Module
M
d l 24:
24
Undriven RLC Circuits
1
Module 24: Outline
Undriven RLC Circuits
p 8
8: Part
a 2:Undriven
U d e RLC
C
Expt.
Circuits
2
Circuits that Oscillate (LRC)
3
Mass on a Spring:
p g
Simple Harmonic Motion
(Demonstration)
4
Mass on a Spring
g
(1)
(2)
What is Motion?
2
((3))
d x
F = −kx = ma = m 2
dt
2
d x
m 2 + kx = 0
dt
((4))
Simple Harmonic Motion
x(t) = x0 cos(ω 0t + φ )
x0: Amplitude of Motion
ω0 =
φ: Phase
Ph
(time
(ti
offset)
ff t)
k
= Angular
g
frequency
q
y
m
5
Mass on a Spring:
g Energy
gy
(1) Spring
(2) Mass
x(t)
( ) = x0 cos((ω 0t + φ )
(3) Spring
(4) Mass
dx
= vx ((t)) = −ω 0 x0 sin((ω 0t + φ )
d
dt
Energy has 2 parts: (Mass) Kinetic and (Spring) Potential
2
1 ⎛ dx ⎞
1 2 2
K = m ⎜ ⎟ = kx0 sin (ω 0t + φ )
2 ⎝ dt ⎠
2
1 2 1 2 2
U s = kx = kx0 cos (ω 0t + )
2
2
Energy
sloshes back
and forth
6
Simple
p Harmonic Motion
1
Period =
frequency
Amplitude (x0)
1
→T =
f
2π
2π
Period =
→T =
angular
l frequency
f
ω
ω 0t + φφ )
x(t)
(t) = x0 cos((ω
−π
Phase Shift (φ ) =
2
7
Electronic
El
t i A
Analog:
l
LC Circuits
8
Analog:
g LC Circuit
Mass doesn
doesn’tt like to accelerate
Kinetic energy associated with motion
2
dv
d
d x
1 2
F = ma = m = m 2 ; E = mv
dt
2
dt
Inductor doesn’t like to have current change
Energ associated
Energy
associated with
ith current
c rrent
2
dI
d q
1 2
ε = − L = − L 2 ; E = LI
dt
2
dt
9
Analog:
g LC Circuit
Spring doesn
doesn’tt like to be compressed/extended
Potential energy associated with compression
1 2
F = −kx; E = kx
2
Capacitor doesn’t like to be charged (+ or -)
Energy associated with stored charge
1
11 2
ε = q; E =
q
C
2C
F → ε ; x → q; v → I; m → L; k → C
−1
10
LC Circuit
1. Set up the circuit above with capacitor, inductor,
resistor, and battery.
2 Let the capacitor become fully charged
2.
charged.
3. Throw the switch from a to b
4. What happens?
11
LC Circuit
It undergoes simple harmonic motion, just like a
mass on a spring,
spring with trade
trade-off
off between charge on
capacitor (Spring) and current in inductor (Mass)
12
Concept Question
Questions:
LC Ci
Circuit
it
13
Concept
p Question: LC Circuit
Consider the LC circuit at
right At the time shown the
right.
current has its maximum
value.
l
At thi
this ti
time
1.
The charge
g on the capacitor
p
has its maximum value
2.
The magnetic field is zero
3.
The electric field has its maximum value
4.
The charge on the capacitor is zero
5.
Don’tt have a clue
Don
14
Concept
p Question: LC Circuit
In the LC circuit at right the
currentt is
i iin th
the di
direction
ti
shown and the charges on
the capacitor have the signs
shown. At this time,
1.
2
2.
3.
4
4.
5.
I is increasing and Q is increasing
I is increasing and Q is decreasing
I is decreasing and Q is increasing
I is decreasing and Q is decreasing
Don’t have a clue
15
LC Circuit
Q
dI
dQ
−L =0 ; I =−
C
dt
dt
2
dQ 1
+
Q=0
2
LC
dt
Simple
p Harmonic Motion
Q(t)) = Q0 cos(( ω 0t + φφ )
Q(
Q0: Amplitude of Charge Oscillation
φ: Phase (time offset)
ω
0
=
1
LC
16
LC Oscillations: Energy
gy
Notice relative phases
⎛ Q02 ⎞
Q
2
UE =
=⎜
ω 0t
cos
⎟
2C ⎝ 2C ⎠
2
⎛ Q02 ⎞ 2
1 2 1 2 2
U B = LI = LI 0 sin ω 0t = ⎜
⎟ sin ω 0t
2
2
⎝ 2C ⎠
2
Q 2 1 2 Q0
U = UE +UB =
+ LI =
2C 2
2C
Total energy is conserved !!
17
Summary:
y The Ideal LC Circuit
Adding D
Addi
Damping:
i
RLC Circuits
19
The Real RLC Circuit: Energy
C
Considerations
id
ti
Include finite resistance:
Q
dI
+ I R+ L =0
C
dt
Multiply by I and after a little work:
2
⎡
d Q
1 2⎤
2
+ L I ⎥= - I R
⎢
dt ⎣ 2C 2
⎦
d
2
⎡⎣ Total
ota Energy
e gy ⎤⎦ = - I R
dt
Damped LC Oscillations
Resistor dissipates
energy and system
rings down over time
Also, frequency decreases:
⎛ R⎞
ω'= ω −⎜ ⎟
⎝ 2L ⎠
2
2
0
21
E
i
t8:
8
Experiment
Part 2 Undriven RLC Circuits
22
Concept
p Question: Expt.
p 8
In today’s lab the battery turns on
and off.
off Which circuit diagram is
most representative of our circuit?
1
1.
2
2.
1.
2.
3.
4.
1
2
3
4
3
3.
4
4.
Load lab while waiting…
Concept Question
Questions:
U di
Undriven
Circuits
Ci
it
24
Concept
p Question: LC Circuit
Current
Charge
Tlag
1.0I0
0.5Q0
0.5I0
0.0Q0
0.0I0
-0.5Q0
-0.5I0
-1.0Q0
0
1. It will increase
2 It will decrease
2.
3. It will stay the same
4 I don
4.
don’tt know
40
80
Time (mS)
-1.0I0
120
Current throu
C
ugh Capacito
or
1.0Q0
Charge on Capacitor
The plot shows the charge
on a capacitor (black curve)
and the current through it
(red curve) after you turn
off the power supply. If you
put a core into the inductor
what will happen to the
time TLag?
Concept
p Question: LC Circuit
Current
Charge
Tlag
1.0I0
0.5Q0
0.5I0
0.0Q0
0.0I0
-0.5Q0
-0.5I0
-1.0Q0
0
40
80
Current throu
C
ugh Capacito
or
Charge on Capacitor
If you increase the
resistance in the circuit
what will happen to rate
of decay of the pictured
amplitudes?
p
1.0Q0
-1.0I0
120
Time (mS)
1. It will increase (decay more rapidly)
1
2. It will decrease (decay less rapidly)
3 It will stay the same
3.
4. I don’t know
26
MIT OpenCourseWare
http://ocw.mit.edu
8.02SC Physics II: Electricity and Magnetism
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.