Sensing differential pressure in form of inductance

Hello Folks,

Right now I am doing an university project which requires me to use an Arduino for sensing atmospheric pressure. I am using a sensor that represents the difference of pressure between his two inputs in form of two different inductances. While the difference rises or fall, one inductance increases and the other decreases. I did all the testing with a pressure generator and could verify that the inductances vary more or less between 7 mH and 10 mH in both sides with 10000 Pa applied to one input. Both inductances will show more or less 8 mH when there's not pressure difference.

So, in order to sense this with an Arduino, the first thing I did was create a small circuit in form of bridge, having the differential inductances and two equal resistances, and feed the system with an AC signal, so I can measure a voltage drop difference between the two bridge branches (Va and Vb), like the circuit image below:

The best result I could find, after testing different resistance/signal setups (which is the same of the image if I'm not mistaken) was to have a sine wave that varies between 4 Vpp (no pressure difference) and 7.2 Vpp (+10000 Pa difference).

Now I got stuck. I need to convert this sine wave into something the Arduino would be able to read (DC signal between 0 and 5 V). What would be the best option?
My advisor recommended using a comparator to convert the sine to a square wave and then have it through a Phase Locked Loop IC (4046)... I tried reading more about this approach but it honestly made no sense to me. I also tried some other options but had no success so far.

I would be thankful if anyone has any idea regarding such problem.

Blue_Phoenix:
The best result I could find, after testing different resistance/signal setups (which is the same of the image if I'm not mistaken) was to have a sine wave that varies between 4 Vpp (no pressure difference) and 7.2 Vpp (+10000 Pa difference).

This is AC. Positive and negative voltage swing on the coil.
You could try zero crossing detector opamps.
That will give you a switch time difference. Look at the LTspice Va,Vb sine waves at 0volt.
Not sure if an Arduino is fast enough to give you enough resolution.
Leo..

Your advisor wants to measure the frequency (a change in the frequency) and you want to measure the amplitude of the sine. That are two different things.

I would use a third option: use the Arduino as a two LC-meters.
Google for : arduino lc meter

Using a sine with the resistors seems odd. If an inductance needs to be measured, then some kind of oscillator with a frequency and a capacitor is needed. It can be a continuous frequency, or it can be a single pulse.

Some users on this forum know about coils and metal detectors. Perhaps they have a better idea.