IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 3, AUGUST 2009 833
A Method for Interfacing Lumped-Circuit Modelsand Transmission-Line System Models With
Application to RailwaysZiya Mazloom, Student Member, IEEE, Nelson Theethayi, Member, IEEE,
and Rajeev Thottappillil, Senior Member, IEEE
Abstract—Transient analysis of lossy multiconductor transmis-sion lines (MTLs) has been studied using the finite-differencetime-domain (FDTD) method with lumped loads/devices connectedat line terminations. In electrified railway networks, series andshunt devices (whose circuit models are derived either from ex-periments/borrowed off the shelf), e.g., transformers, converters,switchgear, signal equipments, etc., are found distributed alongthe MTL system. To simulate such railway systems involving bothtransmission lines (TLs) and lumped circuits, an interface tech-nique between TL systems, which is solved using FDTD, and all thelumped circuits, which are solved using alternative transients pro-gram/electromagnetic transients program software (circuit solver),is proposed. This sufficiently accurate method is simple to ap-ply as only instantaneous voltages and currents are transmittedbetween the stand-alone FDTD routine and circuit solver. More-over, the user avoids coding complex circuit models within theFDTD, while at the same time, efficiently uses the potential ofaccurate frequency-dependent loss models (nonexistent in circuitsoftware) coded in FDTD. The technique is applied on typical elec-trified railway systems to demonstrate how traction transform-ers, track circuits, and line interconnections affect the propagatingvoltages and currents. The method could be beneficial for tran-sient protection and insulation coordination studies in electrifiedrailway systems.
Index Terms—Circuit simulation, crosstalk, power system sim-ulation, transmission-line (TL) modeling.
I. INTRODUCTION
TRANSIENT analysis in multiconductor transmission-line(MTL) systems has been studied in detail for various power
systems and electromagnetic compatibility (EMC) studies [1]–[6]. In all these studies, it is common that the lumped loads(boundary conditions) that exist at the line terminations, e.g.,in typical power systems, transformers, switchgear, generators,etc., are located at the sending and/or receiving end substations.In railway systems, loads and devices exist either in series orshunt along and between lines, and not always only at line ends.In typical Swedish railway systems, track circuits are connected
Manuscript received October 1, 2008; revised February 25, 2009. Firstpublished July 7, 2009; current version published August 21, 2009. This workwas supported by the Swedish National Rail Administration (Banverket).
Z. Mazloom and R. Thottappillil are with the Division for ElectromagneticEngineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm,Sweden (e-mail: [email protected]; [email protected]).
N. Theethayi is with Bombardier Transportation, Mainline and Met-ros, SE-722 14, Vasteras, Sweden (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEMC.2009.2023112
Fig. 1. Typical track circuit configuration used in a Swedish railway system.
between the S- and I-rails forming a three-conductor MTL sys-tem (if other overhead conductors are disregarded) with groundas the return conductor, as shown in Fig. 1.
The purpose of the track circuits is to determine whether aspecific section of the track is occupied or not (train positioncontrol and signaling). The signal control is made using a recti-fier and a relay system. The length of a track section can be upto 2.5 km. The relays are located at the ends of the track section,and the feeding point (rectifier unit) is around the section mid-point [7]. The rectifier unit supplies about 7 V dc [8]. We shalllater use the circuit of the aforementioned relay and rectifierunit [9] in the transient simulations. Further, there are insulationgaps along the I-rail (at regular intervals) forming series opencircuits along the line.
Other devices connected in series along the MTL overheadtraction system consist of booster transformers (BTs) and auto-transformers (ATs). These transformers are connected in series(see Fig. 2) and as shunts between the lines (see Fig. 3). Thetransformers (1 : 1) are required to force (suck) the traction re-turn current from the rail (S-rail) to the distribution transformervia the overhead return conductors.
