Transcript LC Oscillator

LC Oscillators
PH 203
Professor Lee Carkner
Lecture 22
LC Circuit

The capacitor discharges as a current through the
inductor

This plate then discharges backwards through the
inductor
This process will cycle over and over
Like a mass on a swing
LC Oscillations Figure
Circuit Properties
Energy

Sum must be constant
Charge

Electrons switch plates
Current
Current in the circuit will vary
sinusoidally from max one way to
zero to max the other way
Oscillation Frequency

Like all sinusoidal patterns, we can define a
angular frequency
w = 1/(LC)½

There are 2p radians in a complete cycle

The value of w tells us how rapidly the properties
of the circuit cycle
Current and Charge

Similarly, q is the charge at a given time and Q is
the maximum charge

q = Q cos (wt + f)
i = -I sin (wt +f)
Where f is the phase constant

Note that I and Q are related
I = wQ
Energy

UE = q2/2C

UB = Li2/2
We can substitute our expressions for i
and q
UE = Q2/2C cos2 (wt+f)
UB = Q2/2C sin2 (wt+f)
Energy Variations
Unlike q and i, U is always positive

Both energies have the same maximum = Q2/2C

The total amount of energy in the system

When one is a maximum the other is zero
Simple Harmonic Motion

Velocity and position vary sinusoidally

Parameterized by an angular frequency that
depends on two key properties (spring
constant and mass)
Damping

 It will go on forever with total energy
never changing

 Energy, current and charge
decrease with time
 Just like a damped mechanical
oscillator
Damping Factors

Frequency
The frequency of a damped oscillator is less than that
on an undamped one
w’ = (w2 – (R/2L)2)½

The amplitudes are lower by an exponential factor
e(-Rt/L)
Note that the higher the resistance the more
damping

Next Time
Read 31.6-31.8
Problems: Ch 31, P: 13, 17, 18, 28, 29
A switch is closed, starting a clockwise
current in a circuit. What direction is
the magnetic field through the middle
of the loop? What direction is the
current induced by this magnetic field?
A)
B)
C)
D)
E)
Up, clockwise
Down, clockwise
Up, counterclockwise
Down, counterclockwise
No magnetic field is produced
The switch is now opened, stopping the
clockwise current flow. Is there a selfinduced current in the loop now?
A) No, since the magnetic field goes to zero
B) No, self induction only works with constant
currents
C) Yes, the decreasing B field produces a
clockwise current
D) Yes, the decreasing B field produces a
counterclockwise current
E) Yes, it runs first clockwise then
counterclockwise
Consider an inductor connected in series
to a battery and a resistor. If the value
of the resistor is doubled what
happens to the maximum current and
the time it takes to reach the maximum
current?
A)
B)
C)
D)
E)
Both increase
Both decrease
Max current increases, time decreases
Max current decreases, time increases
Neither will change