Finding zeros of intersection of the "x-axis" - Error using "fzero"

Hello,

I am currently facing an issue where I am trying to find the intersection of the different plots on my "x-axis" (horizontal axis).

I have tried using "fsolve" in MATLAB, but they've given me errors such as:

The input to FSOLVE should be either a structure with valid fields or consist of at least two arguments.

For the solve function, it says:

Unable to find explicit solution

See MATLAB code attached:

Zeros_k_p.m

I've tried to use the "solve" function in MATLAB but I just end up getting an empty val ==> Empty sym: 0-by-1

Essentially, I have an equation called "T_M" that has a variable "k_p". I need to find where the equation "T_M" crosses the "x-axis" and find all of the roots for "k_p".

tic
close all; clc;

% For the region k_o < k_p < sqrt(u_r*e_r)*k_o

% Specifying k_p as a symbolic object using the symbolic toolbox
syms k_p;

% Wavelength value lambda
lambda = 1;

% Wavenumber in free space value = k_o
k = (2*pi)/lambda;

% Permeability Value = u_r
u_r = 1;

% Permittivity Value e_r
e_r = 2.2;

% Value of kz_1
kz_1 = sqrt((k)^2*e_r*u_r - (k_p)^2);

% Value of kz_2
kz_2 = sqrt((k_p)^2 - (k)^2);

% Substrate heights = d
d = [0.02 0.04 0.06 0.08 0.10];
    
% Equations for TM(kp)
T_M = kz_1.*sin(d.*kz_1) - e_r*sqrt(kz_2).*cos(d.*kz_1);

fplot(T_M,[k k*sqrt(e_r*u_r)], 'LineWidth',3);
ylim([-5 5]);
xlim([k k*sqrt(e_r*u_r)]);
grid on;
ax = gca;
ax.GridLineWidth = 2;

Answer 1

[Matt J]

tic
close all; clc;

% For the region k_o < k_p < sqrt(u_r*e_r)*k_o

% Wavelength value lambda
lambda = 1;

% Wavenumber in free space value = k_o
k = (2*pi)/lambda;

% Permeability Value = u_r
u_r = 1;

% Permittivity Value e_r
e_r = 2.2;

% Value of kz_1
kz_1 =@(kp) sqrt((k).^2*e_r*u_r - (kp).^2);

% Value of kz_2
kz_2 = @(kp) sqrt((kp).^2 - (k).^2);

% Substrate heights = d
D = [0.02 0.04 0.06 0.08 0.10];
    
for i=1:numel(D)
d=D(i);
% Equations for TM(kp)
T_M = @(kp) kz_1(kp).*sin(d.*kz_1(kp)) - e_r*sqrt(kz_2(kp)).*cos(d.*kz_1(kp));

kp_root=fzero(T_M, [k , k*sqrt(e_r*u_r)])

fplot(T_M,[k k*sqrt(e_r*u_r)], 'LineWidth',3); hold on
plot(kp_root,0,'o','MarkerSize',8,'MarkerFaceColor','k');
ylim([-5 5]);
xlim([k k*sqrt(e_r*u_r)]);
grid on;
ax = gca;
ax.GridLineWidth = 2;

end; hold off

kp_root = 6.2860

kp_root = 6.3292

kp_root = 6.4978

kp_root = 6.8051

kp_root = 7.1518

[Ammar]

Thank You! This is what I needed.