Alternating Current: Step-by-Step Worked Examples (5)

Question

A series LCR circuit has $R = 100\,\Omega$, $L = 0.5$ H, $C = 10\,\mu\text{F}$. It is connected to an AC source of rms voltage $200$ V and frequency $50$ Hz. Find (a) the impedance, (b) the rms current, and (c) the resonant frequency.

Solution — Step by Step

Final answers: (a) $Z \approx 190\,\Omega$, (b) $I_{\text{rms}} \approx 1.05$ A, (c) $f_0 \approx 71$ Hz.

Why This Works

In a series circuit, the current is the same through every element, but the voltages across $L$, $C$, and $R$ have different phases. $V_R$ is in phase with $I$, $V_L$ leads by $90^\circ$, and $V_C$ lags by $90^\circ$. Adding these as phasors gives the impedance triangle: $Z^2 = R^2 + (X_L - X_C)^2$.

At resonance, the inductive and capacitive reactances cancel, leaving $Z = R$. The current is maximum and is in phase with the voltage. That is why resonance is the marquee concept for AC circuits.

Alternative Method

Use the phasor diagram. Draw $V_R$ along the current axis, $V_L$ at $+90^\circ$, $V_C$ at $-90^\circ$. The net phasor’s magnitude gives the source voltage, and the phase angle $\phi = \tan^{-1}((X_L - X_C)/R)$ tells you the lead/lag. Same answers, faster for power-factor questions.

Common Mistake

Forgetting that $X_C = 1/(\omega C)$, not $\omega C$. The capacitor’s reactance decreases with frequency, while the inductor’s increases. Mixing these up is the most common AC error in board exams.