Right, it's an effective approximation because:
1. We don't want loud ass power supplies, we have enough filtering (C) to get the ripple down;
2. The filter cutoff frequency is a small fraction of f_SW.
So during a cycle, very little happens at the filter output (>30dB attenuation, say), and if we like, we can approximate that waveform as a parabolic wave (think: two parabolic sections glued together, so as to approximate a sine wave), and do a small correction to inductor current as a result, and this is fine.
The output will be strongly phase shifted (180°) to the switch node, so we don't even have to worry much about superposition, it's just... expected square voltage making triangle current ripple, plus a small cosine (270°) correction to bring in the output ripple.
And yeah, technically there are corrections to the corrections. It's not corrections all the way down; it's... well, you can write it that way, in fact, but easier to solve the simultaneous equations and be done with it -- if you must.
This is a greatly welcome alternative to calculating the whole thing, which involves a piecewise source (square wave) driving a 2nd order (or worse) differential system; we can do this in the frequency domain fine as well, but neither is going to be particularly simple, and we don't care a damn about the exact waveforms' value over time (i.e. what exact functions and parameters are in v(t), i(t)), as long as it's doing the basics -- so, the approximation also captures our intent, in a sense.
...To say nothing of cap ESR, of course; but that adds some triangle voltage to the output, so, parabolic current to the inductor. This is a higher order effect (1st not 2nd) so could be worth including, but again, the same premise applies: as long as the current is accurate within, like, even 10% is precise for this -- even just, more that the slope is within say 2:1 of what it should be, and is smooth so peak-current control for example is applicable -- and again, as long as output ripple is usefully low, we don't care, it's fine, all those correction terms (or, for exact solution: the simultaneous equations solving for the corrections of corrections) are down in the noise, we don't care.
Aside: the idea of adding correction factors, and corrections on corrections, is a similar process as solving for the value of a geometric series. If we take
S = 1 + x + x^2 + ...
then we have
S - 1 = x + x^2 + x^3 + ...
S - 1 = x(1 + x + x^2 + ...)
S - 1 = xS
Solving,
S = 1 / (1 - x)
(which is true for -1 < x < 1.)
It's like saying, if you know one fraction, say 1/10 = 0.1, and want to know one-off from that say 1/11, and want to avoid doing a whole-ass division, then, you can get closer by subtracting the ratio of the divisor (i.e., 11 is about 1/10th larger than 10, so subtract 1/(10*10), etc.). But that underestimates the result (1/(0.09) ~ 11.1...), so add back in the same but even littler, but then...... and so on. Which, for a simple decimal expansion like this (one off from a nice round power of 10), it's pretty apparent why the decimal expansions of 1/9 and 1/11 repeat so simply (0.111..., 0.0909...).
With respect to differential equations, it might be that we can solve for them easily (in this case, we can -- easy being a relative term!), or it might be that we can't feasibly solve the simultaneous equations at all, and our only tool is an infinite series, an approximation, or a perturbation method. The functions being operated upon, are obviously much more complex, but the basic mechanics are still common to them.
Tim
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