# Travelling-wave modelling of uniform multi-conductor transmission line networks-part I: analytical derivation.

1. INTRODUCTIONProvision of communication and/or smart grid services over the Low Voltage (LV) and Medium Voltage (MV) Distribution Network (DN) by use of the High Frequency (HF) band and also other effects of possible interest such as high-frequency transients resulting, e.g., from lightning or from inductive load switching, depend essentially on the response of unmatched transmission-line (TL) networks. The impedance mismatches found along the power electric network provoke intense multipath fading in the area of MHz [1-9]. Due to structural peculiarities, certain types of triple-pole (phase-neutral-ground) cables follow the two-conductor uniform TL model [10-16], whereas most tricels, as well as any multi-phase (two-phase, three-phase) cable, require a multiconductor transmission line (MTL) model [17-27]. Therefore, a computationally realisable travelling-wave analysis of MTL networks enabling in particular simulation of Power-Line Communication (PLC) channels in the Time Domain (TD) would be of considerable contribution. The principal transmission schemes are Orthogonal Frequency-Division Multiplexing (OFDM) and Code-Division Multiple Access (CDMA) [28-38], so that response to both narrowband and wideband signalling would be desired.

Numerical transition from the Frequency Domain (FD) [39-44], Finite Difference Time Domain (FDTD) analysis [45-52], and the Method of Characteristics (MoC) [53-55] are all common ways of obtaining the TD response of a MTL network. Other established techniques involve discretisation [56, 57], FD solution followed by TD transition [58-61], or polynomial/ rational function FD approximation of MTLs followed by numerical transition to the TD [62-72]. In all such methods, the MTL equations are numerically approximated --in the TD, or FD, or both--so that (i) the TD solution is only a computational result providing no connection with the channel (network) topology, and (ii) it is not possible to isolate either the several paths or the dominant path groups, or to trace back and identify the path routes.

The TD interpretation of modal decomposition introduced in this paper eliminates the above weaknesses (i) and (ii) inherent to all aforementioned methods. Propagation of travelling-wave components along uninterrupted uniform TL segments and reflection effects due to impedance mismatches are the elemental phenomena, which have already been partially addressed in previous articles, e.g., [73-81]. However, no generalised approach has up to now been presented that would reveal the exact correspondence with the standard FD analysis and provide a sound basis for time-response simulation, and here is where novelty of this work lies in, i.e., that the MTL network response and the performance parameters are determined not as the result of any numerical FD-to-TD transformation or/and numerical approximation of the MTL equations, but through analytical elaboration of the multipath propagation process that is the actual (physical) response mechanism, allowing thus (a) interlinking between the actual propagating components or/and the dominant groups of such components and the network topology, as well as (b) tracing back along the route of every single such component. At this point, it should be noted that the travelling-wave approach introduced herein is general and considers all reflections induced by all medium discontinuities (i.e., terminations, serial connections of different lines, line junctions).

The present travelling-wave analysis is directly applicable in the case of narrowband signalling (e.g., OFDM), i.e., when the MTL parameters, and therefore the network response as well, vary along the band of transmission only marginally and can be considered constant, so that upon any potential narrowband (either single- or multi-carrier) transmission scheme it is unrestrictedly valid. Moreover, although not in the scope of this work, treatment of wideband signalling, i.e., if variation of MTL parameters and network' response is considerable and cannot be neglected, is in addition feasible by use of recipes already established to approximate wideband channels via proper expansion of narrowband techniques. This aspect is further discussed in Subsection 3.1 of Part II, entitled "Bandwidth--Variation with Frequency".

In Section 2, the MTL reference model is presented. The modal theory in the FD is briefly referred in Section 3. In Section 4, the time-domain response is derived commencing from the single-mode travelling wave demonstration and then extending to the general multimode response through linear superposition. Verification through coincidence with the FD analysis in terms of (i) identical form of general solution and (ii) equivalent terminal conditions is demonstrated in Section 5. In Section 6, guidelines for computational implementation are suggested, and a concluding summation closes the paper in Section 7.

