Pade Approximation of Line Parameters for the Analysis of Transient Processes in Uniform Lossy Lines

In 21th century, the tasks associated with modeling real systems, whether they are electrical circuits, data transmission lines, or other physical systems, are becoming increasingly complex. Accurate mathematical models are essential for predicting the behavior of such systems, enhancing their efficiency and reliability. In this context, the Pade approximation offers significant advantages due to its ability to accurately describe system behavior based on a limited number of parameters. The main challenge faced by research on Pade approximation is the need to optimize the algorithms used for their application in real systems with limited computational resources. It is crucial to find the optimal balance between the accuracy of the approximation and the computational complexity of the algorithms. Extremely high accuracy may require substantial computational resources, making the model impractical for real-time use.

Allen et al. [1] explored the application of the Pade method for approximating transmission line parameters to streamline the modeling of transient processes. The research demonstrated that the Pade method provides a high level of accuracy in estimating transmission line parameters such as impedance, capacitance, and inductance. Furthermore, the study highlighted the method’s potential in improving the efficiency and precision of transient process modeling, making it a valuable tool for engineers and researchers in the field of electrical engineering.

Başar [2] conducted research applying the Pade method to develop rational functions that effectively modeled the behavior of transmission lines under various conditions, such as signal switching and load variations. The study revealed that the approximations generated by the Pade method surpassed the accuracy of traditional polynomial methods across diverse operational scenarios. Additionally, this approach proved to be robust and versatile, offering significant improvements in the predictive modeling of transmission line behavior under dynamic conditions, thereby enhancing the reliability and performance of electrical systems. Keleş and Tekalp [3] conducted a study on the application of the Pade method for generating rational functions that precisely approximate original functions. The findings indicated that the Pade method markedly enhanced accuracy in comparison to straightforward polynomial approximations. Furthermore, the research highlighted the method’s effectiveness in capturing the intricate behaviors of the original functions, making it a valuable technique for various analytical and computational applications where precision is crucial.

Ali et al. [4] conducted a comparative study on the precision of rational functions versus polynomial approximations of the same order. The study found that rational functions consistently demonstrated significantly higher accuracy than polynomial approximations of equivalent order. Additionally, the research underscored the superiority of rational functions in modeling complex systems, suggesting their greater efficacy in applications requiring high precision and reliability. In the study of Branin [5], the rate of convergence of the Pade method was assessed in comparison to alternative approximation techniques, particularly in scenarios featuring functions with poles. The results validated that the Pade method exhibited superior convergence speed, especially when dealing with functions containing poles, which posed challenges for other approximation methods.

In the investigation conducted by Wedepohl [6], transmission lines in electrical engineering and telecommunications were examined as concrete pathways for the propagation of signals. The study explored the fundamental operational principles and characteristics of these transmission lines, alongside their influence on signal transmission. The conclusions validated the essential role of transmission lines in electrical engineering and telecommunications, emphasizing their effectiveness in reliably transmitting signals over long distances.

In the study by Chang [7], the focus was on the modeling of transient processes within data transmission systems. This research is crucial for comprehending how these systems respond to various changes, such as signal activation or deactivation, load fluctuations, and other influencing factors. Various modeling methods and techniques were explored to capture the dynamic changes within the system, aiming to forecast their impact on overall system performance. The study’s conclusions highlighted the critical importance of modeling transient processes in data transmission systems to understand their behavior under diverse conditions.

In the research conducted by Palusinski and Lee [8], the emphasis was placed on applying the Pade method to develop models capable of predicting system behavior under dynamic conditions. The study evaluated the effectiveness of the Pade method in comparison to other modeling and approximation techniques for predicting dynamic system behavior. The findings from the study showed that models constructed using the Pade method provided more accurate forecasts of system behavior under dynamic conditions than alternative modeling and approximation methods.

