Programmable Logic Controllers. Su Chen Jonathon Lin

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3.9 Gray Codes

      The Gray code is a set of modified binary code that is designed to suit in such applications for which only one bit changes as the counting number increases. This means that the sequential numbers must not have more than one bit changes as the count increments by one. This code is especially useful in working with transducers for linear and angular position sensing. Table 3.11 summarizes the Gray code with binary and decimal equivalents.

3.10 PLC Data Formats

       3.10.1Basic Data Structure

      The data in PLCs is stored in binary word format. A word normally consists of two bytes, each having eight bits. A byte of 8 bits can express a maximum value of 255 decimals (Figure 3.6). The right-most bit in a one-byte word is called the least significant bit and left-most bit, the most significant bit. A two-byte word is used to express numbers larger than 255. It can hold up to 65535 decimals (Figure 3.7). With the signed decimal format, a two-byte word can store up to +32767 decimal numbers and up to –32767 decimal numbers (Figure 3.8).

      PLCs use two words of 16 bits to store decimal numbers larger than +32,767. Two words of 16 bits form a 32-bit decimal format (Figure 3.9). This 32-bit decimal format is referred to as double precision. With the signed numbers, a double precision word can have the maximum decimal value of +2,147,483,647 and the smallest value of –2,147,483,647.

      Table 3.11: Gray code with binary and decimal equivalents.

      Figure 3.6: One-byte word

      Figure 3.7: Two-byte word

      Figure 3.8: Signed decimal numbers

      Figure 3.9: Double precision

       3.10.2The One’s Complement Method

      Two methods are used to represent negative numbers in PLCs: one’s complement and two’s complement. With the one’s complement method, a negative number is obtained by inverting or complementing each bit of the positive binary number. Inverting a bit means to change 1 to 0 and 0 to 1. The procedure of one’s complement is as follows (using 16-bit word format):

      •Express the decimal number in the binary form with a positive sign.

      •Complement each bit.

       Example 3.13: Use the one’s complement method to express the negative number of 150

      The binary number of 150 expressed in 16-bit word is:

      0000000010010110₂

      Convert each bit of this binary number as below.

      The binary number for –150₁₀ in one’s complement becomes

      –150₁₀ = 1111111101101001₂

      The one’s complement of a negative number expressed in binary becomes a positive number with the same quantity. The one’s complement of a negative number is obtained in the same fashion.

       Example 3.14: Find the one’s complement of binary 11010101111110012 (-1075810)

      The procedure is listed below.

      The binary number for the one’s complement of –10758₁₀ becomes

      +10758₁₀ = 0010101000000110₂

       3.10.3The Two’s Complement Method

      The two’s complement method is similar to the one’s complement except that only those bits after the first 1 is detected are inverted. The first 1 is examined from right to left.

       Example 3.15: Find the two’s complement of +5010

      The binary number for 50₁₀ is 110010₂

      Fill in this binary number to a 16-bit word and convert the digits as follows:

      The binary number for the two’s complement of +50₁₀ becomes

      –50₁₀ = 1111111111001110₂

      If a negative number is given in two’s complement, its complement becomes a positive number. Its complement is derived in the same way.

       Example 3.16: Find the two’s complement of a negative number –2110

      The binary number of –21₁₀ given in two’s complement is

      1111111111101011

      Convert the binary number as follows:

      The two’s complement of –21₁₀ becomes:

      +21₁₀ = 0000000000010101₂

3.11 Floating-Point Decimal Numbers

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