Faraday’s Law
Chapter 23:
Induction and
AC Circuits
•
What does this mean?
ε = −N
ΔΦ B
Δt
•  “What good is a baby?”
Brent Royuk
Phys-112
Concordia University
•  “One day, sir, you may tax it.”
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Magnetic Flux
Rotation
•  Water pipe analogy
–  Flux through a butterfly net
•
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Φ B = BAcos θ
–  Can be visualized as the number of
field lines passing through a current
loop
–  Look at orientations of theta
–  Unit: 1 weber (Wb) = 1 T m2
ε = −N
ΔΦ B
Δt
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Changing Flux
Changing Flux
•  Faraday: 1830, an induced emf is
produced by a changing flux in a
circuit loop.
ΔΦ B
•  Demo
ε = −N
•
Ways flux can change:
1.  Relative motion
•
•
ε = −N
Either magnet or loop can move
Not all motion produces a change in flux
ΔΦ B
Δt
–  imagine a large uniform field
2.  Changing field strength
Δt
•
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Demo with two solenoids
3.  Changing orientation
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Rotating loop, for example
4.  Changing area of loop
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Explicit Equation:
⎡
⎛ Δ cos θ
⎛ ΔB ⎞
⎛ ΔA ⎞
⎟⎟ A cos θ + B⎜⎜
⎟⎟ cos θ + BA⎜
⎜ Δt
⎝ Δt ⎠
⎢⎣⎝ Δt ⎠
⎝
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E = −N⎢⎢⎜⎜
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(
)
(
)
(
) ⎞⎟⎤⎥
⎟⎥
⎠⎥⎦
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Changing Field Strength
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Examples
When is current induced?
Demo
•
•
•
The induced emf in a single loop of wire
has a magnitude of 1.48 V when the
magnetic flux is changed from 0.58 Wb
to 0.11 Wb. How much time is required
for this change in flux?
Find the emf for 50 square coils 10 cm
on a side when B changes from 0.02 to
0.18 T in 0.02 seconds.
What if B stays constant at 0.18 T but
the coils expand to 20 cm on a side in
0.02 s?
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Applications
•
Applications
Dynamic (Induction) Microphone
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Guitar Pickups
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Lenz’s Law
ε = −N
•
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Lenz’s Law
ΔΦ B
Δt
What is the meaning of the minus sign in
Faraday’s Law?
Lenz’s Law: An induced current always
flows in a direction that opposes the change
that caused it.
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Lenz’s Law
•
•
Lenz’s Law
Increasing vs. Decreasing strength
Animation
•
The direction of the needle deflection on
a galvanometer.
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Lenz’s Law
•
Eddy Currents
The Sign of the Change
•
•
Magnet in tube
Monstrosity
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Eddy Currents
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Eddy Currents
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Generators
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Generators
A slide-wire generator
Δ cos ωt
= ω sin ωt
Δt
= NBAω sin ωt
εα
ε
ω = 2 πf
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Example: The coil in an electric
generator has 100 turns and an
area of 2.5 x 10-3 m2. It is
desired that the maximum emf
of the coil be 120 V when it
rotates at a rate of 60.0 Hz.
What field strength is required?
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Generators
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Motors
What’s the difference between a
generator and a motor?
Problem: How do you run a motor with
DC electricity?
–  The commutator
Increase the
flux with
multiple
armatures
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Brushless Motors
Self-Inductance
•  Put the coils on the stator and the
magnets on the rotor.
•
•  Or use AC to induce a current on rotating
coils (induction motor).
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A coil in a circuit resists changes in the
amount of current flowing through it.
Inductors are aka chokes- block high
frequencies
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Self-Inductance
L≡
•
•
Φ
I
Φ = LI
ε = −L
ΔI
Δt
Back EMF
Unit: The henry (H)
Example: The coil in an
electromagnet has an
inductance of 2.9 mH
and carries a constant
current of 5.6 A. A
switch is opened and
the current drops to
zero, inducing an emf of
7.2 V. How long does
this take?