Detailed questions about the influence of Pade approximation on line parameters in the case of non-uniform losses and the impact of nonlinear effects on the accuracy of Pade approximation in analyzing transient processes remained unexplored. These include the method’s influence on line parameters under conditions of non-uniform losses, where variations in resistance, inductance, or capacitance challenge accurate modeling of transient behavior. Additionally, the impact of nonlinear effects, such as high signal amplitudes or harmonics, on the precision of the Pade approximation remains insufficiently studied. While the method demonstrates superior accuracy compared to polynomial approaches, its limitations in handling complex systems with highly variable parameters are not fully understood. Addressing these gaps is critical for enhancing the applicability of the Pade approximation in fields like power transmission and telecommunications, where precise transient analysis is essential.

The aim of the research was to assess the effectiveness and limitations of Pade approximation of line parameters in analyzing transient processes in uniform lossy lines. The problematic questions include the impact of Pade approximation on the accuracy of analyzing transient processes in uniform lossy lines and the discussion of limitations and advantages of using Pade approximation compared to other methods.

3.1 The lossy uniform transmission line modelling according to the Thevenine-Norton structure

The analysis of two-conductor uniform lossy lines involves solving generalized equations in the frequency domain, taking into account the boundary conditions of voltages and currents [2]. Figure 1 shows a two-conductor uniform lossy line of length l (where ground is considered the first conductor), the modelling of which is based on these analysis conditions:

1. Boundary conditions in Eqs. (1)-(2):

$v\left( 0,t \right)={{v}_{1}}\left( t \right)$; $\left( l,t \right)={{v}_{2}}\left( t \right)$,                             (1)

$i\left( 0,t \right)={{i}_{1}}\left( t \right)$; $-i\left( l,t \right)={{i}_{2}}\left( t \right)$.                                 (2)

2. The initial conditions in Eq. (3):

$v\left( x,0 \right)=0$; $i\left( x,0 \right)=0$; $0\le x\le l$.                                  (3)

The parameters R, L, C, and G denote the values per unit length of the line, which typically vary with frequency [1, 2]. So, in the frequency domain in Eq. (4):

$R\equiv R\left( s \right)$; $L\equiv L\left( s \right)$; $C\equiv C\left( s \right)$; $G\equiv G\left( s \right)$.                                 (4)

In the complex frequency domain, the solution of the telegraph equations is expressed in terms of complex frequencies and complex amplitudes. This allows for incorporating both active and reactive signal components, alongside phase shifts. In this base the solution of the telegraphic equations can be expressed in the complex frequency domain as follows Eqs. (5)-(6):

$v\left( x,s \right)=\exp \left[ -\gamma \left( s \right)x \right]A\left( s \right)+\exp \left[ +\gamma \left( s \right)x \right]B\left( s \right)$,                                  (5)

$Z\left( s \right)i\left( x,s \right)=\exp \left[ -\gamma \left( s \right)x \right]A\left( s \right)-\exp \left[ +\gamma \left( s \right)x \right]B\left( s \right)$.                                 (6)

V(x, s) and I(x, s) are the voltage and current at a point x along the transmission line in the frequency domain s.

From Figure 1:

$Z\left( s \right)=\sqrt{\frac{R+Ls}{G+Cs}}$, $Y\left( s \right)={{Z}^{-1}}\left( s \right)$,                                (7)

$Z\gamma \left( s \right)=\sqrt{\left( R+Ls \right)\left( G+Cs \right)}$.                                           (8)

Z(s) and Y(s) denote the per-unit-length impedance and admittance matrices in the frequency domain.

Setting:

$v\left( x,s \right)\equiv {{v}_{x}}\left( s \right)$, $i\left( x,s \right)\equiv {{i}_{x}}\left( s \right)$,                           (9)

and successively adding and subtracting Eqs. (5) and (6), the authors obtain the solutions in Eqs. (10)-(11):

${{v}_{1}}\left( s \right)=Z\left( s \right){{i}_{1}}\left( s \right)+{{E}_{1}}\left( s \right)$,                             (10)

${{v}_{2}}\left( s \right)=Z\left( s \right){{i}_{2}}\left( s \right)+{{E}_{2}}\left( s \right)$,                               (11)

where,

${{v}_{2}}\left( s \right)=Z\left( s \right){{i}_{2}}\left( s \right)+{{E}_{2}}\left( s \right)$,                                (12)

