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If you had a resistive coil with an inductance and a resistance, a sinusoidal voltage source with a variable frequency, and a meter that can measure rms voltages and currents, how would you go about determining the inductance and resistance of the coil?

I know that the complex impedance of the coil is $i\omega L + R$, so the current through the coil will be $I = \frac{V}{i\omega L + R}$, but I am not sure what to do from this.

Any help would be appreciated!

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2 Answers 2

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Since $\left (\dfrac{V_{\rm rms}}{I_{\rm rms}}\right )^2=L^2\, \omega^2+R^2 = (4\,\pi^2 L^2)\, f^2+R^2$ is of the form of the general equation of a straight line, $y = m\,x +c$, then taking a series of $V_{\rm rms}$ and $I_{\rm rms}$ readings at different frequencies, $f$, and drawing a graph of $\left (\dfrac{V_{\rm rms}}{I_{\rm rms}}\right )^2$ against $f^2$ should give a straight line of gradient $4\,\pi^2 L^2$ and intercept on the $\left (\dfrac{V_{\rm rms}}{I_{\rm rms}}\right )^2$ axis of $R^2$.

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With your meter you can measure only the rms of voltage and current, but not their phases. Therefore from the complex impedance $$\frac{V}{I}=Z=i\omega L+R$$ you can only measure its absolute value $$\frac{V_\text{rms}}{I_\text{rms}}=|Z|=\sqrt{(\omega L)^2+R^2}.$$

So with your meter you need to measure $|Z|$ for at least two frequencies (or for DC and one frequency). From these $|Z|$ values you can then calculate $R$ and $L$.

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