Numbering Systems - AutomationDirect
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1. E E a E Ng eae A eae eee ee ee eit 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 Sign Exponent SS HHHH Table 7 As seen in Table 7 the BCD and hexadecimal formats are similar although the maximum number for each grouping is different 9 for BCD and F for hexadecimal This allows both formats to use the same display method The unfortunate side effect is that unless the data type is documented it s difficult to know what the data type is unless it contains the letters A F Decimal Bit Value gt xe ke D 2 x lt RNYO Z Z 2 Z Co op lt n D 3 D DL405 User Manual 4th Edition Rev A H 8 Numbering Systems Data Type Mismatch Data type mismatching is a common problem when using an operator interface Diagnosing it can be a challenge until you identify the symptoms Since the PLC uses BCD as the native format many people tend to think it is interchangeable with binary unsigned integer format This is true to some extent but not in this case Table 8 shows how BCD and binary numbers differ Data Type Mismatch Decimal 10 11 BCD 0001 0000 0001 0001 Binary 0000 1010 0000 1011 Table 8 As the table shows BCD and binary share the same bit pattern up until you getto the decimal number 10 Once you get past 10 the bit pattern changes The BCD bit pattern for the decimal 10 is actually equal to a value of 16 in binary causing the number to jump2. whereas only 100 distinct numbers i e 0 99 could be coded using BCD Also note that BCD is a subset of hexadecimal and neither one does negative numbers BCD Bit Pattern 11 110 8 7 Power Bit Value Max Value Table 5 One plus for BCD is that it reads like a decimal number whereas 867 in BCD would mean 867 decimal No conversion is needed DL405 User Manual 4th Edition Rev A H 6 Numbering Systems Real Floating Point Numbering System The terms Real and floating point both describe IEEE 754 floating point arithmetic This standard specifies how single precision 32 bit and double precision 64 bit floating point numbers are to be represented as well as how arithmetic should be carried out on them Most PLCs use the 32 bit format for floating point or Real numbers which will be discussed here Real Floating Point 32 Bit Pattern Bit 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Mantissa continues from above Table 6 Floating point numbers which DirectLOGIC PLCs use have three basic components sign exponent and mantissa The 32 bit word required for the IEEE standard floating point numbers is shown in Table 6 It is represented as a number bits from 0 to 31 left to right The first bit 31 is the sign bit the next eight bits 30 23 are the exponent bits and the final 23 bits 22 0 are the fraction bits In summary The sign bit is either O
3. The BCD and hexadecimal numbering systems do not use signed data types In order to signify that a number is negative or positive we must assign a bit to it Usually this is the Most Significant Bit MSB as shown in Table 10 For a 16 bit number this is bit 15 This means that for 16 bit numbers we have a range of 32767 to 32767 BE eee Table 10 There are two ways to encode a negative number two s complement and Magnitude Plus sign The two methods are not compatible The simplest method to represent a negative number is to use one bit of the PLC word as the sign of anumber while the remainder of the word gives its magnitude It is general convention to use the most significant bit MSB as the sign bit a 1 will indicate a negative and a 0 a positive number Thus a 16 bit word allows numbers in the range 32767 The following tables show a representation of 100 and a representation of 100 in this format Peca Binary Table 11 Two s complement is a bit more complicated Without getting involved with a full explanation a simple formula for two s complement is to invert the binary and add one see Table 12 Basically 1 s are being changed to 0 s and all 0 s are being changed to 1 then a 1 is added Z 5 SS QS J 0 a 9 I gt 3 i Two s Compliment 0000 0000 0110 0100 1111 1111 1001 1100 Table 12 More information about 2 s complement can be found on the Internet however most of
4. for positive or 1 for negative The exponent uses base 2 The first bit of the mantissa is typically assumed to be 1 fff where f is the field of fraction bits The Internet can provide a more in depth explanation of the floating point numbering system One website to look at is http www psc edu general software packages ieee ieee php no Ta x 2D c oop 9 E pS Z DL405 User Manual 4th Edition Rev A H 7 Numbering Systems BCD Binary Decimal Hex Octal What is the Difference Sometimes there is confusion about the differences between the data types used in DirectLOGIC PLCs The PLC s native data format is BCD while the I O numbering system is octal Other numbering formats used are binary decimal and Real Although data is stored inthe same manner 0 s and 1 s there are differences in the way that the PLC interprets it While all of the formats rely in the base 2 numbering system and bit coded data the format of the data is dissimilar The formats discussed are shown in Table 7 below Binary Decimal Bit Pattern Ea ee A ead Bre 5 14 13 2 tt tots e ft 7 te s 4 3 e tt oo Soa sae Weed el espe Weed eal eae Sd Bit Value mavas FT OT O a0 SSS OS ste 6 14 fis fa i fiofs s 7 ets ts 3 2 ii o Sree EE ie eae ee eee Neale Bit Value mavas 0 gt gt T Bie aeaea aa a a e e e BitValue 1 f4 2 t paja i jefa i aje pi fee i
