Series LRC AC Circuits

The most general form of an LRC series AC circuit is one that contains an inductor, a resistor, a capacitor, and an AC oscillator all in series with each other. The order of the components is unimportant.

The oscillator could produce either:
(A) A sinusoidal voltage driven oscillating circuit with voltage or emf represented by,

Most labatory oscillators are voltage driven.

(B) Or a sinusoidal current driven oscillating circuit with the current represented by,

We will focus on current driven oscillator circuits because they are simpler to understand.

In general, the current and the voltage are not in phase with each other. For example, they do not reach their maximum values at the same moment except under special conditions such as the circuit is in resonance.
The time difference (measured in angular terms of the period of oscillation) between when the current peaks and when the voltage peaks is expressed by the phase angle f..

If the circuit is current driven then,

Whereas, if the circuit is voltage driven then,

We will be using the current driven circuit as our frame of reference from which to view the oscillations. If the value of the phase angle is positive then from the current's point of view the voltage peaked ahead by time f /w or a phase angle of f. (From the voltage's point of view the current is lagging behind by f.) The equation for the impedance and phase angle use this frame,

At any one moment the circuit still obeys Kirchhoff's rules. This means the voltage of the oscillator at any given moment is equal to the voltage resistor plus the voltage of inductor plus the voltage of the capacitor as it is in DC series circuit. Also, the current is the same through any component at any moment as it is in a DC series circuit.

However,

because the voltages do not peak at the same moment. Moreover, the maximum voltage across the inductor or the capacitor are often greater than the maximum voltage output of the oscillator. Note that the inductor and the capacitor are 180^o out of phase with each other. This means that when the voltage is peaking in the inductor, the voltage across the capacitor is also peaking but peaking negatively. At resonance the two voltages are the same so that they cancel each other out and the maximum voltage of the oscillator is the same as the maximum voltage of the resistor.

The voltage and current across each component in the circuit are not necessarily in phase with each other. This depends upon the impedance of that component or combination of components.

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