Numerical Solution of the Dielectric Equation for a Coaxial Line

Dielectric measurements using a coaxial transmission line terminated by a shielded open circuit give rise to the transcendental equation z tan z = c, where c = a + jb is a measured quantity. Since a ¿ 0, Newton´s method will rapidly converge to one of the roots unless it gets trapped in an attractive cycle. Regions containing the lowest and second-lowest roots are sketched. Moreover, starting values are given in terms of the quantity c so that the method will converge to the desired root among the first four roots and also avoid attractive cycles. Using high-precision arithmetic, the error in z may be estimated numerically in terms of the error in c.