Translated to English from paper in Japanese, originally published in: Memoirs of the Kakioka
Magnetic Observatory, Vol.20, No.2, 33-44, 1984 with permission of the authors.

Calculation of Magnetic Field Disturbance Produced
by Electric Railway
byTetsuo TOKUMOTO and Satoru TSUNOMURA

Abstract
Magnetic field disturbance by leakage current of electric railways is a serious problem for magnetic
observations. We can get an exact solution in various situations of the route of railway, distributionsof power substations and electric railcars by the method developed herein. The developed method
yields a smaller estimating magnetic field disturbance compared with the old method.

1. Introduction
It is known that the leakage current from a DC
electric railcar forms magnetic fields in thesurrounding environment. Unwanted noises occur
from these magnetic fields, causing interference to theobservation of natural magnetic fields. Because thenoises spread two-dimensionally, they travel over along distance without decaying and cause interference
at remote locations. They have complex waveforms
and it is technically impossible to remove them under
the circumstances.* If the railway track could becompletely insulated from the ground and if thecurrent supplied to drive the electric railcar could bemade to flow only in a closed circuit from a power
substation to a feeder line to an electric railcar, andthen to a railway track, the noises would not propagate
so long a distance even if a DC electric railcar is used.
Because the railway track and the feeder line are
installed close to one another, the effect of the sameamount of current flowing in them cancel each other.
Therefore, at a remote location the circuit is regarded
as if it carried no currents. In reality, however, aconsiderable amount of leakage current flows into theground, because the railway track is not completely＊

insulated from the ground. Not only does the leakage
current itself form magnetic fields, but it also causes
the amount of current through the railway track todiffer from the amount of current through the feeder
line and, as a result, magnetic fields form around therailway track and the feeder line. The imperfectinsulation thus gives rise to two types of noise: thenoise formed by the leakage current and that formedby the current through the railway track and thefeeder.
To observe natural magnetic fields, we must
quantitatively analyze the effects of magnetic field
disturbances caused by the leakage current from a DC
electric railcar.
In this study, we conducted
experiments, made field surveys, performed model
calculations, and attempted to verify the effects ofmagnetic field disturbances caused by the leakage
current from the DC electric railcar.
Inoue (1979) performed model calculations tosimulate the effects of the noises from magnetic fields
around the Joban Line by placing a DC electric railcar
at different positions on the railway track. In his
approach to the model calculations, we noted someshortcomings as pointed out in the following:

In the case of an AC electric railcar, the skin effect prevents the leakage current from flowing deep into the ground,and the frequency of an AC current is higher than the frequencies at which natural magnetic fields are normallyobserved. Therefore, the leakage current from the AC electric railcar does not form magnetic fields, and interfering
noises do not occur.

28

Tetsuo TOKUMOTO and Satoru TSUNOMURA(1) When deriving the amount of current supplied byeach power substation, a certain fixed leak rate wasused, and the model calculations were performed
based on the leak rate.
(2) When obtaining solutions from the results ofanalyses, boundary conditions were not established,
and the scenario that two or more electric railcars
run simultaneously on the railway track was notconsidered. Solutions for each electric railcar were
simply superposed.
(3) Bouncing of the potential distributed across therailway track back to power substations was notconsidered.
Yanagihara (1977) developed approximate
calculations for physically precise solutions andestimated disturbance fields by assuming a simple
model of a railway. He obtained satisfactory resultsby comparing calculated values with measured values.
By further developing the approach taken by
Yanagihara (1977), we studied magnetic field
disturbances, created a new model for simulating realconditions as closely as possible to the extent practicaland physically consistent, and performed calculations
based on several scenarios. This paper describes theresults of our study and calculations. We expect that
our model would give a clear basis for discussion,
reducing the number of unconsidered parameters(such as the non unifomity of the subterranean
structure.)
2. Calculation Method(1) How to calculate the leakage current
As the number of power substations increases, theamount of current that each power substation supplies
to one electric railcar varies depending on the distance
between that power substation and the electric railcar.
Generally, power substations near the electric railcar
supply a larger amount of current than those at somedistance from the electric railcar.
We studied how to calculate the leakage current on
the assumption that the amount of current used by one
electric railcar is always the same and the positions of
the electric railcars, as well as the positions of eachpower substation, are known. As shown in Figure 1,
a railway segment starting at point A and ending at
point B was assumed.
A coordinate "s" wasestablished along this railway segment.
The
direction from A to B was defined as a positive

