Time Varying Circuits

1. Time Varying Circuits 2009 Induction

2. The Final Exam Approacheth • 8-10 Problems similar to Web-Assignments • Covers the entire semester’s work • May contain some short answer (multiple choice) questions. Induction

3. Induction

4. HowjaDo?? • I done good • I done ok • I done not so ok • I screwed up major Induction

5. The Test Itself Was • Fair • Not so fair. • Really Unfair. • The worst kind of unfair in the entire universe. Induction

6. Inductors Sort of like RC circuit issues. Back to Circuits for a bit ….

7. Definition Current in loop produces a magnetic field in the coil and consequently a magnetic flux. If we attempt to change the current, an emf will be induced in the loops which will tend to oppose the change in current. This this acts like a “resistor” for changes in current!

8. Remember Faraday’s Law Lentz

9. Look at the following circuit: • Switch is open • NO current flows in the circuit. • All is at peace!

10. Close the circuit… • After the circuit has been close for a long time, the current settles down. • Since the current is constant, the flux through the coil is constant and there is no Emf. • Current is simply E/R (Ohm’s Law)

11. Close the circuit… • When switch is first closed, current begins to flow rapidly. • The flux through the inductor changes rapidly. • An emf is created in the coil that opposes the increase in current. • The net potential difference across the resistor is the battery emf opposed by the emf of the coil.

12. Close the circuit…

13. Moving right along …

14. Definition of Inductance L UNIT of Inductance = 1 henry = 1 T- m2/A FB is the flux near the center of one of the coils making the inductor

15. Consider a Solenoid l n turns per unit length

16. So…. Depends only on geometry just like C and is independent of current.

17. Inductive Circuit • Switch to “a”. • Inductor seems like a short so current rises quickly. • Field increases in L and reverse emf is generated. • Eventually, i maxes out and back emf ceases. • Steady State Current after this. i

18. THE BIG INDUCTION • As we begin to increase the current in the coil • The current in the first coil produces a magnetic field in the second coil • Which tries to create a current which will reduce the field it is experiences • And so resists the increase in current. Lenz with an ATTITUDE!

19. i Back to the real world… Switch to “a”

20. Solution (See textbook)

21. Switch position “b”

22. Max Current Rate of increase = max emf VR=iR ~current

23. Solve the loop equation.

24. IMPORTANT QUESTION • Switch closes. • No emf • Current flows for a while • It flows through R • Energy is conserved (i2R) WHERE DOES THE ENERGY COME FROM??

25. E=e0A/d +dq +q -q For an answerReturn to the Big C • We move a charge dq from the (-) plate to the (+) one. • The (-) plate becomes more (-) • The (+) plate becomes more (+). • dW=Fd=dq x E x d

26. The calc The energy is in the FIELD !!!

27. What about POWER?? power to circuit power dissipated by resistor Must be dWL/dt

28. So Energy stored in the Capacitor

29. WHERE is the energy?? l

30. Remember the Inductor??

31. So …

32. ENERGY IN THEFIELD TOO!

33. IMPORTANT CONCLUSION • A region of space that contains either a magnetic or an electric field contains electromagnetic energy. • The energy density of either is proportional to the square of the field strength.

34. Solution (From Before)

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

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

37. The Graph of that LR (no emf) circuit I Induction

38. Induction

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

40. Induction

41. LC Circuit High Q/C Low Low High Induction

42. The Math Solution (R=0): Induction

43. 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

44. Variable Emf Applied emf DC Sinusoidal Induction

45. Sinusoidal Stuff “Angle” Phase Angle Induction

46. Same Frequency with PHASE SHIFT f Induction

47. Different Frequencies Induction

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

49. 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

50. The Real World Induction