Device Characterization Techniques Based on Causal Relationships

In this paper, we introduce a novel technique for vector network analyzer (VNA) scattering parameter (S -parameter) device characterization. The presented approach is based on causal relationships that provide a connection between the real and imaginary parts as well as between the magnitude and phase of causal network functions. We discuss the problems encountered in the practical implementation of the dispersion relationships such as numerical evaluation of the singular integrals, finite bandwidth of the experimental data, reconstruction artifacts, and nonuniqueness of dispersion relationships. To reduce these problems, we use the generalized dispersion relationships with reference points (subtractions). We develop integral relationships with subtractions and corresponding estimates for error bounds. We show that an appropriate placement of the reference points can significantly reduce the errors caused by finite spectrum. We analyze the applicability of the developed dispersion relationships to the VNA measurements. We demonstrate that certain parameters can be measured with considerably higher accuracy compared with other parameters and that it is possible to select samples from the measured data that can be used as reference points for dispersive integrals. Based on these observations, we propose a device characterization technique that consists of the causality-constrained reconstruction of the sensitive parameters of the measured devices based on more accurately measured characteristics. The proposed technique allows one to accurately characterize the parameters, like quality factor of a high-quality (high- Q) inductors or loss tangent of low-loss transmission lines, of which direct measurement is normally not possible due to limited sensitivity and dynamic range of the VNA. To demonstrate the effectiveness of the proposed technique, we reconstruct the resistance and the quality factor of a high-Q inductor from the accurately measured inductance.