Modelling the magnetic field caused by a dc-electrified railway with linearly changing leakage currents

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1 Earth Planets Space, 63, , 11 Modelling the magnetic field caused b a dc-electrified railwa with linearl changing leakage currents Risto Pirjola Finnish Meteorological Institute, P. O. Bo 53, FI-11 Helsinki, Finland (Received December 17, 1; Revised Ma 4, 11; Accepted June, 11; Online published Januar 6, 1 Magnetic fields produced b dc railwas can disturb operations at geomagnetic observatories. These magnetic fields consist of two parts: the ideal full-loop field due to the traction current in an overhead wire and in the rails, and the leakage field due to currents leaking from the track into the ground. Pirjola et al. (7 present formulas with numerical eamples for the magnetic field assuming a constant leakage current densit between the train and the feeding substation. Lowes (9 considers this assumption questionable. In this paper, we derive eact formulas for the magnetic field assuming that the leakage current densit changes linearl being largest at the train and decreasing to zero at the substation. The Earth is considered laered. Computer codes enable calculations of the magnetic fields due to an number and configuration of the train-substation pairs at an points of observation. Numerical eamples show that the value of 1 pt can easil be eceeded. An important conclusion is that the magnetic fields in the cases of a linearl changing and a constant leakage current densit do not differ much. Particular attention is paid to the temporal behaviour of the magnetic field when one or two trains travel along a railwa. Ke words: Geomagnetic observator, dc train, dc magnetic field, current leakage in the ground. 1. Introduction Quasi-dc electric currents produce magnetic fields that can seriousl disturb recordings and operations at geomagnetic observatories, at which the maimum allowable noise levels are about 1 nt = 1 pt toda. This maimum value is based on private discussions with representatives of the Ottawa Geomagnetic Observator, Canada, and its choice is also argued b Pirjola et al. (7. Consequentl, advance calculations and estimations of epected magnetic fields at geomagnetic observatories due to possible dc equipment and devices being in the planning stage are ver important. Problematic magnetic fields are created especiall b dc-electrified railwas (e.g., Yanagihara, 1977; Georgescu et al., ; Pirjola et al., 7; Lowes, 9; and references therein. There are historical eamples of the need to close a geomagnetic observator and move it to a more distant location because a dc-electrified railwa or tramwa makes the operation impossible (e.g. Nevanlinna, 4. Ishikawa et al. (7 discuss the noise due to dc trains in geoelectric field data in connection with the development of earthquake prediction techniques. In principle, the traction current needed to move a train should flow in a closed circuit formed b an overhead feeding wire and a return path along the rails. Due to the proimit of the wire and the equal, but opposite, return current, the magnetic field should decrease sharpl with the distance from the railwa. In practice, however, the rails ma have Copright c The Societ of Geomagnetism and Earth, Planetar and Space Sciences (SGEPSS; The Seismological Societ of Japan; The Volcanological Societ of Japan; The Geodetic Societ of Japan; The Japanese Societ for Planetar Sciences; TERRAPUB. doi:1.547/eps electric contacts with the ground enabling a current leakage into the Earth. Thus, the return current does not cancel the magnetic effect of the current in the traction wire. Moreover, the leakage current ma flow to large distances in the ground. Based on the model presented b Georgescu et al. ( for studies of magnetic fields caused b a dc-electrified railwa and b utilising the Biot-Savart law, Pirjola et al. (7 derive the formulas for the ideal full-loop magnetic field, i.e. no leakage, and for the leakage magnetic field, and the total field is naturall the sum of these two contributions. It should be emphasised that the formulas are eact including somewhat complicated algebraic manipulations. This is, however, justifiabl criticised b Frank Lowes in his impressive and significant paper (Lowes, 9 because the geometr and the phsical interpretations are obscured b the algebra. On the other hand, since eact closed-form solutions eist that are eas to handle in computer calculations, the are worth being used in numerical modelling of the magnetic fields in question. Another criticism that Lowes (9 presents concerns the assumption made both b Georgescu et al. ( and b Pirjola et al. (7 that the leakage current (= jd is constant along the track, i.e. the leakage current densit per unit length (= j is the same at each infinitesimal section [, + d] between the substation ( = and the train ( = L. On page viii of Lowes (9, the constantleakage approimation is called a completel unrealistic situation. Lowes proves this statement b careful investigations of the track-to-soil voltages and the associated leakage currents in different cases. To get an idea of the effect of spatial variations of the 991

