Theorem: Undriven RLC oscillators

For an undriven RLC oscillator, the differential equation corresponding to Kirchoff's law is $$\ddot q+\frac{R}{L}\dot q+\frac{1}{LC}q=0$$ Let $\gamma=R/2L$ and $\omega_0^2=1/LC$. The general solution to this differential equation is $$q(t)=q_0e^{-\gamma t}\cos(t\sqrt{\omega_0^2-\gamma^2}+\varphi)$$ See Problem \ref{deriv} below for a derivation.

Theorem: Driven RLC oscillators

A driven oscillator contains a time-dependent sinusoidal voltage source $$\mathcal E(t)=\mathcal E_0\cos\omega t$$ We can substitute this voltage with a complex voltage $\mathcal E_0e^{i\omega t}$, and solve the system in terms of these complex values. The oscillation frequency of the current will match the oscillation frequency of the voltage, so we take $I(t)=\tilde Ie^{i\omega t}$ where $\tilde I$ is the fixed vector $\tilde I=I_0e^{i\varphi}$ in the complex plane. Plugging this into Kirchoff's law yields $$Li\omega\tilde Ie^{i\omega t}+R\tilde Ie^{i\omega t}+\frac{1}{i\omega C}\tilde Ie^{i\omega t}=\mathcal E_0e^{i\omega t}$$Which we can use to find $$I_0=\frac{\mathcal E_0}{\sqrt{R^2+(\omega L-1/\omega C)^2}}\qquad \tan\varphi=\frac{1}{R\omega C}-\frac{\omega L}{R}$$

Theorem: Impedance

We can substitute a capacitor $C$ and an inductor $L$ in a circuit as effective resistors with impedances $$Z_L=i\omega L\qquad Z_C=\frac{1}{i\omega C}$$

Idea: Phasors

A helpful way of analyzing circuits is using impedances and phasors on the complex plane. The general idea is as such:

1. Draw the phasor for the current $I$ at an angle of 0 degrees for ease of analysis.
2. Draw phasors representing the voltages across each of the circuit components by using $V=IZ$.
3. The sum of these phasors will yield the net voltage phasor across the components (if they are connected with series).
4. From this we can extrapolate the phase angle $\varphi$ and the amplitude of $I$ from the voltage amplitude.

Definition: RMS

Since the explicit average of current and voltage in an alternating circuit is 0, we use the root-mean square value to represent an average. The RMS is defined as $$V_{\text{rms}}=\sqrt{\overline{V^2}}$$

Theorem: Power

The average power dissipation in a circuit is given by the RMS voltage on the voltage source and the RMS current passing through it: $$P=\mathcal E_{\text{rms}}I_{\text{rms}}\cos\varphi$$Note $\varphi$ is the phase difference between $\mathcal E(t)$ and $I(t)$. A derivation is found in Problem \ref{powah}.