Resonant frequency is the frequency at which the impedance of an RLC series circuit is minimized, and the circuit oscillates with maximum amplitude. It occurs when the inductive reactance equals the capacitive reactance.
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The resonant frequency $f_0$ can be calculated using $f_0 = \frac{1}{2\pi\sqrt{LC}}$, where L is inductance and C is capacitance.
At resonant frequency, the voltage across the inductor and capacitor are equal in magnitude but opposite in phase.
In an RLC series circuit, at resonance, the total impedance is purely resistive and equals only the resistance (R).
The quality factor Q measures how underdamped an oscillator or resonator is, defined as $Q = \frac{f_0}{\Delta f}$ where $\Delta f$ is the bandwidth over which power falls to half its peak value.
At resonance, power transfer in the circuit is maximized because reactive powers cancel each other out.
Review Questions
How do you calculate the resonant frequency for an RLC series circuit?
What happens to the impedance of an RLC series circuit at its resonant frequency?
Why does maximum power transfer occur at resonant frequency in an RLC series circuit?
Inductive reactance ($X_L$) opposes changes in current in an AC circuit and is given by $X_L = \omega L$, where $\omega$ is angular frequency and L is inductance.
Capacitive reactance ($X_C$) opposes changes in voltage across a capacitor in an AC circuit and is given by $X_C = \frac{1}{\omega C}$, where $\omega$ is angular frequency and C is capacitance.
Quality Factor (Q): $Q$ describes how underdamped a resonator or oscillator is, calculated as $Q = \frac{f_0}{\Delta f}$ where $f_0$ is resonant frequency and $\Delta f$ represents bandwidth.