2 99 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY leakage current densit j, it is assumed in this paper that j is a linear function of the position coordinate between the substation and the train. Referring to section 4 of Lowes (9, a linear variation is not an eact form of the leakage current but, according to his section 3, a linear function is a reasonable approimation. Similarl to Georgescu et al. ( and b Pirjola et al. (7, we assume that the current, which leaks into the ground everwhere along the track between the train and the substation, returns to the sstem at the substation. Thus, the situation considered in this paper corresponds to figure 3(b of Lowes (9, i.e. the leakage current is zero at the substation and increases linearl reaching its maimum at the train. Another reason for the assumption of j being a linear function of is that, similarl to the case of a constant leakage, the magnetic field turns out to have eact closed-form epressions in this case as well. In this paper, we indicate that a linearl changing leakage model results in a larger magnetic field than a constant leakage model. However, the difference between the two cases is small, and so in practice, both models can be applied to estimating magnetic disturbances produced b dc-electrified railwas. In Section of this paper, we present the theoretical model including a linearl changing leakage current densit and give the formulas for the magnetic field. Section 3 is devoted to numerical calculations, and Section 4 provides concluding remarks. It is necessar to emphasise that the numerical values used in the calculations are chosen to be comparable with Pirjola et al. (7, even if the do not correspond to modern railwas precisel.. Theoretical Model Similarl to Pirjola et al. (7, we consider a rectangular loop consisting of a train-substation pair and of the track section and the overhead traction wire between the train and the substation. Let the distance from the substation to the train and the height of the overhead wire be L and h, respectivel. The rectangle thus has two horizontal sides of the length L and two vertical sides of the length h. One horizontal side lies at the Earth s surface and the other at the height h. The train is fed b a dc traction current J 1 flowing along the overhead wire from the substation to the train. A right-handed Cartesian z coordinate sstem is used where the ais is parallel to the above-mentioned rectangular loop with the substation and the train ling at = and = L, respectivel. The plane of the rectangular loop and Earth s surface define the z plane and the plane, respectivel. The z ais points downwards. In an ideal situation, the traction current J 1 returns from the train to the substation along the rails but in practice a certain amount, denoted b J, of the current leaks into the earth and takes a path determined b the ground conductivit before going back to the sstem at the substation, which is assumed to be the location of the return of all the current to the sstem. The leakage at a given infinitesimal section [, + d] ( L is given b j (d where j = j ( is the leakage current densit per unit length. It should be noted that we neglect the track etensions beond the substation and beond the train (at < and > L. This assumption includes an approimation (see Lowes, 9. However, to be consistent and comparable with the stud presented b Pirjola et al. (7, the assumption of neglecting the etensions is also made in this paper. Furthermore, Georgescu et al. ( support the neglect, too. At this point, it is also necessar to emphasise that a theoretical model alwas necessaril suffers from different shortcomings, so that it is not appropriate to tr to look at ever minor detail when a model is considered as a whole. Mathematicall and phsicall the situation discussed in this paper can be described b thinking that in addition to the ideal rectangular full loop we have a set of elementar leakage loops. Each of them carries a current j (d and consists of a part from = to = along the rails and of a ground return path from = to =. Along the rails, this current is opposite to the full-loop current J 1, i.e. it flows to the positive direction. This means that the leakage current flowing towards the train at a point ( L is J = J( = j ( d (1 Thus the real current flowing along the rails towards the substation at this point is J rail = J rail ( = J 1 J( = J 1 j ( d ( We define the total leakage current to be equal to J( = and denote it b J, i.e. J = j ( d (3 Pirjola et al. (7 considered j ( to be independent of, and so from Eq. (3, j = J /L. Now, we assume that j ( is a linear function of, i.e. j = j ( = k (4 where k is a constant, which can be determined in terms of J and L b using Eq. (3 leading to k = J (5 L As concerns the full loop, the resulting magnetic field is, of course, eactl the same as that given b equations (4 (1 of Pirjola et al. (7. Thus we can directl adopt the following formulas for the contribution of the full loop to the total magnetic field B F = B + B 4 (6 + (L B F = B + B h 3 + h L B 4 (7 + (L B F z = B 1 + B 3 + h (8

3 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY 993 Fig. 1. Leakage current J = J( given b Eq. (1 between the substation at = and the train at = L = 3 km. The total leakage current epressed b Eq. (3 equals J = A. The lower (red line and the upper (blue curve correspond to a constant and to a linearl increasing leakage current densit j = j ( per unit length between = and = L, respectivel. where B 1 = µ J 1 4π µ J 1 B = 4π + ( µ J 1 B 3 = 4π + h ( L + (L + + (9 h + + h (1 + + h L + + (L + h (11 µ J 1 B 4 = 4π h (1 + (L + (L + h These B k (k = 1,, 3, 4 quantities are the fields created b the four sides of the full loop. As usual, µ is the vacuum 7 Vs permeabilit (= 4π 1 Am. Similarl to Pirjola et al. (7, we assume that the Earth is laered (and in the etreme case even uniform. Therefore the leakage currents flow radiall smmetricall from the point of injection to all directions in the ground. Based on a modified version of the Fukushima theorem (Fukushima, 1976, such a current distribution creates eactl the same magnetic field at the Earth s surface as a semi-infinite vertical line current starting from the surface downwards. Thus the ground part of the elementar leakage loop that carries a current j ( d can be replaced with a vertical upward current at = =, z and a vertical downward current at =, =, z. The horizontal parts of the leakage loops lie at z = and z =, the latter of which does not contribute to the magnetic field at the Earth s surface. Figure 1 depicts the leakage current J = J( defined b Eq. (1 between the substation located at = and the train located at = L = 3 km. The total leakage current given b Eq. (3 equals J = A. The lower line (with J( = J (1 and the upper curve (with L J( = J (1 correspond to a constant and to a linearl increasing leakage current densit j = j ( per unit L length between = and = L, respectivel. The leakage current J( gives the total amount of current in the elementar leakage loops whose width is larger than or equal to. Therefore, since J( in the linearl increasing case is alwas larger than in the constant case, larger elementar leakage currents are carried b wider loops in the former case. This implies that the magnetic fields are obviousl larger in the linearl increasing case than in the constant case. The same conclusion can also be drawn directl from the growth of the leakage current densit j ( with in the case of the linear increase. Net we derive eact formulas for the magnetic field components B L, B L and B z L produced b all elementar leakage loops, and the total field is the sum of the full and leakage contributions: B TOTAL u = B F u + B L u (13 where u stands for, or z. The magnetic field due to an elementar leakage loop has eactl the same form as equations (11 (13 of Pirjola et al. (7, ecept that j = j ( = k is a function of, i.e. db = µ k d 4π ( + ( + (14

4 994 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY db = µ k d ( 4π + ( + ( L (L ( + 1 (9 db z = µ k d 4π + ( + (16 + We ma thus finall write the formulas for the magnetic The integration of Eqs. (14 (16 from = to = L leads to the following formulas for the leakage contribution to the magnetic field where B L B L B L z = = db = g(c 1 + C (17 = db = g(f 1 + F (18 db z = g (P 1 + P (19 g = µ k 4π = µ J ( π L C 1 = + ( d (1 C = 1 d ( + ( F 1 = + ( d (3 L F = d (4 + ( P 1 = + ( d (5 P = L + d (6 The terms C, F and P can be directl calculated because the integral in them simpl gets the value L. The terms C 1, F 1 and P 1 require more work but the can also epressed in terms of elementar functions in closed forms as follows C 1 = ln + (L + + ( ( ( L arctan + arctan (7 F 1 = L + ln + (L + ( ( ( L arctan + arctan (8 P 1 = L + (L ln L (L field components produced b the leakage currents B L = µ J ln + (L π L + + ( ( ( L arctan + arctan L ( + B L = µ J L + ln + (L π L + ( ( ( L arctan + arctan + L ( + Bz L = µ J L πl + (L ln L L (L + L + (L (3 (31 (3 The three components of the magnetic field due to the trainsubstation sstem with a linearl changing leakage current are now obtained from Eq. (13 with Eqs. (6 (8 and (3 (3. It seems evident that, in practice, the total leakage current depends on the distance L between the train and the substation. Thus, it might be more reasonable to epress J as J = kl in Eqs. (3 (3, i.e. in terms of k and L, in accordance with Eq. (5. The constant k is clearl proportional to the traction current (but not necessaril linearl proportional. Similarl, in the corresponding equations presented b Pirjola et al. (7, it would obviousl be more reasonable to use the quantit j instead of J (= jl. The z coordinate sstem used above is fied to the particular train-substation pair. Thus when investigating the magnetic field produced b several trains operating simultaneousl we actuall have man z sstems, in which onl the z aes are the same, i.e. downward vertical and having the point z = at the Earth s surface. The components of the total magnetic field at a given point of observation have to be epressed in a single fied coordinate

