NAND and NOR gates possess a special property: they are
universal. That is, given enough gates, either type of gate is able to mimic
the operation of any other gate type. For example, it is possible to
build a circuit exhibiting the OR function using three interconnected NAND
gates. The ability for a single gate type to be able to mimic any other gate
type is one enjoyed only by the NAND and the NOR. In fact, digital control
systems have been designed around nothing but either NAND or NOR gates, all
the necessary logic functions being derived from collections of
interconnected NANDs or NORs.
As proof of this property, this section will be divided into subsections
showing how all the basic gate types may be formed using only NANDs or only
NORs.
Constructing the NOT function
As you can see, there are two ways to use a NAND gate as an inverter, and
two ways to use a NOR gate as an inverter. Either method works, although
connecting TTL inputs together increases the amount of current loading to
the driving gate. For CMOS gates, common input terminals decreases the
switching speed of the gate due to increased input capacitance.
Inverters are the fundamental tool for transforming one type of logic
function into another, and so there will be many inverters shown in the
illustrations to follow. In those diagrams, I will only show one method of
inversion, and that will be where the unused NAND gate input is connected to
+V (either Vcc or Vdd, depending on whether the
circuit is TTL or CMOS) and where the unused input for the NOR gate is
connected to ground. Bear in mind that the other inversion method
(connecting both NAND or NOR inputs together) works just as well from a
logical (1's and 0's) point of view, but is undesirable from the practical
perspectives of increased current loading for TTL and increased input
capacitance for CMOS.
Constructing the "buffer" function
Being that it is quite easy to employ NAND and NOR gates to perform the
inverter (NOT) function, it stands to reason that two such stages of gates
will result in a buffer function, where the output is the same logical state
as the input.
Constructing the AND function
To make the AND function from NAND gates, all that is needed is an
inverter (NOT) stage on the output of a NAND gate. This extra inversion
"cancels out" the first N in NAND, leaving the AND function.
It takes a little more work to wrestle the same functionality out of NOR
gates, but it can be done by inverting ("NOT") all of the inputs to a NOR
gate.
Constructing the NAND function
It would be pointless to show you how to "construct" the NAND function
using a NAND gate, since there is nothing to do. To make a NOR gate perform
the NAND function, we must invert all inputs to the NOR gate as well as the
NOR gate's output. For a two-input gate, this requires three more NOR gates
connected as inverters.
Constructing the OR function
Inverting the output of a NOR gate (with another NOR gate connected as an
inverter) results in the OR function. The NAND gate, on the other hand,
requires inversion of all inputs to mimic the OR function, just as we needed
to invert all inputs of a NOR gate to obtain the AND function. Remember that
inversion of all inputs to a gate results in changing that gate's essential
function from AND to OR (or visa-versa), plus an inverted output. Thus, with
all inputs inverted, a NAND behaves as an OR, a NOR behaves as an AND, an
AND behaves as a NOR, and an OR behaves as a NAND. In Boolean algebra, this
transformation is referred to as DeMorgan's Theorem, covered in more
detail in a later chapter of this book.
Constructing the NOR function
Much the same as the procedure for making a NOR gate behave as a NAND, we
must invert all inputs and the output to make a NAND gate function as a NOR.
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