${{E}_{2}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]+\left[ {{v}_{1}}\left( s \right)+Z\left( s \right){{i}_{1}}\left( s \right) \right]$.                                (13)

By comparing Eqs. (10)-(11) with Eqs. (12)-(13), the authors obtain Eqs. (14)-(15):

${{E}_{1}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]+\left[ 2{{v}_{2}}\left( s \right)-{{E}_{2}}\left( s \right) \right]$,                              (14)

${{E}_{2}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]+\left[ 2{{v}_{1}}\left( s \right)-{{E}_{1}}\left( s \right) \right]$.             (15)

Therefore, for Eqs. (10)-(11) is made a correspondence through equivalent network presented in Figure 2, which contains the Thevenine-structure, where E₁(s) and E₂(s) are given in the Eqs. (14)-(15).

ping_mu_jie_tu_2025-03-28_103644.png

Figure 1. Uniform lossy line with one conductor (Ground is the other conductor)

ping_mu_jie_tu_2025-03-28_103648.png

Figure 2. The equivalent network of the loss uniform line with one conductor according to Thevenin – structure

Complex frequencies are represented as the sum of a real part (real frequency) and an imaginary part (complex part). This representation allows for the consideration of both the magnitude and phase of the signal. Analyzing signals in uniform lossy lines is critical due to variations in parameters such as time and distance. By expressing the respective currents, the authors obtain Eqs. (16)-(17):

${{i}_{1}}\left( s \right)=Y\left( s \right){{v}_{1}}\left( s \right)-\exp \left[ -\gamma \left( s \right)l \right]\left[ Y\left( s \right){{v}_{2}}\left( s \right)+{{i}_{2}}\left( s \right) \right]$,                      (16)

${{i}_{2}}\left( s \right)=Y\left( s \right){{v}_{2}}\left( s \right)-\exp \left[ -\gamma \left( s \right)l \right]\left[ Y\left( s \right){{v}_{1}}\left( s \right)+{{i}_{1}}\left( s \right) \right]~$                           (17)

Denoting:

${{J}_{1}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]\left[ Y\left( s \right){{v}_{2}}\left( s \right)+{{i}_{2}}\left( s \right) \right]$,                (18)

${{J}_{2}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]\left[ Y\left( s \right){{v}_{2}}\left( s \right)+{{i}_{2}}\left( s \right) \right]$.                        (19)

Gives:

${{i}_{1}}\left( s \right)=Y\left( s \right){{v}_{1}}\left( s \right)-{{J}_{1}}\left( s \right)$,                      (20)

${{i}_{2}}\left( s \right)=Y\left( s \right){{v}_{1}}\left( s \right)-{{J}_{2}}\left( s \right)$.                        (21)

Comparing the Eqs. (18)-(19) and Eqs. (20)-(21), the authors obtain Eqs. (22)-(23):

${{J}_{1}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]\left[ 2{{i}_{2}}\left( s \right)+{{J}_{2}}\left( s \right) \right]$,                    (22)

${{J}_{2}}\left( s \right)=\exp \left[ -\gamma \left( s \right)l \right]\left[ 2{{i}_{1}}\left( s \right)+{{J}_{2}}\left( s \right) \right]$.                  (23)

The Eqs. (22)-(23) have an equivalent two-port network shown in Figure 3.

ping_mu_jie_tu_2025-03-28_103842.png

Figure 3. The equivalent network of the lossy uniform line with one conductor according to Norton-structure

The use of the complex frequency domain allows for a detailed examination of these systems, providing a comprehensive understanding of signal behavior within the transmission line. Expanding on the theory of telegraphic equations applied to multiconductor transmission lines, the research investigates a uniform lossy line with n-coupled conductors, depicted in Figure 4 [2].

ping_mu_jie_tu_2025-03-28_103848.png

Figure 4. The lossy uniform line with n-conductors and the coordinative system

Assuming the conductor of line parallel with the ground in the medium homogeneous and the initial conditions zero, authors denoted: Ri(s), internal resistance of the i-th line conductor; Li(s), internal inductance of the i-th line conductor; zg(s), earth complete impedance. "In and Dn represent unit matrices of order n×n, with elements defined as follows:"

