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Number
Systems
Before
we get too far ahead of ourselves, let's take a look at the various
number systems used by PLCs.
Many
number systems are used by PLCs. Binary and Binary Coded Decimal are
popular while octal and hexadecimal systems are also common.
Let's
look at each:
As we do, consider the following formula (Math again!):
Nbase= Ddigit * R^unit + .... D1R^1 + D0R^0
where
D=the value of the digit and R= # of digit symbols used in the given
number system.
The "*" means multiplication. ( 5 * 10 = 50)
The "^" means "to the power of".
As you'll recall any number raised to the power
of 0 is 1. 10^1=10, 10^2 is 10x10=100, 10^3 is 10x10x10=1000,
10^4 is 10x10x10x10=10000...
This
lets us convert from any number system back into decimal. Huh? Read
on...
 Decimal
This is the numbering system we use in everyday life. (well most
of us do anyway!) We can think of this as base 10 counting. It can
be called as base 10 because each digit can have 10 different states.
(i.e. 09) Since this is not easy to implement in an electronic
system it is seldom, if ever, used. If we use the formula above
we can find out what the number 456 is. From the formula:
Nbase= Ddigit * R^unit + .... D1R^1 + D0R^0
we have (since we're doing base 10, R=10)
N10= D410^2 + D510^1 + D610^0
= 4*100 + 5*10 + 6*
= 400 + 50 + 6
= 456.
 Binary
This is the numbering system computers and PLCs use. It was far
easier to design a system in which only 2 numbers (0 and 1) are
manipulated (i.e. used).
The binary system uses the same basic principles as the decimal
system. In decimal we had 10 digits. (09) In binary we only have
2 digits (0 and 1). In decimal we count: 0,1,2,3,4,5,6,7,8,9, and
instead of going back to zero, we start a new digit and then start
from 0 in the original digit location.
In other words, we start by placing a 1 in the second digit location
and begin counting again in the original location like this 10,11,12,13,
... When again we hit 9, we increment the second digit and start
counting from 0 again in the original digit location. Like 20,21,22,23....
of course this keeps repeating. And when we run out of digits in
the second digit location we create a third digit and again start
from scratch.(i.e. 99, 100, 101, 102...)
Binary works the same way. We start with 0 then 1. Since there is
no 2 in binary we must create a new digit.
Therefore we have 0, 1, 10, 11 and again we run out of room. Then
we create another digit like 100, 101, 110, 111. Again we ran out
of room so we add another digit... Do you get the idea?
The general conversion formula may clear things up:
Nbase= Ddigit * R^unit + .... D1R^1 + D0R^0.
Since we're now doing binary or base 2, R=2. Let's try to convert
the binary number 1101 back into decimal.
N10= D1 * 2^3 + D1 * 2^2 + D0 * 2^1 + D1 * 2^0
= 1*8 + 1*4 + 0*2 + 1*1
= 8 + 4 +0 +1
= 13
(if you don't see where the 8,4,2, and 1 came from, refer
to the table below).
Now
we can see that binary 1101 is the same as decimal 13. Try translating
binary 111. You should get decimal 7. Try binary 10111. You should get
decimal 23.
Here's
a simple binary chart for reference. The top row shows powers of 2 while
the bottom row shows their equivalent decimal value.
Binary
Number Conversions
2^15 
2^14 
2^13 
2^12 
2^11 
2^10 
2^9 
2^8 
2^7 
2^6 
2^5 
2^4 
2^3 
2^2 
2^1 
2^0 
32768 
16384 
8192 
4096 
2048 
1024 
512 
256 
128 
64 
32 
16 
8 
4 
2 
1 
 Octal
The binary number system requires a ton of digits to represent a
large number. Consider that binary 11111111 is only decimal 255.
A decimal number like 1,000,000 ("1 million") would need
a lot of binary digits! Plus it's also hard for humans to manipulate
such numbers without making mistakes.
Therefore several computer/plc manufacturers started to implement
the octal number system.
This system can be thought of as base8 since it consists of 8 digits.
