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Muhammad Nadeem
School of Electrical Engineering &Computer Sciences
Electromagnetic Oscillations
2
Muhammad Nadeem
School of Electrical Engineering &Computer Sciences
2mcE=
tBE = /
ma=
Physics
20th Century 21st Century
tiV
x =+ hh 22
o/= E
mvP=
PrL = h Px.
0= B
tEjB += /ooo
P
h=
h
G
k
o
o Rc
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Electromagnetic Oscillator
0=+ c
q
dt
diL
Consider a LC circuit with no resistance and zero emf applied. ByKirchhoff's voltage rule,
2qqd
L
C
2 LcdtThe solution of above 2nd order homogenous ODE with constant
coefficients will be
)cos( += tqq m
LC/1=
Where is the maximum charge on capacitor, is the phase
constant and is the angular frequency.
mq
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Electric energy stored in LC circuit is
)(cos22
1 22
2 +== tC
qq
CU mE
2
Electric energy is stored in the
electric field inside capacitor
Magnetic energy stored in LC circuit is
)(sin2
)(sin2
1
22
2
2
222
2
+=
+=
==
tC
q
tqL
dtqLLiU
m
m
B
m
k=
2
Magnetic energy is stored in the
magnetic field inside Inductor
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EK
U
Energy versus time at
)(cos2
2
2
tC
qU mE =
)(sin2
22
tC
qU mB =
C
qE m2
2
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Consider an oscillating LC circuit with L=12mH and C=1.7F.
(a) what value of charge is present on the capacitor when theenergy is shared equally between electric and magnetic field?
(b) At what time t will this condition occur? Assume capacitor is
fully charged at t=0
Since
(a)
C
qE m
2
2
= qUE
2
=
So
C
q
C
qEU mE
22
1
22
1 22
==
2
mqq=
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(b) If capacitor is fully charged at t=0 then
)cos(2
)cos(
tqq
tqq
mm
m
=
=
sLC
t
t
t
1104
4/
2)cos(
==
=
=
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Damped OscillatorA resistance R is always present in LC circuit. When we take thisresistance into account,
0=++ c
q
iRdt
di
L
02
=++ qdq
Rqd
L
C
R
ttThe solution of above 2nd order homogenous ODE with constant
coefficients will be
)cos(2/ += teqq LRtm
22
)2/( LR=
Where is the maximum charge on capacitor, is the phase
constant and is the angular frequency.
mq
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a: Underdamped Oscillations When the damping force is small compared with the maximum
restoring forcethat is, when R is small such that
LR/
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b: Critical Damped Oscillations When the damping force reaches the maximum restoring force
that is, when R is large enough such that
LR/ =2Motion is said to be critical damped
2/ LRt=
In this case the system, once
released from rest at some non
equilibrium position, returns to
equilibrium and then stays there.
The graph of charge versus time
for this case is the red curve in
Figure.
m
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c: Overdamped Oscillations If the damping force is greater than the restoring forcethat is, if,
when R is large such that
t
LR/ >2Motion is said to be overdamped
m
2
2
22
=
L
R
L
R
Again, the system does not oscillate but simply returns to its
equilibrium position. As the damping increases, the time it takes the
system to approach equilibrium also increases, as indicated by the
black curve in Figure.
Where
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Consider RLC circuit having L=12mH, C=1.6F and R=1.5.
After what time t will the amplitude of the charge oscillationsdrop to one half of its initial value?
e LRt
2
12/=
In RLC circuit, amplitude of charge oscillations will be half if
st
R
Lt
LRt
011.0
2ln2
2ln2/
=
=
=
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In any case in which resistance is present, whether thesystem is overdamped or underdamped, the energy of
.
ELECTROMAGNETIC energy dissipates into heat.
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It is possible to compensate for energy
loss in a damped system by applying an
external force that does positive workon the system.
At any instant, energy can be put into
Forced Oscillator
L
C
R
~
the system by an applied force thatacts in the direction of motion of the
oscillator.
The amplitude of motion remains constant if the energy input percycle exactly equals the energy lost as a result of damping. Any
motion of this type is calledforced oscillation.
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A common example of a forced oscillator is a damped oscillator
driven by an external force that varies periodically, such as
Where is the angular frequency of the applied periodic force
)cos(0 t =
=++LC
q
dt
dqR
dt
qd2
2
,
cycle equals the energy lost per cycle, a steady-state condition is
reached in which the oscillations proceed with constant amplitude. At
this time, when the system is in a steady state, the solution of above
Equation is
)cos( += tqq m ( )2
222
/
=
L
R
Lqm
o
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Because an external force is drivin it the motion of the forced
R=0; Undamped
Small R
Large R
oscillator is not damped. The external agent provides the necessaryenergy to overcome the losses due to the retarding force. Note that
the system oscillates at the angular frequency of the driving force.
For small damping, the amplitude becomes very large when the
frequency of the driving force is near the natural frequency of
oscillation. The dramatic increase in amplitude near the natural
frequency is called resonance, and for this reason is sometimes
called the resonance frequency of the system