Physics 2102 Jonathan Dowling

Lecture 20 Ch. 31: Electrical Oscillations, LC Circuits, Alternating Current

What are we going to learn?

A road map

• • • • • • • • •

Electric

charge



Electric

force



Electric

field

on other electric charges , and electric

potential

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



Magnetic force



Magnetic field 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 in Physics

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

All oscillators operate along the same principle: they are systems that can store energy in more than one way and exchange it back and forth between the different storage possibilities. For instance, in pendulums (and swings) one

exchanges energy

between

kinetic and potential

form. In this course we have studied that

coils and capacitors

are devices that can store

electromagnetic energy

. In one case it is stored in a magnetic field, in the other in an electric field.

A mechanical oscillator

E tot



E kin



E pot E tot

 1 2

m v

2  1 2

k x

2

dE tot dt

 0  1 2

m

2

v dv dt

 1 2

k

2

x dx dt v



dx dt m dv dt



k x

 0 2

m d dt

2

x



k x

 0 Solution :

x

(

t

) 

x

0 cos( 

t

  0 ) Newton’s law F=ma!

 

k m x

0 : amplitude  : frequency  0 : phase

An electromagnetic oscillator

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.

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

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 (with opposite polarity) and the cycle restarts.

Electric Oscillators: the math

2 1 1

q E tot



E mag



E elec E tot



L i

2  2 2

C dE tot dt

0   0  

L



di dt

1 2

L

 2

i di dt

 1

C

0 

L d

2

q dt

2 

q C

 1 2

C

2

q dq dt

(the loop rule!) Compare with:

i



dq dt

2

m d dt

2

x



k x

 0 Analogy between electrical and mechanical oscillations:

q



x

1 /

C



k x

(

t

) 

i



v L



M x

0 cos( 

t

  0 )  

k m q



q

0 cos( 

t

  0 )   1

LC

1.5

1 0.5

0 -0.5

Time -1 -1.5

Electric Oscillators: the math

q



q

0 cos( 

t

  0 )

dq i

   

q

0 sin( 

t

  0

dt

Charge ) Current

E mag E ele

 1  2 1

Li

2

q C

2 2   1 2 1

L



q

0 2

C

2 2

q

0 2 sin cos 2 2 ( (  

t t

  0   0 ) ) 1.2

1 0.8

0.6

0.4

0.2

0 Time Energy in capacitor Energy in coil And rememberin g that, cos 2

x

 sin 2

x

 1 , and  

E tot



E mag



E ele

 1 2

C q

0 2 1

LC

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

Example 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.

  1

LC

 1 1  10  6  3 .

18  10  12  5 .

61  10 8

rad

/

s



f

  2  8 .

93  10 7

Hz

 89 .

3

MHz rad

/

s

Example 2

• In an LC circuit, L = 40 mH; C = 4 m F • At t = 0, the current is a maximum; • When will the capacitor be fully charged for the first time?

  1

LC

 1 16

x

10  8

rad

/

s

•  = 2500 rad/s • T = period of one complete cycle = 2/ = 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

Time -1 -1.5

Charge Current

Example 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.

1 mH

b

1 m F

a

E =10 V

i

  

q

0 sin( 

t

  0 ) • Switch in position “a”: charge on cap = (1 m F)(10 V) = 10 m C • Switch in position “b”: maximum charge on C =

q

0 • So, amplitude of oscillating current = = 10 m C 

q

0  1 ( 1

mH

)( 1 m

F

) ( 10 m

C

)  0.316 A

Example 4

In an LC circuit, the maximum current is 1.0 A. If L = 1mH, C = 10 m F what is the maximum charge on the capacitor during a cycle of oscillation?

i



dq q dt



q

0 cos( 

t

  0 )   

q

0 sin( 

t

  0 ) Maximum current is i 0 =  q 0  q 0 =i 0 /  Angular frequency  =1/  LC=(1mH 10 m F) -1/2 = (10 -8 ) -1/2 = 10 4 rad/s Maximum charge is q 0 =i 0 /  = 1A/10 4 rad/s = 10 -4 C

Damped LC Oscillator

Ideal LC circuit without resistance: oscillations go on for ever; 

= (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  = (LC) -1/2 – Peak CHARGE decays with time constant = 2L/R – For small damping, peak ENERGY decays with time constant = L/R

1.0

0.8

0.6

0.4

0.2

0.0

0

C R

4

L

U

max 

Q

2 2

C e



Rt L

8 12

time (s)

16 20

Summary

• Capacitor and inductor combination produces an electrical oscillator, natural frequency of oscillator is  =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).