**Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires**

R. C. Monzani, A. J. Prado, L.S. Lessa and

L. F. Bovolato

[3] Cristina-Maria Dabu, M. D. Nicu, *Calculatoare în Biotehnici – Note de curs pentru uzul*

[4] Jeremy M. Berg, John L. Tymoczko, Lubert Stryer; Web content by Neil D. Clarke (2002). "Protein Structure and Function". Biochemistry. San Francisco: W. H. Free‐

[5] Yanfen Liu, Yihong Ye, Proteostasis regulation at the endoplasmic reticulum: a new perturbation site for targeted cancer therapy, Cell Research (2011) 21:867-883, http://

[6] Ellen Wirawan and colab, Autophagy: for better or for worse, *Cell Research* (2012)

[10] Mitchel L. Model, Bioinformatics programmind using Python, O'Reilly Media, Inc.,

[8] Sal Mangano, Mathematica Cookbook, O'Reilly Media, Inc USA, 2010

[12] http://www.mathworks.com/products/bioinfo/, visited 0ct. 2013

*studentilor*, Universitatea "Politehnica" Bucureşti, 1997

man. ISBN 0-7167-4684-0.

508 MATLAB Applications for the Practical Engineer

www.nature.com/cr, visited 02.09.2013

22:43-61. doi:10.1038/cr.2011.152 [7] www.wolfram.com visited Nov, 2013

[9] www.python.org visited Nov, 2013

[11] Matlab, www.mathworks.com, visited 0ct. 2013

USA, 2009

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/58383

### **1. Introduction**

The analysis of a transmission line allows a complex study that can be used to measure losses in line, to understand the behavior of them in case of a voltage surge and other kinds of phenomena.

This chapter presents the development of a routine that evaluates a method for determining real transformation matrices in three-phase systems considering the presence of ground-wires. Thus, for Z (longitudinal impedance) and Y (transversal admittance) matrices that represent the transmission line, the ground wires are considered not implicit in the values of the phases. This routine was developed using the mathematical tool MatlabTM.

As a proposal, the routine uses a real transformation matrix throughout the frequency range of analysis. This transformation matrix is an approximation of the exact transformation matrix. For elements related to the phase of the system considered, the transformation matrix is composed from the elements of the Clarke's matrix [18].

In parts related to ground wires, the elements of the transformation matrix must establish a relationship with the elements of the phases considering the establishment of a unique, single homopolar reference in mode domain.

In case of three-phase transmission lines with the presence of two ground wires, it is not possible to obtain the complete diagonalization of Y and Z matrices in mode domain. Finally, a correction routine is applied with the goal of minimizing errors obtained for the eigenvalues.

© 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **2. Background**

Different methods can be used in order to analyze electromagnetic transient phenomena in transmission lines. Many mathematical tools can be used. The main tools are: circuit analysis using Laplace transform and Fourier transform, state variables and also differential equations. These tools can be included in a numerical routine in order to obtain the values of voltage and current in electromagnetic transient simulation for any point in the circuit.

The proposed model is based on an approximate modal transformation performed by a single real phase-mode transformation matrix, and frequency independent. This matrix is obtained by linear combination of Clarke's matrix elements. With the implementation of the transfor‐ mation matrix frequency independent, it's obtained a diagonal matrix for transposed lines. In case of a three-phase transmission line not transposed the line parameters matrix can't be diagonalized, with application of a single real phase-mode transformation matrix mentioned. For these cases, the goal is to analyze the relative errors obtained by the establishment

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

511

For the proposed method in this chapter, a similar mathematical basis is used to homopolar hypothesis of a single reference for all phases of the system regardless of the geometrical distribution and organization of the three-phase circuit. Thus, the development is based on the analysis of eigenvectors and eigenvalues, using a linear combination of Clarke's matrix

It's presented two different proposals for real matrices and frequency independent in order to replace the modal transformation matrix of a typical three-phase transmission line in the

A transmission line is represented by a longitudinal impedance Z and the transversal admit‐ tance Y matrices, the characteristics of ground wires are not implied in the values related to the phases. Thus, for a line with two ground wires, the Z and Y matrices are 5-order ones.

For the mentioned systems, the three-phase circuit configurations are considered and the transposed case can be described by a system where each three-phase circuit is ideally transposed and there are coupling impedances and admittances among the three-phase circuits. For EMTP type programs, if each three-phase circuit is considered independently, the transformation matrix is frequency dependent for general cases. For the method proposed in this chapter, a similar mathematical base is used for all considered cases: the assumption of unique ground reference for all phases of the system independently of the geometrical distribution and the organization of the three-phase circuits. The unique ground reference

The relationships between transversal voltages uF and the longitudinal currents iF can be expressed by the following equations, where Z is the longitudinal impedance matrix per unit

*d iF*

Applying the eigenvector and eigenvalue analyses for YZ and ZY product matrices, the λ diagonal eigenvalue matrix and the eigenvector matrices are determined. The eigenvector

*dx* =*Y* ⋅*uF* (1)

length and Y is the transversal admittance matrix per unit length in phase domain.

*<sup>F</sup>* and -

circumstances to use a transformation matrix frequency independent.

elements, and assuming a unique homopolar reference.

leads to a unique homopolar mode in mode domain.

 *d uF dx* =*Z* ⋅ *i*

presence of two ground-wires.

**3. Mathematical model**

The EMTP (ElectroMagnetic Transient Program) identifies a type of program, considering its various versions, which performs simulations of transients in electrical networks [1]. The prototype was developed in the 1960s by professionals in power systems, led by Dr. Hermann Dommel (University of British Columbia, Vancouver, BC, Canada), and Dr. Scott Meyer (Bonneville Power Administration in Portland, Oregon, USA). Currently, the EMTP is the basis of simulations of electromagnetic transients in power systems.

With EMTP-type programs, the following tests may be performed: simulation by switching and lightning surges, transient and temporary overvoltages, transients in electric machines, resonance phenomena, harmonics, power quality and power electronics applications. The most popular type programs are EMTP: MicroTran, PSCAD and ATP.

In analysis of transmission systems, there are simulators that represent different types of systems, from generation, transmission and distribution.

Because it is practically impossible to perform the simulation of electromagnetic transients on real transmission lines, simulations by digital models become useful tools. However, these tools do not provide satisfactory performance as regards the correct representation of the electrical line parameters, as these are dependent on the frequency.

In modal domain, it's possible to represent the transmission line circuits using simple circuits and easily entering frequency dependence of longitudinal parameters.

In general, a system composed of *n-*phases can be transformed into independent modes using a real and unique transformation matrix, if transposition applies to all phases for the frequency range used (ideal transposition). If the analyzed system is not transposed, a mode is obtained for each phase using the frequency dependent phase-mode transformation matrix.

Applying real and unique transformation matrix for the nontransposed lines, approxi‐ mate results can be obtained. Thus, there is an approximate representation of frequency dependence using a real phase-mode transformation matrix [2]-[3]. One possible simplifica‐ tion is to consider the transformation matrix frequency independent, obtaining insignifi‐ cant errors related to the eigenvalues that represent the line. Using the mentioned simplification, the obtained numerical routine may be faster because it avoids using a convolution method [4]-[28].

The objective of this chapter is to analyze the application of a real transformation matrix that is frequency independent in three-phase lines considering the presence of two ground wires. Errors are presented in relation to the exact values obtained from the matrix of eigenvalues.

The proposed model is based on an approximate modal transformation performed by a single real phase-mode transformation matrix, and frequency independent. This matrix is obtained by linear combination of Clarke's matrix elements. With the implementation of the transfor‐ mation matrix frequency independent, it's obtained a diagonal matrix for transposed lines. In case of a three-phase transmission line not transposed the line parameters matrix can't be diagonalized, with application of a single real phase-mode transformation matrix mentioned. For these cases, the goal is to analyze the relative errors obtained by the establishment circumstances to use a transformation matrix frequency independent.

For the proposed method in this chapter, a similar mathematical basis is used to homopolar hypothesis of a single reference for all phases of the system regardless of the geometrical distribution and organization of the three-phase circuit. Thus, the development is based on the analysis of eigenvectors and eigenvalues, using a linear combination of Clarke's matrix elements, and assuming a unique homopolar reference.

It's presented two different proposals for real matrices and frequency independent in order to replace the modal transformation matrix of a typical three-phase transmission line in the presence of two ground-wires.

#### **3. Mathematical model**

**2. Background**

510 MATLAB Applications for the Practical Engineer

Different methods can be used in order to analyze electromagnetic transient phenomena in transmission lines. Many mathematical tools can be used. The main tools are: circuit analysis using Laplace transform and Fourier transform, state variables and also differential equations. These tools can be included in a numerical routine in order to obtain the values of voltage and

The EMTP (ElectroMagnetic Transient Program) identifies a type of program, considering its various versions, which performs simulations of transients in electrical networks [1]. The prototype was developed in the 1960s by professionals in power systems, led by Dr. Hermann Dommel (University of British Columbia, Vancouver, BC, Canada), and Dr. Scott Meyer (Bonneville Power Administration in Portland, Oregon, USA). Currently, the EMTP is the basis

With EMTP-type programs, the following tests may be performed: simulation by switching and lightning surges, transient and temporary overvoltages, transients in electric machines, resonance phenomena, harmonics, power quality and power electronics applications. The

In analysis of transmission systems, there are simulators that represent different types of

Because it is practically impossible to perform the simulation of electromagnetic transients on real transmission lines, simulations by digital models become useful tools. However, these tools do not provide satisfactory performance as regards the correct representation of the

In modal domain, it's possible to represent the transmission line circuits using simple circuits

In general, a system composed of *n-*phases can be transformed into independent modes using a real and unique transformation matrix, if transposition applies to all phases for the frequency range used (ideal transposition). If the analyzed system is not transposed, a mode is obtained

Applying real and unique transformation matrix for the nontransposed lines, approxi‐ mate results can be obtained. Thus, there is an approximate representation of frequency dependence using a real phase-mode transformation matrix [2]-[3]. One possible simplifica‐ tion is to consider the transformation matrix frequency independent, obtaining insignifi‐ cant errors related to the eigenvalues that represent the line. Using the mentioned simplification, the obtained numerical routine may be faster because it avoids using a

The objective of this chapter is to analyze the application of a real transformation matrix that is frequency independent in three-phase lines considering the presence of two ground wires. Errors are presented in relation to the exact values obtained from the matrix of eigenvalues.

for each phase using the frequency dependent phase-mode transformation matrix.

current in electromagnetic transient simulation for any point in the circuit.

of simulations of electromagnetic transients in power systems.

systems, from generation, transmission and distribution.

convolution method [4]-[28].

most popular type programs are EMTP: MicroTran, PSCAD and ATP.

electrical line parameters, as these are dependent on the frequency.

and easily entering frequency dependence of longitudinal parameters.

A transmission line is represented by a longitudinal impedance Z and the transversal admit‐ tance Y matrices, the characteristics of ground wires are not implied in the values related to the phases. Thus, for a line with two ground wires, the Z and Y matrices are 5-order ones.

For the mentioned systems, the three-phase circuit configurations are considered and the transposed case can be described by a system where each three-phase circuit is ideally transposed and there are coupling impedances and admittances among the three-phase circuits. For EMTP type programs, if each three-phase circuit is considered independently, the transformation matrix is frequency dependent for general cases. For the method proposed in this chapter, a similar mathematical base is used for all considered cases: the assumption of unique ground reference for all phases of the system independently of the geometrical distribution and the organization of the three-phase circuits. The unique ground reference leads to a unique homopolar mode in mode domain.

