Lecture 20Lecture 20

Physics 2102

Jonathan Dowling

Ch. 31: Electrical Oscillations, LC Ch. 31: Electrical Oscillations, LC Circuits, Alternating CurrentCircuits, Alternating Current

What are we going to learn?What are we going to learn?A road mapA road map

• Electric charge Electric force on other electric charges Electric field, and electric potential

• Moving electric charges : current • Electronic circuit components: batteries, resistors, capacitors• Electric currents Magnetic field

Magnetic force on moving charges• Time-varying magnetic field Electric Field• More circuit components: inductors. • Electromagnetic waves light waves• Geometrical Optics (light rays). • Physical optics (light waves)

Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system.

All oscillators operate along the sameprinciple: they are systems that can storeenergy in more than one way and exchange it back and forth between thedifferent storage possibilities. For instance, in pendulums (and swings) one exchanges energy between kinetic and potential form.

Oscillators in PhysicsOscillators in Physics

In this course we have studied that coils and capacitors are devicesthat can store electromagnetic energyelectromagnetic energy. In one case it is stored in a magnetic field, in the other in an electric field.

potkintot EEE += 22

21

21

xkvmEtot +=

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛==

dtdxxk

dtdvvm

dtdEtot 2

21

221

0 dtdx

v =

0=+ xkdtdvm

)cos()( :Solution 00 φω += txtx

phase :

frequency :

amplitude :

0

0

φωx

mk=ω

A mechanical oscillatorA mechanical oscillator

02

2

=+ xkdtxdm

Newton’s law F=ma!

The magnetic field on the coil starts to collapse, which will start to recharge the capacitor.

Finally, we reach the same state we started with (withopposite polarity) and the cycle restarts.

An electromagnetic oscillatorAn electromagnetic oscillator

Capacitor discharges completely, yet current keeps going. Energy is all in the inductor.

Capacitor initially charged. Initially, current is zero, energy is all stored in the capacitor.

A current gets going, energy gets split between the capacitor and the inductor.

elecmagtot EEE +=2

2

21

21

Cq

iLEtot +=

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛==

dtdqq

CdtdiiL

dtdEtot 2

21

221

0

( )qCdt

diL

10 +⎟

⎠

⎞⎜⎝

⎛=

dtdq

i =

C

q

dt

qdL += 2

2

0 02

2

=+ xkdtxdmCompare with:

Analogy between electrical and mechanical oscillations:

MLvi

kCxq

→→→→

/1

LC1=ω

)cos( 00 φω += tqq

Electric Oscillators: the mathElectric Oscillators: the math

(the loop rule!)

)cos()( 00 φω += txtx

mk=ω

-1.5

-1

-0.5

0

0.5

1

1.5

Time

Charge

Current

)cos( 00 φω += tqq

)sin( 00 φωω +−== tqdtdq

i

)(sin21

21

0

22

0

22 φωω +== tqLLiEmag

0

0.2

0.4

0.6

0.8

1

1.2

Time

Energy in capacitor

Energy in coil

)(cos21

21

0

22

0

2

φω +== tqCC

qEele

LCxx

1 and ,1sincos

that,grememberin And

22 ==+ ω

2

021q

CEEE elemagtot =+=

The energy is constant and equal to what we started with.

Electric Oscillators: the mathElectric Oscillators: the math

Example 1 : tuning a radio receiverExample 1 : tuning a radio receiver

The inductor and capacitor in my car radio are usually set at L = 1 mH & C = 3.18 pF. Which is my favorite FM station?

(a) QLSU 91.1(b) WRKF 89.3 (c) Eagle 98.1 WDGL

FM radio stations: frequency is in MHz.

srad

srad

LC

/1061.5

/1018.3101

1

1

8

126

×=×××

=

=

−−

ω

MHz

Hz

f

3.89

1093.8

27

=×=

=πω

ExampleExample 22• In an LC circuit,

L = 40 mH; C = 4 F• At t = 0, the current is a

maximum;• When will the capacitor be

fully charged for the first time?

sradxLC

/1016

118−

==ω

• ω = 2500 rad/s• T = period of one complete cycle = 2πω = 2.5 ms• Capacitor will be charged after 1/4 cycle i.e at t = 0.6 ms

-1.5

-1

-0.5

0

0.5

1

1.5

Time

Charge

Current

Example 3Example 3

• In the circuit shown, the switch is in position “a” for a long time. It is then thrown to position “b.”

• Calculate the amplitude of the resulting oscillating current.

• Switch in position “a”: charge on cap = (1 F)(10 V) = 10 C• Switch in position “b”: maximum charge on C = q0 = 10 C• So, amplitude of oscillating current =

== )10()1)(1(

10 C

FmHq μ

μω 0.316 A

)sin( 00 φωω +−= tqi

ba

E=10 V1 mH 1 F

Example 4Example 4In an LC circuit, the maximum current is 1.0 A.

If L = 1mH, C = 10 F what is the maximum charge on the capacitor during a cycle of oscillation?

)cos( 00 φω += tqq

)sin( 00 φωω +−== tqdtdq

i

Maximum current is i0=ωq0 q0=i0/ω

Angular frequency ω=1/LC=(1mH 10 F)-1/2 = (10-8)-1/2 = 104 rad/s

Maximum charge is q0=i0/ω = 1A/104 rad/s = 10-4 C

Damped LC OscillatorDamped LC OscillatorIdeal LC circuit without resistance:

oscillations go on for ever; ω = (LC)-1/2

Real circuit has resistance, dissipates energy: oscillations die out, or are “damped”

Math is complicated! Important points:– Frequency of oscillator shifts away

from ω = (LC)-1/2

– Peak CHARGE decays with time constant = 2L/R

– For small damping, peak ENERGY decays with time constant = L/R 0 4 8 12 16 20

0.0

0.2

0.4

0.6

0.8

1.0

UE

time (s)

LRt

eC

QU

−=2

2

max

C

RL

SummarySummary• Capacitor and inductor combination

produces an electrical oscillator, natural frequency of oscillator is ω=1/LC

• Total energy in circuit is conserved: switches between capacitor (electric field) and inductor (magnetic field).

• If a resistor is included in the circuit, the total energy decays (is dissipated by R).