Non-local telegrapher’s equation as a transmission line model

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Highlights

  • Transmission line displaying non-locality effects is modeled.

  • Non-locality is introduced by coupling inductors appearing in the elementary circuit.

  • Cross-inductivity kernels corresponding to short- and large-scale non-locality are considered.

  • Non-local telegrapher’s equations are solved analytically and numerically.

Abstract

Transmission line displaying non-locality effects is modelled by considering the magnetic coupling of inductors in the series branch of Heaviside’s elementary circuit, so that the magnetic flux is obtained as a superposition of local and constitutively given non-local magnetic flux through the cross-inductivity kernel. Non-local telegrapher’s equations are derived as the continuum limit of corresponding Kirchhoff’s laws and solved for prescribed external excitation analytically by the means of integral transforms method and also numerically. Numerical examples of the mollified impulse responses illustrate the non-local behavior of signal propagation in case of power, exponential, and Gauss type cross-inductivity kernels.

Introduction

Small-scale structures, including transmission lines, may display non-locality effects and the aim of this theoretical study is to: model these effects, formulate the corresponding non-local telegrapher’s equations, and analyze those equations analytically and numerically. The starting point is the k-th Heviside’s elementary circuit, shown in Fig. 1, modeling the transmission line physical properties, while non-local telegrapher’s equationsxv(x,t)=R(x,t)i(x,t)+tϕ(x,t)E(x,t),ϕ(x,t)=L(x,t)i(x,t)+abm(x,ζ,t)i(ζ,t)dζ,xi(x,t)=G(x,t)v(x,t)+t(C(x,t)v(x,t)),to be derived in Section 2.1, mathematically describing the transmission line responses, are obtained from Kirchhoff’s laws applied to the k-th elementary circuit in the limit when circuit length Δxk tends to zero, while the number of circuits tends to infinity.

The elementary circuit, as seen from Fig. 1, consists of: series resistor and inductor, denoted by ΔRk, ΔLk, modeling dominantly conductive properties of transmission line; shunt capacitor and conductor, denoted by ΔCk, ΔGk, modeling its insulative properties; and electromotive force, denoted by ΔEk, modeling the external influence on the line. The current at time-instant t, running through the series branch of the k-th elementary circuit, is denoted by ik=ik(t), while vk=vk(t) denotes the (time dependent) voltage on its shunt branch. The non-locality effects originate from the magnetic coupling of inductors in the series branch, namely by assuming that the magnetic flux within elementary circuit is the consequence of currents running through series branches of all elementary circuits rather than just one elementary circuit.

The survey paper [12] deals with various contributions of Oliver Heaviside in developing classical electromagnetic theory, including the electric transmission line analysis. The diversity of physical processes, including the electric processes in transmission lines, modelled by telegrapher’s equation is considered in [40]. Solutions of classical telegrapher’s equation, derived by using Heaviside’s elementary circuit and passing to the continuum, assumed in the form of solitary waves are studied in [30], [37], [41], while in [32] solutions are sought in the form of traveling waves. Soliton and elliptic solutions of for wave propagation in non-linear transmission line are constructed in [23].

Time fractional telegrapher’s equation, obtained through fractionalization of conductor and inductor in Heaviside’s elementary circuit is studied on bounded domain analytically for the frequency characteristics and numerically by employing the numerical Laplace inversion in [1], [2]. Similar approach is also adopted in [44], [45]. The fractionalization method of replacing the integer order derivatives with the Caputo fractional derivatives for both spatial coordinate and time in telegrapher’s equations for voltage and current is adopted in [16], [17], [18] and solutions are sought in the form of plane waves, either in time or in space, while the integer order derivatives are replaced with the Atangana-Caputo derivatives in [15].

Classical Heaviside’s elementary circuit is topologically generalized in [9] by introducing capacitor in the series branch accounting for the charge accumulation along the transmission line in addition to the generalization of constitutive equations corresponding to capacitive and inductive elements using the fractional integrals. Moreover, fractional telegrapher’s equation corresponding to such a generalization is analytically solved on the semi-infinite domain for the external forcing on the boundary. Frequency characteristics corresponding to the mentioned generalization, as well for a different topological modification of the elementary circuit are studied in [8], [10].