2. THE MTL REFERENCE MODEL

Referring to the (v + 1)-conductor MTL configuration of Fig. 1, the symmetric per-unit-length resistance, inductance, capacitance, and

conductance (v x v) matrices are respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1d)

At every point [z.sub.0] of structural discontinuity--termination, serial connection of different lines, or line junction--the line voltage and current vectors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2a)

are subject to the voltage and current continuity conditions

V([z.sup.-.sub.0], t) = V ([z.sup.+.sub.0], t), I([[z.sup.-.sub.0], t) = I ([z.sup.+.sub.0], t) (2b)

where superscripts (-) and (+) denote respectively longitudinal positioning infinitesimally before and after z0. In the case of line junction, as shown in Fig. 2, the voltage vector V([z.sup.+.sub.0], t) is common to all departure lines, whereas the resultant current vector I([z.sup.+.sub.0], t) equals the sum of the current vectors of all departure lines.

3. REFERENCE THEORY IN THE FREQUENCY DOMAIN

3.1. Frequency-domain Fundamentals

Pursuant to the FD approach, harmonic line voltages [V.sub.k](z, t) and currents [I.sub.k](z, t), k = 1, ..., v, are assumed, and respective phasor vectors are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The per-unit-length impedance and admittance matrices are then formed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4a)

and the Input Admittance Matrix (IAM) [[??].sub.in](z) at any longitudinal position z is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4b)

The continuity conditions (2b) are applied to the phasors as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4c)

In MTL telecommunication networks, excitation is normally applied in the form of a finite-output-impedance voltage source, and no short-circuit termination is expected. Therefore, lumped transverse termination is herein represented by the coupling of a voltage source

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5a)

with output impedance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and output admittance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5b)

between every k-th and k'-th conductor pair (reference conductor: k, k' = 0), and the line termination interaction is expressed through the termination admittance matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5c)

where subscripts "0" and "L" denote line's origin and end, respectively. If no source is coupled between the conductors k and k', [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0 is set. An open circuit between the k-th and k'-th conductor gives [[??].sub.kk'] = 0. No short circuit is expected, as already explained. From the continuity conditions (4c), the excitation/termination conditions at line's origin (z = 0) and end (z = l) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5d)

3.2. Ordinary Modes of Propagation

3.2.1 Decomposition

Performing the modal decomposition (see [82] or equivalently [83]), the similarity transformation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6a)

is applied and the diagonal matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6b)

of the common eigenvalues [[??].sup.2.sub.[xi]], [xi] = 1, ..., v, of the ([??][??]) and ([??][??]) matrices is formed, with the voltage and current transformation matrices chosen to satisfy:

[[??].sub.I] = [([[??].sup.-1.sub.V]).sup.T] (6c)

providing

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6d)

3.2.2 Modal Solution

The [xi]-th "ordinary mode of propagation", or simply "mode", consists of the modal voltage and current quantities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6e)

and the propagation constant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6f)

with [[??].sub.[xi]] [member of] R the attenuation constant and [[??].sub.[xi]] [member of] R the phase constant of the mode.

The general FD solution in modal terms is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7c)

with [[??].sup.+.sub.m](z), [[??].sup.+.sub.m](z) the phasor vectors of the forward or positively evolving stationary modal voltage and current wave components respectively, and [[??].sup.-.sub.m](z), [[??].sup.-.sub.m](z) the phasor vectors of the backward or negatively evolving ones. The modal admittance matrix defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8a)

is diagonal as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8b)

with [[??].sub.m[xi]] the characteristic admittance of the [xi]-th mode.

3.2.3. Line Quantities and Parameters

In terms of the stationary modal components, the forward or positively evolving and backward or negatively evolving phasor vectors of the actual line quantities become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

The Characteristic Admittance Matrix (CAM) Yc of the line is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and the voltage reflection coefficient matrix or simply Reflection Coefficient Matrix (RCM) [rho](z) is defined at any longitudinal position z via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The FD solution for the line voltages and currents is uniquely determined via (6a) combining (i) the general modal solution (7), (ii) interdependence between modal voltages and currents imposed by (6d), and (iii) the applicable continuity (4c) or excitation/termination (5d) conditions.

