AC circuits, Inductance and phase shift.

[knowlewj01]

Homework Statement

An electric motor draws a 1A (rms) from the 230V (rms) mains. and is found to shift the phase of the 50Hz alternating current by 30% of the period. What is the self inductance of the coils in the motor?

Homework Equations

The Attempt at a Solution

i've started by finding the phase shift:

$\delta = \frac{3}{10}\cdot \frac{1}{f} = \frac{3}{10}\cdot\frac{1}{50} = 0.006$

i think the circuit has only an inductor and power source in it, so using kirchhoffs loop rule:

$V_{max}cos(\omega t) - L\frac{dI}{dt} = 0$

separate variables and integrate:

$V_{max} \int cos(\omega t) dt = \int L dI$

$\frac{V_{max}}{\omega}\cdot sin(\omega t) = LI$

sin(wt) = cos(wt - d)

$L = \frac{V_{max}}{\omega I} cos(\omega t - \delta)$

I know everything here apart from t, is this the right way to do this?

 

[knowlewj01]

I just thought of another way to do this, not sure it's right because it doesn't use the phase shift at all:

$I_{max} = \frac{V_{max}}{X_L} = \frac{V_{max}}{\omega L}$

$L = \frac{V_{max}}{\omega I_{max}}$

where

$V_{max} = \sqrt{2}V_{rms}$

$I_{max} = \sqrt{2}I_{rms}$

so:

$L = \frac{V_{rms}}{\omega I_{rms}}$

$L = 0.732 H$