5 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY 995 Fig.. Absolute values of the magnetic field due to a south-north dc-electrified railwa. The thicker upper (black curve and the thinner lower (blue curve refer to a linearl changing and to a constant leakage current densit, respectivel. The horizontal ais gives the distance along a line towards the east from the substation. Additional details of the calculations are given in the tet. The small (red circles show measured data in Calgar, Canada, in March 6 (cf. Pirjola et al., 7. sstem, and our choice is to use, besides the z component, the (geographical north and east components. To achieve this, information of the geographical direction of the line between each train and the corresponding substation(s is needed. A practical wa is to epress all locations in terms of their geographical latitudes and longitudes. In practice, the areas to be considered are small enough to enable simple flat-earth relations between latitudes and longitudes and Cartesian coordinates. Based on the formulas presented above, Octave/MatLab program codes have been prepared to enable numerical computations of the magnetic fields due to a dc-electrified railwa, in which the train-substation pairs and the points of observation ma have an number, configuration and parameter values. 3. Numerical Results We now consider the situation that is associated with the measurements carried out in the vicinit of a (nearl straight south-north dc-electrified railwa in Calgar, Canada, in March 6 (see Pirjola et al., 7, section 4. Unfortunatel, the configuration of the railwa sstem in Calgar and the technical details including the values of the parameters associated with the operation of the sstem are unknown. Therefore theoretical modelling of the situation during the measurements is based on unconfirmed assumptions and conclusions. This emphasises the need for additional magnetic field measurements in Calgar or somewhere else to confirm the validit of the theoretical modelling (or to reveal its shortcomings. Similarl to figure 7 of Pirjola et al. (7, we assume that there is onl one train fed b a current of J 1 = 1 A from a substation located at a distance of 3 km south of the train, i.e. L = 3 km. The total leakage current J and the height of the feeding wire h are A and 5 m, respectivel. Thus, a constant leakage current densit is j = J /L = 6.7 ma/m. For a linearl changing leakage current densit, Eq. (5 implies that the constant k equals k = J /L =.44 ma/m, so that the leakage current densit varies from 13.3 ma/m at the train ( = L to zero at the substation ( =. The upper (black and lower (blue curves in Fig. show the absolute values of the magnetic field for a linearl changing and a constant leakage, respectivel. The horizontal ais gives the distance along a line towards the east from the substation. The small (red circles show the measured data in Calgar. In fact, the lower curve, together with the measurements, re-produces the information included in figure 7 of Pirjola et al. (7. We see that the choice between a linearl changing and a constant leakage does not impl a big difference in the magnetic field, and both models agree well with measurements. However, the lack of knowledge of technical parameters associated with the Calgar railwa has to be kept in mind. The measured data shown in Fig. are concluded from abrupt jumps in the time-dependent signals that otherwise follow the natural variations of the geomagnetic field. Besides, a scaling of the magnetic field is performed based on a reference site (see Pirjola et al., 7. The observation from Fig. that a linearl changing leakage results in a larger magnetic field than a constant leakage can be understood b noting that, in the linear case, the elementar leakage loops with the largest sizes also carr the highest currents, as eplained in Section. Assuming a lin-