3.2.4. Matching

The terminal IAM matrix of a TL of length l terminated at the end (z = l) at admittance matrix [[??].sub.L] is by definition:

[[??].sub.in](l) = [[??].sub.L] (12)

If the line is "matched", i.e., terminated at the end at its characteristic admittance matrix, the backward evolving line and modal electric quantities are cancelled, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

3.2.5. Feasibility of Modal Decomposition

The decoupling is frequency-independent if the transformation matrices result by default constant, as:

[[??].sub.V] = [T.sub.V][nabla]f, [[??].sub.I] = [T.sub.I][nabla]f (14)

which is feasible over the following TL structures [84]:

1) Perfect conductors inside homogeneous dielectric.

2) Perfect conductors inside transversely non-homogeneous dielectric.

3) Cyclic symmetric configurations, defined by cyclic symmetric per-unit-length admittance and impedance matrices as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

In the case of good conductors inside homogeneous dielectric, frequency-independent decoupling can also be feasible. However, there exist MTL structures (fortunately rare) of good conductors inside homogeneous dielectric where modal decomposition fails, i.e., nonsingular transformation matrices cannot be found. This matter is concisely enlightened in [83]. Certain non-diagonalisable situations are described in [85], whereas in [83], the general non-diagonalisable MTL case is presented leading to the generalised modal theory, where the similarity transformations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not diagonal, but quasi-diagonal block matrices in the so-called Jordan form. The present travelling-wave TD analysis applies only if modal decomposition is feasible in the diagonal form of (6b).

4. TIME-DOMAIN RESPONSE DERIVATION

4.1. Single-mode Propagation

4.1.1. Reference Scheme in the Frequency Domain

4.1.1.1 Matched-line Configuration

For a (v + 1)-conductor TL of length l excited at the origin (z = 0) by a voltage source [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (t) of non-zero output impedance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] between every k-th and k'-th conductor pair, as in (5a) and (5b), and matched at the end according to (13), it is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

and from (10) and (16), the terminal conditions (5d) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

so that the forward-evolving standing wave components of the line electric quantities at the origin (z = 0) get:

4.1.1.2 Single-mode Excitation Since the eigenvectors:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

of matrices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively are linearly independent (as the line is assumed diagonalisable), it is feasible to excite a single arbitrary [xi]-th mode as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20a)

by selecting:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20b)

as (i) a total of (v - 1) conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20c)

are applied to every [xi]' [member of] {1, ..., v}, and (ii) up to v! independent voltage sources may be coupled at the origin between the several conductor pairs, up to v of which between the excitation conductors and the reference.

Since only the [xi]-th mode is excited, from (16), (7), (9) and (8), the phasors of the line electric quantities become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21b)

4.1.1.3 Single-mode Response

Supposing excitation to take place at the origin (z = 0), the modal transfer function (TF) [[??].sub.[xi]](z) of every [xi]-th mode is herein defined at any longitudinal position z as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22a)

so that single [xi]-th mode excitation gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22b)

4.1.2. Time-domain Transition

4.1.2.1 Line Excitation and Matching

A (v + 1)-conductor TL of length l is considered, excited at the origin by a voltage source:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23a)

of non-zero output impedance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] between every k-th and k'-th conductor pair, with a narrowband complex envelope [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of bandwidth [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23b)

so that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

where the "[intersection]" sign indicates a pre-envelope quantity, and the "F" operator denotes the Fourier transform.

Due to the narrowband spectral invariability of (24), the line voltages become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25a)

and the termination conditions at the origin become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25b)

Supposing that (i) the line is matched, and (ii) no excitation is applied at line's end, from (25), (24), and (17), the line voltages and currents interrelate at the origin exactly as in the case of harmonic matched-line excitation, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

4.1.2.2 Single-mode Excitation

Imposing over the whole complex-envelope set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the excitation voltages (23) identical conditions as between respective phasors of the excitation voltages (5a)-(5b) for the satisfaction of (20b), it is obtained from (24):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27c)

A single, [xi]-th mode is thus exclusively excited at the origin, i.e., the relation between line voltages and currents is identical as if only harmonic evolution of the [xi]-th mode was present.

