Question

(i) Derive the impedance of an R-C series circuit connected to an AC source v = v_m sin(ωt).
(ii) When does an inductor act as a conductor? Explain.
(iii) A lamp (R) and inductor (L) in series with 220V-50Hz AC give 11A. With 110V DC, it gives 11A. Find L.

Hint:

Impedance, Inductive Reactance, AC circuits

Answer 1

(i) Impedance \(Z = \sqrt{R^2 + X_c^2}\), with \(X_c = 1/\omega C\). (ii) An inductor functions as a conductor under DC conditions because its inductive reactance \(X_L = 2\pi fL\) is 0 when the frequency \(f = 0\). (iii) The resistance is \(R = V_{dc}/I = 10\Omega\). The AC impedance is \(Z = V_{ac}/I = 20\Omega\). Using \(Z = \sqrt{R^2 + X_L^2}\), the inductive reactance is \(X_L = \sqrt{Z^2 - R^2} = 10\sqrt{3} \Omega\). Consequently, the inductance \(L = X_L/\omega = X_L/(2\pi f) \approx 55\) mH.