The load a curve recommends

The source that is not a source draws a real source as an electromotive force behind one resistance, and everything in that essay follows from the terminal characteristic being a straight line: the edge is a current, the resistance nobody drew can be measured from the slope, and the load that takes the most power is equal to it.

That last result carries a coincidence the straight line hides. For a line through $V_\mathrm{oc}$ on one axis and $I_\mathrm{sc}$ on the other, three quantities are the same number — the internal resistance, the negative of the slope, and $V_\mathrm{oc}$/$I_\mathrm{sc}$ — and the maximum power theorem can be stated with any of the three without anybody noticing which one is doing the work.

The three are not the same number for anything whose characteristic bends, and every source that generates rather than stores has a characteristic that bends. This essay solves one: a photocurrent with a junction across it, a shunt beside it and a series resistance in the way. Its answer says which of the three the theorem was about.

What maximising the product actually asks for #

The derivation is two lines and it is worth doing, because the answer is a condition rather than a formula and the difference matters.

The power into the load is the terminal voltage times whatever current the source delivers at that voltage. At a maximum its derivative with respect to the voltage vanishes, which gives the current plus the voltage times the slope of the characteristic equal to zero, and rearranging that gives

$V/I = -\mathrm{d}V/\mathrm{d}I$.

The left side is the load resistance. The right side is the incremental resistance of the source at that operating point — the negative of the characteristic’s own slope there. So the theorem is: match the load to the slope.

For a straight line the slope is the same everywhere and is the internal resistance, so the condition becomes $R_\mathrm{load}$ = $R_\mathrm{internal}$ and can be solved directly. For a curve it cannot: the slope depends on where the source is sitting, and where it is sitting depends on the load. It is a fixed point rather than a recipe, and the way to find it is to search — which is what what a network answers means by a circuit having exactly one answer when the elements are not linear.

Solved on the panel, the search puts the maximum at 17.244 volts and 4.696 amperes. The load there is 3.6717 ohms and the incremental resistance there is 3.6717 ohms — the same number to four figures, which is the condition being satisfied rather than imposed. The solve maximises the power by golden section on the voltage and computes the slope afterwards by central difference; nothing connects the two except the characteristic.

Two resistances that cross once #

Drawing the two sides of the condition along the characteristic turns the fixed point into a picture, and it shows why there is exactly one answer.

The load V/I starts at zero — at short circuit the source is working into nothing — and rises monotonically to infinity at open circuit. The incremental resistance −dV/dI goes the other way: on this panel it is 200 ohms at short circuit, where the only thing limiting the current is the shunt across the junction, and 277 milliohms near open circuit, where the junction itself is conducting hard.

One rising curve and one falling curve cross once, and they cross at 17.244 volts. That is the whole of the maximum power point on a smooth characteristic: an increasing function meeting a decreasing one.

The spread of the second curve is the reason the phrase “the internal resistance of a solar panel” has no referent. Two hundred ohms to a quarter of an ohm, on one device, at one illumination — a factor of seven hundred, and every value in between is the right answer at some operating point. What the source that is not a source could measure from a slope was a constant. Here the slope is a function, and the number a data sheet would have to quote is the one at the maximum, which is the one place it is not a property of the device alone.

The straight line is wrong about the power and nearly right about the load #

The obvious approximation is to draw a line between the two end points and use it, and its two errors are of completely different sizes.

The power it promises is a quarter of $V_\mathrm{oc}$·$I_\mathrm{sc}$, which for this panel is 24.81 watts. The true maximum is 80.99 — 3.26 times more. That factor is the reciprocal of the fill factor, 0.8160, and it is not a small correction: a design sized on the linear model would be built around a third of the power that is available.

The load it recommends is 3.9709 ohms against the right answer of 3.6717 — eight per cent high, which is a great deal better than a factor of three. And the error stays that small across a twentyfold range of illumination: the ratio runs from 1.055 at half an ampere of photocurrent to 0.915 at ten, crossing one on the way.

So the linear model is a bad model of the source and a passable rule for choosing the resistor, which is a strange combination and has a reason. The load at the maximum is where two curves cross, and the crossing point is insensitive to the shape of either curve away from it; the power at the maximum is an area, and it is sensitive to the whole shape. A straight line and a real characteristic that share their two end points have their crossings near each other and their areas nowhere near.