${{\left[ {{I}_{n}} \right]}_{ii}}=1$; ${{\left[ {{I}_{n}} \right]}_{ij}}=0$; $i\ne j$; $i,j=1,2,~...,n$

${{\left[ {{D}_{n}} \right]}_{ii}}=1$; ${{\left[ {{D}_{n}} \right]}_{ij}}=0$; $i\ne j$; $i,j=1,2,~...,n$

${{Z}_{c}}\left( s \right)=\left[ R\left( s \right)+sL\left( s \right) \right]{{1}_{n}}~$is a diagonal n×n order matrix of the line conductor’s complete impedance; Rg(s) is n×n order matrix, which represents the internal earth resistance; ${{Z}_{c}}\left( s \right)=\left[ {{R}_{g}}\left( s \right)+s{{L}_{g}}\left( s \right) \right]$ the Earth impedance matrix of size n×n is computed using Carson’s formulas [2, 3, 6].

The term external inductance (or inductive resistance) refers to the presence of inductance that exists outside the primary components of a circuit, such as wires or PCBs. This phenomenon can occur due to factors such as coil windings or the elongation of wire lengths [9, 10]. L, C are the real, symmetric (independent of frequency) matrices per unit length of external inductance and external capacitance, computed for static conditions and assuming lossless conductor, perfect earth and lossless medium:

$C=CL={{\mu }_{0}}{{\varepsilon }_{0}}{{I}_{n}}$,                        (24)

Therefore

$C={{\mu }_{0}}{{\varepsilon }_{0}}{{L}^{-1}}$,                                 (25)

where, Y(s)=G+sC is a complete admittance symmetric matrix of line V_x(s), I_x(s) are respectively the vectors of phase voltages and currents of the conductors in the point x of the line:

${{V}_{x}}\left( \text{s} \right)=\left[ {{V}_{x1}}\left( s \right),\text{ }\!\!~\!\!\text{ }{{V}_{x2}}\left( s \right),\text{ }\!\!~\!\!\text{ }\ldots \text{ }\!\!~\!\!\text{ }{{V}_{xn}}\left( s \right) \right]$,                    (26)

${{I}_{x}}\left( \text{s} \right)=\left[ {{I}_{x1}}\left( s \right),\text{ }\!\!~\!\!\text{ }{{I}_{x2}}\left( s \right),\text{ }\!\!~\!\!\text{ }\ldots \text{ }\!\!~\!\!\text{ }{{I}_{xn}}\left( s \right) \right]$.                  (27)

Considering the line equation for the TEM mode propagation in line:

$\frac{\partial {{V}_{x}}\left( s \right)}{\partial x}=-Z\left( s \right){{I}_{x}}\left( s \right)$,                     (28)

$\frac{\partial {{I}_{x}}\left( s \right)}{\partial x}=-Y\left( s \right){{V}_{x}}\left( s \right)$.                     (29)

The authors obtain Eqs. (30)-(31):

$\frac{{{d}^{2}}{{V}_{x}}\left( s \right)}{d{{x}^{2}}}=Z\left( s \right)Y\left( s \right){{V}_{x}}\left( s \right)$,                (30)

$\frac{{{d}^{2}}{{I}_{x}}\left( s \right)}{d{{x}^{2}}}=Z\left( s \right)Y\left( s \right){{I}_{x}}\left( s \right)$.                 (31)

By employing variable transformation techniques to generate the general solution for multiconductor transmission line equations [7], the authors converted them into mode quantities as Eqs. (32)-(33):

$\hat{V}\left( x \right)={{\hat{T}}_{V}}{{\hat{V}}_{m}}\left( x \right)$,                 (32)

$\hat{I}\left( x \right)={{\hat{T}}_{I}}{{\hat{I}}_{m}}\left( x \right)$.                    (33)

The n×n complex matrices ${{\hat{T}}_{V}}$ and ${{\hat{T}}_{I}}$ define a change of variables between the actual phasor line voltages and currents $\hat{V}$ and $\hat{I}$, and the mode voltages and currents ${{\hat{V}}_{m}}$ and ${{\hat{I}}_{m}}$. For the case of phase complete transposition line, the matrix of eigenvalues results:

$T={{T}_{V}}={{T}_{I}}$.                   (34)

During the process of linear rearrangement from phase to modal coordinates [6], the authors derived as shown in Eqs. (35)-(36):

${{V}_{x}}\left( s \right)=T{{v}_{mx}}\left( s \right)$,                        (35)

${{I}_{x}}\left( s \right)=T{{i}_{mx}}\left( s \right)$,                       (36)

where, v_mx(s), i_mx(s) are respectively voltage and modal currents vectors at point x of line: m=1, 2, …, n.

