Given the high frequency, 1 kHz, small load current, about 16 mA, and largeish capacitor, 100 μFd... calculate the voltage ripple from cycle-to-cycle assuming there is NO diode voltage drop. Is there a "significant" amount of ripple? If not, assume the capacitor charges to the 8 VDC peak of the input voltage and remains at that voltage for the purpose of solving the problem.

The capacitor will charge to nearly the 8 V peak of each cycle, and the current at that point will be the same as if a constant 8 VDC voltage were applied. Immediately after the peak, the diode is reverse biased (why?) and the voltage across the capacitor begins to decay until, almost a full cycle later, the sinusoidal input voltage becomes greater than the remaining voltage across the capacitor plus the diode forward-conduction voltage drop.

At this point the diode is no longer reverse biased but becomes forward biased and begins to conduct current, recharging the 100 μF capacitor and providing current to the rest of the circuit. The current through the diode, and its forward voltage drop, both continue to increase until the peak of the AC input occurs, but the rate of increase slows down because of the "rounded" peak of the half-cycle AC input. Unless a "significant" amount of current is drawn during the forward conduction phase of the diode, you can assume that the capacitor charges to the peak voltage of the AC input and ignore the forward voltage drop across the diode.

That assumption depends on the amount of ripple you calculated earlier. If there is a heavy load, or the capacitor is too "small," the forward voltage drop across the diode becomes a significant factor and the ripple voltage will be a significant percentage of the output voltage. It becomes a "non-linear" problem then and analytical solutions become much harder to find. When this happens in the real world, I just add more capacitance until the ripple becomes insignificant. There is a "rule of thumb" that many people use to calculate the capacitance value needed for power supplies operating from the full-wave rectified 60 Hz power line. Roughly, 1000 μF per ampere of power supply output current is sufficient for most applications, depending on the amount of tolerable ripple.

Here is an Internet website that provides a more in-depth explanation.