A series R L C circuit has components with the following...

A series $R L C$ circuit has components with the following values: $L=20.0 \mathrm{mH}, C=100 \mathrm{nF}, R=20.0 \Omega,$ and $\Delta V_{\max }=$ $100 \mathrm{V},$ with $\Delta v=\Delta V_{\max } \sin \omega t$ . Find $(\mathrm{a})$ the resonant frequency of the circuit, (b) the amplitude of the current at the resonant frequency, (c) the $Q$ of the circuit, and (d) the amplitude of the voltage across the inductor at resonance.

Key Concepts

Quality Factor (Q Factor)

The quality factor, or Q factor, of a resonant circuit quantifies the sharpness of its resonance and relates the energy stored to the energy dissipated per cycle. In a series RLC circuit, Q can be calculated as Q = ??L/R or equivalently Q = 1/(R)?(L/C). A higher Q indicates a more selective and underdamped circuit, and it is directly related to the voltage magnification seen across the reactive components.

Voltage Amplification in Resonant Circuits

At resonance, the voltage across reactive components like the inductor can be significantly higher than the source voltage due to the voltage amplification effect. This phenomenon occurs because the reactive voltage (either across the inductor or capacitor) is the product of the maximum current and the reactance, and when multiplied by the Q factor, results in a pronounced peak in voltage, even though the overall circuit is driven by a fixed source voltage.

Resonance in RLC Circuits

Resonance in an RLC circuit occurs when the inductive reactance equals the capacitive reactance, resulting in a purely resistive circuit at a specific frequency. This condition maximizes the current flow for a given voltage source and is described by the resonant frequency formula ?? = 1/?(LC) (or f? = 1/(2??(LC))), a fundamental concept that underpins the design and analysis of frequency-selective networks.

Impedance and Current in AC Circuits

In AC circuits, components such as resistors, inductors, and capacitors contribute to a total impedance that determines the current flow according to Ohm’s law (I = V/Z). At resonance in a series RLC circuit, the inductive and capacitive reactances cancel each other, meaning the impedance is minimized and equals the resistance. This condition leads to a maximum current amplitude that is directly determined by the applied voltage and the circuit’s resistive element.

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