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( ω0 t + φ ) where ω0 = 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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