复杂电磁边界下基于曲线坐标的保角变换法
A Conformal Mapping Method Based on Curve Coordinates under Complex Electromagnetic Boundaries

针对具有复杂几何边界条件的二维均匀微波传输线中电磁场难以解析求解从而造成其设计困难的问题,提出了一种基于曲线坐标的保角变换法(CMM)。该方法基于二维传输线的横电磁场(TEM)模型。首先利用保角变换将复杂传输线的边界曲线映射为简单传输线的边界曲线,建立复杂边界传输线的曲线坐标;然后利用曲线坐标把复杂边界传输线所满足的麦克斯韦方程转换到简单边界传输线中,形成两种传输线间的电磁映射;最后论证了两种传输线间电荷、电位分别对应相等,由此得到了与传输特性紧密相关的复杂传输线的电容,进而用以指导传输线的设计。由于以电位为中间量的传统保角变换计算较为复杂,采用CMM方法直接从场出发,省去了关于电位的中间计算过程,简化了计算过程。仿真结果表明,不同尺寸下同轴线特征阻抗仿真值与理论值吻合得很好,验证了CMM方法的正确性。
A conformal mapping method (CMM) based on curve coordinates is proposed to focus on the problem that the electromagnetic field of the two？？dimensional uniform microwave transmission lines with complex geometric boundary conditions is hard to be analytically solved and thus its design is difficult. The method is based on a TEM model of the cross？？dimensional electromagnetic transmission line. First, the conformal mapping is used to map the complex boundary curve of the transmission line into boundary curves of a simple transmission line, and the curvilinear coordinates of the complex border transmission lines are built; Then the curvilinear coordinates are used to convert the Maxwell’s equations for the complex border transmission lines to the simple border transmission lines, and the electromagnetic mapping between the two kinds of transmission line is formed. Finally, it is verified that both the charge and the potential between two kinds of transmission lines are correspondingly equal, so that the capacitance of complex transmission line that is closely related to the transfer characteristic is obtained and can further be used to guide the design of the transmission line. Simulation results and comparisons with the traditional conformal mapping calculation method with the potential being the intermediate amount show that the CMM method starts directly from the field, eliminates the potential on the intermediate calculations, and its results agree well with theoretical calculation