•
•
The braking effect caused by an inductor is
actually a voltage that resists the changing
current, and it’s called Back EMF.
When motors are spun by electricity, they
generate a back EMF
–
–
•
Maximum current occurs during the startup of an
electric motor.
“Cold-cranking amps.”
Generators have a counter torque.
–
Hand-crank generator
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Transformers
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Transformers
Place two solenoids side-by-side.
How can a DC voltage in one produce a
voltage in the other?
How can an AC voltage in one produce a
voltage in the other?
•
•
Get two coils to share the same
changing flux and their voltages will
differ by the number of turns in the
coils.
The transformer relations:
NP
NS
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VS
=
IS
IP
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Transformer Energy Loss
Step-Up vs. Step-Down
Isolation Transformers
Suppose that our neon transformer draws 4 A of current. How
much current does it supply to the discharge tube?
–
VP
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Transformers
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•
•
=
Neon transformers have an inductor in series with the
transformer. Why?
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Losses can come from flux leaks, selfinduction, resistive heating.
•
Eddy currents can be minimized with laminated
cores.
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Mechanical losses: Transformer hum
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The Power Grid
•
The Power Grid
Edison vs. Westinghouse
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The Power Grid
Introduction to AC
•
•
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DC is simple and stolid. AC is complex and
frisky.
How does the charge move through a circuit?
Capacitors and inductors impede current in a
way that depends on the frequency of the
signal.
–
–
•
A 1-ohm inductive circuit can pass practically no
current for 120-V electricity, if the frequency is
high.
A capacitor (which is an open circuit, remember)
can act like a wire for AC, depending on the
frequency.
AC circuits can have natural oscillation
frequencies.
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Resistive AC Circuits
Phasors
•
•
•
()
V t = Vmax sin ωt
The voltage
and current
are in phase.
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()
It =
ω = 2 πf
( ) = V € sin ωt = I
V t
R
max
R
A phasor is a vector that rotates counter-clockwise.
Using phasors to represent voltage and current make
phase relationships easy to see and analyze.
The instantaneous value of a quantity represented by a
phasor is equal to the phasor’s projection onto the yaxis.
max
sin ωt
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Phasors
Power
•  What does P = VI mean?
P = VI is meaningless, use P = I 2 R
I = I = I o sin ωt
So P =
2
= I o2 sin2 ωt = I o2 sin2 ωt =
1
2
I o2
1 2
I o R.
2
Define : I rms =
Similarly, Vrms
Io
2
, so P = I rms
R.
2
V
= o , so Vrms = I rms R, and
2
P = Vrms I rms .
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Home Electricity
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–
–
–
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GFCI Protection
What is home voltage/peak voltage?
•  Ground Fault Circuit Interruptors
Vo = 120 V or Vrms = 120 V?
110, 115, 120,…
How about Europe?
What is the power limit on a typical home circuit with a
15-A circuit breaker?
What if it’s a 20-A breaker with 240-V?
What’s true about
the green and
blue areas?
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Capacitors in AC Circuits
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Capacitors in AC Circuits
How does the current respond to the
voltage in the circuit below?
There’s more “resistance” for:
•
–  Why? Charge has to build up on
the capacitor in order for the
voltage to increase.
–  Low capacitances
•
The voltage lags the current.
The capacitor fills up too quickly
–  Low frequencies
•
•
High frequencies oscillate so quickly that the
capacitor doesn’t have time to fill up.
We call this “resistance” to current
flow the capacitive reactance.
A capacitor stores, then returns energy
but does not consume it.
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Capacitive Reactance
V = IX C
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€ •
•
XC =
1
2 πfC
=
Inductors in AC Circuits
•
1
•
ωC
How does the current respond to the voltage
in the circuit below?
There’s more “resistance” for:
–
High inductances
•
These are rms values.
What is the unit for reactance?
€
How much current would flow in a home
electrical circuit with a 3.7 µF capacitor?