What the model is, and what it is not #

The characteristic solved here is the single-diode model, and it is worth naming its parts because each of them owns one region of the curve.

A photocurrent generates; a junction across it conducts once the terminal voltage is high enough; a shunt resistance across the pair leaks; and a series resistance sits between all of that and the terminals. Near short circuit the shunt is the only thing that limits anything, so the incremental resistance there is the shunt itself — 200 ohms in the panel drawn. Near open circuit the junction is conducting hard and its own dynamic resistance, the thermal voltage over the current, dominates everything. In between, where the maximum is, both matter and so does the series resistance.

That structure is why the fill factor is a useful single number for a cell and a misleading one for a model. Moving the shunt from 20 ohms to a hundred kilohms takes the fill factor from 0.689 to 0.830 and the maximum-power load from 4.24 ohms to 3.62; moving the series resistance from 5 milliohms to 0.3 ohms takes it from 0.819 to 0.755. Two independent defects, one number, and the number cannot say which.

The current in the model is implicit in itself, through the series resistance: the voltage across the junction depends on the current, and the current depends on the voltage across the junction. So it is found by bisection at every point, on a function that is strictly decreasing in the current and therefore bracketed by construction. There is no starting guess to get wrong, which is the reason to prefer bisection here to Newton’s method on a curve with an exponential in it.

What the model does not contain is worth being equally clear about. It has no temperature in it, so the panel it draws is at one temperature and its open-circuit voltage does not fall as it warms. It has no capacitance, so nothing here has a time in it. And it has one junction, where a real cell’s recombination is a second one with a different ideality — which is the same two-mechanism argument the one current a constant is right at makes about a diode, arriving in a device where the current is being generated rather than supplied.

The peak is not symmetric #

Sweeping the load rather than the voltage draws the quantity a designer actually has a knob for, and its shape is the practical result of the whole essay.

Power against load resistance peaks at 80.99 watts at 3.67 ohms, and falls to half of that at 1.54 ohms below and 10.6 ohms above. On a logarithmic axis that is a factor of 2.4 on one side and 2.9 on the other — not symmetric, and the asymmetry is in the direction that matters.

Below the peak the source is being asked for more current than it has, and it cannot supply it: the terminal voltage collapses along the nearly vertical part of the characteristic and the power falls quickly. Above the peak the source is simply being under-used, and the power falls gently along a nearly horizontal part. A straight-line source is symmetric about its peak on that axis, exactly, and therefore cannot show the difference.

That asymmetry is why a maximum-power tracker approaches the point from the high-resistance side. An overshoot towards a smaller load costs more power than an equal overshoot towards a larger one, and the cost is steeper the further it goes — which is the same statement as the terminal voltage collapsing, seen from the other axis.

Where the condition does not apply #

The derivation assumed the characteristic has a slope. A supply with a folded-back current limit does not, and it is worth drawing because the failure is complete rather than approximate.

Such a supply regulates at its nominal voltage through a small output resistance — a source below a frequency measures what that resistance actually is — until its current reaches the limit, and then reduces that limit as the terminal voltage falls — so that a short circuit draws a fraction of the full limit and the pass device survives. The characteristic is two branches meeting at a corner.

The most power is delivered exactly at the corner: 23.67 watts at 11.901 volts and 1.989 amperes, a load of 5.98 ohms. The incremental resistance just above the corner is 50 milliohms, which is the regulation; just below it the characteristic slopes the other way, so the incremental resistance is negative, −9.23 ohms. The condition $V/I = -\mathrm{d}V/\mathrm{d}I$ is satisfied at neither.

Nothing is wrong with the arithmetic. A maximum that sits at a corner is not a stationary point, and a stationary condition says nothing about it. What the corner is saying is that the design’s limit was placed there deliberately, so the maximum power a folded-back supply delivers is a design decision rather than a property of a source — which is exactly the intent.

And $V_\mathrm{oc}$·$I_\mathrm{sc}$ stops being an upper bound. The short-circuit current has been reduced on purpose, so $V_\mathrm{oc}$·$I_\mathrm{sc}$/4 is 2.10 watts against 23.67 actually available: the linear estimate is now low by a factor of eleven, in the same direction as before and for a different reason.