Unlike normal operating conditions, in transient conditionswith lightning or switching transients, series-connected lumpedloads will cause additional reflections and attenuations alongthe MTL system, causing a more complex distribution of volt-ages and currents in the system, causing faults. Statistics fromthe Swedish National Rail Administration (Banverket) havereported that lightning transients have damaged the signalingequipments causing train traffic delays [8]. In order to assessthe failure modes, and later, for surge protection design, thispaper could be beneficial.
834 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 3, AUGUST 2009
Fig. 2. Transformer connections in a BT feeding system.
Fig. 3. Transformer connections in an AT feeding system.
More accurate transient analysis of frequency-dependentlossy MTL systems in time domain has been well studied usingthe finite-difference time-domain (FDTD) method, e.g., [2], [3],and [10]–[12]. On the other hand, alternative transients pro-gram/electromagnetic transients program (ATP/EMTP) [13]–[15] and PSPICE [16] are widely used for steady-state and tran-sient simulations of large power plant/substation equipmentslike switchgear, transformers, generators, etc. Circuit simula-tions in ATP/EMTP are based on Kirchoff’s current law [17],[18]. This feature is advantageous for our study as the FDTDsolution technique is a direct application of both Kirchoff’svoltage and current laws [18].
Earlier MTL simulations, based on solving transmission-line(TL) equations with the FDTD method and involving interfacingwith circuit software, have considered lumped loads typicallyat line terminations, e.g., [19]–[22]. A method for determiningthe effect of lumped components connected to MTL systemsas line terminations using lossless Bergeron lines [23] was pre-sented in [22]. The FDTD method that solves Maxwell’s equa-tions [24] is used in [25] and [26] together with lumped circuitsin SPICE. These studies do not consider components/devices inseries with the line, and hence, are not directly applicable for thekind of MTL systems treated here. In this paper, a more generaland straightforward approach for including series- or shunt-connected components along MTL systems based on solvingTL equations with the FDTD method and interfacing with cir-cuit software will be presented. The solution of TL equationsis based on the FDTD method [12], [18] (code written in FOR-TRAN [27]), and lumped circuits are modeled in ATP/EMTP.The interfacing technique is discussed in the next section withsome validation examples as well.
In Section III, we present the circuit models for the transform-ers and track circuits discussed earlier. Using these circuit mod-
Fig. 4. Voltage node and current section discretization along the TL for solu-tions by the FDTD method.
els, some typical transient simulation cases will be demonstratedin Section IV with the aforementioned interfacing method.
II. INTERFACING TL EQUATIONS SOLVED USING FDTDWITH CIRCUIT SIMULATION SOFTWARE
The algorithms for TL solution based on the FDTD methodcan be found in the literature, e.g., [10], [28], and [29]. Weshall use only the appropriate governing equations of the FDTDmethod for discussion, assuming, for simplicity, a lossless TLsystem. Note that the method is general and applicable forany lossy TL system. Here, the complete system is solved inATP/EMTP, where the FDTD routine is included as a foreignmodel within the ATP/EMTP software [13], [30].
