THEORY OF A MULTICONDUCTOR LINE WITH SPATIAL SYMMETRY

Yu. A. Sher

The theory of multiconductor lines [1 et al.] has by now been studied in detail for two limiting cases: 1) a completely unsymmetrical line, 2) a completely symmetrical line. The quantitative relations of the theory of a completely unsymmetrical multiconductor line are not used in engineering calculations because of the extraordinary computational difficulties involved. The theory of a completely symmetrical multiconductor line leads, as is known, to simple calculation formulas and is successfully used for an approximate description of the phenomenon of propagation of electromagnetic waves arising in the implementation of high-frequency communication over power-transmission lines.

This theory, which assumes equality of the self-resistances of all wires, is unsuitable for describing wave propagation in a bundle of dissimilar (copper and steel) wires.

The widespread introduction of inductive communication with a moving train [5,6] (this new type of communication gives a high economic effect in railway transport) has led to the need to study wave propagation along multiconductor communication lines containing both nonferrous and steel wires.

The theory set forth below takes into account the difference between the self-resistances of copper and steel wires and is quite suitable for an approximate description of wave propagation in a bundle of wires in inductive communication and for obtaining calculation relations. In addition, it can be used in calculations of high-frequency channels over electric-transmission lines, taking into account the influence of steel shield wires.

1. Initial System of Equations and Assumptions

In the general case of an unsymmetrical multiconductor line, the process of propagation of electromagnetic oscillations is described by a system of linear homogeneous differential equations with constant coefficients

\[ \left\{ \begin{aligned} -\,\frac{d}{dx}\,\|U\| &= \|Z\|\,\|I\|,\\ -\,\frac{d}{dx}\,\|I\| &= \|Y\|\,\|U\|, \end{aligned} \right. \tag{1} \]

where

\[ \|U\|=\left\| \begin{array}{c} U_1\\ U_2\\ \ldots\\ U_n \end{array} \right\|,\quad \|I\|=\left\| \begin{array}{c} I_1\\ I_2\\ \ldots\\ I_n \end{array} \right\|,\quad \|Z\|=\left\| \begin{array}{cccc} Z_{11}&Z_{12}&\ldots&Z_{1n}\\ Z_{21}&Z_{22}&\ldots&Z_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ Z_{n1}&Z_{n2}&\ldots&Z_{nn} \end{array} \right\|, \]

\[ \|Y\|=\left\| \begin{array}{cccc} Y_{11}&(-Y_{12})&\ldots&(-Y_{1n})\\ (-Y_{21})&Y_{22}&\ldots&(-Y_{2n})\\ \ldots&\ldots&\ldots&\ldots\\ (-Y_{n1})&(-Y_{n2})&\ldots&Y_{nn} \end{array} \right\|, \tag{2} \]

where
\(Y_{ik}\) is the total mutual conductance between the \(i\)-th and \(k\)-th conductors, equal to

\[ Y_{ik}=G_{ik}+j\omega C_{ik}=Y_{ki}, \tag{3} \]

\(G_{ik}\) is the insulation conductance between the \(i\)-th and \(k\)-th conductors per unit length,
\(C_{ik}\) is the mutual partial capacitance between the \(i\)-th and \(k\)-th conductors per unit length,
\(Y_{ii}\) is the total self-conductance of the \(i\)-th conductor, equal to

\[ Y_{ii}=G_{ii}+j\omega C_{ii}+\sum_{\substack{k=1\\ k\ne i}}^{n} y_{ik}, \tag{4} \]

where
\(G_{ii}\) is the insulation conductance per unit length,
\(C_{ii}\) is the partial capacitance between the \(i\)-th conductor and ground,
\(Z_{ii}\) is the total self-impedance of the \(i\)-th conductor, equal to

\[ Z_{ii}=R+j\omega L_i+\left(j2\omega\ln\frac{2h}{a}+4\omega J_{ii}\right)\cdot 10^{-4}\ \frac{\text{ohm}}{\text{km}}, \tag{5} \]

where
\(R\) is the active resistance of the conductor, taking account of the skin effect, \(L_i\) is the internal inductance of the conductor, \(\omega\) is the angular frequency of oscillation, \(h\) is the suspension height of the conductor, and \(a\) is the radius of the conductor.

\(Z_{ik}\) is the total mutual impedance between the \(i\)-th and \(k\)-th conductors, equal to

\[ Z_{ik}=\left(j2\omega\ln\frac{\rho''}{\rho'}+4\omega J_{ik}\right)\cdot 10^{-4}\ \frac{\text{ohm}}{\text{km}}, \tag{6} \]

\(\rho'\) is the distance between the \(i\)-th and \(k\)-th conductors, \(\rho''\) is the distance from the \(k\)-th conductor to the mirror image of the \(i\)-th conductor, \(j=\sqrt{-1}\), \(J\) is a function depending on the properties of the soil.

Differentiating equations (1) with respect to \(x\) and eliminating the variables leads to the system:

\[ -\frac{d^2}{dx^2}\|U\|=\|Z\|\cdot\|Y\|\cdot\|U\|, \tag{7} \]

\[ -\frac{d^2}{dx^2}\|I\|=\|Y\|\cdot\|Z\|\cdot\|I\|. \]

A necessary condition for solving system (7) is the solution of the characteristic equation \(\Delta_k=0\), where \(\Delta_k\) is the determinant of the matrix

\[ \left\|\Gamma_{ZY}\right\|-\left\|k^2\right\|, \quad \text{where} \quad \left\|\Gamma_{ZY}\right\|=\|Z\|\cdot\|Y\|,\quad \text{and} \quad \left\|k^2\right\|= \left\| \begin{array}{cccc} k^2 & 0 & \ldots & 0\\ 0 & k^2 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & k^2 \end{array} \right\|. \]

The theory of asymmetrical multiconductor lines assumes that all self and mutual resistances \(Z_{ii}\), \(Z_{ik}\) and conductances \(Y_{ii}\), \(Y_{ik}\) are known, and that the characteristic equation \(\Delta_k=0\) has been solved. However, when the number of conductors in a communication line is large, solving the characteristic equation and calculating the primary parameters are associated with great difficulties. Even in calculating the partial capacitances of a conductor located in a bundle of other conductors, one would have to compute determinants of order \(n\), where \(n\) is the number of conductors [2]. Therefore, an assumption is usually introduced that the self and mutual potential coefficients of all conductors are equal. This assumption leads to simple formulas for the partial\(^{1)}\) capacitances.