The negative branch, and what it is for #

The negative incremental resistance in the folded-back region is not an artefact and it is the reason foldback is controversial.

A load whose own characteristic crosses the supply’s twice has two operating points, and the one on the negative branch is stable against small disturbances if the load’s slope is steeper than the supply’s. A motor stalling or a lamp cold is such a load. The supply then sits in a low-voltage state it will not leave when the fault is removed, which is the well-known failure of foldback limiting and which the characteristic predicts without any dynamics being solved at all.

That is a statement about intersections of two curves, which is what what a network answers says a solve is, with the difference that a nonlinear pair can intersect more than once and a linear pair cannot. The refusal that essay describes — a network with no answer — has a companion here: a network with several, and nothing in the solve to say which one the circuit will be found in.

What survives from the straight-line source #

Three of that essay’s results are unchanged and one needs restating.

The edge is still a current. A source stops being a source at a current set by something nobody draws, and on the panel that current is the photocurrent itself — 5 amperes, above which no load at all draws more. The difference is that the approach to it is a curve rather than a straight line, so the “one per cent low” boundary that essay computed has no simple expression here.

The resistance nobody drew can still be measured from the slope, and it is now a different number at every point. Measuring it means stating where, and the only place with a claim to being canonical is the maximum power point — which is a property of the source and the load together.

And maximum power is still at the worst place. That essay’s point was that matching the load for power puts the source at half its open-circuit voltage, which is where its regulation is worst. Here the maximum sits at 0.869 of the open-circuit voltage rather than half, because the characteristic is flat over most of its range; the efficiency argument is weaker and the point still stands, which is the load that takes the most’s subject rather than this one’s.

Two routes, and what they share #

The condition and the maximum are computed by two routines that share only the characteristic, which is what makes their agreement worth anything.

The maximum is found by golden-section search on the terminal voltage, maximising the product of the two directly. It knows nothing about slopes. The incremental resistance is computed afterwards, by a central difference on the same characteristic at the voltage the search returned. The two meet at 3.6717 ohms against 3.6717, to four figures, and the residual is the finite difference’s own error rather than anything about the source.

The same pair on the straight-line source agrees to a part in ten million, which is the check that the the search is not merely finding what it was told to find: there the answer is known in closed form and both routes hit it. And on the folded-back supply they disagree by a factor of sixty, which is the check that the comparison can fail — an agreement that cannot fail is not evidence.

That last case is the one worth carrying out of this essay. Exact outside and wrong within makes the point that a reduction can be exact about one quantity and badly wrong about another; here a theorem is exact where its assumption holds and silent where it does not, and the assumption is differentiability rather than linearity. Linearity was never what the maximum power theorem needed. A slope was.

Still open: two points on the curve, a tracker, and the temperature #

Finding the maximum from two measurements. The fixed point can be reached by iteration from any starting load — measure the power, move, measure again — and that is what every tracker does. What a solve could say and a tracker cannot is how many measurements the fixed point needs: Newton’s method on I + V·dI/dV converges quadratically near the answer, so the question is how far away it has to start before the negative branch or the flat top of the characteristic breaks it. That is a measurement about an algorithm on a curve, and the curve is here.

The characteristic that moves while it is being tracked. A panel’s photocurrent follows the light and its open-circuit voltage follows the temperature, in opposite directions and on different timescales, by the junction coefficient two currents with one name measures. A tracker is therefore hunting a fixed point that is itself moving, and the useful number is how fast it may move before the tracking error costs more than the tracking buys. Nothing here has time in it, and adding it needs the panel’s own capacitance, which this model does not have.

Whether the eight per cent is a coincidence. The straight line through the end points recommends a load within ten per cent of the right one across a twentyfold range of illumination, and the reason offered above — that a crossing is insensitive where an area is not — is an argument rather than a measurement. Sweeping the series resistance and the shunt over the range real cells span would say whether the ten per cent is a property of this characteristic’s shape or of characteristics of this kind, and a rule that held across the family would be worth more than the exact answer for one panel.