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The value of the capacitor depends on the load and on how much voltage ripple you can stand. You could get a circuit simulator, draw up your circuit and play around until you get the result you're after.
The peak of 110VAC is 155.5V. If the rectifier is conducting a few Amps then its voltage drop is 1V. The filter capacitor is quickly charged to 155.5V then slowly discharged before being charged again on the next pulse.
Around here the history of electrical power was that in fact at one time most homes had 110 AC on average. That number was fairly standard through the 1950's until rural electrification became more common place. At that point they are several versions of why the bumped it up another 8 - 10% on average.
One common story is during the long runs of lines needed during rural electrification they cheated it up some to improve the line loss voltage drops. So the standard was that if you were near the power substation you had 120 volts but the guys way out in the country still had around the original 110 volts on average.
Plus many homes have rather poor wiring and the line voltage drops in the house often brought the 120 volt feeds back down to 110 on the higher load items anyway.
Thats why you still see electric motors and many other higher power usage devices rated at a lower voltage than what the source is. Most typical motors are still rated as 110/220 or 115/230. Even the large three phase powered motors have a lower than line voltage ratings well.
190 - 200 on 208 volt systems, 240 - 260 on 277 volt systems, 380 - 400 on 416 volt systems, and 440 - 460 on 480 volt systems.
All standardized appliances and equipment are supposed to be rated to handle -15% to + 10% of their rated input voltage with out any problems. And most good quality electric motors will still handle an even higher present of over voltage without problems.
Another common version of the reason is that many places started out with 50 cycle power with 100 -110 volt systems. When they standardized to the all 60 cycle grid systems that allowed for a slight increase in line voltage being the old 50 cycle equipment could take 60 cycles at a slightly higher voltage without problems and improved power capacity of the old 50 cycle transformers and rotary converters considerably.
But really there are many reasons why it was changed over the years and why some places still say 110/220 instead of 120/240 even though they do in fact have 120/240 more often than not.
Around here we are guaranteed at least 115/230 minimum and no higher than 132/264 maximum. My personal home runs around 122/244 to 126/252 on average.
last week I was checking the voltages of our vacuum which has some issues, and i measured the power coming in and it was 111VAC.
Our house is pretty far out, partially in the woods, and its an old house.
With 30 LEDs as a load then the 1uF filter capacitor is much too small. The voltage will be pulses at double the mains frequency. You might see the LEDs flicker at 100Hz or 120Hz. Since the voltage has such a high amont of AC ripple then the LEDs are turning on and off and will appear dimmed. Be careful that they do not burn out if you add a much bigger and better filter capacitor.
As for calculating the actual needed size of the capacitor, it has to do with ripple factor -- ie, how much ripple is 'ok' for the circuit to see. Somewhere around on this forum is a thread where Hero999 replied with a formula. Ah, right, here it is: