-5 P R O P O RT I O N A L C ONTROLLERS (P M ODE )

1 5 -5 P R O P O RT I O N A L C ONTROLLERS (P M ODE )

A proportional controller adjusts the control variable output in a manner proportional to the error. As shown in Figure 15-24, the controller (Hc) receives feedback information from the process (Hp) in the form of the process variable, which is then compared to the set point. The error created, either positive or negative, tells the controller what percentage of output (CV) to provide to bring the error to zero. Figure 15-25 illustrates a typical proportional controller transfer function for a direct-acting controller (e.g., a cooling system). As the error becomes more negative (PV > SP), the controller will increase the control variable in proportion to the error. This will cause the process variable (from Hp) to decrease, thus pushing the error to zero. If the error becomes more positive, the opposite occurs.

Reaction CV CV

Direct Acting

P mode

PV

Figure 15-24. Proportional closed-loop control.

The control variable output (CV (t) ) of a proportional controller, starting from the set point value, is expressed by:

CV () t = KE P + CV ( E = 0 )

where:

K P = the proportional gain of the controller

E = the current error CV ( E = 0 ) = the controller output when the error equals 0

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Figure 15-25. Direct-acting proportional controller transfer function.

Using this equation, a proportional controller can adjust the value of the control variable according to time and error by replacing the CV (E=0) term with the previous value of CV:

CV new = KE P + CV old

So, if a controller with a control variable output value of 50% senses that the proportional error in the system (K P

E) is 20%, its new output will be 70%:

CV new = 20 % + 50 % = 70 %

This value indicates a linear correspondence between the control variable and the error, as was depicted in the graph in Figure 15-25. This graphic representation is called the proportional band, and it shows the error values associated with the full range of the controller output. The slope of this graph, the proportional gain K P , is computed by dividing the percentage change in output by the percentage change in error:

% change in CV

% change in E

The proportional gain, therefore, is expressed in units of %/%. For example,

a gain of 1 indicates that a 1% change in error will cause a 1% change in controller output. Note that the direction of the slope of the proportional gain (the positive or negative response of the control variable to a change in the error) depends on whether the controller is direct acting or reverse acting.

The proportional gain relationship between the error and the control variable depends on the width of the band upon which the controller is acting. For example, the temperature control system in Figure 15-26 has a temperature

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response that spans from 60 ° F to 180 °

F, equaling a range of 120 ° F (180 ° F–

60 ° F). However, if the controller only needs to exert control from 90 ° F to

150 ° F with the set point at 120 °

F, it will only be controlling a range of 60 ° F

F. Therefore, the proportional band of the controller is 60 ° F over the 120 ° F range. Accordingly, the proportional band (PB) of control as a percentage of the full process variable range is represented as:

(150 ° F – 90 °

F) over the total range of 120 °

PV max − PV min PB = PV (max range ) − PV (min range )

CV 100%

Band of control where proportional output is applied

PV 60 ° F 90 ° F 120 ° F 150 ° F 180 ° F

Proportional Band ( PB )

Gain ( K P )

100% – 0% K P =

K P = 100% = 2%/% 50%

2% change in control output

1% change in error in the band of control

Figure 15-26. Proportional band and gain calculations.

As shown in the calculations in Figure 15-26, the proportional band of the temperature control system is 50%. The proportional gain of the system is defined by how much the control variable output changes for each percent of

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error within the control band. The error percentage range is equal to the proportional band percentage, because both express how much the process variable can deviate from the set point. The gain, according to Figure 15-26, will be 2, meaning that the controller’s output will change 2% for every 1% change in error. This controller is a direct-acting controller, since the control variable will change in the same direction (+ or –) as the percentage of error. Note that the gain and proportional band are inversely related, meaning that: