An R L C series circuit has a resistance of R=325 Ωan inductance L=0.300 mH, and a capacit

step 1 We have frequency f -not equals omega -not over 2 pi equals 1 over 2 pi root lc equals 2 pi squa

An R L C series circuit has a resistance of R=325 Ωan...

An $R L C$ series circuit has a resistance of $R=325 \Omega$ an inductance $\quad L=0.300 \mathrm{mH}, \quad$ and $\quad$ a capacitance $C=33.0 \mathrm{nF} .$ (a) What is the resonant frequency? (b) If the capacitor breaks down for peak voltages in excess of $7.0 \times 10^{2} \mathrm{V},$ what is the maximum source voltage amplitude when the circuit is operated at the resonant frequency?

Key Concepts

Component Voltage Limits in AC Circuits

In practical circuits, components such as capacitors have maximum voltage ratings beyond which they can fail. For a resonant RLC circuit, even though the source voltage might be relatively low because of the low overall impedance at resonance, the voltage across individual reactive components (like the capacitor) can be significantly amplified. Determining the maximum allowable source voltage involves ensuring that this amplification, often related to the Q factor, does not exceed the component’s breakdown voltage, thereby protecting the circuit from damage.

Quality Factor (Q Factor)

The quality factor of an RLC circuit quantifies how underdamped it is and relates to the ratio of stored energy to energy dissipated per cycle. In a series RLC circuit, the Q factor also indicates the voltage magnification across reactive elements relative to the source voltage at resonance. It can be expressed as Q = (1/R)?(L/C), and a higher Q implies a sharper resonance peak, which is crucial when considering how component voltage ratings (such as a capacitor's breakdown voltage) limit the allowable source voltage amplitude.

Resonant Frequency

In an RLC series circuit, resonance occurs when the inductive reactance (?L) equals the capacitive reactance (1/?C). At this frequency, these opposing reactive effects cancel out, leaving only the resistive component to determine the impedance. The resonant frequency is given by the formula f? = 1/(2??(LC)), which is a central concept in AC circuit analysis as it defines the frequency at which the circuit naturally oscillates with minimal impedance.

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