Question

What is the impedance of a series RLC circuit when the frequency of time- varying emf is set to the resonant frequency of the circuit?

Short answer

Answer: At the resonant frequency, the impedance of a series RLC circuit is equal to the resistance R.

Step-by-step solution

Recall the formula for resonant frequency

For a series RLC circuit, the resonant frequency is given by the formula: \[ f_r = \frac{1}{2 \pi \sqrt{LC}} \] where L is the inductance, C is the capacitance, and \(f_r\) is the resonant frequency.

Recall the formula for the impedance

The impedance of a series RLC circuit is given by the formula: \[ Z = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2} \] where R is the resistance, L is the inductance, C is the capacitance, \(\omega\) is the angular frequency, and Z is the impedance.

Calculate the angular frequency at resonance

At resonance, the angular frequency of the time-varying emf is equal to the angular frequency of the resonant frequency. We can use the following relationship between angular frequency and frequency: \[ \omega = 2 \pi f \] Since we know the resonant frequency, we can plug it in to calculate the angular frequency at resonance: \[ \omega_r = 2 \pi f_r \]

Calculate the impedance at resonance

Now, we can find the impedance at the resonant frequency by plugging in the values of R, L, and C and the angular frequency at resonance into the impedance formula: \[ Z_r = \sqrt{R^2 + (\omega_r L - \frac{1}{\omega_r C})^2} \] At resonance, the reactive parts of the impedance, the inductive reactance, and the capacitive reactance are equal and thus cancel each other out. Therefore, the impedance formula simplifies to: \[ Z_r = \sqrt{R^2 + (0)^2} = R \] So, the impedance of the series RLC circuit at the resonant frequency is equal to the resistance R.