Hello everyone,

I’m trying to calculate the *reflection loss (RL)* from complex permittivity and permeability values measured at *8.2 GHz* , but my result from the *free-space transmission line equation* doesn’t match the RL measured directly by the VNA.

*VNA Measured Data (at 8.2 GHz)*

freq (Hz) e′ e″ μ′ μ″ S11 (LogM) S21 (LogM) 8.20E+09 2.62 -3.84E-2 0.774 5.06E-2 -7.80 dB -0.897 dB

*My MATLAB Code:*

clc; clear;
% Constants
f = 8.2e9; % Hz
c = 3e8; % m/s
Z0 = 377; % Ohms
d = 0.00215; % m
% Material properties
epsilon_real = 2.62;
epsilon_imag = -3.84E-02;
mu_real = 0.774;
mu_imag = 5.06E-02;
% Complex values
epsilon_r = complex(epsilon_real, -epsilon_imag);
mu_r = complex(mu_real, -mu_imag);
% Input impedance
term1 = sqrt(mu_r / epsilon_r);
term2 = tanh(1j * 2 * pi * f * d / c * sqrt(mu_r * epsilon_r));
Zin = Z0 * term1 * term2;
% Reflection loss
RL = 20 * log10(abs((Zin - Z0) / (Zin + Z0)));
disp(RL);

*Problem:*

*

VNA measured RL (S11 in LogM) = *-7.80 dB*

*

My calculated RL = (different value, not matching)

*Question:*

*

Am I using the correct formula for free-space reflection loss?

*

Is there an error in my use of tanh or impedance formula?

*

Could this be because the permittivity/permeability values I have are already extracted from S-parameters, making this calculation redundant or incorrect?

*

Is there a more accurate way to compute RL from ε′, ε″, μ′, μ″ for a finite thickness sample?

Any help with identifying what I’m doing wrong (or missing) would be appreciated.

tanh is the hyperbolic tangent function. I believe the tan function should be
used.



I would also recommend testing your code on a half wavelength slab (reflection
should be zero) and a quarter wavelength slab ( reflection should be high)



> On Aug 11, 2025, at 09:30, Saud Aldhafyan via groups.io
<Aldhafyans=gmail.com@groups.io> wrote:
>
>

> 

>

> Hello everyone,

>

> I’m trying to calculate the **reflection loss (RL)** from complex
permittivity and permeability values measured at **8.2 GHz** , but my result
from the **free-space transmission line equation** doesn’t match the RL
measured directly by the VNA.

>

> **VNA Measured Data (at 8.2 GHz)**

>

>
>

> freq (Hz) | e′ | e″ | μ′ | μ″ | S11 (LogM) | S21 (LogM)
> ---|---|---|---|---|---|---
> 8.20E+09 | 2.62 | -3.84E-2 | 0.774 | 5.06E-2 | -7.80 dB | -0.897 dB
>
>
>

>
>

> **My MATLAB Code:**

>

>
>

>
>

>
>

> clc; clear;

>

> % Constants
> f = 8.2e9; % Hz
> c = 3e8; % m/s
> Z0 = 377; % Ohms
> d = 0.00215; % m

>

> % Material properties
> epsilon_real = 2.62;
> epsilon_imag = -3.84E-02;
> mu_real = 0.774;
> mu_imag = 5.06E-02;

>

> % Complex values
> epsilon_r = complex(epsilon_real, -epsilon_imag);
> mu_r = complex(mu_real, -mu_imag);

>

> % Input impedance
> term1 = sqrt(mu_r / epsilon_r);
> term2 = tanh(1j * 2 * pi * f * d / c * sqrt(mu_r * epsilon_r));
> Zin = Z0 * term1 * term2;

>

> % Reflection loss
> RL = 20 * log10(abs((Zin - Z0) / (Zin + Z0)));

>

> disp(RL);

>

>
>

>
>

> **Problem:**

>

> * VNA measured RL (S11 in LogM) = **-7.80 dB**

>

> * My calculated RL = (different value, not matching)

>

>

>

> **Question:**

>

> * Am I using the correct formula for free-space reflection loss?