The relationships between transversal voltages uF and the longitudinal currents iF can be expressed by the following equations, where Z is the longitudinal impedance matrix per unit length and Y is the transversal admittance matrix per unit length in phase domain.

$$\mathbf{-\frac{d\,\,\,d\_F}{dx}} = \mathbf{Z} \cdot \mathbf{i\_F} \text{ and } \mathbf{-\frac{d\,\,i\_F}{dx}} = \mathbf{Y} \cdot \mathbf{u\_F} \tag{1}$$

Applying the eigenvector and eigenvalue analyses for YZ and ZY product matrices, the λ diagonal eigenvalue matrix and the eigenvector matrices are determined. The eigenvector matrices, *TV* and *TI* , correspond to voltage and current mathematical relationships, respec‐ tively. The *TV* and *TI* matrices are related to *λ* based on the following equation:

$$
\lambda = T\_V \cdot Z \cdot Y \cdot T\_V^{\cdot 1} = T\_I \cdot Y \cdot Z \cdot T\_I^{\cdot 1} \tag{2}
$$

element value depends on the number of phase conductors. If this number is identified by n,

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

Considering two ground wires in a three-phase lines system, these matrices are 5-order ones. Therefore, the single real phase-mode transformation matrix has the following structure

0 0

0 0

<sup>5</sup> - <sup>1</sup> 5

<sup>2</sup> - <sup>1</sup> 2

1 5

1 5

*<sup>n</sup>* (5)

http://dx.doi.org/10.5772/58383

513

(6)

(7)

*TSR*-*<sup>n</sup>* <sup>=</sup> <sup>1</sup>

the homopolar mode elements are described by (6).

presented in (6).

**4. Ground wires in three-phase transmission line**


> 1 2

> 1 5

> 1 5

2 <sup>6</sup> - <sup>1</sup> 6

1 5

1 5

<sup>0</sup> - <sup>1</sup> 2

<sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>1</sup>

1 5

1 <sup>5</sup> - <sup>1</sup>

In case of a three-phase lines system is ideally transposed, it creates only one coupling impedance between the lines. The average self-phase impedance value is represented by A. The average coupling impedances are represented by B, within a circuit, and C, between the line and ground wires or other circuits. The average ground wires impedance value is represented by D. The average coupling impedance between the both ground wires is represented by E. For the case of a single three-phase line in the presence of two ground wires

> *A B B C C B A B C C B B A C C C C C D E C C C E D*

The three-phase transmission line circuit tower (Fig. 2) has a height of 36.0 m and is the structure used in this chapter. This is a 400 km length line that operates in 440 kV. It is a system

whose conductors are disposed in such way that there is a vertical symmetry plane.

*TSR*<sup>5</sup> =

(Fig. 1), the structure of the impedance matrix is shown in (7).

*Z*<sup>5</sup> =

If the *TV* and *TI* transformation matrices are used, the eigenvalues can be obtained in mode domain using (1). The per unit length longitudinal impedance matrix (*ZMD*) and transversal admittance matrix (*YMD*) are:

$$\mathbf{Z}\_{\rm MD} = \mathbf{T}\_V \cdot \mathbf{Z} \cdot \mathbf{T}\_I^{\cdot \cdot 1} \text{ and } \quad \mathbf{Y}\_{\rm MD} = \mathbf{T}\_I \cdot \mathbf{Y} \cdot \mathbf{T}\_V^{\cdot \cdot 1} \tag{3}$$

In general, these frequency dependent transformation matrices (*TV* and *TI* ) are different and have complex elements. Using the proposed methodology, the transformation matrices are changed into a single real transformation matrix (*TSR*). The *TSR* matrix is determined from linear combinations of the Clarke's matrix elements [4]-[7]. The determination of exact eigenvalues is approximated and changed into the following:

$$
\lambda\_{\rm SR} = T\_{\rm SR} \cdot Z \cdot Y \cdot T\_{\rm SR}^{\cdot \cdot 1} = T\_{\rm SR} \cdot Y \cdot Z \cdot T\_{\rm SR}^{\cdot \cdot 1} \tag{4}
$$

In case of the EMTP type programs, the transformation matrices are real, if the system is ideally transposed. For this, there is only one self-impedance value for all phase interactions. Con‐ sidering the admittance values, a similar structure to the impedance values is obtained. Applying the EMTP type programs, a system composed by three-phase circuits is analyzed as a non-transposed case, if the each three-phase circuit is considered transposed independently of the ground wires.

Using a single homopolar mode reference, the *λSR* matrix is equal to the exact eigenvalue matrix (*λ*) [8] as well as *TV* and *TI* being equal to a single real transformation matrix for transposed cases [9]-[11]. So, with a single homopolar mode reference, there is a link between the three-phase circuit and the ground wires of the system. With this technique, a transfor‐ mation matrix (*TSR*) is obtained which has interesting characteristics: single, real, frequency independent, line parameter independent and identical to voltages and currents. With these characteristics, phase-mode transformations are carried out using only one matrix multipli‐ cation.

The homopolar or zero sequence components (*Va*0, *Vb*0 and *Vc*0) for a three-phase system are equal and they make the unique ground reference for the three-order phasor system. Using the homopolar reference phasor concept, the application of a single mode reference to the single real phase-mode transformation matrix is proposed. So, the homopolar mode is used as the only mode reference for the analyzed transmission line systems. To compose the *TSR* matrix, each mode must have a unitary modulus. Because of this, each homopolar mode element value depends on the number of phase conductors. If this number is identified by n, the homopolar mode elements are described by (6).

$$T\_{SR-n} = \frac{1}{\sqrt{n}}\tag{5}$$

#### **4. Ground wires in three-phase transmission line**

matrices, *TV* and *TI* , correspond to voltage and current mathematical relationships, respec‐

If the *TV* and *TI* transformation matrices are used, the eigenvalues can be obtained in mode domain using (1). The per unit length longitudinal impedance matrix (*ZMD*) and transversal

In general, these frequency dependent transformation matrices (*TV* and *TI* ) are different and have complex elements. Using the proposed methodology, the transformation matrices are changed into a single real transformation matrix (*TSR*). The *TSR* matrix is determined from linear combinations of the Clarke's matrix elements [4]-[7]. The determination of exact

In case of the EMTP type programs, the transformation matrices are real, if the system is ideally transposed. For this, there is only one self-impedance value for all phase interactions. Con‐ sidering the admittance values, a similar structure to the impedance values is obtained. Applying the EMTP type programs, a system composed by three-phase circuits is analyzed as a non-transposed case, if the each three-phase circuit is considered transposed independently

Using a single homopolar mode reference, the *λSR* matrix is equal to the exact eigenvalue matrix (*λ*) [8] as well as *TV* and *TI* being equal to a single real transformation matrix for transposed cases [9]-[11]. So, with a single homopolar mode reference, there is a link between the three-phase circuit and the ground wires of the system. With this technique, a transfor‐ mation matrix (*TSR*) is obtained which has interesting characteristics: single, real, frequency independent, line parameter independent and identical to voltages and currents. With these characteristics, phase-mode transformations are carried out using only one matrix multipli‐

The homopolar or zero sequence components (*Va*0, *Vb*0 and *Vc*0) for a three-phase system are equal and they make the unique ground reference for the three-order phasor system. Using the homopolar reference phasor concept, the application of a single mode reference to the single real phase-mode transformation matrix is proposed. So, the homopolar mode is used as the only mode reference for the analyzed transmission line systems. To compose the *TSR* matrix, each mode must have a unitary modulus. Because of this, each homopolar mode







tively. The *TV* and *TI* matrices are related to *λ* based on the following equation:

*λ* =*TV* ⋅*Z* ⋅*Y* ⋅*TV*

*ZMD* =*TV* ⋅*Z* ⋅*TI*

eigenvalues is approximated and changed into the following:

*λSR* =*TSR* ⋅*Z* ⋅*Y* ⋅*TSR*

admittance matrix (*YMD*) are:

512 MATLAB Applications for the Practical Engineer

of the ground wires.

cation.

Considering two ground wires in a three-phase lines system, these matrices are 5-order ones. Therefore, the single real phase-mode transformation matrix has the following structure presented in (6).

$$T\_{SR5} = \begin{bmatrix} \frac{1}{\sqrt{6}} & \frac{2}{\sqrt{6}} & -\frac{1}{\sqrt{6}} & 0 & 0\\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0\\ \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}}\\ \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{1}{\sqrt{5}}\\ 0 & 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \tag{6}$$

In case of a three-phase lines system is ideally transposed, it creates only one coupling impedance between the lines. The average self-phase impedance value is represented by A. The average coupling impedances are represented by B, within a circuit, and C, between the line and ground wires or other circuits. The average ground wires impedance value is represented by D. The average coupling impedance between the both ground wires is represented by E. For the case of a single three-phase line in the presence of two ground wires (Fig. 1), the structure of the impedance matrix is shown in (7).

$$Z\_5 = \begin{bmatrix} A & B & B & C & C \\ B & A & B & C & C \\ B & B & A & C & C \\ C & C & C & D & E \\ C & C & C & E & D \end{bmatrix} \tag{7}$$

The three-phase transmission line circuit tower (Fig. 2) has a height of 36.0 m and is the structure used in this chapter. This is a 400 km length line that operates in 440 kV. It is a system whose conductors are disposed in such way that there is a vertical symmetry plane.

**Figure 1.** Coupling impedances for a three-phase transmission line with two ground wires for transposed cases.

The result determined through (4) is a diagonal matrix and the matrix elements are the exact eigenvalues, for the cases where the **Figure 2.** Three phase line tower with two ground wires

initial\_conf.m

ground wires are implicit in the phase values and the line is transposed. The impedance matrix in mode domain () can be calculated as: = ∙ ∙ (8) The result determined through (4) is a diagonal matrix and the matrix elements are the exact eigenvalues, for the cases where the ground wires are implicit in the phase values and the line is transposed. The impedance matrix in mode domain (*ZM* ) can be calculated as:

Figure 2. Three phase line tower with two ground wires

$$Z\_M = T\_{SR} \cdot Z \cdot T\_{SR}^{-1} \tag{8}$$

<sup>0</sup>

A routine with initial configuration was developed in order to set parameters that would be used for other routines. This routine was

Figure 1. Coupling impedances for a three-phase transmission line with two ground wires for transposed cases.

The three-phase transmission line circuit tower (Fig. 2) has a height of 36.0 m and is the structure used in this chapter. This is a 400 km length line that operates in 440 kV. It is a system whose conductors are disposed in such way that there is a vertical symmetry plane.

