VC
UE
0
C
L
0
t
VL
LC
Circuits
0
1
t
UB
0
t
Lecture Outline
• A little review
• Oscillating voltage and current
• Qualitative descriptions:
• LC circuits (ideal inductor)
• LC circuits (L with finite R)
• Quantitative descriptions:
• LC circuits (ideal inductor)
• Frequency of oscillations
• Energy conservation?
Text Reference: Chapter 33: 1 - 5
Review of voltage drops
across circuit elements
I(t)
q
Idt
∫
V =
=
C
C
C
Voltage determined by
integral of current and
capacitance
I(t)
L
dI
d 2q
V =L
=L 2
dt
dt
Voltage determined by
derivative of current and
inductance
Oscillating current and voltage
Q. What does
A.
εosinωt
∼
mean??
It is an “a.c.” voltage source. Output voltage
appears at the terminals and is sinusoidal in time
with an angular frequency ω .
Oscillating circuits have both
ac voltage and current.
I(t)
Simple for resistors, but ….
εosinωt
∼
R
⇒
I (t) =
εo
sin (ωt )
R
What’s next
• Why and how do oscillations occur
in circuits containing capacitors and
inductors
• naturally occurring, not driven
• stored energy
• capacitive <-> inductive
First, a bit of an energy review...
Energy in the Electric Field
Work needed to add charge to capacitor...
+++ +++
q
--- --dW = Vdq =  dq
C 
1Q
1 Q2 1
W = ∫ qdq =
= CV 2 … total work ...
C0
2 C 2
u=
C = εo
A
d
W
1
= ε0 E 2
volume 2
Volume=Ad
… energy density
Energy in the Magnetic Field
“Power” accounting in a LR circuit...
dI
Loop rule x I …
dt
dU
dI
PL =
= LI
Rate of energy flow into L
dt
dt
åI = I 2 R + LI
U
I
U = ∫ dU = ∫ LIdI
B
0
(
Total energy flow
0
)
1 2 1
1
1
1 B2
2
2 2
U = LI = lAµ o n I = volume
(µ o nI ) =
lA
2
2
2
µo
2 µo
… energy stored
Energy Density:
u electric
1
= ε0 E 2
2
u magnetic
1 B2
=
2 µ0
LC Circuits
• Consider the LC and RC
series circuits shown:
++++
----
C R
++++
----
C
L
• Suppose that the circuits are
formed at
t=0 with the capacitor C charged to a value Q. Claim is that
there is a qualitative difference in the time development of the
currents produced in these two cases. Why??
• Consider from point of view of energy!
• In the RC circuit, any current developed will cause energy
to be dissipated in the resistor.
• In the LC circuit, there is NO mechanism for energy
dissipation; energy can be stored both in the capacitor and
the inductor!
RC/LC Circuits
i
Q+++
---
i
Q+++
---
C R
0
i
0
1
0
t
L
LC:
current oscillates
RC:
current decays exponentially
-i
C
0
t
LC Oscillations
(qualitative)
i=0
+ +
- -
C
i = −i0
L
⇒
C
Q = +Q0
Q=0
⇑
⇓
i=0
i = + i0
C
Q=0
L
L
⇐
-
-
+ +
C
L
Q = −Q0
LC Oscillations
(qualitative)
Q
0
VC
0
i
0
1
VL
These voltages are 0
opposite, since the
cap and ind are
traversed in “opposite”
1
directions
di
__
dt
0
1
t
t
How do these change if L
has a finite R?
Lecture 33, ACT 1
t=0
• At t=0, the capacitor in the LC
circuit shown has a total charge
Q0. At t = t1, the capacitor is
+ +
uncharged.
Q = Q0
L
- – What is the value of Vab, the
C
1A voltage across the inductor at
time t1?
(a) Vab < 0
(b) Vab = 0
t=t1
a
Q =0
C
L
b
(c) Vab > 0
1B • What is the relation between UL1, the energy stored in the
inductor at t=t1, and UC1 , the energy stored in the capacitor at
t=t1?
(a) UL1 < UC1
(b) UL1 = UC1
(c) UL1 > UC1
LC Oscillations
(L with finite R)
• If L has finite Resistance, then
– energy will be dissipated in R and
– the oscillations will become damped.
