**2. Multi-conductor transmission line analysis**

#### **2.1. Telegrapher's Equations**

Electromagnetic behavior of transmission lines and cables is described by the Modified Telegrapher Equations, which in frequency domain are expressed as follows:

$$-\frac{d\mathbf{V}}{d\mathbf{x}} = \mathbf{Z}\mathbf{I}.\tag{1}$$

$$-\frac{d\mathbf{I}}{dx} = \mathbf{Y}\mathbf{V}.\tag{2}$$

where *V* is the vector of voltages, *I* is the vector of currents, *Z* and *Y* are the (*N* X *N*) per unitlength series impedance and shunt admitance matrix of a given line with *N* conductors, repectively. To solve equations (1) and (2), let equation (1) be first differentiated with respect to *x*; then, (2) is used to eliminate the vector of currents at the right hand side. The resulting expression is a second order matrix ODE involving only unknown voltages:

$$\frac{d^2\mathbf{V}}{dx^2} = \mathbf{Z}\mathbf{Y}\mathbf{V}.\tag{3}$$

In the same way, equation (2) can be differentiated with respect to *x* and (1) can be used to eliminate the right-hand-side voltage term. The resulting expression involves unknown currents only:

$$\frac{d^2\mathbf{I}}{dx^2} = \mathbf{YZI}.\tag{4}$$

Solution to (4) is:

$$\mathbf{I(x) = C\_1 e^{-\sqrt{\mathbf{Y}\mathbf{Z}}x} + C\_2 e^{\sqrt{\mathbf{Y}\mathbf{Z}}x}},\tag{5}$$

where *C1* and *C2* are vectors of integration constants determined by the line boundary conditions; that is, by the connections at the two line ends. In fact, the term including *C1* represents a vector of phase currents propagating forward (or in the positive *x*–direction) along the line, whereas the one with *C2* represents a backward (or negative *x*–direction) propagating vector of phase currents. Expression (5) is an extension of the well–known solution for the single–conductor line. Note that this extension involves the concept of matrix functions. This topic is explained at section 2.2.

The solution to (3) takes a form analogous to (5) and it is obtained conveniently from (5) and (2) as follows:

$$\mathbf{V}(\mathbf{x}) = -\mathbf{Y}^{-1}\frac{d\mathbf{I}}{d\mathbf{x}} = \mathbf{Z}\_c \left[ \mathbf{C}\_1 e^{-\sqrt{\mathbf{V}\mathbf{Z}}x} - \mathbf{C}\_2 e^{\sqrt{\mathbf{V}\mathbf{Z}}x} \right] \tag{6}$$

where, <sup>1</sup> *C* **Z Y YZ** is the characteristic impedance matrix and its inverse is the characteristic admittance matrix <sup>1</sup> . *<sup>C</sup>* **Y Z YZ**

#### **2.2. Modal analysis and matrix functions**

270 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

**2. Multi-conductor transmission line analysis** 

. *<sup>d</sup>*

**2.1. Telegrapher's Equations** 

examples.

currents only:

Solution to (4) is:

of the ULM, as well as from counting with a ULM–type code that is easy to understand and modify. One problem with this is that the theoretical basis of ULM includes various topics and subjects that are scattered through several dozens of highly specialized papers. Another difficulty with this is the high complexity of the code for a ULM–type model. This chapter aims at providing a clear and complete description of the theoretical basis for this model. Although this description is intended for power engineers with an interest in electromagnetic transient phenomena, it can be of interest also to electronic engineers involved in the analysis and design of interconnects. The chapter includes as well the description of Matlab program of a ULM–type model, along with executable code and basic

Electromagnetic behavior of transmission lines and cables is described by the Modified

. *<sup>d</sup>*

*dx*

where *V* is the vector of voltages, *I* is the vector of currents, *Z* and *Y* are the (*N* X *N*) per unitlength series impedance and shunt admitance matrix of a given line with *N* conductors, repectively. To solve equations (1) and (2), let equation (1) be first differentiated with respect to *x*; then, (2) is used to eliminate the vector of currents at the right hand side. The resulting

. <sup>2</sup>

In the same way, equation (2) can be differentiated with respect to *x* and (1) can be used to eliminate the right-hand-side voltage term. The resulting expression involves unknown

