I’m looking through Lloyd Dixon’s Control Loop Design
paper at http://focus.ti.com/lit/ml/slup098/slup098.pdf,
and I have a few questions:
1. On page 7-4, I don’t fully
get the equation for Gp. There is a –j in the numerator that seems to come out of
nowhere. I think I see where it comes from, and here’s what I’ve got:
-The calculation for vCA makes sense to me, d*Vs.
-vi, on the other hand, confuses me. It should be a voltage
corresponding to the inductor current.
-Rt is in series with L, so iRt = iL. vi =iL * Rt > vi / Rt = iL
-The inductor can be represented by
vL = jwL * iL. Rearrange to iL = vL / jwL
-Plug the above into the one above
that, and you get vi / Rt = vL / jwL, which goes to vi = (vL * Rt) / jwL
-The inductor always has Vout
across it, Q on or Q off. When Q is on, the inductor also has Vin – Vout across
it. So, Vin * d is the voltage across the inductor.
-Substitute the above into the
equation before it, and you get vi = Vin * d * (Rt / jwL)
-vi / vCA then is (Vin / Vs) * (Rt / jwL)
-Plug in the real numbers, and I get 159Vin / (j * fs), or
–j159Vin / fs. This is the equation shown in the paper.
I hope that’s clear. Ultimately, I believe the equation for
vi, right above the equation for Gp, might need a j in the denominator to be
correct. This also affects the intermediate step of calculating Gp which also
needs a j in the denominator. As it is, it seems to me there’s a –j in the
numerator without an explanation when the j should be present in the proceeding calculations.
2. The last paragraph on page 7-4 mentions the closed loop
gain is flat until it rolls off at fci (16.7kHz). Looking at Fig 4, isn’t the
rolloff actually at fci/2, or 8kHz?
3. Figure 5, on page 7-5, shows Vca leaving 7.0V at 20us. The gain of the CA is 15 above fci, where Cfi isn't having any effect, and the gain increases at lower frequencies. I can understand that the difference between the inputs multiplied by the gain will be the output voltage of the CA. And where the CA output is pegged high, it's equivalent to the loop being open, and Lloyd wrote. But if Cfi is storing extra charge and holding high, why does Vca leave 7.0V before IL*Rt crosses Vset? I'm sure there's something simple I'm just missing, so if anyone could add to the description of Figure 5 and point out where I'm confused, that would be great.
Thanks for any help! Evan
4. Then around the equations on page7-6, I can see lots of scribbles which makes me think perhaps the equation is wrong. For one, I can see that dvi/dt in the numerator is written as .05 but there's an arrow to that number from a hand-written 0.025. The third equation on page 7-4 shows dvi/dt = .025V/usec. So perhaps the equation for Gva on page 7-6 is incorrect? What are the hand-written notes (corrections?) around the Gva equation on page 7-6?
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