AND, OR and NOT
Life is full of decisions.
What is true for us is also true of PLCs. We gather information (input) and
based on that we make choices that determine our output. All though I've always
found computers to be quite a bit more logical then human beings.
For an example of how we use
logic in everyday life consider these statements:
If Tommy OR Bob
want to play basketball then I'll play too.
It's 6 o'clock AND I'm NOT hungry
therefore I'm going to keep playing.
If Mom comes out AND orders
me inside OR it get's dark then I'll stop playing.
Now these are pretty simple
decisions especially if you're a ten year old boy. You'll notice that
they all involve three types of comparisions: AND, OR and NOT. Now we
could get more complex but all that we'd be doing is using these simple
building blocks.
In the world of automation
these types of TRUE or FALSE conditions come down to a device being ON or OFF,
CLOSED or OPEN, PRESENT or ABSENT, 24 VOLTS or 0 VOLTS. In the PLC it all
boils down to our now familiar binary
system of a 1 or a 0. Typically having a bit ON represents a
TRUE condition while OFF is FALSE. This is abitrary though as it may make
more sense to use what is called failsafe logic and have an ON bit as a FALSE
condition.
Let's turn again to some
simple statements but this time using automation examples.
When the button is
pressed AND the door is closed then turn on the motor.
If the process is done OR the
emergency stop button signal is NOT on then turn off the motor.
(This is an example of a failsafe operation as the emergency stop button could
be pressed or the wire has been disconnected. In
either case we want to check this for safety reasons. Relying on a signal
to turn on when a wire has fallen off long ago may cause an awkward moment when
we truly have to stop the machine in an emergency.)
If the tank is full OR the
button is pressed AND there are no alarms then start the process.
It would be nice to program
like this but computers like to be a little bit more structured. A series
of graphical objects have been used for years to represent these logic elements
and they can be easily converted to a common ladder logic equivalent.
These functions are also called gates as they act like gate keepers for
different logic.
The simple switching circuits presented further down
should be sufficient to understand practical implementation in hardware of
Boolean Algebra. For clarity, we have attached the correspondent
firmware/software logic equivalent formulas.
BOOLEAN EQUIVALENT OF SWITCHING CIRCUITS  
CIRCUIT  DESCRIPTION 
Fig 1 : Function LED LED = A * B (in Boolean hardware) is equivalent to: LED = A AND B (in firmware logic)  
Fig 2 : Function LED LED = A + B (in Boolean hardware) LED = A OR B (in firmware logic)  
Fig 3 : Function LED LED = A * B * (C + D) (in Boolean hardware) LED = A AND B AND (C OR D) (in firmware logic)  
Fig 4 : Function LED LED = A * D + B * C (in Boolean hardware) LED = (A AND D) OR (B AND C) (in firmware logic)  
Fig 5 : Function LED LED = A * (B + C) (in Boolean hardware) LED = A AND (B OR C) (in firmware logic)  
Fig 6 : Function LED LED = A * B + A * B (in Boolean hardware) LED = (A AND B) OR [(NOT A) AND (NOT B)] (in firmware logic)  
The NOT function
The simplest of all logic
functions is the NOT gate.
It's sole function in life
is to invert of flip the logic state. So an input of 1 will come out as a
0 and visa versa. Shown below is a truth table (it doesn't lie) showing
all possible inputs and the resulting logical output.
Input A

Output

0

1

1

0

The ladder logic equivalent
for a NOT function looks like a normal contact but with a slash through it.
The AND function
The AND gate is associated
with the following symbol that can have any number of inputs but only one
output.
The truth table below shows
that the output is only turned on when all the inputs are true (1). An
easy way to remember this is AND works like multiplication.
Input A

Input B

Output

0

0

0

1

0

0

0

1

0

1

1

1

The ladder logic equivalent for an AND function looks
like two normal contacts side by side.
The OR function
Last but not least the OR
gate is associated with the following symbol that also can have any number of
inputs but only one output.
The truth table below shows
that the output is turned on (1) when any of the inputs are true (1). An
easy way to remember this is OR works like addition.
Input A

Input B

Output

0

0

0

1

0

1

0

1

1

1

1

1

The ladder logic equivalent
for an OR function looks like two normal contacts on top of each other.
Combining AND or OR with NOT
The NOT gate might not look
like much help if you haven't programmed much but you'll find yourself actually
using it frequently. It's very common to use it in combination with AND
and OR. So the engineering gods decided to make some symbols for these
combinations.
Putting the NOT and AND
gates together forms the NAND gate.The truth table below shows that it
is simply an inverted output of the AND gate.
Input A

Input B

Output

0

0

1

1

0

1

0

1

1

1

1

0

A little circle (or if you
like, a bubble) at the end of a AND gate is used to signify the NAND
function. Its symbol and corresponding ladder logic are shown
below. Now pay close attention to the ladder logic because the contacts
are in parallel and not in series like the AND function.
Putting the NOT and OR gates
together forms... you got it... the NOR gate. The truth table below shows
that it is simply an inverted output of the OR gate.
Input A

Input B

Output

0

0

1

1

0

0

0

1

0

1

1

0

Again a little circle is
placed at the end of an OR gate to signify the NOR function. Its symbol
and corresponding ladder logic are shown below. The ladder logic is very
different from the regular OR gate.
But wait! Don't order yet... the XOR gate.
So far with our logic gates
we've covered almost all possible combinations except for one shown by the
truth table below.
Input A

Input B

Output

0

0

0

1

0

1

0

1

1

1

1

0

The logic to produce this
output is called an Exclusive OR gate otherwise known as the XOR gate.
It's a specialized form of the OR gate. So if either one of the inputs
are on then the output is true, otherwise you're out of luck. The symbol
for the XOR gate is shown by added a curved line to the OR gate symbol.
The ladder logic to implement an XOR gate is a little
more complex than the others.
How useful is the XOR
logic? You probably use the XOR gate everyday without thinking about it
if you have a room with a light that works off two switches. If both
switches are in the same position then the light will be off. Therefore just
flipping one switch will turn the light on. In the PLC program this can
be extremely useful for programming alternating actions or gray
codes.
Ok, there is one more logic
gate but I promise it is the last one. It makes sense that there is a
XNOR gate which is the combination of the NOT and XOR logic. It simply
inverts the output of the XOR function.
Input A

Input B

Output

0

0

1

1

0

0

0

1

0

1

1

1

The symbol for the XNOR gate is shown below along with
it's ladder logic equivalent.
While these terms and
symbols may seem a bit esoteric for the PLC beginner they are important in the
long run. A good grasp of these essentials will make PLC programming
easier, simplier and save memory.
Thanks, You wrote awesome, I have learn lots of things from your article. Its really helpful for any readers.
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