Fundamental concept of PLC - Boolean Algebra

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.  

Data inside PLC and Maths behind that

Data Inside PLC

Think of PLC memory as a cabinet with drawers called Registers.  Data is held in these Registers.These Registers come in different sizes, and hold different kinds of data.

Bits:Can have values 1 or 0.  1 typically represents On while 0 represents Off.  Bits are the building block for all other types of data.

Integers:Whole numbers (no decimal points).  Called: Characters (char), Integers (int), long Integers (long int) or Words.  Look for the bit size and whether they are signed or unsigned.   Unsigned are positive numbers, while signed are positive or negative.

Floating point numbers: Numbers with decimal points, and can be positive or negative. They are called floating point numbers (Float), with their larger variety called double floats.

Type	# bits	Signed/Unsigned	Min Value	Max Value
Bit	1		0	1
Int	8	Signed	-128	127
		Unsigned	0	255
	16	Signed	-32,768	32,767
		Unsigned	0	65,535
	32	Signed	-2,147,483,648	2,147,483,647
		Unsigned	0	4,294,967,295
Floating Point	32		1.175 E–38	3.403 E+38
	64		2.225 E–308	1.798 E+308

Addresses/Tags: The Registers are all stacked side by side in the PLC’s memory.  Each location has an Address that lets the PLC know what data you’re talking about.  Older PLC software requires the user to refer to data by this Address (example “x1023” could mean the 1023’d register). Some newer software makes the Addresses transparent.  The user gives a piece of data a name (example “Oven2Temperature”), and the PLC keeps track of where the register is located. If the software uses Addresses to refer to data it’s called “Address Based”, if it uses named data it’s called “Tag Based”.  Some programming packages are Address Based, but allow Addresses to have “Nicknames” or something similar. Their benefits over standard Address Based systems tend to be limited.  Advantages to Tag based systems become evident as programs grow, and remembering what’s stored in x1023 becomes difficult.   If your program is going to have any complexity at all using a Tag Based System simplifies the design.

Diving Deeper

Inside the Data Types: As stated earlier, all Data Types are made of Bits (1′s and 0′s). An 8 bit number is written like bbbbbbbb (where “b” can represent a 0 or 1), so you may have 00000000, 11111111 or 01011110…. any combo will do.  What these Bits mean is determined by the Data Type.

Unsigned Integers: The least valuable bit is the rightmost bit and double in value each position you move left.  The right-most bit is 1 it’s worth 1 next bit to the left 2   4   8 16   32  64 128 256  …… (this goes on for as many bits as the Data Type calls for).  Here’s what its looks like for an Unsigned 8-bit Integer:

If all 8 bits are 0 (00000000) then we get 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0

If all 8 bits are 1 (11111111) then we get 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255

So the range of an 8 bit unsigned Integer is 0 – 255

An example of something in between (00110101)  0 + 0 +  32 + 16 + 0 + 4 + 0 + 1 = 53

Signed Integers:  They’re a tad more complicated check out wikipedia’s article on 2’s compliment for a good understanding.

Hexadecimal Format (Hex):  Instead of writing every bit out, it is common to group sets of 4 bits together. Each group can have a value of 0 – 15, which causes a problem since our number system goes from 0 – 9, so we also use A, B, C, D, E and F to get a total of 16 values. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , A, B, C, D, E, F)

Example 1001 0011 1111 0010 in Hex would be 93F2

To be clear that we’re using Hex format we tack on a ‘0x’ the the start of the number so our last example would be written 0x93F2

Characters: (typically 8 bit unsigned integers) are often used to represent letter or symbols.   Example: you may use them to display text on an operator panel. Use an ASCII table to see how values are mapped to symbols.

What about the bits in Floating Point numbers, how do they break out? It is somewhat complicated, I don’t really want to get into that in this Post.   It is unlikely as a PLC user that you will ever need to know how their bits translate to values, so don’t worry about it.

The Basics - Maths

Addition, Subtraction, Multiplication and Division are what you’d expect.  If you haven’t done math in a programming language before, there are a couple of things you should be aware of.

·         Overflow:  Integers and Floating Point Numbers have size limits.  When you go beyond their size you’ll cause yourself problems.
Example: 8 bit Unsigned Int with value 255 in bits looks like 11111111. If we add 1 to it we get 100000000. There isn’t storage for a 9′th bit, so we’re left with 00000000 or 0. So in this case 255 + 1 = 0
Another Example: 8 bit Unsigned Int with value 0 in bits looks like 00000000. If we subtract 1 to it we get 11111111. So, we can get the opposite problem 0 – 1 = 255

·         Rounding:  Floating Point numbers aren’t perfect.  For this reason, you shouldn’t check to see if Floating point numbers are equal to a value, you should check to see if it’s within a range of numbers.
Example: 2 * 3 performed with Floating Point numbers may not result in exactly 6.
It could be 5.9999999999998 or something similar. So instead of asking is Example = 6, ask is Example greater than 5.9 and less than 6.1. Don’t worry about how to ask these questions, we’ll talk about that later in the tutorial

Boolean Math

Some programming packages allow bit manipulation by using Boolean Math.  The main operations are:

OR (symbol ‘|’ ):  A = B | C.  If either B is 1 or C is 1, then A is 1.  If both B and C are 0, then A is 0.

AND (symbol ‘&’): A = B & C.  If both B and C are 1, then A is 1.  If either B of C are 0, then A is 0.

Exclusive OR (symbol ‘⊕’):  A = B ⊕ C.  If either B is 1 or C is 1, but not both, then A is 1.  If both B and C are 1, or both B and C are 0 then A is 0.