Electronic – Why don’t the coil work

The interesting thing with coils (inductors) is current, not voltage. It might be that the circuit connected to your coil uses some type of current-control. In the most simple case, this would be a resistor in series with the coil.

If this was the case, you might measure something like 3.3 V on the connected coil, and you would now measure something like the full battery voltage (7.2 V) on the wires leading to the coil when the coil is not connected.

Also, it may be that the internal resistance of the batteries plays a big role.

Can you try and measure the coil's DC resistance? Can you try and measure the current flowing into the coil? Can you look into the things connected between the battery and the coil?

Your decision to use a capacitor was the correct idea, but that one is much too small.

It's unclear what model you have, but your RPi probably consumes between 700-1000mA. However, if you calculate that 2200uF capacitor's discharge curve for 5V @ 700mA, it's only going to delay a brownout for maybe a millisecond before it drops below the RPi's minimum 4.75V. And that's assuming your battery is brand-new and fully charged.

That RS-540 motor can draw a massive amount of current for a few seconds when starting or stalled, and that capacitor will be discharged long before it's over. Some motor controllers I've seen designed for the RS-540 are rated to handle 30A for 5 seconds.

Another option is to use a battery with a larger mAh (in addition to a larger capacitor) that can handle higher current draws.

You may still want to consider a second battery so the RPi can remain functional even after the motor battery has been completely exhausted.