. <sup>2</sup>

*dx* **<sup>V</sup> ZI** (1)

**<sup>I</sup> YV** (2)

**<sup>V</sup> ZYV** (3)

**<sup>I</sup> YZI** (4)

1 2 ( ) , *x x xe e* **YZ YZ IC C** (5)

Telegrapher Equations, which in frequency domain are expressed as follows:

expression is a second order matrix ODE involving only unknown voltages:

2

2

*d dx*

*d dx* Matrix functions needed for multi-conductor line analysis are extensions of analytic functions of a one–dimensional variable. Consider the following function and its Taylor expansion:

$$f(\mathbf{x}) = \sum\_{k=0}^{n} a\_k \mathbf{x}^k \tag{7}$$

The application of *f()* to a square matrix *A* of order *NN* as its argument is accomplished as follows:

$$f(\mathbf{A}) = \sum\_{k=0}^{\text{wt}} a\_k \mathbf{A}^k,\tag{8}$$

where *A0* is equal to *U*, the *NN* unit matrix. Consider now the case of a diagonal matrix:

$$\mathbf{A} = \begin{bmatrix} \lambda\_1 & 0 & \dots & 0 \\ 0 & \lambda\_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda\_N \end{bmatrix} . \tag{9}$$

Application of (8) to yields:

$$f\begin{pmatrix}\mathbf{A}\end{pmatrix} = \sum\_{k=0}^{\infty} a\_k \mathbf{A}^k = \begin{bmatrix} \sum\_k a\_k \boldsymbol{\lambda}\_1^k & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \sum\_k a\_k \boldsymbol{\lambda}\_2^k & \dots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \dots & \sum\_k a\_k \boldsymbol{\lambda}\_N^k \end{bmatrix} = \begin{bmatrix} f\begin{pmatrix} \boldsymbol{\lambda}\_1 \end{pmatrix} & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & f\begin{pmatrix} \boldsymbol{\lambda}\_2 \end{pmatrix} & \dots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \dots & f\begin{pmatrix} \boldsymbol{\lambda}\_N \end{pmatrix} \end{bmatrix} \tag{10}$$

This expression thus shows that the function of a diagonal matrix is simply obtained applying the one–dimensional form of the function to the matrix nonzero elements. Consider next the function of a diagonalizable matrix *A*; that is,a matrix *A* that is similar to a diagonal one :

$$\mathbf{A} = \mathbf{M} \mathbf{A} \,\mathbf{M}^{-1}.\tag{11}$$

where *M* is the nonsingular matrix whose columns are the eigenvectors of *A*, while is the matrix whose diagonal elements are the eigenvalues of *A* (Strang, 1988).

Application of *f()* as in (10) to *A* yields:

$$f\left(\mathbf{A}\right) = \sum\_{k=0}^{\infty} a\_k \left(\mathbf{M} \mathbf{A} \mathbf{M}^{-1}\right)^k = \mathbf{M} \left(\sum\_{k=0}^{\infty} a\_k \mathbf{A}^k\right) \mathbf{M}^{-1} = \mathbf{M} f\left(\mathbf{A}\right) \mathbf{M}^{-1} \tag{12}$$

Therefore, the function of a diagonalizable matrix is conveniently obtained first by factoring *A* as in (10), then by applying the function to the diagonal elements of and, finally, by performing the triple matrix product as in (11) and (12).

It is clear from subsection **2.1**, that multi-conductor line analysis requires evaluating matrix functions of *YZ*. To do so, it is generally assumed that *YZ* always is diagonalizable (Wedephol, 1965; Dommel, 1992). Although there is a possibility for *YZ* not being diagonalizable (Brandao Faria, 1986), occurrences of this can be easily avoided when conducting practical analysis (Naredo, 1986).

