lc oscillation

8/4/2019 LC Oscillation

1/20

1

Lecture 41 Electromagnetic oscillations andAlternating current/ Chapter # 31

LC oscillations

Damped oscillations in RLC circuits

Alternating current

Simple circuits Resonance and Power in Alternating

current circuits

Transformers


2/20

Review of Voltage DropsAcross Circuit Elements

Idt QVC C= =

Voltage determined by

integralof current andcapacitance

C

I(t)

2

2= =dI d Q

V L Ldt dt

Voltage determined byderivativeof current andinductance

L

I(t)


3/20

Whats Next?

Why and how do oscillations occurin circuits containing capacitors

and inductors? naturally occurring, not driven for now

stored energy capacitive inductive

Where are we going?

Oscillating circuits radio, TV, cell phone, ultrasound, clocks,

computers, GPS


4/20

LC Circuits

Consider the RC and LCseries circuits shown:

Suppose that the circuits areformed att=0 with thecapacitor charged to value Q.

There is a qualitative difference in the time development of thecurrents produced in these two cases. Why??

Consider from point of view of energy!

In the RC circuit, any current developed will cause energy tobe dissipated in the resistor.

In the LC circuit, there is NO mechanism for energydissipation; energy can be stored both in the capacitor andthe inductor!

LCC R++++

- - - -

++++

- - - -


5/20

5

LC Circuits - Qualitatively

We discussed RC and RL circuits. In an LC

circuit energy oscillates between the capacitor(E field) and inductor (B field)

22

22 iLU

C

qU BE ==


6/20

LC Oscillations

(qualitative)

LC

+ +

- -

0=I

0QQ +=

LC+ +

- -0=I

0QQ =

LC

0II =

0=Q

LC

0II +=

0=Q


7/20

7

Oscillations

RC/LC Ci i


8/20

RC/LC Circuits

RC:

current decays exponentially

C R

-It

0

0

I

Q+++

- - -

LC

LC:

current oscillates

I

0

0t

I

Q+++- - -

LC O ill ti


9/20

LC Oscillations(L with finiteR)

IfL has finiteR, then

energy will be dissipated inR.

the oscillations will become damped.

R = 0

Q

0

t t

0

Q

R 0


10/20

10

The Electrical - MechanicalAnalogy

One can make the analogy with mechanicaloscillations (Table 31-1)

qcorresponds to x 1/C corresponds to k icorresponds to v L corresponds to m

)(1)( circuitLCCL

springblockmk ==


11/20

11

LC Oscillations -

Quantitatively

Block-Spring Oscillator

td

vd

vtd

vd

td

xd

td

xd

td

d

td

vd

td

xdv

2

)( 2

2

2

=

=

=

=

22

2

1

2

1 xkvmUUU sb +=+=

td

dU

=0 td

xd

xktd

xd

td

xd

mtd

xd

xktd

vd

vm +=+= 2

2

)(02

2

nsoscillatiospringblockxk

td

xdm =+

m

kntdisplacemetXx =+= )()cos(

xtXtd

xd 222

2

)cos( =+=


12/20

12

LC Oscillations -

Quantitatively

The LC Oscillator

LCtQq

nsoscillatioLCqCtd

qdL

td

qd

C

q

td

qd

td

qdLtd

qd

C

q

td

idiLtd

Ud

C

qiLUUU EB

1(charge))cos(

)(01

0

22

2

2

2

2

22

=+=

=+

+=+==

+=+=

2

2

td

qd

td

qd

td

d

td

id

tdqdi

=

=

=


13/20

13

LC Oscillations -

Quantitatively

See also Fig 31-4

)cos( += tQq

1cossin 22 =+

CL

LC

11 2 ==

)sin()sin( +=+== tItQtd

qdi

)(sin2

1

2

)(cos22

2222

222

+==

+==

tQLiL

U

tC

Q

C

q

U

B

E

)(sin2

22

+= tC

QUB

C

QUU EB

2

2

=+

LC O ill ti


14/20

LC OscillationsEnergy Check

The other unknowns ( Q0, ) are found from the initialconditions. E.g., in our original example we assumed initialvalues for the charge (Qi) and current (0). For these values:

Q0 = Qi, = 0.