direction. A current flowing in the railway track to bemeasured at point "s" was defined as "Jr(s)," and theleakage current from this railway track was defined as"i(s)." "i(s)" can be expressed as follows:i(s ) = −dJ r (s )ds(1)Because the railway track was not consideredisoelectric, it was thought that the potential is
distributed along the railway track. This distributed
potential was defined as V(s). With the railway trackresistance per unit length and the leakage resistancedefined as "Rr" and "η" respectively, the relationships
between V(s), Jr(s) and i(s) are expressed as follows:dV (s )
= − Rr J r (s )
ds

(2)
(3)

Based on above three equations, the following
ordinary differential equation can be formulated
regarding Jr(s):d 2 J r (s )  Rr 
=   J r (s )
ds 2
η 

(4)

A general solution to this differential equation canbe obtained using equation (5) below:(5)
In this equation,(6)

C1 and C2 in equation (5) are constants to be
determined by boundary conditions.
If there is more than one power substation in the
A-B railway segment and if more than one electricrailcar runs in this railway segment, the number ofboundaries increases, and C1 and C2 must be
determined for each subsegment delimited byboundaries. For example, if the total number of
power substations and electric railcars in the A-B
segment is 10, the number of boundaries includingthat of the A-B segment is 12. This means that theA-B segment has 11 subsegments. Therefore, C1 andC2 must be obtained for each of these 11 subsegments.
The following discussion assumes that neither electricrailcars nor power substations are set at points A andB.

Calculation of Magnetic Field Disturbance Produced by Electric Railway -Fig.1. Leakage current i(s)ds,
rail current Jr(s)at railway s,
and direction of current.

Fig.2.
Circle unit of power
substation, fieder line andrailway.

One subsegment delimited by the boundaries s =sk and s = sk+1 inside the A-B segment as shown in
Figure 2 was established. The railway currents thatflow to the boundary from the left (A side) were
defined as Jk and Jk+1, and the current that branches tothe feeder line at sk was defined as Ik. Based onequation (5), the railway current inside thesubsegment delimited by sk and sk+1 can be calculated
as follows:(7)
where lk = sk+1 − sk
Providing that the total number of power substations
and electric railcars is n:
If sk is a power substation,
If sk is an electric railcar,
Note that J0 = 0 (start point A) and Jn+1 = 0 (end pointB).
In Ik (k = 1, 2, …, n), the number of electric railcars(n−m) is a known constant. Ik for power substations(m) and all Jk(n) are unknowns. These unknowns(n+m) are calculated based on the following three
conditions:
Condition 1: Current conservation law
The total amount of current supplied by power
substations is equal to the amount of electric-car
current, and the current from power substations andthe current in the electric railcar flow in opposite
directions. Therefore, the following equation can beformulated:(8)
Condition 2: Conditions regarding potential V(s) atboundaries
Potential V(s) must be continuous at eachboundary sk (k = 1, 2, …, n).

29

Fig.3. Leakage current i(s)ds, atrailway s, and position vector r.

Therefore,(9)in whichSk =1sinh{α (s k +1 − s k )}1
Tk =tanh{α (s k +1 − s k )}(10)

Condition 3: Kirchhoff's second law
Kirchihoff's second law is applied to the circuit inwhich a current flows out of one power substation,enters another power substation via the feeder line,
and returns to the original power substation via therailway track. Two power substations, s = s1 and sk ,were assumed. The voltage from s1 to sk on therailway track is expressed as follows:

The voltage from a power substation (s1) to thefeeder line, then to another power substation (sk) is
expressed as follows:

Where the power substation voltage is defined as
E0 (constant), the internal resistance of a power
substation as Rf, and the resistance of the feeder line
as Rf .
Because the former voltages are equal to eachother, applying equation (1), (3) and (7) to these, thefollowing equation can be derived.

