**3. A few physical notations**

As mentioned above, one of the drawbacks of AC railway system is the electromagnetic interference to other surrounding electric circuits, particularly if parallel. These interferences can disturbs communication lines, as well as overhead conductors that run parallel.

An example of these phenomena is showed in Figure 1. As can be seen, between the AC railway system and the victim system exist coupled inductors and coupled capacitors that are the cause of the appearance of low frequency disturbances.

So, near an electrified railway AC, two electrical phenomena exist that can disrupt the lower current circuits (González Fernández & Fuentes Losa, 2010. Perticaroli, 2001):


The resulting induced voltages in the victim system, in absence of protection, can cause hazards to personnel, material deterioration, problems in the proper functioning as unexpected operation of railway signaling installations.

Fig. 1. Coupled inductors and coupled capacitors between the disturbing line and the disturbed line

To explain the phenomena the induced voltages in a victim wire will be studied. The development will be done considering the generic catenary pole of Figure 2, composed by:

• 2 catenary wires;

164 Electrical Generation and Distribution Systems and Power Quality Disturbances

these currents will affect – as can be easily seen – to the determination of the boundaries between affected and not affected equipments, facilities or systems. This status quo is changing in the last years, at least in Spain, so current limits are defined taking into account

Conducted disturbances have a major impact in two aspects: the rise of the voltage between the rail and earth and the disturbances that may cause to the 50 Hz track circuits that might be present in the line. The effects on the track circuit are highly affected by the impedance balance of the two rails. If the impedances of both rails are unbalanced a 50 Hz voltage will appear between the rails and depending on the relative phase between the 50Hz signal of the track circuit and the conducted one the operation of the line (if a false track circuit occupation occurs) or even a failure against safety if a shunted track circuit (occupied) due to the disturbance signal will give a false unoccupied information. CITEF, ADIF and the major Spanish 50Hz track circuits suppliers have carried out test to determine these effects and both of the above related cases can occur depending on the rail impedance unbalance and the relative phase of the signals. The location of feeder stations has also an impact in the

distribution of currents along the rail and also affects the propagation of the effects.

These conducted disturbances might appear when a train unit is contacting both lines while crossing through a gauge change facility or a transition zone from AC to DC fed lines or electrified to non electrified lines. The train unit might be fed by AC through the pantograph and some of the returning current might, depending on the train configuration, end returning through the DC rails. In order to avoid this undesired current return isolation rail

There are other signalling systems – like for example electronic block – that might be disturbed by harmonics originated by traction units. Electronic block depending on the used technology can be established between stations using a modulated signal of few kilohertz. Rolling stock modern composition for high speed lines do usually use asynchronous electric engines controlled with frequency shifters in order to enhance the performance and lessen the losses. Asynchronous engines are more efficient for the same weight than other electric engines and very easy to maintain so they are thoroughly used in train units using electronic controllers. These units can emit a series of harmonics that can interfere with such systems

As mentioned above, one of the drawbacks of AC railway system is the electromagnetic interference to other surrounding electric circuits, particularly if parallel. These interferences

An example of these phenomena is showed in Figure 1. As can be seen, between the AC railway system and the victim system exist coupled inductors and coupled capacitors that

So, near an electrified railway AC, two electrical phenomena exist that can disrupt the lower

• electrostatic induction, whose importance is due to the high value of the voltage and to

• Electromagnetic induction due to the nature of alternating current and to the coupling loop between the earth and the catenary, and the one represented by the induced line

can disturbs communication lines, as well as overhead conductors that run parallel.

current circuits (González Fernández & Fuentes Losa, 2010. Perticaroli, 2001):

the capacitances between the system, the earth and the disturbed line;

are the cause of the appearance of low frequency disturbances.

electrification systems information.

joints and impedances are used.

**3. A few physical notations** 

as electronic block.

and the earth.


Fig. 2. Structure of a catenary pole

#### **3.1 Electrostatic induction**

The electric field consists of open field lines starting from the charge generating the field to other charges where field lines end. Considering a wire, the linear charge is calculated using the Gauss Law, stating that the electrical flux coming out of a closed surface is equal to the electrical charge contained in the surface (Arturi, 2008):

$$\mathcal{Y} = \mathbb{Q}\_{\text{int}} \tag{1}$$

Electrical Disturbances from High Speed Railway Environment to Existing Services 167

*<sup>Q</sup> Exy <sup>u</sup>*

Using the electrical field is possible to calculate the potential induced on a wire through the

'

*r*

Where *r* is the generic point where the potential has to be calculated and *r'* is the reference of

0 0

To calculate the influence of all the wires of the catenary pole the image theorem must be applied. In Figure 3 are showed the principal conductor and the respective image conductors in a standard High Speed Line catenary Pole. Wires 1-14 are the main wires and wires 1'-14' are the images of the main wires. Wire 15 and 15' are the victim wire and its image. a2,13 distance between wires 2-13. b2,8 distance between wires 2 to image of wire 8. h2 is the height of wire 2. The wire 15 is the hypothetical conductor exposed to the electrical field produced by all the catenary pole wires and by theirs images. The tracks are

*Q Qr V d*

<sup>1</sup> ln 2 2'

 πε

*l l r* ρ

*r*

πε

'

πε ρ

*r*

Fig. 3. Application of the image method to High Speed Lines

*r*

Where *<sup>i</sup> x* and *<sup>i</sup> y* are the coordinates of each wire.

definition of electrical potential:

From (9) and (11) we obtain:

considered active wires.

the potential.

