Tuning the Series Circuit

Calculate the inductive reactance, the capacitive reactance and the impedance of the circuit at $50.0\ \text{Hz}$. Explain, with reference to the phase relationships of the p.d.s across the three components, why the impedance is not simply $R + X_L + X_C$.

Calculate the rms current and the phase angle between the current and the supply voltage, stating whether the current leads or lags. Describe the phasor diagram.

Show that the current is a maximum at the resonant frequency $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$, and calculate $f_0$ and the maximum rms current.

Sketch a graph of the rms current (vertical axis) against frequency from $0$ to about $400\ \text{Hz}$. Label the peak value and its frequency, the current at $50.0\ \text{Hz}$, and the behaviour as $f \to 0$ and at high frequency. Explain the shape, including why the curve is not symmetrical about $f_0$, and state how the phase angle changes along the curve.