Upload
jasper-austin
View
221
Download
1
Embed Size (px)
Citation preview
ECE 331 – Digital System Design
NAND and NOR Circuits,Multi-level Logic Circuits,
andMultiple-output Logic Circuits
(Lecture #9)
The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
Spring 2011 ECE 331 - Digital System Design 3
CMOS Logic Gates
Inverter
2-input AND
2-input OR
2-input NAND2-input NOR
Spring 2011 ECE 331 - Digital System Design 4
CMOS Logic Gates
# of inputs AND / OR NAND / NOR
2 6 4
3 8 6
4 10 8
5 12 10
Number of transistors per logic gate:
Thus, in terms of transistor count, it is “cheaper” to design logic circuits using NAND and NOR gates.
Spring 2011 ECE 331 - Digital System Design 5
The NAND Gate
Any logic function can be realized using only NAND gates.
It is a functionally complete set of gates.
Spring 2011 ECE 331 - Digital System Design 6
The NOR Gate
Any logic function can be realized using only NOR gates.
It, too, is a functionally complete set of gates.
XX'
AB
(A+B)'
A+B
A
B
A'
B'
(A'+B')' = AB
Spring 2011 ECE 331 - Digital System Design 7
Alternate Logic Gate Symbols Digital circuit designers often find it convenient to use
more than one representation for a given logic gate.
Spring 2011 ECE 331 - Digital System Design 8
NAND and NOR Circuits
A two-level circuit composed of AND and OR gates is easily converted to a circuit composed of NAND or NOR gates only.
AND-OR → NAND-NAND OR-AND → NOR-NOR
To do so algebraically, First use F = (F')' Then apply DeMorgan's Theorem
Spring 2011 ECE 331 - Digital System Design 9
NAND Circuit: Example
Convert the following SOP expression from the AND-OR form to the NAND-NAND form:
F(A,B,C) = A.B' + A'.C' + B.C
Spring 2011 ECE 331 - Digital System Design 10
NOR Circuit: Example
Convert the following POS expression from the OR-AND form to the NOR-NOR form:
F(A,B,C) = (A'+B').(A'+C).(B+C')
Spring 2011 ECE 331 - Digital System Design 11
Design a NAND Circuit
• Find the minimum SOP expression for F.
• Draw the corresponding AND-OR circuit.
• Replace all gates with NAND gates, leaving the gate interconnection unchanged.
• Complement any literals connected directly to the output (OR) gate.
Spring 2011 ECE 331 - Digital System Design 12
NAND Circuit: Example
Design the NAND circuit for the following logic function:
F(A,B,C) = m(0, 3, 4, 5, 6, 7)
Spring 2011 ECE 331 - Digital System Design 13
Design a NOR Circuit
• Find the minimum POS expression for F.
• Draw the corresponding OR-AND circuit.
• Replace all gates with NOR gates, leaving the gate interconnection unchanged.
• Complement any literals connected directly to the output (AND) gate.
Spring 2011 ECE 331 - Digital System Design 14
NOR Circuit: Example
Design the NAND circuit for the following logic function:
F(A,B,C) = m(2, 4)
Spring 2011 ECE 331 - Digital System Design 16
Multi-level Logic Circuits Thus far we have focused on the realization of optimal
logic circuits through the derivation of Minimum Sum of Products (SOP) expressions Minimum Product of Sums (POS) expressions
Both forms of Boolean expressions are realized as two-level logic circuits
SOP ↔ AND-OR (NAND-NAND) circuit POS ↔ OR-AND (NOR-NOR) circuit There are a maximum of two logic gates between every
input and the output(s).
Spring 2011 ECE 331 - Digital System Design 17
Multi-level Logic Circuits A two-level logic circuit is usually efficient for Boolean
expressions of a few variables (i.e. inputs). However, as the number of inputs increases, a two-
level logic circuit may encounter fan-in problems. Fan-in refers to the number of inputs to a logic gate
Whether fan-in is an issue is dependent upon the technology used to implement the logic circuit.
Standard TTL and CMOS chips Complex Programmable Logic Device (CPLD) Field Programmable Gate Array (FPGA)
Spring 2011 ECE 331 - Digital System Design 18
Multi-level Logic Circuits A multi-level logic circuit may require fewer logic gates
than the logically equivalent two-level logic circuit. Reduced (silicon) area Decreased cost
It may require less complex wiring between logic gates Fewer literals results in fewer interconnecting wires
It will have a greater propagation delay than the logically equivalent two-level logic circuit.
Each additional level adds to the propagation delay Decreased speed
Spring 2011 ECE 331 - Digital System Design 19
Multi-level Logic Circuits: Example
Design a logic circuit to realize the following logic function:
F(A,B,C) = m(1, 2, 3, 4, 6)
Given the following criteria:
1. Use AND and OR gates only2. Two- or Three-level circuit only3. Minimize the number of gates4. Minimize the number of gate inputs
Spring 2011 ECE 331 - Digital System Design 20
Multi-level Logic Circuits: Example
Which design would be possible if the following additional criteria was imposed:
5. Two-input logic gates only
Spring 2011 ECE 331 - Digital System Design 22
Design a Multi-level NAND Circuit
• Derive a minimum expression for the logic function.
• Design a multi-level circuit using AND and OR gates. Output gate must be an OR gate.
Gates must alternate: AND, OR, AND, OR, …
• Number the levels starting with the output gate.
• Replace all gates with NAND gates, leaving the interconnection between gates unchanged.
• Leave inputs to gates at the even levels unchanged; complement inputs to gates at the odd levels.
Spring 2011 ECE 331 - Digital System Design 24
Design a Multi-level NOR Circuit
• Derive a minimum expression for the logic function.
• Design a multi-level circuit using AND and OR gates. Output gate must be an AND gate.
Gates must alternate: OR, AND, OR, AND, …
• Number the levels starting with the output gate.
• Replace all gates with NOR gates, leaving the interconnection between gates unchanged.
• Leave inputs to gates at the even levels unchanged; complement inputs to gates at the odd levels.
Spring 2011 ECE 331 - Digital System Design 27
Multiple-output Logic Circuits
Thus far, we have focused on designing logic circuits to realize a single logic function.
A logic circuit with a single output. However, many logic circuits have multiple
outputs. Corresponding to multiple logic functions of
the same input variables.
Spring 2011 ECE 331 - Digital System Design 28
Multiple-output Logic Circuits
An optimal logic circuit may not be realized by simply minimizing each of the logic functions independently.
Rather, it may be necessary to consider which terms (i.e. logic gates), if any, are common to the logic functions to be realized.
Share those logic gates that are in common.
Spring 2011 ECE 331 - Digital System Design 29
Multiple-output Circuits: Example
Design an optimal logic circuit to realize the following logic functions:
F(A,B,C) = m(0, 4, 5)G(A,B,C) = m(0, 2, 6)
Cost = # of logic gates + # of gate inputs