*λSR*<sup>5</sup> =

**5. Initial configuration routine**

initial\_conf.m

xc(1)= 0; xc(2)= 9.27; xc(3)= 18.54;

radius = GMR;

*A* - *B* 0 0 0 0 0 *A* - *B* 0 0 0

0 0 0 0 *D* - *E*

A routine with initial configuration was developed in order to set parameters that would be

frequency %subroutine to call an array of frequencies from 0 to 1 GHz

yc(1)= 24.07-0.7\*13.43; %the subtracted value is the sag of each wire

used for other routines. This routine was developed using Matlab and is presented below:

3*A* + 6*B* - 2*D* - 2*E*

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

3*A* + 6*B* + 12*C* + 2*D* + 2*E*

<sup>5</sup> 0

(9)

515

http://dx.doi.org/10.5772/58383

<sup>5</sup> 0

0 0 <sup>3</sup>*<sup>A</sup>* + 6*<sup>B</sup>* - 12*<sup>C</sup>* + 2*<sup>D</sup>* + 2*<sup>E</sup>*

0 0 <sup>3</sup>*<sup>A</sup>* + 6*<sup>B</sup>* - 2*<sup>D</sup>* - 2*<sup>E</sup>*

%Three-phase line with vertical symmetry ntpc = 1; %three-phase circuit amount

ncond = ntpc\*3+ngw; %conductors amount

%Conductors' position in x axis (in meters)

%Conductors' position in y axis (in meters)

r\_dist = 0.4; %distance between conductors nph = 4; %number of conductors per phase Arg = rc\*r\_dist^3\*sqrt(2); %argument of root GMR = power(Arg,1/nph); %geometric mean radius

sk\_radius = rc-rin; %for skin effect procedure gw\_radius = 0.9144e-2; %ground wire radius

resist = 1000; %Earth resistance (ohms x meters) mizero = 4\*pi\*1e-7; %magnetic permeability (H/m) epszero = 8.8542\*1e-9; %dielectric permittivity (F/km)

sigma= 3.82\*1e7; %conductor's conductivity (mho/m)

ngw = 2; %ground wires amount

xc(4)= 1.76; %ground wire 1 xc(5)= 16.78; %ground wire 2

yc(2)= 27.67-0.7\*13.43; yc(3)= 24.07-0.7\*13.43; yc(4)= 36.00-0.7\*6.40; yc(5)= 36.00-0.7\*6.40; %Radius (in meters)

rc = 2.52e-2; %total radius rin = 0.93e-2; %internal radius

5

5

 = 0 − 0 00 0 0 <sup>0</sup> (9) Considering ground wires, the single real phase-mode transformation matrix does not perfectly diagonalize the impedance matrix.

%Three-phase line with vertical symmetry ntpc = 1; %three-phase circuit amount

0 0

developed using Matlab and is presented below:

5. Initial configuration routine

− 00 00

00 0 0 −

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires http://dx.doi.org/10.5772/58383 515

$$
\lambda\_{SR5} = \begin{bmatrix}
A \cdot B & 0 & 0 & 0 & 0 \\
0 & A \cdot B & 0 & 0 & 0 \\
0 & 0 & \frac{3A + 6B \cdot 12C + 2D + 2E}{5} & \frac{3A + 6B \cdot 2D \cdot 2E}{5} & 0 \\
0 & 0 & \frac{3A + 6B \cdot 2D \cdot 2E}{5} & \frac{3A + 6B + 12C + 2D \cdot 2E}{5} & 0 \\
0 & 0 & 0 & 0 & D \cdot E
\end{bmatrix} \tag{9}
$$

#### **5. Initial configuration routine**

**Figure 1.** Coupling impedances for a three-phase transmission line with two ground wires for transposed cases.

B

GW 1 GW 2

C

Each phase component ACSR-26/7-636 MCM 0.089899 Ω/km Sag: 13.43 m

7.51 m EHS 3/8"

Each phase

Earth resistivity: 1000 Ω⋅m

Figure 2. Three phase line tower with two ground wires

0 0

*ZM* =*TSR* ⋅*Z* ⋅*TSR*

is transposed. The impedance matrix in mode domain (*ZM* ) can be calculated as:

0 0

developed using Matlab and is presented below:

5. Initial configuration routine

 − 00 00 0 − 0 00

00 0 0 −

<sup>0</sup>


<sup>0</sup>

 

A routine with initial configuration was developed in order to set parameters that would be used for other routines. This routine was

(9)

Considering ground wires, the single real phase-mode transformation matrix does not

The result determined through (4) is a diagonal matrix and the matrix elements are the exact eigenvalues, for the cases where the ground wires are implicit in the phase values and the line

%Three-phase line with vertical symmetry ntpc = 1; %three-phase circuit amount

(8)

as:

**Figure 2.** Three phase line tower with two ground wires

=

perfectly diagonalize the impedance matrix.

 

initial\_conf.m

= ∙ ∙

24.07 m A

9.27 m

36.00 m

0.9144 cm

514 MATLAB Applications for the Practical Engineer

Figure 1. Coupling impedances for a three-phase transmission line with two ground wires for transposed cases.

4.188042 Ω/km Sag: 6.40 m

3.60 m

Square design 0.4 m

0.93 cm

2.52 cm

The three-phase transmission line circuit tower (Fig. 2) has a height of 36.0 m and is the structure used in this chapter. This is a 400 km length line that operates in 440 kV. It is a system whose conductors are disposed in such way that there is a vertical symmetry plane. A routine with initial configuration was developed in order to set parameters that would be used for other routines. This routine was developed using Matlab and is presented below:

```
The result determined through (4) is a diagonal matrix and the matrix elements are the exact eigenvalues, for the cases where the 
ground wires are implicit in the phase values and the line is transposed. The impedance matrix in mode domain () can be calculated 
Considering ground wires, the single real phase-mode transformation matrix does not perfectly diagonalize the impedance matrix. 
                                                                                                          initial_conf.m
                                                                                                          %Three-phase line with vertical symmetry
                                                                                                          ntpc = 1; %three-phase circuit amount
                                                                                                          ngw = 2; %ground wires amount
                                                                                                          ncond = ntpc*3+ngw; %conductors amount
                                                                                                          frequency %subroutine to call an array of frequencies from 0 to 1 GHz
                                                                                                          %Conductors' position in x axis (in meters)
                                                                                                          xc(1)= 0;
                                                                                                          xc(2)= 9.27;
                                                                                                          xc(3)= 18.54;
                                                                                                          xc(4)= 1.76; %ground wire 1
                                                                                                          xc(5)= 16.78; %ground wire 2
                                                                                                          %Conductors' position in y axis (in meters)
                                                                                                          yc(1)= 24.07-0.7*13.43; %the subtracted value is the sag of each wire
                                                                                                          yc(2)= 27.67-0.7*13.43;
                                                                                                          yc(3)= 24.07-0.7*13.43;
                                                                                                          yc(4)= 36.00-0.7*6.40;
                                                                                                          yc(5)= 36.00-0.7*6.40;
                                                                                                          %Radius (in meters)
                                                                                                          rc = 2.52e-2; %total radius
                                                                                                          rin = 0.93e-2; %internal radius
                                                                                                          r_dist = 0.4; %distance between conductors
                                                                                                          nph = 4; %number of conductors per phase
                                                                                                          Arg = rc*r_dist^3*sqrt(2); %argument of root
                                                                                                          GMR = power(Arg,1/nph); %geometric mean radius
                                                                                                          radius = GMR;
                                                                                                          sk_radius = rc-rin; %for skin effect procedure
                                                                                                          gw_radius = 0.9144e-2; %ground wire radius
                                                                                                          resist = 1000; %Earth resistance (ohms x meters)
                                                                                                          mizero = 4*pi*1e-7; %magnetic permeability (H/m)
                                                                                                          epszero = 8.8542*1e-9; %dielectric permittivity (F/km)
                                                                                                          sigma= 3.82*1e7; %conductor's conductivity (mho/m)
```
sk\_radius = rc-rin; %for skin effect procedure gw\_radius = 0.9144e-2; %ground wire radius

longitudinal inductance in Fig. 3 (b) and capacitance in Fig. 3 (c).

radius = GMR;

#### **6. Eigenvalue analyses for non-transposed three-phase transmission line** Eigenvalue analyses for non-transposed three-phase transmission line

initial\_conf

if j==k

else

 end end end

end

for j = 1:ncond, xi(j) = xc(j); yi(j) = -yc(j); for k = j:ncond

fid(j,k) = fopen(str,'w');

fprintf(fid(j,k),str);

\n clear x\n x = [\n']; fprintf(fid(j,k),str);

%Image conductor coordinates

%External inductance (H/km)

if j < ntpc\*3+1

dx = xc(j) - xc(k);

 dx = xcondut(j) - xi(k); dy = ycondut(j) - yi(k); bd = sqrt(dx^2 + dy^2);

z = i\*2\*pi\*freq(j)\*induct(k,m);

induct(j,k) = 1000\*(1/(2\*pi))\*mizero\*log(dezao/dezinho);

fprintf(fid(k,m), '%30.20f %30.20f\n',real(z),imag(z));

for j = 1: ncond, xi(j) = xc(j); yi(j) = -yc(j);

for j = 1:ncond, for k = j:ncond, if j == k

ld = radius;

ld = gw\_radius;

 dy = yc(j) - yc(k); ld = sqrt(dx^2 + dy^2);

for j = 1:length(freq), for k = 1:ncond for m = k:ncond

else

 end else

 end end

end

 end end

str = ['Zext' int2str(j) '\_' int2str(k) '.m'];

str = ['%% External impedance of phase ' int2str(j) ' \n clear x\n x = [\n'];

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

517

str = ['%% External impedance between phase ' int2str(j) ' and ' int2str(k) '

resist = 1000; %Earth resistance (ohms x meters)

sigma= 3.82\*1e7; %conductor's conductivity (mho/m)

Considering the three-phase line tower with two ground wires shown in Fig. 2, the total longitudinal impedance value is composed by earth effect, calculated by Carson`s method [28], external effects and skin effect. For phase 1, the longitudinal resistance is shown in Fig. 3 (a), the longitudinal inductance in Fig. 3 (b) and capacitance in Fig. 3 (c). Considering the three-phase line tower with two ground wires shown in Fig. 2, the total longitudinal impedance value is composed by earth effect, calculated by Carson`s method [28], external effects and skin effect. For phase 1, the longitudinal resistance is shown in Fig. 3 (a), the

**Figure 3.** (a) Longitudinal Resistance, (b) Longitudinal inductance, (c) Capacitance : phase 1.

The longitudinal resistance and inductance and the capacitance of phase 1 were obtained using a routine developed in MatLab, which is commented and shown below:

Fig. 3 (a) Longitudinal Resistance, (b) Longitudinal inductance, (c) Capacitance : phase 1.

```
external_impedance.m
clear all
'Calculating Z due to external effect'
%File Reading (calling subroutine of initial configuration)
```
**6. Eigenvalue analyses for non-transposed three-phase transmission line**

Considering the three-phase line tower with two ground wires shown in Fig. 2, the total longitudinal impedance value is composed by earth effect, calculated by Carson`s method [28], external effects and skin effect. For phase 1, the longitudinal resistance is shown in Fig. 3 (a),

longitudinal impedance value is composed by earth effect, calculated by Carson`s method [28], external effects and skin effect. For phase 1, the longitudinal resistance is shown in Fig. 3 (a), the

> (a) (b)

<sup>10</sup>-2 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>4</sup> <sup>10</sup><sup>6</sup> <sup>10</sup>-4

FREQUENCY [Hz]

(c) Fig. 3 (a) Longitudinal Resistance, (b) Longitudinal inductance, (c) Capacitance : phase 1.

The longitudinal resistance and inductance and the capacitance of phase 1 were obtained using

<sup>10</sup>-2 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>4</sup> <sup>10</sup><sup>6</sup> 1.2283

FREQUENCY [Hz]

Considering the three-phase line tower with two ground wires shown in Fig. 2, the total

10-3

INDUTANCE [H]

10-2

the longitudinal inductance in Fig. 3 (b) and capacitance in Fig. 3 (c).

longitudinal inductance in Fig. 3 (b) and capacitance in Fig. 3 (c).

