We can construct simply logic functions for our hypothetical lamp circuit, using multiple contacts, and document these circuits quite easily and understandably with additional rungs to our original "ladder." If we use standard binary notation for the status of the switches and lamp (0 for unactuated or de-energized; 1 for actuated or energized), a truth table can be made to show how the logic works:
Now, the lamp
will come on if either contact A or contact B is actuated, because all it takes
for the lamp to be energized is to have at least one path for current from wire
L1 to wire 1. What we have is a simple OR logic function,
implemented with nothing more than contacts and a lamp.
We can mimic
the AND logic function by wiring the two contacts in series instead of
parallel:
Now, the lamp
energizes only if contact A and contact B are simultaneously actuated. A
path exists for current from wire L1 to the lamp (wire 2) if and
only if both switch contacts are closed.
The logical
inversion, or NOT, function can be performed on a contact input simply by using
a normally-closed contact instead of a normally-open contact:
Now, the lamp
energizes if the contact is not actuated, and de-energizes when the
contact is actuated.
If we take our OR
function and invert each "input" through the use of normally-closed
contacts, we will end up with a NAND function. In a special branch of
mathematics known as Boolean algebra, this effect of gate function
identity changing with the inversion of input signals is described by DeMorgan's
Theorem, a subject to be explored in more detail in a later chapter.
The lamp will
be energized if either contact is unactuated. It will go out only if both
contacts are actuated simultaneously.
Likewise, if
we take our AND function and invert each "input" through the use of
normally-closed contacts, we will end up with a NOR function:
- Parallel contacts are equivalent to an OR
gate.
- Series contacts are equivalent to an AND
gate.
- Normally-closed contacts are equivalent to
a NOT gate (inverter).
We can build
combinational logic functions by grouping contacts in series-parallel
arrangements, as well. In the following example, we have an Exclusive-OR
function built from a combination of AND, OR, and inverter (NOT) gates:
The top rung
(NC contact A in series with NO contact B) is the equivalent of the top NOT/AND
gate combination. The bottom rung (NO contact A in series with NC contact B) is
the equivalent of the bottom NOT/AND gate combination. The parallel connection
between the two rungs at wire number 2 forms the equivalent of the OR gate, in
allowing either rung 1 or rung 2 to energize the lamp.
To make the
Exclusive-OR function, we had to use two contacts per input: one for direct
input and the other for "inverted" input. The two "A"
contacts are physically actuated by the same mechanism, as are the two
"B" contacts. The common association between contacts is denoted by
the label of the contact. There is no limit to how many contacts per switch can
be represented in a ladder diagram, as each new contact on any switch or relay
(either normally-open or normally-closed) used in the diagram is simply marked
with the same label.
Sometimes,
multiple contacts on a single switch (or relay) are designated by a compound
labels, such as "A-1" and "A-2" instead of two
"A" labels. This may be especially useful if you want to specifically
designate which set of contacts on each switch or relay is being used for which
part of a circuit. For simplicity's sake, I'll refrain from such elaborate
labeling in this lesson. If you see a common label for multiple contacts, you
know those contacts are all actuated by the same mechanism.
If we wish to
invert the output of any switch-generated logic function, we must use a
relay with a normally-closed contact. For instance, if we want to energize a
load based on the inverse, or NOT, of a normally-open contact, we could do
this:
We will call
the relay, "control relay 1," or CR1. When the coil of CR1
(symbolized with the pair of parentheses on the first rung) is energized, the
contact on the second rung opens, thus de-energizing the lamp. From
switch A to the coil of CR1, the logic function is noninverted. The
normally-closed contact actuated by relay coil CR1 provides a
logical inverter function to drive the lamp opposite that of the switch's
actuation status.
Applying this
inversion strategy to one of our inverted-input functions created earlier, such
as the OR-to-NAND
From the
switches to the coil of CR1, the logical function is that of a NAND
gate. CR1's normally-closed contact provides one final inversion to
turn the NAND function into an AND function.
- REVIEW:
- Parallel contacts are logically equivalent
to an OR gate.
- Series contacts are logically equivalent
to an AND gate.
- Normally closed (N.C.) contacts are
logically equivalent to a NOT gate.
- A relay must be used to invert the output
of a logic gate function, while simple normally-closed switch contacts are
sufficient to represent inverted gate inputs.
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