4.1.2.3 Demonstration of Modal Propagation

Being from (24) the line and modal characteristic parameters constant along the bandwidth [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of every excitation voltage [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined in (23), upon exclusive [xi]-th mode excitation at the origin (z = 0) as in (27), each one of the excited line voltages [V.sub.k](0, t), k = 1, ..., v, defined in (27) is along the line subject to the [xi]-th mode TF of (22) for f = [f.sub.c], as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28a)

Similarly, the line currents become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28b)

Therefore, the narrowband excitation (27) applied at the origin propagates along the line forming voltage and current travelling waves, which cover distance z within time (z/2[pi][f.sub.c]/[[??].sub.[xi]]), attenuated by the factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and having the complex envelope shifted by (-[[??].sub.[xi]]z).

4.1.3. Travelling-wave Quantification

4.1.3.1 Modal Parameters

The propagation velocity [v.sub.[xi]] of the [xi]-th mode is defined as the ratio of the angular velocity or cyclic frequency w to the phase constant of the mode, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

and represents along the line the travelling velocity of the electric quantities excited pursuant to (27).

The wavelength [[lambda].sub.[xi]] of the [xi]-th mode is defined as the ratio of the modal propagation velocity to the excitation frequency

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

and represents the travel distance of the electric quantities excited pursuant to (27), along which the angle of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] harmonic term is shifted by (2[pi]).

The propagation delay [[tau].sub.[xi]]([DELTA]l) of the [xi]-th mode along length [DELTA]l is defined as the ratio of the covered length to the propagation velocity

[[tau].sub.[xi]]([DELTA]l) [??] [DELTA]l/[v.sub.[xi]] (31)

and represents the time during which the electric quantities excited pursuant to (27) cover (arrive at) distance [DELTA]l towards the travelling direction.

4.1.3.2 Travelling Quantities--Characteristic Matrices

All the travelling components defined in (28) are positively or forward travelling paths as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32a)

and the corresponding vectorial paths are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32b)

Defining the time-variant travelling phasors of the line voltage and current paths as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33a)

and the corresponding vectorial phasors as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33b)

which travel along the line as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33c)

it is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

The Travelling Admittance Matrix (TAM) [[??].sub.c] is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35a)

and from (10), (27c), and (33c), it equals the CAM as:

[[??].sub.c] = [[??].sub.c] (35b)

Analogously to the FD, the voltage and current modal travelling components, or equivalently modal paths are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36a)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36b)

respectively, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the phasors, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the modal travelling excitation phasors.

At this point, it should be noted that the concept of a time-variant phasor quantity carries in fact the meaning of a complex-envelope, which is obvious from the above definitions. However, for the sake of (i) term consistency, and (ii) term analogy with the FD approach, the term "phasor" is herein adopted to denote both the FD phasor parameter, and the aforementioned TD components.

4.2. Aggregate Response

4.2.1. Superposition of Modal Excitations

4.2.1.1 Superposition of Modal Quantities Excited at Origin

If the single-mode excitation conditions set in (27) for the complex envelopes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the excitation voltages applied according to (23) are not satisfied, the whole set of line's modes are in the general case excited, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37c)

where subscript "0" denotes the initial line voltages and currents excited at the origin--[V.sup.+.sub.k0](0, t) and [I.sup.+.sub.k0] respectively, k = 1, ..., v -- as well as the consequent positively travelling paths--as such "travelling" concept is defined in Sub-section (4.1.2.3) hereof--arisen as soon as the excitation voltages of (23) are applied. These positively travelling paths are the modal voltages [V.sup.+.sub.k[xi]0](0,t) and currents [I.sup.+.sub.k[xi]0](0,t), k = 1, ..., v, [xi] = 1, ..., v, which constitute the components of the voltage [V.sup.+.sub.k[xi]0](0,t) and the current [I.sup.+.sub.k[xi]0](0,t) excited respectively at the line's origin between every k-th conductor and the reference (conductor "0"), and which start right upon excitation to travel (propagate) along the line covering distance z within time (z/2[pi][f.sub.c]/[[??].sub.[xi]]), attenuated by the factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and having their complex envelope shifted by (- [[??].sub.[xi]]z).