(0,1,2,3,4,5,6,7)
So we count like 0,1,2,3,4,5,6,7,10,11,12...17,20,21,22...27,30,...
Using the formula again, we can convert an octal number to decimal
quite easily.
Nbase= Ddigit * R^unit + .... D1R^1 + D0R^0
So octal 654 would be: (remember that here R=8)
N10= D6 * 8^2 + D5 * 8^1 + D4 * 8^0
= 6*64 + 5*8 + 4*1
= 384 +40 +4
= 428
(if you don't see where the white 64,8 and 1 came from,
refer to the table below).
Now
we can see that octal 321 is the same as decimal 209. Try translating
octal 76. You should get decimal 62. Try octal 100. You should get decimal
64.
Here's
a simple octal chart for your reference. The top row shows powers of
8 while the bottom row shows their equivalent decimal value.
Octal
Number Conversions
8^7 
8^6 
8^5 
8^4 
8^3 
8^2 
8^1 
8^0 
2097152 
262144 
32768 
4096 
512 
64 
8 
1 
Lastly,
the octal system is a convenient way for us to express or write binary
numbers in plc systems. A binary number with a large number of digits
can be conveniently written in an octal form with fewer digits. This
is because 1 octal digit actually represents 3 binary digits.
Believe
me that when we start working with register data or address locations
in the advanced chapters it becomes a great way of expressing data.
The following chart shows what we're referring to:
Binary
Number with its Octal Equivalent
1 
1 
1 
0 
0 
1 
0 
0 
1 
1 
1 
0 
0 
1 
0 
1 
1 
6 
2 
3 
4 
5 
From
the chart we can see that binary 1110010011100101 is octal 162345. (decimal
58597) As we can see, when we think of registers, it's easier to think
in octal than in binary. As you'll soon see though, hexadecimal is the
best way to think. (really)
 HexadecimalThe
binary number system requires a ton of digits to represent a large
number. The octal system improves upon this. The hexadecimal system
is the best solution however, because it allows us to use even less
digits. It is therefore the most popular number system used with
computers and PLCs. (we should learn each one though)
The hexadecimal system is also referred to as base 16 or just simply
hex. As the name base 16 implies, it has 16 digits. The digits are
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
So we count like
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13,...
1A,1B,1C,1D,1E,1F,20,21... 2A,2B,2C,2D,2E,2F,30...
Using the formula again, we can convert a hex number to decimal
quite easily.
Nbase= Ddigit * R^unit + .... D1R^1 + D0R^0
So hex 6A4 would be:(remember here that R=16)
N10= D6 * 16^2 + DA * 16^1 + D4 * 16^0
= 6*256 + A(A=decimal10)*16 + 4*1
= 1536 +160 +4
= 1700
(if you don't see where the 256,16 and 1 came from,
refer to the table below)
Now
we can see that hex FFF is the same as decimal 4095. Try translating
hex 76. You should get decimal 118. Try hex 100. You should get decimal
256.
Here's
a simple hex chart for reference. The top row shows powers of 16 while
the bottom row shows their equivalent decimal value. Notice that the
numbers get large rather quickly!
Hex
Number Conversions
16^8 
16^7 
16^6 
16^5 
16^4 
16^3 
16^2 
16^1 
16^0 
4294967296 
268435456 
16777216 
1048576 
65536 
4096 
256 
16 
1 
Finally,
the hex system is perhaps the most convenient way for us to express
or write binary numbers in plc systems. A binary number with a large
number of digits can be conveniently written in hex form with fewer
digits than octal. This is because 1 hex digit actually represents 4
binary digits.
Believe
me that when we start working with register data or address locations
in the advanced chapters it becomes the best way of expressing data.
The following chart shows what we're referring to:
Binary
Number with its Hex Equivalent
0 
1 
1 
1 
0 
1 
0 
0 
1 
0 
1 
0 
0 
1 
0 
1 
7 
4 
A 
5 
From
the chart we can see that binary 0111010010100101 is hex 74A5. (decimal
29861) As we can see, when we think of registers, it's far easier to
think in hex than in binary or octal.
4
digits go a long way after some practice!
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