CHAP. 8]

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

171

s-domain where only magnitudes and phase angles are shown. In the s-domain, inductances are expressed by sL and capacitances by 1=ðsCÞ. The impedance in the s-domain is ZðsÞ ¼ VðsÞ=IðsÞ. A network function HðsÞ is deﬁned as the ratio of the complex amplitude of an exponential output YðsÞ to the complex amplitude of an exponential input XðsÞ If, for example, XðsÞ is a driving voltage and YðsÞ is the output voltage across a pair of terminals, then the ratio YðsÞ=XðsÞ is nondimensional. The network function HðsÞ can be derived from the input-output diﬀerential equation an

dny d n1 y dy d mx d m1 x dx þ a0 y ¼ bm m þ bm1 m1 þ    þ b1 þ b0 x þ an1 n1 þ    þ a1 n dt dt dt dt dt dt

When xðtÞ ¼ Xest and yðtÞ ¼ Yest , ðan sn þ an1 sn1 þ    þ a1 s þ a0 Þest ¼ ðbm sm þ bm1 sm1 þ    þ b1 s þ b0 Þest Then, HðsÞ ¼

YðsÞ a sn þ an1 sn1 þ    þ a1 s þ a0 ¼ nm XðsÞ bm s þ bm1 sm1 þ    þ b1 s þ b0

In linear circuits made up of lumped elements, the network function HðsÞ is a rational function of s and can be written in the following general form HðsÞ ¼ k

ðs  z1 Þðs  z2 Þ    ðs  z Þ ðs  p1 Þðs  p2 Þ    ðs  p Þ

where k is some real number. The complex constants zm ðm ¼ 1; 2; . . . ; Þ, the zeros of HðsÞ, and the pn ðn ¼ 1; 2; . . . ; Þ the poles of HðsÞ, assume particular importance when HðsÞ 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 ¼ zm , the response will be zero, no matter how great the excitation; whereas, when s ¼ pn , the response will be inﬁnite, no matter how small the excitation. EXAMPLE 8.8 A passive network in the s-domain is shown in Fig. 8-13. current IðsÞ due to an input voltage VðsÞ.

IðsÞ 1 ¼ VðsÞ ZðsÞ    5s 20 s2 þ 8s þ 12 3 s ZðsÞ ¼ 2:5 þ ¼ ð2:5Þ 5s 20 s2 þ 12 þ 3 s HðsÞ ¼

Since

Obtain the network function for the

we have HðsÞ ¼ ð0:4Þ

s2 þ 12 ðs þ 2Þðs þ 6Þ

Fig. 8-13

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An