15 Reputation

One Badge

0 years, 237 days

MaplePrimes Activity

These are replies submitted by tangentspace


If I take a step back, is it possbile to solve these 2 equations analytically for X, Y, treating all other terms as parameters?

f := (X, Y) -> -(T1 - X)/Rthhot + (X - Y)/Rth + S^2*X*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

g := (X, Y) -> (Y - T2)/Rthcold - (X - Y)/Rth - S^2*Y*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

Please see attached worksheet:



Thank you very much for your help!


Maybe I posed the question incorrectly.  The real variables in this problem is Rolad and Rth.

X and Y are functions of Rload and Rth given by the 2 equations:    

0=-(T1 - X)/Rthhot + (X - Y)/Rth + S^2*X*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2

0=(Y - T2)/Rthcold - (X - Y)/Rth - S^2*Y*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

and S is a function of Rth  is given by 

0=S^2*Rth/R - Z

then we are trying to maximize power as a function of Rload and Rth

Power=S^2*(X - Y)^2*Rload/(R + Rload)^2;

So, fundamentally there are only 2 variables: Rth and Rload.  Can you see if it's still a saddle point if we look at it this way?  When I vary Rth and Rload numerically, it seems the solution is a maximum.

By the way, when I tried to run your code, I got some errors:



That's right, the variables are X, Y, Rload, Rth, S,  and Rth does not directly enter the expression for power, it enters through the constraint equations.

Here is an example:  given  T1 = 310 ,  T2 = 300 , Rthhot = 0.1 , Rthcold = 6,   R = 6 , Z=1/305

The solution I found using Mathematica's FindMaximum or Lgarange multiplier methods is:

Power=0.00230234, X=309.918, Y =304.91, Rload=8.48585, Rth= 8.66666, S= 0.0476431

I checked other solutions near this point, it seems to be the maximum.


Thanks for your heip, but if I assign numerical values to T1, T2, R, Rthhot, Rthcold, and Z, I can solve this problem and get good solutions.


  • "maximum of p" : what is p? is it power?    yes, p is power.  I've corrected it in the main question.
  • Are  [Rth, Rload, S, X, Y] the unknowns of the maximization  problem?  Yes that's correct
  • In such a case why does the expression of power not contain Rth but R? A typo?  Not a typo, R is a fixed parameter.  power contains Rth through the contstraints in equations f,g,h
  • Are the remaining variables a kind of  "parameters" you would like to express the maximiser in terms of?  yes, that's correct:  T1, T2, R, Rthhot, Rthcold, and Z are the paramaters.

Thanks for your help!

Page 1 of 1