Maximum power delivered by a solar cell to a resistive load

[pc2-brazil]

Homework Statement

The current-voltage characteristic of a photovoltaic energy converter (solar cell) shown in the attached figure can be approximated by:

$$i = I_1(e^{v/V_{TH}} - 1) - I_2$$

where the first term characterizes the diode in the dark and I₂ is a term that depends on light intensity.

Assume $I_1 = 10^{-9}$ and assume light exposure such that $I_2 = 10^{-3}\ A$.

If it is desired to maximize the power that the solar cell can deliver to a resistive load, determine the optimum value of the resistor. How much power can this cell deliver?

Homework Equations

The Attempt at a Solution

This question doesn't provide any value for V_TH, but the book mentions that diodes tipically have V_TH = 0.025 V, so I assume it is the value to be used here.

Applying KVL to the attached figure:

$$v+Ri=0$$

The question asks for the value of R for which $P=Ri^2$ is a maximum, so I suppose I should differentiate P with respect to R. However, I first need to solve $v+Ri=0$ and $i = I_1(e^{v/V_{TH}} - 1) - I_2$ simultaneously in order to find i in terms of R, and it doesn't appear to be possible analytically.

Any hint on how to continue?

Thank you in advance.

 

[Simon Bridge]

Your problem is that you want to do $\frac{d}{dR}i(R)=0$ on $$i(R)=I_1\left ( e^{-iR/V_{TH}} -1 \right ) -I_2$$ ... won't it do that implicitly by the chain rule?
(could be wrong - in which case I'd look for approximations)

 

[pc2-brazil]

Your problem is that you want to do $\frac{d}{dR}i(R)=0$ on $$i(R)=I_1\left ( e^{-iR/V_{TH}} -1 \right ) -I_2$$ ... won't it do that implicitly by the chain rule?
(could be wrong - in which case I'd look for approximations)

Thank you for the answer.

I think that I arrived at a solution. I tried to do it as follows:

Since the voltage across the resistor is $-v$ (in which I put a minus sign to follow the associated variables convention) and the current through the resistor is $i$, the power consumed by the resistor is $P = -vi$.

$$\frac{dP}{dR} = -v\frac{di}{dR} - i\frac{dv}{dR}$$

I look for a maximum when the above expression is zero:

$$v\frac{di}{dR} = -i\frac{dv}{dR}$$

$\frac{di}{dR}$ can be expressed as:

$$\frac{di}{dR} = \frac{di}{dv}\frac{dv}{dR} = \frac{I_1e^{v/V_{TH}}}{V_{th}}\frac{dv}{dR}$$

Substituting that into $v\frac{di}{dR} = -i\frac{dv}{dR}$, I get:

$$v\frac{I_1e^{v/V_{TH}}}{V_{th}}\frac{dv}{dR} = -i\frac{dv}{dR}$$

So, the $\frac{dv}{dR}$ ends up cancelling out. Since i is expressed in terms of v, I got an expression that involves v only, which can be solved for v. Doing this, I get that v is approximately 0.28 V. Plugging this into the expression for i, I get that i is approximately -0.00092 A.

Thus, from v = -iR, the resistance is approximately 307.64 ohms, and power consumed is P = -vi = +0.26 mW.

Now I'm left with proving that this is actually the maximum value for P. But I guess that this is reasonable, because P = 0 for R = 0, and R is not bounded to the right, so it appears the maximum can't be at an extremum.

Is this correct?

 

[rude man]

Happy to corroborate your answer. Nice work! My approach was a bit different: I solved for R(i), then extremalized (i^2)R(i) using d(i^2)R(i)/dt = 0. Had to use Excel to arrive at your number for i (transcendental equation).

It's a maximum for exactly the reason you gave.

 

[pc2-brazil]

Happy to corroborate your answer. Nice work! My approach was a bit different: I solved for R(i), then extremalized (i^2)R(i) using d(i^2)R(i)/dt = 0. Had to use Excel to arrive at your number for i (transcendental equation).

It's a maximum for exactly the reason you gave.

OK, thank you for confirming it and for providing an alternative approach.

 

[rude man]

OK, thank you for confirming it and for providing an alternative approach.

OK, and of course I meant to say (d[i²R(i)/dR = 0.

Also BTW I used the approximation exp(v/V_Th) - 1 ~ exp(v/V_Th). In this case exp(0.28v/0.026V) = 47535 which certainly >> 1. You can almost always do that for a diode i-V characteristic.