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STEP FUNCTION. Mathematically, the step function of magnitude A can be ex-
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pressed as X(t) = Au(t) where u(t) is the unit-step function defined in Chap. 2. A graphical representation is shown in Fig. 5.3.
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x=0; t<o
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X=A; t10
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FIGURE 5-3 Step input.
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The transform of this function is X(S) = A/s. A step function can be approximated very closely in practice. For example, a step change in flow rate can be obtained by the sudden opening of a valve.
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IMPULSE F UNCTION.
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Mathematically, the impulse function of magnitude A is X(t) = Ati
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defined as where s(t) is the unit-impulse function defined and discussed in Chap. 4. A graphical representation of this function, before the limit is taken, is shown in Fig. 5.4. The true impulse function, obtained by letting b I 0 in Fig. 5.4, has Laplace transform A. It is used more frequently as a mathematical aid than as an actual input to a physical system. For some systems it is difficult even to approximate an impulse forcing function. For this reason the representation of Fig. 5.4 is valuable, since this form can usually be approximated physically by application and removal of a step function. If the time duration b is sufficiently small, we shall see in Chap. 6 that the forcing function of Fig. 5.4 gives a response that closely resembles the response to a true impulse. In this sense, we often justify the use of A as the Laplace transform of the physically realizable forcing function of Fig. 5.4.
SINUSOIDAL INPUT. This function is represented mathematically by the equa-
tions x=0 t<O tzo X =Asinwt where A is the amplitude and w is the radian frequency. The radian frequency w is related to the frequency f in cycles per unit time by w = 2~f. Figure 5.5 shows the graphical representation of this function. The transform is/X(s) = Aol(s* + o*). This forcing function forms the basis of an important branch of control theory known as frequency response. Historically, a large segment of the development of control theory was based on frequency-response methods, which will be presented in Chaps. 16 and 17. Physically, it is mote difficult to obtain a sinusoidal forcing function in most process variables than to obtain a step function. This completes the discussion of some of the common forcing functions. We shall now devote our attention to the transient response of the first-order system to each of the forcing functions just discussed.
RESPONSE
FIRST-ORDER
SYSTEMS
x=0; t<o X==A shut; t;rO
FIGURE 5-5 Sinusoidal input.
Step Response
If a step change of magnitude A is introduced into a first-order system, the transform of X(t) is X(s) = + The transfer function, which is given by Eq. (5.7), is 1 Y(s) -=7s + 1 X(s) Combining Eqs. (5.7) and (5.9) gives Y ( s ) A 1 = - s7s+l (5.10) (5.7) (5.9)
This can be expanded by partial fractions to give Y(s) = A/r (s)(s + l/T)
Cl c2 =-++ S
s + l/T
(5.11)
Solving for the constants Ci and C2 by the techniques covered in Chap. 3 gives Cl = A and Cz = -A. Inserting these constants into Eq. (5.11) and taking the inverse transform give the time response for Y: Y(t)= 0 Y(t)= A(1 - emr ) t<O tro (5.12)
Hereafter, for the sake of brevity, it will be understood that, as in Eq. (5.12), the response is zero before t = 0. Equation (5.12) is plotted in Fig. 5.6 in terms of the dimensionless quantities Y(r)/A and t/r. Having obtained the step response, Eq. (5.12), from a purely mathematical approach, we should consider whether or not the result seems to be correct from physical principles. Immediately after the thermometer is placed in the new environment, the temperature difference between the mercury in the bulb and the bath temperature is at its maximum vaIue. With our simple lumped-parameter model,
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