平行多导线系统的相模变换矩阵和波阻抗计算

Calculation of Phase-Mode Transformation Matrix and Surge Impedance for Parallel Multi-Conductor Systems

电力系统中大量使用的输电线路是平行多导线系统。一个平行多导线系统的基本参数为单位长度串联阻抗矩阵Z 和导纳矩阵 Y,相似变换矩阵 T_(v)、T_(i)分别将 ZY 和 YZ 对角化。利用线性矩阵代数理论,通过严格数学证明,讨论了 T_(v)和 T_(i)的内在关系,说明被大量文献引用的 T_(i)= T_(v)^(-T)是一种人为规定,而不是将 ZY 和 YZ 对角化的必要条件。证明 T_(v)、T_(i)也能将 Z 和 Y 对角化,澄清了不能对角化的不实观点。平行多导线系统波阻抗是基本参数的导出参数,明确了相域波阻抗的定义,给出了计算方法和几种计算公式。证明模域波阻抗随变换矩阵的不同而不同,但相域波阻抗不随变换矩阵的变化而变化。还阐述了单导线系统与多导线系统波阻抗的区别,不可将前者的计算公式直接应用于后者。最后,通过实例示范了 ZY、YZ 的对角化和波阻抗计算。

A large amount of parallel multi-conductor systems is used in power industry.The basic parameters of such a system are the per-unit-length series impedance matrix Z and shunt admittance matrix Y.Similarity transformation matrixes Tv and Ti diagonalize ZY and YZ,respectively.By applying linear matrix algebraic theory,this paper discusses the inherent relationship between Tv and Ti through strict mathematic deduction.The equation T_(i)=T_(v)^(-T),which has been cited in a variety of literature,is just a rule made purposely to facilitate the mathematic process,but not the necessary condition for the diagonalization of ZY and YZ.It is proved that T_(v) and T_(i) can also diagonalize Z and Y,which has been otherwise stated in some reference.Surge impedances,parameters resulted from the basic ones,are clearly defined for the phase domain.Methods and formulae to evaluate them are presented.The surge impedances in the modal domain change with the tranforamtion matrix,but those in the phase domain do not.The difference between the surge impedances of a single conductor and those of a multi-conductor system is clarified,and the formulae for the former should not be adopted directly for the latter.At the end,the diagnolizing of ZY and YZ and the calculating of surge impedances are demonstrated.