<sup>10</sup>-2 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>4</sup> <sup>10</sup><sup>6</sup> <sup>10</sup>-3

FREQUENCY [Hz]

1.2283 x 10-8

1.2283 1.2283 1.2283 1.2283 1.2283 1.2283

**Figure 3.** (a) Longitudinal Resistance, (b) Longitudinal inductance, (c) Capacitance : phase 1.

a routine developed in MatLab, which is commented and shown below:

%File Reading (calling subroutine of initial configuration)

'Calculating Z due to external effect'

CAPACITANCE [F]

external\_impedance.m

clear all

Eigenvalue analyses for non-transposed three-phase transmission line

resist = 1000; %Earth resistance (ohms x meters) mizero = 4\*pi\*1e-7; %magnetic permeability (H/m) epszero = 8.8542\*1e-9; %dielectric permittivity (F/km)

sigma= 3.82\*1e7; %conductor's conductivity (mho/m)

sk\_radius = rc-rin; %for skin effect procedure gw\_radius = 0.9144e-2; %ground wire radius

radius = GMR;

516 MATLAB Applications for the Practical Engineer

10-2

10-1

10<sup>0</sup>

RESISTANCE [ohm]

10<sup>1</sup>

10<sup>2</sup>

10<sup>3</sup>

```
initial_conf
for j = 1:ncond,
xi(j) = xc(j);
yi(j) = -yc(j);
for k = j:ncond
 str = ['Zext' int2str(j) '_' int2str(k) '.m'];
 fid(j,k) = fopen(str,'w');
 if j==k
 str = ['%% External impedance of phase ' int2str(j) ' \n clear x\n x = [\n'];
 fprintf(fid(j,k),str);
 else
 str = ['%% External impedance between phase ' int2str(j) ' and ' int2str(k) ' 
\n clear x\n x = [\n'];
 fprintf(fid(j,k),str);
 end
end
end
%Image conductor coordinates
for j = 1: ncond,
 xi(j) = xc(j);
 yi(j) = -yc(j);
end
%External inductance (H/km)
for j = 1:ncond,
 for k = j:ncond,
 if j == k
 if j < ntpc*3+1
 ld = radius;
 else
 ld = gw_radius;
 end
 else
 dx = xc(j) - xc(k);
 dy = yc(j) - yc(k);
 ld = sqrt(dx^2 + dy^2);
 end
 dx = xcondut(j) - xi(k);
 dy = ycondut(j) - yi(k);
 bd = sqrt(dx^2 + dy^2);
 induct(j,k) = 1000*(1/(2*pi))*mizero*log(dezao/dezinho);
 end
end
for j = 1:length(freq),
 for k = 1:ncond
 for m = k:ncond
 z = i*2*pi*freq(j)*induct(k,m);
 fprintf(fid(k,m), '%30.20f %30.20f\n',real(z),imag(z));
 end
 end
```

```
end
for j = 1:ncond
 for k = j:ncond
 fprintf(fid(j,k), ']; \n');
 str = ['ze' int2str(j) '_' int2str(k) ' = x(:,1) + i*x(:,2);'];
 fprintf(fid(j,k), str);
 end
end
fclose('all');
z_skins.m
%Internal impedance (skin effect)
clear all
'Calculatin Z due to skin effect'
%File reading
initial_conf
fid10 = fopen('zskin.m','w');
fprintf(fid10, '%% Internal impedance \n clear x\n x = [\n');
%===========================================================
%Bessel Formula
%==========================================================
%mi = mizero/1000 ;
%radiuso = raio/1000 ;
for j = 1:length(freq),
 m = sqrt(i*2*pi*freq(j)*mizero*sigma);
 mr = sk_radius * sqrt(i*2*pi*freq(j)*mizero*sigma);
 I0 = BESSELI(0,(mr),1);
 I1 = BESSELI(1,(mr),1);
 %Impedance calculus (number 4 appears because there are 4 subconctors)
 z = (1/4)*1000*((1/sigma)*m)/(2*pi*sk_radius)*(I0/I1);
 fprintf(fid10, '%30.20f %30.20f\n',real(z),imag(z));
end
fprintf(fid10, ']; \n');
fprintf(fid10, 'zskin = x(:,1) + x(:,2)*i;');
fclose('all');
z_carson.m
%Impedance due to earth effect (Carson's method)
clear all
'Calculating Z due to earth effect - Carson's method
%File reading
initial_conf
nt = 120; %amount of terms to be used (number multiple of 4)
v = -1; %variable used for signal
%Image conductors coordinates
```
for j = 1:ncond, xi(j) = xc(j); yi(j) = -yc(j);

for k = j:ncond

fprintf(fid(j,k),str);

fprintf(fid(j,k),str);

if j==k

= [\n'];

 end end end

for j = 1:ncond, for k = j:ncond, dx = xc(j) - xi(k); dy = yc(j) - yi(k);

else

 end end

n = 0;

 end end

end

for j = 1:nt/4, v = -v; for k = 1:4, n = n+1; signal\_b(n) = v;

fid(j,k) = fopen(str,'w');

Earth effect \n clear x\n x = [\n'];

 bd(j,k) = sqrt(dx^2 + dy^2); cat = yc(j) + yc(k); cossine = cat/bd(j,k); ang(j,k) = acos(cossine);

%signal change each 4 terms

%Calculus of bi element b(1) = sqrt(2)/6; b(2) = 1/16; for j = 3:nt,

%Calculus of ci element c(2) = 1.3659315;

str = ['Zcarson' int2str(j) '\_' int2str(k) '.m'];

% Calculus of terms b, c and d of Carson's series

b(j) = abs(b(j-2))\*(1/(j\*(j+2)))\*signal\_b(j);

str = ['%% Phase impedance ' int2str(j) ' due to earth effect \n clear x\n x

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

519

str = ['%% Impedance between phases ' int2str(j) ' and ' int2str(k) ' due to

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

end

 end end

z\_skins.m

clear all

%File reading initial\_conf

%Bessel Formula

end

%mi = mizero/1000 ; %radiuso = raio/1000 ;

for j = 1:length(freq),

 I0 = BESSELI(0,(mr),1); I1 = BESSELI(1,(mr),1);

fprintf(fid10, ']; \n');

fclose('all');

z\_carson.m

clear all

%File reading initial\_conf

m = sqrt(i\*2\*pi\*freq(j)\*mizero\*sigma);

fprintf(fid10, 'zskin = x(:,1) + x(:,2)\*i;');

%Impedance due to earth effect (Carson's method)

v = -1; %variable used for signal

%Image conductors coordinates

'Calculating Z due to earth effect - Carson's method

nt = 120; %amount of terms to be used (number multiple of 4)

mr = sk\_radius \* sqrt(i\*2\*pi\*freq(j)\*mizero\*sigma);

 z = (1/4)\*1000\*((1/sigma)\*m)/(2\*pi\*sk\_radius)\*(I0/I1); fprintf(fid10, '%30.20f %30.20f\n',real(z),imag(z));

for j = 1:ncond for k = j:ncond

518 MATLAB Applications for the Practical Engineer

fclose('all');

fprintf(fid(j,k), ']; \n');

%Internal impedance (skin effect)

'Calculatin Z due to skin effect'

fid10 = fopen('zskin.m','w');

fprintf(fid(j,k), str);

str = ['ze' int2str(j) '\_' int2str(k) ' = x(:,1) + i\*x(:,2);'];

fprintf(fid10, '%% Internal impedance \n clear x\n x = [\n');

%===========================================================

%==========================================================

%Impedance calculus (number 4 appears because there are 4 subconctors)

```
for j = 1:ncond,
 xi(j) = xc(j);
 yi(j) = -yc(j);
 for k = j:ncond
 str = ['Zcarson' int2str(j) '_' int2str(k) '.m'];
 fid(j,k) = fopen(str,'w');
 if j==k
 str = ['%% Phase impedance ' int2str(j) ' due to earth effect \n clear x\n x 
= [\n'];
 fprintf(fid(j,k),str);
 else
 str = ['%% Impedance between phases ' int2str(j) ' and ' int2str(k) ' due to 
Earth effect \n clear x\n x = [\n'];
 fprintf(fid(j,k),str);
 end
 end
end
for j = 1:ncond,
 for k = j:ncond,
 dx = xc(j) - xi(k);
 dy = yc(j) - yi(k);
 bd(j,k) = sqrt(dx^2 + dy^2);
 cat = yc(j) + yc(k);
 cossine = cat/bd(j,k);
 ang(j,k) = acos(cossine);
 end
end
%*****************************************************************************
% Calculus of terms b, c and d of Carson's series
%*****************************************************************************
%signal change each 4 terms
n = 0;
for j = 1:nt/4,
 v = -v;
 for k = 1:4,
 n = n+1; 
 signal_b(n) = v;
 end
end
%Calculus of bi element
b(1) = sqrt(2)/6;
b(2) = 1/16;
for j = 3:nt,
 b(j) = abs(b(j-2))*(1/(j*(j+2)))*signal_b(j);
end
%Calculus of ci element
c(2) = 1.3659315;
```

```
for j = 4:nt,
 c(j) = c(j-2) + (1/j) + (1/(j+2));
end
%Calculus of di element
d = (pi/4)*b;
%*****************************************************************************
for f = 1:length(freq),
 for j = 1:ncond,
 for k = j:ncond,
 phi = ang(j,k);
 a = sqrt(mizero*2*pi*freq(f)/resist)*bd(j,k);
 subrotine_carson_delta_r;
 subrotine_carson_delta_x;
 fprintf(fid(j,k), '%30.20f %30.20f\n',delta_r(j,k), delta_x(j,k));
 end
 end
end
for j = 1:ncond,
 for k = j:ncond,
 fprintf(fid(j,k), ']; \n');
 fprintf(fid(j,k), ['zsolo' int2str(j) '_' int2str(k) ' = x(:,1) + i*x(:,
2);']);
 end
end
fclose('all');
subrotine_carson_delta_r.m
if a < 5,
%'ok'
 r1 = b(1)*a*cos(phi);
 for nj = 1:(nt/4) -1,
 term1 = b(4*nj +1)*(a^(4*nj +1))*cos((4*nj + 1)*phi);
 r1 = term1 + r1;
 end
 parc1 = (c(2) - log(a))*(a^2)*cos(2*phi);
 parc2 = (phi*(a^2)*sin(2*phi));
 r2 = b(2)*(parc1 + parc2);
 for nj = 1:(nt/4) -1,
 parc1 = (c(4*nj +2) - log(a))*(a^(4*nj +2))*cos((4*nj +2)*phi);
 parc2 = (phi*(a^(4*nj +2))*sin((4*nj +2)*phi));
 r2 = r2 + b(4*nj +2)*(parc1 + parc2);
 end
 r3 = b(3)*(a^3)*cos(3*phi);
 for nj = 1:(nt/4) -1,
 term3 = b(4*nj + 3)*(a^(4*nj + 3))*cos((4*nj + 3)*phi);
 r3 = r3 + term3;
 end
 r4 = d(4)*(a^4)*cos(4*phi);
 for nj = 1:(nt/4) -1,
 term4 = d(4*nj + 4)*(a^(4*nj + 4))*cos((4*nj + 4)*phi);
```
r4 = r4 + term4;

t1 = cos(phi)/a;

 t3 = cos(3\*phi)/(a^3); t4 = 3\*cos(5\*phi)/(a^5); t5 = 5\*cos(7\*phi)/(a^7);

subrotine\_carson\_delta\_r.m

 x2 = d(2)\*(a^2)\*cos(2\*phi); for nj = 1:(nt/4) -1,

 x3 = b(3)\*(a^3)\*cos(3\*phi); for nj = 1:(nt/4) -1,

(phi)\*(a^(4\*nj + 4))\*sin((4\*nj + 4)\*phi);

note that the routines are simple to implement and to understand.