The time-variant voltage and current travelling excitation phasors are respectively defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38a)

and the vectorial travelling excitation phasors as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38b)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

the modal vectorial travelling excitation phasors.

4.2.1.2 Excitation Terminal Conditions at Origin

From (25b), (26), and (37), excitation at line's origin satisfies the following terminal conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39b)

no matter whether the end is matched or not, since no excitation effect appears and no related reaction arises at the end before arrival of the initial paths generated at the origin.

4.2.1.3 Travelling Quantities--Superposition

By virtue of (i) linear independency between the eigenvectors defined in (19), and (ii) linearity of the TL under consideration, all the several excited paths travel independently along the line, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40c)

each one with the propagation velocity and the propagation constant [[??].sub.[xi]] of its respective [xi]-th mode.

4.2.2. Medium Discontinuity--Terminal Conditions

4.2.2.1 Line-multitude Continuity

4.2.2.1.1 Formulation

Incidence of a vectorial path (incident path)--modal by definition, since, in the general case, each mode travels at its own propagation velocity--against a longitudinal structural-discontinuity point [z.sub.0]--where by definition TAM discontinuity is as well observed--causes paths travelling away from the discontinuity to be raised, and in particular (i) the "transmitted" paths, propagating in line with the direction of incidence, and (ii) the "reflected" ones, opposite to the incident path. Let [[??].sub.i[xi]](t) and [[??].sub.i[xi]] be the [xi]-th--mode (with no loss of generality) voltage and current respectively incident vectorial phasors, [[??].sub.t](t) and [[??].sub.t](t) the travelling-excitation vectorial phasors of the transmitted paths at the point of discontinuity, and [[??].sub.r](t) and [[??].sub.r](t) the travelling-excitation vectorial phasors of the reflected ones.

Due to (i) the infinitesimal longitudinal dimension of the medium-discontinuity point, and (ii) the consequent instantaneous generation of transmitted and reflected paths, the terminal conditions (2b) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41a)

and satisfy (41b), where [[??].sub.ci], [[??].sub.ct] are respectively the TAM matrices right before and after the point of discontinuity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41b)

4.2.2.1.2 Characteristic Matrices

In the general case of incidence upon a terminated junction as in Fig. 2, i.e., a longitudinally infinitesimal TL point where three (3) or more TL sections intersect and passive lumped elements are as well connected in between one or more pairs of the TL conductors, covering thus also the simple cases of TL termination and TL junction, the infinitesimal longitudinal dimension of the junction point makes from (35b) the resultant TAM matrix right after the medium discontinuity equal to the sum of the TAM matrices of all lumped and distributed terminations, as:

[[??].sub.ct] = [[??].sub.0] + [n.summation over (i=1)] [[??].sub.ci] (42)

The above expression stems directly from the TAM matrix definition, but is still quoted to assure full comprehension of the developed approach.

4.2.2.1.3 Line Discontinuity Matrices

The travelling voltage reflection coefficient matrix, or simply the travelling reflection coefficient matrix (TRCM) [??] is defined from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43a)

The travelling voltage transmission coefficient matrix, or simply the travelling transmission coefficient matrix (TTCM) [??], is defined from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43b)

Combination of (41) and (43) provides:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44b)

where the TAM matrices are supposed to be symmetric.

4.2.2.1.4 Modal Discontinuity Matrices

Since the TAM matrices from one to the other side of the medium discontinuity are not identical, in the general case, incidence causes generation of the whole set of line's modes, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

along the line of incidence, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

along the i-th departure line as in Fig. 2, where [[??].sub.Vi] and [[??].sub.Ii] are respectively the voltage and current transformation matrices before the point of discontinuity, and [[??].sub.Vti], [[??].sub.Iti] the homologous matrices of the i-th departure line.