5. system was developed as a convenience for humans since the PLC computer only understands pure binary The hexadecimal system is useful because it can represent every byte 8 bits as two consecutive hexadecimal digits It is easier for us to read hexadecimal numbers than binary numbers Hexadecimal Numbering System The hexadecimal numbering system uses 16 characters base 16 to represent values The first ten characters are the same as our decimal system 0 9 and the first six letters of the alphabet A F Table 3 lists the first eighteen decimal numbers 0 17 in the left column and the equivalent hexadecimal numbers are shown in the right column Decimal Hex Decimal Hex 9 gt xe Ss D J 2 x T Z Z 2 Z Co op lt 2 D 3 D 9 A B C D E F he oO Table 3 Note that 10 and 11 in hex are not the same as 10 and 11 in decimal Only the first ten numbers 0 9 are the same in the two representations For example consider the hex number D8AF To evaluate this hex number use the same method used to write decimal numbers Each digit ina decimal number represents a multiple of a power often base 10 Powers of ten increase from right to left For example the decimal number 365 means 3x1 02 6x10 5 In hex each digit represents a multiple of a power of sixteen base 16 Therefore the hex number D8AF translated to decimal means
6. 13x168 8x162 10x16 15 55471 However going through the arithmetic for hex numbers in order to evaluate them is not really necessary The easier way is to use the calculator that comes as an accessory in Windows It can convert between decimal and hex when in Scientific view Note that a hex number such as 365 is not the same as the decimal number 365 Its actual value in decimal terms is 3x162 6x16 5 869 To avoid confusion hex numbers are often labeled or tagged so that their meaning is clear One method of tagging hex numbers is to append a lower case h at the end Another method of labeling is to precede the number with Ox Thus the hex number D8AF can also be written D8AFh where the lower case h at the end is just a label to make sure we know that it is a hex number Also D8AF can be written with a labeling prefix as OxD8AP DL405 User Manual 4th Edition Rev A H 4 Numbering Systems no Ta x 2D c oop 9 E pS Z Octal Numbering System Many of the early computers used the octal numbering system for compiled printouts Today the PLC is about the only device that uses the Octal numbering system The octal numbering system uses 8 values to represent numbers The values are 0 7 being Base 8 Table 4 shows the first 32 decimal digits in octal Note that the octal values are 0 7 10 17 20 27 and 30 37 Octal Decimal Octal Decimal 20 16 22 18 2
7. 3 19 24 20 35 29 36 30 7 31 0 0 1 1 2 2 3 3 4 4 5 5 7 7 10 8 11 9 12 1 13 1 14 1 15 1 16 1 17 1 Le Table 4 This follows the DirectLOGIC PLCs Refer to the bit maps in Chapter 3 and notice that the memory addresses are numbered in octal as well as each bit The octal system is much like counting in the decimal system without the digits 8 and 9 being available The general format for four digits of the octal number is d x 8 d x 81 d x 82 d x 83 where d means digit This is the same format used in the binary decimal or hexadecimal systems except that the base number for octal is 8 0 1 2 TEREE 4 5 DL405 User Manual 4th Edition Rev A Numbering Systems H 5 Using the powers of expansion the example below shows octal 4730 converted to decimal positional value gt 512 64 8 1 Ox8 0x 1 0 3x8 3x 8 24 7x8 7x 64 448 4x 8 4 x 512 2048 2520 decimal equivalent oi gt xe ke D J 2 x T Binary Coded Decimal BCD Numbering System Z Z 2 Z Co op lt 2 D 3 D BCD is a numbering system where four bits are used to represent each decimal digit The binary codes corresponding to the hexadecimal digits A F are not used in the BCD system For this reason numbers cannot be coded as efficiently using the BCD system For example a byte can represent a maximum of 256 different numbers i e 0 255 using normal binary
8. Numbering Systems In This Appendix Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal BCD Numbering System Real Floating Point Numbering System BCD Binary Decimal Hex Octal What is the Difference Data Type Mismatch Signed vs Unsigned Integers AutomationDirect com Products and Data Types H 2 Numbering Systems wn z x lt Z0 cD Oo i Z Introduction As almost anyone who uses a computer is somewhat aware the actual operations of a computer are done with a binary number system Traditionally the two possible states for a binary system are represented by the digits for zero 0 and one 1 although off and on or sometimes no and yes are closer to what is actually involved Most of the time a typical PC user has no need to think about this aspect of computers but every now and then one gets confronted with the underlying nature of the binary system A PLC user should be more aware of the binary system specifically the PLC programmer This appendix will provide an explanation of the numbering systems most commonly used by a PLC Binary Numbering System Computers including PLCs use the Base 2 numbering system which is called Binary or Boolean Like in a computer there are only two valid digits in Base 2 a PLC relies on zero and one or off and on respectively You
9. ore In the C more and C more Micro Graphic HMI operator panels the 16 bit BCD Micro Graphic format is listed as BCD int 16 Binary format is either Unsigned int 16 or Signed Panels int 16 depending on whether or not the value can be negative Real number format is Floating PT 32 More available formats are BCD int 32 Unsigned int 32 and Signed int 32 DL405 User Manual 4th Edition Rev A
