19电阻矩阵+地损耗文档格式.docx

1、and are the complex current amplitude vector and complex voltage amplitude vector, respectively, and are the current and voltage at the i-th conductor, respectively,is the propagation constant of the transmission line,is the attenuation constant, and is the phase constant. Substitution of these two

2、expressions into the telegraph equations gives or That will lead to If the transmission line is lossless, i.e. then giving the following eigenvalue equations: where and represent, respectively, the complex current amplitude vector and the complex voltage amplitude vector for the lossless line, and t

3、hese two vectors should be real since the transmission line does not have any losses.denotes the phase velocity of the guided wave along the line. Since the transmission line is composed of conductors, these eigenvalue equations will produce eigenvectors for current, , and eigenvectors for voltage,:

4、 Since and are real symmetrical, the matrices and should be mutually transpose and complex conjugate. In fact, where the superscript T symbolizes the transpose and * the conjugate. It follows that the eigenvectors of and the eigenvectors of should be mutually orthogonal, namely bi-orthogonal, where

5、the is Kronecker Delta， It also follows that the eigenvalues of and the eigenvalues of should be mutually complex conjugate. Since the phase velocity is real, these two matrices, and, possess the same eigenvalues, Therefore, this transmission line system composed of conductors and one ground plane p

6、ossess propagating modes. In a lossless line system, all eigenvectors, either and, should be real. For a lossless system, and the earlier mentioned formulasare reduced to If dissipation is involved in a transmission line, then the complex power propagated along the longitudinal direction, i.e. axis,

7、 is and the average power is The power loss per unit length of the line system is where and denote, respectively, the conducting loss per unit length and the dielectric loss per unit length, It will be proven below that the attenuation constant in a dissipated transmission line is given by where and

8、 indicate the attenuation constant for conductors and the attenuation constant for dielectrics , respectively. Proof Sincethen hence To calculate the resistance matrix, the attenuation constant for conductors is consider only, The numerator in this quotient is the dissipated conducting power per uni

9、t length, , which, according to the perturbation theory, is approximately equal to is the number of conductors, is the contour of cross section of the i-th conductor, is the surface resistance of conductors, is the current density of the j-th conductor, which is approximately equal to， is the free c

10、harge density on the j-th conductor. If each of conductors is driven by the elements of the i-th eigenvector, , here, , represents the voltage between the j-th conductor and the ground, then the free charge density on the j-th conductor, , can be determined in a way given earlier, and the surface cu

11、rrent density flowing on the j-th conductor, , can be found in an above-mentioned formula, The average power transmitted along the transmission line, , appeared in the quotient expression of is approximately equal to Therefore the attenuation constant for conductors, , becomes Since there exist curr

12、ent and voltage modes in the transmission line, and, then there are attenuation constants, , given by To calculate the resistance matrix alone, a transmission line is assumed to have conducting loss only, viz., the telegraph equation for voltage is reduced to For a low loss transmission line, , the

13、above equation becomes and the equality of imaginary parts of both sides gives rise to namely It is inferred from that If the dielectric loss is not considered for the time being, , then the attenuation constant for dielectrics becomes zero, and the attenuation constant is simplified as As a result,

14、 the equation is transferred as As mentioned above, there are modes for, , which give rise to equations, are modal currents given by are modal voltages given by are modal attenuation constants given byare modal current densities given byare modal charge densities determined by putting each element of modal voltage, , to the each conductor. Each of the following m