First rule with small single phase fan motors is the the colours of the internal motor wiring don't necessarily conform to the usual standards of blue/black = neutral and brown/red = live etc, depending on the manufacturer you'll find green or yellow wires as different speed windings and you'll find reds and even browns used as commons and neutrals inside the motor itself.
I'm not familiar with the woods fan you're working on....especially since you still haven't given a model number but I can tell you there's a good chance you'll need to move the capacitor onto different terminals if you remove the speed controller. Likewise when you install one you might also need to relocate the cap wiring.
What type of motor is it? Give us a model number or at least run a google images search and give us a photo of something similar.
What type of speed control is it? Is it a transformer type or is it a PWM type?
Whatever changes you make it's critical that you check the run current when you've finished and make sure it's equal to or less than the current on the motor info plate. It's also a good idea to listen to the motor, if there's any signs of iron noises such as rumbling or grating sounds then switch off before the motor is magically transformed into scrap. If you have little or no experience with this type of motor and speed control arrangement then there's a good chance you'll damage the motor beyond repair so if you can't afford a new replacement then rather get someone with knowledge to assist you.
Just for info, if it's a single speed fan U1 will be the common point of both windings. Test between U1&U2 for one winding and test between U1 and Z2 for the other. There should also be other wires in the motor that are for safety, they might be labeled as 'K' or 'S' or even 'T' depending on whether a thermistor/klixon/thermostat is used on the windings. It would be a very shrewd move to incorporate these into your DOL control or switching contactor as an interlock.
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