2 2 <sup>0</sup> <sup>1</sup> (,) <sup>2</sup> ( )( ) *<sup>i</sup>*

*<sup>l</sup> xx yy* = ⋅ − +−

*<sup>i</sup> i i*

(10)

*V E dl* =− ⋅ (11)

=− ⋅ ⋅ =− ⋅ (12)

Where the electrical flux in equal to the integral of the electrical flux density in a surface:

$$d\Psi = \oint\_{S} d\Psi = \oint D \cdot dS = D\_{\rho} \int dS \tag{2}$$

Considering a cylindrical conductor of radius ρand length *l* the electrical flux is:

$$\Psi = \mathbf{D}\_{\rho} \cdot \mathbf{2}\pi\text{pl} \tag{3}$$

The electrical charge is the integral of the linear charge in a line:

$$Q\_{\rm int} = \int\_{\cdot} \mathcal{A}\_{\rm t} dl = \mathcal{A}\_{\rm t} \cdot \mathbf{l} \tag{4}$$

Where λ*<sup>l</sup>* is the linear charge density of the wire. Comparing the equation (3) and (4) the electrical flux is:

$$D\_{\rho} = \frac{Q}{2\pi\rho \cdot l} = \frac{\lambda\_l \cdot l}{2\pi\rho \cdot l} = \frac{\lambda\_l}{2\pi\rho} \tag{5}$$

And the electrical flux vector is:

$$
\overline{D} = D\_{\rho} \overline{a}\_{\rho} = \frac{\mathcal{\lambda}\_{t}}{2\pi\rho} \overline{a}\_{\rho} \tag{6}
$$

The electrical field is proportional to the electrical flux vector through the permittivity of the material:

$$
\overline{D} = \varepsilon\_0 \overline{E} \tag{7}
$$

Where <sup>12</sup> <sup>0</sup> 8.85 10 *<sup>F</sup> m* ε <sup>−</sup> = ⋅ is the permittivity of vacuum. Consequently the electrical field is:

$$
\overline{E} = \frac{\overline{D}}{\mathcal{E}\_0} \tag{8}
$$

$$\overline{E} = \frac{\overline{D}}{\varepsilon\_0} = \frac{\mathcal{\lambda}\_l \cdot l}{2\pi\rho\varepsilon\_0 \cdot l} \overline{a}\_\rho = \frac{Q}{2\pi\rho\varepsilon\_0 \cdot l} \overline{a}\_\rho \tag{9}$$

Through the image theorem and considering the position of the catenary pole conductors, the electric field intensity inducted in a generic point (*x,y*) is due to the following relation expressing the influence of all the wires and its images to the point:

$$\overline{E}(\mathbf{x}, y) = \frac{1}{2\pi\varepsilon\_0 l} \sum\_{i} \frac{Q}{\sqrt{(\mathbf{x} - \mathbf{x}\_i)^2 + (y - y\_i)^2}} \cdot \overline{\boldsymbol{\mu}}\_i \tag{10}$$

Where *<sup>i</sup> x* and *<sup>i</sup> y* are the coordinates of each wire.

Using the electrical field is possible to calculate the potential induced on a wire through the definition of electrical potential:

$$V = -\oint\_{r^\*} \overline{E} \cdot dl \tag{11}$$

Where *r* is the generic point where the potential has to be calculated and *r'* is the reference of the potential.

From (9) and (11) we obtain:

166 Electrical Generation and Distribution Systems and Power Quality Disturbances

The electric field consists of open field lines starting from the charge generating the field to other charges where field lines end. Considering a wire, the linear charge is calculated using the Gauss Law, stating that the electrical flux coming out of a closed surface is equal

= *Q*int (1)

and length *l* the electrical flux is:

(2)

(3)

(4)

<sup>⋅</sup> === ⋅ ⋅ (5)

= = (6)

(7)

<sup>=</sup> (8)

ψ

*S*

ψ

ψ = ⋅ ρ πρ

Where the electrical flux in equal to the integral of the electrical flux density in a surface:

Ψ= = ⋅ = *d D dS D dS*

ρ

int *l l l Q dl l* = =⋅ λ

222 *<sup>D</sup> Q l l l l l*

λ

πρ

2 *D Da a <sup>l</sup>* ρ ρ

The electrical field is proportional to the electrical flux vector through the permittivity of the

*D E*<sup>0</sup> = ε

0 *<sup>D</sup> <sup>E</sup>* ε

> *l l* ρ

 πρε  ρ

<sup>⋅</sup> == = ⋅ ⋅ (9)

00 0 2 2 *Dl Q <sup>l</sup> E aa*

Through the image theorem and considering the position of the catenary pole conductors, the electric field intensity inducted in a generic point (*x,y*) is due to the following relation

λ

 πρε

ε

expressing the influence of all the wires and its images to the point:

λ

πρ

 λ

ρ

πρ

 λ ρ

to the electrical charge contained in the surface (Arturi, 2008):

Considering a cylindrical conductor of radius

2 *D l*

*<sup>l</sup>* is the linear charge density of the wire. Comparing the equation (3) and (4) the electrical flux is:

The electrical charge is the integral of the linear charge in a line:

ρ

πρ

**3.1 Electrostatic induction** 

Where

material:

Where <sup>12</sup>

ε

λ

And the electrical flux vector is:

<sup>0</sup> 8.85 10 *<sup>F</sup>*

Consequently the electrical field is:

*m*

<sup>−</sup> = ⋅ is the permittivity of vacuum.

$$V = -\oint\_{r} \frac{Q}{2\pi\varepsilon\_{0}l} \cdot \frac{1}{\rho} \cdot d\rho = -\frac{Q}{2\pi\varepsilon\_{0}l} \cdot \ln\frac{r}{r} \tag{12}$$

To calculate the influence of all the wires of the catenary pole the image theorem must be applied. In Figure 3 are showed the principal conductor and the respective image conductors in a standard High Speed Line catenary Pole. Wires 1-14 are the main wires and wires 1'-14' are the images of the main wires. Wire 15 and 15' are the victim wire and its image. a2,13 distance between wires 2-13. b2,8 distance between wires 2 to image of wire 8. h2 is the height of wire 2. The wire 15 is the hypothetical conductor exposed to the electrical field produced by all the catenary pole wires and by theirs images. The tracks are considered active wires.

