3. LC & LCR Series Circuit

Definitions

LC Series Circuit

• An inductor and capacitor are connected in Series

• No resistance

LCR Series Circuit

• An inductor, capacitor and resistor are connected in Series

Instananeous Charge $q$

• The charge on the capacitor at any given time during the oscillation.

Phase Constant $\phi$

• Sets where in the oscillation the system starts

 

Derivations

LC Series Circuit

If the capacitor is charged, then put in series with the inductor

$$\sum V = V_L + V_C = 0$$

subbing in $V_L$ and $V_C$

$$L\dfrac{di}{dt} + \dfrac{q}{C} = 0$$

since $i = \dfrac{dq}{dt}$

$$\dfrac{d^2q}{dt^2} + \dfrac{1}{LC}q = 0$$

this takes the form of SMH so

$$\omega = \dfrac{1}{\sqrt{LC}}, \quad T = 2\pi \sqrt{LC}$$

we can now form a wave equation for this

$$q = Qcos(wt + \phi)$$

where $Q$ is the maxium charge on the capacitor