x4 = x4 + b(4\*nj + 4)\*term4;

t2 = sqrt(2)\*cos(2\*phi)/(a^2);

 x1 = b(1)\*a\*cos(phi); for nj = 1:(nt/4) -1,

x1 = term1 + x1;

x2 = x2 + term2;

x3 = x3 + term3;

 x4 = b(4)\*term4; for nj = 1:(nt/4) -1,

t1 = cos(phi)/a;

 t3 = cos(3\*phi)/(a^3); t4 = 3\*cos(5\*phi)/(a^5); t5 = 5\*cos(7\*phi)/(a^7);

t2 = sqrt(2)\*cos(2\*phi)/(a^2);

delta\_r(j,k) = 4\*2\*pi\*freq(f)\*(1e-4)\*((pi/8) - r1 + r2 + r3 - r4);

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

521

delta\_r(j,k) = (4\*2\*pi\*freq(f)\*(1e-4)/sqrt(2))\*(t1 -t2 + t3 +t4 +t5);

%Carson's series to calculate reactance of conductors due to Earth effect

term1 = b(4\*nj +1)\*(a^(4\*nj +1))\*cos((4\*nj + 1)\*phi);

term2 = d(4\*nj + 2)\*(a^(4\*nj + 2))\*cos((4\*nj + 2)\*phi);

term3 = b(4\*nj + 3)\*(a^(4\*nj + 3))\*cos((4\*nj + 3)\*phi);

term4 = (c(4) - log(a))\*(a^4)\*cos(4\*phi) + (phi)\*(a^4)\*sin(4\*phi);

delta\_x(j,k) = (4\*2\*pi\*freq(f)\*(1e-4)/sqrt(2))\*(t1 - t3 + t4 + t5);

The theoretical procedure of routines presented above can be fully found in [29]-[30]. The above routines show how useful Matlab is in order to perform calculus and link routines. It's easy to

term4 = (c(4\*nj + 4) - log(a))\*(a^(4\*nj + 4))\*cos((4\*nj + 4)\*phi) +

delta\_x(j,k) = 4\*2\*pi\*freq(f)\*(1e-4)\*(0.5\*(0.6159315 - log(a)) + x1 - x2 + x3

end

else

end

if a < 5,

end

end

end

end


end

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires http://dx.doi.org/10.5772/58383 521

for j = 4:nt,

520 MATLAB Applications for the Practical Engineer

d = (pi/4)\*b;

%Calculus of di element

for f = 1:length(freq), for j = 1:ncond, for k = j:ncond, phi = ang(j,k);

end

 end end end

2);']); end end

if a < 5, %'ok'

end

end

end

for j = 1:ncond, for k = j:ncond,

fclose('all');

fprintf(fid(j,k), ']; \n');

subrotine\_carson\_delta\_r.m

 r1 = b(1)\*a\*cos(phi); for nj = 1:(nt/4) -1,

r1 = term1 + r1;

r3 = r3 + term3;

c(j) = c(j-2) + (1/j) + (1/(j+2));

a = sqrt(mizero\*2\*pi\*freq(f)/resist)\*bd(j,k);

term1 = b(4\*nj +1)\*(a^(4\*nj +1))\*cos((4\*nj + 1)\*phi);

parc1 = (c(4\*nj +2) - log(a))\*(a^(4\*nj +2))\*cos((4\*nj +2)\*phi);

term3 = b(4\*nj + 3)\*(a^(4\*nj + 3))\*cos((4\*nj + 3)\*phi);

term4 = d(4\*nj + 4)\*(a^(4\*nj + 4))\*cos((4\*nj + 4)\*phi);

parc1 = (c(2) - log(a))\*(a^2)\*cos(2\*phi);

r2 = r2 + b(4\*nj +2)\*(parc1 + parc2);

parc2 = (phi\*(a^(4\*nj +2))\*sin((4\*nj +2)\*phi));

 parc2 = (phi\*(a^2)\*sin(2\*phi)); r2 = b(2)\*(parc1 + parc2); for nj = 1:(nt/4) -1,

 r3 = b(3)\*(a^3)\*cos(3\*phi); for nj = 1:(nt/4) -1,

 r4 = d(4)\*(a^4)\*cos(4\*phi); for nj = 1:(nt/4) -1,

 subrotine\_carson\_delta\_r; subrotine\_carson\_delta\_x;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

fprintf(fid(j,k), '%30.20f %30.20f\n',delta\_r(j,k), delta\_x(j,k));

fprintf(fid(j,k), ['zsolo' int2str(j) '\_' int2str(k) ' = x(:,1) + i\*x(:,

```
 r4 = r4 + term4;
 end
 delta_r(j,k) = 4*2*pi*freq(f)*(1e-4)*((pi/8) - r1 + r2 + r3 - r4);
else
 t1 = cos(phi)/a;
 t2 = sqrt(2)*cos(2*phi)/(a^2);
 t3 = cos(3*phi)/(a^3);
 t4 = 3*cos(5*phi)/(a^5);
 t5 = 5*cos(7*phi)/(a^7);
 delta_r(j,k) = (4*2*pi*freq(f)*(1e-4)/sqrt(2))*(t1 -t2 + t3 +t4 +t5);
end
subrotine_carson_delta_r.m
%Carson's series to calculate reactance of conductors due to Earth effect
if a < 5,
 x1 = b(1)*a*cos(phi);
 for nj = 1:(nt/4) -1,
 term1 = b(4*nj +1)*(a^(4*nj +1))*cos((4*nj + 1)*phi);
 x1 = term1 + x1;
 end
 x2 = d(2)*(a^2)*cos(2*phi);
 for nj = 1:(nt/4) -1,
 term2 = d(4*nj + 2)*(a^(4*nj + 2))*cos((4*nj + 2)*phi);
 x2 = x2 + term2;
 end
 x3 = b(3)*(a^3)*cos(3*phi);
 for nj = 1:(nt/4) -1,
 term3 = b(4*nj + 3)*(a^(4*nj + 3))*cos((4*nj + 3)*phi);
 x3 = x3 + term3;
 end
 term4 = (c(4) - log(a))*(a^4)*cos(4*phi) + (phi)*(a^4)*sin(4*phi);
 x4 = b(4)*term4;
 for nj = 1:(nt/4) -1,
 term4 = (c(4*nj + 4) - log(a))*(a^(4*nj + 4))*cos((4*nj + 4)*phi) + 
(phi)*(a^(4*nj + 4))*sin((4*nj + 4)*phi);
 x4 = x4 + b(4*nj + 4)*term4;
 end
 delta_x(j,k) = 4*2*pi*freq(f)*(1e-4)*(0.5*(0.6159315 - log(a)) + x1 - x2 + x3 
else
 t1 = cos(phi)/a;
 t2 = sqrt(2)*cos(2*phi)/(a^2);
 t3 = cos(3*phi)/(a^3);
 t4 = 3*cos(5*phi)/(a^5);
 t5 = 5*cos(7*phi)/(a^7);
 delta_x(j,k) = (4*2*pi*freq(f)*(1e-4)/sqrt(2))*(t1 - t3 + t4 + t5);
end
```
The theoretical procedure of routines presented above can be fully found in [29]-[30]. The above routines show how useful Matlab is in order to perform calculus and link routines. It's easy to note that the routines are simple to implement and to understand.

The principal point to be observed is that all the data generated by the routines are stored into m-files to be used by other routines. To perform this action, first it's necessary to open a file (if doesn't exist, it'll be automatically created) and link with a variable, for this operation, function *fopen* shall be used, the arguments of this function are the name of file and the kind of action to be performed by the file, in this case, *w* was used in order to *write* in the file. To write in the file, the function *fprintf* must be used, the arguments are the name of file, the text (which can be variables of decimal point (%d), float point (%f), and so forth, followed by the variables name. Finally, it's necessary to close the file, with the function *fclose*. In this case the argument *all* is used in order to close all opened files.

pot(k,j) = pot(j,k);

cap = 2\*pi\*epslonzero\*(inv(pot));

fprintf(fid10, '%30.20f ',cap(j,k));

A first analysis is based on Eq. (9). Through iterative process the exact eigenvectors and eigenvalues, and also the eigenvectors and eigenvalues, from Clark`s matrix are calculated. At

eigenvalues, and also the eigenvectors and eigenvalues, from Clark`s matrix are calculated. At

(a) (b) Fig. 4 Relative differences between the exact modes and the quasi-modes.

RELATIVE DIFERENCE (%)

the relative difference of mode γ is high (Fig. 4 (b)). To minimize the error shown for mode 4, a correction procedure for non-transposed three-phase transmission line cases [31]-[35] shall be

The relative difference between modes α, β, 0 and δ is relatively low (Fig. 4 (a)), however, the relative difference of mode γ is high (Fig. 4 (b)). To minimize the error shown for mode 4, a correction procedure for non-transposed three-phase transmission line cases [31]-[35] shall be

In order to verify the limits of this method, a frequency range from 10 *Hz* to 1 *GHz* was applied, from the results of resistance, inductance and capacitance obtained, it was verified that the method could converge until 1 *MHz* , after this range the method is not valid and

applied, from the results of resistance, inductance and capacitance obtained, it was verified that the method could converge until 1 [���], after this range the method is not valid and need a new

The relative difference between modes α, β, 0 and δ is relatively low (Fig. 4 (a)), however,

In order to verify the limits of this method, a frequency range from 10 [��] to 1 [���] was

The numeric routine used to obtain the results shown above was developed with Matlab and

A first analysis is based on Eq. (9). Through iterative process the exact eigenvectors and

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

*<sup>λ</sup>ex* <sup>⋅</sup><sup>100</sup> (10)

∙ 100 (10)

http://dx.doi.org/10.5772/58383

523

γ

<sup>10</sup><sup>1</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>3</sup> <sup>10</sup><sup>4</sup> <sup>10</sup><sup>5</sup> <sup>10</sup><sup>6</sup> <sup>0</sup>

FREQUENCY [Hz]

the end of process the relative difference is calculated for each mode with Eq. (10).

the end of process the relative difference is calculated for each mode with Eq. (10).

�

*err*(%)= *<sup>λ</sup>cl* - *<sup>λ</sup>ex*

����%� <sup>=</sup> � <sup>−</sup> �

 end end

end

end

used in a future work.

used in a future work.


RELATIVE DIFERENCE (%)

approach will be requested.

need a new approach will be requested.

is described below in details: evaluation\_routine.m

clear all; Zfull

capacitance; %Clarke's matrix

for j = 1:ncond, for k = 1:ncond,

fclose('all');

α δ

0

β

fprintf(fid10, '\n');

fprintf(fid10, ']; \n');

*Evaluation of proposed real transformation matrix*

<sup>10</sup><sup>1</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>3</sup> <sup>10</sup><sup>4</sup> <sup>10</sup><sup>5</sup> <sup>10</sup><sup>6</sup> -2.5

FREQUENCY [Hz]

**Figure 4.** Relative differences between the exact modes and the quasi-modes.

Evaluation of proposed real transformation matrix

As the procedure for calculus in all impedances cases shown above are the same, a lasso function can be used in order to make the routine shorter. Thus, structure *for* is implemented together structure *if* in order to make a loop and to decide what kind of operation shall be performed.