In the FDTD method, the line is discretized into NDZ+1 volt-age nodes and NDZ current sections separated by the distance∆z, as shown in Fig. 4. In Fig. 4, an ideal current source feedsthe line at one end, and RS and RL are the source and loadterminations, respectively. The voltages and currents along theline are solved by the leap-frog scheme [10]. In the time domain,lossy TL equations for aforementioned ground conductors aregiven by
∂V (z, t)∂z
+ Le∂I (z, t)
∂t+
∫ t
0ς (t − τ)
∂I (z, τ)∂τ
dτ = 0 (1a)
∂I (z, t)∂z
+ Ce∂V (z, t)
∂t= 0 (1b)
where V and I are the voltages and currents on the line atany point, the per-unit-length external inductance Le and lineexternal capacitance Ce are calculated from the line geome-try [10], [28], and ς(t) is the transient ground impedance due tothe penetration of electromagnetic fields into the finitely con-ducting ground in the time domain [2], [31], [32]. Note that thetotal internal impedance of the conductors is negligible as com-pared to the total external impedance in the presence of finitelyconducting ground [12]. The transient ground admittance termis omitted in (1) since its contribution is negligible for above-ground wires [12], [28], [31]. Under lossless conditions, theeffect of ς can be disregarded. In the FDTD method, as shownin Fig. 4, for every time step, the voltages along the MTL aresolved first, and later, the currents at each line section are solvedfor. The recursive relations, corresponding to (1b), for the volt-age at the first node V1 , any node along the MTL Vk , and the
MAZLOOM et al.: METHOD FOR INTERFACING LUMPED-CIRCUIT MODELS AND TL SYSTEM MODELS 835
last node VN DZ+ 1 are given by
V n+11 =
(Ce
2∆z
∆t+
12RS
)−1
(Ce
2∆z
∆t− 1
2RS
)V n
1
−In1 +
InS + In−1
S
2
(2a)
V n+1k =
(Ce
∆t
)−1 [Ce
∆tV n
k −Ink − In
k−1
∆z
](2b)
V n+1N DZ+1 =
(Ce
2∆z
∆t+
12RL
)−1
×
(Ce
2∆z
∆t− 1
2RL
)V n
N DZ+1
+InN DZ
. (2c)
The superscripts n in (2) are time steps. Using the voltagesat the nodes, the currents in each section between the nodesof lossless and lossy TL systems are solved by (3a) and (3b),respectively
In+1k =
(Le
∆t
)−1[
Le
∆tInk −
V n+1k+1 + In+1
k
∆z
](3a)
In+1k =
(Le
∆t+
ς (0)2
)−1
(Le
∆t+
ς (0)2
− ς (1)2
)Ink
+ς (1)
2In−1k −
V n+1k+1 + In+1
k
∆z−CIn
k
.
(3b)
In (3b), ς(0) and ς(1) are the transient ground impedances attimes 0 and ∆t, respectively. The transient ground impedancecan be approximated by (4) with N exponential terms (AA con-stants and χ exponentials). Then, the recursive convolution termCI is iteratively calculated, as shown in (5), for Nc conductors.The solutions of this recursive convolution terms are well ex-plained in [12]
ς (t) = AA1e−χ1 t + AA2e
−χ2 t + · · · + AAN e−χN t (4)
CIm =Nc∑j=1
N∑r=1
CImj,r (5a)
CImj,rn + 1k
= eχm j , r ∆t
×
CImj,rn
k+
12AAmj,r
[I
n+(1/2)k − I
n−(1/2)k
]+
12AAmj,r e
χm j , r ∆t[I
n−(1/2)k − I
n−(3/2)k
].
(5b)
In the interface scheme between the FDTD method of solv-ing the MTL with any lumped-circuit model, for a given timestep, all the voltage nodes and section currents along the MTL,except the voltage nodes on which the lumped components areconnected, are solved first. Those nodes, to which the lumpedcomponents are connected, are solved by the circuit solver. Theappropriate section currents entering these nodes are modeled
Fig. 5. Interface scheme between the FDTD method of solving TL and thelumped-circuit models: single conductor line with a component connected atline termination.
Fig. 6. Interface scheme between the FDTD method of solving TL and thelumped-circuit models: single conductor line with a component connected inseries with the line.
as current sources connected to the lumped device, as shown inFigs. 5 and 6. The calculated node voltages (which are the samefor the FDTD method at the lumped load under consideration)are returned from the circuit solver to the FDTD routine and theprocedure is repeated for the next time step.
Consider, for simplicity, a lossless single conductor line abovea perfectly conducting ground and two possible cases: case 1,when the component is at the end of the line (line termination),as shown in Fig. 5, and case 2, when the component is in middleof the line, as shown in Fig. 6.
The left section in Fig. 5 is solved by the FDTD method forthe TL and the right section is solved by the circuit solver. Thevariable Z is any lumped component connected at the load endof the line. C′ is the line node capacitance to the ground of theTL in the FDTD method, i.e., in the circuit form, it has a valuegiven by 1/2C ∆z [see the capacitance terms in (2c)]. IN DZ
is the current entering the last node that is held at a potentialVN DZ+1 . The current source in the right section of Fig. 5 hasthe value IN DZ (values received from FDTD), and the voltageat VN DZ+1 is calculated by the circuit solver, which is laterused in the FDTD recursive current expression.