Fig. 3. Application of the image method to High Speed Lines

To apply the image method it is assumed that the earth is a perfect conductor. Consequently the earth should have an electrical charge to verify the following relation:

$$q\_{earth} = -\sum\_{i=1}^{n=i\_{wics}} q\_i \tag{13}$$

Electrical Disturbances from High Speed Railway Environment to Existing Services 169

1

*pp*

*c*

*j j i*

= ≠ <sup>⋅</sup> = − 

In the equation (20) can be seen that the electrostatic induction relies only on the voltages of

The magnetic field, unlike the electric field, consists of closed field lines that can pass

As for electrostatic induction, the various conductors which form the railway system have a magnetic coupling between them. For the development of the following method the Figure 2

The study of inductive coupling refers to the characteristic impedance of a conductor with earth return and the mutual impedance between two insulated conductors that have both earth return. Instead of using the inductance value, the impedance value is used. The relation between impedance *Z*, self-inductance *Lii* and mutual inductance *Mij* , for

ω

ω

Expressions defining the impedance have real and imaginary components whose magnitudes vary greatly with the distance between the wires. Imaginary component of *Z* predominates near the inducing line. The electric field induced in this area along a parallel conductor and the

At great distance from the inducing line real component dominates Z. The electric field induced along a parallel conductor and the current flow at earth level, are almost in phase

The variance based on the phase difference between the magnitudes of the two currents is continuous and fairly uniform. For the model the following assumptions have been

• the earth is homogeneous, with finite resistivity and unitary relative magnetic

In Figure 4 is showed the circuit for the calculation of the self-impedance. Electrical parameters associated with self and mutual impedances of conductors with earth return are:

• inducing line is a straight horizontal conductor of infinite length in both directions;

• the mutual impedance is calculated between parallel lines;

• *<sup>i</sup> I* : Current in conductor *i* (inductor) with earth return.

**3.3 Calculation of self and mutual impedances** 

current density at earth level has an offset of almost 90 º respect the inducing current.

*p*

*V*

the wires and not on frequency power.

alternating current of frequency *f* is given by:

• Self-impedance: *Z jL ii ii* =

• Mutual impedance: *Zij ij* = *j*

opposition with the inducing current.

**3.2 Electromagnetic induction** 

through any material.

will be used.

considered:

permeability.

• Self-impedance of the wire.

*ij j*

(20)

(21)

*M* (22)

*c V*

*n wires*

So the voltage induced in the victim wire can be calculated through the following relation:

Being the whole system neutral:

$$\sum\_{i=1}^{\text{Unwaves}\cdot\text{earth}} q\_i = 0 \tag{14}$$

The following notation will be applied:


Applying the image theorem the following system is obtained:

$$V\_i = \left(\frac{1}{2\pi\varepsilon\_0 l} \text{Im}\frac{2h\_i}{d\_i}\right) q\_i + \sum\_{j=1 \atop j\neq i}^{n=i\text{wires}} \left(\frac{1}{2\pi\varepsilon\_0 l} \text{Im}\frac{b\_{ij}}{a\_{ij}}\right) q\_j \tag{15}$$

Where *n wires* includes also the victim conductor.

The system can be also rewritten in the following way:

$$V\_i = s\_{ii} \cdot q\_i + \sum\_{j=1 \atop j \neq i}^{waves} s\_{ij} \cdot q\_j \tag{16}$$

Where *ii s* and *ij s* are the Maxwell's potential coefficients.

From (16) it could be deduced the coupling capacitances of the systems:

$$\begin{bmatrix} \mathbf{c} \end{bmatrix} = \begin{bmatrix} \mathbf{s} \end{bmatrix}^{-1} \tag{17}$$

Consequently the system can be rewritten as a function of the capacitances:

$$Q\_i = \mathbf{c}\_{ii} \cdot V\_i + \sum\_{j=1 \atop j \neq i}^{n \text{waves}} \mathbf{c}\_{ij} \cdot V\_j \tag{18}$$

Using the system (16) is easy to evaluate the inducted voltage in the victim conductor *p*, just applying the boundary condition that the victim wire has a null charge (floating conductor) and using the voltages of the catenary pole wires:

$$\mathcal{Q}\_p = \mathcal{c}\_{pp} \cdot V\_p + \sum\_{j=1 \atop j \neq i}^{n wires} \mathcal{c}\_{ij} \cdot V\_j = \mathbf{0} \tag{19}$$

So the voltage induced in the victim wire can be calculated through the following relation:

$$\boldsymbol{V}\_p = -\frac{\left(\sum\_{\substack{i=1 \\ j\neq i}}^{n \text{wives}} \mathbf{c}\_{ij} \cdot \mathbf{V}\_j\right)}{\mathbf{c}\_{pp}} \tag{20}$$

In the equation (20) can be seen that the electrostatic induction relies only on the voltages of the wires and not on frequency power.

