alternating-current circuits and machines chapter 22
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Alternating-Current Circuits and Machines
Chapter 22
DC Circuit SummaryDC circuits
DC stands for direct currentSource of electrical energy is generally a batteryIf only resistors are in the circuit, the current is
independent of timeIf the circuit contains capacitors and resistors, the
current can vary with time but always approaches a constant value a long time after closing the switch
AC Circuit IntroductionAC stands for alternating current
The power source is a device that produces an electric potential that varies with time
There will be a frequency and peak voltage associated with the potential
Household electrical energy is supplied by an AC sourceStandard frequency is 60 Hz
AC circuits have numerous advantages over DC circuits
DC vs. AC Sources
Generating AC VoltagesMost sources of AC voltage employ a generator
based on magnetic inductionA shaft holds a coil with many loops of wireThe coil is positioned between the poles of a
permanent magnetThe magnetic flux through the coil varies with time
as the shaft turnsThis changing flux induces a voltage in the coil
Section 22.1
Generators Generators of electrical energy convert the
mechanical energy of the rotating shaft into electrical energy
The principle of conservation of energy still appliesThe source of electrical energy in a circuit enables
the transfer of electrical energy from a generator to an attached circuit
Section 22.1
AC Circuits and Simple Harmonic MotionThe voltage variation of an AC circuit is reminiscent
of a simple harmonic oscillatorThere is also a close connection between circuits
with capacitors and inductors and simple harmonic motion
Section 22.1
Resistors in AC Circuits Assume a circuit
consisting of an AC generator and a resistor
The voltage across the output of the AC source varies with time according to
V = Vmax sin (2 π ƒ t)V is the instantaneous
potential difference
Section 22.2
Resistors, cont.Applying Ohm’s Law:
Since the voltage varies sinusoidally, so does the current
VI R
max
max
maxmax
VI sin πƒt or
RI I sin πƒt where
VI
R
2
2
Section 22.2
RMS VoltageTo specify current and
voltage values when they vary with time, rms values were adoptedRMS stands for Root
Mean SquareFor the voltage
maxrms avg
VV V 2
2
Section 22.2
RMS CurrentThe root-mean-square value can be defined for any
quantityFor the current
The root-mean-square values of the voltage and current are typically used to specify the properties of an AC circuit
maxrms max max
II I . I 21
0 712 2
Section 22.2
PowerThe instantaneous
power is the product of the instantaneous voltage and instantaneous currentP = I V
Since both I and V vary with time, the power also varies with time
P = Vmax Imax sin2 (2πƒt)
Section 22.2
Power, cont.Devices come with a power rating
A single number that tells you about the power usage of the device
The instantaneous power varies between Vmax Imax and 0
The average power is ½ the maximum powerPavg = ½ (Vmax Imax ) = Vrms Irms
Ohm’s Law can again be used to express the power in different ways
rmsave rms
VP I R
R
22
Section 22.2
LC Circuit
Most useful circuits contain multiple circuit elementsWill start with an LC circuit, containing just an
inductor and a capacitorNo AC generator is included, but some excess
charge is placed on the capacitor at t = 0Section 22.5
LC Circuit, cont.After t = 0, the charge moves from one capacitor plate to
the other and current passes through the inductorEventually, the charge on each capacitor plate falls to
zeroThe inductor again opposes change in the current, so the
induced emf now acts to maintain the current at a nonzero value
This current continues to transport charge from one capacitor plate to the other, causing the capacitor’s charge and voltage to reverse sign
Eventually the charge on the capacitor returns to its original value
Section 22.5
LC Circuit, finalThe voltage and current in the circuit oscillate
between positive and negative valuesThe circuit behaves as a simple harmonic oscillatorThe charge is q = qmax cos (2πƒt)
The current is I = Imax sin (2πƒt)
Section 22.5
Energy in an LC CircuitCapacitors and inductors
store energyA capacitor stores energy
in its electric field and depends on the charge
An inductor stores energy in its magnetic field and depends on the current
As the charge and current oscillate, the energies stored also oscillate
Section 22.5
Energy CalculationsFor the capacitor,
For the inductor,
The energy oscillates back and forth between the capacitor and its electric field and the inductor and its magnetic field
The total energy must remain constant
maxcap
qqPE cos πƒt
C C
2221 1
22 2
ind maxPE LI LI sin πƒt 2 2 21 12
2 2
Section 22.5
Energy, finalThe maximum energy in the capacitor must equal
the maximum energy in the inductor From energy considerations, the maximum value of
the current can be calculated
