Designing a simple dual power supply actually involves a just a center tap power supply and a bridge rectifier along with a couple of high value filter capacitors as shown below.
Simple 12v power supply circuit without transformer.
As the name defines a transformerless power supply circuit provides a low dc from the mains high voltage ac without using any form of transformer or inductor.
Here is the circuit diagram and explanation for transformerless power supply.
We can convert the high voltage ac into low voltage such as 5v 6v 9v 12v dc without using the transformer which is called transformerless power supply.
The transformer is one of the main components in a power supply circuit because we need to convert the higher voltage available in the power point into a lower voltage we needed.
And this series of works using capacitive reactance and resistance are not using this will reduce the heat on the circuit.
To be safe when there is a short circuit series always use a fuse on the input voltage is 220v.
Circuit is simple voltage dropping capacitor of 225j is connected in series with phase line of ac this is a non polarised capacitors so it can be connected.
Here is a series of power supply without a simple transfomer.