How to solve a complex-valued equation

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Ajay Gupta
Ajay Gupta 2018-12-2
编辑： Bruno Luong 2018-12-2
I'm trying to use fzero, but it seems that it can't handle solving for a variable that's complex. Here's my code:
E = 169e9; % Young's Modulus of Silicon [N/m^2] (Assuming one that's isotropic)
W = 2e-6; % Width of Cantilever Beam [m]
H = 2e-6; % Height of Cantilever Beam [m]
L = 5e-6; % Length of Cantilever Beam [m]
V_dc = 5; % DC Voltage [V]
V_ac = 2; % AC Voltage Magnitude [V]
omega = [80e6, .01e6, 110e6]; % Angular Frequency of AC Voltage [rad/s]
g = 5e-6; % Initial Gap between Cantilever Beam and Substrate/Switch Terminal [m]
A = W * H; % Overlap Area between Cantilever Beam and Electrodes [m^2]
e_0 = 8.854e-12; % Permittivity of Free Space [F/m]
k = (E * W * H ^ 3) / (4 * L ^ 3); % Spring Constant of Cantilever Beam [N/m]
% Setting dU/dx equal to kx, and solving for x
for k1 = 1:length(omega)
omega_val = omega(k1);
f = @(x) (1 / 2) * 1i * omega_val * ((e_0 * A) / (g - x) ^ 2) * (V_dc + V_ac) ^ 2 - k * x; % Setting up equation to solve
x(k1) = fzero(f, 0);
end
Basically, omega is a vector of some length with real values, and all the other variables are real as well. But since the imaginary number is present in the equation, the value for x that I'm looking for should be imaginary as well. How would I go about doing this? Thanks in advance.
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Bruno Luong
Bruno Luong 2018-12-2
"But since the imaginary number is present in the equation, the value for x that I'm looking for should be imaginary as well. "
Not really, you barely multiply your entire relevant expression by 1/2*1i.
If you solve f(x) = 0, it does not change anything fundamental after multiplying the equation by constant that is not equal to 0, where as it's complex, imaginary or real.
Secondly
((e_0 * A) / (g - x) ^ 2)
never vanishes unless |x| = infinity
Ajay Gupta
Ajay Gupta 2018-12-2
I'm subtracting by k*x at the end, so I think x should almost always be some complex number. I also realized that I forgot to include the initialization for k in my code, so I've corrected it. I removed it before from my original code because I thought it wasn't necessary for the question being asked.

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采纳的回答

Bruno Luong
Bruno Luong 2018-12-2
编辑：Bruno Luong 2018-12-2
So your equation is of the form
a / (g - x)^2 = k* x
Just multiply by (g - x)^2 in both sides, you'll get
a = (g - x)^2 * k* x
k*x^3 - 2*k*g*x^2 + k*g^2*x - a = 0
The 3 solutions (for each omega) can be obtained by
for k1 = 1:length(omega)
omega_val = omega(k1);
a = (1 / 2) * 1i * omega_val * (e_0 * A) * (V_dc + V_ac)^2;
x = roots([k,-2*g*k,k*g^2,-a])
end
This gives you 3 solutions, I don't know which one you want to pick or not.

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Stephan
Stephan 2018-12-2
Hi,
use fsolve to handle complex valued problems. Fzero only can handle real valued problems.
Best regards
Stephan

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