complement
Instr: Dr. Awais M. Kamboh.
17
The NAND Gate
A
X
B
Example waveforms:
A
B
X
The NAND gate is particularly useful because it is a
“universal” gate – all other basic gates can be constructed
from NAND gates.
How to build a NOT gate from a 2-input NAND gate?
Instr: Dr. Awais M. Kamboh.
18
The NOR Gate
A
B
X
The NOR gate produces a LOW (0) output if any input is
HIGH (1); if all inputs are HIGH (1), the output is LOW(0)
Think of it as OR followed by NOT
Inputs
For a 2-input gate, the truth table is
A
B
X
0
0
1
1
0
1
0
1
1
0
0
0
Input
NOR
A
B
OR
OR-NOT
X
0
0
0
1
1
0
1
1
0
1
1
1
1
0
0
0
1
0
0
0
The NOR operation is written as X = A + B = (A+B)’
pronounced as A+B whole complement
Instr: Dr. Awais M. Kamboh.
19
Output
The NOR Gate
A
B
X
Example waveforms:
A
B
X
The NOR operation will produce a LOW if any input is HIGH.
The NOR gate is also a “universal” gate – all other basic
gates can be constructed from NOR gates.
Only NAND and NOR are called universal gates
Instr: Dr. Awais M. Kamboh.
20
The XOR Gate
A
B
X
The XOR Exclusive-OR gate produces a HIGH output only
when both inputs are at opposite logic levels. The truth table is
Inputs
Output
A
B
X
0
0
1
1
0
1
0
1
0
1
1
0
The XOR operation is written as X = AB + AB.
Alternatively, it can be written with a circled plus sign
between the variables as X = A + B.
Instr: Dr. Awais M. Kamboh.
21
The XOR Gate
A
B
X
Example waveforms:
A
B
X
Notice that the XOR gate will produce a HIGH only when exactly one
input is HIGH.
If the A and B waveforms are both inverted for the above
waveforms, how is the output affected?
There is no change in the output.
Instr: Dr. Awais M. Kamboh.
22
The XNOR Gate
A
B
X
The XNOR Exclusive NOR gate produces a HIGH output
only when both inputs are at the same logic level.
Inputs
Think of it as XOR followed by NOT
Input
XNOR
A
B
XOR
XOR-NOT
X
0
0
0
1
1
0
1
1
0
1
1
0
1
0
0
1
1
0
0
1
The XNOR operation shown as X = AB + AB.
It can be shown as X = A
Instr: Dr. Awais M. Kamboh.
.
B.
23
Output
A
B
X
0
0
1
1
0
1
0
1
1
0
0
1
The XNOR Gate
A
B
X
Example waveforms:
A
B
X
Notice that the XNOR gate will produce a HIGH when both inputs are the
same. This makes it useful for comparison functions.
If the A waveform is inverted but B remains the same, how is
the output affected?
The output will be inverted.
Instr: Dr. Awais M. Kamboh.
24
Three Input AND Gate
A
C
B
Inputs
X
Output
A
B
C
X=ABC
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
0
0
0
0
0
0
0
1
Notice that
1
1
• A 1-input inverter has two possible input values: 0, 1.
1
• A 2-input gate has 4 possible input combinations: 00, 01, 10, 11.
• A 3-input gate has 8 possible input combinations, for each input the
number of combinations is doubled. c
Instr: Dr. Awais M. Kamboh.
25
Three Input OR Gate
Inputs
Outputs
A
B
C
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
Notice that
X=A+B+C
1
1
1
To create the input combinations, we use alternating 0s and 1s.
• Input C has one 0 followed by one 1.
• Input B has two 0s followed by two 1s.
• Input A has four 0s followed by four 1s, and so on…
Instr: Dr. Awais M. Kamboh.
26
0
1
1
1
1
1
1
1
Three Input XOR Gate
X= A + B + C
Inputs
XOR
A
B
C
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
D=A + B
0
0
1
1
1
1
0
0
For ODD number of 1s, the output is 1.
Instr: Dr. Awais M. Kamboh.
27
D+C
X
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1