Hi all,

I'm working with electromagnetic materials and trying to reverse-engineer or verify complex permittivity (ε) and permeability (μ) values from S-parameter data (S11, S21). I have a dataset like this for each frequency point:

Here’s a sample of the kind of data I have (per row):

frequency(Hz), Tr1 (ε'), (ε''), Tr2 (ε'), (ε''), Tr3 (ε'), (ε''),
Tr4 (μ'), (μ''), Tr5 (μ'), (μ''), Tr6 S11(Re), (Im), Tr7 S21(Re), (Im)
8.20E+09, 2.32, -0.0456, 2.32, -0.0456, 2.32, -0.0456, 0.832, 0.0824, 0.832, 0.0824, -0.128, -0.310, 0.809, -0.456

And here is the MATLAB code I currently have:

clc; clear;
% Constants
c = 3e8;                % Speed of light (m/s)
freq_Hz = 8.20e9;       % Frequency (Hz)
d = 0.002;              % Sample thickness (m)
omega = 2*pi*freq_Hz;
% Given S-parameters from Keysight
S11 = -0.128 - 1i*0.310;
S21 = 0.809 - 1i*0.456;
% Step 1: Calculate reflection coefficient Gamma
X = (S11.^2 - S21.^2 + 1) ./ (2*S11);
root = sqrt(X.^2 - 1);
Gamma_candidates = [X + root, X - root];
Gamma = Gamma_candidates(abs(Gamma_candidates) < 1);
% Step 2: Calculate e^(-gamma*2d)
e_neg_gamma_2d = S21 ./ (1 - Gamma .* S11);
% Step 3: Phase unwrap for proper branch selection
mag = abs(e_neg_gamma_2d);
phase = unwrap(angle(e_neg_gamma_2d));
e_neg_gamma_2d_unwrapped = mag .* exp(1i*phase);
% Step 4: Calculate gamma (propagation constant)
gamma = -log(e_neg_gamma_2d_unwrapped) / (2*d);
% Step 5: Refractive index n
n = gamma * c / (1i * omega);
% Step 6: Impedance Z
Z = (1 + Gamma) ./ (1 - Gamma);
% Step 7: Relative permeability and permittivity
mu_r = n .* Z;
epsilon_r = n ./ Z;
% Step 8: Display results
fprintf('Results at %.2f GHz:\n', freq_Hz/1e9);
fprintf('Real Permittivity (ε\')   : %.6f\n', real(epsilon_r));
fprintf('Imaginary Permittivity (ε\'\') : %.6f\n', -abs(imag(epsilon_r)));
fprintf('Real Permeability (μ\')    : %.6f\n', real(mu_r));
fprintf('Imaginary Permeability (μ\'\')  : %.6f\n', -abs(imag(mu_r)));

What I’m looking for:

*

A way to *adjust or verify this code so that it gives me ε and μ matching the values in my data* (as shown above).

*

Possibly suggestions on which assumptions or approximations in NRW might need tweaking (e.g., sample thickness, sign convention, root choice, etc.).

*

Bonus: how to handle this across multiple frequencies from Excel.

Any insights or examples would really help. Let me know if you want the full dataset or script I’m using.

Thanks a lot!

Best regards,

I'm not sure the reflection coefficient is calculated correctly. Isn't S11 basically the reflection coefficient (in dB) - S11 is the reflected voltage divided by the incident voltage, which is the very definition of reflection coefficient.
Granted, taking typical mag/phase data you'd have to convert from dB magnitude and angle to complex, but you've already got that done in your numbers from Keysight.


Why are you subtracting S21^2, etc

*Hi Jim,*

Thanks for your input — you’re absolutely right that S11 represents the
reflection coefficient, and in many contexts, especially for simple
transmission lines, it can be used directly.

In this case though, I’m applying the Nicolson-Ross-Weir (NRW) method to
extract the complex permittivity and permeability of a material from
S-parameter data (S11 and S21). The method involves calculating the
internal reflection coefficient and propagation constant to back out ε and
μ, so it goes beyond treating S11 as the total reflection coefficient
alone. It also accounts for multiple internal reflections and phase shifts
through the sample thickness.

I initially thought simplifying the calculation using S11 directly might
work, but the results ended up diverging quite a bit from the reference ε
and μ values I’m trying to match.

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On Tue, Jun 10, 2025 at 6:59 PM Jim Lux via groups.io <jimlux=
earthlink.net@groups.io> wrote:

> I'm not sure the reflection coefficient is calculated correctly. Isn't
> S11 basically the reflection coefficient (in dB) - S11 is the reflected
> voltage divided by the incident voltage, which is the very definition of
> reflection coefficient.
>
> Granted, taking typical mag/phase data you'd have to convert from dB
> magnitude and angle to complex, but you've already got that done in your
> numbers from Keysight.
>
>
>
>
>
> Why are you subtracting S21^2, etc
>
>
>
>
>
>

*Hi Jim,*

Thanks for your input — you’re absolutely right that S11 represents the reflection coefficient, and in many contexts, especially for simple transmission lines, it can be used directly.

In this case though, I’m applying the Nicolson-Ross-Weir (NRW) method to extract the complex permittivity and permeability of a material from S-parameter data (S11 and S21). The method involves calculating the internal reflection coefficient and propagation constant to back out ε and μ, so it goes beyond treating S11 as the total reflection coefficient alone. It also accounts for multiple internal reflections and phase shifts through the sample thickness.

I initially thought simplifying the calculation using S11 directly might work, but the results ended up diverging quite a bit from the reference ε and μ values I’m trying to match.