20 5 Generators n Alternating Current AC generator

20. 5 Generators n Alternating Current (AC) generator n Converts mechanical energy to electrical energy n Consists of a wire loop rotated by some external means n There a variety of sources that can supply the energy to rotate the loop n For example, these may include falling water or heat by burning coal to produce steam

AC Generators, cont. n Basic operation of the generator n As the loop rotates, the magnetic flux through it changes with time n This induces an emf and a current in the external circuit n The ends of the loop are connected to slip rings that rotate with the loop n Connections to the external circuit are made by stationary brushes in contact with the slip rings

AC Generators, cont. Area A=ℓa The emf generated in wire BC is Bℓv where ℓ is the length of the wire and v is the velocity component perpendicular to the B field (v has no effect on the charges in the wire ). An emf of Bℓv is also generated in the wire DA with the same sense as in BC. Because v =v sinq, the total emf is ε =2 Bℓv sinθ.

AC Generators, cont. Since v=r (tangential speed=radius times angular speed), it follows v=(a/2) and t ε = 2 Bℓv sinθ = 2 Bℓ (a/2) sinθ Therefore, e=Bℓa sin t and with A=ℓa e=NBA sin t For a coil with N turns

Generator equation from Faraday’s law t e=-ND B/Dt e=-NBA[D(cosq)/Dt] Consider: d(cos t)/dt=- sin t e=NBA sin t emax = NBA (maximum value of the emf) e=emaxsin t=emaxsin 2 ft 2 f

AC Generators, final n The emf generated by the rotating loop can be found by ε =2 Bℓv sinθ If the loop rotates with a constant angular speed, ω, and N turns ε=NBAω sinωt n ε = εmax when loop is parallel to the field n ε = 0 when the loop is perpendicular to the field n

Direct current (DC) Generators Components are essentially the same as that of an ac generator n The major difference is the contacts to the rotating loop are made by a split ring, or commutator n

DC Generators, cont The output voltage always has the same polarity n The current is a pulsating current n To produce a steady current, many loops and commutators around the axis of rotation are used n The multiple outputs are superimposed and the output is almost free of fluctuations n

Motors n Motors are devices that convert electrical energy into mechanical energy n A motor is a generator run in reverse n A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

Motors and Back 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 e.

Motors and Back emf The phrase back emf is used for an emf that tends to reduce the current due to an applied voltage current through the motor: I=V-eb/R, where V is the line voltage, eb is the back emf and R is the coil resistance n When a motor is turned on, there is no back emf initially n The current is very large because it is limited only by the resistance of the coil n

Motors and Back emf, cont. n As the coil begins to rotate, the induced back emf opposes the applied voltage n The current in the coil is reduced n The power (i. e. , current) requirements for starting a motor and for running it under heavy loads are greater than those for running the motor under average loads

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. n (a) I=V/R=120 V/10 n I=12 A n (b) I=(V-eb)/R n I=(120 V-70 V)/10 n I=50 V/10 =5 A

20. 6 Self-inductance n Self-inductance occurs when the changing flux through a circuit arises from the circuit itself n n As the current increases, the magnetic flux through a loop due to this current also increases The increasing flux induces an emf that opposes the current As the magnitude of the current increases, the rate of increase lessens and hence the induced emf decreases This opposing emf results in a gradual increase in 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. n The self-induced emf is given by Faraday’s law and must be proportional to the time rate of change of the current n n L is a proportionality constant called the inductance of the device The negative sign indicates that a changing current induces an emf in opposition to that change

Self-inductance, final n The inductance of a coil depends on geometric factors n The SI unit of self-inductance is the Henry n 1 H = 1 (Vs)/A n The equation for L

20. 7 RL Circuits Inductor has a large inductance (L) and consist of closely wrapped coil of many turns n Inductance can be interpreted as a measure of opposition to the rate of change in the current n n n Remember resistance R is a measure of opposition to the current As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current n Therefore, the current doesn’t change from 0 to its maximum instantaneously

Comparison of R and L in a simple circuit e=-IR R is a measure of opposition to the current e=-L(DI/Dt) L is a measure of opposition to the rate of change in current

RL Circuit When the current reaches its maximum, the rate of change and the back emf are zero n The time constant, , for an RL circuit is the time required for the current in the circuit to reach 63. 2% of its final value n

RL Circuit, cont n The time constant depends on R and L n The current at any time can be found by

QUICK QUIZ 20. 5 The 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 Magnetic Field n The emf induced by an inductor prevents a battery from establishing an instantaneous current in a circuit n The battery has to do work to produce a current n This work can be thought of as energy stored by the inductor in its magnetic field

Energy stored, final n The increment of work done by a battery to move DQ through an inductor is: DW=DQe n DW=DQ [L(DI/Dt)] n Since I=DQ/Dt, the work done is: DW=LI (DI) Energy stored by an inductor