Fractional telegrapher’s equation on unbounded and semibounded domain also finds application in modelling electrodiffusion of ions in nerve cells, as shown in [25], while solutions of fractional telegrapher’s equation on finite domain for mixed Robin boundary conditions model properties of spiny dendritic branch segments, as discusses in [26]. Anomalous diffusion phenomena, like leakage and signal conduction in spiny dendrites, are also modelled by fractional telegrapher’s equation in [20], [39], while fractional telegrapher’s equation, modelling same type phenomena, is solved on bounded domain for non-local boundary conditions in [4]. Space-fractional telegrapher’s equation, analyzed for existence and uniqueness in [43], also found its application in image denoising.

Numerical methods for solving telegrapher’s equation, used for modelling diffusive movement of ions in neuronal system, include the finite differences method discretizing both time and spatial domain, as well as its combination either with the spectral method or the Galerkin finite element method for the spatial coordinate, as done in [28], [29], [46]. Space and/or time fractional telegrapher’s equations on bounded domain are solved in [11], [21], [31], [35] using finite differences for discretizing time coordinate and either Galerkin technique, or finite differences for spatial coordinates yielding implicit and explicit schemes. Two-dimensional variable and constant order fractional telegrapher’s equations are solved analytically in [22] and numerically in [5], [42].

Space and/or time fractional telegrapher’s equations are solved analytically using the method of Laplace and Fourier integral transforms having solution expressed through Green’s function in terms of the Fox H-function in [6], [34], [36], while in [33] telegrapher’s equation is written in terms of time derivatives of Hilfer and Hadamard type and in terms of the Riesz-Feller type spatial derivatives. For the applications of operational calculus in various problems of the electric engineering see [27].

Section 2 is devoted to the derivation of non-local telegrapher’s equations regardless of the type of non-locality and including the possibility of model parameters to be dependent on the coordinate in the spatial domain, which is followed by the discussion on the suitable types of non-locality kernels and finished by formulating non-local telegrapher’s equations in terms of either voltage or current in the case of constant model parameters. Non-local telegrapher’s equations are solved in Section 3 using the method of integral transformations regardless of the form of non-locality kernel (as far as it obeys some requirements), so that the voltage is obtained as a convolution of the solution kernel and electromotive force. Asymptotic properties of the solution kernel are examined as well. Classical telegrapher’s equation is solved in Section 4 in order to emphasize the similarities (both equations have the same asymptotics in infinity) and differences (finite signal propagation speed in the case of classical telegrapher’s equation) of local and non-local telegrapher’s equations. In Section 5 a numerical scheme is developed in order to solve non-local telegrapher’s equation, while Section 6 is devoted to numerical examples of the impulse responses, illustrating the non-local behavior of signal propagation, as well as the agreement of analytical and numerical methods. Finally, concluding remarks are given in Section 7.

Section snippets

Model

Non-local telegrapher’s equations are derived as the continuum limit of Kirchhoff’s laws applied to Heaviside’s elementary circuit, taking into account the magnetic coupling between inductors in all elementary circuits. Further, properties and several choices of non-locality kernels are discussed and finally non-local telegrapher’s equations are written either in terms of voltage, or in terms of current.

Analytical solution of non-local telegrapher’s equations

Non-local telegrapher’s equations (5) - (7), modeling transmission line subject to external forcing, for infinite spatial domain and constant model parameters can be rewritten asxv(x,t)=Ri(x,t)+tϕ(x,t)E(x,t),xi(x,t)=Gv(x,t)+Ctv(x,t),ϕ(x,t)=Li(x,t)+m(|x|)*xi(x,t),since in this case the integral in non-local magnetic flux (8) is represented by the convolution with respect to the spatial coordinateϕNL(x,t)=m(|x|)*xi(x,t)=Mm¯(|xζ|)i(ζ,t)dζ,with m¯ denoting the normalized

Classical telegrapher’s equation with excitation

Non-local telegrapher’s equation expressed either in terms of voltage (11), or in terms current (12), become classical telegrapher’s equations containing the forcing termLC2t2v(x,t)+(RC+LG)tv(x,t)+RGv(x,t)2x2v(x,t)=xE(x,t)andLC2t2i(x,t)+(RC+LG)ti(x,t)+RGi(x,t)2x2i(x,t)=(G+Ct)E(x,t)by excluding the non-locality effects, i.e., by setting m=0 in (11) and (12).