#### **3.2 Electromagnetic induction**

168 Electrical Generation and Distribution Systems and Power Quality Disturbances

To apply the image method it is assumed that the earth is a perfect conductor. Consequently

1

0 0 1 12 1 ln ln

*V sq sq* = ≠

> [] [] <sup>1</sup> *c s* <sup>−</sup>

*Q cV cV* = ≠

Using the system (16) is easy to evaluate the inducted voltage in the victim conductor *p*, just applying the boundary condition that the victim wire has a null charge (floating conductor)

*V q q*

*i i j*

 = +

*n wires ij <sup>i</sup>*

*h b*

*<sup>j</sup> <sup>i</sup> ij j i*

≠

1

1

1

= ≠

*n wires p pp p ij j j j i*

*Q c V cV*

0

*n wires i ii i ij j j j i*

*n wires i ii i ij j j j i*

πε*l a*

(15)

=⋅+ ⋅ (16)

= (17)

=⋅+ ⋅ (18)

= ⋅+ ⋅= (19)

2 2

*l d* <sup>=</sup>

*i*

=

*n wires earth*

1

*i*

*q* +

0

= − (13)

<sup>=</sup> (14)

*n wires earth i i q q* =

the earth should have an electrical charge to verify the following relation:

• *ij b* : distance between main wire *i* and the image of the main wire *j*;

• the main wires will have a charge of *<sup>i</sup>* +*q* and theirs images a charge of *<sup>i</sup>* −*q* ;

Being the whole system neutral:

The following notation will be applied: • *ij a* : distance between main wires *i* and *j*;

• *<sup>i</sup> h* : height of wire conductor *i* ; • *<sup>i</sup> d* : diameter of wire conductor *i* ;

• the victim wire will have a charge equal to zero.

Where *n wires* includes also the victim conductor. The system can be also rewritten in the following way:

Where *ii s* and *ij s* are the Maxwell's potential coefficients.

and using the voltages of the catenary pole wires:

Applying the image theorem the following system is obtained:

πε

From (16) it could be deduced the coupling capacitances of the systems:

Consequently the system can be rewritten as a function of the capacitances:

The magnetic field, unlike the electric field, consists of closed field lines that can pass through any material.

As for electrostatic induction, the various conductors which form the railway system have a magnetic coupling between them. For the development of the following method the Figure 2 will be used.

The study of inductive coupling refers to the characteristic impedance of a conductor with earth return and the mutual impedance between two insulated conductors that have both earth return. Instead of using the inductance value, the impedance value is used. The relation between impedance *Z*, self-inductance *Lii* and mutual inductance *Mij* , for alternating current of frequency *f* is given by:


Expressions defining the impedance have real and imaginary components whose magnitudes vary greatly with the distance between the wires. Imaginary component of *Z* predominates near the inducing line. The electric field induced in this area along a parallel conductor and the current density at earth level has an offset of almost 90 º respect the inducing current.

At great distance from the inducing line real component dominates Z. The electric field induced along a parallel conductor and the current flow at earth level, are almost in phase opposition with the inducing current.

The variance based on the phase difference between the magnitudes of the two currents is continuous and fairly uniform. For the model the following assumptions have been considered:


#### **3.3 Calculation of self and mutual impedances**

In Figure 4 is showed the circuit for the calculation of the self-impedance. Electrical parameters associated with self and mutual impedances of conductors with earth return are:

	- *<sup>i</sup> I* : Current in conductor *i* (inductor) with earth return.

The self-impedance can be defined as the sum of the following impedances:

$$\mathbf{Z}\_{\rm sil} = \mathbf{Z}\_{0i} + \mathbf{Z}\_{ei} \tag{23}$$

Electrical Disturbances from High Speed Railway Environment to Existing Services 171

According to the law of reciprocity, the values of impedance for the wires *ith* and *jth* can be

The impedance of the conductors can the calculated easily using the expressions developed by Carson (Carson, 1926; Dommel et al., 1992). These expressions are based on series that

In a conductor flowing current flows is distributed over the entire surface of its cross section. Consequently magnetic fields exist inside and outside. To properly evaluate the internal field is necessary to know the geometric properties of the cross section of conductor and internal distribution of the current. For the study, the following hypotheses are assumed:

μ

<sup>4</sup> 2 10

4

*<sup>r</sup> Z Rj r km*

2 10 ln *<sup>i</sup>*

( ) <sup>0</sup> <sup>2</sup> ln

( ) <sup>0</sup> ln 2

*i ei ii ii i <sup>h</sup> Z j R X*

*ij ij ii ii ij D Z j R X*

 <sup>Ω</sup> = ⋅ ⋅ ⋅ + Δ +Δ 

<sup>Ω</sup> = ⋅ ⋅ ⋅ + Δ +Δ

ω <sup>−</sup> <sup>Ω</sup> = +⋅⋅ ⋅ ⋅

*<sup>r</sup> Z Rj oi i km*

*<sup>r</sup>* is constant.

4

'

*i*

*r km*

*d km*

= +⋅⋅ ⋅ ⋅ (28)

= ⋅ (29)

(30)

(31)

(32)

μ ω<sup>−</sup> <sup>Ω</sup>

*ij i U*

*Z*

**3.4 Calculation of the impedances of the basic Carson's formulas** 

The starting point is the same image method used in 3.1 applied in Figure 3.

• current is distributed throughout the cross section of the conductor;

The internal impedance of a conductor per unit length is defined by the formula:

' /4 *i i r re*<sup>−</sup>μ

*oi i*

**3.4.2 External impedance of the self-impedance and mutual impedance**  The external impedance of the self-impedance is due to the following relation:

2

2

π

μ ω

The mutual impedance can be calculated using the equation:

π

μ ω

*ij*

*I l* <sup>=</sup> <sup>⋅</sup> (27)

exchanged, i.e. *Z Z ij ji* = .

depend on the position of these wires.

