Transcript Welcome to the Vanderbilt Center for Radiation Oncology

Chapter 20
Induced Voltages and
Inductance
Inductors & RL Circuits
Sections 5–8
General
Physics
Generators
 Alternating Current (AC) and Direct Current
(DC) generators
– Converts mechanical energy to electrical energy
– Consists of a wire loop rotated through a magnetic
field by some external means
– There are a variety of sources that can supply the
energy to rotate the loop
• These may include falling water, heat by burning coal to
produce steam
General
Physics
AC Generators
 As the loop rotates (θ changes), the
magnetic flux through the loop
changes with time
 This induces an emf and a current in
the external circuit (toaster)
 The ends of the loop are connected to
slip rings that rotate with the loop
 Connections to the external circuit are
made by stationary brushes in contact
with the slip rings
 The output voltage oscillates
between positive and negative
polarity
 The current is an AC current
General
Physics
AC Generators – Rotating Loop
 Wires BC and AD act as bars moving
vertically through the horizontal magnetic
field between the N and S poles.
 An emf is generated in wires BC and AD
 The total emf produced in these 2 wires is
ε = 2 B ℓ v= 2 B ℓ v sin θ
 If the loop rotates with a constant angular
speed, ω, the emf generated by the
rotating loop is
ε =2 B ℓ (a / 2) ω sin ωt = B A ω sin ωt
 If a coil has N turns, the emf is N times as
large
ε = N B A ω sin ω t
Active Figure: AC Generator
General
Physics
DC Generators
 Components are essentially the
same as that of an AC generator
 The major difference is the
contacts to the rotating loop are
made by a split ring, or
commutator
 The output voltage always has the
same polarity
 The current is a DC pulsing
current
Active Figure: DC Generator
General
Physics
Motors
 Motors are devices that
convert electrical energy
(through magnetic forces)
into mechanical energy
– A motor is a generator run
in reverse
 A motor can perform useful
mechanical work when a
shaft connected to its
rotating coil is attached to
some external device
General
Physics
Motors and Back emf
 As the motor rotates, the magnetic flux
through the loop changes with time
 This induces a back emf that tends to
reduce the current applied to the motor
from the external source
 When a motor is first turned on, the
current is very large because there is
no back emf initially
 As the coil begins to rotate, the
induced back emf opposes the applied
voltage
 The current in the coil is reduced
 The power requirements for starting a motor and for running it
under heavy loads are greater than those for running the motor
under average loads
General
Physics
Joseph Henry
 1797 – 1878
 First director of the
Smithsonian
 First president of the Academy
of Natural Science
 First to produce an electric
current with a magnetic field
 Improved the design of the
electro-magnetic and
constructed a motor
 Discovered self-inductance
General
Physics
Self-inductance
 Self-inductance occurs when the
changing flux through a circuit
arises from the circuit itself
– When the switch is closed, the
–
–
–
–
current increases from zero
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 change in
magnetic flux
As the magnitude of the current increases, the rate of
increase lessens and the induced emf decreases
This opposing emf results in a gradual increase of the
current rather than a sharp increase
General
Physics
Self-inductance, cont
 The self-induced emf is proportional to
the time rate of change of the current
I
  L
t
– L is a proportionality constant called the
self-inductance of the circuit or device
– The SI unit of self-inductance is the Henry
1 H = 1 (V · s) / A
– The negative sign indicates that a changing
current induces an emf in opposition to that
change – Lenz’s law
General
Physics
Self-inductance, cont
 The inductance of a coil depends on
geometric factors
 You can determine L from the
expression
N B
L
I
 For a solenoid the inductance is
L
0 N  r
2
2
l
General
Physics
Self-Inductance and Lenz’ Law
 Consider an increasing current
through the inductor
 The self-induced emf has a
direction so as to oppose the
increase in the current
 Consider a decreasing current
through the inductor
 The self-induced emf has an
opposite direction so as to oppose
the decrease in the current
General
Physics
Inductor in a Circuit – RL Circuit
 When the switch is closed, the current
in the RL circuit increases from zero
 The increasing current induces an
emf in the inductor that opposes the
change in the current
 As the magnitude of the current
increases, the rate of increase lessens
and the self-induced emf decreases
 When the current reaches its
maximum, the rate of change and the
self-induced emf become zero
 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
General
Physics
RL Circuit, cont
 The time constant depends on R and L
L
 
R
 The current at any time can be found by
I 

1e


R
t / 
Active Figure: An RL Circuit
General
Physics
Energy Stored in a Magnetic
Field
 The emf induced by an inductor prevents a
battery from establishing an instantaneous
current in a circuit
 The battery has to do work to produce a current
– This work results in energy being stored by the
inductor in its magnetic field
PEL = ½ L I2
– Note that this result is similar to the expression for the
energy stored by a capacitor in its electric field
PEC = ½ C ΔV2
General
Physics