Black Box Impedance Sleuthing

Suppose there is a box containing an ideal sinusoidal AC voltage source (RMS magnitude of VV) in series with a resistance RR and an inductive reactance XX (for a total complex impedance Z=R+jX)Z = R + jX). The circuit is incomplete, and a pair of terminals are brought outside the box. Aside from the availability of the terminals, the box is closed and its contents are inaccessible.

You want to find out the values of the resistance and inductive reactance, so you devise a clever strategy to determine them. You take three measurements, each time connecting an ideal AC ammeter in series with a test impedance ZTZ_T and connecting the series combination across the box terminals.

For the first measurement, ZTZ_T is a perfect short-circuit with zero impedance, and the measured RMS current magnitude is Vα\Large{\frac{V}{\sqrt{\alpha}}}.

For the second measurement, ZTZ_T is a 1Ω1 \Omega resistance, and the measured RMS current magnitude is Vα+5\Large{\frac{V}{\sqrt{\alpha+5}}}.

For the third measurement, ZTZ_T is a 1Ω1 \Omega inductive reactance, and the measured RMS current magnitude is Vα+9\Large{\frac{V}{\sqrt{\alpha+9}}}.

What is the value of R+XR + X?

Note: In electrical engineering, the letter j''j'' is commonly used to denote 1\sqrt{-1}, since the letter i''i'' is commonly used to represent current.


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