electromagnetic oscillation

8/10/2019 Electromagnetic Oscillation

1/16

Muhammad Nadeem

School of Electrical Engineering &Computer Sciences

Electromagnetic Oscillations

2

Muhammad Nadeem

School of Electrical Engineering &Computer Sciences

[email protected]

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


2/16

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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3/16

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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4/16


5/16

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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6/16

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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7/16

(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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8/16

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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9/16

a: Underdamped Oscillations When the damping force is small compared with the maximum

restoring forcethat is, when R is small such that

LR/


10/16

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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11/16

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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12/16

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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13/16

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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14/16

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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15/16

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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16/16

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

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electromagnetic oscillation

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