How can I use the NRW method in MATLAB to calculate permittivity and permeability from S-parameters at 8.2 GHz and get results matching ε′ ≈ 2.32, ε″ ≈ -0.0456, μ′ ≈ 0.832, an

I am using the NRW method in MATLAB to calculate the real and imaginary parts of permittivity (ε′, ε″) and permeability (μ′, μ″) from S-parameters measured in a waveguide at 8.2 GHz with a 2 mm thick sample. My goal is to obtain results close to the expected values:

• ε′ ≈ 2.32
• ε″ ≈ -0.0456
• μ′ ≈ 0.832
• μ″ ≈ 0.0824

How can I modify or verify my MATLAB code to ensure it produces these values accurately from the given S11 and S21 data?

% Constants
c = 3e8;
mu0 = 4*pi*1e-7;
eps0 = 8.854e-12;
a = 22.86e-3;
d = 2e-3;
f = 8.2e9;

% Waveguide parameters
k0 = 2*pi*f/c;
fc = c/(2*a);
kc = pi/a;
gamma0 = 1i * sqrt(k0^2 - kc^2);

% Given S-parameters
S11 = -0.1278677 - 1i*0.3101051;
S21 = 0.8092633 - 1i*0.4556378;

% Calculate reflection coefficient R
V1 = (S11^2 - S21^2 + 1) / (2*S11);
R1 = V1 + sqrt(V1^2 - 1);
R2 = V1 - sqrt(V1^2 - 1);
if abs(R2) < abs(R1)
    R = R2;
else
    R = R1;
end

% Calculate transmission coefficient T
T = (S11 + S21 - R) / (1 - (S11 + S21)*R);

% Calculate complex propagation constant gamma
gamma = -log(T) / d;

% Calculate impedance Z
Z = sqrt((1 + R)/(1 - R));

% Calculate relative permeability mu_r
mu_r = (Z * gamma) / gamma0;

% Calculate relative permittivity epsilon_r
epsilon_r = gamma / (1i * 2 * pi * f * sqrt(mu0 * eps0) * sqrt(mu_r));

% Display results
fprintf('epsilon_real = %.4e\n', real(epsilon_r));

epsilon_real = 1.0049e+00

fprintf('epsilon_imag = %.4e\n', imag(epsilon_r));

epsilon_imag = -2.7589e-02

fprintf('mu_real = %.4e\n', real(mu_r));

mu_real = 1.1658e+00

fprintf('mu_imag = %.4e\n', imag(mu_r));

mu_imag = -2.8260e-01

[Mr.DDWW]

This is the correct result. I am trying to get these results

[Torsten]

We don't know the physical background - so we cannot tell whether S11, S21 and your formulae to compute epsilon_r and mu_r are correct.

The equations reported here are in part different from yours:

https://en.wikipedia.org/wiki/Nicolson%E2%80%93Ross%E2%80%93Weir_method

[Mr.DDWW]

Here is the complete data with S-parameters and the calculated permittivity and permeability.

I am developing a MATLAB code based on the Nicolson-Ross-Weir (NRW) method to calculate the complex permittivity and permeability of materials from measured S-parameter data.

[Mr.DDWW]

here it is

Ports: 1 & 2 Measurement Model: Reflection/Transmission Mu and Epsilon Sample Holder Type: Waveguide Cutoff Frequency: 6.560010 GHz Sample Holder Length: 2.000000 mm Distance to Sample: 0.000000 mm Sample Thickness: 2.000000 mm IFBW: 300.000000 Hz Power: 0.000000 dBm