The T matrix in case of n=3 and for the preliminary condition has the form of Clark rearrangement:

$T=\frac{1}{\sqrt{6}}\left[ \begin{matrix}   \sqrt{2} & \sqrt{3} & 1  \\   \sqrt{2} & 0 & -2  \\   \sqrt{2} & -\sqrt{3} & 1  \\  \end{matrix} \right]$;   $T^{-1}=\frac{1}{\sqrt{6}}\left[ \begin{matrix}   \sqrt{2} & \sqrt{2} & \sqrt{2}  \\   \sqrt{3} & 0 & \sqrt{3}  \\   1 & -2 & 1  \\  \end{matrix} \right]$.  (37)

After diagonalized the symmetric matrices Z(s) and Y(s); expressing the characteristic resistances in modal coordinates and diagonalized too, admitting for m=1, 2, 3:

$z\left( s \right)\equiv {{z}_{m}}\left( s \right)$; $y\left( s \right)\equiv {{y}_{m}}\left( s \right)$,                      (38)

${{V}_{1}}\left( s \right)\equiv V_{m}^{\left( 1 \right)}\left( s \right); {{V}_{2}}\left( s \right)\equiv V_{m}^{\left( 2 \right)}\left( s \right);   {{\text{E}}_{1}}\left( \text{s} \right)\equiv \text{E}_{\text{m}}^{\left( 1 \right)}\left( \text{s} \right);    {{\text{E}}_{2}}\left( \text{s} \right)\equiv \text{E}_{\text{m}}^{\left( 2 \right)}\left( \text{s} \right)$,                  (39)

${{i}_{1}}\left( s \right)\equiv i_{m}^{\left( 1 \right)}\left( s \right)$; ${{i}_{2}}\left( s \right)\equiv i_{m}^{\left( 2 \right)}\left( s \right)$; ${{J}_{1}}\left( s \right)\equiv J_{m}^{\left( 1 \right)}\left( s \right)$; ${{J}_{2}}\left( s \right)\equiv J_{m}^{\left( 2 \right)}\left( s \right)$.                        (40)

The conditions of frequency-independent imply that the external inductance and external capacitance remain unchanged over a wide frequency range, which can be crucial for various applications, especially in high-frequency circuits. The authors obtain the two-port equivalent networks in modal coordinates according to Thevenin and Norton structure [11]. In the same way, using T and T-1 eigenvalue matrix, can be the transform in modal coordinates of the source network and the load at the end of the line.

3.2 Pade approximation of the characteristic impedance (admittance) and exponential propagation function

Each function F(y), which is expanded in Mac Lauren series in relation with y, around point y=0, formed Eq. (41):

${{F}_{k}}\left( y \right)=1+{{m}_{1}}y+{{m}_{2}}{{y}^{2}}+...+{{m}_{k}}{{y}^{k}}$.                           (41)

Putting in correspondence with rational function of (n, n) order by Pade approximation formed Eqs. (42)-(43):

${{F}_{p}}\left( y \right)=\frac{{{a}_{n}}{{y}^{n}}+{{a}_{n-1}}{{y}^{n-1}}+...+{{a}_{1}}y+1}{{{b}_{n}}{{y}^{n}}+{{b}_{n-1}}{{y}^{n-1}}+...+{{b}_{1}}y+1}$,                  (42)

${{F}_{p}}\left( y=\infty  \right)=\frac{{{a}_{n}}}{{{b}_{n}}}$; ${{F}_{p}}\left( y=0 \right)=1$,                        (43)

where, the moments are computed as below:

