volume | PIER Journals

Abstract

A methodology for estimating the complex permittivity of liquid dielectrics is presented in a rectangular waveguide using the Ku-band (10-15 GHz, WR75). A two-dimensional finite difference time-domain (FDTD) model with perfectly matched layers (PMLs) serves as the forward solver, and TE₁₀ modal projection provides the simulated scattering parameters. Subsequently, a gradient-free Nelder-Mead inversion extracts the real and imaginary parts of the permittivity from the measured S₁₁ and S₂₁ parameters. This approach is implemented in a multilayer fixture, which enables leak-tight loading while remaining analytically simple. Validation on air and water shows good agreement between simulation and measurement across 10-15 GHz, and results at 12 GHz are consistent with independent X-band extractions. This approach is computationally efficient, practical for experimentation, and can be extended to other liquids and multilayers.

1. Krupka, Jerzy, "Microwave measurements of electromagnetic properties of materials," Materials, Vol. 14, No. 17, 5097, 2021.
doi:10.3390/ma14175097

2. Tao, Yuan, Bowen Yan, Daming Fan, Nana Zhang, Shenyan Ma, Liyun Wang, Yejun Wu, Mingfu Wang, Jianxin Zhao, and Hao Zhang, "Structural changes of starch subjected to microwave heating: A review from the perspective of dielectric properties," Trends in Food Science & Technology, Vol. 99, 593-607, 2020.
doi:10.1016/j.tifs.2020.02.020

3. Pakkathillam, Jayaram Kizhekke, Balamurugan T. Sivaprakasam, Jayaprakash Poojali, C. V. Krishnamurthy, and Kavitha Arunachalam, "Tailoring antenna focal plane characteristics for a compact free-space microwave complex dielectric permittivity measurement setup," IEEE Transactions on Instrumentation and Measurement, Vol. 70, 1-12, 2021.
doi:10.1109/tim.2020.3013997

4. Ebara, Hidetoshi, Takao Inoue, and Osamu Hashimoto, "Measurement method of complex permittivity and permeability for a powdered material using a waveguide in microwave band," Science and Technology of Advanced Materials, Vol. 7, No. 1, 77, 2006.
doi:10.1016/j.stam.2005.11.019

5. Saad-Falcon, Alex, Zijian Zhang, David Ryoo, James Dee, Ryan S. Westafer, and James C. Gumbart, "Extraction of dielectric permittivity from atomistic molecular dynamics simulations and microwave measurements," The Journal of Physical Chemistry B, Vol. 126, No. 40, 8021-8029, 2022.
doi:10.1021/acs.jpcb.2c05260

6. Ma, Jialu, Jingchao Tang, Kaicheng Wang, Lianghao Guo, Yubin Gong, and Shaomeng Wang, "Complex permittivity characterization of liquid samples based on a split ring resonator (SRR)," Sensors, Vol. 21, No. 10, 3385, 2021.
doi:10.3390/s21103385

7. Talmoudi, Omaima, Lahcen Ait Benali, Jaouad Terhzaz, Abdelwahed Tribak, and Tomás Fernández-Ibáñez, "A novel hybrid 2D-FDTD-PML and Nelder–Mead methods for estimating liquid complex permittivity using a rectangular waveguide," Journal of Applied Physics, Vol. 137, No. 14, 144501, 2025.
doi:10.1063/5.0264173

8. Talmoudi, Omaima, Álvaro Gómez-Gómez, Oscar Fernandez, Jaouad Terhzaz, Abdelwahed Tribak, and Tomás Fernández-Ibáñez, "Substrate integrated waveguide resonator sensor for X-band dielectric constant characterization," Journal of Applied Physics, Vol. 138, No. 7, 074503, 2025.
doi:10.1063/5.0280603

9. Hasar, U. C. and A. Cansiz, "Simultaneous complex permittivity and thickness evaluation of liquid materials from scattering parameter measurements," Microwave and Optical Technology Letters, Vol. 52, No. 1, 75-78, 2010.
doi:10.1002/mop.24837

10. Hasar, Ugur Cem, "Permittivity measurement of thin dielectric materials from reflection-only measurements using one-port vector network analyzers," Progress In Electromagnetics Research, Vol. 95, 365-380, 2009.
doi:10.2528/pier09062501

11. Bois, K. J., L. F. Handjojo, A. D. Benally, K. Mubarak, and R. Zoughi, "Dielectric plug-loaded two-port transmission line measurement technique for dielectric property characterization of granular and liquid materials," IEEE Transactions on Instrumentation and Measurement, Vol. 48, No. 6, 1141-1148, 1999.
doi:10.1109/19.816128

12. Bois, Karl, Aaron Benally, and Reza Zoughi, Two-port network analyzer dielectric constant measurement of granular or liquid materials for the study of cement based materials, 291-296 Springer, 1998.
doi:10.1007/978-1-4615-4847-8_46

13. Terhzaz, Jaouad, Hassan Ammor, Abdelhadi Assir, and Ahmed Mamouni, "Application of the FDTD method to determine complex permittivity of dielectric materials at microwave frequencies using a rectangular waveguide," Microwave and Optical Technology Letters, Vol. 49, No. 8, 1964-1968, 2007.
doi:10.1002/mop.22611

14. Krupezevic, D. V., V. J. Brankovic, and F. Arndt, "The wave-equation FD-TD method for the efficient eigenvalue analysis and S-matrix computation of waveguide structures," IEEE Transactions on Microwave Theory and Techniques, Vol. 41, No. 12, 2109-2115, 1993.
doi:10.1109/22.260694

15. Mosavirik, Tahoura, Vahid Nayyeri, Mohammad Hashemi, Mohammad Soleimani, and Omar M. Ramahi, "Direct permittivity reconstruction from power measurements using a machine learning aided method," IEEE Transactions on Microwave Theory and Techniques, Vol. 71, No. 10, 4437-4448, 2023.
doi:10.1109/tmtt.2023.3267390

16. Qin, Jincheng, Zhifu Liu, Mingsheng Ma, and Yongxiang Li, "Machine learning approaches for permittivity prediction and rational design of microwave dielectric ceramics," Journal of Materiomics, Vol. 7, No. 6, 1284-1293, 2021.
doi:10.1016/j.jmat.2021.02.012

17. Python Software Foundation "Python Optimization User’s Guide," 2022.

18. Nelder, J. A. and R. Mead, "A simplex method for function minimization," The Computer Journal, Vol. 7, No. 4, 308-313, 1965.
doi:10.1093/comjnl/7.4.308

19. Ellison, W. J., "Permittivity of pure water, at standard atmospheric pressure, over the frequency range-25 THz and the temperature range-100 C," Journal of Physical and Chemical Reference Data, Vol. 36, No. 1, 1-18, 2007.
doi:10.1063/1.2360986

Ms. Omaima Talmoudi

Dr. Lahcen Ait Benali

Prof. Jaouad Terhzaz

Professor Abdelwahed Tribak