03 nand nor conversions

7/31/2019 03 Nand Nor Conversions

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Page 1

NAND/NOR Only Implementations

Although we can implement any circuit with AND/OR/NOT, we can also implement

any circuit with only NAND or NOR gates.

We might want to do this because of technology considerations; That is, these gates

might be cheaper to implement in silicon or they might be the only type of gates we

have available.

Since we can always use only NAND or NOR gates, these gates are sometimes called

universal gates.

The trick (if you want to call it that) is to see that we can implement the three basic

gates (AND, OR, NOT) in terms of NAND or NOR gates.

ECE124 Digital Circuits and Systems


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NOT-gate via NAND

ECE124 Digital Circuits and Systems Page 2

We can implement a NOT-gate using a NAND by applying the same signal to

both inputs.


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AND-gate via NAND


We can implement an AND-gate by using a NAND gate and (then) another NAND gate

to invert the output.


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OR-gate via NAND


We can implement an OR-gate by using a NAND gate and (then) other NAND gates to invert the

inputs.

The ability to implement INV/AND/OR with NAND means we can implement anycircuit using only NANDs (if only my replacing the AND, OR and NOT gates withtheir NAND implementations).


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Comment


Before continuing, we should make note that a NAND gate performs the same function as an OR

gate with inverts at the inputs. This is useful to remember.

We can illustrate for a 3-input gate:


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2-level NAND only implementations (1)

If we are given a 2-level SOP, obtaining a NAND only implementation is trivial; we

simply use apply a double inversion and use the DeMorgan theorem being careful ofwhere we leave the inversions in the final expression.

For example:

f = x1 !x2 + !x1 x2 + x3

= !(!f)

= !( !(x1 !x2 + !x1 x2 + x3) )

= !( !(x1 !x2) !(!x1 x2) !(x3) )



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We can also make the conversion graphically. Again, given a 2-level SOP the procedure

is straightforward:

Insert double inverters at input of OR gate (input will be either from an AND gate, or

from an input literal).

Convert OR gate plus one level of inverters to NAND gate.

Convert AND-gates plus the other level of inverters to NAND gates.

Some literals might get inverted.


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Multi-level NAND gate implementations (1)

2x

1x

1x

2x

3x

4x

5x


Circuits are not always in 2-level (or SOP) form.

Example:

Conversion method is essentially the same:

Insert double inverters before OR gates. Use one set of inverters plus the OR gates to convert to NAND.

Use other inverts with AND gates to convert to NAND.

Occasionally, might need to implement an inverter using NAND.


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Multi-level NAND gate implementations (2)

2x

1x

1

x

2x

3x

4x

5x


2x

1x

1x

2x

3x

4x

5x

Notice the extra inverters


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NOT-gate via NAND


We can implement a NOT-gate using a NOR by applying the same signal to both

inputs.


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OR-gate via NOR


We can implement an OR-gate using a NOR and (then) another NOR to perform

inversion.


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AND-gate via NOR


We can implement an AND-gate by using a NOR gate and (then) other NOR gates to invert the

inputs.

The ability to implement INV/AND/OR with NOR means we can implement anycircuit using only NORs (if only my replacing the AND, OR and NOT gates withtheir NOR implementations).


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Comment


Before continuing, we should make note that a NOR gate performs the same function as an AND

gate with inverters at the inputs. This is useful to remember.

We can illustrate for a 3-input gate:


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2-level NOR only implementations (1)

If we are given a 2-level POS, obtaining a NOR only implementation is trivial; we

simply use apply a double inversion and use the DeMorgan theorem being careful ofwhere we leave the inversions in the final expression.

For example:

f = (x1 + x2)(x3 + x4)(x5)

= !(!f)

= !( !((x1 + x2)(x3 + x4)(x5)) )

= !( !(x1 + x2) + !(x3 + x4) + !(x5) )



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2-Llevel NOR gate implementations (2)


Can also make the conversion graphically and it is straightforward:

Insert double inverters at input of AND gate (input will be either from an OR gate, or froman input literal).

Convert AND gate plus one level of inverters to NOR gate.

Convert OR gates plus the other level of inverters to NOR gates.

Some literals might get inverted.


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2-Llevel NOR gate implementations (3)



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Multi-level NOR gate implementations (1)

2x

1x

1x

2x

3x

4x

5x


Circuits are not always in 2-level (or POS) form.

Example:

Conversion method is essentially the same:

Insert double inverters before AND gates.

Use one set of inverters plus the AND gates to convert to NOR.

Use other inverters with OR gates to convert to NOR.

Occasionally, might need to implement an inverter using NOR.


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Multi-level NOR gate implementations (2)

2x

1x

1x

2x

3x

4

x

5x


2x

1x

1x

2x

3x

4x

5x

Notice the extra inverters

03 nand nor conversions

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