**3.4.1 Internal impedance of the self-impedance** 

• the relative permeability of the material

Considering the following relations,

*r*' GMR (geometric mean radius)

*r* Radius of the wire

relation (28) can be rewritten as:

Where the meaning of the terms is:

$$Z\_{sli} = \frac{\mathcal{U}\_s}{I\_i \cdot l} \tag{24}$$

$$Z\_{0i} = \frac{\mathcal{U}\_{0i}}{I\_i \cdot I} \tag{25}$$

$$Z\_{el} = \frac{\mathcal{U}\_{el}}{I\_i \cdot l} \tag{26}$$

Fig. 4. Calculation of the self-impedance

• Mutual impedance between two wires.

In Figure 5 is showed the circuit for the calculation of the mutual impedance. *<sup>i</sup> I* is the current for the conductor *I* and*Uij* in the voltage in conductor *j* due to the current for the current *<sup>i</sup> I* .

Fig. 5. Mutual impedance calculation

According to the law of reciprocity, the values of impedance for the wires *ith* and *jth* can be exchanged, i.e. *Z Z ij ji* = .

$$Z\_{ij} = \frac{\mathbf{U}\_{ij}}{I\_i \cdot \mathbf{l}} \tag{27}$$

#### **3.4 Calculation of the impedances of the basic Carson's formulas**

The impedance of the conductors can the calculated easily using the expressions developed by Carson (Carson, 1926; Dommel et al., 1992). These expressions are based on series that depend on the position of these wires.

The starting point is the same image method used in 3.1 applied in Figure 3.

#### **3.4.1 Internal impedance of the self-impedance**

In a conductor flowing current flows is distributed over the entire surface of its cross section. Consequently magnetic fields exist inside and outside. To properly evaluate the internal field is necessary to know the geometric properties of the cross section of conductor and internal distribution of the current. For the study, the following hypotheses are assumed:


The internal impedance of a conductor per unit length is defined by the formula:

$$Z\_{ol} = \left(R\_i + j \cdot 2 \cdot 10^{-4} \cdot o \cdot \frac{\mu\_r}{4}\right) \frac{\Omega}{km} \tag{28}$$

Considering the following relations,

*r* Radius of the wire

170 Electrical Generation and Distribution Systems and Power Quality Disturbances

• *Uei* : External voltage in the circuit formed by the conductor *i* and earth, caused by

*s*

0

*ei*

*i*

*i <sup>U</sup> <sup>Z</sup>*

*sii*

0

*ei i <sup>U</sup> <sup>Z</sup>*

In Figure 5 is showed the circuit for the calculation of the mutual impedance. *<sup>i</sup> I* is the current for the conductor *I* and*Uij* in the voltage in conductor *j* due to the current for the

*i i <sup>U</sup> <sup>Z</sup>*

*Z ZZ sii i ei* = + 0 (23)

*I l* <sup>=</sup> <sup>⋅</sup> (24)

*I l* <sup>=</sup> <sup>⋅</sup> (25)

*I l* <sup>=</sup> <sup>⋅</sup> (26)

• *U*<sup>0</sup>*<sup>i</sup>* : Voltage in the outer surface of conductor *i* caused by the current *i*.

The self-impedance can be defined as the sum of the following impedances:

the current *i*.

Where the meaning of the terms is:

Fig. 4. Calculation of the self-impedance

Fig. 5. Mutual impedance calculation

current *<sup>i</sup> I* .

• Mutual impedance between two wires.

*r*' GMR (geometric mean radius)

$$
\dot{r\_i} = r\_i \cdot e^{-\mu/4} \tag{29}
$$

relation (28) can be rewritten as:

$$Z\_{oi} = \left(R\_i + j \cdot 2 \cdot 10^{-4} \cdot o \cdot \ln \frac{r\_i}{r\_i}\right) \frac{\Omega}{km} \tag{30}$$

#### **3.4.2 External impedance of the self-impedance and mutual impedance**

The external impedance of the self-impedance is due to the following relation:

$$Z\_{el} = \left( j \cdot oo \cdot \frac{\mu\_0}{2\pi} \cdot \ln \frac{2h\_i}{r\_i} + \left( \Delta R\_{\bar{\imath}i} + \Delta X\_{\bar{\imath}i} \right) \right) \frac{\Omega}{km} \tag{31}$$

The mutual impedance can be calculated using the equation:

$$Z\_{\bar{\imath}} = \left( j \cdot \alpha \cdot \frac{\mu\_0}{2\pi} \cdot \ln \frac{D\_{\bar{\imath}}}{d\_{\bar{\imath}}} + 2 \left( \Delta R\_{\bar{\imath}i} + \Delta X\_{\bar{\imath}i} \right) \right) \frac{\Omega}{km} \tag{32}$$

The correction terms Δ*R* and Δ*X* take into account the effect of earth return and they are a function of angle θ (between conductors) and the parameter p, which is:


$$\text{Where} \tag{3}$$

$$\alpha = 4\pi\sqrt{5} \cdot 10^{-4} \cdot \sqrt{\frac{f}{\rho}} \tag{3}$$

Electrical Disturbances from High Speed Railway Environment to Existing Services 173

Once the matrix [*Z*] containing the self-impedances and the mutual impedances has been calculated, the inducted voltage in the victim wire can be evaluated using the following

1 1,1 1,13 1

*V Z ZI*

*V Z ZI* <sup>Δ</sup> <sup>=</sup> <sup>Δ</sup> 

> 12 13 13, 1

*i V ZI* =

It is possible to prevent the problems caused by the electrostatic and electromagnetic

As seen in section 3.1, the electric field produced by the conductors of the railway system induces a voltage in an eventual wire or in any conductive part not directly connected to the earth. Furthermore, the possible presence of electric field, due to atmospheric electrical charge present in that region of space, must be also the cause of the appearance of induced

For safety reasons any mass, in principle, is directly connected to the earth, to prevent a

Generally to allow the proper operation of the signaling systems the wires connecting the different elements are wires having a protective shield. This shield must be connected to earth in order to fix the potential to earth. If the shield is earthed, voltage *Ushield = 0*. For electrostatic reduction is enough to connect just an extreme of the shield to earth. In this case of connection, to guarantee the condition *Ushield = 0*, the shield must be earthed in more points along the line, as close as possible. In this way in the extreme not connected to earth, the voltage to earth will be low and it can be assumed *Ushield ≈ 0*. If the section is too long there will be fewer guarantees to maintain *Uscreen = 0*. Considering equation (15), (16) and (17) it can be seen that capacitances depend on the length of the shielding: decreasing the

length decreases the capacitance, and consequently the electrostatic effect is reduced.

