Can a 26 Inch Controller Drive a 20 Inch Wheel? My situation: 


A Bakfiets with a bad controller (hills too much for it I think)
Supplier has offered to send me a new controller 
Supplier has also offered to supply a whole new separate battery - controller - motor system to mount on the front wheel (to prevent a recurrence of controller failure). This is a unit he has unused and it is designed for a rear 26 inch wheel.


This sounds great, but I am worried that his second controller won't properly drive my 20 inch front wheel. 

My question - will this system work? Do controllers need to be configured to drive different wheel sizes? I feel the answer is yes, otherwise, how can a controller know the speed of the bike - it only has two ways of knowing this - the pedelec or the motor.
You may be able to reprogram the wheel size controller from the lcd control panel (or usb reprogramming cable). But this assumes a certain level of intelligence in the controller and yours might not have this capacity. You’d have to check with the vendor.

Even if it were reprogrammable you have the following challenges:


Motors have optimum torque and speed ranges. Your motor may be wound for slower rotations of a 26” wheel and won’t give you enough speed that a 20” needs
controllers may obey the ebike assist speed limit by limiting max RPMs. If your controller isn’t wheel size reprogrammable you’ll find yourself limited to a much reduced speed  and even if you can, the physical/electronic max rpm limit may still limit your top speed (my Bafang geared hub motor/controller doesn’t seem to go above a certain rpm irregardless of the wheel size; this may be either a motor or controller limitation) 
you’ll need to build the wheel and you might find some of the spoke angles are going to be very tight
a rear ehub will not fit normally in the front forks because of the hubspacing and freewheel clearance.