Power factor correction is a fairly understandable process for normal sine wave mains devices. Most loads are a combination of inductive and resistive, the supply has to supply an in phase current to the resistive bit and a reactive current to the inductive bit. The total line current is the vector sum of both and the increased current causes increased I^R losses in the supply cables and the power station alternator. A capacitor can neutralise the inductive component and just leave the in phase resistive bit at unity power factor.
Once you get into the realms of non linear loads such as rectifiers then power factor still remains as the definition of true power/ VA but the more common description of V x I x cos ( phi) no longer applies. The low power factor with non linear loads results from waveform distortion rather than a phase shift between volts and amps. This can't be corrected by just adding capacitors across the line. This is really where it becomes a black art.
There can sometimes be an improvement even in this case but the improvement comes about in various other ways and it doesn't always work out. Capacitors are not going to directly correct the power factor in this case of distorted waveforms but they will correct any residual inductive components that are in the system from transformer magnetising current or other stray things.
The other thing that capacitors do across an alternator is increase its excitation and this may be the factor that brings you most benefit. Rectifiers produce large current spikes that chop off the peaks of the alternator waveform resulting in a high volt drop during this critical period depending on the impedance of the alternator. Anything that can make the alternator stiffer during this current peak will help.
The scope is not going to tell you a lot but you should be able to see the chopped top to the waveform and anything that holds the peak up is going to benefit you, the rest of the waveform is of no relevance and improving it to make it more pretty will have no effect.
You can make a far better single phase battery charger for running from alternators by including a large choke in the dc output from the rectifier. This forces the conduction to take place over the complete half cycle instead of on the peak and the impedance has less effect.
To do this you need a higher voltage transformer as you are going from peak volts to mean. You will need a transformer capable of supplying about 1.5 times as many volts
and the choke needs a lot of inductance and negligible resistance ( a big one).
Hope this helps.
Flux