In the *z\_skins.m* file it's noted the use of Bessel function; this is a special function which can be found in any advanced mathematical calculus. This function is used in order to calculate the impedance due to skin effect, as described in [29].

```
calc_capacitance.m
clear all
'Calculating capacitances'
%File reading
initial_conf
fid10 = fopen('capacitance.m','w');
fprintf(fid10, '%% Capacitances \n clear x\n cap = [\n');
%Image conductors coordinates
for j = 1: ncond,
 xi(j) = xc(j);
 yi(j) = -yc(j);
end
%Potential matrix coefficients
for j = 1:ncond,
 for k = j:ncond,
 if j == k
 if j < ntpc*3+1
 ld = radius;
 else
 ld = gw_radius;
 end
 else
 dx = xc(j) - xc(k);
 dy = yc(j) - yc(k);
 ld = sqrt(dx^2 + dy^2);
 end
 dx = xc(j) - xi(k);
 dy = yc(j) - yi(k);
 bd = sqrt(dx^2 + dy^2);
 pot(j,k) = log(bd/ld);
```

```
 pot(k,j) = pot(j,k);
 end
end
cap = 2*pi*epslonzero*(inv(pot));
for j = 1:ncond,
 for k = 1:ncond,
 fprintf(fid10, '%30.20f ',cap(j,k));
 end
 fprintf(fid10, '\n');
end
fprintf(fid10, ']; \n');
fclose('all');
```
#### *Evaluation of proposed real transformation matrix* Evaluation of proposed real transformation matrix

The principal point to be observed is that all the data generated by the routines are stored into m-files to be used by other routines. To perform this action, first it's necessary to open a file (if doesn't exist, it'll be automatically created) and link with a variable, for this operation, function *fopen* shall be used, the arguments of this function are the name of file and the kind of action to be performed by the file, in this case, *w* was used in order to *write* in the file. To write in the file, the function *fprintf* must be used, the arguments are the name of file, the text (which can be variables of decimal point (%d), float point (%f), and so forth, followed by the variables name. Finally, it's necessary to close the file, with the function *fclose*. In this case the argument

As the procedure for calculus in all impedances cases shown above are the same, a lasso function can be used in order to make the routine shorter. Thus, structure *for* is implemented together structure *if* in order to make a loop and to decide what kind of operation shall be

In the *z\_skins.m* file it's noted the use of Bessel function; this is a special function which can be found in any advanced mathematical calculus. This function is used in order to calculate the

*all* is used in order to close all opened files.

522 MATLAB Applications for the Practical Engineer

impedance due to skin effect, as described in [29].

'Calculating capacitances'

%Image conductors coordinates

%Potential matrix coefficients

if j < ntpc\*3+1

dx = xc(j) - xc(k);

 dx = xc(j) - xi(k); dy = yc(j) - yi(k); bd = sqrt(dx^2 + dy^2); pot(j,k) = log(bd/ld);

 dy = yc(j) - yc(k); ld = sqrt(dx^2 + dy^2);

fid10 = fopen('capacitance.m','w');

fprintf(fid10, '%% Capacitances \n clear x\n cap = [\n');

calc\_capacitance.m

for j = 1: ncond, xi(j) = xc(j); yi(j) = -yc(j);

for j = 1:ncond, for k = j:ncond, if j == k

 ld = radius; else ld = gw\_radius;

clear all

end

 end else

end

%File reading initial\_conf

performed.

A first analysis is based on Eq. (9). Through iterative process the exact eigenvectors and eigenvalues, and also the eigenvectors and eigenvalues, from Clark`s matrix are calculated. At the end of process the relative difference is calculated for each mode with Eq. (10). A first analysis is based on Eq. (9). Through iterative process the exact eigenvectors and eigenvalues, and also the eigenvectors and eigenvalues, from Clark`s matrix are calculated. At

the end of process the relative difference is calculated for each mode with Eq. (10).

**Figure 4.** Relative differences between the exact modes and the quasi-modes.

approach will be requested.

is described below in details: evaluation\_routine.m

clear all; Zfull

capacitance; %Clarke's matrix

The relative difference between modes α, β, 0 and δ is relatively low (Fig. 4 (a)), however, the relative difference of mode γ is high (Fig. 4 (b)). To minimize the error shown for mode 4, a correction procedure for non-transposed three-phase transmission line cases [31]-[35] shall be The relative difference between modes α, β, 0 and δ is relatively low (Fig. 4 (a)), however, the relative difference of mode γ is high (Fig. 4 (b)). To minimize the error shown for mode 4, a correction procedure for non-transposed three-phase transmission line cases [31]-[35] shall be used in a future work.

Fig. 4 Relative differences between the exact modes and the quasi-modes.

used in a future work. In order to verify the limits of this method, a frequency range from 10 [��] to 1 [���] was applied, from the results of resistance, inductance and capacitance obtained, it was verified that In order to verify the limits of this method, a frequency range from 10 *Hz* to 1 *GHz* was applied, from the results of resistance, inductance and capacitance obtained, it was verified that the method could converge until 1 *MHz* , after this range the method is not valid and need a new approach will be requested.

the method could converge until 1 [���], after this range the method is not valid and need a new

The numeric routine used to obtain the results shown above was developed with Matlab and

The numeric routine used to obtain the results shown above was developed with Matlab and is described below in details:

end

 end end end

 end end

z\_skins.m

clear all

%File reading initial\_conf

%Bessel Formula

end

%mi = mizero/1000 ; %radiuso = raio/1000 ;

for j = 1:length(freq),

 I0 = BESSELI(0,(mr),1); I1 = BESSELI(1,(mr),1);

fprintf(fid10, ']; \n');

fclose('all');

z\_carson.m

clear all

m = sqrt(i\*2\*pi\*freq(j)\*mizero\*sigma);

fprintf(fid10, 'zskin = x(:,1) + x(:,2)\*i;');

%Impedance due to earth effect (Carson's method)

mr = sk\_radius \* sqrt(i\*2\*pi\*freq(j)\*mizero\*sigma);

 z = (1/4)\*1000\*((1/sigma)\*m)/(2\*pi\*sk\_radius)\*(I0/I1); fprintf(fid10, '%30.20f %30.20f\n',real(z),imag(z));

for j = 1:ncond for k = j:ncond

fclose('all');

for j = 1:length(freq), for k = 1:ncond for m = k:ncond

fprintf(fid(j,k), ']; \n');

%Internal impedance (skin effect)

'Calculatin Z due to skin effect'

fid10 = fopen('zskin.m','w');

fprintf(fid(j,k), str);

z = i\*2\*pi\*freq(j)\*induct(k,m);

fprintf(fid(k,m), '%30.20f %30.20f\n',real(z),imag(z));

str = ['ze' int2str(j) '\_' int2str(k) ' = x(:,1) + i\*x(:,2);'];

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

525

fprintf(fid10, '%% Internal impedance \n clear x\n x = [\n');

%===========================================================

%==========================================================

%Impedance calculus (number 4 appears because there are 4 subconctors)

```
external_impedance.m
clear all
'Calculating Z due to external effect'
%File Reading (calling subroutine of initial configuration)
initial_conf
for j = 1:ncond,
xi(j) = xc(j);
yi(j) = -yc(j);
for k = j:ncond
 str = ['Zext' int2str(j) '_' int2str(k) '.m'];
 fid(j,k) = fopen(str,'w');
 if j==k
 str = ['%% External impedance of phase ' int2str(j) ' \n clear x\n x = [\n'];
 fprintf(fid(j,k),str);
 else
 str = ['%% External impedance between phase ' int2str(j) ' and ' int2str(k) ' 
\n clear x\n x = [\n'];
 fprintf(fid(j,k),str);
 end
end
end
%Image conductor coordinates
for j = 1: ncond,
 xi(j) = xc(j);
 yi(j) = -yc(j);
end
%External inductance (H/km)
for j = 1:ncond,
 for k = j:ncond,
 if j == k
 if j < ntpc*3+1
 ld = radius;
 else
 ld = gw_radius;
 end
 else
 dx = xc(j) - xc(k);
 dy = yc(j) - yc(k);
 ld = sqrt(dx^2 + dy^2);
 end
 dx = xcondut(j) - xi(k);
 dy = ycondut(j) - yi(k);
 bd = sqrt(dx^2 + dy^2);
 induct(j,k) = 1000*(1/(2*pi))*mizero*log(dezao/dezinho);
 end
```
The numeric routine used to obtain the results shown above was developed with Matlab and

str = ['%% External impedance of phase ' int2str(j) ' \n clear x\n x = [\n'];

str = ['%% External impedance between phase ' int2str(j) ' and ' int2str(k) '

%File Reading (calling subroutine of initial configuration)

is described below in details:

524 MATLAB Applications for the Practical Engineer

clear all

if j==k

else

 end end end

end

initial\_conf

for j = 1:ncond, xi(j) = xc(j); yi(j) = -yc(j); for k = j:ncond

fid(j,k) = fopen(str,'w');

fprintf(fid(j,k),str);

\n clear x\n x = [\n']; fprintf(fid(j,k),str);

%Image conductor coordinates

%External inductance (H/km)

if j < ntpc\*3+1

dx = xc(j) - xc(k);

 dx = xcondut(j) - xi(k); dy = ycondut(j) - yi(k); bd = sqrt(dx^2 + dy^2);

induct(j,k) = 1000\*(1/(2\*pi))\*mizero\*log(dezao/dezinho);

for j = 1: ncond, xi(j) = xc(j); yi(j) = -yc(j);

for j = 1:ncond, for k = j:ncond, if j == k

ld = radius;

ld = gw\_radius;

 dy = yc(j) - yc(k); ld = sqrt(dx^2 + dy^2);

else

 end else

end

end

external\_impedance.m

'Calculating Z due to external effect'

