CHAP. 8]

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

181

At the input terminals, KVL gives   15 VðsÞ ¼ 2sIðsÞ þ V 0 ðsÞ ¼ 2s þ IðsÞ s Then

8.15

HðsÞ ¼

VðsÞ 2s2 þ 15 ¼ IðsÞ s

For the two-port network shown in Fig. 8-24 ﬁnd the values of R1 , R2 , and C, given that the voltage transfer function is Hv ðsÞ 

Vo ðsÞ 0:2 ¼ Vi ðsÞ s2 þ 3s þ 2

Fig. 8-24 The impedance looking into xx 0 is Z0 ¼

ð1=sCÞðR1 þ R2 Þ R1 þ R2 ¼ ð1=sCÞ þ R1 þ R2 1 þ ðR1 þ R2 ÞCs

Then, by repeated voltage division,       Vo Vo Vxx 0 R2 Z0 R2 =ðR1 þ R2 ÞC ¼ ¼ ¼ 0 1 2 Vi Vxx 0 Vi R1 þ R2 Z þ s1 s þ sþ 1 ðR1 þ R2 ÞC C Equating the coeﬃcients in this expression to those in the given expression for Hv ðsÞ, we ﬁnd: C¼

8.16

1 F 2

R1 ¼

3  5

R2 ¼

1  15

Construct the pole-zero plot for the transfer admittance function HðsÞ ¼

Io ðsÞ s2 þ 2s þ 17 ¼ 2 Vi ðsÞ s þ 3s þ 2

In factored form, HðsÞ ¼

ðs þ 1 þ j4Þðs þ 1  j4Þ ðs þ 1Þðs þ 2Þ

Poles exist at 1 and 2; zeros at 1  j4. See Fig. 8-25.

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An