Generators and MotorsGenerators and Motors

Lightning ReviewLightning Review

Last lecture:

1.1. Induced voltages and inductionInduced voltages and induction Induced EMFInduced EMF Faraday’s lawFaraday’s law Motional EMFMotional EMF

Review Problem: Two very long, fixed wires cross each other perpendicularly. They do not touch but are close to each other, as shown. Equal currents flow in the wires, in the directions shown. Indicate the locus of points where the net magnetic field is zero.

cosBA

Nt

E

BlvE

0

2

IB

r

GeneratorsGenerators

A generator is something that converts mechanical energy A generator is something that converts mechanical energy

to electrical energy.to electrical energy.

Generators and motors are two of the most important Generators and motors are two of the most important applications of induced emf (magnetic inductance). applications of induced emf (magnetic inductance).

Alternating Current (AC) generatorAlternating Current (AC) generator Direct Current (DC) generatorDirect Current (DC) generator

A motor does the opposite, it converts electrical energy to A motor does the opposite, it converts electrical energy to

mechanical energymechanical energy. .

AC GeneratorsAC Generators Basic operation of the generatorBasic operation of the generator

– As the As the loop rotatesloop rotates, the , the magnetic flux through it magnetic flux through it changeschanges with time with time

– This This induces an emfinduces an emf and a and a current in the external circuitcurrent in the external circuit

– 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 stationary circuit are made by stationary brushed in contact with the brushed in contact with the slip ringsslip rings

AC generatorAC generator

Compute EMFCompute EMF– It is only generated in BC It is only generated in BC

and DA wiresand DA wires– EMF generated in BC and EMF generated in BC and

DA would beDA would be

– Thus, total EMF isThus, total EMF is

– If the loop is rotating with If the loop is rotating with

B

vv sin

A

AB

CD

Blv BC DAE E

2 2 sinBlv Blv E

2 sin 2 sin2

aBlv t Bl t

E as v=r=a/2

AC generator (cont)AC generator (cont)

Generalize the result to N Generalize the result to N loopsloops

where we also noticed that where we also noticed that A=laA=la

Note: Note: is reached is reached when when t=90t=90˚̊ or 270 or 270˚̊ (loop parallel (loop parallel to the magnetic field)to the magnetic field)

sinNBA t E

NBAmaxE

EMF generated by the AC generator

AC generatorAC generator

Compute EMFCompute EMF– It is only generated in BC It is only generated in BC

and DA wiresand DA wires– EMF generated in BC and EMF generated in BC and

DA would beDA would be

– Thus, total EMF isThus, total EMF is

– If the loop is rotating with If the loop is rotating with

B

vv sin

A

AB

CD

Blv BC DAE E

2 2 sinBlv Blv E

2 sin 2 sin2

aBlv t Bl t

E as v=r=a/2

DC generatorDC generator By a clever change to the rings and brushes By a clever change to the rings and brushes

of the ac generator, we can create a dc of the ac generator, we can create a dc generator, that is, a generator where the generator, that is, a generator where the polarity of the emf is always positive. polarity of the emf is always positive.

The basic idea is to use The basic idea is to use a single split ring a single split ring instead of two complete ringsinstead of two complete rings. The split ring is . The split ring is arranged so that, just as the emf is about to arranged so that, just as the emf is about to change sign from positive to negative, the change sign from positive to negative, the brushes cross the gap, and the polarity of the brushes cross the gap, and the polarity of the contacts is switched. contacts is switched.

The polarity of the contacts changes in phase The polarity of the contacts changes in phase with the polarity of the emf -- the two changes with the polarity of the emf -- the two changes essentially cancel each other out, and the emf essentially cancel each other out, and the emf remains always positive. remains always positive.

The emf still varies The emf still varies sinusoidallysinusoidally during each during each half cycle, but every half cycle is a positive half cycle, but every half cycle is a positive emf. emf.

MotorsMotors

A motor is basically a generator running in A motor is basically a generator running in reverse. A current is passed through the reverse. A current is passed through the coil, producing a torque and causing the coil coil, producing a torque and causing the coil to rotate in the magnetic field. Once turning, to rotate in the magnetic field. Once turning, the coil of the motor generates a the coil of the motor generates a back emfback emf, , just as does the coil of a generator. The just as does the coil of a generator. The back emf cancels some of the applied emf, back emf cancels some of the applied emf, and limits the current through the coil. and limits the current through the coil.

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 tends to reduce the applied currentapplied current

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

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

Example: coil in magnetic fieldExample: coil in magnetic field

A coil of area 0.10 m² is rotating at 60 rev/s with its axis of rotation A coil of area 0.10 m² is rotating at 60 rev/s with its axis of rotation perpendicular to a 0.20T magnetic field. (a) If there are 1000 turns on perpendicular to a 0.20T magnetic field. (a) If there are 1000 turns on the coil, what is the maximum voltage induced in the coil? (b) When the the coil, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the coil maximum induced voltage occurs, what is the orientation of the coil with respect to the magnetic field? with respect to the magnetic field?

Eddy currents (application)Eddy currents (application) Magnetic Levitation (Maglev) TrainsMagnetic Levitation (Maglev) Trains

– Induced surface (Induced surface (“eddy”“eddy”) currents produce field in opposite direction) currents produce field in opposite direction

– Maglev trains today can travel up to 310 mphMaglev trains today can travel up to 310 mph

Twice the speed of Amtrak’s fastest conventional train!Twice the speed of Amtrak’s fastest conventional train!

– May eventually use superconducting loops to produce B-fieldMay eventually use superconducting loops to produce B-field

No power dissipation in resistance of wires!No power dissipation in resistance of wires!

N

S

rails“eddy” current

Self-inductanceSelf-inductance When a current flows through a loop, the magnetic field

created by that current has a magnetic flux through the area of the loop.

If the current changes, the magnetic field changes, and so the flux changes giving rise to an induced emf. This phenomenon is called self-induction because it si the loop's own current, and not an external one, that gives rise to the induced emf.

Faraday’s law states

Nt

E

The magnetic flux is proportional to the magnetic field, The magnetic flux is proportional to the magnetic field, which is proportional to the current in the circuitwhich is proportional to the current in the circuit

Thus, the self-induced EMF must be proportional to the Thus, the self-induced EMF must be proportional to the time rate of change of the currenttime rate of change of the current

where where LL is called the is called the inductance inductance of the deviceof the device

Units: SI: henry (H)Units: SI: henry (H)

If flux is initially zero,If flux is initially zero,

IL

t

E

1 1V sH A

NL N

I I

Example: solenoidExample: solenoid

A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75mV. through it to produce an emf of 75mV.