${{m}_{k}}=\frac{\frac{{{d}^{k}}F\left( y \right)}{d{{y}^{k}}}}{k!}{{|}_{y=0}}$.                        (44)

On this basis, the authors can transform the characteristic impedance (admittance) into the form Eq. (42), after expanding it as shown in Eq. (41) [12, 13]. Therefore, if the authors perform expansion around point s=∞ in the relation with 1/s:

$Y\left( s \right)=\sqrt{\frac{C}{L}}{{\left( \frac{1+\frac{G}{Cs}}{1+\frac{R}{Ls}} \right)}^{1/2}}$,                 (45)

$F\left( y \right)={{\left( \frac{1+ay}{1+by} \right)}^{1/2}}$ for $a=\frac{G}{C}$, $b=\frac{R}{L}$.                      (46)

For y=1/s, calculating the (2n-1) moments of function as shown in Eq. (44), the Pade approximation of impedance and admittance has the form Eqs. (47)-(48):

${{Z}_{P}}\left( s \right)=\sqrt{\frac{L}{C}}\frac{{{s}^{n}}+{{a}_{1}}{{s}^{n-1}}+...+{{a}_{n}}}{{{s}^{n}}+{{b}_{1}}{{s}^{n-1}}+...+{{b}_{n}}}$,                       (47)

${{Y}_{P}}\left( s \right)=\sqrt{\frac{C}{L}}\frac{{{s}^{n}}+{{a}_{1}}{{s}^{n-1}}+...+{{a}_{n}}}{{{s}^{n}}+{{b}_{1}}{{s}^{n-1}}+...+{{b}_{n}}}$,                       (48)

from where the authors can determine the functions if impedance (admittance) as below Eqs. (49)-(50):

${{Z}_{P}}\left( s \right)=Z\left( s=0 \right)={{\left( \frac{R}{G} \right)}^{1/2}}$,                       (49)

${{Y}_{P}}\left( s \right)=Y\left( s=0 \right)={{\left( \frac{G}{R} \right)}^{1/2}}$.                             (50)

The coefficients a_j, b_j in Eqs. (47)-(48), depending by the sort of F(y) and the regarding moments are computed Eqs. (51)-(52):

$\begin{matrix}   \frac{1-{{a}_{n}}}{{{b}_{n}}} & {{m}_{1}} & {{m}_{2}} & ... & {{m}_{n-1}}  \\   {{m}_{1}} & {{m}_{2}} & {{m}_{3}} & ... & {{m}_{n}}  \\   {{m}_{2}} & {{m}_{3}} & {{m}_{4}} & ... & {{m}_{n+1}}  \\   ... & ... & ... & ... & ...  \\   {{m}_{n-1}} & {{m}_{n}} & {{m}_{n+1}} & ... & {{m}_{2n-2}}  \\  \end{matrix}\begin{matrix}  {{b}_{n}}  \\   {{b}_{n-1}}  \\   {{b}_{n-2}}  \\   ...  \\   {{b}_{1}}  \\ \end{matrix}=-\begin{matrix}   {{m}_{n}}  \\   {{m}_{n+1}}  \\   {{m}_{n+2}}  \\   ...  \\   {{m}_{2n-1}}  \\ \end{matrix}$,                      (51)

$\begin{matrix}   {{a}_{1}}& 1 & 0 & 0 & ... & 0 & {{b}_{1}}   \\   {{a}_{2}} &  {{m}_{1}} & 1 & 0 & ... & 0 & {{b}_{2}}  \\ {{a}_{3}} = & {{m}_{2}} & {{m}_{1}} & 1 & ... & 0 & {{b}_{3}} +  \\   ... & ... & ... & ... & ... & ... &  ... \\{{a}_{n-1}} & {{m}_{n-2}} & {{m}_{n-3}} & {{m}_{n-4}}& ...&1 &{{b}_{n-1}}   \\ &&&{{m}_{1}} &&& \\  &&& {{m}_{2}}&&&  \\   &&&{{m}_{3}} &&& \\ &&&  ... &&& \\   &&&{{m}_{n-1}} &&& \\  \end{matrix}$.                        (52)

