Application and Limitation of Fourier Analysis

[I_am_learning]

Suppose I have a non-linear load in my home (A half wave rectifier supplied DC load, say).
Since it will consume current from the source only during +ve cycle of the Voltage, the current will be half-wave too. The current isn't sinusoidal.
We can mathematically say that
Distored Current = Algebraic Sum of Various Sine Wave currents with various Frequencies.
Which we call the Fourier Analysis.
But where can we apply this?
I don't think we can say,
Power Consumed By load = Sum of (Irms ^ 2 ) * R of all sine wave components.
(This we can't do because, superposition principle don't work when we square the quantity)
We can however do,
Poser Consumed by load = Sum of (Vsource rms * Irms * Cos(phi)) of all sine wave components.
(Because superposition holds here, because the current isn't squared)

Now, How can this harmonic analysis aid me in finding the losses in the source and cables.
For the same reason, I don't think we can say,
Power lost in source and cable = Sum of (Irms ^ 2)* (Resistance of source+cable) for all sine wave components.
How do we find the losses then?
To summarize,
Two load current waveforms can have same RMS value, but one with lots of harmonics and other with very less harmonics.
From the point of minimizing the losses in source and cables, which one would be better.
Why and how?

 

[sophiecentaur]

Not a trivial theoretical problem when the system is non linear and dispersive unless you can characterise the system accurately..
The best answer would be to sample V and I, instantaneously over a cycle and integrate V times I. That would be doing what you really want - i.e. summing the infinitesimal values of energy over the cycle, to give the total energy. Actually, that would be one of the easiest solutions, too because DSP at that sort of frequency is a piece of cake.

 

[I_am_learning]

Not a trivial theoretical problem when the system is non linear and dispersive unless you can characterise the system accurately..
The best answer would be to sample V and I, instantaneously over a cycle and integrate V times I. That would be doing what you really want - i.e. summing the infinitesimal values of energy over the cycle, to give the total energy. Actually, that would be one of the easiest solutions, too because DSP at that sort of frequency is a piece of cake.

Thanks for the reply. I agree, that's the best method to find the true power.
But my real question here is quite different. Perhaps It would be better if I rephrase it like this,
I have a resistive DC load.
I want to dissipate say 10 KW in it.
If I directly hook this resister to AC or Full wave bridge rectifier, it will dissipate way lot more power than 10KW.
So I want controlled voltage source, that will reduce effective Vrms across this load so that I can control its power.
I can use Power Electronics to switch the device on/off. The switching can be done in various patterns yet all of them giving the same Vrms.
For eg. see this

By carefully designing the controlled rectifier, it can be so made that both scheme 1 and scheme 2 provide the same Irms or Vload rms,
In the point of view of effects on source and cables, which scheme is better?
It appreas that, the losses in source and cables resistance should be equal in both scheme because, Irms is same.
But, if we do Fourier analysis harmonics contents are different on those two schemes.
Has harmonics got anything to do here regarding the losses?
If they have nothing to do, what's their significance, and why reduce them?
Thanks