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I would like to place a non-polarized capacitor in between V1 and R1. Is it possible to replace the non-polarized capacitor(ceramic capacitor) with two polarized (electrolytic capacitors) C1 and C2 shown in the diagram above?
It looks like you need to flip one of the diodes and put sign indicators on the capacitors so that one is active during the positve half cycle, and the other during the negative half cycle. You could simply use one capacitor and a bridge rectifier like most power supply designs. But the previous comment is on-target. What is it you are trying to accomplish?
No. You have drawn both capacitors in identical roles. In both cases they will be biased with the positive electrode beside the diode. But that is fine, the circuit is inherently polarized. As you have drawn it the circuit will simply top up the capacitors to near the positive peak of the incoming sine wave, minus the 0.6V or so forward bias of a typical silicon diode. Then it will sit there doing nothing except for the faint trickle of capacitor leakage balancing the faint clipped peak of the input. The leakage is so faint that the 0.6V will slowly reduce and the capacitors will get close to the positive incoming voltage until the trickle through the diode balances the capacitor leakage, which might be as little as 0.3V if the capacitor is good and the diode is large.
This is what the circuit would do with a ceramic cap too, if you could find one that large. MilliF caps in ceramic would be gigantic, if anyone makes them.