Question

A source of alternating e.m.f. is connected to a series combination of a resistor R, an inductor L, and a capacitor C. Obtain with the help of a vector diagram and impedance diagram, an expression for (i) the effective voltage (ii) the impedance (iii) the phase relationship between the current and the voltage.

Answer 1

Refer IMAGE $01$

$1.$ $V=iR+i(j{X}_L-j{X}_C)$

Refer IMAGE $02$

$2.$ $|V_{i}n|=i\sqrt{R^2+(X_L-X_C{)}^2}$

$i_o=\frac{V_m}{\sqrt{R^2+(X_L-X_C{)}^2}}Z=\sqrt{R^2+(X_L-X_C{)}^2}$

$i=i_{o}sin(\omega t-\varphi )$

$\varphi ={\tan }^{-1}\left(\frac{X_L-X_C}{iR}\right)$

$3.$ $i=\frac{V_m}{\sqrt{R^2+(X_L-X_C{)}^2}}\sin (\omega t-\varphi )$