Q.Find the impedance of a series RLC circuit, if the inductive reactance, capacitive reactance and resistance are 184 $\Omega$, 144 $\Omega$ and 30 $\Omega$ respectively. Also calculate the phase angle between voltage and current.

Setting up. For a resistor R, inductor L and capacitor C all in series across $v=V_m\sin\omega t$, the SAME current i flows through every element, so $V_R$ is in phase with i, $V_L$ leads i by $90^{\circ}$, and $V_C$ lags i by $90^{\circ}$ -- meaning $V_L$ and $V_C$ point in exactly OPPOSITE directions on a phasor diagram. Combining $V_R$ with the net $(V_L-V_C)$ by the parallelogram law gives the applied voltage's magnitude,

Dividing through by $I_m^2$ (since each voltage is current times its own R or reactance) gives

where Z, the impedance, is the circuit's total effective opposition (in ohms) to AC, combining resistance and net reactance.

Phase angle -- from the similar voltage/impedance triangles. $\tan\phi = (V_L-V_C)/V_R = (X_L-X_C)/R$. …