The shield could also be earthed in both ends. In this case it is very important to earth the shield along the line often, to reduce the magnetic coupling. Closing the shield it forms a

Protection against electromagnetic induction is hard to achieve. As explained in section 3.2, the magnetic coupling between the railway system and the victim conductor produces

*i i*

Voltage Δ*V*<sup>13</sup> can be evaluated considering the border condition that <sup>13</sup> *i* = 0 :

coupling using some criteria that will be explained in next sections.

potential that can be dangerous for people, animals and things.

spire and consequently it can be victim of electromagnetic coupling.

13 13,1 14,13 13

[Δ= ⋅ *V ZI* ] [ ] [ ] (44)

Δ= ⋅ (46)

(45)

**3.5 Voltage calculation** 

**4. Field reduction** 

**4.1 Electrostatic reduction** 

voltage in the victim conductor.

**4.2 Electromagnetic reduction** 

relation:

with *f* :frequency of the current.

ρ: resistivity of the earth.

Expressions of Δ*R* and Δ*X* series given by Carson are:

$$\begin{aligned} \Delta R &= \alpha \cdot 2 \cdot 10^{-4} \left\{ \frac{\pi}{8} - b\_1 p \cos \theta + b\_2 p^2 \left[ (c\_2 - \ln p) \cos 2\theta + \theta \sin 2\theta \right] + \\\\ &+ b\_3 p^3 \cos 3\theta - d\_4 p^4 \cos 4\theta - b\_5 p^5 \cos 5\theta + \dots \right\} \end{aligned} \tag{36}$$

$$
\Delta \mathbf{X} = a \boldsymbol{\theta} \cdot \mathbf{2} \cdot \mathbf{1} \mathbf{0}^{-4} \left\{ \frac{1}{2} (0.6159315 - \ln p) + b\_1 p \cos \theta - d\_2 p^2 \cos 2\theta + b\_3 p^3 \cos 3\theta - \dots \right\} \tag{37}
$$

$$
$$

Where the coefficients *a*, *b*, *c* and *d* are:

$$b\_1 = \frac{\sqrt{2}}{6} \tag{38}$$

$$b\_i = b\_{i-2} \frac{sign}{i(i+2)}\tag{39}$$

With the starting value:

$$b\_2 = \frac{1}{16} \tag{40}$$

$$\mathbf{c}\_{i} = \mathbf{c}\_{i-2} + \frac{1}{i} + \frac{1}{\left(i + 2\right)}\tag{41}$$

With the starting value:

$$c\_2 = 1.3659315\tag{42}$$

$$d\_i = \frac{\pi}{4} b\_i \tag{43}$$

#### **3.5 Voltage calculation**

172 Electrical Generation and Distribution Systems and Power Quality Disturbances

The correction terms Δ*R* and Δ*X* take into account the effect of earth return and they are a

α

α

ρ

( )

( ) }

 θ

θ

θθ

12 3

2*h* (33)

*b* (34)

}

 θ  θ

θθ

(36)

(37)

<sup>−</sup> = ⋅⋅ (35)

θθ

*b* = (38)

*i i* <sup>=</sup> <sup>−</sup> <sup>+</sup> (39)

*b* = (40)

= ++ <sup>−</sup> <sup>+</sup> (41)

<sup>2</sup> *c* = 1.3659315 (42)

= (43)

function of angle θ (between conductors) and the parameter p, which is:

α π

4 2

*R bp bp c p*

( )

1 22

cos3 cos 4 cos5 ...

*X p bp dp bp*

ln cos 4 sin 4 cos5 ...

1

2

2

<sup>4</sup> *i i d b* π

θ

θ

345 345

+− − +

4 2 3

<sup>1</sup> 2 10 0.6159315 ln cos cos2 cos3

4 5 4 4 5

> 2 6

( ) <sup>2</sup> 2 *i i sign b b*

> 1 16

1 1 ( 2) *i i c c i i*

θθ

−− + − + 

*bp c p b p*

<sup>−</sup> Δ = ⋅⋅ −+ − + −

*bp dp bp*

<sup>−</sup> Δ= ⋅⋅ − + − + +

2 10 cos ln cos2 sin 2

• For self-impedance: *p* =

with *f* :frequency of the current.

: resistivity of the earth.

