Lecture 6 Universal Gates

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Universal Gates

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Review of Boolean algebra

• Just like Boolean logic• Variables can only be 1 or 0

– Instead of true / false

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Review of Boolean algebra• Not is a horizontal bar above the number

– 0 = 1– 1 = 0

• Or is a plus– 0+0 = 0– 0+1 = 1– 1+0 = 1– 1+1 = 1

• And is multiplication– 0*0 = 0– 0*1 = 0– 1*0 = 0– 1*1 = 1

__

__

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Basic logic gates

• NOT

• AND

• OR

• NAND

• NOR

• XOR

x x

xy

xy xyxyz

zx+yx

yxy

x+y+zz

xy

xy

x+yxy

xÅyxy

Universal Gate – NANDThis presentation will demonstrate• The basic function of the NAND gate.• How a NAND gate can be used to replace an AND gate, an OR

gate, or an INVERTER gate.• How a logic circuit implemented with AOI(And, OR, Inverter)

logic gates can be re-implemented using only NAND gates.• That using a single gate type, in this case NAND, will reduce

the number of integrated circuits (IC) required to implement a logic circuit.

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AOI Logic NAND Logic

More ICs = More $$ Less ICs = Less $$

NAND Gate

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X Y Z0 0 1

0 1 1

1 0 1

1 1 0

XY

Y X YXZ +

NAND Gate as an Inverter Gate

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X Z0 1

1 0

X XZ

XXX (Before Bubble)

Equivalent to Inverter

NAND Gate as an AND Gate

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X Y Z0 0 0

0 1 0

1 0 0

1 1 1

XY

YX YXZ

YX

NAND Gate Inverter

Equivalent to AND Gate

NAND Gate as an OR Gate

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X Y Z0 0 0

0 1 1

1 0 1

1 1 1

Equivalent to OR Gate

X

YYXY X Y XZ ++

X

NAND GateInverters

Y

NAND Gate Equivalent to AOI Gates

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INVERTERORAND

Process for NAND Implementation1. If starting from a logic expression, implement the design with

AOI logic.

2. In the AOI implementation, identify and replace every AND,OR, and INVERTER gate with its NAND equivalent.

3. Redraw the circuit.

4. Identify and eliminate any double inversions (i.e., back-to-back inverters).

5. Redraw the final circuit.

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NAND Implementation

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Example:Design a NAND Logic Circuit that is equivalent to the AOI circuit shown below.

CA C B +

NAND Implementation

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Solution – Step 2

Identify and replace every AND,OR, and INVERTER gate with its NAND equivalent.

NAND Implementation

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Redraw the circuit.

Solution – Step 3

NAND Implementation

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Identify and eliminate any double inversions.

Solution – Step 4

NAND Implementation

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Solution – Step 5

Redraw the circuit.

Proof of Equivalence

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C C B

C AC A C BZ

C AC BZ

CA C B

+

+

AOI vs. NAND

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IC Type Gates Gate / IC # ICs

74LS04 1 6 1

74LS08 2 4 1

74LS32 1 4 1

Total Number of ICs → 3


74LS00 4 4 1


Universal Gate – NORThis presentation will demonstrate…• The basic function of the NOR gate.• How an NOR gate can be using to replace an AND gate, an OR

gate or an INVERTER gate.• How a logic circuit implemented with AOI logic gates could be

re-implemented using only NOR gates• That using a single gate type, in this case NOR, will reduce the

number of integrated circuits (IC) required to implement a logic circuit.

AOI Logic NOR Logic

More ICs = More $$ Less ICs = Less $$ 19

NOR Gate

X Y Z0 0 1

0 1 0

1 0 0

1 1 0

XY

Y X Y XZ +

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NOR Gate as an Inverter Gate

X Z0 1

1 0

X XZ

XXX + (Before Bubble)

Equivalent to Inverter

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NOR Gate as an OR Gate

X Y Z0 0 0

0 1 1

1 0 1

1 1 1

XY

YX Y XZ ++

Y X +

NOR Gate “Inverter”

Equivalent to OR Gate

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NOR Gate as an AND Gate

X Y Z0 0 0

0 1 0

1 0 0

1 1 1

Equivalent to AND Gate

X

YY XY X Y XZ +

X

NOR Gate“Inverters”

Y

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NOR Gate Equivalent of AOI Gates

INVERTERORAND

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Process for NOR Implementation1. If starting from a logic expression, implement the design with

AOI logic.

2. In the AOI implementation, identify and replace every AND,OR, and INVERTER gate with its NOR equivalent.

3. Redraw the circuit.

4. Identify and eliminate any double inversions. (i.e. back-to-back inverters)

5. Redraw the final circuit.

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NOR ImplementationExample:

Design a NOR Logic Circuit that is equivalent to the AOI circuit shown below.

CA C B +

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NOR ImplementationSolution – Step 2

Identify and replace every AND,OR, and INVERTER gate with its NAND equivalent.

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NOR ImplementationRedraw Circuit.

Solution – Step 3

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NOR ImplementationIdentify and eliminate any double inversions.

Solution – Step 4

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NOR ImplementationSolution – Step 5

Redraw Circuit.

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Proof of Equivalence

B

CA C AC A +

C A C BZ

C A C B

+

+

C

C

A

C BC BCB +

C A C B +

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AOI vs NOR


74LS04 1 6 1

74LS08 2 4 1

74LS32 1 4 1



74LS02 7 4 2


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Lecture 6 Universal Gates

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Transcript of Lecture 6 Universal Gates