Now consider a purely resistive load Z = RL . In accordancewith Fig. 5, the current entering the last node along the line canbe expressed in the time domain as
IN DZ (t) =VN DZ+1 (t)
RL+
C∆z
2∂VN DZ+1 (t)
∂t. (6)
The aforementioned expression (6) that expanded in centraldifference yields (7), and is similar to (2c)
InN DZ + In−1
N DZ
2=
V n+1N DZ+1 + V n
N DZ+1
2RL
+C∆z
2V n+1
N DZ+1 − V nN DZ+1
∆t. (7)
836 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 3, AUGUST 2009
Now let us extend the same to loads connected anywherebetween the line, as shown in Fig. 6. In such cases, thecircuit solved in ATP-EMTP will have two current sourcesIk−1 and Ik+1 (values received from FDTD correspondingto currents in the left and right sections of the line). Let usagain consider purely resistive loads Z1 = R1 , Z2 = R2 , andZ3 = R3 for the sake of demonstration. For the circuit sectionin Fig. 6, the currents entering the nodes VL and VR can beexpressed as
IL (t) =C∆z
2∂VL (t)
∂t+
VL (t) − VR (t)R1
+VL (t)R2
(8a)
−IR (t) =C∆z
2∂VR (t)
∂t+
VR (t) − VL (t)R1
+VR (t)
R3. (8b)
Discretizing the aforementioned equations gives (9a) and(9b), respectively
InL +In−1
L
2=
C∆z
2V n+1
L − V nL
∆t− V n+1
R + V nR
2R1
+V n+1
L + V nL
2
(1
R1+
1R2
) ≈ In
L
(9a)
−InR+In−1
R
2=
C∆z
2V n+1
R − V nR
∆t− V n+1
L + V nL
2R1
+V n+1
R + V nR
2
(1
R1+
1R3
) ≈ −In
R .
(9b)
Consider a special case of R1 → 0 and R2 = R3 → ∞.With this boundary condition, in the aforementioned equations,the node voltages are the same, i.e., VL = VR = Vk , and thecurrents are IL = Ik−1 and IR = Ik . With this boundary con-dition, adding (9a) and (9b), (2b) is obtained. The aforemen-tioned analogy can be extended to MTL systems, wherein theequivalent circuit section will contain two current sources foreach line.
III. CIRCUIT MODELS FOR TRACTION TRANSFORMERS
AND TRACK CIRCUITS
For terminal model extraction, a short circuit (SC) test is madeassuming the traction transformer to be a two-port network [18].One port of the network is shorted and a source is connectedto the other port. The voltages and currents are measured at theinjection port, and the SC current is measured at the shortedport. The driving point admittances of the two-port network areobtained in Laplace domain as (10) and (11) from the SC testsas (
I1I2
)=
(y11 y12y21 y22
) (V1V2
)(10)
y11 (s) =(
I1 (s)V1 (s)
)V2 (s)=0
(11a)
Fig. 7. π admittance model used to derive the transformer models.
Fig. 8. RLCG circuit equivalent admittance term.
y22 (s) =(
I2 (s)V2 (s)
)V1 (s)=0
(11b)
y12 (s) = y21 (s) =(
I2 (s)V1 (s)
)V2 (s)=0
. (11c)
The driving point admittance can be related to an equivalentπ model, as shown in Fig. 7. Each component in the π modelis realized as circuit components consisting of resistance, in-ductance, capacitance, and conductance (RLCG model), as dis-cussed in [33]–[35].
The π model admittances are related to the driving pointadmittances as
y11 = y11π + y12π (12a)
y22 = y22π + y12π (12b)
y12 = y21 = −y12π . (12c)
The vector fitting method [36] can be used to determine theequivalent circuit components by fitting the branch of the πnetwork from corresponding admittance functions of (12) inpole–residue form as
yfit (s) =n∑
i=1
ci
s − ai+ s × e + d. (13)
Based on [33]–[38], for every real pole, an RL branch, and forevery complex pole, an RLCG branch can be realized, as seenin Fig. 8. The parameters G0 and C0 , and the RL and RLCGbranches can be directly obtained by the methods proposedin [35]–[38].
