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Self-inductance

Self-inductance occurs when the changing
flux through a circuit arises from the
circuit itself
• 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 the induced emf
decreases
• This opposing emf results in a gradual increase
of the current
Self-inductance cont

The self-induced emf must be proportional
to the time rate of change of the current
i
  L
t
• 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


The inductance of a coil depends on
geometric factors
The SI unit of self-inductance is the
Henry
• 1 H = 1 (V · s) / A
For infinitely long solenoid
B = monI
n: number of turns/m
http://www.bugman123.com/Physics/Physics.html
Winding density: n
Loop area: A
Length of coil: ℓ
Increase current through the coil
from 0 to i in t.
Causes flux change for each loop
from F = 0 to F = AB where B = moni
Vind = - n ℓ F/t
= - n ℓ A(mon)i/t
= - n2moA ℓ (i/t)
L: self inductance
Purely geometrical quantity
[L] = Henry
r = 1 cm
= 0.01 m
ℓ = 2.5 cm
= 0.025 m
n = 10 turns/cm = 1000 turns/m
L = n2moA ℓ
= (1000)2(4 p x 10-7)(p x 0.0012)0.025
= 9.9 x 10-6 H
= 9.9 mH
Example 24.5
A steady current of 5.0 A is flowing through a coil with 2 H
Self inductance. The current is suddenly stopped in 0.01 s.
How large an emf induced in the coil?
Vind = -L (i/t)
Di = if – ii = -5 A
= - (2.0) (-5/0.01)
= 1000 V
Inductors in an AC Circuit



Consider an AC circuit
with a source and an
inductor
The current in the
circuit is impeded by
the back emf of the
inductor
The voltage across the
inductor always leads
the current by 90°
(T/4)
V
v
A
AC
i lags 90 deg behind v.
i
v= vosin(wt)
v = -L i/t
i = -io cos(wt)
vosin(wt) = wLiosin(wt)
vo = wLio
Inductive Reactance and Ohm’s
Law

The effective resistance of a coil in
an AC circuit is called its inductive
reactance and is given by
• XL = wL=2pƒL


When ƒ is in Hz and L is in H, XL will be in
ohms
Ohm’s Law for the inductor
• V = I XL
Inductor in a Circuit

Inductance can be interpreted as a
measure of opposition to the rate of
change in the current
• 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
• Therefore, the current doesn’t change from 0
to its maximum instantaneously
RL Circuit


When the current
reaches its maximum,
the rate of change and
the back emf are 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
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  t /  
R
Transformer
Iron Core
V2
i1
= - M21 (i1/t)
Mutual inductance
To ch. circuit
N1
N2
AC
The RLC Series Circuit


The resistor,
inductor, and
capacitor can be
combined in a
circuit
The current in the
circuit is the same
at any time and
varies sinusoidally
with time
Current and Voltage
Relationships in an RLC Circuit



The instantaneous
voltage across the
resistor is in phase
with the current
The instantaneous
voltage across the
inductor leads the
current by 90°
The instantaneous
voltage across the
capacitor lags the
current by 90°
VC
v = vc + vL + vR
VL
A
i= iosin(wt)
VR
i
vR
i= iosin(wt)
vo = Xio
Riosin(wt)
AC
i lags 90 deg behind v.
vL
vC
V lags 90 deg behind i.
wLiocos(wt)
- (1/wC)iocos(wt)
RLC Series Resonance Circuit
v = vc + v L + v R
= -(1/wC)io cos(wt) + wLio cos(wt) + Rio sin(wt)
= {(wL-1/wC)io} cos(wt) + Rio sin(wt)
= {(wL-1/wC)2 + R2}1/2(io) sin(wt+q)
XRLC
vo = Xio
Resonance: wL = 1/wC
XRLC = {(wL-1/wC)2 + R2}0.5
R
w = 1/(LC)0.5
io = vo/XRLC
w
RLC Parallel Resonance Circuit
vo
i= iosin(wt)
V
A
AC
w = 1/(LC)0.5
Power in an AC Circuit

No power losses are associated with
capacitors and pure inductors in an AC
circuit
• In a capacitor, during one-half of a cycle
energy is stored and during the other half the
energy is returned to the circuit
• In an inductor, the source does work against
the back emf of the inductor and energy is
stored in the inductor, but when the current
begins to decrease in the circuit, the energy is
returned to the circuit
Power in an AC Circuit, cont

