Second-Order Circuits

1. Second-Order Circuits Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

2. Contents • Introduction • Finding Initial and Final Values • The Source-Free Series RLC Circuit • The Source-Free Parallel RLC Circuit • Step Response of a Series RLC Circuit • Step Response of a Parallel RLC Circuit • General Second-Order Circuits • Duality • Applications

3. Introduction • A second-order circuit is characterized by a second-order differential equation • It consists of resistors and the equivalent of two energy storage elements

4. i _ v + Finding Initial and Final Values • v and i are defined according to the passive sign convention • Continuity properties • Capacitor voltage • Inductor current

5. Example

6. Example (Cont’d)

7. Example (Cont’d)

8. The Source-Free Series RLC Circuit

9. Cont’d Natural frequencies Damping factor Resonant frequency Characteristic equation

10. Summary • Three cases discussed • Overdamped case :  > 0 • Critically damped case :  = 0 • Underdamped case :  < 0

11. i(t) t Overdamped Case ( > 0)

12. Critically damped Case ( = 0)

13. i(t) t Critically damped Case (Cont’d)

14. Underdamped Case ( < 0)

15. i(t) t Underdamped Case (Cont’d)

16. Finding The Constants A1,2

17. Conclusions • The concept of damping • The gradual loss of the initial stored energy • Due to the resistance R • Oscillatory response is possible • The energy is transferred between L and C • Ringing denotes the damped oscillation in the underdamped case • With the same initial conditions, the overdamped case has the longest settling time. The critically damped case has the fastest decay.

18. Example Find i(t). t > 0 t < 0

19. Example (Cont’d) t < 0 t > 0

20. The Source-Free Parallel RLC Circuit

21. Summary • Overdamped case :  > 0 • Critically damped case :  = 0 • Underdamped case :  < 0

22. Finding The Constants A1,2

23. Comparisons • Series RLC Circuit • Parallel RLC Circuit

24. Example 1 Find v(t) for t > 0. v(0) = 5 V, i(0) = 0 Consider three cases: R = 1.923  R = 5  R =6.25 

25. Example 1 (Cont’d)

26. Example 1 (Cont’d)

27. Example 2 Find v(t). Get x(), dx(0)/dt, s1,2, A1,2. Get x(0). t < 0 t > 0

28. Example 2 (Cont’d) t > 0 t < 0

29. Step Response of A Series RLC Circuit

30. Characteristic Equation

31. Summary (Overdamped) (Critically damped) (Underdamped)

32. Example Find v(t), i(t) for t > 0. Consider three cases: R = 5  R = 4  R =1  Get x(), dx(0)/dt, s1,2, A1,2. Get x(0). t < 0 t > 0

33. Case 1: R = 5 

34. Case 2: R = 4 

35. Case 3: R = 1 

36. Example (Cont’d)

37. Step Response of A Parallel RLC Circuit

38. Characteristic Equation

39. Summary (Overdamped) (Critically damped) (Underdamped)

40. General Second-Order Circuits • Steps required to determine the step response • Determine x(0),dx(0)/dt, andx() • Find the transient response xt(t) • Apply KCL and KVL to obtain the differential equation • Determine the characteristic roots (s1,2) • Obtain xt(t) with two unknown constants (A1,2) • Obtain the steady-state response xss(t) = x() • Usex(t) = xt(t) + xss(t) to determine A1,2 from the two initial conditions x(0) and dx(0)/dt

41. Example Find v, i for t > 0. Get x(), dx(0)/dt, s1,2, A1,2. Get x(0). t < 0 t > 0

42. Example (Cont’d) t < 0 t > 0

43. Example (Cont’d) t > 0

44. Duality • Duality means the same characterizing equations with dual quantities interchanged. Table for dual pairs

45. Example 1 • Series RLC Circuit • Parallel RLC Circuit

46. Example 2

47. Application: Smoothing Circuits Output from a D/A v0 vs