The size of the transformer depends on two things: maximum power and AC frequency. The grid (both 110 and 230 V) has very low frequency (50 or 60 Hz), so powerful transformers must be big and heavy, if they want to plug directly into the grid.
But modern switching power supplies do not plug transformers directly into the grid. They convert the grid AC into 110/230 V DC, and then back into AC, but with a much larger frequency (~10000 Hz). That allows them to use small transformers.
The 110 V or 230 V is irrelevant to the problem. In fact, many 230 V switching supplies can work with 110 V as well – they will automatically regulate the output voltage. For ex., my laptop supply says “INPUT: 100-240 V, 50-60 Hz”. So both 110 V and 230 V are within its working range.
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