Nicholson-Ross-Weir Method (Python)

Nicholson-Ross-Weir (NRW) method in python for calculating refractive index, permittivity and permeability from scattering parameters (S11/S21). The algorithm is also taking into account the phase ambiguity of the transmission coefficient and selects the best n based on the group velocity.

Content

• Function for NRW Method: Calculates the permittivity and permeability. Takes frequency values, complex S11/S21, the sample length and a cutoff frequency (optional).
• Function for reverse NRW Method: Calculates the scattering parameters S11/S21 from the permittivity and the permeability. Takes frequency values, complex permittivity/permeability, the sample length and a cutoff frequency (optional).
• High precision forwards-backwards NRW calculation: This was done as an instability analysis. One can specify the permittivity/permeability to calculate the scattering parameters S11/S21 and back to permittivity/permeability using NRW method. By introducing noise it is possible to test a measurement result.

# before install the dependencies in `requirements.txt`
from nrw import nrw, nrw_reverse
from nrw_gmpy2 import nrw2, nrw_reverse2

Analysis

Basis

The NRW method starts using the complex scattering parameters $S_{11}$ and $S_{21}$. The reflection coefficient is defined as:

$$X = \frac{S_{11}^2-S_{21}^2+1}{2S_{11}}$$ $$\Gamma = X \pm \sqrt{(X^2-1)}$$

Choose the sign of the root for $\Gamma$ such that $|\Gamma|\le1$. The transmission coefficient is defined as.

$$T = \frac{S_{11}+S_{21}-\Gamma}{1-(S_{11}+S_{21})\Gamma}$$

Using $\Gamma$ and $T$ one can calculate $\Lambda$:

$$\frac{1}{\Lambda^2} = \left(\frac{\varepsilon_r\mu_r}{\lambda_0}-\frac{1}{\lambda_c^2}\right) = -\left(\frac{1}{2\pi L}ln\left(\frac{1}{T}\right)\right)^2$$

The equation for $\Lambda$ is not well defined since the logarithm of $1/T$ has multiple solutions which are equal to $ln(1/|T|) + i(\Theta + 2\pi n)$ where $n$ is an integer value. In the NRW method one can estimate the value $n$ by using the group delay.

The idea is that the group delay is the derivative of the phase and therefore is independent of the $2\pi n$. One can compare the measured group delay and the calculated group delay and the best value for $n$ is found where:

$$\tau_{meas} - \tau_{calc} \approx 0$$

The group delay for both values is defined as:

$$\tau = \frac{d\phi}{\omega} = -\frac{1}{2\pi}\frac{d\phi}{df}$$ $$\tau_{meas} = -\frac{1}{2\pi}\frac{d\phi_{meas}}{df} = -\frac{1}{2\pi}\frac{d}{df}arg(T)$$ $$\tau_{calc} = -\frac{1}{2\pi}\frac{d\phi_{calc}}{df} = \frac{d}{df}\frac{L}{\Lambda} =$$ $$=L\frac{d}{df}\sqrt{\frac{\varepsilon_r\mu_r f^2}{c^2}-\frac{1}{\lambda_c^2}} = \frac{1}{c^2}\frac{f\varepsilon_r\mu_r+f^2\frac{1}{2}\frac{d(\varepsilon_r\mu_r)}{df}}{\sqrt{\frac{\varepsilon_r\mu_r f^2}{c^2}-\frac{1}{\lambda_c^2}}}L$$

With this the permittivity and permeability is defined:

$$\mu_r = \frac{1+\Gamma}{\Lambda(1-\Gamma)\sqrt{\frac{1}{\lambda_0^2}-\frac{1}{\lambda_c^2}}}$$ $$\varepsilon_r = \frac{\lambda_0^2}{\mu_r}\left(\frac{1}{\lambda_c^2} - \left[\frac{1}{2\pi L}\ln\left(\frac{1}{T}\right)\right]^2\right)$$

Literature

The analysis explained in literature.

• O. Luukkonen, S. I. Maslovski and S. A. Tretyakov, "A Stepwise Nicolson–Ross–Weir-Based Material Parameter Extraction Method," in IEEE Antennas and Wireless Propagation Letters, vol. 10, pp. 1295-1298, 2011, doi: 10.1109/LAWP.2011.2175897.
• A. M. Nicolson and G. F. Ross, "Measurement of the Intrinsic Properties of Materials by Time-Domain Techniques," in IEEE Transactions on Instrumentation and Measurement, vol. 19, no. 4, pp. 377-382, Nov. 1970, doi: 10.1109/TIM.1970.4313932.
• W. B. Weir, "Automatic measurement of complex dielectric constant and permeability at microwave frequencies," in Proceedings of the IEEE, vol. 62, no. 1, pp. 33-36, Jan. 1974, doi: 10.1109/PROC.1974.9382.
• Measurement of Dielectric Material Properties Application Note R&S.

Instabilities

Some instabilities were observed when:

• The cutoff frequency is near the lowest measurement frequency
• Long sample lengths or high permittivity/permeability values (short wavelengths media)