Improved Technique for Determining Complex Permittivity with the Transmission/Reflection Method

The transmission/reflection method for complex permittivity and permeability determination is studied. The special case of permittivity measurement is examined in detail. New robust algorithms for permittivity determination that eliminate the ill-behaved nature of the commonly used procedures at frequencies corresponding to integer multiples of one-half wavelength in the sample are presented. An error analysis is presented which yields estimates of the errors incurred due to the uncertainty in scattering parameters, length measurement, and reference plane position. In addition, new equations are derived for determining complex permittivity independent of reference plane position and sample length. ,B ROAD-BAND measurements of complex permittivity and permeability are required for a multitude of applications. Due to its relative simplicity, the transmission/reflection (TR) method is presently a widely used broad-band measurement technique. The relevant literature in this area is copious and no attempt is made in this paper to review it exhaustively. A measurement using the TR method proceeds by placing a sample in a section of waveguide or coaxial line and measuring the two-port complex scattering parameters, preferably by an automatic network analyzer (ANA). The scattering equations then relate the measured scattering parameters to the permittivity and permeability of the material. The system of equations contains as unknowns the complex permittivity and permeability, the calibration reference plane positions, and, in some applications, the sample length. This system of equations is generally overdetermined and therefore can be solved in various ways. From a pragmatic viewpoint, what is needed are equations that are stable over the frequency range of interest and equations that do not depend on the position of the calibration reference plane. With the development of modern network analyzer systems there is generally no paucity of data; thus correct and efficient numerical algorithms for the reduction of the scattering data are of paramount importance. To accommodate modern network analyzer acquisition sysManuscript received November 2, 1989; revised March 30, 1990. The authors are with the Broadband Microwave Metrology Group, National Institute of Standards and Technology, Boulder, CO 80303. IEEE Log Number 9036432. tems, Nicolson and Ross [I] and Weir [2] introduced procedures for obtaining broad-band measurements in both time and frequency domains. In the Nicolson-Ross procedure, the equations for the scattering parameters are combined in such a fashion as to allow the system of equations to decouple, yielding an explicit equation for the permittivity and permeability as a function of the scattering parameters. This solution has formed the basis of the commonly utilized technique for obtaining permittivity and permeability from scattering measurements [3]-[5]. The compact form of these equations, while elegant, is not well behaved for low-loss materials at frequencies corresponding to integer multiples of one-half wavelength in the sample. At these frequencies the solution for low-loss materials is divergent. Many researchers have bypassed this problem by using samples which are less than one-half wavelength long at the highest measurement frequency. However, this approach, as shown later in the paper, severely limits the viability of the TR method since short samples increase measurement uncertainty. Stuchly and Matuszewski [6] presented a slightly different derivation from that of Nicolson and Ross and obtained two explicit equations for the permittivity. They showed that one of these equations was ambiguous;, the other equation was similar to the Nicolson-Ross equation which is unstable for low-loss materials at multiples of one-half integer wavelengths. Ligthardt [7], in a detailed analysis, presented a method for shorted line measurements where the scattering equations for the permittivity were solved over a calculated uncertainty region and the results were then averaged. The equations used by Ligthardt are useful for high-loss materials, but for lowloss materials they suffer the same pathologies as the Nicolson-Ross [I] and Weir [2] equations at multiples of one-half wavelength. It can be shown that the NicolsonRoss-Weir solution for combined permeability and permittivity for low-loss materials is inherently divergent at integer multiples of one-half wavelength in the sample. In this paper we present a procedure for obtaining complex permittivity from the scattering equations which is stable over the frequency spectrum. This procedure minimizes the instability of the Nicolson-Ross-Weir equations by U.S. Government work not protected by U.S. copyright BAKER-JARVIS et al.: IMPROVED TECHNIQUE FOR DETERMINING COMPLEX P E R M I ~ I V I T Y I I I I where I I I + L 4 I