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