Using fminsearch to solve an equation - matlab

Question

(vgb-phy_s)^2=G^2*phy_t*((exp(-x)+x-1)+exp(-(2*phi_b/phi_t))*(exp(x)-x-1))
where
x=phy_s/phy_t
phy_t=0.0288; % phy_t=k*T/q; (k=1.3806503*10^-23, T=300 K, q=1.6*10^-19)
phy_b=0.5267; % phy_b=phy_t*ln(Na/ni)
G=(sqrt(2*q*es*Na)/cox);
Here I need to plot phy_s for different values of vgb.
I tried many ways but since I'm new to matlab I'm on my learning process, I'm not able to find a proper solution.
Few people suggested me to use fminsearch but its quite confusing and I'm getting lot of errors.

Answer 1

fminsearch is a function for finding a minimum of a function, not for finding a solution of an equation. Further, here you don't have one equation but an equation group of at least 5 equations. You can use solve to solve equations and equation groups. However, an equation group of the following equations 1-5 does not have explicit solution. Another issue is that the constant values you propose seem to be imprecise values, and if you have more than one rounded or otherwise imprecise value, you can not find a solution even if the equation group was solvable (however, this equation group does not have [an explicit] solution)).
So, I'll show the steps to solve this but there seems to be something wrong with this equation group, even if the [probably imprecise] constant definitions (phy_t=0.0288; phy_t=k*T/q; (k=1.3806503*10^-23; T=300; q=1.6*10^-19; phy_b=0.5267;`) were left out.
Equations (without constant definitions):
1. (vgb-phy_s)^2 = G^2*phy_t*((exp(-x)+x-1)+exp(-(2*phi_b/phi_t))*(exp(x)-x-1))
2. x = phy_s/phy_t
3. phy_t = k*T/q
4. phy_b=phy_t*ln(Na/ni)
5. G=(sqrt(2*q*es*Na)/cox)
To solve eg. equation group of equations 1, 2 & 3:
Solution = solve('(vgb-phy_s)^2 = G^2*phy_t*((exp(-x)+x-1)+exp(-(2*phi_b/phi_t))*(exp(x)-x-1))', 'x = phy_s/phy_t', 'phy_t = k*T/q');
Solution.q
ans =
(T*k)/phy_t
(T*k)/phy_t
Solution.vgb
ans =
phy_s + (G*phy_t^(1/2)*(exp((2*phi_b)/phi_t) - exp(phy_s/phy_t) + exp((2*phy_s)/phy_t) - exp((2*phi_b)/phi_t)*exp(phy_s/phy_t) - (phy_s*exp(phy_s/phy_t))/phy_t + (phy_s*exp((2*phi_b)/phi_t)*exp(phy_s/phy_t))/phy_t)^(1/2))/(exp((2*phi_b)/phi_t)^(1/2)*exp(phy_s/phy_t)^(1/2))
phy_s - (G*phy_t^(1/2)*(exp((2*phi_b)/phi_t) - exp(phy_s/phy_t) + exp((2*phy_s)/phy_t) - exp((2*phi_b)/phi_t)*exp(phy_s/phy_t) - (phy_s*exp(phy_s/phy_t))/phy_t + (phy_s*exp((2*phi_b)/phi_t)*exp(phy_s/phy_t))/phy_t)^(1/2))/(exp((2*phi_b)/phi_t)^(1/2)*exp(phy_s/phy_t)^(1/2))
Solution.x
ans =
phy_s/phy_t
phy_s/phy_t
Do note that this solution is valid only for equation group of equations 1-3. For example equation group of equations 1, 2, 4 or 1, 2, 5 gives a different solution.
To solve the equation group of all 5 equations you could use this:
Solution = solve('(vgb-phy_s)^2 = G^2*phy_t*((exp(-x)+x-1)+exp(-(2*phi_b/phi_t))*(exp(x)-x-1))', 'x = phy_s/phy_t', 'phy_t = k*T/q', 'phy_b = phy_t*ln(Na/ni)', 'G = sqrt(2*q*es*Na)/cox');
However, there is no solution:
Warning: Explicit solution could not be found.
In solve at 160
Solution =
[ empty sym ]
So, I suggest that you try to find out what is wrong with your equations and then try solve again with corrected equations.