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This depends on the circuit in question. If the circuit only has resistors and maybe incandescent light bulbs, then with an equvalent RMS voltage of AC, to the previous DC, the circuit will behave almost the same. If the circuit has components such as capacitors and inductors, then the current will be shifted to flow at a waveform which no longer matches the voltage waveform.

If you're talking about a circuit which was designed to run on a 12 volt battery, then you go and plug it into the wall, it will probably break, as the equivalent voltage causes a much higher current than these components were designed to handle.

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Q: What will happen to the circuit if the power supply is change from AC to DC?
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