str = ['Zext' int2str(j) '\_' int2str(k) '.m'];

```
end
for j = 1:length(freq),
 for k = 1:ncond
 for m = k:ncond
 z = i*2*pi*freq(j)*induct(k,m);
 fprintf(fid(k,m), '%30.20f %30.20f\n',real(z),imag(z));
 end
 end
end
for j = 1:ncond
 for k = j:ncond
 fprintf(fid(j,k), ']; \n');
 str = ['ze' int2str(j) '_' int2str(k) ' = x(:,1) + i*x(:,2);'];
 fprintf(fid(j,k), str);
 end
end
fclose('all');
z_skins.m
%Internal impedance (skin effect)
clear all
'Calculatin Z due to skin effect'
%File reading
initial_conf
fid10 = fopen('zskin.m','w');
fprintf(fid10, '%% Internal impedance \n clear x\n x = [\n');
%===========================================================
%Bessel Formula
%==========================================================
%mi = mizero/1000 ;
%radiuso = raio/1000 ;
for j = 1:length(freq),
 m = sqrt(i*2*pi*freq(j)*mizero*sigma);
 mr = sk_radius * sqrt(i*2*pi*freq(j)*mizero*sigma);
 I0 = BESSELI(0,(mr),1);
 I1 = BESSELI(1,(mr),1);
 %Impedance calculus (number 4 appears because there are 4 subconctors)
 z = (1/4)*1000*((1/sigma)*m)/(2*pi*sk_radius)*(I0/I1);
 fprintf(fid10, '%30.20f %30.20f\n',real(z),imag(z));
end
fprintf(fid10, ']; \n');
fprintf(fid10, 'zskin = x(:,1) + x(:,2)*i;');
fclose('all');
z_carson.m
%Impedance due to earth effect (Carson's method)
clear all
```

```
'Calculating Z due to earth effect - Carson's method
%File reading
initial_conf
nt = 120; %amount of terms to be used (number multiple of 4)
v = -1; %variable used for signal
%Image conductors coordinates
for j = 1:ncond,
 xi(j) = xc(j);
 yi(j) = -yc(j);
 for k = j:ncond
 str = ['Zcarson' int2str(j) '_' int2str(k) '.m'];
 fid(j,k) = fopen(str,'w');
 if j==k
 str = ['%% Phase impedance ' int2str(j) ' due to earth effect \n clear x\n x 
= [\n'];
 fprintf(fid(j,k),str);
 else
 str = ['%% Impedance between phases ' int2str(j) ' and ' int2str(k) ' due to 
Earth effect \n clear x\n x = [\n'];
 fprintf(fid(j,k),str);
 end
 end
end
for j = 1:ncond,
 for k = j:ncond,
 dx = xc(j) - xi(k);
 dy = yc(j) - yi(k);
 bd(j,k) = sqrt(dx^2 + dy^2);
 cat = yc(j) + yc(k);
 cossine = cat/bd(j,k);
 ang(j,k) = acos(cossine);
 end
end
%*****************************************************************************
% Calculus of terms b, c and d of Carson's series
%*****************************************************************************
%signal change each 4 terms
n = 0;
for j = 1:nt/4,
 v = -v;
 for k = 1:4,
 n = n+1; 
 signal_b(n) = v;
 end
end
%Calculus of bi element
```
b(1) = sqrt(2)/6; b(2) = 1/16; for j = 3:nt,

%Calculus of ci element c(2) = 1.3659315; for j = 4:nt,

%Calculus of di element

for f = 1:length(freq), for j = 1:ncond, for k = j:ncond, phi = ang(j,k);

d = (pi/4)\*b;

end

end

 end end end

2);']); end end

if a < 5, %'ok'

end

end

for j = 1:ncond, for k = j:ncond,

fclose('all');

fprintf(fid(j,k), ']; \n');

subrotine\_carson\_delta\_r.m

 r1 = b(1)\*a\*cos(phi); for nj = 1:(nt/4) -1,

r1 = term1 + r1;

b(j) = abs(b(j-2))\*(1/(j\*(j+2)))\*signal\_b(j);

a = sqrt(mizero\*2\*pi\*freq(f)/resist)\*bd(j,k);

term1 = b(4\*nj +1)\*(a^(4\*nj +1))\*cos((4\*nj + 1)\*phi);

parc1 = (c(4\*nj +2) - log(a))\*(a^(4\*nj +2))\*cos((4\*nj +2)\*phi);

parc1 = (c(2) - log(a))\*(a^2)\*cos(2\*phi);

r2 = r2 + b(4\*nj +2)\*(parc1 + parc2);

parc2 = (phi\*(a^(4\*nj +2))\*sin((4\*nj +2)\*phi));

 parc2 = (phi\*(a^2)\*sin(2\*phi)); r2 = b(2)\*(parc1 + parc2); for nj = 1:(nt/4) -1,

 subrotine\_carson\_delta\_r; subrotine\_carson\_delta\_x;

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

527

fprintf(fid(j,k), '%30.20f %30.20f\n',delta\_r(j,k), delta\_x(j,k));

fprintf(fid(j,k), ['zsolo' int2str(j) '\_' int2str(k) ' = x(:,1) + i\*x(:,

c(j) = c(j-2) + (1/j) + (1/(j+2));

'Calculating Z due to earth effect - Carson's method

str = ['Zcarson' int2str(j) '\_' int2str(k) '.m'];

% Calculus of terms b, c and d of Carson's series

v = -1; %variable used for signal

fid(j,k) = fopen(str,'w');

Earth effect \n clear x\n x = [\n'];

 bd(j,k) = sqrt(dx^2 + dy^2); cat = yc(j) + yc(k); cossine = cat/bd(j,k); ang(j,k) = acos(cossine);

%signal change each 4 terms

%Calculus of bi element

%Image conductors coordinates

nt = 120; %amount of terms to be used (number multiple of 4)

str = ['%% Phase impedance ' int2str(j) ' due to earth effect \n clear x\n x

str = ['%% Impedance between phases ' int2str(j) ' and ' int2str(k) ' due to

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

%File reading initial\_conf

526 MATLAB Applications for the Practical Engineer

for j = 1:ncond, xi(j) = xc(j); yi(j) = -yc(j);

for k = j:ncond

fprintf(fid(j,k),str);

fprintf(fid(j,k),str);

if j==k

= [\n'];

 end end end

for j = 1:ncond, for k = j:ncond, dx = xc(j) - xi(k); dy = yc(j) - yi(k);

else

 end end

n = 0;

 end end

for j = 1:nt/4, v = -v; for k = 1:4, n = n+1; signal\_b(n) = v;

```
b(1) = sqrt(2)/6;
b(2) = 1/16;
for j = 3:nt,
 b(j) = abs(b(j-2))*(1/(j*(j+2)))*signal_b(j);
end
%Calculus of ci element
c(2) = 1.3659315;
for j = 4:nt,
 c(j) = c(j-2) + (1/j) + (1/(j+2));
end
%Calculus of di element
d = (pi/4)*b;
%*****************************************************************************
for f = 1:length(freq),
 for j = 1:ncond,
 for k = j:ncond,
 phi = ang(j,k);
 a = sqrt(mizero*2*pi*freq(f)/resist)*bd(j,k);
 subrotine_carson_delta_r;
 subrotine_carson_delta_x;
 fprintf(fid(j,k), '%30.20f %30.20f\n',delta_r(j,k), delta_x(j,k));
 end
 end
end
for j = 1:ncond,
 for k = j:ncond,
 fprintf(fid(j,k), ']; \n');
 fprintf(fid(j,k), ['zsolo' int2str(j) '_' int2str(k) ' = x(:,1) + i*x(:,
2);']);
 end
end
fclose('all');
subrotine_carson_delta_r.m
if a < 5,
%'ok'
 r1 = b(1)*a*cos(phi);
 for nj = 1:(nt/4) -1,
 term1 = b(4*nj +1)*(a^(4*nj +1))*cos((4*nj + 1)*phi);
 r1 = term1 + r1;
 end
 parc1 = (c(2) - log(a))*(a^2)*cos(2*phi);
 parc2 = (phi*(a^2)*sin(2*phi));
 r2 = b(2)*(parc1 + parc2);
 for nj = 1:(nt/4) -1,
 parc1 = (c(4*nj +2) - log(a))*(a^(4*nj +2))*cos((4*nj +2)*phi);
 parc2 = (phi*(a^(4*nj +2))*sin((4*nj +2)*phi));
 r2 = r2 + b(4*nj +2)*(parc1 + parc2);
 end
```

```
 r3 = b(3)*(a^3)*cos(3*phi);
 for nj = 1:(nt/4) -1,
 term3 = b(4*nj + 3)*(a^(4*nj + 3))*cos((4*nj + 3)*phi);
 r3 = r3 + term3;
 end
 r4 = d(4)*(a^4)*cos(4*phi);
 for nj = 1:(nt/4) -1,
 term4 = d(4*nj + 4)*(a^(4*nj + 4))*cos((4*nj + 4)*phi);
 r4 = r4 + term4;
 end
 delta_r(j,k) = 4*2*pi*freq(f)*(1e-4)*((pi/8) - r1 + r2 + r3 - r4);
else
 t1 = cos(phi)/a;
 t2 = sqrt(2)*cos(2*phi)/(a^2);
 t3 = cos(3*phi)/(a^3);
 t4 = 3*cos(5*phi)/(a^5);
 t5 = 5*cos(7*phi)/(a^7);
 delta_r(j,k) = (4*2*pi*freq(f)*(1e-4)/sqrt(2))*(t1 -t2 + t3 +t4 +t5);
end
subrotine_carson_delta_r.m
%Carson's series to calculate reactance of conductors due to Earth effect
if a < 5,
 x1 = b(1)*a*cos(phi);
 for nj = 1:(nt/4) -1,
 term1 = b(4*nj +1)*(a^(4*nj +1))*cos((4*nj + 1)*phi);
 x1 = term1 + x1;
 end
 x2 = d(2)*(a^2)*cos(2*phi);
 for nj = 1:(nt/4) -1,
 term2 = d(4*nj + 2)*(a^(4*nj + 2))*cos((4*nj + 2)*phi);
 x2 = x2 + term2;
 end
 x3 = b(3)*(a^3)*cos(3*phi);
 for nj = 1:(nt/4) -1,
 term3 = b(4*nj + 3)*(a^(4*nj + 3))*cos((4*nj + 3)*phi);
 x3 = x3 + term3;
 end
 term4 = (c(4) - log(a))*(a^4)*cos(4*phi) + (phi)*(a^4)*sin(4*phi);
 x4 = b(4)*term4;
 for nj = 1:(nt/4) -1,
 term4 = (c(4*nj + 4) - log(a))*(a^(4*nj + 4))*cos((4*nj + 4)*phi) + 
(phi)*(a^(4*nj + 4))*sin((4*nj + 4)*phi);
 x4 = x4 + b(4*nj + 4)*term4;
 end
 delta_x(j,k) = 4*2*pi*freq(f)*(1e-4)*(0.5*(0.6159315 - log(a)) + x1 - x2 + x3 
else
 t1 = cos(phi)/a;
 t2 = sqrt(2)*cos(2*phi)/(a^2);
 t3 = cos(3*phi)/(a^3);
 t4 = 3*cos(5*phi)/(a^5);
```
t5 = 5\*cos(7\*phi)/(a^7);

method it performs calculus as shown before.

end

**7. Conclusion**

delta\_x(j,k) = (4\*2\*pi\*freq(f)\*(1e-4)/sqrt(2))\*(t1 - t3 + t4 + t5);

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

529

The above routines show the procedure to calculate the correct and proposed method values. The first routine call subroutines capacitance and Zfull (which is the sum of impedances shown before), thus for the correct value it uses the eigenvalue function (*eig*) and for the proposed

The second routine gets all processed data and plot the information considering the relative

The objective of this project was to analyze the application of modal transformation matrix that is independent of frequency in analyses of three-phase lines considering the presence of 2 ground wires. Through analysis, both the limits of this approach and the possible errors in

The model proposed in this project uses approximate modal transformation, accomplished through a transformation matrix independent of frequency. This matrix is obtained by linear combination of elements of Clarke's matrix. With application of this transformation matrix independent of frequency, it obtains diagonal matrices for the cases of transposed three-phase lines. For non-transposed three-phase lines, matrices of parameters are not diagonal with the application of the transformation matrix mentioned. For those cases not implemented, the proposal is to analyze the relative errors obtained by establishing circumstances in which one

This chapter presented a method that can be used for analyzing electromagnetic transients using real transformation matrices in three-phase systems considering the presence of ground wires. This method was implemented using Matlab, and then the routines used to develop it were presented and commented. The proposal analyzed used a real transformation matrix for the entire frequency range considered in this case. For those elements related to the phases of the considered system, the transformation matrix was composed of the elements of Clarke's matrix. In part related to the ground wires, the elements of the transformation matrix had to establish a relationship with the elements of the phases considering the establishment of a single homopolar reference in the mode domain. In the case of three-phase lines with the presence of two ground wires, it was unable to get the full diagonalization of the matrices Z and Y in the mode domain. The relative errors between the proposed routine and the correct values of eigenvalues were shown by graphs plotted using Matlab. Thus, for a future work, a correction routine will be used for non-transposed three-phase transmission line cases for the

difference between the correct value and the proposed one as could be seen in Fig. 4.

relation to the exact values obtained from eigenvalues and eigenvectors.

can use a transformation matrix independent of frequency.

transformation matrix presented.

```
 t5 = 5*cos(7*phi)/(a^7);
 delta_x(j,k) = (4*2*pi*freq(f)*(1e-4)/sqrt(2))*(t1 - t3 + t4 + t5);
end
```
The above routines show the procedure to calculate the correct and proposed method values. The first routine call subroutines capacitance and Zfull (which is the sum of impedances shown before), thus for the correct value it uses the eigenvalue function (*eig*) and for the proposed method it performs calculus as shown before.

