T.I. Tretyakov1, M.E. Komnatnov2

1,2 Tomsk State University of Control Systems and Radioelectronics (Tomsk, Russia)

1 timur.i.tretiakov@tusur.ru; 2 maksim.e.komnatnov@tusur.ru

Ensuring signal and power integrity in the design of electronic equipment depends on the electrical characteristics of transmission lines, one of which is inductance. Calculations of per-unit-length inductance matrices for multi-conductor transmission lines, based on magnetic field energy and vector potential, are considered more universal and are widely used in modern computer-aided design (CAD) systems. At the same time, calculations based on magnetic flux density are used much less frequently due to the complexity and unwieldiness of the expressions involved. However, such methods can improve the accuracy of the results. The aim of the present work is to propose a simplified method for calculating the per-unit-length inductance matrix based on magnetic flux density.  To achieve this, the calculation of inductance matrices for various transmission lines is based on determining the flux linkages of the conductors and their segments using the Biot-Savart-Laplace law. The computation is simplified by a proposed straightforward expression for flux linkage. This expression is derived under the assumption that the lengths of the conductor segments approach infinity, whereby their mutual and self flux linkages per unit length are defined solely by the distance between the conductor segments in the cross-sectional plane. An algorithm and a corresponding software module have been developed to automate the calculation of the inductance matrices. Using the proposed approach, the elements of the per-unit-length inductance matrices for 1-, 2-, and 3-conductor transmission lines have been calculated. An analysis of the convergence of the results with an increasing number of segments in the transmission line cross-section has been performed. The results from the developed software module agree with those obtained from ANSYS Q3D simulations, with a relative deviation not exceeding 1.5%. A simplified method for calculating the inductance matrix L of multi-conductor transmission lines with high accuracy has thus been presented. An algorithm and a software module based on it have been developed. Using this method, the L matrices for various transmission lines were computed, showing a relative deviation of no more than 1.5% from those calculated in ANSYS Q3D. The proposed method is applicable for calculating the L matrix of various transmission lines and other conductive structures within CAD systems.

The development of a method for calculating the L matrix was funded by the Russian Science Foundation, grant 23-79-10165, https://rscf.ru/project/23-79-10165/, at TUSUR. The development of the algorithm and the implementation of a software module based on it were carried out under project FEWM-2024-0005 of the Ministry of Education and Science of the Russian Federation.

Tretyakov T.I., Komnatnov M.E. Calculation of per-unit-length inductance matrices for multi-conductor transmission lines based on magnetic flux density. Radiotekhnika. 2026. V. 90. № 5. P. 109−119. DOI: https://doi.org/10.18127/j00338486-202605-14
(In Russian)

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