Lecture 09

NAND and XOR Implementations

Overview

° Developing NAND circuits

° Two-level implementations• Convert from AND/OR to NAND (again!)

° Multi-level NAND implementations• Convert from a network of AND/ORs

° Exclusive OR • Comparison with SOP

° Parity checking and detecting circuitry• Efficient with XOR gates!

Gates

A X X = (A + B)’B

AND A X = A • B X orB X = AB

0 0 0 0 1 0 1 0 0 1 1 1

0 0 0 0 1 1 1 0 1 1 1 1

OR A X X = A + BB

I A X X = A’0 11 0

Buffer A X X = A A X0 01 1

NANDA X X = (AB)’B

0 0 10 1 11 0 11 1 0

NOR0 0 10 1 01 0 01 1 0

XORExclusive OR

A X = A B X orB X = A’B + AB’

0 0 00 1 11 0 11 1 0

A X = (A B)’ X orB X = A’B’+ AB

0 0 10 1 01 0 01 1 1

XNORExclusive NORor Equivalence

A B X

A B X

A X

A B X

A B X

A B X

A B X

Axioms and Graphical representation of DeMorgan's Law

Y X Y X 14B)

Y X YX 14A)

YXYXX 13D)

YXYXX 13C)

YXXYX 13B)

YXYXX 13A)

YZYWXZXWZWYX 12B)

XZXYZYX 12A)

ZYXZY X 11B)

ZXYYZX 11A)

X Y Y X 10B)

X Y Y X 10A)

Commutative

Law

Associative Law

Distributive Law

Consensus Theorem

nand

nor

and

or

not

• Recall that symbolic DeMorgan’s duals exist for all gate primitives

• The above alternate symbols can be used to facilitate the analysis and design of NAND and NOR gate networks.

Network Conversion Using Alternate Gate Symbols

NAND-NAND & NOR-NOR Networks

DeMorgan’s Law:

(a + b)’ = a’ b’ (a b)’ = a’ + b’

a + b = (a’ b’)’ (a b) = (a’ + b’)’

push bubbles or introduce in pairs or remove pairs.

= =

==

NAND-NAND & NOR-NOR Networks

= =

==

NAND-NAND Networks

° Mapping from AND/OR to NAND/NAND

ab

cd

a) b)

c) d)

NAND-NAND Networks

Implementations of Two-level Logic

° Sum-of-products• AND gates to form product terms

(minterms)

• OR gate to form sum

° Product-of-sums• OR gates to form sum terms

(maxterms)

• AND gates to form product

Two-level Logic using NAND Gates

° Replace minterm AND gates with NAND gates

° Place compensating inversion at inputs of OR gate

Two-level Logic using NAND Gates (cont’d)

° OR gate with inverted inputs is a NAND gate• de Morgan's: A' + B' = (A • B)'

° Two-level NAND-NAND network• Inverted inputs are not counted

• In a typical circuit, inversion is done once and signal distributed

Two-level Logic using NAND Gates (cont’d)

Conversion Between Forms

° Convert from networks of ANDs and ORs to networks of NANDs and NORs

• Introduce appropriate inversions ("bubbles")

° Each introduced "bubble" must be matched by a corresponding "bubble"

• Conservation of inversions

• Do not alter logic function

° Example: AND/OR to NAND/NAND

A

B

C

D

Z

A

B

C

D

Z

NAND

NAND

NAND

Z = [ (A • B)' • (C • D)' ]'

= [ (A' + B') • (C' + D') ]'

= [ (A' + B')' + (C' + D')' ]

= (A • B) + (C • D)

Conversion Between Forms (cont’d)

° Example: verify equivalence of two forms

A

B

C

D

Z

A

B

C

D

Z

NAND

NAND

NAND

Conversion to NAND Gates

° Start with SOP (Sum of Products)• circle 1s in K-maps

° Find network of OR and AND gates

ABC

DE

FG

X

Multi-level Logic

° x = A D F + A E F + B D F + B E F + C D F + C E F + G• Reduced sum-of-products form – already simplified

• 6 x 3-input AND gates + 1 x 7-input OR gate (may not exist!)

• 25 wires (19 literals plus 6 internal wires)

° x = (A + B + C) (D + E) F + G• Factored form – not written as two-level S-o-P

• 1 x 3-input OR gate, 2 x 2-input OR gates, 1 x 3-input AND gate

• 10 wires (7 literals plus 3 internal wires)

Level 1 Level 2 Level 3 Level 4

originalAND-OR network

A

CD

B

BC ’

F

introduction andconservation of

bubblesA

CD

B

BC ’

F

redrawn in termsof conventional

NAND gates A

CD

B ’

BC ’

F

Conversion of Multi-level Logic to NAND Gates° F = A (B + C D) + B C'

A

XBCD

F(a)

Original circuit

A

XBC

D

F(b)

Add double bubbles at inputs

D ’

A

X ’

BC

F(c)

Distribute bubblessome mismatches

D ’

AX

BC

FX ’

(d)

Insert inverters to fix mismatches

Conversion Between Forms

° Example

Making NAND circuits (Ex)° The easiest way to make a NAND circuit is to start with

a regular, primitive gate-based diagram.

° Two-level circuits are trivial to convert, so here is a slightly more complex random example.

Converting to a NAND° Step 1: Convert all AND gates to NAND gates and

convert all OR gates to NAND gates.

AND

AND

AND

AND

OR

OR

Converting to NAND° Step 2: Cancel all pairs of inverters ((x’)’ = x)..

Exclusive-OR and Exclusive-NOR Circuits

Exclusive-OR (XOR) produces a HIGH output whenever the two inputs are at opposite levels.

Exclusive-NOR (XNOR) :

Exclusive-NOR (XNOR) produces a HIGH output whenever the two inputs are at the same level.

Exclusive-NOR Circuits

XNOR gate may be used to simplify circuit implementation.

Exclusive-NOR Circuits

XOR Function° XOR function can also be implemented with AND/OR gates (also NANDs).

FIGURE 4-25 XOR gates used to implement the parity generator and the parity checker for an even-parity system.

Parity Generation and Checking

Summary

° Follow rules to convert between AND/OR representation and symbols

° Conversions are based on DeMorgan’s Law

° NOR gate implementations are also possible

° XORs provide straightforward implementation for some functions

° Used for parity generation and checking• XOR circuits could also be implemented using AND/Ors

° Next time: Hazards