Analysis of partially filled waveguides for the synthesis of microwave devices as an inverse and optimization problem

Waveguides has great importance in microwave engineering due to their high power handling capacity and low loss features to be designed as microwave devices as well as their ability to determine electrical properties of materials with good accuracy in the wideband frequency range. Despite their advantages of having low-loss property, ease of manufacturability, circular waveguides takes less place in literature compared to rectangular cross-sectioned waveguides. The main reason lies behind this is considered due to the complexity of modal analysis with circular waveguides. Examining circular waveguides has the potential for further possibilities in biomedical applications, non-destructive testing, design of microwave devices etc. To this aim, inverse scattering problem related to circular waveguides are studied in the first place. Inverse scattering problem in electromagnetic theory interests to determine the shape, location and electrical properties of a target utilizing the scattering field data generated by the target in the most general case. In this study, since the location and shape of the object is \ey{a priori} known, only the electrical properties of the target are aimed to be determined. The first problem to be considered in this thesis is to determine the dielectric profile for inhomogeneous, arbitrarily shaped, non-magnetic and lossy dielectric objects loaded in circular waveguides. Direct scattering problem in electromagnetic theory interests to determine the scattered field at any point in the geometry of the problem due to an obstacle in the presence of an incident field. In this study, synthetic scattered field obtained by direct scattering problem is utilized to test the inverse algorithm instead of real measurement results. In direct problem, scattered field is related to the electrical properties of the object via integral equation consisting the electrical type Dyadic Green's Function of hollow circular waveguide. The volume of the object is discretized which gives a set of linear equations by converting the integral equation which is constituted inside the object. First integral equation relating the field quantities inside the object is called as \ey {object equation}. Equivalent volume current density is related to the total electrical field in terms of electrical properties of the medium according to the volume equivalence principle and assumed to be constant in each subcell by applying point matching method. Obtained linear equations are formed as a matrix equation and the unknown current density values are obtained through the moment procedure. In the second integral equation known as \ey {data equation} is solved to calculate the scattered field at any point exterior to the object and interior to the waveguide. Calculated scattered field is then used in place of real measurement data in the process of inverse algorithm. The data and object equations are used to determine the complex permittivity by implementing a Newton type inverse algorithm. The system of equations defining the problem becomes nonlinear in nature and a first-order Taylor expansion is applied to the data equation for the linearization of the problem. Newton algorithm is an iterative optimization approach of which the ability of the convergence increases with closer initial values to the solution. Therefore, \ey {back-propagation algorithm} is employed which maps the scattered field back into the domain of the object. The problem is severely ill posed in nature and regularization in the sense of Tikhonov is applied to obtain a stable solution. It is observed that the applied inverse algorithm is more capable for smoothly varying profiles than abruptly changing inhomogeneities. The integral-equation-based formalism of circular waveguide analysis also has the potential of being used for design purposes by reverting the procedure to an inverse problem algorithm. Similar philosophy is applied to the rectangular waveguide geometry for the purpose of designing microwave devices in the scope of this thesis. Measurement of the dielectric properties of a material differs from the problem of extracting the dielectric profile of an inhomogeneity by the expectation of a good accuracy. In this study, two different approaches are implemented to determine the dielectric permittivity of homogeneous materials via one-port waveguide measurements. In both methods, Newton-Raphson method is applied in optimization routine and the direct problem in each iteration of optimization is solved through moment procedure as explained above. In first approach, dielectric permittivity of the material is measured at each single frequency in the interested band. However, it is observed that the success of the method decreases as the material enlarges and the measured reflection coefficient approaches to zero. To obtain a stable solutions in a wideband range, the same problem is examined using multi-frequency data. Hence, the dielectric permittivity requires to be modeled as a function of frequency. At the first attempt, real part of the permittivity is assumed as frequency invariant while the imaginary part of complex permittivity is accepted as having a frequency invariant conductivity which gives a linearly decreasing variation with increasing frequency. To construct a more reliable analytical model, the frequency behavior of the material in terms of dielectric permittivity is modeled by one-pole Debye relaxation model. In that case, the problem is converted to obtain the Debye parameters for electrical behavior of the material. The verification of both methods are done by comparing the measurement data and the simulated data for extracted frequency model of the material. Microwave passive devices utilizing waveguide structures are first designed by corrugating the shape of or inserting metal discontinuities in the waveguide. Since waveguides are mainly preferred for high frequency applications, minor faults in the fabrication process may drastically affect the operation properties of the device. One another reason for operating waveguides in microwave systems is their high power handling ability. Therefore, breakdown voltage of hollow waveguide may cause problems in some high power applications. Besides, these type of waveguide devices have a bulky and heavy geometry. A possible