Precision calculations of the characteristic impedance of complex coaxial waveguides used in wideband thermal converters of AC voltage and current

Krzysztof Kubiczek, Marian Kampik


The article presents precision and numerically stable method of calculation of the characteristic impedance of cylindrical multilayer waveguides used in high-precision wideband measuring instruments and standards, especially calculable thermal converters of AC voltage and precision wideband current shunts. Most of currently existing algorithms of characteristic impedance calculation of such waveguides are based upon approximations. Unfortunately, application of such methods is limited to waveguides composed of a specific, usually low number of layers. The accuracy of approximation methods as well as the number of layers is sometimes not sufficient, especially when the coaxial waveguide is a part of precision measurement equipment. The article presents the numerically stable matrix analytical formula using exponentially scaled modified Bessel functions to compute characteristic impedance and its components of the cylindrical coaxial multilayer waveguides. Results obtained with the developed method were compared with results of simulations made using the Finite Element Method (FEM) software simulations. Very good agreement between results of those two methods were achieved.

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Y. Shan, Y. Meng and P. Filipski, „Evaluation of a Calorimetric Thermal Voltage Converter for RF–DC Difference up to 1 GHz”, IEEE Transactions on Instrumentation and Measurement, vol. 63, no. 2, pp. 467-472, 2014.

P. Filipski, C. van Mullem, D. Janik, M. Klonz, J. Kinard, T. Lipe and B. Waltrip, „Comparison of high-frequency AC-DC voltage transfer standards at NRC, VSL, PTB, and NIST”, IEEE Transactions on Instrumentation and Measurement, vol. 50, no. 2, pp. 349-352, 2001.

M. Malinowski, K. Kubiczek and M. Kampik, „A precision coaxial current shunt for current AC-DC transfer”, Measurement, vol. 176, p. 109126, 2021.

M. Malinowski et al., „A Precision Coaxial Low-Current Shunt with Improved Mathematical Model”, 2021 13th International Conference on Measurement, 2021.

G. Kyriazis, R. de Souza, E. Yasuda and L. Di Lillo, „Modeling the AC–DC Transfer Difference of Wideband Cage-Type Current Shunts”, IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 7, pp. 4436-4444, 2020.

A. Andreychenko, H. Kroeze, D. Klomp, J. Lagendijk, P. Luijten and C. van den Berg, „Coaxial waveguide for travelling wave MRI at ultrahigh fields”, Magnetic Resonance in Medicine, vol. 70, no. 3, pp. 875-884, 2012.

H. Bao, K. Nielsen, O. Bang and P. Jepsen, „Dielectric tube waveguides with absorptive cladding for broadband, low-dispersion and low loss THz guiding”, Scientific Reports, vol. 5, no. 1, 2015.

J. Melzer, M. Navarro-Cía, O. Mitrofanov and J. Harrington, „Silver-coated Teflon hollow waveguides for the delivery of terahertz radiation”, Optical Fibers and Sensors for Medical Diagnostics and Treatment Applications XIV, 2014.

S. Khalili, M. Botshekanan Dehkordi and E. Carrera, „A nonlinear finite element model using a unified formulation for dynamic analysis of multilayer composite plate embedded with SMA wires”, Composite Structures, vol. 106, pp. 635-645, 2013.

Z. Mikno, M. Stepien and B. Grzesik, „Optimization of resistance welding by using electric servo actuator”, Welding in the World, vol. 61, no. 3, pp. 453-462, 2017.

M. Stepien, S. Krosny and B. Grzesik, „Analysis of quench propagation using coupled electrical-thermal FEM model”, Journal of Physics: Conference Series, vol. 234, no. 2, p. 022036, 2010.

G. Aiello, S. Alfonzetti, E. Dilettoso and N. Salerno, „Eddy Current Computation by the FEM-SDBCI Method”, IEEE Transactions on Magnetics, vol. 52, no. 3, pp. 1-4, 2016.

S. Krosny, M. Woźniak, S. Hopkins, M. Stȩpień, B. Grzesik and B. Glowacki, „Modelling of transient state phenomena of composite superconducting conductors during pulse Ic(B) measurements”, Journal of Physics: Conference Series, vol. 234, no. 2, p. 022019, 2010.

