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Time Varying Circuits

Time Varying Circuits. 2008. A look into the future. We have one more week after today (+ one day) Time Varying Circuits Including AC Some additional topics leading to waves A bit of review if there is time. There will be one more Friday morning quiz.

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Time Varying Circuits

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  1. Time Varying Circuits 2008 Induction

  2. A look into the future • We have one more week after today (+ one day) • Time Varying Circuits Including AC • Some additional topics leading to waves • A bit of review if there is time. • There will be one more Friday morning quiz. • I hope to be able to return the exams on Monday at which time we will briefly review the solutions. Induction

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

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

  5. We Solved the loop equation. Induction

  6. We also showed that Induction

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

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

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

  10. Induction

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

  12. Induction

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

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

  15. 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

  16. Variable Emf Applied emf DC Sinusoidal Induction

  17. Sinusoidal Stuff “Angle” Phase Angle Induction

  18. Same Frequency with PHASE SHIFT f Induction

  19. Different Frequencies Induction

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

  21. 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

  22. The Real World Induction

  23. A Induction

  24. Induction

  25. The Flux: Induction

  26. problems … Induction

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. Source Voltage: Induction

  35. Average value of anything: h T Area under the curve = area under in the average box Induction

  36. Average Value For AC: Induction

  37. 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

  38. Average Value Induction

  39. RMS Induction

  40. Usually Written as: Induction

  41. R E ~ Example: What Is the RMS AVERAGE of the power delivered to the resistor in the circuit: Induction

  42. Power Induction

  43. More Power - Details Induction

  44. Resistive Circuit • We apply an AC voltage to the circuit. • Ohm’s Law Applies Induction

  45. Consider this circuit CURRENT ANDVOLTAGE IN PHASE Induction

  46. A protion of next group of slides is stolen from Dr. Braunstein. Induction

  47. V(t) Vp wt p 2p fv -Vp An “AC” circuit is one in which the driving voltage and hence the current are sinusoidal in time. Alternating Current Circuits V = VP sin (wt - fv ) I = IP sin (wt - fI ) wis theangular frequency (angular speed) [radians per second]. Sometimes instead ofwwe use thefrequency f [cycles per second] Frequency f [cycles per second, or Hertz (Hz)]w= 2p f Induction

  48. V(t) Vp wt p 2p fv -Vp Phase Term V = VP sin (wt - fv ) Induction

  49. V(t) Vp wt p 2p fv -Vp V = VP sin (wt - fv ) I = IP sin (wt - fI ) Alternating Current Circuits I(t) Ip Irms Vrms t fI/w -Ip Vp and Ip are the peak current and voltage. We also use the “root-mean-square” values: Vrms = Vp / and Irms=Ip / fv andfI are called phase differences (these determine when V and I are zero). Usually we’re free to set fv=0 (but not fI). Induction

  50. Example: household voltage In the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. Induction

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