20.5 Generators20.5 Generators Alternating Current (AC) generatorAlternating Current (AC) generator

Converts mechanical energy to electrical Converts mechanical energy to electrical energyenergy

Consists of a wire loop rotated by some Consists of a wire loop rotated by some external meansexternal means

There are a variety of sources that can supply There are a variety of sources that can supply the energy to rotate the loopthe energy to rotate the loop For example, these may include falling water For example, these may include falling water

or heat by burning coal to produce steamor heat by burning coal to produce steam

AC Generators, cont.AC Generators, cont. Basic operation of the Basic operation of the

generatorgenerator As the loop rotates, the As the loop rotates, the

magnetic flux through it magnetic flux through it changes with timechanges with time

This induces an emf and a This induces an emf and a current in the external current in the external circuitcircuit

The ends of the loop are The ends of the loop are connected to slip rings that connected to slip rings that rotate with the looprotate with the loop

Connections to the external Connections to the external circuit are made by circuit are made by stationary brushes in stationary brushes in contact with the slip ringscontact with the slip rings

AC Generators, cont. AC Generators, cont.

The emf generated in wire BC is Bℓvℓv where ℓℓ is the length of the wire and vv is the velocity component perpendicular to the B field (v has no effect on the charges in the wire ). An emf of Bℓvℓv is also generated in the wire DA with the same sense as in BC. Because vv=v sin, the total emf is ε ε =2=2Bℓv Bℓv sinsinθ.θ.

Area A=ℓaℓa

AC Generators, cont.AC Generators, cont. SinceSince v v==rr(tangential (tangential

speed=radius times angular speed), speed=radius times angular speed), it follows it follows vv=(=(aa/2)/2)andand

ε ε = 2= 2Bℓv Bℓv sinsinθθ = 2 = 2Bℓ Bℓ ((aa/2)/2) sinsinθθ Therefore, Therefore, ==BℓBℓaa sinsint and with t and with

A=ℓaℓa=NBA sint

t

For a coil with N turns

Generator equation from Generator equation from Faraday’s lawFaraday’s law

=-=-NNBB//tt=-=-NBANBA[[(cos(cos)/)/tt]]Consider: d(cosConsider: d(costt)/d)/dtt=-=-sinsintt==NBANBAsinsintt

==maxmaxsinsint=t=maxmaxsin2sin2ftft

maxmax NBANBA (maximum value of the (maximum value of the emf) emf)   

t

22ff

AC Generators, finalAC Generators, final The emf generated by The emf generated by

the rotating loop can be the rotating loop can be found byfound byε ε =2=2BℓvBℓv=2=2BℓvBℓv sin sinθθ

If the loop rotates with a If the loop rotates with a constant angular speed, constant angular speed, ωω, and , and NN turns turnsεε==NBAωNBAω sin sinωtωt

εε = = εεmaxmax when loop is when loop is parallel to the fieldparallel to the field

εε = 0 when when the = 0 when when the loop is perpendicular to loop is perpendicular to the fieldthe field

Direct current (DC) Direct current (DC) GeneratorsGenerators

Components are Components are essentially the essentially the same as that of an same as that of an ac generatorac generator

The major The major difference is the difference is the contacts to the contacts to the rotating loop are rotating loop are made by a split made by a split ring, or commutatorring, or commutator

DC Generators, contDC Generators, cont The output voltage The output voltage

always has the same always has the same polaritypolarity

The current is a The current is a pulsating currentpulsating current

To produce a steady To produce a steady current, many loops and current, many loops and commutators around the commutators around the axis of rotation are usedaxis of rotation are used The multiple outputs The multiple outputs

are superimposed and are superimposed and the output is almost the output is almost free of fluctuationsfree of fluctuations

MotorsMotors Motors are devices that convert Motors are devices that convert

electrical energy into mechanical electrical energy into mechanical energyenergy A motor is a generator run in reverseA motor is a generator run in reverse

A motor can perform useful mechanical A motor can perform useful mechanical work when a shaft connected to its work when a shaft connected to its rotating coil is attached to some rotating coil is attached to some external deviceexternal device

Motors and Back emfMotors and Back emfBack emf

The applied voltage V supplies the current I to drive the motor. The circuit shows V along with the electrical equivalent of the motor, including the resistance R of its coil and the back emf .

Motors and Back emfMotors and Back emf The phrase The phrase back emfback emf is used for an emf that is used for an emf that

tends to reduce the current due to an applied tends to reduce the current due to an applied voltage voltage current through the motor: current through the motor: I=V-I=V-bb/R, /R, where where VV is the line voltage, is the line voltage, bb is the back emf is the back emf and and RR is the coil resistance is the coil resistance

When a motor is turned on, there is no back When a motor is turned on, there is no back emf initiallyemf initially

The current is very large because it is limited The current is very large because it is limited only by the resistance of the coilonly by the resistance of the coil

Motors and Back emf, Motors and Back emf, cont.cont.