ω

ω

With the starting value:

With the starting value:

Where the coefficients *a*, *b*, *c* and *d* are:

ρ

• For mutual impedance: *ij p* =

Where <sup>4</sup> 4 5 10 *<sup>f</sup>*

Expressions of Δ*R* and Δ*X* series given by Carson are:

2

8

π

Once the matrix [*Z*] containing the self-impedances and the mutual impedances has been calculated, the inducted voltage in the victim wire can be evaluated using the following relation:

$$\begin{bmatrix} \Delta V \end{bmatrix} = \begin{bmatrix} Z \end{bmatrix} \cdot \begin{bmatrix} I \end{bmatrix} \tag{44}$$

$$
\begin{bmatrix}
\Delta V\_1 \\
\vdots \\
\Delta V\_{13}
\end{bmatrix} = \begin{bmatrix}
Z\_{1,1} & \cdots & Z\_{1,13} \\
\vdots & \ddots & \vdots \\
Z\_{13,1} & \cdots & Z\_{14,13}
\end{bmatrix} \begin{bmatrix}
I\_1 \\
\vdots \\
I\_{13}
\end{bmatrix} \tag{45}
$$

Voltage Δ*V*<sup>13</sup> can be evaluated considering the border condition that <sup>13</sup> *i* = 0 :

$$
\Delta V\_{13} = \sum\_{i=1}^{12} Z\_{13,i} \cdot I\_i \tag{46}
$$

#### **4. Field reduction**

It is possible to prevent the problems caused by the electrostatic and electromagnetic coupling using some criteria that will be explained in next sections.

#### **4.1 Electrostatic reduction**

As seen in section 3.1, the electric field produced by the conductors of the railway system induces a voltage in an eventual wire or in any conductive part not directly connected to the earth. Furthermore, the possible presence of electric field, due to atmospheric electrical charge present in that region of space, must be also the cause of the appearance of induced voltage in the victim conductor.

For safety reasons any mass, in principle, is directly connected to the earth, to prevent a potential that can be dangerous for people, animals and things.

Generally to allow the proper operation of the signaling systems the wires connecting the different elements are wires having a protective shield. This shield must be connected to earth in order to fix the potential to earth. If the shield is earthed, voltage *Ushield = 0*. For electrostatic reduction is enough to connect just an extreme of the shield to earth. In this case of connection, to guarantee the condition *Ushield = 0*, the shield must be earthed in more points along the line, as close as possible. In this way in the extreme not connected to earth, the voltage to earth will be low and it can be assumed *Ushield ≈ 0*. If the section is too long there will be fewer guarantees to maintain *Uscreen = 0*. Considering equation (15), (16) and (17) it can be seen that capacitances depend on the length of the shielding: decreasing the length decreases the capacitance, and consequently the electrostatic effect is reduced.

The shield could also be earthed in both ends. In this case it is very important to earth the shield along the line often, to reduce the magnetic coupling. Closing the shield it forms a spire and consequently it can be victim of electromagnetic coupling.

#### **4.2 Electromagnetic reduction**

Protection against electromagnetic induction is hard to achieve. As explained in section 3.2, the magnetic coupling between the railway system and the victim conductor produces induced voltages depending on the position of the different conductors. In the case of highspeed lines, the inducing circuit is composed by the loop power substation-overhead wiretrain-rails. The current that is circulating varies in time with a given frequency and produces a magnetic field of the same frequency. This magnetic field induces an electromotive force in the circuit composed by the victim conductor and the earth, and consequently an induced current in the victim circuit will be generated.

To protect the signaling system from voltages induced by electromagnetic coupling in general in railways are used cables with a factor reduction. Besides of the electrostatic shields, these kinds of cables have another shield against the electromagnetic effects. Using these kind of shields it is possible to reduce the magnetic field and consequently to reduce the induced voltages and currents.

The reduction factor or factor shielding as the mitigation of electromagnetic fields produced by a power line by the presence of shields or other metal objects:

$$K = \frac{V\_1}{V\_2} \tag{47}$$

Electrical Disturbances from High Speed Railway Environment to Existing Services 175

create a magnetic field contrary to the one that is induced by the high speed line. In this way the two electromotive forces are summed up reducing the original effect. In order to give a

The two extremes of the shields must be earthed to create a closed loop where creating the opposite induced voltage. There are -in the case of connecting the shield to earth at both ends- the possibility of occurrence of earth loops in the event that land is not equipotential. This potential difference would generate a disturbing current on the shield. This effect can be reduced by a good earthing, improving the land and making the earthing as often as possible so that the potential difference between both ends of the cable is as close to zero as

Fig. 7. Interaction between the magnetic field produced by the AC railway system and the

When both ends are connected to earth, the effect of stray currents or earth return currents must be considered. These currents depend on the quality of insulation of the ballast and earth conductivity. Presumably, earthing the shield at both ends of the return current will

One way of addressing this problem is the connection of capacitors in an end of the screen. A capacitor acts like an open circuit for DC preventing its movement along the screen. Moreover, the capacitor value can be calculated for a resonance frequency of 50Hz if the value of the inductance of the shield for a given field is known. The inductance of the shield is a value that depends of a nonlinear magnetic field as it is composed of a ferromagnetic material. This makes the calculation of this capacitor complex because it should be calculated for a magnetic field value undetermined. In addition, the inductance of the cable depends on

the type of cable, the thickness of it, the reduction factor and the length of the span.

conventional DC railway system in presence of magnetic shield

ease the conduction by the shield.

more visual approach, Figure 7 shows how this kind of shielding works.

possible.

Where *V*1 is the voltage induced in presence of these shields and *V*<sup>2</sup> is the voltage that would be induced in their absence. Thus the reduction factor will always be less than the unit.

In general there are 2 families of magnetic screen:


Magnetic screen with magnetic material are based on the Snell's Law. If a magnetic material with high permeability is put in the interface between two zones (Figure 6), the magnetic flux lines change direction and reduce the magnitude according to the magnetic permeability of the material. Reducing the magnetic field, the induced voltages will be lower.

