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It has been pointed out that perfect derivative action is not available in conventional controllers. Derivative gain is limited to about 10, and the maximum available phase lead is in the vicinity of 45 . In effect, derivative is therefore accompanied by a lag, whose time constant is ><c the value of the derivative time. In most controllers, derivative act,s on the output. Physically, it is a lag introduced in the feedback path around the controller amplifier. Therefore, if he output of the controller is constant, no derivative action will take place no matter what the controlled variable may be doing. This situation occurs whenever the controller s output has reached one of its limits. The output of a pneumatic controller, for example, can go as low as zero or as high as 20 psi (the supply pressure), although the range of the control valve-hence the proportional band-is 3 to 15 psi. Ko derivative action will take place, then, until the controlled variable approaches the proportional band as in Fig. 4.2. In the discussion on two-capacity processes, this property placed a limitation on the width of the proportional band required for critical damping. In most controllers, derivative action does not distinguish between measurement and set point. The purpose of derivative is to speed the response of the closed loop, but the set point lies outside the loop. The controlled variable cannot change instantaneously, because of the lags inherent in the process. But it is normal to introduce set-point changes instantaneously, which derivative action amplifies into gross output

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We can break the vector a up in the following manner: a = (xa,ya,za) = (xa,0,0) + (0,ya,0) + (0,0,za) = xa(1,0,0) + ya(0,1,0) + za(0,0,1) = xai + yaj + zak

Question 8-8

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Suppose we have two standard-form vectors in Cartesian xyz space, defined as a = (xa,ya,za) and b = (xb,yb,zb) How can we calculate the dot product a b How can we calculate the dot product b a How do they compare

Answer 8-8

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We can calculate a b as a real number using the formula a b = xaxb + yayb + zazb Alternatively, it is a b = rarb cos qab where ra is the magnitude of a, rb is the magnitude of b, and qab is the angle between the vectors as determined in the plane containing them both, rotating from a to b In the same fashion, we can calculate b a using the formula b a = xbxa + ybya + zbza Alternatively, it is b a = rbra cos qba where rb is the magnitude of b, ra is the magnitude of a, and qba is the angle between the vectors as determined in the plane containing them both, rotating from b to a The dot product is commutative In other words, for all vectors a and b in Cartesian xyz space, we can be sure that

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fluctuations. Consequently, properly adjusted derivative acting upon the error or the output is hypersensitive to set-point variations. Ideally, derivative should act on neither the output nor the error, but on the measurement alone. Figure 4.3 shows the effect of derivative acting on the set point. Response can be improved markedly by introducing a lag in the setpoint circuit. Figure 4.4 indicates the sort of results that can be obtained with such an arrangement. Set-point changes are often introduced automatically, by the output of another controller in cascade (see Chap. 6). The controller whose set point is adjusted in this way cannot tolerate derivative, especially if the primary (adjusting) controller has it.

Review Questions and Answers Question 8-9

How can we find the cross product of two standard-form vectors a and b in three-space if we know their magnitudes and the angle between them

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