-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.
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