What is the purpose of these coils in contactor auxiliary contacts?
I think you have omitted to mention that these contactors are part of a power-factor control panel and so they are switching a purely capacitive load.

If so, those coils are inductors.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Power-factor correction capacitor soft-start.

Remember that a discharged capacitor appears as a short-circuit when connected to a voltage source. A huge inrush current occurs and this is hard on the contacts.

The auxiliary contact block on the contactors is designed to close before the main contacts. This allows a charging current to flow into the capacitors to partially charge them before the main contacts make and reduces wear and tear on the them. When I first saw them on an installation I was very surprised that such fine wires could do that work.

According to a couple of online calculators including MustCalculate the inductance for my estimate of Ø = 10 mm, L = 40 mm and 9 turns the inductance is about 180 nH. This has an impedance of only 56 &mu;&Omega; at 50 Hz but it's effectively a transient that's being switched so the impedance will be much higher.