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# Resonance

Because both inductive and capacitive reactance are frequency-dependent, so is the impedance *Z* of an oscillating circuit. The current *I* in the circuit is at a maximum when the impedance is at a minimum. This happens when the inductive and capacitive reactances are equal, i.e. *X*_{L} = *X*_{C}. The second added component under the root sign becomes zero, leaving the impedance as:

In this case, the two reactances cancel each other out precisely and the series oscillating circuit acts purely as an active resistance. This condition is termed *resonance*, hence resonant circuit. The corresponding *resonant frequency f*_{r} is determined as follows:

The vector diagram below represents the condition of resonance. The voltages across the capacitance and inductance in this instance cancel each other out exactly, so that even a small supply voltage *U* = *U*_{R} = *I*·*R* is sufficient to achieve the high values for *U*_{C} and *U*_{L}. As indicated in the vector diagram, the current *I* at resonance is in phase with the supply voltage *U*.

A series resonant circuit's resistance reaches a minimum at resonance. In this case, the coil and capacitor both experience a voltage peak (voltage resonance). |

The ratio between either of the voltage components and the total voltage at resonance is termed the resonant circuit's *quality Q*:

At resonance, each of the two voltage components here amounts to *Q* times the applied voltage. Because the voltage components in a series circuit are proportional to the associated resistances, the quality is also determined as follows:

Substituting the formula for the resonant frequency results in the following:

This equation permits the quality to be determined directly from component parameters. Series resonant circuits are used, for instance, to suppress the resonant component of a mixture of frequencies. According to the equation above, an oscillating circuit's quality increases as the ohmic (loss) resistance *R* decreases.