1. Week 6 Second Order Transient Response

2. Topics Second Order Definition Dampening Parallel LC Forced and homogeneous solutions

3. 2nd Order Circuits Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2. Any voltage or current in such a circuit is the solution to a 2nd order differential equation.

4. A 2nd Order RLC Circuit The source and resistor may be equivalent to a circuit with many resistors and sources. R Cv s (t) i (t) L +–+–

5. Applications Modeled by a 2nd Order RLC Circuit Filters –A low-pass filter with a sharper cutoff than can be obtained with an RC circuit.

6. 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

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

8. Second-Order Circuits: In previous work, circuits were limited to one energy storage element, which resulted in first-order differential equations. Now, a second independent energy storage element will be added to the circuits to result in second order differential equations:

9. Second –Order Circuits Dampening coefficient Un-damped resonant frequency Define: Forcing function

10. Solution of the Second-Order Equation

11. Important Concepts 1.The differential equation 2.Forced and homogeneous solutions 3.The natural frequency and the damping ratio

12. The Differential Equation KVL around the loop: v r (t) + v c (t) + v l (t) = v s (t) R Cv s (t) + – v c (t) + – v r (t) L +– v l (t) i (t) +–+–

13. Differential Equation

14. The Differential Equation Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form:

15. Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

16. The Particular or Forced Solution The particular (or forced) solution i p (t) is usually a weighted sum of f(t) and its first and second derivatives. If f(t) is constant, then i p (t) is constant. If f(t) is sinusoidal, then i p (t) is sinusoidal.

17. The Complementary Solution The complementary (homogeneous) solution has the following form: K is a constant determined by initial conditions. This may be a starting voltage or current value including zero. s is a constant determined by the coefficients of the differential equation.

18. Solution of the Complementary Equation Try: Factoring Characteristic equation

19. Solution of the Complementary Equation Dampening ratio (Zeta) Roots of the characteristic equation:

20. Characteristic Equation To find the complementary solution, we need to solve the characteristic equation: The characteristic equation has two roots- call them s 1 and s 2.

21. Characteristic Equation

23. Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

24. Natural Solutions Find characteristic equation from homogeneous equation:  Convert to polynomial by the following substitution: to obtain  Based on the roots of the characteristic equation, the natural solution will take on one of three particular forms. Roots given by:

25. Damping

26. Damping Ratio (  ) and Natural Frequency (  0 ) The damping ratio is . The damping ratio determines what type of solution we will get: –Exponentially decreasing (  >1) –Exponentially decreasing sinusoid (  < 1) The natural frequency is w 0 –It determines how fast sinusoids wiggle.

27. Damped Harmonic Motion

28. 1. Over-damped case (ζ > 1). If ζ > 1 (or equivalently, if α > ω 0 ), the roots of the characteristic equation are real and distinct. Then the complementary solution is: In this case, we say that the circuit is over-damped.

29. 2. Critically damped case (ζ = 1). If ζ = 1 (or equivalently, if α = ω 0 ), the roots are real and equal. Then the complementary solution is In this case, we say that the circuit is critically damped.

30. 3. Underdamped case (ζ < 1). Finally, if ζ < 1 (or equivalently, if α < ω 0 ), the roots are complex. (By the term complex, we mean that the roots involve the square root of –1.) In other words, the roots are of the form: in which j is the square root of -1 and the natural frequency is given by:

31. Damping Summary

32. Examples follow for later reference

33. Comparisons Series RLC CircuitParallel RLC Circuit

34. 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 

35. Example 1 (Cont’d)

37. Example 2 t < 0t > 0 Find v(t). Get x(0). Get x(  ), dx(0)/dt, s 1,2, A 1,2.

38. Example 2 (Cont’d) t > 0t < 0

39. Step Response of A Series RLC Circuit

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

41. Case 1: R = 5 

42. Case 2: R = 4 

43. Case 3: R = 1 

44. Example (Cont’d)

45. Step Response of A Parallel RLC Circuit