CHAPTER 1 SAFETY

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Programmable Logic Controllers Third Edition
Frank D. Petruzella McGraw-Hill
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Chapter 3
Number Systems And Codes
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Decimal System
The radix or base of a number system
determines the total number of different symbols
or digits used by the system.
The decimal system has a base of 10.
In the decimal system, 10 unique numbers or
digits ( 0 through 9) are used the total
number of symbols is the same as the base, and
the symbol with the largest value is 1 less than
the base.
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Decimal System
The decimal system can be summarized as
follows Ten digits 0, 1, 2, 3, 4, 5, 6, 7,
8, 9 Base 10 Weights 1, 10, 100, 1000,
(powers of base 10)
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Decimal System
Weighted value in the decimal system
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Binary System
The binary system has a base of 2.
The only allowable digits are 0 and 1
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Binary System
The binary system can be summarized as
follows Two digits 0, 1 Base 2 Weights
1, 2, 4, 8, 16, 32, (powers of base 2)
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Binary System
Since the binary system uses only two digits,
each position of a binary number can go through
only two changes, and then a 1 is carried to the
immediate left position.
To express the number in the binary system
requires many more digits than in the
decimal system.
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Converting For Binary To Decimal
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Converting For Binary To Decimal
Another Method
In the binary number when you see a 1,
multiply that 1 times the value that is directly
over it. Where you see a 0 in the box, just
ignore it.
If we add only those numbers which have a binary
1 in the box under them, we come up with
12832841 which equals 173.
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Bits Bytes - Words
Each digit of a binary number is known as a
bit. A group of 8 bits is known as a byte. A
group of bits that occupies one or more storage
locations and is treated as a unit is known as a
word. A 16-bit word is made up of two bytes
(Upper and Lower). The least significant bit
(LSB) is the digit that represents the smallest
value. The most significant bit (MSB) is the
digit that represents the largest value.
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PLC Processor Memory Size
The size of the programmable controller relates
to the amount of user program that can be stored.
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Converting For Decimal To Binary
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One 16-bit output file word is reserved for each
slot in the chassis.
Each bit represents the on or off state of
one output point. These points are numbered 0
through15.
A word will be created in the table only if the
processor finds an output module residing in a
particular slot.
Made up of single bits grouped into 16-bit words
The column on the right lists the output module
address.
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1. The binary number system has a base of
8. (True/False)
2. The decimal number 7 would be written
in binary as 1011. (True/False)
3. To express a number in decimal requires fewer
digits than in the binary system. (True/False)
4. For a base 2 number system, the weight value
associated with the 3rd digit would be 4.
(True/False)
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5. What is the decimal value of binary 110000
? a. 48 c. 13 b. 26
d. 7
6. The decimal number 15 would be written
in binary as a. 1111 c. 4C b.
1000 d. 00011001
7. Data can be stored in one 16-bit word as two
separate groups of 8-bit data. (True/False)
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8. A group of 8 bits is known as a byte.
(True/False)
9. The MSB is the digit that represents the
smallest value. (True/False)
10. Since the binary system has only two
digits, it is limited as far as representing very
large quantities. (True/False)
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Negative Numbers
In the binary system it is not possible to use
positive and negative symbols to represent the
polarity of a number.
One method is of representing a binary number as
either a positive or negative value is to use an
extra digit, or sign bit, at the MSB of the
number. In the sign bit position, a 0 indicates
that the number is positive, and a 1 indicates
a negative number.
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Negative Numbers
Another method of expressing a negative number in
a digital system is by using the complement of a
binary number. To represent a negative number in
1's complement you simply take the numbers
magnitude and flip all the bits (i.e. 1 becomes
0, and 0 becomes 1).
3 binary representation 0011 -3 binary
representation 1100 (1s complement)
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Negative Numbers
The most common way to express a negative
binary number is to show it as a 2s complement
number. The 2s complement is the binary number
that results when 1 is added to the 1s
complement.
3 binary representation 0011 -3 binary
representation 1100 (1s complement) -3 binary
representation 1101 (2s complement)
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Octal System
The octal numbering system can be summarized as
follows Eight digits 0, 1, 2, 3, 4, 5, 6,
7 Base 8 Weights 1, 8, 64, 512, (powers
of base 8)
The octal number system is sometimes used
because 8 data bits make up a byte of
information that can be easily addressed by the
PLC user or programmer.
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Octal System
The Allen-Bradley PLC-5 family of PLCs uses the
octal numbering systems for addressing of I/O
modules.
I2/16
O3/22
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Octal System
The digits range from 0 to 7 therefore,
numbers 8 and 9 are not allowed!
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Converting Octal To Decimal
As in all other numbering system, each digit in
an octal number has a weighted decimal value
according to its position.
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Converting Octal-to-Binary
Octal is used to handle large binary numbers. One
octal digit is used to express three binary
digits.
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Hexadecimal System
The hexadecimal (hex) numbering system can be
summarized as follows Sixteen digits 0, 1,
2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Base
16 Weights 1, 16, 256, (powers of base 16)
The hex numbering system is used in PLCs
because a word of data often consists of 16 data
bits, or two 8-bit bytes.
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Hexadecimal-to-Decimal Conversion
To convert a hexadecimal number to its decimal
equivalent, the hexadecimal digits in the
columns are multiplied by the base 16 weight.
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Hexadecimal-to-Binary Conversion
Using the hex numbering system allows the
status of a large number of binary bits to be
represented in a small space such as a PLC
programming display.
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BCD (Binary-Coded Decimal) System
The BCD (Binary-Coded Decimal) numbering system
provides a convenient way of handling large
numbers that need to be inputted to or outputted
from a PLC.
There is no easy way to go from binary to
decimal and back. The BCD system provides a means
of converting a code readily handled by humans
(decimal) to a code readily handled by equipment
(binary).
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Examples Of Numeric Values In Decimal, Binary, BC
D, and Hexadecimal, Representation
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BCD Representation Of Decimal Number
Conversion from Decimal to BCD is
straightforward. You merely use 4 bits to
represent each decimal digit.
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BCD Thumb-Wheel Switch Interface
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Typical PLC Number Conversion Instruction
Convert To Decimal Instruction
This instruction will convert the binary bit
pattern at the source address N723, into a BCD
bit pattern of the same decimal value as the
destination address, O20. The instruction
executes every time it is scanned and the
instruction is true.
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Gray Code
The Gray code is a special type of binary code
that does not use position weighting.
It is set up so that as we progress from one
number to the next, only one bit changes. For
this reason, the Gray code is considered to be an
error-minimizing code.
Because only one bit changes at a time, the speed
of transition for the Gray code is considerably
faster than that for codes such as BCD.
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Gray Code
Gray codes are used with with position encoders
for accurate control of the motion of robots,
machine tools, and servomechanisms.
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ASCII Code
ASCII stands for American Standard Code
for Information Interchange. It is an
alphanumeric code because it indicates letters as
well as numbers.
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Parity Bit
Some PLC communications systems use a parity
bit to check the accuracy of data transmission.
For example, when data are transferred between
PLCs, one of the binary bits may accidentally
change states.
Parity is a system where each character
transmitted contains one additional bit known as
a parity bit. The bit may be binary 0 or binary
1, depending on the number of 1s and 0s in the
character itself.
Two systems of parity are normally used odd and
even.
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Parity Bit
Odd parity means that the total number of binary
1 bits in the character, including the parity
bit, is odd.
Even parity means that the total number of binary
1 bits in the character, including the parity
bit, is even.
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Binary Addition
When adding with binary numbers, there are
only four conditions that can occur.
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Binary Addition
When adding larger binary numbers, the
resulting 1s are carried into higher-order
columns.
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Binary Subtraction
To subtract from larger binary numbers,
subtract column by column, borrowing from the
adjacent column when necessary. Remember that
when borrowing from the adjacent column, there
are two digits, i. e., 0 borrow 1 gives 10.
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Binary Subtraction