(11)

30

Tetsuo TOKUMOTO and Satoru TSUNOMURAThis relationship is applied to the power substation on
the left end (defined as s = s1 ) and other power
substations (m−1).
The numbers of equations obtained based on the three
conditions shown on the previous page are 1, n, andm−1, respectively. The total number is (n+m).
Therefore, the unknown number matches the numberof equations, and the simultaneous equation can besolved. Ik and Jk calculated this way are substituted
into equation (7) to obtain Jr(s). Jr(s) obtained thisway is substituted into equation (1) to obtain theleakage current i(s).
(2) How to calculate the amplitude of magnetic field
disturbances
Although the leakage current spreads radially
underground, the effect of the leakage current onmagnetic fields can be defined as the effect of asemi-infinite line current that flows perpendicular to
the ground. As Yanagihara pointed out in 1977,
magnetic field (H) formed by the leakage current
from the A-B segment and currents through therailway track and the feeder line at external point P
can be calculated as follows:(12)
in whichis the unit vector in the vertical direction,is the position vector from point s in the A-B
segment to point P, and Jf (s) is the feeder current.
The first term represents magnetic field formed by theleakage current, and the second term represents afield formed by currents through the railway track andthe feeder line. We performed the calculations using
the method of successive approximation based on
Simpson's rule.
3. Calculating the Amplitude of Magnetic Field
Disturbances
3.1
Comparison with the Values from theGenerally Used Model(1) Amount of current supplied by power substations
To simulate real conditions as closely as possible, itis necessary to extend the length of the railway track
and to establish more than one power substation. It
is thought that power substations near the electricrailcar supply a larger amount of current than those atsome distance from the electric railcar. Althoughquantitative calculations were made to clarify the
relationships between the amount of current and the
distance between power substations and the electricrailcar, the appropriateness of the establishedpreconditions for calculations was questioned, as
mentioned in Chapter 1. Introduction (the study madeby Inoue in 1979). In this study, we compared theresults of calculations made using the new model withthose of calculations made using the generally usedmodel under the same conditions as when Inoue
performed the calculations.
Figure 4 shows the amount of current supplied byeach power substation in a case where an electricrailcar was set at the extreme left of the segment (0km) and ten power substations were set at 1 km
intervals. In the case of the generally used model,
the amount of current supplied by each power
substation was determined only by the resistance ofthe feeder line. In the case of the new model, theamount of current supplied by each power substation
decreased rapidly as the distance from the electricrailcar increased, because not only the resistance ofthe feeder line but also the resistance of the railway
track and the ground leakage resistance were taken
into consideration. However, it is inappropriate toconclude from this that the magnetic field is disturbed
only by the current around the electric railcar. For
example, the amount of current that leaks to theground and then returns again to the railway track will
increase if power substations are set at long intervals.
Therefore, it should be noted that there are cases inwhich the amount of current being supplied by power
substations will increase though they are set apart at
distant intervals.
(2) Relationships between the position of the electricrailcar and the amplitude of magnetic field
disturbances
Figure 5 shows the position of Kakioka relative tostations on the Joban Line, assuming that the line is
DC electrified as far as Tsuchiura Station.
In
performing the calculations, the railway track of the
Joban Line was limited to the segment from Tsuchiura
Station to Abiko Station, and this segment wasapproximated by a line graph, as shown in Figure 5.
Figures 6 and 7 show the amplitudes of horizontal
magnetic field disturbances calculated under the same
conditions as in Figure 5. The intervals between

Calculation of Magnetic Field Disturbance Produced by Electric Railway -power substations shown in these figures are forpower substations for the railway section as far as
Fujishiro Station.
In the railway section from
Fujishiro Station to other stations in the up-line
direction of the Joban Line, one power substation isestablished each at Fujishiro, Toride and Abiko.
Figure 6 shows the values calculated using
the generally used model, and Figure 7 shows thoseFig.4. Comparison of the supply current throughpower substations between the new model andthe old one.
This figure shows currents through power
substations in the condition that ten power
substations are ordered at 1 Km intervals.

calculated using the new model. Positions of theelectric railcar are shown along the abscissa axis with
Tsuchiura Station as the starting point.
Theamplitudes of horizontal magnetic field disturbances
measured at Kakioka are shown along the ordinate
axis.
Because calculations were performed bychanging the interval between power substations, eachcurve represents the state of disturbance relative toeach specific interval.

Fig.5. Map of KAKIOKA and Joban Line.
The railway from TSUCHIURA to ABIKO isapproximated by segments of straightline.

