FIRST-ORDER CIRCUITS

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The input voltage in the circuits of 7-45 is a weighted sum of sinusoids with the highest frequency f0 Hz. Assuming that RC < 1= 360 f0 , nd v2 t . Ans: v2 t % v1 t 2RC

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Find the relationship between v2 and v1 in the circuit of Fig. 7-46. Ans: v2 RC dv2 2v1 dt

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Fig. 7-47 Fig. 7-46 7.46 In the circuit of Fig. 7-47, nd the di erential equation relating v2 to v1 . Fig. 7-45(a) of Example 7.17. dv 1 dv v1 RC 1 Ans: v2 RC 2 2 dt dt In the circuit of Fig. 7-48, nd the relationship between v2 and v1 . dv2 C1 dv1 v R2 C2 Ans: v2 R1 C1 dt C2 1 dt Compare with the circuit of

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Fig. 7-48

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Fig. 7-49

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FIRST-ORDER CIRCUITS

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[CHAP. 7

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In the circuit of Fig. 7-49, let k 0. Ans: v e t ; i 1 0:5e t

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Find v and i after the switch is closed at t 0.

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Show that the segment of the circuit enclosed by the dashed box in the circuit of Fig. 7-49 is equivalent to an inductor with value L 1= 1 k H. Hint: Write KVL between terminals AB of the dashed box.

The switch in the circuit of Fig. 7-49 is closed at t 0. Find v at t > 0 for the following values of k: (a) 0.5, (b) 1, (c) 2. Ans: a v e t=2 ; b v 1; c v et

Find i, the current drawn from the battery, in Problem 7.50. Ans: a i 1 0:5e t=2 ; b i 0:5; c i 1 0:5et

Higher-Order Circuits and Complex Frequency

8.1 INTRODUCTION In 7, RL and RC circuits with initial currents or charge on the capacitor were examined and rst-order di erential equations were solved to obtain the transient voltages and currents. When two or more storage elements are present, the network equations will result in second-order di erential equations. In this chapter, several examples of second-order circuits will be presented. This will then be followed by more direct methods of analysis, including complex frequency and pole-zero plots.

SERIES RLC CIRCUIT

The second-order di erential equation, which will be examined shortly, has a solution that can take three di erent forms, each form depending on the circuit elements. In order to visualize the three possibilities, a second-order mechanical system is shown in Fig. 8-1. The mass M is suspended by a spring with a constant k. A damping device D is attached to the mass M. If the mass is displaced from its rest position and then released at t 0, its resulting motion will be overdamped, critically damped, or underdamped (oscillatory). Figure 8-2 shows the graph of the resulting motions of the mass after its release from the displaced position z1 (at t 0).

Fig. 8-1

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HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

[CHAP. 8

Fig. 8-2

The series RLC circuit shown in Fig. 8-3 contains no voltage source. the closed loop after the switch is closed is vR vL vC 0 di 1 i dt 0 Ri L dt C

Kirchho s voltage law for

Di erentiating and dividing by L yields d 2 i R di 1 i 0 2 L dt LC dt A solution of this second-order di erential equation is of the form i A1 es1 t A2 es2 t . solution in the di erential equation obtains Substituting this

Fig. 8-3

R 1 R 1 A1 es1 t s2 s1 A2 es2 t s2 s2 0 1 2 L LC L LC that is, if s1 and s2 are the roots of s2 R=L s 1=LC 0, R s1 2L where R=2L, s 2 R 1 2L LC R s2 2L s 2 R 1 2L LC

q p 2 !2 , and !0 1= LC . 0

CHAP. 8]

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

Overdamped Case > !0 In this case, both and are real positive numbers. i A1 e t A2 e t e t A1 et A2 e t