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HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
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EXAMPLE 8.6 A series RL circuit, with R 10
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and L 2 H, has an applied voltage v 10 e 2t cos 10t 308 . Obtain the current i by an s-domain analysis. v 10 Since i Iest , 10 308 est 10Iest 2sIest Substituting s 2 j10, I 10 308 10 308 0:48 43:38 10 2 2 j10 6 j20 or I 10 308 10 2s 308 est Ri L di di 10i 2 dt dt
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Then, i Iest 0:48e 2t cos 10t 43:38 (A). EXAMPLE 8.7 A series RC circuit, with R 10
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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
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Then, i 1:01e 2t cos 10t 32:88 (A).
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Note that the s-domain impedance for the capacitance is 1= sC . a series RLC circuit will be Z s R sL 1= sC
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Thus the s-domain impedance of
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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
Fig. 8-12
CHAP. 8]
HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
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
When x t Xest and y t Yest , an sn an 1 sn 1 a1 s a0 est bm sm bm 1 sm 1 b1 s b0 est Then, H s Y s a sn an 1 sn 1 a1 s a0 nm X s bm s bm 1 sm 1 b1 s b0
In linear circuits made up of lumped elements, the network function H s is a rational function of s and can be written in the following general form H s k s z1 s z2 s z s p1 s p2 s p
where k is some real number. The complex constants zm m 1; 2; . . . ;  , the zeros of H s , and the pn n 1; 2; . . . ;  the poles of H s , assume particular importance when H s is interpreted as the ratio of the response (in one part of the s-domain network) to the excitation (in another part of the network). Thus, when s zm , the response will be zero, no matter how great the excitation; whereas, when s pn , the response will be in nite, no matter how small the excitation.
EXAMPLE 8.8 A passive network in the s-domain is shown in Fig. 8-13. current I s due to an input voltage V s . H s Obtain the network function for the
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