Complex Analytical Conformal Mapping Application In The Electromagnetic Engineering

Potential calculation is the main content of the electromagnetic theory. Due to the complexity of the actual electromagnetic engineering problem, the complex conformal mapping can be used to map complicated boundary into simple boundary. Thus the complex conformal mapping method which plays an important role in electromagnetic region is the foundational method of solving potential. This paper discusses the analytical conformal mapping application in the electromagnetic engineering. The work mainly focuses on:1. The significance of the complex conformal mapping application in electromagnetic theory is introduced. The development and the application of the complex conformal mapping are briefly reviewed.2. This thesis discusses the analytic function property, the relationship between potential function and the complex analytic function, and the application of the common elementary analytic functions in two-dimensional static field.3. The application and influence of fractional linear mapping in static field and microwave transmission line are summarized in details. Typical application examples of Smith chart, Weissfloch chart and the circle mapping theorem in microwave system are introduced. The relationship between the angle?of Weissfloch chart and the length lof transmission line is deduced.4. The application of Schwarz - Christopher mapping in static field and microwave transmission line are deduced respectively. The solution process of an actual physical problem can be concluded as follows:(1) original modelThe practical problem is replaced by a polygon problem which can be analyzed in electrostatic theory.(2) z?tmappingThe given polygon is mapped into the real axis in t-plane.(3)t?W mappingThe real axis in t-plane is mapped into the boundary of the classic problem which is solved already.(4) z?W mappingThe demanded solution is achieved.(5) Further analysis based on computing solutions will be given. After dealing the mapping method this way, we can get the easier result and convenient for practical engineers.All examples are derived in details. Two parameters in [1] are corrected in deriving of the potential between two infinite conducting plates. The deduction in [32,33] is completed during the application of symmetric stripline and microstrip transmission line.5. The further research of the analytical conformal mapping is given.(1) The plane mirror is considered as the basic model. Various typical applications of the active conformal mapping are discussed. It is pointed out that the solution region exist image charge in infinity after mapping, which exceeded the criteria of the generalized image method. The original charge and the image charge are in pairs.(2) This thesis discusses the inverse Joukowski mapping W=z+(z²-c²)^1/2(c>0), which can be classi?ed into active and passive inverse transformation. By using the active inverse Joukowski mapping, the generalized image problems that the line charge?_l is located outside the elliptical conducting cylinder or the finite conducting plate can be solved. By using the passive logarithmic inverse Joukowski mapping, the capacitance C of a finite conducting plate placed vertically above the infinite conducting plate can be solved.(3) Thus the conformal mapping method can replace the image method and electrical axis method and become the uniform method to solve the electrostatic problems.The unified conformal mapping has two meanings.The first is pointed out from kinds of calculating methods that both the image method in two-dimensional static field and the electric axis method can be unified as complex conformal mapping. The former corresponds to active conformal mapping which contains the line charge. The latter corresponds to passive conformal mapping which does not contain the line charge. Then, we can get the unified method both in concept and idea.The other meaning is that, for the electric axis method, the issues of eccentric coaxial line and parallel pairs of cylindrical are unified by the unified conformal mapping W=(z-d)/(z+d). The difference is as follows. The eccentric cluster is mapped into the unit circle of W-plane and the parallel pairs of cylindrical are mapped outside the unit circle of W-plane.According to conformal mapping method, the approximation solution of capacitance C of three-wire transmission line can be derived by equal area method. The characteristic, position and the relationship with the unit circle of the line 3 which is mapped into W-plane are deduced in details. The regulation of the lines in image axis of z-plane is derived. By using the unified conformal mapping properties of conformal, conformal circle, conformal symmetric and the mapping characteristics of Line 3 in W-plane, the mutual capacitance approximation solution of symmetric four-line can be easily deduced by the symmetric three-wire transmission line. The conclusion can be deduced to n-wire transmission line.