Time Varying Circuits

1. Time Varying Circuits April 10, 2006 Induction - Spring 2006

2. What is going on? • There are only 7 more classes and the final is 3 weeks away. • Scotty, beam me somewhere else! • Exam Issues • Look Ashamed! • Inductor Circuits • Quiz Friday • AC Next week & Following Monday Induction - Spring 2006

3. B h a a Other Wire Second Problem q q Both Currents are going into the page. Induction - Spring 2006

4. DV First Problem Induction - Spring 2006

5. The Last two Problems were similar to WebAssigns that were also reviewed in class. Circular Arc – Easy Biot-Savart Moving Rod Induction - Spring 2006

6. Question • What about these problems was “unfair”? • Why so many blank or completely wrong pages? Induction - Spring 2006

7. And Now ….. From the past Induction - Spring 2006

8. Max Current Rate of increase = max emf VR=iR ~current Induction - Spring 2006

9. Solve the loop equation. Induction - Spring 2006

10. We also showed that Induction - Spring 2006

11. i LR Circuit Steady Source Induction - Spring 2006

12. Time Dependent Result: Induction - Spring 2006

13. R L Induction - Spring 2006

14. At t=0, the charged capacitor is now connected to the inductor. What would you expect to happen?? Induction - Spring 2006

15. The math … For an RLC circuit with no driving potential (AC or DC source): Induction - Spring 2006

16. The Graph of that LR (no emf) circuit .. Induction - Spring 2006

17. Induction - Spring 2006

18. Mass on a Spring Result • Energy will swap back and forth. • Add friction • Oscillation will slow down • Not a perfect analogy Induction - Spring 2006

19. Induction - Spring 2006

20. LC Circuit High Q/C Low Low High Induction - Spring 2006

21. The Math Solution (R=0): Induction - Spring 2006

22. New Feature of Circuits with L and C • These circuits produce oscillations in the currents and voltages • Without a resistance, the oscillations would continue in an un-driven circuit. • With resistance, the current would eventually die out. Induction - Spring 2006

23. Variable Emf Applied emf DC Sinusoidal Induction - Spring 2006

24. Sinusoidal Stuff “Angle” Phase Angle Induction - Spring 2006

25. Same Frequency with PHASE SHIFT f Induction - Spring 2006

26. Different Frequencies Induction - Spring 2006

27. Note – Power is delivered to our homes as an oscillating source (AC) This makes AC Important! Induction - Spring 2006

28. Producing AC Generator x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Induction - Spring 2006

29. The Real World Induction - Spring 2006

30. A Induction - Spring 2006

31. Induction - Spring 2006

32. The Flux: Induction - Spring 2006

33. April 12, 2006 Induction - Spring 2006

34. Schedule • Today • Finish Inductors • Friday • Quiz on this weeks material • Some problems and then AC circuits • Monday • Last FULL week of classes • Following Monday is last day of class • FINAL IS LOOMING! Induction - Spring 2006

35. Some Problems Induction - Spring 2006

36. 14. Calculate the resistance in an RL circuit in which L = 2.50 H and the current increases to 90.0% of its final value in 3.00 s. Induction - Spring 2006

37. 16. Show that I = I0e – t/τ is a solution of the differential equation where τ = L/R and I0 is the current at t = 0. Induction - Spring 2006

38. 17. Consider the circuit in Figure P32.17, taking ε = 6.00 V, L = 8.00 mH, and R = 4.00 Ω. (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250 μs after the switch is closed. (c) What is the value of the final steady-state current? (d) How long does it take the current to reach 80.0% of its maximum value? Induction - Spring 2006

39. 18. In the circuit shown in Figure P32.17, let L = 7.00 H, R = 9.00 Ω, and ε = 120 V. What is the self-induced emf 0.200 s after the switch is closed? Induction - Spring 2006

40. 27. A 140-mH inductor and a 4.90-Ω resistor are connected with a switch to a 6.00-V battery as shown in Figure P32.27. (a) If the switch is thrown to the left (connecting the battery), how much time elapses before the current reaches 220 mA? (b) What is the current in the inductor 10.0 s after the switch is closed? (c) Now the switch is quickly thrown from a to b. How much time elapses before the current falls to 160 mA? Induction - Spring 2006

41. 32. At t = 0, an emf of 500 V is applied to a coil that has an inductance of 0.800 H and a resistance of 30.0 Ω. (a) Find the energy stored in the magnetic field when the current reaches half its maximum value. (b) After the emf is connected, how long does it take the current to reach this value? Induction - Spring 2006

42. 52. The switch in Figure P32.52 is connected to point a for a long time. After the switch is thrown to point b, what are (a) the frequency of oscillation of the LC circuit, (b) the maximum charge that appears on the capacitor, (c) the maximum current in the inductor, and (d) the total energy the circuit possesses at t = 3.00 s? Induction - Spring 2006

43. Back to Variable Sources Induction - Spring 2006

44. Source Voltage: Induction - Spring 2006

45. Average value of anything: h T Area under the curve = area under in the average box Induction - Spring 2006

46. Average Value For AC: Induction - Spring 2006

47. So … • Average value of current will be zero. • Power is proportional to i2R and is ONLY dissipated in the resistor, • The average value of i2 is NOT zero because it is always POSITIVE Induction - Spring 2006

48. Average Value Induction - Spring 2006

49. RMS Induction - Spring 2006

50. Usually Written as: Induction - Spring 2006