>

> * Is there an error in my use of `tanh` or impedance formula?

>

> * Could this be because the permittivity/permeability values I have are
already extracted from S-parameters, making this calculation redundant or
incorrect?

>

> * Is there a more accurate way to compute RL from ε′, ε″, μ′, μ″ for a
finite thickness sample?

>

>

>

> Any help with identifying what I’m doing wrong (or missing) would be
appreciated.

>

>
>

>
>

>
>

>

_._,_._,_

* * *

This may be totally "all the wall", but: Your VNA measurement is in a
50-ohm system. Your calculations are in a free space impedance, 120π ohms.

Dave - WØLEV

On Tue, Aug 12, 2025 at 4:59 AM gary miller via groups.io <gary_e_miller=
yahoo.com@groups.io> wrote:

> tanh is the hyperbolic tangent function. I believe the tan function should
> be used.
>
> I would also recommend testing your code on a half wavelength slab
> (reflection should be zero) and a quarter wavelength slab ( reflection
> should be high)
>
> On Aug 11, 2025, at 09:30, Saud Aldhafyan via groups.io <Aldhafyans=
> gmail.com@groups.io> wrote:
>
> 
>
> Hello everyone,
>
> I’m trying to calculate the *reflection loss (RL)* from complex
> permittivity and permeability values measured at *8.2 GHz*, but my result
> from the *free-space transmission line equation* doesn’t match the RL
> measured directly by the VNA.
>
> *VNA Measured Data (at 8.2 GHz)*
>
>
> freq (Hz) e′ e″ μ′ μ″ S11 (LogM) S21 (LogM)
> 8.20E+09 2.62 -3.84E-2 0.774 5.06E-2 -7.80 dB -0.897 dB
>
>
>
> *My MATLAB Code:*
>
>
>
> clc; clear;
> % Constants
> f = 8.2e9; % Hz
> c = 3e8; % m/s
> Z0 = 377; % Ohms
> d = 0.00215; % m
> % Material properties
> epsilon_real = 2.62;
> epsilon_imag = -3.84E-02;
> mu_real = 0.774;
> mu_imag = 5.06E-02;
> % Complex values
> epsilon_r = complex(epsilon_real, -epsilon_imag);
> mu_r = complex(mu_real, -mu_imag);
> % Input impedance
> term1 = sqrt(mu_r / epsilon_r);
> term2 = tanh(1j * 2 * pi * f * d / c * sqrt(mu_r * epsilon_r));
> Zin = Z0 * term1 * term2;
> % Reflection loss
> RL = 20 * log10(abs((Zin - Z0) / (Zin + Z0)));
> disp(RL);
>
>
>
> *Problem:*
>
> -
>
> VNA measured RL (S11 in LogM) = *-7.80 dB*
> -
>
> My calculated RL = (different value, not matching)
>
> *Question:*
>
> -
>
> Am I using the correct formula for free-space reflection loss?
> -
>
> Is there an error in my use of tanh or impedance formula?
> -
>
> Could this be because the permittivity/permeability values I have are
> already extracted from S-parameters, making this calculation redundant or
> incorrect?
> -
>
> Is there a more accurate way to compute RL from ε′, ε″, μ′, μ″ for a
> finite thickness sample?
>
> Any help with identifying what I’m doing wrong (or missing) would be
> appreciated.
>
>
>
>
>
>
>
>

--

*Dave - WØLEV*

Saud

It’s the finite thickness of the sample.  You have to sum multiple reflections from each side of the sample, or the sample has to be thick enough that there is enough loss that they are negligible.  In real world there are errors from reflections from clutter behind the sample and from how you launch into the sample too (directional antenna, waveguide flange et cetera).  The incident wave isn’t an idealized plane wave.  It’s easier if the sample properties are very different from free space and if one way loss is significant because it makes the errors less significant.  If mu is very close to free space you will have a hard time measuring it.

Jim