#### **3. Line modelling**

Figure 1 shows the representation of a multi-conductor transmission line (or cable) of length *L*, with one of its ends at *x = 0* and the other at *x = L*. Let *I0* be the vector of phase currents being injected into the line and *V0* the vector of phase voltages, both at *x=0*. In the same form, *IL* and *VL* represent the respective vectors of injected phase currents and of phase voltages at *x=L*. Line equation solutions (5) and (6) are applied to the line end at *x=0*:

$$\mathbf{I}\_0 = \mathbf{I}(0) = \mathbf{C}\_1 + \mathbf{C}\_2 \tag{13}$$

$$\mathbf{V}\_0 = \mathbf{V}(0) = \mathbf{Z}\_{\mathbb{C}} (\mathbf{C}\_1 + \mathbf{C}\_2). \tag{14}$$

Then, the value of *C1* is determined from (13) and (14):

$$\mathbf{C}\_{1} = \frac{\mathbf{I}\_{0} + \mathbf{Y}\_{\mathbb{C}}\mathbf{V}\_{0}}{2}. \tag{15}$$

Expressions (5) and (6) are applied to the line end conditions at *x = L*:

$$\mathbf{I}\_L = -\mathbf{I}(L) = -\mathbf{C}\_1 e^{-\sqrt{\mathbf{Y}\mathbf{Z}}L} - \mathbf{C}\_2 e^{\sqrt{\mathbf{Y}\mathbf{Z}}L} \tag{16}$$

and

$$\mathbf{V}\_{\perp} = \mathbf{V}(L) = \mathbf{Z}\_{\mathbb{C}} \left[ \mathbf{C}\_{\mathbf{1}} e^{-\sqrt{\mathbf{Y} \mathbf{Z}} L} - \mathbf{C}\_{\mathbf{2}} e^{\sqrt{\mathbf{Y} \mathbf{Z}} L} \right]. \tag{17}$$

After doing this, (17) is pre–multiplied by *YC* and subtracted from (16) to obtain:

$$\mathbf{I}\_L - \mathbf{Y}\_{\mathbf{C}} \mathbf{V}\_L = -2\mathbf{C}\_1 e^{-\mathbf{Q}\mathbf{Y}\mathbf{Z}L}.\tag{18}$$

Finally, (15) is introduced in (18) rendering

272 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

diagonal one

:

Application of *f()* as in (10) to *A* yields:

This expression thus shows that the function of a diagonal matrix is simply obtained applying the one–dimensional form of the function to the matrix nonzero elements. Consider next the function of a diagonalizable matrix *A*; that is,a matrix *A* that is similar to a

where *M* is the nonsingular matrix whose columns are the eigenvectors of *A*, while

 0 0 ( )*k k k k k k fa a f*

Therefore, the function of a diagonalizable matrix is conveniently obtained first by factoring

It is clear from subsection **2.1**, that multi-conductor line analysis requires evaluating matrix functions of *YZ*. To do so, it is generally assumed that *YZ* always is diagonalizable (Wedephol, 1965; Dommel, 1992). Although there is a possibility for *YZ* not being diagonalizable (Brandao Faria, 1986), occurrences of this can be easily avoided when

Figure 1 shows the representation of a multi-conductor transmission line (or cable) of length *L*, with one of its ends at *x = 0* and the other at *x = L*. Let *I0* be the vector of phase currents being injected into the line and *V0* the vector of phase voltages, both at *x=0*. In the same form, *IL* and *VL* represent the respective vectors of injected phase currents and of phase

> 0 0 <sup>1</sup> . <sup>2</sup>

1 2 ( ) *L L*

voltages at *x=L*. Line equation solutions (5) and (6) are applied to the line end at *x=0*:

matrix whose diagonal elements are the eigenvalues of *A* (Strang, 1988).

*A* as in (10), then by applying the function to the diagonal elements of

performing the triple matrix product as in (11) and (12).

Then, the value of *C1* is determined from (13) and (14):

Expressions (5) and (6) are applied to the line end conditions at *x = L*:

conducting practical analysis (Naredo, 1986).

**3. Line modelling** 

and

1

**<sup>1</sup> 1 1 A MΛM M <sup>Λ</sup> M M <sup>Λ</sup> <sup>M</sup>** (12)

. **A M Λ M** (11)

0 12 **II CC** (0) (13)

*<sup>C</sup>* **I YV <sup>C</sup>** (15)

*<sup>L</sup> Le e* **YZ YZ II C C** (16)

[ <sup>0</sup> 1 2 (0) ]. *<sup>C</sup>* **V V ZC C** (14)

is the

and, finally, by

$$\mathbf{I}\_L - \mathbf{Y}\_{\mathbb{C}} \mathbf{V}\_L = -e^{-\sqrt{\mathbf{Y} \mathbf{Z}} L} \left[ \mathbf{I}\_0 + \mathbf{Y}\_{\mathbb{C}} \mathbf{V}\_0 \right] \tag{19}$$

Expression (19) establishes the relation among voltages and currents at the terminals of a multi-conductor line section. Its physical meaning follows from realizing that the term *I0+YCV0* at its right hand side represents a traveling wave of currents leaving the line end at *x = 0* and propagating in the positive *x–*axis direction, whereas *IL–YCVL* at the left hand side is the traveling wave of currents leaving the line end at *x = L*.

