Solved Examples (Dedicated Problem Set)

How to Use This Problem Set

Your full workout for Alternating Current, grouped by theme: rms values and resistors, inductors, capacitors, series LCR impedance, resonance, AC power, and transformers.

Keep these handy:

• $I = i_m/\sqrt 2$, $V = v_m/\sqrt 2$; $P = I^2R = IV$ (resistor)
• $X_L = \omega L = 2\pi\nu L$; $X_C = 1/\omega C$; current lags (L) / leads (C) by 90 degrees; both lossless
• $Z = \sqrt{R^2 + (X_C - X_L)^2}$; $\tan\phi = (X_C - X_L)/R$; $\cos\phi = R/Z$
• Resonance: $\omega_0 = 1/\sqrt{LC}$, $Z = R$, $I_{max} = V/R$, $\cos\phi = 1$
• $P = VI\cos\phi = I^2R$; wattless current at $\phi = 90$ degrees
• Transformer: $v_s/v_p = N_s/N_p = i_p/i_s$ (ideal)

Work in rms unless peaks are explicitly asked, and carry units.

Solved Examples - RMS Values & Resistors

Example 1. $i = 14.14\sin\omega t$ A. The rms current?

Solution: $I = i_m/\sqrt 2 = 14.14/1.414 = 10$ A.

Example 2. A source has peak 400 V. Its rms voltage?

Solution: $V = 400/\sqrt 2 = 283$ V.

Example 3. A 60 W bulb on 220 V mains: resistance and rms current?

Solution: $R = V^2/P = 48400/60 = 807\ \Omega$; $I = P/V = 60/220 = 0.27$ A.

Example 4. (NCERT 7.1) 100 W bulb on 220 V: R, peak voltage, rms current?

Solution: $R = 484\ \Omega$; $v_m = 311$ V; $I = 0.454$ A.

Example 5. An rms current of 4 A in an 11 $\Omega$ resistor: average and peak power?

Solution: $P = I^2R = 16 \times 11 = 176$ W; peak $= 2P = 352$ W.

Example 6. At what first instant does $v = v_m\sin(100\pi t)$ equal its rms value?

Solution: $\sin = 1/\sqrt2$ at $100\pi t = \pi/4$: $t = 1/400$ s = 2.5 ms.

Solved Examples - Inductors on AC

Example 7. (NCERT 7.2) 25 mH on 220 V, 50 Hz: $X_L$ and I?

Solution: $X_L = 2\pi(50)(0.025) = 7.85\ \Omega$; $I = 220/7.85 = 28$ A.

Example 8. Reactance of 0.5 H at 100 Hz?

Solution: $X_L = 2\pi \times 100 \times 0.5 = 314\ \Omega$.

Example 9. What L gives $X_L = 100\ \Omega$ at 50 Hz?

Solution: $L = X_L/2\pi\nu = 100/314 = 0.318$ H.

Example 10. Average power consumed by a pure inductor carrying 28 A from a 220 V source?

Solution: Zero — $\phi = 90$ degrees, wattless current; energy only sloshes.

Solved Examples - Capacitors on AC

Example 11. (NCERT 7.4) 15 $\mu$F on 220 V, 50 Hz: $X_C$, I, $i_m$?

Solution: $X_C = 212\ \Omega$; $I = 1.04$ A; $i_m = 1.47$ A (leading by 90 degrees).

Example 12. Reactance of 10 $\mu$F at 50 Hz?

Solution: $X_C = \frac{1}{2\pi(50)(10^{-5})} = 318\ \Omega$.

Example 13. What C gives $X_C = 212\ \Omega$ at 100 Hz?

Solution: $C = \frac{1}{2\pi(100)(212)} = 7.5\ \mu$F — half the NCERT value, since frequency doubled.

Example 14. A capacitor's current at 50 Hz is 1.0 A. At 200 Hz (same source voltage)?

Solution: $I \propto \nu$ (since $X_C \propto 1/\nu$): $I = 4.0$ A.

Solved Examples - Series LCR Impedance

Example 15. R = 60 $\Omega$, $X_L = 120\ \Omega$, $X_C = 40\ \Omega$ on 200 V: Z, I, $\cos\phi$?

Solution: $Z = \sqrt{60^2 + 80^2} = 100\ \Omega$; $I = 2$ A; $\cos\phi = 0.6$ (current lags).

Example 16. (NCERT 7.6) 200 $\Omega$ + 15 $\mu$F on 220 V/50 Hz: I, $V_R$, $V_C$?

