1. Copyright © 2009 Pearson Education, Inc. Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits

2. Copyright © 2009 Pearson Education, Inc. Recap:

3. Copyright © 2009 Pearson Education, Inc. A circuit consisting of an inductor and a resistor will begin with most of the voltage drop across the inductor, as the current is changing rapidly. With time, the current will increase less and less, until all the voltage is across the resistor. 30-4 LR Circuits

4. Copyright © 2009 Pearson Education, Inc. If the circuit is then shorted across the battery, the current will gradually decay away: 30-4 LR Circuits

5. Copyright © 2009 Pearson Education, Inc. 30-4 LR Circuits Example 30-6: An LR circuit. At t = 0, a 12.0-V battery is connected in series with a 220-mH inductor and a total of 30-Ω resistance, as shown. (a) What is the current at t = 0? (b) What is the time constant? (c) What is the maximum current? (d) How long will it take the current to reach half its maximum possible value? (e) At this instant, at what rate is energy being delivered by the battery, and (f) at what rate is energy being stored in the inductor’s magnetic field?

6. Copyright © 2009 Pearson Education, Inc. 30-5 LC Circuits and Electromagnetic Oscillations An LC circuit is a charged capacitor shorted through an inductor.

7. Copyright © 2009 Pearson Education, Inc. 30-5 LC Circuits and Electromagnetic Oscillations The charge and current are both sinusoidal, but with different phases.

8. Copyright © 2009 Pearson Education, Inc. 30-5 LC Circuits and Electromagnetic Oscillations The total energy in the circuit is constant; it oscillates between the capacitor and the inductor:

9. Copyright © 2009 Pearson Education, Inc. 30-5 LC Circuits and Electromagnetic Oscillations Example 30-7: LC circuit. A 1200-pF capacitor is fully charged by a 500-V dc power supply. It is disconnected from the power supply and is connected, at t = 0, to a 75-mH inductor. Determine: (a) the initial charge on the capacitor; (b) the maximum current; (c) the frequency f and period T of oscillation; and (d) the total energy oscillating in the system.

10. Copyright © 2009 Pearson Education, Inc. 30-6 LC Oscillations with Resistance ( LRC Circuit) Any real (nonsuperconducting) circuit will have resistance.

11. Copyright © 2009 Pearson Education, Inc. 30-6 LC Oscillations with Resistance ( LRC Circuit) Now the voltage drops around the circuit give A current flowing through a resistor means energy is dissipated so the solution must die out over time. The solutions to this equation are damped harmonic oscillations.

12. Copyright © 2009 Pearson Education, Inc. 30-6 LC Oscillations with Resistance ( LRC Circuit) The system will be underdamped for R 2 4L/C. Critical damping will occur when R 2 = 4L/C. This figure shows the three cases of underdamping, overdamping, and critical damping.

13. Copyright © 2009 Pearson Education, Inc. 30-6 LC Oscillations with Resistance ( LRC Circuit) The angular frequency for critical and under damped oscillations is given by and the charge in the circuit as a function of time is The over damped case, is more complicated but the solutions look like ( don’t worry about this)

14. Copyright © 2009 Pearson Education, Inc. 30-6 LC Oscillations with Resistance ( LRC Circuit) Example 30-8: Damped oscillations. At t = 0, a 40-mH inductor is placed in series with a resistance R = 3.0 Ω and a charged capacitor C = 4.8 μF. (a) Show that this circuit will oscillate. (b) Determine the frequency. (c) What is the time required for the charge amplitude to drop to half its starting value? (d) What value of R will make the circuit nonoscillating?

15. Copyright © 2009 Pearson Education, Inc. Resistors, capacitors, and inductors have different phase relationships between current and voltage when placed in an ac circuit. The current through a resistor is in phase with the voltage. 30-7 AC Circuits with AC Source

16. Copyright © 2009 Pearson Education, Inc. Therefore, the current through an inductor lags the voltage by 90°. 30-7 AC Circuits with AC Source The voltage across the inductor is given by or.

17. Copyright © 2009 Pearson Education, Inc. 30-7 AC Circuits with AC Source The voltage across the inductor is related to the current through it: The quantity X L is called the inductive reactance, and has units of ohms:.

18. Copyright © 2009 Pearson Education, Inc. 30-7 AC Circuits with AC Source Example 30-9: Reactance of a coil. A coil has a resistance R = 1.00 Ω and an inductance of 0.300 H. Determine the current in the coil if (a) 120-V dc is applied to it, and (b) 120-V ac (rms) at 60.0 Hz is applied.

19. Copyright © 2009 Pearson Education, Inc. Therefore, in a capacitor, the current leads the voltage by 90°. 30-7 AC Circuits with AC Source The voltage across the capacitor is given by.

20. Copyright © 2009 Pearson Education, Inc. 30-7 AC Circuits with AC Source The voltage across the capacitor is related to the current through it: The quantity X C is called the capacitive reactance, and (just like the inductive reactance) has units of ohms:.

21. Copyright © 2009 Pearson Education, Inc. 30-7 AC Circuits with AC Source Example 30-10: Capacitor reactance. What is the rms current in the circuit shown if C = 1.0 μF and V rms = 120 V? Calculate (a) for f = 60 Hz and then (b) for f = 6.0 x 10 5 Hz.

22. Copyright © 2009 Pearson Education, Inc. 30-7 AC Circuits with AC Source This figure shows a high-pass filter (allows an ac signal to pass but blocks a dc voltage) and a low-pass filter (allows a dc voltage to be maintained but blocks higher-frequency fluctuations).

23. Copyright © 2009 Pearson Education, Inc. Analyzing the LRC series AC circuit is complicated, as the voltages are not in phase – this means we cannot simply add them. Furthermore, the reactances depend on the frequency. 30-8 LRC Series AC Circuit

24. Copyright © 2009 Pearson Education, Inc. We calculate the voltage (and current) using phasors – these are vectors representing the individual voltages. Here, at t = 0, the current and voltage are both at a maximum. As time goes on, the phasors will rotate counterclockwise. 30-8 LRC Series AC Circuit

25. Copyright © 2009 Pearson Education, Inc. Some time t later, the phasors have rotated. 30-8 LRC Series AC Circuit

26. Copyright © 2009 Pearson Education, Inc. The voltage across each device is given by the x -component of each, and the current by its x -component. The current is the same throughout the circuit. 30-8 LRC Series AC Circuit

27. Copyright © 2009 Pearson Education, Inc. We find from the ratio of voltage to current that the “effective resistance,” called the impedance, of the circuit is given by BUT – only an actual resistance dissipates energy. The inductor and capacitor store it then release it. 30-8 LRC Series AC Circuit

28. Copyright © 2009 Pearson Education, Inc. 30-8 LRC Series AC Circuit The phase angle between the voltage and the current is given by The factor cos φ is called the power factor of the circuit. or

29. Copyright © 2009 Pearson Education, Inc. 30-8 LRC Series AC Circuit Example 30-11: LRC circuit. Suppose R = 25.0 Ω, L = 30.0 mH, and C = 12.0 μF, and they are connected in series to a 90.0-V ac (rms) 500-Hz source. Calculate (a) the current in the circuit, (b) the voltmeter readings (rms) across each element, (c) the phase angle , and (d) the power dissipated in the circuit.

30. Copyright © 2009 Pearson Education, Inc. The rms current in an ac circuit is Clearly, I rms depends on the frequency. 30-9 Resonance in AC Circuits

31. Copyright © 2009 Pearson Education, Inc. We see that I rms will be a maximum when X C = X L ; the frequency at which this occurs is f 0 = ω 0 /2π is called the resonant frequency. 30-9 Resonance in AC Circuits

32. Copyright © 2009 Pearson Education, Inc. 30-10 Impedance Matching When one electrical circuit is connected to another, maximum power is transmitted when the output impedance of the first equals the input impedance of the second. The power delivered to the circuit will be a maximum when dP/dR 2 = 0 ; this occurs when R 1 = R 2.

33. Copyright © 2009 Pearson Education, Inc. Summary of Chapter 30 LR circuit: Inductive reactance: Capacitive reactance:..

34. Copyright © 2009 Pearson Education, Inc. Summary of Chapter 30 LRC series circuit: Resonance in LRC series circuit:.