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There are two reasons for your results. The first relates to the operation of the transformer:

All Transformers are rated to deliver a maximum current. At that current, the output voltage will be the rated voltage. A transformer that has an output rating of 12V and 1A will deliver a voltage of 12 volts when the current drawn is 1 amp. If the current drawn is less than 1 amp, you can expect the voltage to rise by up to 10% when no current is drawn.

The second reason relates to the way AC voltage is measured. 12V AC is an RMS voltage. This represents the "average" voltage. (note that the term average is incorrect but will suffice for this answer). The waveform actually peaks higher than the average level. For a 12V AC voltage the peak voltage is close to 17V.

When the smoothing capacitor is added to the output of the bridge rectifier, current will flow into the capacitor to charge it until the capacitor voltage reaches the peak voltage from the rectifier. In this case, the peak is 16.8V minus 1.2V (the voltage drop across two diodes in the rectifier) to give a DC voltage of around 15.5V DC.

As soon as a load is added to the power supply, the capacitor voltage will discharge on each cycle. At each peak voltage from the rectifier, it will charge up again. The DC voltage will actually look like a shallow sawtooth when examined on an oscilloscope.

There are several ways to obtain a 12 volt DC output from the power supply. The simplest is to use a 3 pin regulator IC called a 7812. There are several versions for different output current and input voltage variants. The output of a 7812 will be held at 12 volts regardless of the input voltage variation.

When using a voltage regulator, look up the data sheet for it. Each one has guidelines on using them. The guidelines are simple but important so don't ignore them.

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Q: I am building a simple 12VDC power supply with a 12VAC transformer and full wave bridge rectifier Whenever I add the filter capacitor I get a voltage of around 15VDC too far from 12V why is that?
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