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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
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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.
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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:
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ω =2π f
• Ohm’s Law for the inductor
∆VL,rms = Irms XL
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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
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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.
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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?
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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:
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Copyright © 2010 Pearson Education, Inc.
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