Solve two equations for one variable

Hello everyone!

I am having problems to solve a simple system of two equations for a single variable. The equations look like:

It is possible to analytically calculate the solution for through the following steps:

• Isolate in the second equation and replace it in the first equation:

• Then using the fact that

• Then, isolating :

• Finally:

Using MATLAB Symbolic Toolbox, I wrote the following piece of code in order to find the same analytical solution:

syms P_st vdq_st_abs Vabs delta_st Vangle Xg Q_st
eq1 = P_st - vdq_st_abs * Vabs * sin(delta_st - Vangle) / Xg == 0;
eq2 = (Q_st * Xg - Vabs^2)/(Vabs * cos(delta_st - Vangle)) - vdq_st_abs == 0;
eqs = [eq1; eq2];
sol = solve(eqs, delta_st);

Warning: Solutions are parameterized by the symbols: z. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

disp(sol)

z

However, the output is parameterized, and I do not understand it very well. Could someone explain me this parametrization? Also, why can't MATLAB output the same analytical solution I just mentioned?

Thank you very much in advance!

Answer 1

[Torsten]

syms P_st vdq_st_abs Vabs delta_st Vangle Xg Q_st real
eq1 = P_st - vdq_st_abs * Vabs * sin(delta_st - Vangle) / Xg == 0;
eq2 = (Q_st * Xg - Vabs^2)/(Vabs * cos(delta_st - Vangle)) == vdq_st_abs;
eqn = lhs(eq1) * lhs(eq2) == 0;
sol = solve(eqn,delta_st,'ReturnConditions',1)

sol = struct with fields:
      delta_st: [2×1 sym]
    parameters: k
    conditions: [2×1 sym]

sol.delta_st

ans = 

sol.conditions

ans = 

[Vitor Eiti Uekawa]

Thank you very much for your help! However, it is still not clear why does the output solution is not the same as the analytical solution that I provided in my question.

[Torsten]

sin^2 and 1-cos^2 also look different, but are equal ...

Did you test the results with numerical values for the parameters ?

I wonder how the two equations can have a common solution for delta_st. You can solve each equation separately and you get two different results. I think you must have misunderstood something:

P_st = 1.0;
vdq_st_abs = 1.0;
Vabs = 1.0;
Vangle = 1.0;
Xg = 1.0;
Q_st = 1.0;
delta_st = Vangle + atan(P_st/(Q_st+Vabs^2/Xg))

delta_st = 1.4636

P_st - vdq_st_abs * Vabs * sin(delta_st - Vangle) / Xg

ans = 0.5528

(Q_st * Xg - Vabs^2)/(Vabs * cos(delta_st - Vangle)) - vdq_st_abs

ans = -1

Thus none of the equations gives 0 for delta_st = Vangle + atan(P_st/(Q_st+Vabs^2/Xg))

[Torsten]

My guess is this is what you want:

syms P_st vdq_st_abs Vabs delta_st Vangle Xg Q_st real
eq1 = P_st - vdq_st_abs * Vabs * sin(delta_st - Vangle) / Xg == 0;
eq2 = (Q_st * Xg - Vabs^2)/(Vabs * cos(delta_st - Vangle)) == vdq_st_abs;
S = solve([eq1,eq2],[delta_st,vdq_st_abs])

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

S = struct with fields:
      delta_st: [2×1 sym]
    vdq_st_abs: [2×1 sym]

simplify(S.delta_st(1))

ans = 

simplify(S.delta_st(2))

ans = 

[Vitor Eiti Uekawa]

Thank you very much for your help. I didn't try to test the results with numerical values, but since you already stated it doesn't give the expected solution, it is very likely I don't have enough understanding of my equations. However, I edited the original post to show that the expression I mentioned is indeed a solution for both equations. Maybe there is a calculation mistake, or this is one of the multiple possible solutions?

Of course, if I use the last equation of my step-by-step, MATLAB is able to find the desired solution, but I wonder if this is the only way of finding this solution.

syms P_st vdq_st_abs Vabs delta_st Vangle Xg Q_st real
eq1 = P_st - (Q_st + Vabs^2/Xg)*tan(delta_st - Vangle) == 0;
sol = solve(eq1,delta_st)

sol =