10. six digits by when viewing it as the BCD With larger numbers the error multiplies Binary values from 10 to 15 Decimal are actually invalid for the BCD data type Looking at a larger number such as the value shown in Table 9 both the BCD bit pattern and the decimal bit pattern correspond to a base 10 value of 409510 If bit patterns are read or interpreted in a different format than what is used to write them the data will not be correct For instance if the BCD bit pattern is interpreted as a decimal binary bit pattern the result is a base 10 value of 1653310 Similarly if you try to view the decimal binary bit pattern as a BCD value it is not a valid BCD value at all but could be represented in hexadecimal as OxFFF 1 o x Z0 c oop o E pS Z Base 10 Value BCD Bit Pattern Binary Bit Pattern 4095 0100 0000 1001 0101 1111 1111 1111 Table 9 Look atthe following examples and note the same value represented by the different numbering systems The decimal values of 67 and 4660 are used 6 7 Decimal 4 6 6 Q Decimal 0110 0111 BCD 0100 0110 0110 0000 BCD 0100 0011 Binary 0001 0010 0011 0100 Binary 4 3 Hex 1 2 3 4 Hex 1 0 3 Octal 1 1 0 6 4 Octal DL405 User Manual 4th Edition Rev A Numbering Systems H 9 Signed vs Unsigned Integers So far we have dealt with unsigned data types only Now we will deal with signed data types negative numbers
11. the websites deal with 8 bit examples Two fairly good websites are listed below you can also do a search and checkout more websites http www evergreen edu biophysics technotes program 2s_comp htm http www website masmforum com tutorials fptute fouchap2 htm DL405 User Manual 4th Edition Rev A H 10 Numbering Systems AutomationDirect com Products and Data Types DirectLOGIC PLCs The DirectLOGIC PLC family uses the octal numbering system for all addressing which includes inputs outputs internal V memory locations timers counters internal control relays bits etc Most data in the PLC including timer and counter current values is in BCD format by default User data in V memory locations may be stored in other data types if it is changed by the programmer or comes from some external source such as an operator interface Any manipulation of data must use instructions appropriate for that data type which includes Load instructions Math instructions Out box instructions comparison instructions etc In many cases the data can be changed from one data type to another but be aware of the limitations of the various data types when doing so For example to change a value from BCD to decimal binary use a BIN instruction box To change from BCD to a real number use a BIN and a BTOR instruction box When using Math instructions the data types must match For example a BCD or decimal binary number cannot be added to a real n
12. umber and a BCD number cannot be added to a decimal binary number If the data types are mismatched the results of any math operation will be meaningless To simplify making number conversions Intelligent Box IBox Instructions are available with DirectSOFT 5 These instruction descriptions are located in the DL405 IBox Instructions PLC User Manual Supplement page 30 in the Math IBox group Most DirectLOGIC analog modules can be setup to give the raw data in decimal binary format or in BCD format so it is necessary to know how the module is being used DirectLOGIC PID is another area where not all values are in BCD In fact nearly all of the PID parameters are stored in the PLC memory as decimal binary numbers wn z x lt Z0 cD Oo a Z NOTE The PID algorithm uses magnitude plus sign for negative decimal binary numbers whereas the standard math functions use two s complement This can cause confusion while debugging a PID loop When using the Data View in DirectSOFT be certain that the proper format is selected for the element to be viewed The data type and length is selected using the drop down boxes atthe top of the Data View window Also notice that BCD is called BCD Hex Remember that BCD is a subset of hexadecimal so they share a display format even though the values may be different This is where good documentation of the data type stored in memory is crucial C more C m
13. would think that it would be hard to have a numbering system built on Base 2 with only two possible values but the secret is by encoding using several digits Each digit in the base 2 system when referenced by a computer is called a bit When four bits are grouped together they form what is known as a nibble Eight bits or two nibbles would be a byte Sixteen bits or two bytes would be a word as in Thirty two bits or two words is a double word as in Table 1 Double Word Bye Bye ejojojojojojojojojojojojojojojojojojojojojojojojojojojojojojojo Table 1 Binary is not natural for us to use since we have grown up using the base 10 system Base 10 uses the numbers 0 9 as we are all well aware From now on the different bases will be shown as a subscripted number following the number Example 10 decimal would be 1040 Table 2 shows how base 2 numbers relate to their decimal equivalents A nibble of 1001 would be equal to a decimal number 9 1 23 1 20 or 840 110 A byte of 110101012 would be equal to 213 1 27 1 26 1 24 1 22 1 20 or 12840 6446 1640 440 140 Binary Decimal Bit Pattern Bit Power 4 13 12 1 1 9 s8f7 6 513 pi gii 10 Decimal Bit Value g Max Value DL405 User Manual 4th Table 2 Edition Rev A Numbering Systems H 3 The binary numbering system can be difficult and cumbersome to interpret for some users Therefore the hexadecimal numbering
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