Chapter 3: Number Systems and Codes Textbook: Petruzella, Frank D., Programmable Logic Controllers. McGraw Hill Companies Inc., 5 th edition
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.
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)
Decimal System Weighted value in the decimal system
Binary System The binary system has a base of 2. The only allowable digits are 0 and 1 Digital Signal Waveform: with digital circuits it is easy to distinguish between two voltage levels - +5 V and O V, which can be related to the binary digits 1 and 0. Volts +5 0 Time High (H) (1) Low (L) (0)
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)
Binary System Decimal Binary 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. 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000
Converting For Binary To Decimal
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. 128 64 32 16 8 4 2 1 If we add only those numbers which have a binary 1 in the box under them, we come up with 128+32+8+4+1 which equals 173.
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 16-Bit Word one or more storage locations MSB and is treated as a unit is known Bit as a word. A 16-bit word is made up of two bytes (Upper and Lower). The least Upper significant Byte bit (LSB) is the digit that represents LSB the smallest value. The most significant bit (MSB) is the digit that represents the largest value.
PLC Processor Memory Size The size of the programmable controller relates to the amount of user program that can be stored. The 1 K word memory size shown can store 1,024 words, or 16,380 (1,024 x 16) bits of information using 16-bit words or 32,768 (1,024 x 32) using 32 bit words.
Converting For Decimal To Binary
Binary Representation Of Data Even though the binary system has only two digits, it can be used to represent any quantity that can be represented in the decimal system. Computer memory is then a series of binary 1s and 0s. SLC 500 Modular Chassis Output Status File A Each One word 16-bit will represents be output created file the in word on the table is or reserved off only state if for the of each processor one slot The Made column up of on single the right bits lists grouped the output into 16-bit module words address. finds output the an point. chassis. output These module points residing are numbered in a particular 0 through15. slot.
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)
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)
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)
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. Sign Sign Bit Bit Magnitude Bits Bits Decimal Value
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 (1 s complement)
Negative Numbers The most common way to express a negative binary number is to show it as a 2 s complement number. The 2 s complement is the binary number that results when 1 is added to the 1 s complement. +3 binary representation: 0011-3 binary representation: 1100 (1 s complement) -3 binary representation: 1101 (2 s complement)
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.
Octal System The Allen-Bradley PLC-5 family of PLCs uses the octal numbering systems for addressing of I/O modules. I:2/16 O:3/22
Octal System The digits range from 0 to 7; therefore, numbers 8 and 9 are not allowed!
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.
Converting Octal-to-Binary Octal is used to handle large binary numbers. One octal digit is used to express three binary digits.
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.
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.
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.
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).
Examples Of Numeric Values In: Decimal, Binary, BCD, and Hexadecimal, Representation
BCD Representation Of Decimal Number Conversion from Decimal to BCD is straightforward. You merely use 4 bits to represent each decimal digit.
BCD Thumb-Wheel Switch Interface The circuit board has one connection for each bit s weight plus a common A decimal number is selected The thumb-wheel switch outputs the equivalent 4-bits of BCD data
Typical PLC Number Conversion Instruction Convert To Decimal Instruction This instruction will convert the binary bit pattern at the source address N7:23, into a BCD bit pattern of the same decimal value as the destination address, O:20. The instruction executes every time it is scanned and the instruction is true.
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.
Gray Code Gray codes are used with with position encoders for accurate control of the motion of robots, machine tools, and servomechanisms. Typical Encoder Disk The encoder disk is attached to a rotating shaft and outputs a digital Gray code signal that is used to determine the position of the shaft.
ASCII Code ASCII stands for American Standard Code for Information Interchange. It is an alphanumeric code because it indicates letters as well as numbers. The keystrokes on the keyboard of a computer are converted directly into ASCII for processing by the computer.
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 1 s and 0 s in the character itself. Two systems of parity are normally used: odd and even.
Odd parity means that the total number of binary 1 bits in the character, including the parity bit, is odd. Parity Bit Even parity means that the total number of binary 1 bits in the character, including the parity bit, is even.
Binary Addition When adding with binary numbers, there are only four conditions that can occur.
Binary Addition When adding larger binary numbers, the resulting 1 s are carried into higher-order columns.
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.
Binary Subtraction To subtract using the 1 s complement: 1. Change the subtrahend to 1 s complement 2. Add the two numbers 3. Remove the last carry and add it to the number 1 s complement
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
Binary Multiplication To multiply numbers with more than one digit, form partial products and add them together. 101 x 110 000 101 101 11110
Binary Division The process for dividing one binary number by another is the same for both binary and decimal numbers. 111 10 1110 10 11 10 10 10 00
Typical PLC Add, Subtract, Multiply, and Divide Instructions
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. A = B (A equals B) A > B (A is greater than B) A < B (A is less than B)
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
14. The hexadecimal (hex) numbering system is a base system. (a) 2 (b) 8 (c) 10 (d) 16 15. What is the decimal equivalent for the BCD number 1000 0100 0010 0001? (a) 8421 (b) 7863 (c) 1234 (d) 3728
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? (a) BCD (b) HEX (c) ASCII (d) OCTAL
18. A parity bit is used to check the of data transmission. (a) speed (b) type (c) accuracy (d) time 19. What is the sum of binary numbers 1100 and 1011? a. 10110 c. 10111 b. 11100 d. 00111
20. Which instruction is used to compare the relative magnitude of two quantities? (a) Add (b) Subtract (c) Multiply (d) Less Than