A novel method to solve the complex transcendental equation for the permittivity determination in short-circuited line

The precise permittivity determination of dielectric materials has been a very important task for ever-increasing microwave and millimeter-wave applications. For these reasons, various microwave techniques, each with its unique advantage and constraints, are introduced to characterize the electrical properties of materials. Owing to its relative simplicity, broad frequency coverage and higher accuracy, short-circuited line method, a type of non-resonant method, is widely utilized for characterization of materials. However, obtaining the electrical permittivity from the experimental measurements requires solving a transcendental equation of the form (tanhz)/z = c on the complex plane, where c is obtained experimentally. In solving these transcendental equations, iterative numerical methods are employed, of which the more commonly used is the Nowton-Raphson method. However, the main problem is that the transcendental equation has many roots, and it is difficult for its solution to converge to the correct value unless a very good initial guess is provided using some extra approach. In this presentation, a mathematical procedure is proposed which combines contour integral method, full-label triangles method, and simplex method for solving the transcendental equation on the complex plane which arises in the short-circuited line method. Although this equation possesses infinite solutions, only one of them is physical. Before deciding which of the solutions is the physical one it is better to find all possible permittivity in the given range. In our procedure, a formula of contour integral is used to determine the root number of the transcendental equation in the given range after the singularities are eliminated. And the equation is solved by simplex method from the initial values found by full-label triangles method. The main advantage of this procedure is that all roots can be solved only one time. The inherent ambiguity in the transcendental equation is avoided with the aid of measurement results at adjacent frequency. By using this procedure, a precise characterization of the complex permittivity is possible, thus overcoming some limitations of previous methods. Simulation and measurement of reference material have been carried out to validate the method.