6 996 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY Fig. 3. Absolute values of the full-loop (lowermost, blue, leakage (middle, red and total (uppermost, black magnetic fields at a point of observation located at 5 km east of a south-north railwa as functions of time produced b a train travelling from south to north. The leakage current densit is assumed to be constant. The maimum allowable noise level at a geomagnetic observator (1 pt is shown b the horizontal (light blue line. The small (black circles and the small (red crosses are related to the schedule of the train. Additional details of the calculation are given in the tet. earl changing leakage improves the agreement with measured magnetic data at larger distances whereas the agreement becomes slightl worse at smaller distances. Considering disturbing magnetic fields to be accepted at an observator, small distances are, however, less important because there the magnetic fields are definitel too high and thus need not be investigated precisel. Net we consider the time dependence of the magnetic field at a point of observation when one train travels or two trains travel along the railwa. Such calculations have been performed for the planned Light Rail Transit (LRT sstem in Ottawa, Canada, but in this paper, we simpl continue considering a straight south-north railwa. It is assumed to consist of five sections between substations, each of them 3 km in length, so that the total length of the railwa is 15 km and the number of substations is si. Let the point of observation be located at 5 km east of the third substation from south. The trains are alwas assumed to get the traction current from the nearest substation south of the train b feeding wires at the height of 6 m. The trains are assumed to stop at ever substation (and nowhere else with the following schedule: - Acceleration from zero to the (maimum speed of v ma = 8 km/h during T a = s. - Constant speed of v ma = 8 km/h. - Deceleration from v ma = 8 km/h to zero during T d = s. - Stop at the substation for T s = 3 s. During the acceleration deceleration phases, the traction current is set to J 1 = 3 A, and during the constant speed phase to J 1 = 5 A. (These values are based on private discussions in North America in 1. The choice of an appropriate value for the leakage current densit seems to involve a lot of discrepancies. In the above calculation, we have the constant value of 6.7 ma/m. Some unpublished information indicates leakage current densities of ma/1 feet.67 ma/m, i.e. a difference of a factor of 1. In the computations presented now, we use the value of 95 ma/3 m = 3.17 ma/m ( 95 ma/1 feet for a constant leakage current densit. (It is based on a private discussion in North America in 1. When the leakage current densit is constant, the total leakage current at a track length L is jl. To make the cases of a constant and of a linearl changing leakage current densit comparable, we want that, in the case of a linearl changing leakage current densit, the total leakage current at a track length L is also jl, which implies from Eq. (5 that k = j/l with j = 95 ma/3 m. For a constant leakage current densit, the lowermost (blue, middle (red and uppermost (black curves in Fig. 3 present the absolute values of the full-loop magnetic field (B F, of the leakage magnetic field (B L and of the total magnetic field (B TOTAL = B F + B L at the point of observation as functions of time, respectivel. (Note that since we consider the absolute values of three-dimensional vectors the sum of the blue and red curves is generall not equal to the black curve. Figure 4 corresponds to Fig. 3 but a linearl changing leakage current densit is assumed. Similarl to Fig., Figs. 3 and 4 show that a linearl changing leakage current densit leads to somewhat higher magnetic fields at the point of observation, whose eplanation results from high currents in large elementar leakage loops in the

7 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY 997 Fig. 4. Similar to Fig. 3, but a linearl changing leakage current densit is assumed. Additional details of the calculation are given in the tet. Fig. 5. Absolute value of the (total magnetic field a point of observation located at 5 km east of a south-north railwa as a function of time produced b two identical trains travelling from south to north at 9-s separation. The thicker lower (black and the thinner upper (blue curves correspond to a constant and a linearl changing leakage current densit, respectivel. The maimum allowable noise level at a geomagnetic observator (1 pt is shown b the horizontal (light blue line. The small (black circles and the small (red crosses are related to the schedule of the first train. Additional details of the calculations are given in the tet. case of a linearl changing leakage current densit, as mentioned above. Both in Fig. 3 and in Fig. 4, the maimum allowable noise level at a geomagnetic observator (= the horizontal line at 1 nt is clearl eceeded during most of the travel time of the train. We also see that the leakage magnetic field is usuall larger than the full-loop magnetic field. So far we have onl considered the magnetic field created b one train but in practice there are usuall several trains travelling simultaneousl. In Fig. 5, it is assumed that a train is followed b another identicall operating train after 9 s. The lower (black and the upper (blue curves depict