• To subtract using the 1s complement
• Change the subtrahend to 1s complement
• Add the two numbers
• Remove the last carry and add it to the number

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Binary Multiplication
When multiplying binary numbers, there are
only four conditions that can occur.
0 x 0 0
0 x 1 0
1 x 0 0
1 x 1 1
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Binary Multiplication
To multiply numbers with more than one
digit, form partial products and add them
together.
101 x 110
000
101
101
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Binary Division
The process for dividing one binary number
by another is the same for both binary and
decimal numbers.
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Typical PLC Add, Subtract, Multiply, and Divide
Instructions
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PLC Data Comparison Instructions
Are used to compare the relative magnitude of
two quantities.
At times devices may need to be controlled when
they are less than, equal to or greater than
other data values or set points used in the
application, like timer and counter values.
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11. In the binary system and symbols are used
to indicate whether a number is positive
or negative. (True/False)
12. Numbers 8 and 9 are not used in the
octal number system. (True/False)
13. The octal number 153 would be written in
binary as a. 011 101 001 c. 011 111
101 b. 001 101 011 d. 010 100 011
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• 14. The hexadecimal (hex) numbering system is a
• base ______ system.
• 2
• 8
• 10
• 16

• 15. What is the decimal equivalent for the BCD
• number 1000 0100 0010 0001?
• 8421
• 7863
• 1234
• 3728

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16. The Gray code is set up so that as we
progress from one number to the next, only one
bit changes. (True/False)

• 17. Which code is used to convert the keystrokes
• on the keyboard of a computer for the
• processor?
• BCD
• HEX
• ASCII
• OCTAL

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• 18. A parity bit is used to check the _______ of
  data
• transmission.
• speed
• type
• accuracy
• time

19. What is the sum of binary numbers 1100 and
1011? a. 10110 c. 10111 b. 11100
d. 00111
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20. Which instruction is used to compare
the relative magnitude of two quantities? (a)
Add (b) Subtract (c) Multiply (d) Less Than