A. BT and AT Model
An SC test was performed on a BT unit. The voltage at theinjection point, and the corresponding primary and secondarycurrents were used to derive the π model. For the BT unit,each of the admittances in the π model, as explained previously,consists of two RL branches. The values for the componentsfor y11π and y22π are, as shown in Fig. 7, RS1 = 550 Ω,
MAZLOOM et al.: METHOD FOR INTERFACING LUMPED-CIRCUIT MODELS AND TL SYSTEM MODELS 837
Fig 9. Comparison between experimental and simulated voltage and currentwaveforms for a BT unit.
LS1 = 0.01 H, RS2 = −0.255 Ω, and LS2 = 0.512 mH. Thecomponents of y12π branch are RS1 = 900 Ω, LS1 = 0.1 H,R S2 = −0.271 Ω, and LS2 = 0.481 mH.
The circuit with aforementioned values was injected with thesame experimental injected voltage to see the possible differ-ences in simulated and experimentally measured current andvoltage waveforms, and reasonably good agreement was found,as shown in Fig. 9.
A similar SC test was conducted on an AT unit. For thistransformer, the y11π and y22π branches had four RL branches,and the component values were G0 = 0.700 mS, C0 =3.66 pF, RS1 = 64.0 Ω, LS1 = 3.11 mH, RS2 = −2.48 Ω,LS2 = −1.15 mH, RS3 = 0.831 Ω, LS3 = 0.599 mH,RS4 = −0.405 Ω, and LS4 = −2.25 mH. Similarly, the y12π
The injected voltage and obtained current waveforms in theprimary and secondary terminals of the AT, corresponding tosimulated waveforms based on the π model circuit, are shownin Fig. 10. Again, reasonably good agreement was found. In theexperiments with the AT, higher oscillations were found in oneof the windings in the early time, but not in the other (comparemiddle and bottom windows in Fig. 10). This could be due tothe switching noise carried from the impulse generator.
In the BT tests, a Schaffner NSG 650 impulse generator, twoPearson 411 current probes, and a Tektronix P6015A voltageprobe were used. In the AT tests, a MIG 1206 impulse generator,Pearson 5046 and 101 current probes, and a Tektronix P6015Avoltage probe were used.
B. Track Circuits
Both the track circuits, i.e., relay and rectifier units are con-nected as shunts between the rails. The relay circuit across therail consists of an inductor of 4 H in series with a relay coil
Fig 10. Experimental and simulated waveforms for a SC test on an AT unit.
Fig. 11. Two-conductor MTL system above ground with boundary conditions,as shown in Table I.
whose resistance is 1 kΩ and inductance is about 2 H. The rec-tifier unit supplies a 15-Ω resistance to maintain a potential of7 V across it. This 15-Ω resistor in series with an inductor of1 H and a 2-Ω resistor appear across the rail.
IV. FDTD-ATP/EMTP INTERFACE VALIDATION
Before demonstrating simulations with traction transformersand track circuits, we demonstrate a validation of the FDTD-ATP/EMTP interface discussed earlier for an MTL case assum-ing simple series RL components. Simulations by a direct FDTDmethod (solving all relevant differential equations representingthe lines and circuit equations, as discussed earlier) and theinterface method were compared.
The considered MTL case consists of two lossless overheadconductors, with radii 5 mm and located 10 m above a perfectlyconducting ground, with a horizontal spacing of 1 m betweenthe conductors, as shown in Fig. 11. The length of the lines wasset as 3 km. A double-exponential impulse current source, givenby IS (t) = 1.09(e−104 t − e−5.8×105 t), feeds the MTL at one ofthe ends. Such a current source waveform represents lightningor some of the switching transient waveforms. The components(arbitrary values chosen only for the sake of demonstrations)and their positions along the line are described in Table I, basedon Fig. 11. The simulated currents are shown at the middle ofthe line and at the line ends (points marked in Fig. 11). Theresistance RS is kept as an open circuit.
