An R L C circuit includes a 1.7-H inductor and a 270-μF capacitor rated at 350 V. The circuit i

step 1 since the resistance is present in the circuit present in the circuit the circuit acts as circui

An R L C circuit includes a 1.7-H inductor and a 270-μF...

An $R L C$ circuit includes a $1.7-\mathrm{H}$ inductor and a $270-\mu \mathrm{F}$ capacitor rated at $350 \mathrm{~V}$. The circuit is connected across a sine-wave generator with $V_{\mathrm{p}}=29 \mathrm{~V}$. What minimum resistance will ensure that the capacitor voltage does not exceed its rated value when the circuit is at resonance?

Key Concepts

Resonance in RLC Circuits

Resonance occurs in RLC circuits when the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase, effectively canceling each other out. This results in the circuit having a purely resistive impedance at the resonant frequency, with the source voltage directly driving the resistive element, and typically produces a large current as well as significant voltage drops across the reactive components.

Quality Factor (Q Factor)

The quality factor in an RLC circuit is a dimensionless parameter that indicates the selectivity or sharpness of the resonance. It quantifies the ratio of the energy stored to the energy dissipated per cycle and directly influences the voltage magnification across reactive components during resonance. A high Q implies that even a modest source voltage can produce a substantially higher voltage across the inductor or capacitor.

Voltage Magnification in Resonant Circuits

In a resonant RLC circuit, because the inductive and capacitive reactances cancel, the current reaches its maximum value, and the voltages across the individual reactive elements can become much larger than the applied source voltage. This voltage magnification, determined by the Q factor, is critical when considering component ratings; components like capacitors must be selected or protected to ensure that their voltage ratings are not exceeded despite the high voltages that develop at resonance.

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