⚡ AC Voltage Across a Series L-C-R Circuit

1. The Circuit at a Glance 🔍

A resistor R, inductor L, and capacitor C sit one after the other on the same loop. The source voltage is \(v = v_m \sin\omega t\). Applying Kirchhoff’s rule gives

\[ L\,\frac{di}{dt} \;+\; R\,i \;+\; \frac{q}{C} \;=\; v_m \sin\omega t \]:contentReference[oaicite:13]{index=13}

2. Riding the Phasor Ferris Wheel 🎡

  • The current in every element is identical at each instant: \[ i = i_m \sin(\omega t + \phi) \]:contentReference[oaicite:14]{index=14}
  • Peak (rms) voltages on each element are tied to the same current: \[ v_{Rm}= i_m R,\qquad v_{Lm}= i_m X_L,\qquad v_{Cm}= i_m X_C \] with \(X_L = \omega L\) and \(X_C = 1/(\omega C)\).:contentReference[oaicite:15]{index=15}
  • Impedance triangle: vectors \(V_R\) (in step with I), \(V_L\) (ahead by 90°), and \(V_C\) (behind by 90°) add tip-to-tail. The math yields \[ Z = \sqrt{\,R^{2} + (X_L – X_C)^{2}} \]:contentReference[oaicite:16]{index=16}
  • The source–current phase angle is \[ \tan\phi = \frac{X_C – X_L}{R}. \] • If \(X_C > X_L\) ➜ circuit behaves like a capacitor and current leads voltage. • If \(X_L > X_C\) ➜ circuit behaves like an inductor and current lags.:contentReference[oaicite:17]{index=17}

3. Current & Impedance Equations 🎯

Combining everything, the peak current is

\[ i_m = \frac{v_m}{Z} =\frac{v_m}{\sqrt{R^{2}+(X_L-X_C)^{2}}}. \]:contentReference[oaicite:18]{index=18}

4. Resonance — The Sweet Spot 🎶

  • Condition: \(X_L = X_C\) ⇒ \( \omega_0 = \dfrac{1}{\sqrt{LC}}\).:contentReference[oaicite:19]{index=19}
  • At resonance, the imped­ance collapses to \(Z = R\) and current peaks at \(i_m = v_m/R\).:contentReference[oaicite:20]{index=20}
  • The current-versus-frequency curve looks like a sharp mountain whose height falls as R gets larger.:contentReference[oaicite:21]{index=21}
  • Real-life uses: radio/TV tuning, metal detectors, and any gadget that “picks out” one frequency while rejecting others.:contentReference[oaicite:22]{index=22}
  • No resonance occurs if either L or C is missing (pure RL or RC chains).:contentReference[oaicite:23]{index=23}

5. Worked-Out Example 📝

A 200 Ω resistor and 15 µF capacitor in series with 220 V-50 Hz supply:

  1. \(Z = 291.6\;Ω\) ➜ \(I_{\text{rms}} ≈ 0.755\;A\).
  2. Volt­ages: \(V_R = 151\;V\), \(V_C = 160\;V\). Their vector (not algebraic!) sum equals the 220 V source.:contentReference[oaicite:24]{index=24}

Important Concepts for NEET 🎯

  • Impedance of an LCR chain: \(Z = \sqrt{R^{2}+(X_L – X_C)^{2}}\).
  • Phase angle formula: \(\tan\phi = (X_C – X_L)/R\) (lead vs. lag).
  • Resonant frequency: \(\omega_0 = 1/\sqrt{LC}\); at this point, \(Z = R\) and current maximizes.
  • Capacitive vs. inductive nature decided by comparing \(X_C\) and \(X_L\).
  • Resonance is impossible without both L and C; RL or RC pairs can’t resonate.

😊 Happy studying! You’ve got this! ✨