basic gates 3.1 basic digital logic: nand and nor gates ©paul godin created september 2007 last...
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Basic Gates 3.1
Basic Digital Logic:NAND and NOR Gates
©Paul GodinCreated September 2007Last Update Sept 2009
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Basic Gates 3.2
The Universal Gates:NAND and NOR
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Basic Gates 3.3
Combinational logic
◊ How would your describe the output of this combinational logic circuit?
A
B
Z
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Basic Gates 3.4
NAND Gate
◊ The NAND gate is the combination of an NOT gate with an AND gate.
The Bubble in front of the gate is an inverter.
A
B
Z
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Basic Gates 3.5
NAND Gate
◊ IEEE Symbol
◊ Boolean equations for the NAND gate:
&
A●B = x
AB = x
The triangle is the same as the bubble.
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Basic Gates 3.6
Combinational logic
◊ How would your describe the output of this combinational logic circuit?
B
AZ
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Basic Gates 3.7
NOR gate
◊ The NOR gate is the combination of the NOT gate with the OR gate.
The Bubble in front of the gate is an inverter.
B
AZ
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Basic Gates 3.8
NOR Gate
◊ IEEE Symbol for a NOR gate:
◊ Boolean equation for a NOR gate:
≥1
A+B = x
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Basic Gates 3.9
Exercise 1
Complete the Truth Table for the NAND and NOR Gates
Input Output
0 0
0 1
1 0
1 1
Input Output
0 0
0 1
1 0
1 1
NAND NOR
Hint: Think of the AND and OR truth tables. The outputs for the NAND and NOR are inverted.
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Basic Gates 3.10
Exercise 2
◊ Turn the NAND and NOR gates into inverter (NOT) gates. Hint: Look at the Truth Table.
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Basic Gates 3.11
Exercise 3
Complete the following timing diagram
A
B
Z
A
B
Z
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Basic Gates 3.12
Exercise 4
Complete the following timing diagram
A
B
Z
A
B
Z
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Basic Gates 3.13
Exercise 5
◊ Draw the following circuit: AB + BC
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Basic Gates 3.14
Exercise 6
◊ Describe the function of a NAND gate, starting with the term “If any input...”
◊ Describe the function of a NOR gate, starting with the term “If any input...”
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Basic Gates 3.15
DeMorgan
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Basic Gates 3.16
DeMorgan Theorem
◊ The DeMorgan Theorem describes a method for converting between AND/NAND and OR/NOR operations.
◊ The theorem states:
A ● B = A + B
A + B = A ● B
“Break the bar and change the sign”
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Basic Gates 3.17
DeMorgan
An example of DeMorgan:
AB + BC
ABC
1
2
3
4
Original Equation
DeMorgan applied to NOR expression
Double inversions cancel
Simplified expression
AB ● BC
AB ● BC
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Basic Gates 3.18
DeMorgan Exercise 1
Use DeMorgan to simplify the following expressions
A+B+C
AB
AB + C+D
“Break the bar and change the sign”
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Basic Gates 3.19
Universal Gates
◊ The NAND and NOR gates are considered Universal Gates. They can be used to create any other gate.
◊ Using universal gates is an important aspect of digital logic design.
Examples provided in class.
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Basic Gates 3.20
NAND and NOR as NOT
ZA
VccA Z
Method 1
A Z
Method 1
A Z
Method 2
A
Z
Method 2
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Basic Gates 3.21
NAND and NOR as AND
ZB
A
ZB
A
AB = AB = A+B (DeMorgan)
A
ZB
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Basic Gates 3.22
NAND and NOR as OR
A+B = A+B = AB (DeMorgan)
ZB
A
A
ZB
ZB
A
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Basic Gates 3.23
Universal Gates Exercise 1
◊ Convert NAND as NOR
◊ Convert NOR as NAND
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Basic Gates 3.24
Example: Universal Gates
Convert the following circuit to NAND only:
Convert each of the gates in the circuit to its NAND equivalent and progressively re-draw the circuit.
Additional Examples given in class
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Basic Gates 3.25
Active Input States
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Basic Gates 3.26
Digital 1’s and 0’s
◊ In a binary system, the logic 1’s are as important as the logic 0’s. A “0” is a signal also.
◊ When the “0” forces a change it is called Active Low (the low causes the action).
◊ When the “1” forces a change it is called Active High (the high causes the action).
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Basic Gates 3.27
Comparison of Active States
A logic 0 causes the LED
to light up
A logic 1 causes the LED
to light up
Vcc
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Basic Gates 3.28
Comparison of Active Inputs
A ZB
Active Low InputsActive High Output
Active High InputsActive Low Output
ZBA
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Alternate Gate Representations
Basic Gates 3.29
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Comparison of Active Inputs
A ZB
Active Low InputsActive High Output
Active High InputsActive Low Output
ZBA
Basic Gates 3.30
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Bubble to Bubble
◊ Alternate gate representations can make circuit analysis faster.
◊ A bubble attached to a bubble means the bubbles cancel.
=Bubbles Cancel
Basic Gates 3.31
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Bubble to Bubble Cancellation Example
2 Z1ABC
In this example with bubble-to-bubble representation, the output bubble from gate 1
cancels with the input bubble from gate 2.
This makes it easy to quickly determine that if either A or B inputs are low, outcome Z is low.
2 Z1ABC
Basic Gates 3.32
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Alternate Representations
◊ The “bubble” on a gate represents inversion.
◊ In many cases it is easier to follow the circuit logic if “bubble” outputs are linked to “bubble” inputs
◊ Cancelled bubbles helps make the active input state more easily visible for troubleshooting
Basic Gates 3.33
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Example use of Alternate Representation
◊ The output is active when the input state is a 101
◊ Note how much easier it is to see the active input at a glance using a bubble instead of a NOT gate.
Z
CBA
Basic Gates 3.34
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Basic Gates 3.35
Example of Alternate Representation
A
ZB
ZB
A
A
ZB
Equals
Equals
A+B = AB (DeMorgan)
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Basic Gates 3.36
Alternate Representation
ZBA
Z
A Z A Z
AZB
AB
ZBA
ZBA Z
A ZB
AB
ZBA
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Basic Gates 3.37
Alternate Representation Exercise
Z
A ZB
AB
Z
A ZB
AB
Identify the gate that is alternately represented
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Basic Gates 3.38
END
101101101
©Paul R. Godinprgodin°@ gmail.com