It'd be much easier if you could reach across that diode and just apply your 120VAC to a 2:1 step-down transformer.
How small a motor? How much current is available on the half-wave rectified DC?
I'm not sure that there's an off the shelf device that could do this (unless the suggestion of a UPS pans out). Filtering the DC and then converting it to AC may work. How you'd do that without building most of a UPS, or hacking it, I don't know.
Filter the DC, find a DC-DC converter that can go from that raw DC to
12V, plug the output of that converter into the battery port of a UPS, plug your motor into that. It's way overkill, but it may work.
Use the DC to actuate a relay, that switches a 2:1 stepdown transformer, plug the thing into the wall...
If it's really half an AC wave and it stays that way no matter how you load it, then putting in a honking big blocking cap and applying it directly to the motor may work. It may not, and if it did it may be more expensive than custom electronics in quantities more than 10 or so, but it may work.
For a small enough motor, I think it is. (This would not perhaps be practical for production.) :)
Push the half wave into a parallel 'tuned tank' composed of a power transformer primary in parallel with a lot of 'motor run' capacitors. You have a resonant transformer that synthesizes the negative half cycles to create a full A.C. sine wave output on the secondary.
Plug 7.036201 microfarads and 1 Henry into this calculator to see how close you can get to 60 Hz:
Wots your REAL Problem? This sounds like a silly thing to want to do so you will only get silly answers. If your starting SIGNAL is only a SIGNAL i.e. no real power, then you are not going to get anywhere without some form of amplification to get the power and in that you could generate whatever type of output you need.
Hypothetical questions are very difficult to answer. Please tell us your REAL problem. :-?
I suppose you could construct a bridge that would pass the ON time of the 120v to the motor lead while the other lead is connected to the common. Mean while as this is taking place, it is also charging a large cap.. WHen the incoming sine wave drops near 0 volts you then would have a triggered circuit to switch that bridge so that now the charged cap is in a reverse manner and thus generating a (-) signal. The start of the sine wave regenerating from the incoming will simply shut this switch off and go back to repeat the cycle.
Actually, as I think about it, you wouldn't even need a full bridge, a half bridge would do it, you only need to short the (+) of the cap to the common lead of the motor and open the common of the cap from the common of the system coming in. You can do this because the DIODE you are using will block the (-) coming in and there for, shorting this will not cause any unwanted blue smoke!
If you look at charge pump voltage generating circuits, you can see where opposite polarities of power can be obtained. Many times
Usualy you need an inverter to get AC from DC. someone else suggested a UPS, Common Uninterruptable power supplies use an inverters to convert DC from a secondary battery to AC in th event of a power falure.
If your DC supply is also poweing a DC load you may be able to extract a small amount of AC by puttign a choke in series with the rectifier || +---||---------- | || | AC load | ----->|-+--wwwww--+----- ===== | | in ===== DC load | | -----------------+-----
The waveform that the AC load sees will have lots of harmonics and the voltage will vary depending on the size of the DC load and the AC load
it may be a easier idea change to a motor that can handle DC (eg BLDC, PM, or universal)