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Define "work"
Bob
It should 'work' but it will depend on the component values as to what you get.
R3 (1k) will limit the power and you may get out less than you put in.
A better circuit would use a full H-bridge. DC to AC inverter.
Bob
You will need to specify the power supply and the components.
Fets will be better than npn transistors if they are being driven by logic circuits since they take little power to drive. They are also more efficient.
You can look up H-bridge yourself if you wish. It is a circuit which can change the polarity of the output, often used to reverse DC motors. It will have protection from the case where the two transistors are on for a very short time (shoot through) during switch over. Your circuit does not have this protection which may be a problem.
Yes, shoot through is a term used where one transistor turns on before the other is fully turned off. This may not be a problem if the current is limited and it does not occur often.
An invertor running at 100kHz with a 1microsecond overlap will have problems, one running at 10kHz with the same overlap may not have a problem.
I imagine you're referring to a design that comprises more than your basic 2 NPN's, 2 PNP's.
Chris