I N T E R P R E TAT I O N OF E RROR

I N T E R P R E TAT I O N OF E RROR

The representation of error as the difference between the set point and the process variable provides a “natural” value; that is, a value expressed in the units being measured. For example, a PLC that receives process variable and set point data in the form of analog counts will express the error as a function of analog counts. Likewise, a system with a set point of 125 ° C and a process variable of 120 ° C will have an error of 5 °

C. However, the system cannot determine if 5 ° C is an acceptable error because it does not know how close the error is to zero relative to the variable range. Therefore, another way for the controller to calculate error is as a percentage of the target set point. This is expressed as:

SP − PV E =

SP

Using this equation, the error for the previous temperature example would be:

125 °− C 120 ° C

E = 125 ° C

This indicates that the 5 ° C system error is within 4% of the set point target. This percentage value provides more information than the 5 ° C error value; however, the controller requires even more information to adjust the process correctly.

The expression of error as a percentage of the process variable range provides an even more indicative value of error. The range of PV indicates the maximum and minimum values that the process value can have. Figure 14-

10 illustrates the error as a percentage of the process variable range. Math- ematically, this can be expressed as:

E % = the error as a percentage of PV range SP = the set point PV = the process variable PV max = the maximum value of PV PV min = the minimum value of PV

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S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

Error PV

Figure 14-10. Error as a percentage of the process variable range.

Note that, in this equation, the sign of the terms PV min and PV max (positive and negative, respectively) are such that the error will be positive if the process variable is above the set point and negative if it is below the set point. This error representation provides additional information about the magnitude of the error.

E X AM PLE 1 4 -1

A process with a temperature set point of 180 °

C has a process variable

input of 168 °

C (see Figure 14-11). Express the error as a percentage of range given that the process variable has a range of (a) 100 °

Process PV = 168 ° Σ C

E Controller

Figure 14-11. Process control loop for Example 14-1 given (a) a 100–200 ° C process variable range and (b) a 50–350 °

C range.

S OLU T I ON

(a) The value of the error as a percentage of range ( E %) is expressed

as: SP PV −

E % = PV min − PV max

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S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

For a process variable range of 100 to 200 °

C, the error is:

(b) For a process variable range of 50 to 350 °

C, the error is:

Although the actual natural value of the error is the same in both parts (a) and (b)—i.e., 12 ° C—the magnitude of the error in the first case (12%) is three times greater than the magnitude of the error in the second case (4%). The negative sign of the error calculations indi- cates that the process variable is lower than the set point.