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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. XL = XC. The second added component under the root sign becomes zero, leaving the impedance as:

Electric Resonant Circuits !

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 fr is determined as follows:

Electric Resonant Circuits !

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 = UR = I·R is sufficient to achieve the high values for UC and UL. As indicated in the vector diagram, the current I at resonance is in phase with the supply voltage U

Electric Resonant Circuits !

 

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

Electric Resonant Circuits !

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:

Electric Resonant Circuits !

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

Electric Resonant Circuits !

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. 

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