The modal travelling voltage reflection coefficient or simply the modal travelling reflection coefficient (MTRC) [[[??]mi].sub.[xi]'[xi]] from the [xi]-th mode to the [xi]'-th one is defined upon the medium discontinuity from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47a)

where [[[[??].sub.im](t)].sub.[xi]] the modal phasor of the incident line voltage path. The modal travelling voltage transmission coefficient or simply the modal travelling transmission coefficient (MTTC) [[[[??].sub.mi]].sub.[xi]'[xi]] from the [xi]-th mode to the [xi]'-th one along the i-th departure line is defined upon the medium discontinuity from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47b)

The v x v modal travelling reflection coefficient matrix (MTRCM) and the v x v modal travelling transmission coefficient matrix (MTTCM) are then extracted as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47c)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47d)

respectively, combining (45), (46), and (44).

4.2.2.2 Transition to Line-multitude Discontinuity

4.2.2.2.1 Analytical TL Model Expansion

A medium discontinuity point in considered of (n + 1) intersecting TLs (n [greater than or equal to] 1), each one in general with its own multitude of conductors as ([v.sub.0] + 1) the line of incidence, ([v.sub.1] + 1) the 1st departure line, ...,([v.sub.n] + 1) the n-th departure line. Assuming (i) continuity of the reference conductor from one to the other side of the discontinuity, and (ii) numbering of all other conductors (excitation conductors) of the intersecting TLs according to the homologous in terms of conduction continuity excitation conductors of the line with the maximum multitude v' as

v' = max{[v.sub.0],[v.sub.1], ..., [v.sub.n]} (48)

the TAM matrix [([[??].sub.ci]).sub.vi x vi] of every i-th line (i = 0, 1, ..., n) expands its dimension up to v' as [([[??].sup.E.sub.ci]).sub.v' x v'] setting by default equal to zero all additional lines and columns that correspond to non-existing conductors. Similarly, the vectorial phasors [[[??].sub.i[xi]](t)].sub.vo] and [[[??].sub.i[xi](t)].sub.vo] of the incident voltage and current paths are expanded as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] setting by default equal to zero all additional elements (voltage/current phasor quantities) that correspond to non-existing conductors.

Then, formulae (41) and (42) apply by default between the expanded matrices and vectors, whence the related v' x v' applicable TRCM and TTCM matrices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are obtained from (44a) and (44b) respectively.

4.2.2.2.2 Modal Discontinuity Matrices

From the definition (43a), and since all expansion elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [[[[??].sup.E.sub.i[xi]](t)].sub.v'] are equal to zero, the [v.sub.o] x [v.sub.o] TRCM matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] relating the vectorial voltage phasor of the incident path with the resultant of all consequent reflected ones on the TL of incidence infinitesimally before the discontinuity as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49a)

equals the sub-matrix remaining from the TRCM matrix [([??]).sub.(v' x v')] that comes out of (44a) after omitting all lines and columns of the conductors that do not exist along the line of incidence. The respective MTRCM matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is then obtained from definition (47a) combined with (45), as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49b)

For any one of the departure lines, let it be the i-th one (i = 1, ..., n), and since all expansion elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [[[[??].sup.E.sub.i[xi]](t)].sub.v'] are by default equal to zero, the [v.sub.i] x [v.sub.0] applicable MTTCM matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] relating the vectorial voltage phasor of the incident path with the resultant of all consequent transmitted ones on the i-th departure line infinitesimally after the discontinuity as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50a)

equals the sub-matrix remaining from the TTCM matrix [([??]).sub.(v' x v')] that comes out of (44b) after omitting all lines and columns of the conductors that do not exist along the i-th departure line. The respective MTTCM matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is then obtained from the definition (47b) combined with (46), as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50b)

4-2.3. Resultant Path Aggregation

4.2.3.1 Generation Process

Incidence of any of the paths generated at the origin as soon as the excitation (23) is applied upon the end of the line generates the consequent reflected and transmitted paths, as described under Subsection 4.2.2. Incidence of any path travelling along a TL upon a horizontally infinitesimal discontinuity of the line (point of medium discontinuity) causes anew travelling-wave excitation, as also described in Subsection 4.2.2. As thus time elapses from the initial excitation (23) applied at the origin, the number of paths travelling over the TL network increases continuously with accelerating rate.

Denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the excitation time of the [[kappa].sup.[+ or -].sub.[xi]]- th positively/negatively travelling path of the [xi]-th mode, and numbering the paths consecutively on the basis of excitation sequence as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

the vectorial paths and phasors are represented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52a)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52b)

Formulae (51) and (52) hold also identically for respective current quantities and vectors, by substituting:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53a)

and applying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53b)

4.2.3.2 End-to-end Aggregation

All the excited paths of (52) superpose along the TL to the following resultant response, of which the phasor (resultant complex envelope) indicates consecutive non-continuous step variations that are due to the above-described discrete path-generation process:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54b)

5. COINCIDENCE WITH FREQUENCY-DOMAIN ANALYSIS

5.1. Physical Interpretation

The FD analysis of a TL provides the response in terms of electric line quantities against harmonic excitation of line's origin according to (5a)-(5b), or equivalently after infinite time from application of the excitation defined in (23) with complex step phasor as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55a)

so that the actual excitation applied is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55b)

Despite the Fourier spectrum expression of (55b), whence the harmonic step [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] seems to lie infinitely along the FD, the excitation of (55a) is indeed narrowband in the sense of (23b), as (i) its spectrum fades rapidly when moving away from the centre (peak) value [f.sub.c], and (ii) assuming a narrowband modulation and transmission scheme, any residual spectral content outside the transmission band is normally expected to be suppressed.

Since correspondence between the TD response of any system and its respective TF in the FD is by default unique, in order to prove coincidence of the well established TL analysis in the FD as summarised in Section 3 above with the travelling-wave TD approach introduced herein, it is sufficient to demonstrate that the response against the harmonic excitation of (5a)-(5b) equals the response against the harmonic step of (55) for:

t [right arrow] [infinity] (56a)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56b)

5.2. Resulting Time-domain Response

Since (i) the phasor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the harmonic step excitation (55) is constant (time invariable), and (ii) the excitation terminal conditions (39) and the medium-discontinuity terminal conditions (41)-(44) involve all merely linear interdependence between the complex envelopes of the excitation applied and the consequent travelling line quantities, the phasors of all travelling modal and line quantities are as well time-invariant, and path aggregation along the line results in consecutive step variations of the complex envelope of the resultant response, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57b)

and analogously for the related current quantities via the substitutions of (53a) after setting:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57c)

5.3. End-to-end Line Quantities

As time elapses towards infinite, the electric line quantities of (54) become harmonic, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58a)

the line voltages, whereas respective line currents are similarly extracted as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58b)

and from (35b), (52), (53), (57), and (58) it results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

5.4. Terminal Conditions at Line's Origin

The phasors of the travelling excitation at line's origin determined in (38) satisfy the terminal conditions (39). From (57a), (57b), (35b), and (41b), the vectorial phasors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of any [k.sup.-.sub.[xi]]-th negatively travelling path in line with the [xi]-th mode (with no loss of generality) incident upon the TL origin excited according to (55), and the vectorial phasors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of respective reflected paths

as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60b)

satisfy:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60c)

Combination of (43a) and (44a) gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As all positively travelling paths except the zero-indexed ones of (37) are generated from reflection following incidence of negatively travelling paths upon the excited origin, from (38a), (39), and (61), the terminal conditions of the harmonic electric line quantities of (58) at line's origin (z = 0) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

5.5. Terminal Conditions at Line's End

For the vectorial phasors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of any positively-travelling [[kappa].sup.+.sub.[xi]] -th path of the [xi]-th mode (with no loss of generality) that arrives at the passively terminated end of the line (i.e., with no excitation source in between any pair of line conductors) and the related travelling excitation vectorial phasors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the reflected paths

consequent upon incidence, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63b)

satisfying through (52a), (35b), and (41b) the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63c)

from (43a) and (44a) it holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

As all negatively travelling paths are generated from reflection following incidence of positively travelling paths upon line's end, from (38a), (39), and (61), the terminal conditions of the harmonic electric line quantities in (58) at line's end (z = l) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

5.6. Resulting Coincidence

The response (58) against the asymptotically harmonic excitation deriving from (55) and (56a) is absolutely equivalent to the FD response set out in Section 3, since (i) all quantities are identically dependent one-to-one after substituting:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

and (ii) all homologous voltage and current phasors are identically associated along the line by virtue of (10), (8a), and (59), and (iii) from (5d), (62), and (65), fall under identical as well terminal conditions.

