Study On Two Kinds Of Electromagnetic Field Eigenvalue Problems

The electromagnetic field eigenvalue problem remains an important topic in the electromagnetism project domain, especially in the microwave technology area, which is not only because the analysis of electromagnetic field and transmission characteristic of modes in many structures such as waveguide, resonant cavity and dielectric, e.g., periodic dielectric, are related to eigenvalue problems, but also because the solution of eigenvalue problem is the foundation of many microwave components and systems design. Two kinds of eigenvalue problems, namely the dielectric eigenvalue problem and the structure eigenvalue problem, are studied in the paper. In the theoretical research of the dielectric eigenvalue problem, stress is emphasized on the longitudinal characteristic modes in two-dimensionally periodic dielectric. The general two-dimensionally periodic dielectric theory is introduced, and the control equation governing the characteristic modes is deduced in detail. The reason why it is difficult to study longitudinal propagation in two-dimensionally periodic dielectric by the general two-dimensionally periodic dielectric theory is given. A novel coordinate system called the propagation coordinate is presented, and correspondingly an effective direction is found. The control equations for longitudinal TE and TM modes are firstly separated from the initial control equation on the special direction in the propagation coordinate. The eigenvalues and corresponding eigenvectors of longitudinal TE or TM modes in periodic dielectric are obtained. The separation of the initial control equation contributes significantly in both simplifying the analysis of longitudinal modes in two-dimensionally periodic dielectric and improving the general periodic dielectric theory. In the theoretical analysis of the structure eigenvalue problem, the complex guided wave structure eigenvalue problems are mainly investigated, which include three-dimensionally complex coaxial resonant cavity and arbitrary waveguide eigenvalue problems. An equivalent circuit and boundary element mixed method is presented to resolve complex coaxial resonant cavity. The three-dimensionally resonant cavity eigenvalue problem is transformed into a two-dimensionally problem in the cross section of the cavity and an equivalent circuit problem. By applying the boundary element method to the analysis two-dimensionally electromagnetic field problem, the circuit parameters are extracted. The transverse resonant method is adopted to calculate the resonant frequency. The mixed method showed outstanding efficiency and precision in the analysis of the coaxial resonant cavity. Furthermore, the formulae of transmitted power, loss power, attenuation constant and equivalent impedance for TM and TE modes are deduced. This is an extended application of the boundary element method in electromagnetic field eigenvalue problem.The dispersion relation of longitudinal modes in two-dimensionally periodic dielectric is approximately analyzed. The separated control equations for longitudinal TM and TE modes are also used to analyze rectangular and circular dielectric waveguide array. The Brillouin diagrams of longitudinal TM and TE modes with different unperturbed circles and the dispersion relation in the stop bands are given. We find some interesting phenomena. First, stopbands appear in the Brillouin diagrams; Furthermore, the wave numbers of longitudinal modes turn into complex in the stopbands; Finally, modes purely propagate on the longitudinal direction. Besides, as the eigenvectors, the amplitudes of longitudinal TE and TM modes are calculated and analyzed. The instructive law of harmonics coupling and interaction is generalized. The harmonics interactions are intensive inside stopbands and weak outside stopbands. The mixed equivalent circuit and boundary element method are used to study 30Mev cyclotron cavity. The resonant frequency is computed and primary optimization design is presented. The theoretical result agrees with the measured result on the model or on the enti...