Amplitude of horizontal magnetic field disturbances (nT)

Characteristics of horizontal magnetic field disturbances at Kakioka(if the interval between power substations is used as a parameter)

Interval

Tsuchiura

Arakawaoki

UshikuFujishiro

Fig.6. Horizontal magnetic field disturbanceat KAKIOKA as a function of position of one
electric car calculated by the old model.
(After CHIGIKI KANSOKUSHO MONDAI KENKYUKAIHOKOKUSHO by IBARAKI‑KEN)

31

Fig.7. Horizontal magnetic field distubance at
KAKIOKA calculated by the new model.

32

Tetsuo TOKUMOTO and Satoru TSUNOMURAFor a railway segment in which a specific interval
between power substations is known, the total leakage
current can be calculated as follows:(13)
It should be noted that in this equation Jr is notalways uniform throughout the railway track;
specifically Jr differs largely in segments si (1, 2, …, n).
Because Jr (si) in the segment (si, si+1) is different fromJr (si) in the segment (si−1, si ), different Jr values must
be used as integrands for each segment when
performing integrations on the leakage current. The
above equation represents rail current variations, andthe longer the interval between power substations, thelonger the rail length and, as a result, the larger theamount of the leakage current becomes. Therefore,
the tendency should be noted that the amplitude of themagnetic field disturbances increases as the interval
between power substations increases. This tendency
is demonstrated by the results of calculations using
both models.
What should be noted about the results ofcalculations using the new model is that the amplitudeof horizontal magnetic disturbances becomes small
when the electric railcar is at a point about 10 km froma power substation. We termed this point a singularpoint.
In performing calculations using both old and newmodels, only one electric railcar was used. We foundthat the amount of the leakage current was large atpoints near the electric railcar and that the leakage
current returned to the railway track at distant points
from the electric railcar. If the railway track ran
almost in a straight line as viewed from theobservation point, we found that magnetic field
disturbances caused by the leakage current near theelectric railcar and those caused by the returningcurrent near the observation point cancelled eachother out after the electric railcar traveled a certain
distance.
We also found that components
perpendicular to the railway track dominated othercomponents and that the orientation of thesecomponents was reversed at the above mentioned
singular point where the amplitude of horizontal
magnetic field disturbances dropped markedly. If therailway track is straight and if the observation point islocated on an extension of the straight line, there

should be a position of the electric railcar where theamplitude of the magnetic field disturbances becomeszero because the horizontal components of magnetic
field disturbances is perpendicular to the railway track.
To verify the correctness of this assumption, we useda simple arrangement of the railway track and theobservation point shown in Figure 8, and performed
the calculations.
Figure 9 is a graph showing the amplitude of themagnetic field disturbances relative to the positions of
the electric railcar. As is apparent from this figure,there is the singular point where the horizontal
components of magnetic field disturbances become
extremely small, as in Figure 7. If the observation
point is on an extension of the railway track, magnetic
field disturbances consist of only the components
perpendicular to the railway track; they do not containvertical components at all. The railway track was
divided into three segments: a 2-km long segment
before and after the electric railcar, which was defined
as segment B, a segment near the observation point,
which was defined as segment C, and a segment far
from the observation point, which was defined as
segment A. Calculations were performed to verify
the specific contributions of these segments to theamplitude of the magnetic field disturbances. Figure
10 is a graph plotted based on the results of thecalculations.
Because the leakage current dominates in segment
B and the returning current dominates in segments A
and C, the orientation of magnetic field disturbances
was reversed. The amplitude of the magnetic field
disturbances in segment B decreased moderately as
the electric railcar was going away from theobservation point (as the source of the leakage current
was going away from the observation point).
The current that leaked near the electric railcar
returned to a near railway track, not to a far railway
track. Therefore, in the A and C segments, theamplitude of the magnetic field disturbances waslarger at points near the electric railcar than at distantpoints from it, on the condition that the observation
point was at some distance from the railway track.

Calculation of Magnetic Field Disturbance Produced by Electric Railway -33

Fig.8. Simple arrangement of observation point and railway.
Railway is divided three segments A, B and C. Power
substations are ordered at 2 Km intervals.

Fig.10. Horizontal magnetic field disturbance produced byeach segment or railway.
Segments A, B and C are shown in Fig.8. Be careful thatthe direction of magnetic field disturbance produced bysegment B is reverse to those by segments A and C.
Fig.9. Horizontal magnetic field disturbance in the caseshown by Fig.8.

