When a relay is connected to a purely capacitive load and an AC supply voltage is turned on, it will experience inrush current. Since the potential difference of an AC voltage can be represented by $$V_0sin(\omega*t)$$ then I wonder if the relay is turned on only during the zero crossing, where $$sin(w*t)=0$$
Does this help to reduce inrush current as compared to the instance $$sin(\omega*t)=1 or -1$$
A Zero Crossing Switch, ZCS circuit is a common solution to minimize the stress of step voltage in random phase for rectified voltage into large electrolytic capacitors. The surge current is limited by instant voltage / the loop series R will be significantly reduced unless mitigated by some other means. Although AC power relays are defined by the MUST switch voltage threshold, the nominal threshold might be 8 ms but the tolerance is not reliable for use as a ZCS.
There exists Optocoupled Triacs for low current triggers to power Triacs that are inexpensive and Solid State Relays which include the power Triac and heatsink, which tend to be expensive. There are many other ways such as using a diode bridge to transistor switch for a negative ZCS pulse using an NPN. With a 2nd inverting switch a +ve ZCS pulse from a low power DC voltage can be made. In logic levels an XOR gate with the sine wave squared and a few us delay in the other input will also create a ZCS pulse.
Here is a simulation for a relay closing at zero degrees, 125 A peak:
The peak current is limited because the input AC voltage starts at zero and charges the capacitor with a lower current over a quarter cycle (5 ms time for 50 Hz)
And for 90 degrees, 423 A peak:
Initial surge current is limited only by the total series resistance of the circuit (about 800 mohms) and the full peak applied voltage (350 V). This includes diode resistance as well as ESR of the capacitor.
