Chapter 21: Alternating Currents

1. Chapter 21: Alternating Currents • Sinusoidal Voltages and Currents • Capacitors, Resistors, and Inductors in AC Circuits • Series RLC Circuits • Resonance • AC to DC Conversion

2. §21.1 Sinusoidal Currents and Voltage A power supply can be set to give an EMF of form: This EMF is time dependent, has an amplitude 0, and varies with angular frequency .

3. angular frequency in rads/sec frequency in cycles/sec or Hz The current in a resistor is still given by Ohm’s Law: The current has an amplitude of I0=0/R.

4. The instantaneous power dissipated in a resistor will be: The power dissipated depends on t (where in the cycle the current/voltage are).

5. What is the average power dissipated by a resistor in one cycle? The average value sin2t over one cycle is 1/2. The average power is

6. What are the averages of V(t) and I(t) over one cycle? The “problem” here is that the average value of sin t over one complete cycle is zero! This is not a useful way to characterize the quantities V(t) and I(t). To fix this problem we use the root mean square (rms) as the characteristic value over one cycle.

7. In terms of rms quantities, the power dissipated by a resistor can be written as:

8. Example (text problem 21.4): A circuit breaker trips when the rms current exceeds 20.0 A. How many 100.0 W light bulbs can run on this circuit without tripping the breaker? (The voltage is 120 V rms.) Each light bulb draws a current given by: If 20 amps is the maximum current, and 0.83 amps is the current drawn per light bulb, then you can run 24 light bulbs without tripping the breaker.

9. Example (text problem 21.10): A hair dryer has a power rating of 1200 W at 120 V rms. Assume the hair dryer is the only resistance in the circuit. (a) What is the resistance of the heating element?

10. Example continued: (b) What is the rms current drawn by the hair dryer? (c) What is the maximum instantaneous power that the resistance must withstand? Pmax = 2Pav = 2400 Watts

11. §21.3-4 Capacitors, Resistors and Inductors in AC circuits For a capacitor: In the circuit: Slope of the plot V(t) vs. t

13. The current in the circuit and the voltage drop across the capacitor are 1/4 cycle out of phase. Here the current leads the voltage by 1/4 cycle. Here it is true that VCI. The equality is Vc = IXC where XC is called capacitive reactance. (Think Ohm’s Law!) Reactance has units of ohms.

14. For a resistor in an AC circuit, The voltage and current will be in phase with each other.

15. For an inductor in an AC circuit: Slope of an I(t) vs. t plot Also, VL = IXL where the inductive reactance is:

16. The current in the circuit and the voltage drop across the inductor are 1/4 cycle out of phase. Here the current lags the voltage by 1/4 cycle.

17. Plot of I(t), V(t), and P(t) for a capacitor. The average power over one cycle is zero. An ideal capacitor dissipates no energy.

18. A similar result is found for inductors; no energy is dissipated by an ideal inductor.

19. §21.5 Series RLC Circuits

20. Applying Kirchhoff’s loop rule:

21. y VL 0 VR X VC To find the amplitude (0) and phase () of the total voltage we add VL, VR, and VC together by using phasors. Z is called impedance.

22. y VL 0  VR VC The phase angle between the current in the circuit and the input voltage is: X >0 when XL> XC and the voltage leads the current (shown above). <0 when XL< XC and the voltage lags the current.

23. Example (text problem 21.79): In an RLC circuit these three elements are connected in series: a resistor of 20.0 , a 35.0 mH inductor, and a 50.0 F capacitor. The AC source has an rms voltage of 100.0 V and an angular frequency of 1.0103 rad/sec. Find… (a) The reactances of the capacitor and the inductor. (b) The impedance.

24. Example continued: (c) The rms current: (d) The current amplitude:

25. Example continued: (e) The phase angle: (Or 37°) (f) The rms voltages across each circuit element:

26. y VL rms  VR X VC Example continued: (g) Does the current lead or lag the voltage? Since XL>XC,  is a positive angle. The voltage leads the current. (h) Draw a phasor diagram.

27. The power dissipated by a resistor is: where cos is called the power factor (compare to slide 7; Why is there a difference?).

28. §21.6 Resonance in RLC Circuits A plot of I vs.  for a series RLC circuit has a peak at  = 0.

29. The peak occurs at the resonant frequency for the circuit. The current will be a maximum when Z is a minimum. This occurs when XL = XC (or when Z=R). This is the resonance frequency for the circuit.

30. At resonance: The phase angle is 0; the voltage and the current are in phase. The current in the circuit is a maximum as is the power dissipated by the resistor.

31. §21.7 Converting AC to DC; Filters A diode is a circuit element that allows current to pass through in one direction, but not the other.

32. The plot shows the voltage drop across the resistor. During half a cycle, it is zero. Putting a capacitor in the circuit “smoothes” out VR, producing a nearly constant voltage drop (a DC voltage).

33. A capacitor may be used as a filter. Low-pass filter. When XC << R ( is large) the output voltage will be small compared to the input voltage. When XC >> R ( is small), the output voltage will be comparable to the input voltage. This circuit will allow low frequency signals to pass through while filtering out high frequency signals.

34. A high-pass filter. This will allow high frequency signals to pass through while filtering out low frequency signals.

35. Summary • Difference Between Instantaneous, Average, and rms Values • Power Dissipation by R, L, and C • Reactance for R, L, and C • Impedance and Phase Angle • Resonance in an RLC Circuit • Diodes • High- and Low-Pass Filters