838 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 3, AUGUST 2009
TABLE ILUMPED LINE TERMINATION AND LUMPED SERIES COMPONENTS
CONNECTED ALONG THE LINES IN FIG. 11
Fig. 12. Simulations corresponding to boundary condition 1, as shown inTable I.
The currents at the middle of the lines and line ends corre-sponding to boundary conditions 1 and 2 are shown in Figs. 12and 13, respectively. It is seen that the proposed interface methodis in excellent agreement with the direct solutions by the FDTDmethod. Note that we have shown simple cases of resistive andinductive loads for which the governing circuit equations canbe easily incorporated with the FDTD code itself. Such an ap-proach will be difficult if complex linear or nonlinear circuits,e.g., parts of a substation, need to be included with the FDTDmethod through coded integro-differential equations. Concern-ing solution stability, the authors have verified that for typicalTL systems, it is better to adopt a time step less than the timestep given by the Courant condition (wavelength to line lengthrelationship [10], [28]). This is because without any lumpedcomponents on the line, a time step (dtc) of an order less thanthe time step calculated from Courant condition usually givesstable and ringing-free solutions. When the components on theline are added, these circuits bring additional reflections, at-tenuations, and dispersions (decided by circuit time constants).The resulting waveforms at the lumped loads may not sup-port Courant time step dtc for propagation along lines; hence,a trial and error method for finding a suitable final time stepmuch lower than dtc is to be sought for stable and ringing-freesolutions.
V. MTL SIMULATIONS TO STUDY THE INFLUENCE OF
AFOREMENTIONED LUMPED COMPONENTS
The MTL system used for simulations with and withoutseries- and shunt-connected devices is shown in Fig. 14. The
Fig. 13. Simulations corresponding to boundary condition 2, as shown inTable I.
Fig. 14. MTL system representative of a typical railway traction conductorfeeding system used for simulations.
TABLE IICONDUCTOR RADII AND CHARACTERISTIC IMPEDANCES
FOR LINE TERMINATIONS IN FIG. 14
system is representative of a typical railway traction feedingsystem with five-key above-ground conductors, namely, an aux-iliary wire, a catenary wire, a return conductor/negative feeder(addressed as return conductor as of here), and S- and I-rails.The conductor radii and the values of characteristic impedancesused for line terminations, as in Fig. 14, are shown in Table II.In order to fit more than one BT or AT, as shown in Figs. 2 and3, a line length of 15 km was chosen for simulations.
In Fig. 14, RS is kept as an open circuit. The same source, asdiscussed in the previous section, is connected to the auxiliarywire in all calculations, as shown in Fig. 14. Three case simula-tions, as shown in Table III, were made to show the influence ofseries- and shunt-connected transformers and track circuits onthe currents propagating along the conductors shown in Fig. 14.In case 1, the I-rail was continuous, but in the other cases, theI-rail was discontinuous and track circuits were connected, asshown in Fig. 1. The return conductor interconnections witheither the S-rail or the transformers are as in Figs. 2 and 3.
MAZLOOM et al.: METHOD FOR INTERFACING LUMPED-CIRCUIT MODELS AND TL SYSTEM MODELS 839
TABLE IIISYSTEM TOPOLOGY FOR THE CASES SIMULATED
Fig. 15. Current distribution along the MTL system of Fig. 14, correspondingto case 1. The top window shows currents in the rails (solid lines for S-railand dashed lines for I-rail). The bottom window shows currents in the catenary(solid lines) and return conductors (dashed lines).
In the simulations, a finitely conducting ground with conduc-tivity and relative permittivity of 1 mS/m and 10 was used tosimulate wave propagation over lossy ground [2], [12], [28].
Currents propagating along MTL systems corresponding tocases 1, 2, and 3 (see Table III) are shown in Figs. 15–17.