K. Kubiczek and M. Kampik, „Highly accurate and numerically stable computations of double-layer coaxial waveguides", Engineering Computations, 2019. Available:

C. Paul, Introduction to electromagnetic compatibility. Hoboken, N.J.: Wiley, 2006, pp. 871-900.

J. Jackson, Classical electrodynamics. New Delhi: Wiley India, 2011, pp. 69-75.

H. W. Dommel, “Overhead line parameters from handbook formulasand computer programs”, with discussions by A. Deri and G. Tevan, Adam Semlyen, and F. L. Alvarado, IEEE Trans., Power Appar. Syst., 1985, 104, (2), pp. 366–372

H. W. Dommel, “EMTP Theory Book”, Microtran Power System Analysis Corporation, Vancouver, British Columbia:, 1992, 2nd Edn.

S. Vujević, V. Boras, P. Sarajčev, „A novel algorithm for internal impedance computation of solid and tubular cylindrical conductors”, International Review of Electrical Engineering (IREE). Vol.4, Part B, pp. 1418-1425, 2009.

N. Nahman and D. Holt, „Transient analysis of coaxial cables using the skin effect approximation „, IEEE Transactions on Circuit Theory, vol. 19, no. 5, pp. 443-451, 1972.

A. Semlyen and A. Deri, „Time Domain Modeling of Frequency Dependent Three-Phase Transmission Line Impedance”, IEEE Power Engineering Review, vol. -5, no. 6, pp. 64-65, 1985.

S. Vujevic and D. Lovric, „On the numerical computation of cylindrical conductor internal impedance for complex arguments of large magnitude”, Facta universitatis - series: Electronics and Energetics, vol. 30, no. 1, pp. 81-91, 2017.

S. Vujevic, D. Lovric and V. Boras, „High-Accurate Numerical Computation of Internal Impedance of Cylindrical Conductors for Complex Arguments of Arbitrary Magnitude”, IEEE Transactions on Electromagnetic Compatibility, vol. 56, no. 6, pp. 1431-1438, 2014.

J. Carson and J. Gilbert, „Transmission characteristics of the submarine cable”, Journal of the Franklin Institute, vol. 192, no. 6, pp. 705-735, 1921.

J. A. Brandao Faria, „A matrix approach for the evaluation of the internal impedance of multilayered cylindrical structures”, Progress In Electromagnetics Research B, Vol. 28, 351-367, 2011.

K. Kubiczek and M. Kampik, „Highly Accurate and Numerically Stable Matrix Computations of the Internal Impedance of Multilayer Cylindrical Conductors”, IEEE Transactions on Electromagnetic Compatibility, pp. 1-8, 2019. Available:

ANSYS Academic Multiphysics Campus Solution 2017. „Engineering Simulation & 3-D Design Software | ANSYS”,, 2019. [Online]. Available: Available:

U. Bakshi and A. Bakshi, Transmission lines and waveguides. 70p.: Ill., pp. 24, ch.1.

P. Peres, C. de Souza and I. Bonatti, „ABCD Matrix: A Unique Tool for Linear Two-Wire Transmission Line Modelling”, International Journal of Electrical Engineering Education, vol. 40, no. 3, pp. 220-229, 2003.

S. Schelkunoff, "The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields", Bell System Technical Journal, vol. 13, no. 4, pp. 532-579, 1934.

J. H. Poynting, “On the transfer of energy in the electromagnetic field”, Philosophical Transactions of the Royal Society of London. 175: 343–361, 1884.

D. Lovric and S. Vujevic, „Accurate Computation of Internal Impedance of Two-Layer Cylindrical Conductors for Arguments of Arbitrary Magnitude”, IEEE Transactions on Electromagnetic Compatibility, vol. 60, no. 2, pp. 347-353, 2018.

A. Gil, J. Segura and N. Temme, „Computing solutions of the modified bessel differential equation for imaginary orders and positive arguments”, ACM Transactions on Mathematical Software, vol. 30, no. 2, pp. 145-158, 2004.

N. Marcuvitz, Waveguide handbook. New York: McGraw-Hill, 1951, pp. 72-80.


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