As a result, the travelling-wave TD approach developed herein is identically equivalent with the established TL theory in the FD.

6. IMPLEMENTATION GUIDELINES

6.1. Channel Response Estimation

The TD response against any narrowband excitation applied according to (23) at the origin of any TL is determined in the TD through the following sequence of steps:

1) Calculate the initial positively travelling paths generated as soon as excitation is applied, from the terminal conditions (39) and interrelation (37) between line and modal excitations.

2) Apply the proper form of the generic terminal conditions (49) and (50) every time an incident path arrives at a point of structural medium discontinuity.

3) Aggregate all paths existing at the point and time of observation according to (54).

If excitation is applied to more than one TL within the observed network, since any uniform TL is a linear system, the aforementioned calculation sequence is performed separately for every single excitation, and the overall response is the resultant of all several responses.

6.2. Structural Reductions

6.2.1. Open Circuit between Excitation Conductor Origins

If no excitation or termination branches are connected except between the excitation conductors and the reference one, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67a)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67b)

the voltage excitation phasors vector

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68a)

is defined, and the line's origin excitation terminal conditions (39) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68b)

6.2.2. Two-conductor Transmission Line

In the case of a single two-conductor TL, the travelling-wave approach turns directly out scalar, as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (69)

From the scalar equivalent (69) of the terminal conditions (68b), the initial positively-travelling path is determined, which is raised as soon as excitation is applied at line's origin. The medium-discontinuity terminal conditions (41)-(44) are replaced by the scalar equivalent according to (69), and all paths raised travel in line with the same propagation constant Vu and the related propagation velocity vu, wavelength An, and propagation delay [[tau].sub.II]([DELTA]l), as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (70)

6.3. Signalling Circuitry--Reference Transposition

Regarding PLC signalling over power cables, transmission is normally expected from an excitation conductor to the reference one, whereas reception is observed either in between the same pair of conductors to determine response of the transmission circuit (channel), or from a different excitation conductor to the common reference one, if interference over another channel (crosstalk) of the MTL structure examined is required. However, transmission and/or reception may also be respectively performed and observed in between a pair of excitation conductors, when differential TL excitation is applied, or if interference from Electromagnetic (EM) field(s) penetrating the TL is required. In such a case, transposition of the reference conductor is necessary in order to estimate the response on travelling-wave basis.

As regards reference transposition from the 0-th conductor of a (v + 1)-conductor transmission line to the k-th one, the electric line quantities are by default transformed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (71a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (71b)

where the initial reference conductor is transposed with the final, and final quantities are indicated by the accentuation mark "'".

7. CONCLUSION

In this paper, travelling-wave analysis of uniform multi-conductor transmission line networks has been elaborated determining the time-domain response to narrowband excitation, developed in direct equivalence with the well established transmission line theory in the frequency domain. Applicability in respect of transmission-line structure has been clarified, and guidelines have been provided for computational implementation, along with further clarifications towards treatment of Power-Line Communication signalling, a well suitable field for application, whereas potential applicability is also extended to other related topics such as surge propagation resulting, e.g., from lightning or transient overvoltage.

APPENDIX A. DERIVATION OF EQ. (28a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX B. DERIVATION OF EQ. (58a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX C. DERIVATION OF EQ. (61)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX D. DERIVATION OF EQ. (62)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX E. DERIVATION OF EQ. (64)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX F. DERIVATION OF EQ. (65)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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Ioannis C. Papaleonidopoulos *, Nickolas J. Theodorou, Christos N. Capsalis

School of Electrical and Computer Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., Zografou, Athens 15780, Greece

Received 20 January 2013, Accepted 10 May 2013, Scheduled 10 June 2013

* Corresponding author: Ioannis C. Papaleonidopoulos (jpap@central.ntua.gr).

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Author: | Papaleonidopoulos, Ioannis C.; Theodorou, Nickolas J.; Capsalis, Christos N. |
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Publication: | Progress In Electromagnetics Research B |

Article Type: | Abstract |

Geographic Code: | 1USA |

Date: | Jul 1, 2013 |

Words: | 8933 |

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