1. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 9 The RLC Circuit

2. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.2  an RLC circuit has both an inductor and a capacitor  these circuits have a wide range of applications, including oscillators and frequency filters  they also can model automobile suspension systems, temperature controllers, airplane responses, and so forth

3. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.3 Apply KCL and differentiate to show:

4. To solve, assume v=Ae st. The solution must then satisfy which is called the characteristic equation. If s 1 and s 2 are the solutions, then the natural response is Answer: i 3 (t) = 1.333 sin t V Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.4

5. The solutions to the characteristic equation are Define ω 0 the resonant frequency: and α the damping coefficient: Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.5

6. With these definitions, the solutions can be expressed as: The constants A 1 and A 2 are determined by the initial conditions. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.6

7. If α>ω 0 the solutions are real, unequal and the response is termed overdamped. If α<ω 0 the solutions are complex conjugates and the response is termed underdamped. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.7 [If α=ω 0 the solutions are real and equal and the response is termed critically damped. ]

8. Show that v(t) = 84(e −t − e −6t ) when i(0 + )=10 A and v(0 + )=0 V. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.8

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10. Show that v C (t) = 80e −50,000t − 20e −200,000t V for t>0. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.10

11. Find R 1 such that the circuit is critically damped for t>0 and R 2 so that v(0)=2 V. Answer: R 1 = 31.63 kΩ, R 2 =0.4Ω Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.11

12. If α<ω 0, define and the solution is or equivalently Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.12

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14. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.14

15. Show for t>0 i L = e −1.2t (2.027 cos 4.75t + 2.561 sin 4.75t) Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.15

16. For the series RLC circuit, This circuit is the dual of the parallel RLC circuit. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.16

17. The characteristic equation is and the solution is where Answer: i 3 (t) = 1.333 sin t V Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.17

18. Define and Then if α>ω 0 (overdamped): α=ω 0 (critically damped): α<ω 0 (underdamped): Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.18

19. The response of RLC circuits with dc sources and switches will consist of the natural response and the forced response: v(t) = v f (t)+v n (t) The complete response must satisfy both the initial conditions and the “final conditions” or the forced response. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.19

20. Find the labeled voltages and currents at t=0 - and t=0 +. Answer: i R (0 − ) = −5 Av R (0 − ) = −150 V i R (0 + ) = −1 Av R (0 + ) = −30 V i L (0 − ) = 5 Av L (0 − ) = 0 Vi L (0 + ) = 5 Av L (0 + ) = 120 V i C (0 − ) = 0 Av C (0 − ) = 150 Vi C (0 + ) = 4 Av C (0 + ) = 150 V Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.20

21. Find the first derivatives of the labeled voltages and currents at t=0 +. Answer: di R /dt(0 + ) = −40 A/sdv R /dt(0 + ) = -1200 V/s di L /dt(0 + ) = 40 A/sdv L /dt(0 + ) = -1092 V/s di C /dt(0 + ) = -40 A/sdv C /dt(0 + ) = 108 V/s Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.21

22. Show that for t>0 v C (t) = 150 + 13.5(e −t − e −9t ) volts Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.22

23.  The resistor in the RLC circuit serves to dissipate initial stored energy.  When this resistor becomes 0 in the series RLC or infinite in the parallel RLC, the circuit will oscillate. Example: for t>0, v(t) =2 sin 3t if i(0)=-1/6 A and v(0)=0 V Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.23