Study On Analytical Solution Of Transmission Line Equations

In the past ten years with the rapid development of VLSI (very large scale integration) circuits and continual spread of electric power system scale, transient response of transmission lines has been regarded as an important research field. There are large numbers of literatures about the research on transmission lines in internal and overseas periodicals, but there is few literature about the research on analytical solution of lossy transmission line equations. The reason is that voltage and current on lines include many direct and return waves. It is quite difficult to solve transmission line equations in time-domain or in complex frequency-domain. So using numerical calculation method becomes a research field to a number of researchers. In this thesis, methods of solving transmission line equations in time-domain and in complex frequency-domain are discussed. A few results were established by using the method in complex frequency-domain to solve uniform transmission line equations as follows: 1) The complex frequency-domain solution with certain excitation under certain boundary conditions are obtained from complex frequency-domain models of transmission line equations. Two Laplace transforms, which can be propitious to calculate the impulse responses of distortionless lines and infinite lines, are educed by using Laplace transform theory's differentiation theorem, frequency-shift theorem and time-shift theorem. 2) The impulse responses of distortionless lines terminated with resistance at terminal can be got by using the first obtained Laplace transform. The zero-state responses of distortionless lines excitated with arbitrary signal can be got according to the impulse responses.DC steady-state responses and sinusoidal steady-state responses of distortionless lines are obtained from the analytical solution with step and sinusoidal excitatons. Based on DC and sinusoidal steady-state responses, the steady-state responses to distortionless lines with nonsinusoidal periodic signal excitation can be got. At the same time the condition of insuring transmitted signal distortionless is discussed. The zero-input responses of distortionless lines are analyzed and the analytical expressions are got. Based on travelling wave's propagation and reflection rules, the analytical solution of distortionless lines terminated with inductance or capacitance are educed. 3) The impulse responses of lossy infinite uniform transmission lines can be found by using the second obtained Laplace transform when the internal resistance of voltage source is zero. 4) The sinusoidal steady-state responses can be got directly from the complex frequency-domain solution,and the DC steady-state solution can be found by using the final value theorem of Laplace transform theory. 5) Transient responses of distortionless transmission lines with step and sinusoidal excitations, transient and steady responses of distortionless transmission lines excited with rectangle voltage signal, zero-input responses of distortionless lines, transient response of distortionless transmission lines with step excitation and terminated with inductance, transient response of distortionless transmission lines excited with exponential signal and terminated with capacitance, impulse responses of infinite transmission lines, DC steady-state responses and sinusoidal steady-state responses of lossy transmission lines, steady-state responses of lossy transmission lines excited with square voltage signal are respectively calculated by examples and the analytical solution educed in this thesis, and curve of each response is protracted. The results in literatures and experimental waveforms demonstrated that the proposed method is valid and conclusions are correct.