**Figure 1.** Multi-conductor line segment of length L.

By a similar process as the previous one for deriving (19), it is possible to show also that the following relation holds as line equation solutions (5) and (6) are applied to line end conditions at x=0:

$$\mathbf{I}\_0 - \mathbf{Y}\_\mathbf{C} \mathbf{V}\_0 = -e^{-\sqrt{\mathbf{Y}\mathbf{Z}}L} \left[ \mathbf{I}\_L + \mathbf{Y}\_\mathbf{C} \mathbf{V}\_L \right] \tag{20}$$

Note however that this relation can also be obtained by simply exchanging at (19) sub– indexes *0* and *L*. This exchange is justified by the input/output symmetry of the line section. Expressions (19) and (20) provide a very general mathematical model for a multi-conductor transmission line. This is a model based on traveling wave principles. Let (19) and (20) be rewritten as follows:

$$\mathbf{I}\_L = \mathbf{I}\_{sh,L} - \mathbf{I}\_{aux,L} \tag{21}$$

where, *Ish,L* =*YCVL* is the shunt currents vector produced at terminal *L* by injected voltages *VL*. *Iaux,L* =*HIrfl,0* is the auxiliary currents vector consisting of the reflected currents at terminal *0, Irfl,0*= *I0+ YCV0* and the transfer functions matrix *H*=*e-(YZ)L*.

In the same way as it has been previously done for (19), expression (20) is conveniently represented as follows:

$$\mathbf{I}\_0 = \mathbf{I}\_{sh,0} - \mathbf{I}\_{\text{aux},0} \tag{22}$$

with, *Ish,0* =*YCV0* , *Iaux,0* =*HIrfl,L* ,and *Irfl,L*= *IL+ YCVL* .

Expressions (21) and (22) constitute a traveling wave line model for the segment of length *L* depicted in figure 1. The former set of expressions represents end *L* of the line segment, while the latter set represents end *0*. A schematic representation for the whole model is provided by figure 2. Note that the coupling between the two line ends is through the auxiliary sources *Iaux,0* and *Iaux,L*.

**Figure 2.** Frequency domain circuit representation of a multi-conductor line.

The line model defined by expressions (21) and (22) is in the frequency domain. Power system transient simulations require this model to be transformed to the time domain. For instance, the transformation of (21) to the time domain yields:

$$
\dot{\mathbf{i}}\_0 = \dot{\mathbf{i}}\_{sh,0} - \dot{\mathbf{i}}\_{amx,0} \tag{23}
$$

with

,0 \* *sh C L* **i yv** (24)

and

$$\mathbf{i}\_{aux,0} = \mathbf{h}^\* \mathbf{i}\_{rf,0} \tag{25}$$

Note that at (23), (24), (25) the lowercase variables represent the time domain images of their uppercase counterparts at (22) and that the symbol \* represents convolution. Reflected currents can be represented as

$$\mathbf{i}\_{r\emptyset,L} = \mathbf{2}\mathbf{i}\_{sh,L} - \mathbf{i}\_{aux,L} \tag{26}$$

Expressions (23)-(26) constitute a general traveling–wave based time–domain model for line end *0*. The model corresponding to the other end is obtained by interchanging sub–indexes "*0*" and "*L*" at (23)-(26). Equation (23) essentially provides the interface of the line–end *0* model to the nodal network solver that, for power system transient analysis, usually is the EMTP (Dommel, 1996). Expressions (24) and (25) require the performing of matrix–to–vector convolutions that are carried out conveniently by means of State–Space methods (Semlyen & Abdel-Rahman, 1982). State–Space equivalents of (23) and (24) arise naturally as *YC* and *H* are represented by means of fitted rational functions (Semlyen & Dabuleanu, 1975).