In analogue manner is performed the Pade expanding Eq. (42) of the characteristic impedance (admittance) in series y=s for s=0, where the coefficients a_j, b_j are computed according the above matrices in Eqs. (53)-(54):

${{Z}_{P}}\left( s \right)=\sqrt{\frac{R}{G}}\frac{{{a}_{n}}{{s}^{n}}+{{a}_{n-1}}{{s}^{n-1}}+...+{{a}_{1}}s+1}{{{b}_{n}}{{s}^{n}}+{{b}_{n-1}}{{s}^{n-1}}+...+{{b}_{1}}s+1}$,                                (53)

${{Y}_{P}}\left( s \right)=\sqrt{\frac{G}{R}}\frac{{{a}_{n}}{{s}^{n}}+{{a}_{n-1}}{{s}^{n-1}}+...+{{a}_{1}}s+1}{{{b}_{n}}{{s}^{n}}+{{b}_{n-1}}{{s}^{n-1}}+...+{{b}_{1}}s+1}.$                         (54)

Let’s denote:

$a=\frac{R}{L}$; $b=\frac{G}{C}$; $\tau =l{{\left( LC \right)}^{\frac{1}{2}}}$; ${{\tau }_{1}}=\frac{\tau }{l}$; $A=\gamma \left( s=0 \right)\cdot l=l\cdot {{\left( RG \right)}^{\frac{1}{2}}}$; $\gamma \left( s=\infty  \right)\cdot l=s\cdot \tau$; ${{\alpha }_{1}}=-{{m}_{i+1}}{{\left( LC \right)}^{\frac{1}{2}}};i=1,2,~...$

Let's transform the function:

$\gamma \left( s \right)={{\left[ \left( R+Ls \right)\left( G+Cs \right) \right]}^{\frac{1}{2}}}=s{{\tau }_{1}}{{\left[ \left( 1+\frac{a}{s} \right)\left( 1+\frac{b}{s} \right) \right]}^{\frac{1}{2}}}=s{{\tau }_{1}}+s{{\tau }_{1}}\left[ {{\left( 1+\frac{a}{s} \right)}^{\frac{1}{2}}}{{\left( 1+\frac{b}{s} \right)}^{\frac{1}{2}}}-1 \right]$,                            (55)

By setting $y=\frac{1}{s}$ for $s\to \infty$ and y=0, and expanding in a Maclaurin series with respect to y [5], the authors computed the moments in Eq. (44) and obtained Eq. (56):

$\sqrt{\left( 1+\frac{a}{s} \right)\left( 1+\frac{b}{s} \right)}=1+\mathop{\sum }_{i=1}^{\infty }\frac{{{m}_{i}}}{{{s}^{i}}}$,                        (56)

In the same way as in paragraph 2, the authors calculate respectively α_j, β_j coefficients for the exponential propagation function by Pade approximation:

$\exp \left[ -\gamma \left( s \right)\cdot l \right]=\exp \left( -s\tau  \right)\exp \left( {{\alpha }_{0}} \right)\frac{{{s}^{n}}+{{a}_{1}}{{s}^{n-1}}+...+{{a}_{n}}}{{{s}^{n}}+{{b}_{1}}{{s}^{n-1}}+...+{{b}_{n}}}$,                     (57)

Solving by the boundary conditions and transforming further as function of impedance [14], the authors obtain the expression:

$\exp \left[ -\gamma \left( s \right)\cdot l \right]=\exp \left( -s\tau  \right)\frac{\left( \frac{1}{2} \right){{Z}_{0}}\exp \left( {{\alpha }_{0}} \right)F\left( s \right)}{\left( \frac{1}{2} \right){{Z}_{0}}}=\exp \left( -s\tau  \right)\left[ \left( \frac{1}{2} \right){{Z}_{0}} \right]{{y}_{e}}\left( s \right)$,                                (58)

where,

${{Z}_{0}}=Z\left( s=\infty  \right)={{\left( \frac{L}{C} \right)}^{\frac{1}{2}}}$, ${{Z}_{e}}\left( s \right)=2{{Z}_{0}}\exp \left( {{\alpha }_{0}} \right)F\left( s \right)$, ${{Y}_{e}}\left( s \right)=\frac{1}{\frac{1}{2}{{Z}_{0}}}\cdot \exp \left( {{\alpha }_{0}} \right)F\left( s \right)$.                              (59)

The line model according to Thevenine structure, together with data of characteristic impedance synthesis are shown in Figure 5.