Question: Does this solution conserve energy?

)(cos21)(

21)( 0

22

0

2 +== tQ

CCtQtUE

)(sin2

1)(2

1)( 0

22

0

2

0

2

+== tQLtLitUB

Oscillation frequency has been found from theloop equation. LC

10 =

E Ch k


15/20

UE

t

0

Energy Check

UB

0t

Energy in Capacitor

)(cos2

1)( 0

22

0 += tQC

tUE

Energy in Inductor

)(sin

2

1)( 0

22

0

2

0 += tQLtUB

LC

10 =

)(sin21)( 0

22

0 += tQC

tUB

CQtUtU BE2

)()(2

0=+Therefore,

P bl


16/20

Problem At t=0 the current flowing through the circuit is

1/2 of its maximum value.

Which of the following plots bestrepresents UB, the energy stored in the

inductor as a function of time?

3A

LC+ +

- -

I

Q

Which of the following is a possible value for the phase , when thecharge on the capacitor is described by: Q(t) = Q0cos(t + )

3B

(a) (b) (c)

00

UB

time

00

UB

time

00

UB

time

(a) = 30 (b) = 45 (c) = 60

P bl ( t)

I


17/20

Problem (cont) At t=0 the current flowing through the circuit is

1/2 of its maximum value.Which of the following plots best represents UB, the

energy stored in the inductor as a function of time?3A

(a) (b) (c)

00

UB

time

00

UB

time

00

UB

time

The key here is to realize that the energy stored in the inductor isproportional to the CURRENT SQUARED.

Therefore, if the current att=0 is 1/2 its maximum value, the energystored in the inductor will be 1/4 of its maximum value!!

LC

+ +

- -

I

Q

P bl ( t)


18/20

Problem (cont) At t=0 the current flowing through the circuit is

1/2 of its maximum value. Which of the following is a possible value for the

phase , when the charge on the capacitor isdescribed by: Q(t) = Q0cos(t + )(a) = 30 (b) = 45 (c) = 60

3B

We are given a form for the charge on the capacitor as a function oftime, but we need to know the current as a function of time.

)sin()( 000 tQdt

dQtI +==

At t = 0, the current is given by: QI sin)0( 00=1 1

max 0 02 2( )I Q= =

Therefore, the phase angle must be given by:21sin = 30=

LC+ +

- -

I

Q


19/20

19

Checkpoint 31-2

A capacitor in an LC oscillator has a maximum potential difference of17 V and a maximum energy of 160 J. When the capacitor has apotential difference of 5 V and energy of 10 J, what are (a) the emf

across the inductor and (b) the energy stored in the magnetic field?

C

qVC =

( )max2

2EEB UC

QUU ==+

td

idLa L =E)(

LE

CV

)sin(

)cos(

+=

+=

tQi

tQq

VVCL 5==E

qLtQLL22 ))cos(( =+=E

CL

12 =

CVCq ==

( ) JJJUUUb EEB 15010160)( max ===

S l P bl )sin( += tQi


20/20

20

Sample Problem

A 1.5 F capacitor is charged to 57 V. The charging battery is thendisconnected, and a 12 mHcoil is connected in series with the

capacitor so that LC oscillations occur. (a) Assuming that the circuit

contains no resistance, what is the maximum current in the coil?

)sin(

)sin(

+=

+=

tI

tQi

QIa =)(

CVCQ =

CVC= CL1

=CC V

L

CVC

CL== 1

AVHFV

LCI C 637.0)57(

1012105.1 3

6

===

(b) What is the maximum rate (di/dt)max at which the current i changes?

)cos())sin(( +=+= tItd

tIdtdid

sA

FH

A

CL

II

td

id/4750

)105.1)(1012(

637.01

63max

=

===

lc oscillation

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