Haha. The relay will survive but that's definitely a good way to shorten the life of your HIDs.
I see 2 solutions:
1) put a capacitor + flyback diode across the relay coils and hope that it doesn't smooth the ripple so much that it restricts the output current. I assume this will work since the HIDs are already of sufficient brightness correct?
2) use something else to trigger the relay. The coil portion of the relay doesn't consumes a negligible amount of power but you should be safe putting the coil portion in parallel with any other relay (make sure the relay output/HID wires don't connect with the other relays output) or in series with another wire that uses a fair amount of power and isn't pulse with modulated (PWMd) eg. fog lights?
3) build a 'pwm filter' like this guy did
BONUS: if you can set your multimeter to measure frequency, tell me what frequency appears at the relays terminals. This will tell me how quickly Hyundai pulses the signal.
By knowing the coils resistance I can figure out the maximum size diode + capacitor you need. Telling me the frequency will let me narrow the selection of diodes and capacitor to the smallest ones necessary....although not necessary it'll save you a few dollars and be more compact.
Shame, couldn't find a data sheet for that relay, but that is alright.
Alternative bonus: not necessary but it's an alternative way for me to figure out the frequency. If you can measure DC voltage, while the car is running, what is the voltage across the battery terminals and across the relay coil terminals.
I'll get back to you with diode + capacitor recommendations when I'm back home later tonight.
The PWM unit is 95% most likely in the main computer.
That's a good idea, anything that isn't PWMd will work for the trigger. Your relay only pulls 140mA to operate.
...........
Just ran the proper simulations.
Couldn't find the switching frequency online so use a worst case scenario (80Hz pulses from Volvo). And after calculating the parts.... you can't get them from TheSouce so here are the digikey links.
Resistor: 2.5+ ohm resistor. This will limit the surge current (start up current) that the capacitor draws to ~4.3A...since your headlight fuse is probably rated for 5A.
http://www.digikey.ca/product-detail/en/PNP300JR-73-2R7/2.7AECT-ND/2059185
(put the resistor in series either before or after the diode)
Capacitor: 4,700 uF will work. The noise will be reduced considerably, but ~20,000 uF will reduce the vibrations 5x in magnitube (...not sure how much quiter that would be sound wise though). Since there are other noises the car makes while on 4,700 uF should be fine. If you want though, you can order both sizes and use the larger one if you can still hear the noise. Without testing it's hard to tell is 4,700 uF would even be audible, but
4,700 uF: http://www.digikey.ca/product-detail/en/EGPA250ELL472MM30S/565-3383-ND/3528520
22,000 uF: http://www.digikey.ca/product-detail/en/ECO-S1EA223DA/P6591-ND/131411
If you care less about the buzz noise (because you already can't hear it when your hood is closed) and just want to make sure your relay is operating within spec: 500uF at a minimum will work. Regardless, 500 uF will still reduce the noise significantly from what you have now and it may not even be audible. How this translates into audible sound is hard to guess without testing.
680 uF: http://www.digikey.ca/product-detail/en/MAL214651681E3/4502PHBK-ND/2410367
If the buzzing doesn't bother you much now, and you only somewhat hear it when the hood is closed, I'd just go with the 680 uF. It is the small and will get the job done. Otherwise since they aren't to expensive, you can get the 22,000 uF as well and just see how much of a diference it makes.
(the negaticve arrows on the case of the capacitors point towards the negative leg)
Diode:
This will be plenty: http://www.digikey.ca/product-detail/en/1N4002/1N4002FSCT-ND/1532743
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From digikey you can checkout as a guest. $8 shipping flat rate.
AWESOME. That's great to hear and overall a clean look.
And you're right, electrical takes a while to get comfortable with but it's easier to learn when you're motivated to do your own work.
I wouldn't worry much about the 0.25W resistor, if it holds up after a month of use then you're golden.... the only reason I gave you the link to a 3W one was because it had stress testing documentation that made me happy.
P.S. capacitors are usually the things that fail first (the top will burst open; so if your relay turns off (the cap shorted) or starts buzzing again (the cap internally disconnected) you know what to expect, most likely from vibrations or overheating) but a standard capacitor is rated for 2000 hours minimum (almost 3 months of constant use) which is already plenty.
P.S.S. If you're super unlucky and the cap shorts when it dies then it'll take out the resistor, then if the resistor shorts when it dies, it'll take out the diode, then if that shorts it'll take out your PWM unit LOL but that's a cascade of unlikely failures before reaching the PWM unit.
+1 for velcro tape