The amplitude of the magnetic field disturbances insegment A decreased as the electric railcar went awayfrom the observation point. This is because thelength of the segment became short and the point near
the electric railcar where the amplitude of themagnetic field disturbances was largest in segment A
also went away from the observation point. In
segment C, although the length of the segment
became long as the electric railcar went away from theobservation point, the position near the electric railcar
where the amplitude of the magnetic field disturbances
was largest in segment C went away and, as a result,the amplitude of the magnetic field disturbances did
not change noticeably. In this case, the amplitude
magnetic field disturbance became zero when theelectric railcar was at a point about 13 km from the endof the railway track. When the electric railcar
traveled further beyond this 13-km point, theorientation of the magnetic field disturbances wasreversed.

3.2 Comparison with the Results of Field
Experiments
It is inappropriate to discuss whether the newmodel has achieved an improvement over thegenerally used model only by comparing both
calculation results.
Furthermore, there is noavailable data on the results of a comparison between
measured values and the values calculated using thegenerally used model. In this study, we comparedmeasured values and the values calculated using thegenerally used model in order to verify theeffectiveness of the new model.
We measured the amplitudes of magnetic field
disturbances caused by the effects of the DC
electrified segment on the JR Joban Line (to thesouthwest of an AC - DC switching power substation,
which is located between Toride Station and Fujishiro
Station) at observation points shown in Figure 11
during the period from Jan. 7 to Jan. 9, 1983. Figure
12 shows the state of observed magnetic field
disturbances.
Figure 13 shows a comparison
between measured values and calculated values.

34

Tetsuo TOKUMOTO and Satoru TSUNOMURA

Fig.11. Map of observation points for field experiment and
Joban‑Line.

In performing the calculations, the railway track
was approximated to a line graph, and contributions ofeach straight portion were totaled. Two electricrailcars were set near Matsudo Station, and one
electric railcar was set near Toride Station, based on
the timetable.
The magnetic field in observation point Nos. 1, 2,
4 and 5 and observation point Nos. 3, 6, 7 and 8 were
measured on different dates. When Nos. 1, 2, 4 and 5
were measured, it rained, so the leakage resistancemight be different from the other day’s.
Inobservation point 3, 6, 7 and 8, the amplitude of themagnetic field disturbances could not be detected
because the observation point was too far from therailway track. Therefore, only observation point Nos.
1, 2, 4 and 5 were used to compare them withcalculated values.
Although it was thought that a current of
thousands of amperes is supplied to the electric railcar,
an exact ampere value could not be measured.
Because the amplitude of the magnetic field
disturbances is proportional to the amount of current
supplied to the electric railcar, calculated values were
evaluated at each observation point based on thevalues obtained by normalizing the amplitude of themagnetic field disturbances at the observation point
No. 1 with the calculated amplitude values (measured

values/ calculated values).
Concerning the orientation of magnetic field
disturbances, it was verified that all measured values
were in good agreement with calculated values. As
shown in Figure 13, the amplitude of horizontal
magnetic field disturbances is almost the same at allobservation points, except for at the observation point
No. 1, where the amplitude was slightly larger. It wasthought that satisfactory results had been obtained
regarding all vertical components of magnetic field
disturbances.
Vertical components of a magnetic field, which areproduced by horizontal components of an electric
current, flow through the rail and the feeder line, andthe route of current flow is fixed. Considering thathorizontal components of a magnetic field are
produced by the leakage current, we assumed that inperforming the calculations, they would flow in thevertical direction. However, if the subterraneanelectric conductivity is nonuniform, this assumption
does not stand. If it is nonuniform, it was thoughtthat the observation point near the railway track, suchas the observation point No.1, would be most strongly
affected, while distant observation points where theleakage current could hardly reach would be minimallyaffected.
3.3 Distribution of Magnetic Field Disturbances
in a Simple Arrangement of Observation Points
and the Railway Track
The results of calculations using the new model
were in good agreement with measured values, as
described in section 3.2. Using the new model and asimple arrangement of observation points and therailway track, we performed the calculations to verify
the two-dimensional distribution of magnetic field
disturbances, which was considered to be the firstattempt of its kind in Japan.
A straight railway track was established, power
substations were set at equal intervals, and electricrailcars were randomly assigned on the track, asshown in Figure 14. In Figure 14, the contour lines
distribution, which represents the total magnetic forceof magnetic field disturbances, is also shown. In thissimple arrangement, the amplitude of the magnetic
field disturbances was about 0.15 nT at a point close to
the end of the railway track and it was 0.2 to 0.5
nT around the mid-point of the railway track, when

