What is difference between half-wave and full-wave rectifier ?
Half-wave only show the positive part of the sine wave?
After I add a capacitor in parallel to the diode circuit, why I get this result(figure)?

Thank you

 

Full wave uses the positive and negative portion of the AC input. Half wave only uses one portion of the AC input and blocks the other half.

 

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After I add a capacitor in parallel to the diode circuit, why I get this result(figure)?

The capacitor charges to the peak voltage (minus the diode forward drop) and retains the charge when the diode stops conducting.
During the time the diode is not conducting, the capacitor provides current to the load, causing the observed droop in the voltage as the capacitor discharges.
The amount of the droop is determined by the capacitor size and the load current.