#### Fig. 6. Snell's Law

Magnetic screen with conductive material are the most used in railways. The principle is to create a spire in the victim conductor where inducing an electromotive force in order to

induced voltages depending on the position of the different conductors. In the case of highspeed lines, the inducing circuit is composed by the loop power substation-overhead wiretrain-rails. The current that is circulating varies in time with a given frequency and produces a magnetic field of the same frequency. This magnetic field induces an electromotive force in the circuit composed by the victim conductor and the earth, and consequently an induced

To protect the signaling system from voltages induced by electromagnetic coupling in general in railways are used cables with a factor reduction. Besides of the electrostatic shields, these kinds of cables have another shield against the electromagnetic effects. Using these kind of shields it is possible to reduce the magnetic field and consequently to reduce

The reduction factor or factor shielding as the mitigation of electromagnetic fields produced

Where *V*1 is the voltage induced in presence of these shields and *V*<sup>2</sup> is the voltage that would be induced in their absence. Thus the reduction factor will always be less than the

Magnetic screen with magnetic material are based on the Snell's Law. If a magnetic material with high permeability is put in the interface between two zones (Figure 6), the magnetic flux lines change direction and reduce the magnitude according to the magnetic permeability of the material. Reducing the magnetic field, the induced voltages will be

Magnetic screen with conductive material are the most used in railways. The principle is to create a spire in the victim conductor where inducing an electromotive force in order to

1 2 *<sup>V</sup> <sup>K</sup> V*

= (47)

current in the victim circuit will be generated.

In general there are 2 families of magnetic screen: • Magnetic screen with magnetic material; • Magnetic screen with conductive material.

by a power line by the presence of shields or other metal objects:

the induced voltages and currents.

unit.

lower.

Fig. 6. Snell's Law

create a magnetic field contrary to the one that is induced by the high speed line. In this way the two electromotive forces are summed up reducing the original effect. In order to give a more visual approach, Figure 7 shows how this kind of shielding works.

The two extremes of the shields must be earthed to create a closed loop where creating the opposite induced voltage. There are -in the case of connecting the shield to earth at both ends- the possibility of occurrence of earth loops in the event that land is not equipotential. This potential difference would generate a disturbing current on the shield. This effect can be reduced by a good earthing, improving the land and making the earthing as often as possible so that the potential difference between both ends of the cable is as close to zero as possible.

Fig. 7. Interaction between the magnetic field produced by the AC railway system and the conventional DC railway system in presence of magnetic shield

When both ends are connected to earth, the effect of stray currents or earth return currents must be considered. These currents depend on the quality of insulation of the ballast and earth conductivity. Presumably, earthing the shield at both ends of the return current will ease the conduction by the shield.

One way of addressing this problem is the connection of capacitors in an end of the screen. A capacitor acts like an open circuit for DC preventing its movement along the screen.

Moreover, the capacitor value can be calculated for a resonance frequency of 50Hz if the value of the inductance of the shield for a given field is known. The inductance of the shield is a value that depends of a nonlinear magnetic field as it is composed of a ferromagnetic material. This makes the calculation of this capacitor complex because it should be calculated for a magnetic field value undetermined. In addition, the inductance of the cable depends on the type of cable, the thickness of it, the reduction factor and the length of the span.

Electrical Disturbances from High Speed Railway Environment to Existing Services 177

These orders are usually 50 Hz signals. They can be of a different range of voltages that can vary between 50 to 220 volts. So there is a chance that induced voltages can either turn off one aspect of the signal, which would only affect operation of the line, or can turn it on what

Communication wires are laid along the line from one station to their collaterals. Voice services are transmitted via these wires and also data. 50 Hz disturbances in these wires can

Block wires connect collateral stations. This is a coded signal that is not likely to be affected by 50 Hz disturbances, but can be affected by harmonic disturbances caused by the rolling stock. In order to protect these services from disturbances several actions can be carried out: • Wires can be substituted by fiber optic that is immune to electrical disturbances. This is a good solution, especially for communications and services between stations. • Wires can be shielded. This shield has to be earthed on both ends in order to provide effective shielding against both inductive coupling and capacitive coupling. This way of shielding is very effective against disturbances, but gives a good return for DC

• Wires can be shielded with both ends earthed, but in one side earthing is made through a capacitor (Koopal & Evertz, 2008). Thus no DC would return using the shield. This is an expensive way because each capacitor has to be calculated depending on the length of wire and the resistivity of the earth in order to obtain a minimum impedance at 50

Railway level crossings are particular signalling systems that can be or not be connected to other signalling systems such as interlocking. These systems are in charge of protecting cars and citizens from railways. Level crossings electronic systems can be triggered by the interlocking or by track pedal that gives a train announcement to the system. These facilities

The solutions to immunize these systems from disturbances are the same one that those

• Rail potential (EN 50122-1): The voltage occurring under operating conditions when the running rails are utilised for carrying the traction return current or under fault

• Accessible voltage (EN 50122-1):That part of the rail potential under operating conditions which can be bridged by persons, the conductive path being conventionally

• Touch voltage (EN 50122-1): Voltage under fault conditions between parts when

As was described above, usually DC rails used in traction lines are isolated from earth in order to prevent stray currents. This measure can raise rail potential. This can happen even without tractor trains in the line if there's a parallel high speed line nearby. Passenger might want to cross from one platform to other crossing the rails even if it's forbidden. This is

Hz frequency so shielding against 50 Hz electrical disturbances is effective.

affect communications, but are not likely to have an influence in safety issues.

circulating through earth back to the feeder station.

have also wires that can have a length of more than two kilometres.

from hand to both feet through the body or from hand to hand.

stated above for signalling and communication wires.

conditions between running rails and earth.

more common in small stations with low traffic density.

According to EN 50122-1 we can define the following terms:

**5.4 Accessible metallic elements** 

touched simultaneously.

can affect safety.

**5.3 Level crossings** 