The second routine gets all processed data and plot the information considering the relative difference between the correct value and the proposed one as could be seen in Fig. 4.

#### **7. Conclusion**

 r3 = b(3)\*(a^3)\*cos(3\*phi); for nj = 1:(nt/4) -1,

 r4 = d(4)\*(a^4)\*cos(4\*phi); for nj = 1:(nt/4) -1,

t2 = sqrt(2)\*cos(2\*phi)/(a^2);

r3 = r3 + term3;

528 MATLAB Applications for the Practical Engineer

r4 = r4 + term4;

t1 = cos(phi)/a;

 t3 = cos(3\*phi)/(a^3); t4 = 3\*cos(5\*phi)/(a^5); t5 = 5\*cos(7\*phi)/(a^7);

subrotine\_carson\_delta\_r.m

 x2 = d(2)\*(a^2)\*cos(2\*phi); for nj = 1:(nt/4) -1,

 x3 = b(3)\*(a^3)\*cos(3\*phi); for nj = 1:(nt/4) -1,

(phi)\*(a^(4\*nj + 4))\*sin((4\*nj + 4)\*phi);

x4 = x4 + b(4\*nj + 4)\*term4;

t2 = sqrt(2)\*cos(2\*phi)/(a^2);

 x1 = b(1)\*a\*cos(phi); for nj = 1:(nt/4) -1,

x1 = term1 + x1;

x2 = x2 + term2;

x3 = x3 + term3;

 x4 = b(4)\*term4; for nj = 1:(nt/4) -1,

t1 = cos(phi)/a;

 t3 = cos(3\*phi)/(a^3); t4 = 3\*cos(5\*phi)/(a^5);

end

end

else

end

if a < 5,

end

end

end

end


term3 = b(4\*nj + 3)\*(a^(4\*nj + 3))\*cos((4\*nj + 3)\*phi);

term4 = d(4\*nj + 4)\*(a^(4\*nj + 4))\*cos((4\*nj + 4)\*phi);

term1 = b(4\*nj +1)\*(a^(4\*nj +1))\*cos((4\*nj + 1)\*phi);

term2 = d(4\*nj + 2)\*(a^(4\*nj + 2))\*cos((4\*nj + 2)\*phi);

term3 = b(4\*nj + 3)\*(a^(4\*nj + 3))\*cos((4\*nj + 3)\*phi);

term4 = (c(4) - log(a))\*(a^4)\*cos(4\*phi) + (phi)\*(a^4)\*sin(4\*phi);

term4 = (c(4\*nj + 4) - log(a))\*(a^(4\*nj + 4))\*cos((4\*nj + 4)\*phi) +

delta\_x(j,k) = 4\*2\*pi\*freq(f)\*(1e-4)\*(0.5\*(0.6159315 - log(a)) + x1 - x2 + x3

delta\_r(j,k) = 4\*2\*pi\*freq(f)\*(1e-4)\*((pi/8) - r1 + r2 + r3 - r4);

delta\_r(j,k) = (4\*2\*pi\*freq(f)\*(1e-4)/sqrt(2))\*(t1 -t2 + t3 +t4 +t5);

%Carson's series to calculate reactance of conductors due to Earth effect

The objective of this project was to analyze the application of modal transformation matrix that is independent of frequency in analyses of three-phase lines considering the presence of 2 ground wires. Through analysis, both the limits of this approach and the possible errors in relation to the exact values obtained from eigenvalues and eigenvectors.

The model proposed in this project uses approximate modal transformation, accomplished through a transformation matrix independent of frequency. This matrix is obtained by linear combination of elements of Clarke's matrix. With application of this transformation matrix independent of frequency, it obtains diagonal matrices for the cases of transposed three-phase lines. For non-transposed three-phase lines, matrices of parameters are not diagonal with the application of the transformation matrix mentioned. For those cases not implemented, the proposal is to analyze the relative errors obtained by establishing circumstances in which one can use a transformation matrix independent of frequency.

This chapter presented a method that can be used for analyzing electromagnetic transients using real transformation matrices in three-phase systems considering the presence of ground wires. This method was implemented using Matlab, and then the routines used to develop it were presented and commented. The proposal analyzed used a real transformation matrix for the entire frequency range considered in this case. For those elements related to the phases of the considered system, the transformation matrix was composed of the elements of Clarke's matrix. In part related to the ground wires, the elements of the transformation matrix had to establish a relationship with the elements of the phases considering the establishment of a single homopolar reference in the mode domain. In the case of three-phase lines with the presence of two ground wires, it was unable to get the full diagonalization of the matrices Z and Y in the mode domain. The relative errors between the proposed routine and the correct values of eigenvalues were shown by graphs plotted using Matlab. Thus, for a future work, a correction routine will be used for non-transposed three-phase transmission line cases for the transformation matrix presented.

#### **Acknowledgements**

This work was supported by FAPESP.

### **Author details**

R. C. Monzani2 , A. J. Prado1 , L.S. Lessa2 and L. F. Bovolato2

1 Departamento de Sistemas e Energia (DSE) – Faculdade de Engenharia Elétrica e Computação (FEEC) – Campinas State University (UNICAMP), Brazil

2 Departamento de Engenharia Elétrica (DEE) – Faculdade de Engenharia de Ilha Solteira (FEIS) – Paulista State University (UNESP) , Brazil

[8] Nguyen, H. V.; Dommel, H. W.; Marti, J. R. Direct phase-domain modeling of fre‐ quency-dependent overhead transmission lines. IEEE Transactions on Power Deliv‐

Eigenvalue Analysis in Mode Domain Considering a Three-Phase System with two Ground Wires

http://dx.doi.org/10.5772/58383

531

[9] Noda, T.; Gagaoka, N.; Ametani, A. Phase domain modeling of frequency dependent transmission lines by means of an ARMA model. IEEE Transactions on Power Deliv‐

[10] Tavares, M. C.; Pissolato, J.; Portela, C. M. Six-Phase transmission line propagation characteristics and new three-phase representation. IEEE Transactions on Power,

[11] Semlyen, M. H.; Abdel-Rahman, M. H. State equation modeling of untransposed three phase lines. IEEE Transaction on Power Apparatus and Systems Pas, New

[12] Bhatt, N. B.; Venkata, S. S. Venkata; Guyker, W. C.; Booth, W. H., Sixphase (multiphase) power transmission systems: fault analysis, IEEE Transactions on Power Ap‐

[13] Ryan, H. M. High voltage engineering and testing. London: Peter Peregrinus on be‐

[15] Brandão, J. A.; Faria, J. Overhead three-phase transmission lines – nondiagonalizable situations. IEEE Transactions on Power Delivery, New York, v. 3, n. 4, p. 1348–1355,

[16] Brandão, J. A. F.; Briceño Mendez, J. H. Modal analysis of unstransposed bilateral three-phase lines – a perturbation approach. IEEE Transactions on Power Delivery,

[17] Brandão, J. A. F.; Briceño Mendez, J. H. On the modal analysis of asymmetrical threephase transmission lines using standard transformation matrices. IEEE Transaction

[19] Prado, A. J.; Pissolato Filho, J.; Kurokawa, S.; Bovolato, L. F. Eigenvalue analyses of two parallel lines using a single real transformation matrix, in: IEEE/Power Engineer‐ ing Society General Meeting, 2005, San Francisco. Conference of the… San Francisco:

[20] Prado, A. J.; Pissolato Filho, J.; Kurokawa, S.; Bovolato, L. F. Nontransposed threephase line analyses with a single real transformation matrix. in: IEEE/Power Engi‐ neering Society General Meeting, 2005, San Francisco. Conference of the… San

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[8] Nguyen, H. V.; Dommel, H. W.; Marti, J. R. Direct phase-domain modeling of fre‐ quency-dependent overhead transmission lines. IEEE Transactions on Power Deliv‐ ery, New York, v. 12, n. 3, p. 1335-1344, 1997.

**Acknowledgements**

530 MATLAB Applications for the Practical Engineer

**Author details**

R. C. Monzani2

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This work was supported by FAPESP.

, A. J. Prado1

(FEIS) – Paulista State University (UNESP) , Brazil

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1 Departamento de Sistemas e Energia (DSE) – Faculdade de Engenharia Elétrica e

2 Departamento de Engenharia Elétrica (DEE) – Faculdade de Engenharia de Ilha Solteira

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**Chapter 19**

**Analysis of Balancing of Unbalanced Rotors and Long**

Rotating machinery is commonly used in mechanical and electromechanical systems that include rotors of motors and engines, machining tools, industrial turbomachinery, etc. In case of unbalanced distribution of rotating masses around an axis of rotation the rotor unbalance arises. This presents a serious engineering problem because it is a major cause of excessive vibrations, esp. at higher speeds. Arising large centrifugal unbalanced forces can lead to damage of bearings and finally to destruction of machines. This is the reason why solving of

Vibration of the rotating machinery is suppressed by eliminating the root cause of vibration – the system unbalance. Practically, vibrations cannot reach zero values but usually it is acceptable to decrease them to a value lower than that one prescribed for a certain quality class of the machinery [1]. Balancing of the rotor increases the bearing life, minimizes vibrations,

The problems arising when dealing with unbalanced rotating bodies have been analyzed in many references. The exceptional positions among them hold two references [1, 2]. Due to its importance, numerous references have dealt with vibrations and their eliminations, e.g. [3] – [6] and also serious companies are facing the vibrations, just to mention few of them – [7, 8]. If the vibrations are below a normal level, they may indicate only normal wear but when they are increasing, it may signal the need to take an immediate maintenance action. A level of unbalance that is acceptable at low speeds is completely unacceptable at a higher speed. This is because the unbalance condition produces centrifugal force, which increases with square of the speed [9] – if the speed doubles, the force quadruples; etc. For this reason it is important to determine the critical speed at which excessive oscillations present a direct serious danger.

> © 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the unbalance is a basic concern in design and operation of the machinery.

audible noise, power losses, and finally it results in increased quality of products.

**Shafts using GUI MATLAB**

http://dx.doi.org/10.5772/58378

**1. Introduction**

Viliam Fedák, Pavel Záskalický and Zoltán Gelvanič

Additional information is available at the end of the chapter
