Analytical expressions for electrodynamic parameters of the shielded microstrip line

On the basis of an electrodynamic model of a screened microstrip line, built on the basis of the projection method using the Chebyshev basis, which explicitly takes into account the edge features of the field, a mathematical model of a microstrip line with a strip conductor was developed. The line width does not exceed the height of the substrate. In this case, the current density on the strip conductor is approximated by only one basis function. Analytical expressions are presented in the form of a sum of slowly and rapidly converging series to determine the main electrodynamic parameters of the line – wave resistance and deceleration coefficient. Due to logarithmic features, slowly converging series are summed up and transformed into rapidly converging power series. In addition, limit expressions in the form of improper integrals are given for the main electrodynamic parameters of an open microstrip line in the quasi-static approximation. Due to the logarithmic features, these integrals are also converted to rapidly converging power series. As a result, simple approximate formulas were obtained. They allow calculating the deceleration coefficient and wave impedance of the line with an error not exceeding 1%, when the width of the strip conductor is less than twice the thickness of the substrate. The results of calculating the electrodynamic parameters obtained on the basis of the developed mathematical model and on the basis of the projection method with an accuracy of up to 5 significant digits are presented. These results make it possible to establish the limits of applicability of the quasi-static approximation and to determine the error in calculating the deceleration coefficient and wave resistance using the obtained analytical expressions. The error does not exceed 0.1%, if the width of the strip conductor is less than twice the thickness of the substrate in a wide range of changes in the substrate dielectric constant and frequency.

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