An alternative model for aerial multiconductor transmission lines excited by external electromagnetic fields based on the method of characteristics

Abstract

In this paper, a model of aerial multiconductor transmission lines with frequency-dependent electrical parameters for the simulation of time-domain electromagnetic transients due to incident electromagnetic fields is presented. The frequency dependency of the electrical parameters is taken into account using the penetration impedance; this allows applying the method of characteristics without having to resort to modal transformations. With this alternative approach, transforming the transmission line partial differential equations to ordinary differential equations is performed in the phase domain, which simplifies the mathematical development of the method and also the numerical solution. The validity of the proposed model is shown with five study cases involving different transmission systems and excitations.

Appendix

In this Appendix, the recursive convolution algorithm based on the method of the auxiliary differential equation is revisited. Using (4) and (5b) can be expressed in the Laplace domain as follows

$$ {\varvec{\uppsi}}(x,s) = \sum\limits_{i = 1}^{N} {{\varvec{\uppsi}}_{i} (x,s)} $$

(25a)

where

$$ {\varvec{\uppsi}}_{i} (x,s) = \left( {\frac{{{\mathbf{K}}_{i} }}{{s - p_{i} }}} \right)\,{\mathbf{I}}(x,s) $$

(25b)

From (23b), it follows that

$$ s{\varvec{\uppsi}}_{i} (x,s) - p_{i} {\varvec{\uppsi}}_{i} (x,s) = \,{\mathbf{K}}_{i} {\mathbf{I}}(x,s) $$

(26)

Transforming (24a, 27) to the time domain gives

$$ \frac{d}{dt}{\varvec{\uppsi}}_{i} (x,t) - p_{i} {\varvec{\uppsi}}_{i} (x,t) = \,{\mathbf{K}}_{i} {\mathbf{i}}(x,t) $$

(27)

The auxiliary differential Eq. (27) can be solved applying several finite-difference schemes; here, it is used the backward-difference formula; therefore, it can be written

$$ \frac{{{\varvec{\uppsi}}_{i} (x,t) - {\varvec{\uppsi}}_{i} (x,t - \Delta t)}}{\Delta t} - p_{i} {\varvec{\uppsi}}_{i} (x,t) = {\mathbf{K}}_{i} {\mathbf{i}}(x,t) $$

(28)

Solving for Ψ_i(x,t) gives

$$ {\varvec{\uppsi}}_{i} (x,t) = \frac{1}{{1 - p_{i} \Delta t}}{\varvec{\uppsi}}_{i} (x,t - \Delta t) + \frac{{\Delta t{\kern 1pt} }}{{1 - p_{i} \Delta t}}{\mathbf{K}}_{i} {\mathbf{i}}(x,t) $$

(29)

Finally, using (29) expression (5b) becomes

$$ {\varvec{\uppsi}}(x,t) = \sum\limits_{i = 1}^{N} {\frac{1}{{1 - p_{i} \Delta t}}{\varvec{\uppsi}}_{i} (x,t - \Delta t)} + \left( {\sum\limits_{i = 1}^{N} {\frac{\Delta t}{{1 - p_{i} \Delta t}}{\mathbf{K}}_{i} } } \right){\mathbf{i}}(x,t). $$

(30)