Comparing Figs. 15–17, it is seen that the series- and shunt-connected components do affect the current wave shapes (corre-spondingly voltages). For one-to-one comparisons, the verticalaxes of all the aforementioned figure windows are kept constant.The following observations can be made for a 1-A-current-peakdouble-exponential impulse in the auxiliary wire (currents inthe auxiliary wire not shown), as shown in Fig. 14.
A. Source End
At source end, for all cases up to 10 µs, the induced cur-rents in the rails are similar (compare Figs. 15–17). After thistime, unlike case 1, for cases 2 and 3, S-rail currents amplitudeincreases with the same polarity, while the I-rail polarity haschanged and the amplitude is the same as was the peak in the0–10 µs interval. For case 2 (see Fig. 16), after 40 µs, the S-railcurrent peak reduces as a function of time but for case 3 (seeFig. 17), the S-rail current peak increase by three times com-pared to the peak at around 20 µs. Note that in cases 2 and 3,the current peak from which both the polarity and amplitudesdeviate for the rails compared to case 1 (see Fig. 15) is at about
Fig. 16. Current distribution along the conductors in the MTL system ofFig. 14 with three BTs connected, corresponding to case 2. The top windowshows currents in the rails (solid lines for S-rail and dashed lines for I-rail).The bottom window shows currents in the catenary (solid lines) and returnconductors (dashed lines).
Fig. 17. Current distribution along the conductors in the MTL system ofFig. 14 with two ATs connected, corresponding to case 3. The top windowshows currents in the rails (solid lines for S-rail and dashed lines for I-rail). Thebottom window shows currents in the catenary (solid lines) and return conductor(dashed lines).
6 µs due to the location of the track circuit, and is at about900 m from the source end.
Interestingly, the currents at the source end in the catenary andreturn conductors have similar amplitudes all through (compareFigs. 15–17). It is found that current amplitudes for catenaryand return conductors, in general, have been affected in cases 2and 3 from about 17 µs, as the BT or AT are located at a distanceof 2.5 km from the source end. The currents in the catenary aresomewhat higher than the return conductor currents possiblydue to the finitely conducting ground [39].
B. Middle of the Line
The currents in the middle of the lines generally have higheramplitudes and also show dispersion when the currents are prop-agating along a TL in dispersive media [39]. The current in the
840 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 3, AUGUST 2009
Fig. 18. Voltages across the rails at the relay unit positions, R-1, R-3, R-6,and R-8, for the cases, as shown in Table III.
rails are initially of positive polarity (see Figs. 15–17), and af-ter about 8–10 µs, the polarity is the same as that for the caseof the source end. The currents in the rails for cases 2 and 3show polarity variable oscillations (see Figs. 16 and 17) withdifferent amplitudes due to reflections from both track circuitsand interconnections. Variable amplitudes are also attributed tosharing of the currents between wires through interconnections.The currents in the catenary and return conductors do not showany polarity variable oscillations (see Figs. 16 and 17), andthey largely resemble case 1 (see Fig. 15). The peak amplitudesfor the currents in the catenary and return conductors have notbeen affected; however the waveforms, at later times, presentamplitude oscillations due to BTs, ATs, and interconnections.
C. Load End
The currents at the load end are similar to the currents inthe middle of the line for case 1 (see Fig. 15). The currents inthe rails for cases 2 and 3 (see Figs. 16 and 17) are oscillatory,and vary depending on the number of reflections that arrivefrom track circuits, interconnections, and source end. These areseen for the S-rail currents that are increasing in the late times(see Figs. 16 and 17). The currents in the catenary and returnconductors follow a similar trend as was the case in the middleof the lines (see Figs. 15–17).
D. Voltage Across Rails
The voltages across the rails are of interest, in general, forEMC studies. This includes providing protection for track cir-cuits against lightning and switching transients. It is evidentfrom the previous analyses that currents in the rails vary signifi-cantly due to the presence of lumped loads and interconnectionsalong the MTL. The voltages for four different relay positionsfor the cases, as per Table III, are shown in Fig. 18.
