-8 D E R I VAT I V E C ONTROLLERS (D M ODE ) S TANDARD D E R I V AT I V E C ONTROLLERS

1 5 -8 D E R I VAT I V E C ONTROLLERS (D M ODE ) S TANDARD D E R I V AT I V E C ONTROLLERS

The output of a derivative controller is proportional to the rate of change of the error in the system, which is expressed as dE dt (see Figure 15-52). This derivative action, also referred to as rate mode, is expressed mathematically as:

dE

CV new = K D + CV old

dt where:

CV new = the control variable CV old = the previous value of CV K D = the derivative gain constant in %(sec/%) dE =

the rate of change of error over the duration of change in %/sec

Figure 15-52. Derivative controller action.

The derivative gain constant (K D ) is also referred to as the rate time. It can be

expressed in seconds or minutes as: K D = T D seconds (rate time)

or

minutes (if is given in seconds) T D

In Laplace form, the derivative controller transfer function takes the form:

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Figure 15-53 illustrates the derivative gain transfer function in a direct- acting system by indicating the corresponding controller outputs for differ-

ent rates of change ( dE dt ) in error. Like in the integral mode, the rates of error change form several family curves (see Figure 15-53b). For example, if the error increases at a rate of 1.0%/sec, the controller will apply a derivative action that makes its output jump from 50% to 70% (see Figure 15-53a). If the rate of increase slows down to 0.5%/sec, the controller will decrease its output to 60%. When the rate of change of error equals zero, the controller will decrease its output to 50% again (see Figure 15-54). Note that the derivative action is based on the rate at which the error changes, not the actual value of the error.

-2.5 -2.0 -1.5 -1.0 -0.5

2.5%/sec rate

2.0%/sec rate

(b)

1.5%/sec rate

1.0%/sec rate

0.5%/sec rate t (sec)

Figure 15-53. (a) Derivative controller transfer function and (b) its family of curves.

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Rate: 0% 0 Rate: 0.5%

Rate: –0.5%

Rate: 1%

Figure 15-54. Derivative controller response for the transfer function shown in

Figure 15-53a.

Derivative action is not used by itself in a controller; rather, it is used in combination with proportional and proportional-integral actions. There are several reasons for this. First, the derivative action response to a step change

(see Figure 15-55a) creates an infinite change in error over time ( dE dt =∞ ), causing the output of the controller to have 100% saturation for an instant (point 1 in Figure 15-55b). If the error remains at its stepped up value, the controller will sense no change and will return the control variable to 50% (between points 1 and 2). At point 2, when the error drops in a step fashion (see Figure 15-55a), the control variable will again have an infinite change over time, thus causing a 0% output (point 2 in Figure 15-55b).

The second reason why derivative action is not used alone is that it only produces a change in output if there is a change in the rate of error (points 3,

5, 6, and 7 in Figure 15-55). If a large error remains constant, the controller will maintain the control variable at 50% of its range (point 8), thus the error will not be corrected.

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t Figure 15-55. (a) Step changes and (b) their corresponding derivative responses.