Electronic – How does the QTR-8RC capacitor charging works

Capacitors do store charge. In fact, that's basically what a capacitor does, with the added characteristic that its voltage will be proportional to the amount of charge it has stored.

A capacitors doesn't store AC, but it does store whatever charge is on it given the voltage at the time it was disconnected. Since the AC voltage can vary from zero to fairly high peaks, it is somewhat random what the capacitor will be charged to. The peaks of the AC line is the square root of 2 times the RMS voltage. For example, 115 VAC has peaks of ±163 V. A capacitor could get charged to anywhere in that range.

The current when charging a capacitor is not based on voltage (like with a resistive load); instead it's based on the rate of change in voltage over time, or ΔV/Δt (or dV/dt).

The formula for finding the current while charging a capacitor is:

$$I = C\frac{dV}{dt}$$

The problem is this doesn't take into account internal resistance (or a series current-limiting resistor if you include one) or if the capacitor already has some charge.

You have to account for the continually changing charge being applied to the capacitor. In other words, at the very beginning, it looks like a short circuit to your power supply (barring resistance, again). Thus, whatever maximum current your power supply can handle is the theoretical max current. As the capacitor charges, this current decreases exponentially, until the capacitor reaches max charge Q.

The formula for this is:

$$I = \frac{V_b}{R}e^{-t/RC}$$

Where \$V_b\$ is the source voltage, R is resistance, t is time and RC is the time constant (product of resistance and capacitance).

Let's say you don't use a current-limiting resistor and your power supply has an internal resistance of 4Ω:

$$I = \frac{12}{4}e^{-0/0.0132}$$

At time 0 s, the current is 3A. If we figure for, say, 1 ms later:

$$I = \frac{12}{4}e^{-0.001/0.0132}$$

Now the current is ~1 A.

So, how long will it take to charge the capacitor? If you take the time constant, RC (the 0.0132 in the exponent) as a value in seconds, there's a rule of thumb that a capacitor will be charged in 5 times this duration:

$$5\cdot0.0132 = 0.066s$$

The initial current (or the current during some portion of this duration) is referred to as the inrush current. You may want to reduce it by adding a series current-limiting resistor to protect your power supply.