Solution: $Z = 291.5\ \Omega$; $I = 0.755$ A; $V_R = 151$ V; $V_C = 160.3$ V; phasor sum $= 220$ V.

Example 17. Z of a coil (R = 8 $\Omega$, $X_L = 6\ \Omega$)?

Solution: $Z = \sqrt{64+36} = 10\ \Omega$ (cos phi = 0.8, lagging).

Example 18. In Example 15, the element voltages?

Solution: $V_R = 120$ V, $V_L = 240$ V, $V_C = 80$ V; check $\sqrt{120^2 + 160^2} = 200$ V.

Example 19. A series circuit on DC (steady state) with R, L, C carries:

Solution: Zero current — the capacitor blocks ($X_C \to \infty$ at $\omega = 0$).

Example 20. R = 100 $\Omega$, L = 0.5 H, C = 10 $\mu$F at 50 Hz: net reactance and behaviour?

Solution: $X_L = 157\ \Omega$, $X_C = 318\ \Omega$: net $X_C - X_L = 161\ \Omega$ capacitive — current leads; $Z = \sqrt{100^2 + 161^2} \approx 190\ \Omega$.

Solved Examples - Resonance

Example 21. (NCERT exercise pattern) L = 2.0 H, C = 32 $\mu$F, R = 10 $\Omega$: resonant angular frequency?

Solution: $\omega_0 = \frac{1}{\sqrt{2 \times 32\times10^{-6}}} = \frac{1}{8\times10^{-3}} = 125$ rad/s.

Example 22. Its resonant frequency in Hz?

Solution: $\nu_0 = 125/2\pi \approx 19.9$ Hz.

Example 23. On a 230 V source at resonance, the current (R = 10 $\Omega$)?

Solution: $I = V/R = 23$ A — impedance has collapsed to R.

Example 24. L = 1 mH, C = 1 nF (NCERT's plot): $\omega_0$?

Solution: $\omega_0 = 1/\sqrt{10^{-12}} = 10^6$ rad/s.

Example 25. At resonance with $X_L = 50\ \Omega$, R = 5 $\Omega$, V = 100 V: voltage across C?

Solution: $I = 20$ A; $V_C = IX_C = IX_L = 1000$ V — tenfold magnification, cancelled by $V_L$.

Solved Examples - Power & Power Factor

Example 26. (NCERT 7.8) R = 3, $X_L = 8$, $X_C = 4\ \Omega$, peak 283 V: Z, phase, P, pf?

Solution: $Z = 5\ \Omega$; $\phi = -53.1^{\circ}$ (lagging); $I = 40$ A; $P = 4800$ W; pf $= 0.6$.

Example 27. (NCERT 7.9) Same circuit at resonance: I and P?

Solution: $Z = 3\ \Omega$; $I = 66.7$ A; $P = I^2R \approx 13.35$ kW.

Example 28. V = 230 V, I = 5 A, $\phi = 37$ degrees: power?

Solution: $P = VI\cos\phi = 230 \times 5 \times 0.8 = 920$ W.

Example 29. A circuit draws 2 A from 220 V consuming 220 W. Power factor?

Solution: $\cos\phi = \frac{P}{VI} = \frac{220}{440} = 0.5$ ($\phi = 60$ degrees).

Example 30. Average power drawn by a pure capacitor carrying 1.04 A from 220 V?

Solution: Zero — wattless current ($\cos 90^{\circ} = 0$).

Solved Examples - Transformers

Example 31. $N_p$ = 3000, $N_s$ = 150, input 220 V: output voltage?

Solution: $v_s = 220 \times \frac{150}{3000} = 11$ V — a step-down by 20.

Example 32. That transformer (ideal) delivers 2.2 A at 11 V. Primary current?

Solution: $i_p = \frac{i_s v_s}{v_p} = \frac{2.2 \times 11}{220} = 0.11$ A.

Example 33. A step-up transformer (1:10) takes 220 V at 5 A. Output (ideal)?

Solution: $v_s = 2200$ V, $i_s = 0.5$ A; power 1100 W both sides.

Example 34. 44 kW must cross a 20 $\Omega$ line. Line loss at 220 V vs 22 kV?

Solution: At 220 V: $I = 200$ A, loss $= 200^2 \times 20 = 800$ kW (absurd). At 22 kV: $I = 2$ A, loss $= 80$ W. The transformer's case, closed.