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10/23/2011 Units of Chapter 24 Chapter 24 • Alternating Voltages and Currents Alternating-Current Circuits • Capacitors in AC Circuits • RC Circuits • Inductors in AC Circuits • RLC Circuits • Resonance in Electrical Circuits Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 24-1 Alternating Voltages and Currents Wall sockets provide current and voltage that vary sinusoidally with time. 24-1 Alternating Voltages and Currents Since this circuit has only a resistor, the current is given by: Here, the current and voltage have peaks at the same time – they are in phase. The voltage and current in an ac circuit both average to zero, making the average useless in describing their behavior. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 24-1 Alternating Voltages and Currents By calculating the power and finding the average, we see that: Irms = Imax 2 = 0.707 Imax The root mean square (rms) value: Square the value, find the mean value, and then take the square root. 2 ℘av = Irms R Copyright © 2010 Pearson Education, Inc. Ohm’s Law in an AC Circuit • rms values will be used when discussing AC currents and voltages – AC ammeters and voltmeters are designed to read rms values. – 120V hosehold voltage is the rms value. – Many of the equations will be in the same form as in DC circuits • Ohm’s Law for a resistor, R, in an AC circuit VR,rms = Irms R • Also applies to the maximum values of v and i Example: An AC voltage source has an output of ∆V=(200V)sin 2πft. This source is connected to a 100-ohm resistor. Find the rms voltage, rms current, and power consumption in the resistor. Copyright © 2010 Pearson Education, Inc. 1 10/23/2011 Capacitors in an AC Circuit • Consider a circuit containing a capacitor and an AC source, the current starts out at a large value and charges the plates of the capacitor • As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases. The voltage lags the current by 90° • The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by Inductors in an AC Circuit • Consider an AC circuit with a source and an inductor • The current in the circuit is impeded by the back emf of the inductor • The voltage across the inductor always leads the current by 90° • The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by: ω =2πf • Ohm’s Law for a capacitor in an AC circuit: Copyright © 2010 Pearson Education, Inc. ω =2π f • Ohm’s Law for the inductor ∆VL,rms = Irms XL Copyright © 2010 Pearson Education, Inc. 24-5 RLC Circuits The RLC Series Circuit • The current in the circuit is the same at any time and varies sinusoidally with time • The instantaneous voltage across the resistor is in phase with the current • The instantaneous voltage across the inductor leads the current by 90° • The instantaneous voltage across the capacitor lags the current by 90° Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. Power in an AC Circuit • No power losses are associated with pure capacitors and pure inductors in an AC circuit – In a capacitor, during one-half of a cycle energy is stored and during the other half the energy is returned to the circuit – In an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit • The average power delivered by the generator is converted to internal energy in the resistor Pav = Irms∆VR = Irms∆Vrms cos φ cos φ is called the power factor of the circuit Copyright © 2010 Pearson Education, Inc. The phase angle φ between the current and the voltage: Positive φ (0 to 90°) means voltage leads current. Negative φ (0 to -90°) means current leads voltage. Summary of Circuit Elements, Impedance and Phase Angles Example: A 50.0-Ω resistor is connected in series with a 15.0-μF capacitor and a 60.0-Hz, 120-V source. Find the current and the power consumption in the circuit. Copyright © 2010 Pearson Education, Inc. 2 10/23/2011 24-6 Resonance in Electrical Circuits Resonance in an AC Circuit • Resonance occurs at the frequency, ƒo, where the current has its maximum value – To achieve maximum current, the impedance must have a minimum value. This occurs when XL = XC – Then, ƒo = If a charged capacitor is connected across an inductor, the system will oscillate indefinitely in the absence of resistance. 1 2π LC Theoretically, if R = 0 the current would be infinite at resonance. But real circuits always have some resistance Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. Summary of Chapter 24 Resonance, Applications • Tuning a radio – A varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received • Metal Detector – The portal is an inductor, and the frequency is set to a condition with no metal present – When metal is present, it changes the effective inductance, which changes the current – The change in current is detected and an alarm sounds • The voltage from an ac generator varies sinusoidally: • Phasor represents voltage or current in ac circuit; as it rotates, its y component gives the instantaneous value. • Root mean square (rms) of a sinusoidally varying quantity: Example: An RLC circuit has a capacitance of 0.26 μF. (a) What inductance will produce a resonance frequency of 81 MHz? (b) It is desired that the impedance at resonance be one-fifth the impedance at 20 KHz. What value of R should be used to obtain this result? Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. Summary of Chapter 24 • rms current in a capacitor: Summary of Chapter 24 • Inductive reactance: • Impedance of an RL circuit: • Capacitive reactance: • Voltage across capacitor lags current by 90° • Impedance of an RLC circuit: • Impedance in an RC circuit: • Resonant frequency of an LC circuit: • Average power: Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 3