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Combinational Logic : NAND – NOR Gates CS370 – Spring 2003 Multi-Level Logic: Advantages Reduced sum of products form: x = A D F + A E F + B D F + B E F + C D F + C E F + G 6 x 3-input AND gates + 1 x 7-input OR gate (may not exist!)

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multi level logic advantages
Multi-Level Logic: Advantages

Reduced sum of products form:

x = A D F + A E F + B D F + B E F + C D F + C E F + G

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

25 wires (19 literals plus 6 internal wires)

Factored form:

x = (A + B + C) (D + E) F + G

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)

multi level logic conversion of forms

A

A

B

B

A

+

B

A

·

B

A

+

B

A

·

B

0

1

0

1

0

0

1

1

0

1

1

0

1

1

1

1

1

0

0

1

1

1

1

1

1

0

1

0

1

1

0

0

Multi-Level Logic: Conversion of Forms

NAND-NAND and NOR-NOR Networks

DeMorgan's Law: (A + B)' = A' · B'; (A ·B)' = A' + B'

Written differently: A + B = (A' · B')'; (A · B) = (A' + B')'

In other words,

OR is the same as NAND with complemented inputs

AND is the same as NOR with complemented inputs

NAND is the same as OR with complemented inputs

NOR is the same as AND with complemented inputs

OR/NAND

Equivalence

A

A

OR

OR

B

B

A

A

Nand

Nand

B

B

mult level logic conversion between forms

A

A

B

B

A

·

B

A

+

B

A

·

B

A

+

B

0

1

0

1

0

0

1

1

0

1

1

0

0

0

0

0

1

0

0

1

0

0

0

0

1

0

1

0

1

1

0

0

Mult-Level Logic: Conversion Between Forms

AND/NOR

Equivalence

A

A

AND

AND

B

B

A

A

NOR

NOR

B

B

Introduce appropriate inversions(“bubbles”)to convert from networks

with ANDs and ORs to networks with NANDs and NORs.

Each introduced "bubble" must be matched with a

corresponding "bubble“ to preserve logic levels

multi level logic conversion of forms5
Multi-Level Logic: Conversion of Forms

Example: Map AND/OR network to NAND/NAND network

(A)

(B)

AND

OR

AND

NAND

(C)

(D)

NAND

NAND

NAND

NAND

NAND

multi level logic conversion of forms6
Multi-Level Logic: Conversion of Forms

Example: Map AND/OR network to NAND/NAND network

NAND

NAND

A

A

NAND

B

B

Z

Z

C

C

D

D

Z = [(A · B)' + (C · D)']'

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

= [(A' + B')' ·(C' + D')']

= (A · B) + (C · D)

Equivalence of the

two forms can be verified

This is the easy conversion!

multi level logic mapping between forms
Multi-Level Logic: Mapping Between Forms

Example: Map AND/OR network to NOR/NOR network

A

\A

NOR

NOR

\B

NOR

B

Z

Z

C

\C

NOR

NOR

D

\D

Step 2

Step 1

Conserve

"Bubbles"

Conserve

"Bubbles"

Z = {[(A' + B')' + (C' + D')']'}'

= {(A' + B') ·(C' + D')}'

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

= (A · B) + (C · D)

Equivalence

of the two forms can be verified

This is the hard conversion!

AND/OR to NAND/NAND more natural

multi level logic mapping between forms8
Multi-Level Logic: Mapping Between Forms

Example: Map OR/AND network to NOR/NOR network

A

A

NOR

B

B

NOR

C

C

NOR

D

D

Conserve

Bubbles

Z

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

= {(A + B)'}' · {(C + D)'}'

= (A + B) · (C + D)

Equivalence

of the two forms Can be verified

This is the easy conversion!

multi level logic mapping between forms9
Multi-Level Logic: Mapping Between Forms

Example: Map OR/AND network to NAND/NAND network

Nand

Nand

Nand

Nand

Nand

Step 2

Step 1

Conserve

Bubbles!

Conserve

Bubbles!

{[(A' ·B')' · (C' ·D')']'}'

= {(A' ·B') + (C' ·D')}'

= (A' ·B')' · (C' ·D')'

= (A + B) ·(C + D)

Z =

Verify equivalence

of the two forms

OR/AND to NOR/NOR more natural

This is the hard conversion!

multi level logic more than two levels

Add double bubbles at inputs

Original circuit

Distribute bubbles

some mismatches

Insert inverters to fix

mismatches

Multi-Level Logic: More than Two-Levels

Conversion Example