Question

Describe a series L, C, R resonant circuit.

Hint:

For a series LCR circuit, resonance occurs when the inductive and capacitive reactances are equal in magnitude. The resonant frequency is given by \(f_0 = \frac{1}{2\pi \sqrt{LC}}\).

Answer 1

A series L, C, R resonant circuit consists of three basic components connected in series: an inductor (L), a capacitor (C), and a resistor (R). In such a circuit, the total impedance of the circuit is affected by the frequency of the applied AC signal. The behavior of the circuit changes as the frequency varies, and resonance occurs at a specific frequency known as the resonant frequency. Impedance of the Series LCR Circuit: The total impedance \(Z\) of a series L, C, R circuit is given by: \[ Z = R + j\left(\omega L - \frac{1}{\omega C}\right) \] where: - \(R\) is the resistance, - \(L\) is the inductance, - \(C\) is the capacitance, - \(\omega = 2\pi f\) is the angular frequency of the AC signal. At resonance, the inductive reactance \( \omega L \) and the capacitive reactance \( \frac{1}{\omega C} \) cancel each other out, and the total impedance of the circuit becomes purely resistive: \[ Z_{\text{resonance}} = R \] Thus, at resonance, the impedance is minimized, and the current in the circuit reaches its maximum value. Resonant Frequency: The resonant frequency \(f_0\) is the frequency at which the inductive and capacitive reactances are equal in magnitude but opposite in sign. At this frequency, the circuit exhibits purely resistive behavior, and the impedance is equal to the resistance \(R\). The resonant frequency is given by: \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \] At resonance, the voltage across the inductor and the capacitor is maximum, while the total impedance of the circuit is at a minimum. Power at Resonance: At resonance, the power delivered to the circuit is maximized. The power in a series LCR circuit is given by: \[ P = I^2 R \] where \(I\) is the current through the circuit. At resonance, the current reaches its maximum value, and the power dissipated in the resistor is at its peak.