Complex Analytical Conformal Mapping Application in the Electromagnetic Engineering
|School||Xi'an University of Electronic Science and Technology|
|Course||Electromagnetic Field and Microwave Technology|
|Keywords||Conformal transformation Electromagnetic Theory Static field solution Capacitance Characteristic impedance Fractional linear transformations Unified conformal transformation Electric axis method Three-wire transmission line Plane mirror Active conformal transformation ∞ at the mirror The Khodorkovsky confucianist transform Minkowski mapping inverse Confucianism The active inverse mapping and passive inverse mapping Generalized method of images Schwarz - Christoph mapping|
The calculation of the potential field of electromagnetic theory, due to the complexity undone conformal transformation of the actual electromagnetic engineering problems can be complex boundary transform into simple easy-to-solve boundary, thus complex conformal mapping method as a variety of other potential field the basis of the solution of the electromagnetic fields play a pivotal role. This paper studies resolve complex conformal mapping applications in electromagnetic theory, which can be summarized as follows: 1. Outlined the significance of the study of complex conformal mapping applications in electromagnetic theory, a brief review of the brief history of the complex conformal transformation and Applied Research Overview. 2. Discuss the characteristics of analytic functions, the relationship between the bit functions and complex analytic functions and commonly used in elementary analytic functions in the two-dimensional static field. 3 summarizes in detail the application of fractional linear transformations in static field and microwave transmission line, fractional linear transformation of the static field and microwave transmission line system. More detailed introduction of the Smith chart, Weissfloch, circular diagram, circle transformation theorem in microwave systems for the typical application instance, and deduced out of the Weissfloch circle on the angle φ and transmission line length l of the relationship and the transmission line length l of the calculation formula. Christoph Schwarz - mapping the static field and microwave transmission line. Summarized in detail. A physical problem solving process is summarized as the following process: (1) the original model. Practical problems found by the electrostatic field theory one can be used for analysis of the polygon. (2) z → t transform. Transformation given polygon t plane of the real axis (3) t → W transform t plane of the real axis transform the perimeter of the typical problems of the typical questions are known. (4) z → W conversion. Given the demand answers. (5) for further analysis operation according to the answers. Such procedural treatment of the mapping method can make it by hard into the simple, easy engineering staff to learn to use. Application examples are enumerated, are derived in detail. In the derivation of the potential distribution in the two infinite conducting plates instance, to correct the literature the two parameter error . 5 gives the results of in-depth study of analytic conformal transformation: (1) the plane mirror as the basic model, in-depth discussion of a variety of typical applications of active conformal transformation that in solving regional transformed, allow ∞ at the image charge of the original problem. The conclusion breakthrough principles of general image method, also see the image charge paired with the original charge principle. (2) in-depth study of the inverse of Confucianism Minkowski mapping W = z (z 2 sup> the-c 2 sup>) 1/2 sup> (c gt ; 0), for the conformal mapping in the electromagnetic field can be divided into two categories: the active inverse Confucianism Minkowski mapping to solve the elliptic conductor line outside the column and the limited width of the conductor plate charge ρ l generalized two-dimensional image problems; the passive inverse logarithmic Confucianism can Minkowski mapping gives the the infinite conductor above the plane of the vertical finite conductor plate capacitor C approximation solution. (3) a bit of a unified field analytical solving method, to replace the existing image method and electric axis method using conformal mapping. The unified conformal mapping has two meanings: one from the various calculations of the potential field methods: is a two-dimensional image method or electric axis method can be unified to complex conformal mapping: a two-dimensional image method corresponds to the active Paul conformal mapping (mapping contains the line charge); electric axis method corresponding passive conformal mapping (mapped) does not contain a line charge. Thus, conceptually and ideologically there will be a unified approach. Another layer of electrical axis, with uniform conformal mapping W = (zd) / (zd), the the eccentric circle clusters and double the parallel conductors column system unify. The difference is: the eccentric circle cluster mapped to the W-plane unit circle; the dual parallel conductors column system to expand to outside the unit circle. Unified conformal mapping, plot method for approximating solutions of the mutual capacitance of three-wire. Line 3 is mapped circle in the W plane characteristics detailed derivation, the relationship between the position of the unit circle, and thus launched the imaginary axis of the other n lines in the z-plane image regularity. Such unified security angle mapping insurance angle, Paul Yuan, Paul symmetry and Line 3 mapping circle on the W plane image of the characteristics, symmetrical three-wire transmission line mutual capacitance C of the approximation solution, can be easily obtained symmetrical four lines Mutual capacitance C approximation solution. This conclusion can also be extended to the case of the n lines.