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Figure 5. Thevenine-structure line model with characteristic impedance data, synthesis, and exponential propagation function

The synthesis of function in Eq. (58) can be achieved using lossless lines with characteristic impedance Z₀ by incorporating the time delay $\tau$  and synthesizing the function Z_e(s) or Y_e(s) through Foster’s or Cauer’s methods [8, 9].

3.3 Case study

Example 1. A theoretical model of two-conductor transmission line. Let`s have a line with parameters (PUL R=2.5 Ω/cm; C=4×10<sup>-12</sup> F/cm, L=108 H/cm, G=5×10<sup>-4</sup> Ω<sup>-1</sup>/cm and its length l=5 cm. An active resistance of 100Ω is connected at the line’s output, while at the input, there is a periodic voltage source, shown in Figure 6, in series with an internal resistance of 100Ω [15, 16]. It`s performed the Caurer synthesis using the Pade approximation of the characteristic impedance, then is performed the Z_RC(s) function synthesis, in form of lumped RC-parameters given in Figure 5. The synthesis of the exponential propagation function is done for n=2 in an admittance y_e(s), which is performed according to Foster [17, 18]. After performing the transient simulation through ATP software (Figures 6 and 7), in Figure 8 are shown the input and output voltages in line, and the current in line also.

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Figure 6. The model of line in ATP software

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Figure 7. The impuls of voltage source

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a)

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b)

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c)

Figure 8. a) Voltage at the beginning of line; b) Voltage at the end of line; c) Current in line

Example 2. A model of an overhead 110 kV multiconductor transmission line. It`s analyzed a 110 kV transmission line with l=150 km, in the case of short circuit with ground of phase A, as shown in Figure 9.

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Note: The parameters of line are: $\text{Z}_{1}^{\text{ }\!\!'\!\!\text{ }}$=0.12+0.3871 Ω/km, $\text{C}_{1}^{\text{ }\!\!'\!\!\text{ }}$=9 nF/km (positive sequence) and $\text{Z}_{0}^{\text{ }\!\!'\!\!\text{ }}$=0.27+1.3927 Ω/km, $C_{0}^{'}$=4.912 nF/km (zero sequence). The source network: V_Q=110/√3 kV (r.m.s values); R_Q=0.331 Ω; X_Q=3.331 Ω.

Figure 9. Three phase overhead 110 kV transmission line

Using the Pade approximation, there are calculated the modal parameters and the exponential propagation function of line. The model of line and the results of simulations in ATP software are shown in Figures 10 and 11.

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Figure 10. The model of line in ATP software

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Figure 11. The results of simulations for voltages and currents: a) Voltages at the beginning of line; b) Voltages at the end of line for the damage phase C and the other phases A and B; c) Current at the point of short-circuit in phase C of line

In the context of analyzing transient processes in homogeneous lossy lines, the Pade method is employed to approximate the parameters of these lines [19, 20]. This method accounts for losses due to the resistance of the line material, as well as other factors affecting the dynamic processes within the line.

The Pade approximation method, while offering superior accuracy compared to polynomial methods, has certain limitations and challenges. One significant drawback is its sensitivity to the order of approximation, as higher-order expansions can increase computational complexity and potentially lead to numerical instability. The method may also struggle with accurately modeling systems with non-uniform losses or nonlinear effects, as these conditions introduce complexities not inherently addressed by the standard Pade formulation. Furthermore, its reliance on rational functions can result in inaccuracies when applied to highly dynamic or chaotic systems with abrupt parameter variations. Additionally, the integration of the Pade approximation into existing modeling frameworks can pose implementation challenges, particularly in real-time applications where computational efficiency is critical. Understanding and addressing these limitations is essential to maximize the method’s utility in practical scenarios.