8 998 R. PIRJOLA: MAGNETIC FIELD CAUSED BY A DC RAILWAY the absolute value of the magnetic field at the point of observation for a constant and a linearl changing leakage current densit, respectivel. In agreement with Figs. 3 and 4, the linearl changing leakage results in a larger magnetic field. The magnetic field is above the maimum allowable noise level practicall all the time. An important conclusion obtained from Figs. 3, 4 and 5 and alread drawn from Fig. is that the magnetic fields in the cases of a linearl changing and a constant leakage current densit do not differ much from each other. Consequentl, both leakage current models can be equall utilised for practical estimation of magnetic fields due to dc railwas. In the numerical computations discussed in this section, we onl consider the absolute values of the magnetic field. The theor and formulas presented in Section would naturall enable studies of individual components of the magnetic field separatel. However, regarding practical investigations of disturbances at observatories due to magnetic fields, the absolute value is the quantit of importance. Moreover, the absolute value is larger than the components and thus represents the worst case. 4. Concluding Remarks Railwas operating b dc electricit create disturbing magnetic fields, which ma cause problems at geomagnetic observatories. Of particular significance in this respect is the amount of currents leaking from the track into the ground. As a supplement to the paper b Pirjola et al. (7, eact formulas for the magnetic field due to a dcelectrified railwa are presented in this paper b assuming that the leakage current densit per unit length decreases linearl from the train to the substation. Pirjola et al. (7 consider a spatiall constant leakage current densit, which is an assumption criticised b Lowes (9. The theoretical model and formulas introduced in this paper, as well as those discussed b Pirjola et al. (7, together with the pertinent computer programs, allow the investigation of the magnetic fields created b an number and configuration of train-substation pairs with an parameter values at an points of observation. Numerical eamples presented in this paper demonstrate the usefulness of the model as a practical tool for estimating magnetic disturbances due to dc-electrified railwas. The also indicate that the maimum allowable noise level set to 1 pt nowadas is easil eceeded at a nearb geomagnetic observator. It is indicated in this paper that a linearl changing leakage current densit results in a larger magnetic field than a constant leakage, which is eplained b noting that, in the linear case, the elementar leakage loops with the largest sizes also carr the highest currents. The difference between the cases of a linearl changing and a constant leakage current densit is, however, so small that it would certainl vanish in errors due to man approimations included in theoretical models. This means that, in practical estimations of magnetic fields caused b dc-electrified railwas, both models are equall applicable. In the computations considered in this paper, the traction current is assumed to be taken to trains from onl one station in a railwa section. However, it is important to realise that, in practice, a train ma also be fed from both ends of a section (Ishikawa et al., 7; private discussions in Ottawa, Canada, in 1. Our numerical calculations not presented in this paper, whose principal objective is to introduce the model and formulas associated with a linearl changing leakage current densit and to compare the models of linearl changing and constant leakage currents, show that a two-end feeding significantl reduces the magnetic fields as compared to a one-end feeding. Thus the question about the feeding sstem plas an important role when magnetic disturbances due to a dc-electrified railwa are considered. There are also man other items in railwa technolog including the leakage currents that have to be known eactl to obtain correct estimates of the resulting magnetic fields. All this emphasises the need for interaction between geophsical scientists and railwa engineers. Acknowledgments. The author wishes to thank the personnel of the Geomagnetic Laborator of the Natural Resources Canada in Ottawa, Canada, for introducing the topic of magnetic fields due to dc-electrified railwas and for ecellent collaboration on the subject. Thanks go especiall to David Boteler, Lorne McKee, Larr Newitt and Benoit St-Louis, and to Peter Fernberg and Gerrit Jansen van Beek, who carried out the measurements in Calgar in 6. The author also wants to acknowledge Frank Lowes (Newcastle Universit, UK for man interesting and ver useful discussions about modelling dc railwas. Finall, the author is obliged to the two unknown reviewers of the manuscript for man constructive and helpful comments and suggestions. 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