Testing hall effect sensor of a motor [Beginer question]

I've used the Allegro 3503 ratiometric linear sensor. It's discontinued, but I suspect its similar to the 1301/1302 sensors. It is very linear when positioned between two opposing magnets. This particular config (push-push mode) is shown on page 27 of the guide that you can download at http://www.allegromicro.com/en/Design-Center/Technical-Documents/Hall-Effect-Sensor-IC-Publications/Hall-Effect-IC-Application-Guide.aspx, but there are some other interesting modes. Give that ref a good read. You are talking mm here, and not cm. I think if you have a small 1 mm range, you can probably get 0.5 mm resolution, but I can't say I've pushed it that far. Strong neodymium magnets would certainly help.

They're even easier to use than you describe. The output voltage simply changes as a function of flux density.

As a design alternative, you might consider an optical position sensitive detector from Hamamatsu. You'd just need a 1-D sensor. They're good to about 50 microns over two millimeters.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. Reorganised schematic.

Your schematic shows the sensor pulling the Pi input to 3.3 V. In fact it pulls it to ground.

The relay armature forms a closed circuit when pulled in. The magnetic flux will take the easiest route through the iron which has a permeability hundreds of times that of the air.

^{simulate this circuit}

Figure 2. Air-gap core DIY hall current transducer.

Figure 3. Air-gap in torroidal ferrite core. Credit: SoftSolder.com.

You could make your own current transducer as shown above. (The link is for a different application.) The hall transducer is popped into the gap in the ferrite core. The conductor to be monitored passes through the core and, if not sensitive enough, additional turns can be wound to increase sensitivity. In your case you would use PVC insulated wire rather than the enamel coated wire shown here. Cutting the core is problematic and the article addresses that. I've seen other similar posts on the web.

This suggestion would only work for your application if there is current flowing all the time the power source is on.