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Transformer is used to step down the voltage. Your normal supply voltage is 230V but diodes used in rectifier can't handle that huge voltage hence this voltage needs to be reduced. This is done by transformer. Transformer brings down 230V voltage to say 10V. If you are talking about step down transformer than you should know that step down transformer produces equal voltages although opposite in polarity but equal in magnitude(ie +V and -V) across both diodes. If you don't use it the voltage across one diode may be greater than voltage across other diode. Say non-center tapped transformer produces 10V across diode D1 & -8V across diode D2. In next half cycle, -10V will be produced across d1 & +8V across D2. So in first half cycle current due to forward biased diode D1(as voltage across it is greater) will be greater than the current produced due to diode D2 in next half cycle. So rectified current waveform won't be equal.

The transformer is for isolation so the full wave rectifier can float free of the line hot & neutral wires. The power supply can then define its own ground node without "fighting" the line, this is also much safer for the user. Step-up, Step-down, or one-to-one transformer types are irrelevant to full wave rectifier but are selected by the needs of the application. There are solid state diodes that can handle several thousand volts if the application needs it and vacuum tube diodes that can go even higher.

Full wave bridge rectifiers do not use center tapped transformer secondaries, but require the isolation the transformer provides or diodes will blow out in operation!

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