Final answer:
Three-phase half-wave rectifiers provide a smoother output voltage than single-phase rectifiers due to phase differences reducing voltage ripple. For a transformer with 12 input coils, 6 output coils are needed to halve the output voltage.
Step-by-step explanation:
Regarding the comparison between the output voltage of three-phase half-wave rectifiers and that of single-phase rectifiers, the output voltage of three-phase half-wave rectifiers is indeed smoother than the output voltage of single-phase rectifiers. This is due to the fact that a three-phase half-wave rectifier combines the outputs from three different phases which, in their nature, already have a phase difference between each other. This phase difference allows for a reduction in voltage ripple and results in a smoother output voltage in comparison to the single-phase rectifier, which only rectifies a single-phase AC voltage and thus has larger ripples.
For the laminated-coil transformer problem, where one side has a wire coiled 12 times, to get a voltage output that is one half of the input voltage, you would need six coils on the opposite side. This is option a - six coils because the ratio of output to input output voltage is the same as the ratio of the number of output coils to input coils.