As mains supplies are sinusoidal AC waveforms, an inverter must generate this as an accurate 50Hz clock signal which is then amplified using a PA stage. A square wave is easily generated but undesirable. A simple resistive load such as a kettle element or tungsten bulb filiments cannot care less about the waveform, it cares only about the RMS (root mean square) voltage. However inductive loads will pruduce high voltage transients (as Fourier analysis will model mathematically), caue a lot of RFI (radio frequency interference) and can even damage other devices. A florescent light run on square wave AC will make a lot of noise! So, you need a sinewave, the nearer a true sinewave - the better. This emulates the waveform produced by an AC generator.
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