Need help designing power supply

[ja421]

I have a vehicle with twin LED headlights.  The LED's have a cooling fan that runs when the LED's are on.
The bike operates at 13.8v -- due to other projects on the bike, I have a stable 12v, 5v, and 3.3v available (all limited to 3amp tho).
The lights have a low & high mode. When switched to high, the fan spins faster.  The difference between high and low is voltage (about 10 on low and 20 on high).  Fast switching between high and low results in the led slowly heating up, as the fan doesn't instantly spin faster.

The goal is to build a headlight modulator to cycle (ala sine wave) between 10v and 20v about 2 to 3 times a second while leaving the fan on high to keep it cool enough.

I've done various projects on my own, but I have no idea how to start designing the power supply that i can manipulate via an arduino PWM pin (5v).  I don't want to randomly pick some opamp & hope it works with trial and error until it does.

Would someone be able to help me?

[picandmix]

You could sort the problem out using the method you suggest, but it seems a little complex.

My own take would be to add a little delay timer so when the circuit goes from hi to lo he fan in kept running at high speed for a minute or two longer, so ensuring the led is fully cooled ..?
Or cannot the fan simply be wired for full speed all the time ..|?

A circuit diagram of the fan, led and their controllers wiring  would help for a more detailed reply.

[ja421]

There's not much to see regarding the circuit -- 12v+/gnd goes into a power supply that bumps it up to 20v.
4 leads come out (two pair, both V/gnd) One pair goes to the fan, one goes to the led.  Circuitry in fan & led is not documented.
I want to power the fan 100% on "high"; but ramp up/down the led.  Regulations say motorcycle can 'modulate' the high beams during the day -- BUT they must not "flick on and off" they *must* ramp up/down.. it's a fine line I guess... I can buy a pre-made one, but  it will ramp up/down 12v not 20v...
I had a guy who thought he could build it, but he flaked on me.  I can read a schematic & solder it up -- but I don't know how to build the opamp circuit needed... or even how to begin -- I was not a science major....

Thanks for your time.