As the coil begins to rotate, the As the coil begins to rotate, the induced back emf opposes the induced back emf opposes the applied voltageapplied voltage

The current in the coil is reducedThe current in the coil is reduced The power (i.e., current) The power (i.e., current)

requirements for starting a motor requirements for starting a motor and for running it under heavy loads and for running it under heavy loads are greater than those for running are greater than those for running the motor under average loadsthe motor under average loads

(a)(a) II==VV//RR=120 V/10 =120 V/10

II=12 A=12 A

(b)(b) II=(=(VV--bb)/)/RR II=(120 V-70 V)/10 =(120 V-70 V)/10 II=50 V/10 =50 V/10 =5 A=5 A

Example: A motor has a 10 coil. When running at its maximum speed, the back emf is 70 V. Find the current (a) when the motor starts and (b) when the motor has reached its maximum speed.

20.6 Self-inductance20.6 Self-inductance Self-inductanceSelf-inductance occurs when the changing occurs when the changing

flux through a circuit arises from the circuit flux through a circuit arises from the circuit itselfitself As the current increases, the magnetic flux As the current increases, the magnetic flux

through a loop due to this current also increasesthrough a loop due to this current also increases The increasing flux induces an emf that opposes The increasing flux induces an emf that opposes

the currentthe current As the magnitude of the current increases, the As the magnitude of the current increases, the

raterate of increase lessens and hence the induced of increase lessens and hence the induced emf decreasesemf decreases

This opposing emf results in a gradual increase This opposing emf results in a gradual increase in the currentin the current

Self-inductance, cont.

(a) A current in the coil produces a magnetic field directed to the left. (b) If the current increases, the coil acts as a source of emf directed as shown by the dashed battery. (c) The induced emf in the coil changes its polarity if the current decreases.

Self-inductance, cont.Self-inductance, cont. The self-induced emf is given by Faraday’s law The self-induced emf is given by Faraday’s law

and must be proportional to the time rate of and must be proportional to the time rate of change of the currentchange of the current

LL is a proportionality constant called the is a proportionality constant called the inductanceinductance of the device of the device

The negative sign indicates that a changing The negative sign indicates that a changing current induces an emf in opposition to that current induces an emf in opposition to that changechange

tIL

tN

B

Self-inductance, finalSelf-inductance, final The inductance of a coil depends The inductance of a coil depends

on geometric factorson geometric factors The SI unit of self-inductance is the The SI unit of self-inductance is the

HenryHenry 1 H = 1 (V1 H = 1 (Vs)/As)/A

The equation for The equation for LL

IΦNL

B

20.7 RL Circuits20.7 RL Circuits InductorInductor has a large inductance ( has a large inductance (LL) and consist ) and consist

of closely wrapped coil of many turns of closely wrapped coil of many turns Inductance can be interpreted as a measure of Inductance can be interpreted as a measure of

opposition to the rate of change in the currentopposition to the rate of change in the current Remember resistance Remember resistance RR is a measure of opposition is a measure of opposition

to the currentto the current As a circuit is completed, the current begins to As a circuit is completed, the current begins to

increase, but the inductor produces an emf increase, but the inductor produces an emf that opposes the increasing currentthat opposes the increasing current Therefore, the current doesn’t change from 0 to its Therefore, the current doesn’t change from 0 to its

maximum instantaneouslymaximum instantaneously

Comparison of Comparison of RR and and LL in in a simple circuit a simple circuit

=-IR =-L(I/t)L is a measure of opposition to the rate of change in current

R is a measure of opposition to the current

RL CircuitRL Circuit When the current When the current

reaches its maximum, reaches its maximum, the rate of change and the rate of change and the back emf are zerothe back emf are zero

The time constant, The time constant, , , for an for an RLRL circuit is the circuit is the time required for the time required for the current in the circuit to current in the circuit to reach 63.2% of its final reach 63.2% of its final valuevalue

RL Circuit, contRL Circuit, cont The time constant depends on The time constant depends on RR

and and LL

The current at any time can be The current at any time can be found byfound by

RL

t/eR

I 1

QUICK QUIZ 20.5The switch in the circuit shown in the figure below is closed and the lightbulb glows steadily. The inductor is a simple air-core solenoid. An iron rod is inserted into the interior of the solenoid, which increases the magnitude of the magnetic field in the solenoid. As the rod is inserted into the solenoid, the brightness of the lightbulb (a) increases, (b) decreases, or (c) remains the same.

20.8 Energy Stored in a 20.8 Energy Stored in a Magnetic FieldMagnetic Field The emf induced by an inductor The emf induced by an inductor

prevents a battery from establishing prevents a battery from establishing an instantaneous current in a circuitan instantaneous current in a circuit

The battery has to do work to The battery has to do work to produce a currentproduce a currentThis work can be thought of as This work can be thought of as

energy stored by the inductor in energy stored by the inductor in its magnetic fieldits magnetic field

Energy stored, finalEnergy stored, final The increment of work done by a The increment of work done by a

battery to move battery to move QQ through an through an inductor is: inductor is: WW==QQ

WW==Q Q [[L(L(I/I/tt)])] Since Since II==Q/Q/t, t, the work done is: the work done is: WW==LI LI ((II))

2

21 LIIdILW

Energy stored by an inductor