Waves and heaviside propagator in transmission lines

Abstract

We characterize all plane and modal waves of lossy multiconductor lines. They are obtained by solving matrix dierential equations in closed-form. First-order and second-order telegraph formulations are considered. Wavesare described in terms of a scalar basis coupled with a nite number of matrixiterates. Modal waves are decomposed in forward and backward propagators. This decomposition can be applied in obtaining transmission and reection matrices at junctions. Eects due to initial, boundary or external inputs are described in terms of an integral contribution due to initial data, a convolution due to external disturbances and a bilinear term involving boundary values. For a two conductor transmission line satisfying the Heaviside condition, the fundamental response is obtained for the rst-order system and for the telegraph formulation. Numerical examples are given for lossy and lossless transmission lines and for determination of initial values of steady solutions.

Keywords:

• Lossy multiconductor transmission lines
• plane waves
• modal waves
• heaviside propagator

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In the mathematical physics literature, harmonic analysis usually transforms in space in order to have a time differential equation of a lumped model to deal with. In signal processing, time information can be causal or anti-causal [45].

2 In free space, it is assumed that $\mathbf{h}$ $\to$ 0 when $|z|$  $\to$  ∞. In general, $\mathbf{h}=\mathbf{h}(t,t^{{}^{\mathrm{\prime }}},z,\xi )$ but due to invariance, it turns out $\mathbf{h}(t,t^{{}^{\mathrm{\prime }}},z,\xi )=\mathbf{h}(t-t^{{}^{\mathrm{\prime }}},z-\xi )$.

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior– Brasil (CAPES) – Finance Code 001.