188

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

[CHAP. 8

Fig. 8-40

8.41

A series RLC circuit contains R ¼ 1 , L ¼ 2 H, and C ¼ 0:25 F. Simultaneously apply magnitude and frequency scaling, with Km ¼ 2000 and Kf ¼ 104 . What are the scaled element values? Ans: 2000 ; 0:4 H; 12:5 mF

8.42

At a certain frequency !1 , a voltage V1 ¼ 25 08 V applied to a passive network results in a current I1 ¼ 3:85 308 (A). The network elements are magnitude-scaled with Km ¼ 10. Obtain the current which results from a second voltage source, V2 ¼ 10 458 V, replacing the ﬁrst, if the second source frequency is !2 ¼ 103 !1 . Ans: 0:154 158 A

8.43

In the circuit of Fig. 8-41 let R1 C1 ¼ R2 C2 ¼ 103 . Find v2 for t > 0 if: (a) v1 ¼ cos ð1000tÞuðtÞ, (b) v1 ¼ sin ð1000tÞuðtÞ. Ans: ðaÞ v2 ¼ sin ð1000tÞ; ðbÞ v2 ¼ 1  cos ð1000tÞ

8.44

In the circuit of Fig. 8-42 assume R ¼ 2 k, C ¼ 10 nF, and R2 ¼ R1 and v1 ¼ cos !t. following frequencies: (a) !0 ¼ 5  104 rad/s, (b) !1 ¼ 105 rad/s. Ans: ðaÞ v2 ¼ 2 sin !0 t; ðbÞ v2 ¼ 0:555 cos ð!1 t  146:38Þ

Fig. 8-41

Find v2 for the

Fig. 8-42

8.45

Noninverting integrators. In the circuits of Fig. 8-43(a) and 8-43(b) ﬁnd the relationship between v2 and v1 . Ans: ðaÞ v1 ¼ ðRC=2Þdv2 =dt; ðbÞ v1 ¼ 2RCdv2 =dt

8.46

In the circuit of Fig. 8-44 ﬁnd the relationship between v2 and v1 . v2 ¼ R2 v1 =ðR1 þ R2 Þ. Ans:

R1 R2 ðC1 þ C2 Þ

dv2 dv þ ðR1 þ R2 Þv2 ¼ R1 R2 C1 1 þ R2 v1 dt dt

Show that for R1 C1 ¼ R2 C2 we obtain

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An