12.2 RLC Circuits: Damped Oscillations
12.2.1
Unforced Oscillations
We saw in the last section that a simple LC circuit will oscillate just like a spring with no friction.
What will happen now if we introduce a resistor to the circuit? By recognizing that a resistor acts
like a frictional force in circuits we can take our spring analogy further and predict that the circuit
will oscillate, but that the oscillations will eventually die out just like a spring.
We will ﬁrst think about this circuit with no driving voltage. That is, we will charge up the
capacitor and then let it go. As we did with the LC circuit, lets use Kirhoﬀ’s loop rule to write
down a diﬀerential equation for the circuit.
L
d
2
q
dt
2
+
R
dq
dt
=
1
C
q
= 0
Unlike the other diﬀerential equations which we have encountered, the solution to this diﬀer
ential equation is nontrivial though not impossible to discern.
q
=
Qe

Rt
2
L
cos(
ω
0
t
+
φ
) where
ω
0
=
s
ω
2

±
R
2
L
²
2
(44)
where
ω
=
1
√
LC
just like the undamped circuit. Notice that this solution has two components.
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 Spring '08
 Any
 Physics, Force, Friction, Oscillations, RLC, 2L

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