This shows how the amplitudes of the current and charge oscillations in the LC circuits are related
max maxI qLC
1
Section 22.5
RL Circuit ExampleWhen the input frequency is
very low, the reactance of the inductor is smallThe inductor acts as a wire Voltage drop will be 0
At high frequencies, the inductor acts as an open circuitNo current is passedThe output voltage is equal
to the input voltageThis circuit acts as a high-
pass filter
Section 22.8
RC Circuit ExampleWhen the input frequency is
very low, the reactance of the capacitor is largeThe current is very smallThe capacitor acts as an
open circuitThe output voltage is equal
to the input voltageAt high frequencies, the
capacitor acts as a short circuitThe inductor acts as a wire The output voltage is 0
This circuit acts as a low-pass filter
Section 22.8
Transformers
Transformers are devices that can increase or decrease the amplitude of an applied AC voltage
A simple transformer consists of two solenoid coils with the loops arranged so that all or most of the magnetic field lines and flux generated by one coil pass through the other coil
Section 22.9
Transformers, cont.The wires are covered with a nonconducting layer so
that current cannot flow directly from one coil to the other
An AC current in one coil will induce an AC voltage across the other coil
An AC voltage source is typically attached to one of the coils called the input coil
The other coil is called the output coil
Transformers, EquationsFaraday’s Law applies to both coils
If the input coil has Nin coils and the output coil has Nout turns, the flux in the coils is related by
The voltages are related by
outinin outV and V
t t
outout in
in
N
N
outout in
in
NV V
N
Section 22.9
Transformers, finalThe ratio of the turns can be greater than or less
than oneTherefore, the input voltage can be transformed to a
different valueTransformers cannot change DC voltages
Since they are based on Faraday’s Law
Section 22.9
Practical TransformersMost practical
transformers have central regions filled with a magnetic material
This produces a larger flux, resulting in a larger voltage at both the input and output coils
The ratio Vout / Vin is not affected by the presence of the magnetic material
Section 22.9
Applications of TransformersTransformers are used in the transmission of electric
power over long distancesMany household appliances use transformers to
convert the AC voltage at a wall socket to the smaller voltages needed in many devicesTwo steps are needed – converting 120 V to 9 V then
AC to DC
Section 22.9
Transformers and PowerThe output voltage of a transformer can be made
much larger by arranging the number of coilsAccording to the principle of conservation of energy,
the energy delivered through the input coil must either be stored in the transformer’s magnetic field or transferred to the output circuit
Over many cycles, the stored energy is constantThe power delivered to the input coil must equal the
output power
Section 22.9
Power, cont.Since P = V I, if Vout is greater than Vin, then Iout must
be smaller than Iin
Pin = Pout only in an ideal transformer
In real transformers, the coils always have a small electrical resistance
This causes some power dissipationFor a real transformer, the output power is always
less than the input powerUsually by only a small amount
Section 22.9
Motors
An AC voltage source can be use to power a motorThe AC source is connected to a coil wound around
a horseshoe magnetThe input coil induces a magnetic field that circulates
through the horseshoe magnet
Section 22.10
Motors, cont.A second coil is mounted between the poles of the
horseshoe magnet and attached to a rotating shaftThe forces acting on the second coil produce a
torque on the coilThis causes the shaft to rotate
As the AC current in the input coil changes direction, so do the forces
The torques continue to produce a rotation that is always in the same direction
The oscillations of the AC current and field make the shaft rotate
Section 22.10
Advantages of AC vs. DCBiggest advantage is in
the systems that distribute electric power across long distances
The power generated at a power plant must be distributed to distance places
The power plant acts as an AC generator
Section 22.11
Advantages, cont.There is power dissipated in the power linesPave = (Irms )2 Rline
The power company wants to minimize these power losses, so they want to make Irms as small as possible
The voltage is increased in order to decrease the current
A transformer is used to drop the high voltages in the power lines to the lower voltages at the house
Section 22.11
Advantages, finalThe power lines have typical maximum voltages of
500,000 VThe transformer reduces the voltage to a maximum
voltage of 170 VTypically 5% to 10% of the energy that leaves the
power plant is dissipated in the resistance of the power lines
Section 22.11
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