AC Circuit Principles

The LRC circuit

A LRC circuit consist of an inductor, resistor, capacitor, and AC voltage source all connected in series, as shown below.

Here, we assume that the frequency generator outputs a voltage V = V_maxsin(ωt). Because the capacitor C and inductor L are reactive, a phase difference φ exists between the current and applied voltage. Thus the current in the circuit is given by I = I_maxsin(ωt - φ)

Reactance

Capacitors and inductors inhibit current flow via the capacitive and inductive reactance:

Both capacitive and inductive reactance has units of ohms, and just like conventional resistance, they determine the voltage drops across the corresponding circuit elements. Thus, the capacitor experiences a sinusoidal voltage drop with maximum amplitude V_Cmax = I_maxΧ_C, and the inductor has a sinusoidal voltage drop with amplitude V_Lmax = I_maxΧ_L.

Impedance

Capacitive and inductive reactance combine with conventional resistance R using the formula for impedance:

The impedance represents the combined resistance to current flow due to all reactances and resistances in the circuit, and thus I_max = V_max/Z.

Resonance

It is evident that the impedance Z is a minimum when Χ_L = Χ_C. In this case, the current in the circuit has its maximum amplitude. Solving for ω gives the resonant frequency

Phase

Although the voltages and current in an AC circuit are all sinusoidal, in general they are all out of phase, meaning that they all reach their maximum values at different times. The exception is I and V_R (voltage across the resistor). Because of Ohm's Law, these two quantities are always perfectly in phase.

Current lags V_L by π/2 rad, or 90 degrees.

Current leads V_C by π/2 rad, or 90 degrees.

Thus, V_L and V_C are out of phase by 180 degrees.

In general, the current is also out of phase with the applied voltage (i.e., voltage from the frequency generator). In this case, the phase angle between the current and applied voltage is

Example

By adjusting the frequency slider, you can observe the voltages and currents in a typical AC circuit.

In particular, notice that at resonance, the voltages across the inductor and capacitor can be quite large - much larger than the applied voltage. They are out of phase, so Kirchhoff's loop rule of course still applies at each instant of time.