Why modified sine wave? The disadvantage of rectangular inverters is (among other things) that the amplitude of the output voltage is only 230VDC, not 325VDC like in the 230VAC sine wave. Effective (RMS) value therefore corresponds to the network, but the amplitude differs considerably. The problem may occur especially in case of appliances with rectifier, where the rectified voltage will be about 30% smaller than when operating from the mains. Sine wave inverters are very complicated. Therefore a compromise between the rectangle and sinus is usually chosen, which is so called modified sine wave. It's a little misleading term because it differs from the sine wave a lot :). But the point is, that modified sine wave has the same effective (RMS) value and also the amplitude as a sine. Hence it is used by most factory manufactured inverters. The simplest way to generate the proper modified sine wave is using two halfbridges with a phase shift of 25%. The output voltage is therefore tri-state, switching in the order: +325V, 0V, -325V, 0V. The transformer is AC 230V / 9V and must be designed for the maximum required load. For higher power it is necessary to use adequate heat sinks (and MOSFETs that can be attached to them :)). If you use common heat sink for all MOSFETs it is necessary to use insulating pads. You can also connect multiple MOSFETs in parallel. The 12V supply must be sufficiently hard, the supply voltage should be in the range of 9 - 14V. If the supply voltage is lower than 9V, the control circuit IR2153 shuts down, preventing damage to the battery, inverter or powered appliance. At the input it is necessary to use a proper fuse.