Calculation of Magnetic Field Disturbance Produced by Electric Railway -Fig.12. Magnetic field disturbance produced by Joban‑Line.
Magnetic field disturbances at No. 1, 2, 4, 5 and at No.3, 6, 7, 8 were observed at different date. Dashed lines
show the assumed records without magnetic field disturbance. The difference between line and dashed line shows
magnitude of magnetic field disturbance produced by the electric railway.

35

Tetsuo TOKUMOTO and Satoru TSUNOMURA

36

Fig.13. Comparison of magnetic field disturbance between observed values and calculated values.
Bar graphs refer left ordinate, line graphs refer right ordinate. Line graphs show the ratio of observed value
and calculated one which is normalized by such ratio at No.1 observation point, so the ratio at No.1 is exactly 1.

Fig.14.

Distribution of magnetic total force disturbance in the simple case such as this figure.

Calculation of Magnetic Field Disturbance Produced by Electric Railway -measured at a location 30 km from the railway track.
This state of amplitude distribution was thought to be
associable with where each electric railcar is locatedon the railway track.
Sharply curved contour lines are due to mutual
canceling out of the effects from the two lower electricrailcars and one upper electric railcar shown in Figure
14. It is estimated that as the number of electricrailcars increases, the distribution of magnetic field
disturbances around the railway track becomescomplicated.
4. Conclusion
We found that the amplitude of the magnetic field
disturbances calculated with the new model was
generally smaller than that calculated with theoperational model of the Railway Technical ResearchInstitute. Our model can be applied to various
situations flexibly, keeping the strictness of theassumption of uniformity of subterranean electric
conductivity. In all these calculations, calculated
values were in good agreement with measured values.
On the other hand, we noted the following difficulties:
・It was difficult to identify η (leakage resistance), a
parameter of the new model. This parameter was
unstable and varied greatly depending on theweather. It may also vary depending on railway
laying conditions. Therefore, it is inappropriate toassume that this parameter is uniform through thewhole railway track.
・It was assumed that the leakage current would flow
over an infinite distance in the vertical direction on
the condition that the subterranean electric
conductivity was uniform.
However, thesubterranean distribution of currents is not reallyuniform if the subterranean electric conductivity is
distributed anisotropically. For example, if theelectric conductivity at a depth more than a few
kilometers from the ground surface is small, theleakage current stops there and does not flow
deeper.
These difficulties, however, can be somewhat
overcome by using the following techniques:

37

・Concerning η, equations (1) through (6) can be used
as is if specific values are assigned to each interval
between power substations.
・Concerning the route of the leakage current flow,
because the leakage current generally flows only a
few kilometers deep, it can be assumed that theleakage current first flows vertically a few
kilometers, then turns its direction, and continues toflow at the same depth along the same route as therailway track.
Under this assumption, thedifference between the amount of currents in therailway track and that of currents in the feeder line
would decrease further due to the current flowing
underground horizontally and, as a result, verticalcomponents of magnetic field disturbances would
become even smaller.
Strictly speaking, thisassumption requires that a three-dimensional
structure be prepared and a relational expression be
solved. In this case, current integrations must be
performed over an infinite range and the conditionsof the subterranean distribution of electric
conductivity must be grasped, which is practically
impossible.
Acknowledgments
We would like to express our sincere appreciation to
Mr. Kawamura, former director of the Observatory,
and Mr. Harada, incumbent director of the Observatory,
for giving us the opportunity to make this study. In
performing the calculations, we used the M-200
computer at the Meteorological Research Institute,using the remote batch processing method.
References
Inoue, H., Effects of DC Electric Railways on the
Observation of Magnetic Field Disturbances.
Materials concerning railway engineering, 37 (No.
7), 26-27, 1979.
Yanagihara, K., Magnetic Field Disturbances Produced
by Electric Railway. Report on Magnetic Field
Observations, separate volume No. 7, 1977.