solution to overcome drawbacks of the corrugated or metal inserted hollow waveguide structures is to fill the apparatus with dielectric materials which provides reduced size devices with a high breakdown voltage. Although difficulties due to the fabrication process of the metallic structure remain. In view of these facts, uniform waveguides with dielectric loading is a possible and efficient solution to the design of microwave devices utilizing waveguides. However, there is rare literature for dielectric loaded uniform waveguides to be used as microwave devices and practically realizable. In the scope of this thesis, two types of dielectric loading are considered. In the first case, homogeneous dielectric layers are entirely filled to the cross-section of the hollow waveguide and a longitudinally inhomogeneous structure is obtained. This type of loading is named as \ey{Longitudinally Inhomogeneous Waveguide} which gives two optimization parameters: dielectric and thickness values of layers. For a practical approach, dielectric permittivity values of layers should be predefined. Hence, the only optimization parameter remains as thickness values of layers. The aim of the optimization process is to determine the thickness values of layers so that the desired scattering parameters could be obtained. For implementation of the proposed method, microwave filters are chosen which requires assembling a plenty of layers. It is observed that the success of optimization process decreases as the targeted optimization parameters increase. For that reason, the whole structure is divided into equal unit cells in which one is interested to optimize much less number of parameters compared to the optimization process of the whole structure. For optimization tool, Trust Region Reflective algorithm is used which is an iterative and built-in function in MATLAB software. Iterative optimization process requires initial values which affect the success of optimization. Therefore, reflection coefficient of the multilayer structure is approximated by regarding the first reflected and transmitted waves at each layer. Afterwards, initial values are generated by solving the approximated overdetermined system in the manner of an inverse problem. A second step in the optimization routine is applied by cascading the unit cells which gives the frequency characteristic of the whole structure. Direct problem in the optimization process of this type of loading is formulated by the Mode Matching method in which only the dominant mode interaction is regarded. Longitudinally Inhomogeneous Waveguide structures require the use of more than one material and layers to be prepared one by one. Preparation of the whole structure is cumbersome and prone to the cutting errors affecting the measurement results. Besides, mechanical stability of the structure is another issue to be tackled. A second type of design possibility is proposed to solve these drawbacks. That type of loading allows one type of material use (one may utilize more than one material as well) and inhomogeneity is provided transversely and longitudinally. Hence, the structure is named as \ey{Transversely Inhomogeneous Waveguide} (TIW) for which the scattering parameters can be obtained through a direct problem solver (e.g. Method of Moments). Although, the large size of the structure gives an impossible task to achieve if one integrates such a direct problem approach into the optimization process. Therefore, the structure is formed by cascaded equal cells as in the previous design possibility, but this time the analysis of modal interaction between propagating and evanescent modes should be regarded to obtain the frequency response of the whole structure. Since the interaction between each mode is regarded, classical scattering matrix constructed for two-port network is replaced by an $N$ port network where $N$ stands for the number of regarded modes and the constituted scattering matrix is named as \ey{Generalized Scattering Matrix} (GSM). The GSM of the whole structure can be obtained by cascading individual GSMs of each cell. For clarity, the GSM for subcells are named as \ey{Generalized Block Scattering Matrix} (GBSM). Moment procedure is carried out to obtain the GBSM of each cell by exciting the structure with each regarded mode and then extracting the modes from the scattered field data. However, integrating the GSM method as the direct problem solver into the optimization procedure gives a complexity of computational cost due to the manipulation of large matrices. Accordingly, GSM is used as the final verification for frequency response of the designed structure. For design procedure, one dimensional LIW model of the two dimensional TIW structure is optimized. The mapping between two structures is provided by equalizing propagation constants to each other. For that purpose, transverse inhomogeneities in TIW are replaced by homogeneous layers modeled with the effective dielectric permittivity of the original medium. Effective dielectric constant of the TIW medium is calculated by observing the total electric field on the dielectric-air interface in the direction of propagation. Firstly, the propagation constant is determined and then effective dielectric constant is simply derived by considering as if only the dominant mode propagates. The experiments are conducted in order to test the proposed methods both for LIW and TIW structures. Devices are designed with FR4 material for which the frequency behavior of dielectric properties are determined by the method proposed in this thesis. Design process is carried out by ignoring the losses of the material, then the measured and expected frequency responses of lossy microwave devices are compared. It has been observed that the proposed methods give good accuracy for experimental cases as well as simulations. The success of the design increases by TIW compared to LIW since the errors arising from manufacturing process decrease for TIW structures. In this thesis, microwave filters are selected for the implementation of the proposed methods which are based on formulating the problem as an optimization and inverse problem by controlling the frequency response. Hence, proposed methods have the potential to be applied to different types of microwave devices of which the device parameters are defined in terms of scattering parameters (e.g. phase shifters, antenna feeds, impedance matching devices etc.).