–
High frequencies
•
•
The strong inductance resists changes in flux, thus
current.
Low frequencies oscillate slowly so that the
inductor allows the current to flow.
We call this “resistance” to current flow the
inductive reactance.
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Inductors in AC Circuits
•
Inductive Reactance
The voltage leads the current.
–
Why? The current builds up more
slowly than the voltage.
V = IX L
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X L = 2 πfL = ωL
•  How much current would flow in a
home €
electrical circuit with a 3.7
mH inductor?
An inductor stores, then returns energy
but does not consume it.
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ELI the ICE Man
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Impedance
•  What happens when we make a
series combination of LRC
elements?
•  Four possibilities:
–  RC, LR, RLC, LC
Graphic from Rod
Nave
The
Hyperphysics
Archive
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•  The combined effect of RLC circuit
elements on the current’s response
to the voltage is called the
impedance, Z.
•  V = IZ
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Impedance vs. Reactance
Finding Impedance
•
What are impedance and reactance?
Circuits in which current is proportional to voltage are called
linear circuits. (As soon as one inserts diodes and transistors,
circuits cease to be linear, but that's another story.) The ratio of
voltage to current in a resistor is its resistance. Resistance does not
depend on frequency, and in resistors the two are in phase.
However, circuits with only resistors are not very interesting.
In general, the ratio of voltage to current does depend on
frequency and in general there is a phase difference. So impedance
is the general name we give to the ratio of voltage to current. It has
the symbol Z. Resistance is a special case of impedance. Another
special case is that in which the voltage and current are out of phase
by 90°: this is an important case because when this happens, no
power is lost in the circuit. In this case where the voltage and
current are out of phase by 90°, the ratio of voltage to current is
called the reactance, and it has the symbol X.
Taken from: Physclips from UNSW
•
The big picture: we can combine reactance
and resistance “phasors” as vectors to find the
total impedance.
The phase angle ϕ tells how much the voltage
leads or lags the current in the total circuit.
http://www.physclips.unsw.edu.au/jw/AC.html
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RC Circuit
RL Circuit
R
ϕ
XC
Z
XL
Z
ϕ
R
R2 + XC2
Z =
tan ϕ =
XC
Z = R2 + X L2
R
ϕ < 0 (ICE)
tan ϕ =
ϕ > 0 (ELI)
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RLC Circuit
RLC Example
•
Z
XL
XL –XC
ϕ
R
XC
Z =
(
R2 + X L − X C
2
)
tan ϕ =
XL
R
Find the rms voltage across each element in an
RLC circuit with R = 9.9 kΩ, C = 0.15 mF, and
L = 25 mH. The generator supplies an rms
voltage of 115 V at a frequency of 60.0 Hz.
What is the phase angle between the voltage
and the current in the circuit?
X L − XC
R
ϕ> 0 means inductive circuit
ϕ< 0 means capacitive circuit
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Power Factor
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High/Low Frequency Behavior
Does the phase angle have any physical
meaning?
Power Factor = cos ϕ
cos ϕ =
R
Z
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Filters
•
“Equivalent” circuits
P = Vrms I rms cos ϕ
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Filters
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High-Pass and Low-Pass Filters
A Crossover Network
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LC Circuits
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LC Circuits
If there’s no resistance in a circuit, what is true?
Kirchoff’s Loop Theorem:
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•
IX L − IX C = 0
The Mass on a Spring Analogy
x⇒q, v⇒I, m⇒L, k⇒1/C
X L = XC
1
= ω oL
ω oC
ωo =
1
LC
; f =
1
ωo =
k
vs. ω o =
m
1
LC
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2 π LC
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Resonance
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Tunable Resonant Circuits
For a given R, L and C, at
what frequency is current
maximized?
Sound familiar? That’s
just like the LC circuit.
Demo gadget
When an RLC circuit is
driven at resonance, does
the voltage lead or lag
the current?
ωo =
1
LC
fo =
1
2 π LC
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