Q
Q
0
0
t
R=0
t
R≠ 0
LC Oscillations
(quantitative)
• What do we need to do to turn our
qualitative knowledge into quantitative
knowledge?
• What is the frequency ω of the
oscillations (when R=0)?
– (it gets more complicated when R
finite…and R is always finite)
+ +
- -
C
L
LC Oscillations
(quantitative)
i
• Begin with the loop rule:
Q
d 2Q
Q
L 2 + =0
C
dt
+ +
- -
C
L
• Guess solution: (just harmonic oscillator!)
Q = Q 0 cos(ω 0 t + φ)
where:
remember:
− kx = m
• ω0 determined from equation
d2x
dt 2
• φ, Q0 determined from initial conditions
• Procedure: differentiate above form for Q and substitute into
loop equation to find ω0.
LC Oscillations
(quantitative)
• General solution:
+ +
- -
Q = Q 0 cos(ω 0 t + φ)
• Differentiate:
dQ
= − ω 0 Q 0 sin( ω 0 t + φ )
dt
d2Q
dt 2
C
∴
L
d 2Q
Q
+
=0
2
C
dt
= − ω 20 Q 0 cos( ω 0 t + φ )
• Substitute into loop eqn:
1
L( − ω 20 Q 0 cos( ω 0 t + φ ) ) + (Q 0 cos( ω 0 t + φ )) = 0
ω0 =
L
C
1
LC
⇒
− ω 20 L +
1
=0
C
which we could have determined
from the mass on a spring result:
ω0 =
k
1/ C
1
=
=
m
L
LC
2
Lecture 33, ACT 2
t=0
• At t=0 the capacitor has charge Q0; the resulting
oscillations have frequency ω 0. The maximum
current in the circuit during these oscillations has
value I0 .
2A – What is the relation between ω0 and ω2 , the
frequency of oscillations when the initial charge
= 2Q0 ?
(a) ω 2 = 1/2 ω 0
(b) ω 2 = ω 0
+ +
Q = Q0
- -
C
L
(c) ω 2 = 2 ω 0
2B • What is the relation between I0 and I2 , the maximum current in
the circuit when the initial charge = 2Q0 ?
(a) I2 = I0
(b) I2 = 2 I0
(c) I2 = 4 I0
LC Oscillations
Energy Check
1
• Oscillation frequency ω 0 =
has been found from the
LC
loop eqn.
• The other unknowns ( Q0, φ ) are found from the initial
conditions. eg in our original example we took as given,
initial values for the charge (Qi) and current (0). For these
values: Q0 = Qi , φ = 0 .
• Question: Does this solution conserve energy?
1 Q2 (t)
1 2
UE (t) =
=
Q 0 cos2 (ω 0 t + φ )
2 C
2C
U B ( t) =
1 2
1
Li ( t ) = Lω 20 Q 20 sin2 (ω 0 t + φ )
2
2
Energy Check
Energy in Capacitor
UE
1 2
Q 0 cos2 ( ω 0 t + φ )
2C
UE ( t) =
Energy in Inductor
U B ( t) =
1
L ω 20 Q 20 sin 2 ( ω 0 t + φ )
2
U B ( t) =
Therefore,
1
LC
⇒
ω0 =
1 2
Q 0 sin 2 ( ω 0 t + φ )
2C
Q 20
U E ( t) + U B ( t ) =
2C
0
t
3
UB
0
t
Lecture 33, ACT 3
i
• 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 φ , when the charge on the capacitor is
ω t + φ ).
2A described by: Q(t) = Q 0cos(ω
(b) φ = 45°°
(a) φ = 30°°
Q
+ +
- -
C
(c) φ = 60°°
2B • Which of the following plots best represents UB, the energy
stored in the inductor as a function of time ?
(b)
(a)
UB
(c)
UB
0
UB
0
0
time
0
0
2
time
4
6
0
0
time
L
Appendix: LCR Damping
For your interest, I do not derive here, but only illustrate the
following behavior
R = R0
R
Q
+ +
- -
C
L
Q = Q 0 e − β t cos( ω 'o t + φ )
β=
R
2L
 1
R2 
ω'o = 
− 2
 LC 4 L 
0
t
In a LRC circuit, ω depends also on R !
Q
R=
R0
4
0
t