An LC circuit is composed of an inductor and a capacitor. A capacitor stores energy in an electric field between its plates, and an inductor stores energy in its magnetic field.
Suppose an inductor was connected to a charged capacitor in a circuit. First, the capacitor would drive a current through an inductor because of the voltage across the capacitor. The inductor builds up a magnetic field from the current flow. After the voltage comes to 0 for the capacitor, the built up magnetic field starts to induce a voltage in the coil since it tries to resist the change in current. The induced voltage causes a current to flow in the opposite direction, and the capacitor starts to recharge, with a voltage of opposite polarity. Over time, this will produce an oscillating effect, where the energy is traded between the capacitor and inductor. In an ideal circuit with no resistance, this means the oscillating energy would go on forever, however, in practical purposes there is always internal resistance which makes the oscillations die out.
A RLC circuit is composed of a resistor, inductor, and capacitor either in series or parallel. This note section will concern it self with the series RLC circuits. A RLC circuit is like an LC circuit in that it has that property of resonance. Though, the oscillation dies out quicker because of the resistor.
The formulas in this section for RLC circuits will be for series RLC circuits.
Maximum charge of Capacitor:
\[ Q_{max} = CV \]
Energy in Capacitor of LC circuit:
\[ U_{C} = \frac{1}{2C} Q_{max}^{2} cos^{2}( \omega t + \phi ) = \frac{q^{2}(t)}{2C} \]
Energy of an LC circuit at an arbitrary time:
\[ \frac{q^{2}(t)}{2C} + \frac{Li^{2}}{2} \]
Where:
q(t) is the capacitor charge as function of timei(t) is the current as function of timeL is the inductanceC is the capacitanceSince there is no energy disspiated in the ideal LC circuit (no resistance), the following holds true:
\[ U = \frac{1}{2} \frac{q^{2}}{C} + \frac{1}{2} Li^{2} = \frac{1}{2} \frac{q_{0}^{2}}{C} = \frac{1}{2} LI_{0}^{2} \]
In other words, the energy that both the capacitor and inductor store (as a sum) is equal to any of the individual components (inductor/capacitor) if they held maximum charge.
Current of an LC circuit:
\[ i = \pm \frac{1}{\sqrt{LC}} \sqrt{Q^{2} - q(t)^{2}} \]
This can be derived using Kirchoff’s voltage law.
Charge in capacitor of an LC circuit at time t:
\[ q(t) = Q cos( \omega t + \phi ) \]
Resonance frequency of LC / RLC circuit:
\[ f_{0} = \frac{\omega_{0}}{2 \pi} \]
\[ \omega_{0} = \frac{1}{\sqrt{LC}} \]
Notes:
Capacitive Reactance for a capacitor in an AC circuit:
\[ X_{c} = \frac{1}{2 \pi f C} \]
Inductive Reactance for a inductor in an AC circuit:
\[ X_{L} = 2 \pi f L \]
Impedance (Measure of opposition from a circuit when voltage applied):
\[ Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}} \]
RMS current (root mean square) current. This is the time-averaged voltage or current in an AC circuit.
\[ I_{rms} = \frac{V_{S}}{Z} \]
Where:
Voltage of a RLC circuit
\[ V_{S} = \sqrt{V_{R}^{2} + (V_{L} - V_{C})^{2}} \]
Where:
Power consumed by RLC circuit:
\[ P = I^{2} R \]
Notes:
Power consumed by RLC circuit (2):
\[ P = V_{rms} I_{rms} cos( \phi ) \]
Where: