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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 21: Alternating Currents Sinusoidal Voltages and Currents Capacitors, Resistors, and Inductors in AC Circuits Series RLC Circuits Resonance AC to DC Conversion

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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 .

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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 I 0 = 0 /R.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 The instantaneous power dissipated in a resistor will be: The power dissipated depends on t (where in the cycle the current/voltage are).

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 What is the average power dissipated by a resistor in one cycle? The average value sin 2 t over one cycle is 1/2. The average power is

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 In terms of rms quantities, the power dissipated by a resistor can be written as:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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?

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 (b) What is the rms current drawn by the hair dryer? (c) What is the maximum instantaneous power that the resistance must withstand? P max = 2P av = 2400 Watts Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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 V C I. The equality is V c = IX C where X C is called capacitive reactance. (Think Ohm’s Law!) Reactance has units of ohms.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 For a resistor in an AC circuit, The voltage and current will be in phase with each other.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 For an inductor in an AC circuit: Also, V L = IX L where the inductive reactance is: Slope of an I(t) vs. t plot

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 A similar result is found for inductors; no energy is dissipated by an ideal inductor.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 §21.5 Series RLC Circuits

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Applying Kirchhoff’s loop rule:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 To find the amplitude ( 0 ) and phase ( ) of the total voltage we add V L, V R, and V C together by using phasors. Z is called impedance. X y VRVR VLVL VCVC 00

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 The phase angle between the current in the circuit and the input voltage is: >0 when X L > X C and the voltage leads the current (shown above). <0 when X L < X C and the voltage lags the current. X y VRVR VLVL VCVC 00

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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 10 3 rad/sec. Find… (a) The reactances of the capacitor and the inductor. (b) The impedance.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 (c) The rms current: (d) The current amplitude: Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 (e) The phase angle: (f) The rms voltages across each circuit element: (Or 37°) Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 (g) Does the current lead or lag the voltage? (h) Draw a phasor diagram. Since X L >X C, is a positive angle. The voltage leads the current. y X VRVR VLVL VCVC rms Example continued:

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 The power dissipated by a resistor is: where cos is called the power factor (compare to slide 7; Why is there a difference?).

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 §21.6 Resonance in RLC Circuits A plot of I vs. for a series RLC circuit has a peak at = 0.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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 X L = X C (or when Z=R). This is the resonance frequency for the circuit.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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 V R, producing a nearly constant voltage drop (a DC voltage).

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 A capacitor may be used as a filter. Low-pass filter. When X C << R ( is large) the output voltage will be small compared to the input voltage. When X C >> 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.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 A high-pass filter. This will allow high frequency signals to pass through while filtering out low frequency signals.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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

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