The main observation is that the value of the peak voltagesfor cases 2 and 3 are about 50–100 times higher as comparedto case 1. Further, the shapes of voltages have been signifi-cantly affected. This indicates that, while deciding protection
for equipment, realistic simulations with all possible lumpeddevices along the MTL should be considered.
VI. CONCLUSION
In this paper, it is shown that while dealing with MTL sys-tems representative of electrified railway networks, the lumpedcomponents like BTs, ATs, track circuits, line interconnections,etc., along the MTL system should be considered for realisticcurrent/voltage distribution assessments. Due to the practicaldifficulties associated in simulating MTL systems in conjunc-tion with complex circuits (representative of lumped compo-nents and interconnections), an interfacing method between theFDTD technique (which solves TL equations) and a circuitsolver (any circuit solver software) is proposed. This methodis validated through examples as well. The extraction of circuitmodels for aforementioned lumped components by means of SCtests and terminal modeling based on vector fitting technique isalso discussed.
Simulations are shown for three cases, namely, case 1: with-out any lumped devices or interconnections or discontinuities;case 2: BT system with all interconnections and discontinuities;and case 3: AT system with all interconnections and disconti-nuities. It is demonstrated that both peak current and voltagedistributions in all the aforementioned cases are different fromeach other, and that in cases 2 and 3, reflections are very sig-nificant to bring about attenuation, amplification, or dispersionof propagating voltages and currents along the MTL system.The methods and observations presented in this paper could beuseful for the protection and mitigation studies for distributedelectrical systems such as electrified railways.
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Ziya Mazloom (S’09) was born in Yazd, Iran, in 1982. He received the M.S.degree in applied physics from Uppsala University, Uppsala, Sweden, in 2005,and the Ph.D. degree from the Division of Electricity, Uppsala University, in2009. He is currently working toward the Ph.D. degree at the Division for Elec-tromagnetic Engineering, Royal Institute of Technology, Stockholm, Sweden.
Nelson Theethayi (S’04–M’06) was born in India, in 1975. He received theB.E. degree (Hons.) in electrical and electronics from the University of Mysore,Mysore, India, in 1996, the M.Sc. (engineering) degree in high-voltage engi-neering from the Indian Institute of Science, Bangalore, India, in 2001, and thePh.D. degree in electricity from Uppsala University, Uppsala, Sweden, in 2005.
From 2005 to 2008, he was a Researcher at the Electromagnetic Compati-bility (EMC) Group, Division for Electricity, Uppsala University. Since 2008,he has been with Bombardier Transportation, Mainline and Metros, Vasteras,Sweden. His current research interests include EMC, high-voltage engineering,and electrical power systems.
Dr. Theethayi is a member of the IEEE EMC Society, the IEEE Dielectricsand Electrical Insulation Society, the IEEE Power Engineering Society, and theIEEE Industry Applications Society.
Rajeev Thottappillil (S’88–M’88–SM’06) receivedthe B.Sc. degree in electrical engineering from theUniversity of Calicut, Malappuram, India, in 1981,and the M.S. and Ph.D. degrees in electrical engi-neering from the University of Florida, Gainesville,in 1989 and 1992, respectively.
During 2000–2008, he was Professor at UppsalaUniversity, Uppsala, Sweden. Since September 1,2008, he has been the Chair of the Electric Power En-gineering and Design, Royal Institute of Technology,Stockholm, Sweden. He has authored or coauthored
more than 150 scientific papers in journals and international conferences. Hiscurrent research interests include lightning phenomenon, electromagnetic inter-ference, and electromagnetic field theory.
Prof. Thottappillil is the Chairman of the European Cooperation in Scienceand Technology (EU COST) Action P18 “Physics of Lightning Flash and ItsEffects.” He is a member of the SC 77 C, International Electrotechnical Com-mission on High Power Transients. He is also a member of the InternationalCouncil on Large Electric Systems (CIGRE) Committee WG C4.407 on light-ning parameters for engineering applications.
