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and C 0:2 F, has the same applied voltage as in Example 8.6. Obtain the current by an s-domain analysis. As in Example 8.6, 1 i dt 10i 5 i dt v 10 308 est Ri C Since i Iest , 10 308 est 10Iest 5 st Ie s from which I 10 308 1:01 10 5=s 32:88

Comparison of (8.5) and (8.7) reveals that the network phase angle is the mirror image of the phase angle for the network of Example 8.1. (As ! increases, ranges from 08 to 908.) Further, (8.8) shows that the amplitude ratio is the mirror image of the amplitude ratio of Example 8.1. (As ! increases, Mdb ranges from 0 to positive values.) Thus, the complete Bode plot consists of the mirror images about zero of Mdb and of Fig. 8-2. Since here the phase angle is everywhere positive, this network is called a lead network.

36 60

Then, i 1:01e 2t cos 10t 32:88 (A).

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21. (a) 12; (b) 8; (c) 60 22. (a) 15; (b) 15; (c) 61 2 24. (a) 35 ; (b) 53 25. (a) a 27. (a) (b) (c) (d) (e) (f) 16; (b) b 15; (c) c 126 jABE > jEDC, jBAE > jDCE (also vert. at E) jBAF > jFEC, jB > jD (also jEAD > jBFA) jA > jEDF, jF > jBCA jA > jA, jB > jC jC > jD, jCAB > jCAD jA > jA, jC > jDBA

A break frequency or corner frequency is the frequency 1= . For a simple lag or lead network, it is the frequency at which M 2 jA j! j2 has changed by 50 percent from its value at ! 0; at that frequency, Mdb has changed by 3 db from its value at ! 0. Corner frequencies serve as key points in the construction of Bode plots.

Note that the s-domain impedance for the capacitance is 1= sC . a series RLC circuit will be Z s R sL 1= sC

28. (a) jD > jB, jAED > jFGB; (b) jADB > jABC, jA > jA; (c) jABC > jAED, jBAE > jEDA 29. (a) jC > jF, 14 20

A driving voltage of the form v Vest applied to a passive network will result in currents and voltages throughout the network, each having the same time function est ; for example, Ie j est . Therefore, only the magnitude I and phase angle need be determined. We are thus led to consider an s-domain where voltages and currents are expressed in polar form, for instance, V , I , and so on. Figure 8-12 suggests the correspondence between the time-domain network, where s j!, and the

CHAP. 8]

6 30. (a) 18 8 24

Fig. 8-12

The network has Laplace-domain transfer function A s s and frequency transfer function A j! j!. Converting A j! to polar form shows that tan 1 and ! 908 0 8:9 (8.10)

20 35

CHAP. 8]

(c) 12 18

Mdb 20 log !

s-domain where only magnitudes and phase angles are shown. In the s-domain, inductances are expressed by sL and capacitances by 1= sC . The impedance in the s-domain is Z s V s =I s . A network function H s is de ned as the ratio of the complex amplitude of an exponential output Y s to the complex amplitude of an exponential input X s If, for example, X s is a driving voltage and Y s is the output voltage across a pair of terminals, then the ratio Y s =X s is nondimensional. The network function H s can be derived from the input-output di erential equation an dny d n 1 y dy d mx d m 1 x dx a0 y bm m bm 1 m 1 b1 b0 x a1 n an 1 n 1 dt dt dt dt dt dt

21 4

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