The average power delivered by the
generator is converted to internal
energy in the resistor
• Pav = IVR = IV cos 
• cos  is called the power factor of the
circuit

Phase shifts can be used to maximize
power outputs
Resonance in an AC Circuit

Resonance occurs at
the frequency, ƒo,
where the current has
its maximum value
• To achieve maximum
current, the impedance
must have a minimum
value
• This occurs when XL =
XC
1
ƒo 
2p LC
Resonance, cont

Theoretically, if R = 0 the current would be
infinite at resonance
• Real circuits always have some resistance

Tuning a radio
• A varying capacitor changes the resonance
frequency of the tuning circuit in your radio to
match the station to be received

Metal Detector
• The portal is an inductor, and the frequency is set
to a condition with no metal present
• When metal is present, it changes the effective
inductance, which changes the current which is
detected and an alarm sounds
Simple Radio
http://www.tricountyi.net/~randerse/xtal.htm
Summary of Circuit Elements,
Impedance and Phase Angles
James Clerk Maxwell


Electricity and
magnetism were
originally thought to
be unrelated
in 1865, James Clerk
Maxwell provided a
mathematical theory
that showed a close
relationship between
all electric and
magnetic phenomena
Maxwell’s Starting Points




Electric field lines originate on positive
charges and terminate on negative
charges
Magnetic field lines always form closed
loops – they do not begin or end
anywhere
A varying magnetic field induces an emf
and hence an electric field (Faraday’s Law)
Magnetic fields are generated by moving
charges or currents (Ampère’s Law)
Maxwell’s Predictions




Maxwell used these starting points and a
corresponding mathematical framework to
prove that electric and magnetic fields play
symmetric roles in nature
He hypothesized that a changing electric field
would produce a magnetic field
Maxwell calculated the speed of light to be
3x108 m/s
He concluded that visible light and all other
electromagnetic waves consist of fluctuating
electric and magnetic fields, with each
varying field inducing the other
Hertz’s Confirmation of
Maxwell’s Predictions

Heinrich Hertz was
the first to
generate and
detect
electromagnetic
waves in a
laboratory setting
James Clerk Maxwell’s Equations
(1867)
E-field comes
out from p-charge and
Pre-Maxwell
terminates at negative charge.
 
E 
B-field cannot do like E-field.
o
No magnetic monopole!

B  0

Faraday’s law

B
 E  
t
Ampere’s law


  B  mo J
 
E 
o

B  0


B
 E  
t



E
  B  m o J  m o o
t
The velocity of transverse undulations in our hypothetical
medium, calculated from the electromagnetic experiments,
agrees so exactly with the velocity of light calculated from
the optical experiments, that we can scarcely avoid the
inference that light consists in the transverse undulation
of same medium which is the cause of electric and magnetic
Phenomena.
v
1
 o mo

1
(8.85 10 12 C 2 / N  m2)( 4p 10 7 N / A2 )
2.9986 x 108 m/s
Hertz’s Experiment
(1887)
Hertz’s Basic LC Circuit


When the switch is
closed, oscillations
occur in the current
and in the charge on
the capacitor
When the capacitor is
fully charged, the total
energy of the circuit is
stored in the electric
field of the capacitor
• At this time, the current
is zero and no energy is
stored in the inductor
LC Circuit, cont



As the capacitor discharges, the energy
stored in the electric field decreases
At the same time, the current increases and
the energy stored in the magnetic field
increases
When the capacitor is fully discharged, there
is no energy stored in its electric field
• The current is at a maximum and all the
energy is stored in the magnetic field in the
inductor


The process repeats in the opposite direction
There is a continuous transfer of energy
between the inductor and the capacitor
Hertz’s Experimental Apparatus



An induction coil is
connected to two
large spheres
forming a capacitor
Oscillations are
initiated by short
voltage pulses
The inductor and
capacitor form the
transmitter
Hertz’s Experiment

Several meters away from the
transmitter is the receiver
• This consisted of a single loop of wire
connected to two spheres
• It had its own inductance and
capacitance

When the resonance frequencies of
the transmitter and receiver
matched, energy transfer occurred
between them
Hertz’s Conclusions

Hertz hypothesized the energy
transfer was in the form of waves
• These are now known to be
electromagnetic waves

Hertz confirmed Maxwell’s theory by
showing the waves existed and had
all the properties of light waves
• They had different frequencies and
wavelengths