Classical telegrapher’s equation (29), expressed in terms of attenuation coefficient and time constants (19), by using

Numerical solution of non-local telegrapher’s equations

Model of transmission line subject to external forcing, that is represented by non-local telegrapher’s equations (5) - (7), for constant model parameters and assuming cross-inductivity kernel to be dependent on the distance |xζ| between neighboring points ζ of point x, i.e., m(x,ζ)=m(|xζ|), as it is the case for cross-inductivity kernels from Table 1, can be rewritten asxv(x,t)=Ri(x,t)+t(Mi)(x,t)E(x,t),xi(x,t)=Gv(x,t)+Ctv(x,t),where the operator M is defined by(Mi)(x,t)=Li(x,t)+abm

Numerical examples

In order to illustrate signal propagation in transmission line displaying non-locality effects, the analytical solution (22) to non-local telegrapher’s equations (13) - (15), assuming impulse-type external electromotive forcing, i.e.,E(x,t)=δ(x)δ(t),is used to produce spatial signal profiles at specified time-instants depending on the cross-inductivity models of non-locality. Moreover, the signal responses, calculated according to the analytical solution formula, are checked against the ones

Conclusion

Non-local telegrapher’s equations (5) - (7), modeling transmission line subject to external forcing, are derived as the continuum limit of Kirchhoff’s laws applied to Heaviside’s elementary circuit from Fig. 1, with the non-locality effects introduced by considering the magnetic coupling of inductors in the series branch of Heaviside’s circuit. More precisely, the magnetic flux within elementary circuit is assumed to be the consequence of currents running through series branches of all

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgments

This work was supported by the Serbian Ministry of Science, Education and Technological Development under Grants 451-03-68/2020-14 (SMC), TR32018, TR33013 (MRR), through Mathematical Institute of the Serbian Academy of Arts and Sciences and Grant No. 451-03-68/2020-14/200125 (DZ).

References (46)

  • N. Al-Zubaidi R-Smith et al.

    Application of numerical inverse Laplace transform methods for simulation of distributed systems with fractional-order elements

    J. Cir. Sys. Compu.

    (2018)
  • G.B. Arfken et al.

    Mathematical Methods for Physicists

    (2013)
  • E. Bazhlekova et al.

    Exact solution for the fractional cable equation with nonlocal boundary conditions

    Cent. Eur. J. Phys.

    (2013)
  • A.H. Bhrawy et al.

    Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

    Nonlinear Dyn.

    (2015)
  • L. Can et al.

    Analytical solutions, moments, and their asymptotic behaviors for the time-space fractional cable equation

    Commun. Theor. Phys.

    (2014)
  • S.M. Cvetićanin et al.

    Frequency analysis of generalized time-fractional telegrapher’s equation

    European Conference on Circuit Theory and Design (ECCTD), Catania, Italy

    (2017)
  • S.M. Cvetićanin et al.

    Generalized time-fractional telegrapher’s equation in transmission line modeling

    Nonlinear Dyn.

    (2017)
  • S.M. Cvetićanin et al.

    Frequency characteristics of two topologies representing fractional order transmission line model

    Circ. Syst. Signal Process.

    (2020)
  • C. Donaghy-Spargo

    On Heaviside’s contributions to transmission line theory: waves, diffusion and energy flux

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

    (2018)
  • D.G. Duffy

    Transform Methods for Solving Partial Differential Equations

    (2004)
  • A.C. Eringen

    Nonlocal Continuum Field Theories

    (2002)
  • J.F. Gómez-Aguilar

    Analytical and numerical solutions of the telegraph equation using the Atangana-Caputo fractional order derivative

    J. Electromag. Waves Appl.

    (2018)
  • J.F. Gómez-Aguilar et al.

    Fractional transmission line with losses

    Z. Nat. A

    (2014)
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