Modern Characterization of Electromagnetic Systems and its Associated Metrology. Magdalena Salazar-Palma

      Preface

The area of electromagnetics is an evolutionary one. In the earlier days the analysis in this area was limited to 11 separable coordinate systems for the solution of Helmholtz equations. The eleven coordinate systems are rectangular, circular cylinder, elliptic cylinder, parabolic cylinder, spherical, conical, parabolic, prolate spheroidal, oblate spheroidal, ellipsoidal and paraboloidal coordinates. However, Laplace’s equation is separable in 13 coordinate systems, the additional two being the bispherical and the toroidal coordinate systems. Outside these coordinate systems it was not possible to develop a solution for electromagnetic problems in the earlier days. However, with the advent of numerical methods this situation changed and it was possible to solve real practical problems in any system. This development took place in two distinct stages and was primarily addressed by Prof. Roger F. Harrington. In the first phase he proposed to develop the solution of an electromagnetic field problem in terms of unknown currents, both electric and magnetic and not fields by placing some equivalent currents to represent the actual sources so that these currents produce exactly the same desired fields in each region. From these currents he computed the electric and the magnetic vector potentials in any coordinate system. In the integral representation of the potentials in terms of the unknown currents, the free space Green’s function was used which simplified the formulation considerably as no complicated form of the Green’s function for any complicated environment was necessary. From the potentials, the fields, both electric and magnetic, were developed by invoking the Maxwell‐Hertz‐Heaviside equations. This made the mathematical analysis quite analytic and simplified many of the complexities related to the complicated Green’s theorem. This was the main theme in his book “Time Harmonic Electromagnetic Fields”, McGraw Hill, 1961. At the end of this book he tried to develop a variational form for all these concepts so that a numerical technique can be applied and one can solve any electromagnetic boundary value problem of interest. This theme was further developed in the second stage through his second classic book “Field Computations by Moment Methods”, Macmillan Company, 1968. In the second book he illustrated how to solve a general electromagnetic field problem. This gradual development took almost half a century to mature. In the experimental realm, unfortunately, no such progress has been made. This may be partially due to decisions taken by the past leadership of the IEEE Antennas and Propagation Society (AP‐S) who first essentially disassociated measurements from their primary focus leading antenna measurement practitioners to form the Antenna Measurements Techniques Association (AMTA) as an organization different from IEEE AP‐S. And later on even the numerical techniques part was not considered in the main theme of the IEEE Antennas and Propagation Society leading to the formation of the Applied Computational Electromagnetic Society (ACES). However, in recent times these shortcomings of the past decisions of the AP‐S leadership have been addressed.

The objective of this book is to advance the state of the art of antenna measurements and not being limited to the situation that measurements can be made in one of the separable coordinate systems just like the state of electromagnetics over half a century ago. We propose to carry out this transformation in the realm of measurement first by trying to find a set of equivalent currents just like we do in theory and then solve for these unknown currents using the Maxwell‐Hertz‐Heaviside equations via the Method of Moments popularized by Prof. Harrington. Since the expressions between the measured fields and the unknown currents are analytic and related by Maxwell‐Hertz‐Heaviside equations, the measurements can be carried out in any arbitrary geometry and not just limited to the planar, cylindrical or spherical geometries. The advantage of this new methodology as presented in this book through the topic “Source Reconstruction Method” is that the measurement of the fields need not be done using a Nyquist sampling criteria which opens up new avenues particularly in the very high frequency regime of the electromagnetic spectrum where it might be difficult to take measurement samples half a wavelength apart. Secondly as will be illustrated these measurement samples need not even be performed in any specified plane. Also because of the analytical relationship between the sources that generate the fields and the fields themselves it is possible to go beyond the Raleigh resolution limit and achieve super resolution in the diagnosis of radiating structures. In the Raleigh limit the resolution is limited by the uncertainty principle and that is determined by the length of the aperture whose Fourier transform we are looking at whereas in the super resolution system there is no such restriction. Another objective of this book is to outline a very simple procedure to recover the non‐minimum phase of any electromagnetic system using amplitude‐only data. This simple procedure is based on the principle of causality which results in the Hilbert transform relationship between the real and the imaginary parts of a transfer function of any linear time invariant system. The philosophy of model order reduction can also be implemented using the concepts of total least squares along with the singular value decomposition. This makes the ill‐posed deconvolution problem quite stable numerically. Finally, it is shown how to interpolate and extrapolate measured data including filling up the gap of missing measured near/far‐field data.

      The book contains ten chapters. In Chapter 1, the mathematical preliminaries are described. In the mathematical field of numerical analysis, model order reduction is the key to processing measured data. This also enables us to interpolate and extrapolate measured data. The philosophy of model order reduction is outlined in this chapter along with the concepts of total least squares and singular value decomposition.

      In Chapter 2, we present the matrix pencil method (MPM) which is a methodology to approximate a given data set by a sum of complex exponentials. The objective is to interpolate and extrapolate data and also to extract certain parameters so as to compress the data set. First the methodology is presented followed by some application in electromagnetic system characterization. The applications involve using this methodology to deembed device characteristics and obtain accurate and high resolution characterization, enhance network analyzer measurements when not enough physical bandwidth is available for measurements, minimize unwanted reflections in antenna measurements and, when performing system characterization in a non‐anechoic environment, to extract a single set of exponents representing the resonant frequency of an object when data from multiple look angles are given and compute directions of arrival estimation