Since, further, the mutual inductance coefficients \(M_{ik}\) and the external self-inductance \(L_{ii}\) can be expressed through the potential coefficients

\[ L_{ii}=\frac{\alpha_{ii}}{c^2}; \qquad M_{ik}=\frac{\alpha_{ik}}{c^2}, \tag{9} \]

it follows from the indicated assumption that these quantities are equal for all conductors.

Equality of the self and mutual potential coefficients can hold only for a multiconductor line with special transposition, the simplest example of which is the transposition of three-conductor power-transmission lines. With such transposition, the quantities \(J_{ii}\) and \(J_{ik}\) (expressions (5) and (6)) must also be equal for all conductors.

The magnitude of the total internal resistance of a conductor \(R+j\omega L_i\) depends on the conductor material and differs sharply for copper and steel conductors. Thus, for example, at a frequency of \(2.5\) MHz, for a copper conductor, \(d=4\) mm,

\[ R+j\omega L_i=34(1+j)\,\frac{\text{ohm}}{\text{km}}, \]

whereas for a steel conductor \(d=5\) mm,

\[ R+j\omega L_i=(800 \div 900)(1+j)\,\frac{\text{ohm}}{\text{km}}. \]

If we assume that all steel conductors (as well as all copper conductors) have the same internal resistance, then the idealized line under consideration will have the following properties:

1) equality of the mutual resistances and mutual conductances of all conductors:

\[ Z_{ik}=Z, \tag{10} \]

\[ Y_{ik}=Y; \]

\(^{1)}\) Self partial capacitance:

\[ C_{ii}=\frac{\alpha}{(\alpha_s-\alpha)\,[\alpha_s+(n-1)\alpha]}. \]

Mutual partial capacitance

\[ C_{ik}=\frac{1}{\alpha_s+(n-1)\alpha}, \]

where \(\alpha_s\) is the mean self potential coefficient,
\(\alpha\) is the mean mutual potential coefficient.

2) equality of the self-conductances of all wires

\[ Y_{ii}=Y_s; \tag{11} \]

3) equality of the self-resistances of all steel wires and of all copper wires:

\[ \begin{aligned} Z_{ii}&=Z_s^c &&\text{for steel ones,}\\ Z_{ii}&=Z_s^m &&\text{for copper ones.} \end{aligned} \tag{12} \]

2. Determination of the propagation constants of independent waves

Let us assume that the bundle consists of \(n\) steel and \(m\) copper wires. Under assumptions (10–12), system (1) is simplified:

\[ \|Z\|= \left\| \begin{array}{ccccc} Z_s^c&Z&\ldots&Z&Z\\ Z&Z_s^c&\ldots&Z&Z\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ Z&Z&\ldots&Z_s^m&Z\\ Z&Z&\ldots&Z&Z_s^m \end{array} \right\| \begin{array}{l} \}n\\[2.5em] \}m \end{array} \quad \underbrace{\hspace{7em}}_{m+n} ;\qquad \|y\|= \left\| \begin{array}{ccccc} Y_s&(-Y)&\ldots&(-Y)&(-Y)\\ -Y&Y_s&\ldots&(-Y)&(-Y)\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ (-Y)&(-Y)&\ldots&Y_s&(-Y)\\ (-Y)&(-Y)&\ldots&(-Y)&Y_s \end{array} \right\|. \tag{13} \]

Multiplication of the matrices gives:

\[ \|\gamma\|_{zy}=\|Z\|\cdot\|y\|= \left\| \begin{array}{cccc} \gamma_s^c&\gamma^c&\ldots&\gamma^c\\ \gamma^c&\gamma_s^c&\ldots&\gamma^c\\ \ldots&\ldots&\ldots&\ldots\\ \gamma^m&\gamma^m&\ldots&\gamma_s^m\\ \gamma^m&\gamma^m&\ldots&\gamma_s^m \end{array} \right\|. \tag{14} \]

The matrix \(\|\gamma\|_{yz}=\|y\|\cdot\|Z\|\) is obtained from \(\|\gamma\|_{zy}\) by a permutation of rows and columns.

Here the notation is:

\[ \begin{aligned} \gamma_s^c&=Z_s^cY_s-(m+n-1)ZY;\\ \gamma^c&=-Z_s^cY+ZY_s-(m+n-2)ZY;\\ \gamma_s^m&=Z_s^mY_s-(m+n-1)ZY;\\ \gamma^m&=-Z_s^mY+ZY_s-(m+n-2)ZY. \end{aligned} \tag{15} \]

It is necessary to solve system (7), where \(\|\gamma_{zy}\|\) and \(\|\gamma_{yz}\|\) are determined by expressions (14).

The solution should be sought in the form:

\[ \|U\|=\|P'\|e^{-kx}+\|P''\|e^{kx}. \]

\[ \|I\|=\|Q'\|e^{-kx}+\|Q''\|e^{kx}, \]

where

\[ \|P\|= \left\| \begin{array}{c} P_1\\ P_2\\ \ldots\\ P_{n+m} \end{array} \right\|; \qquad \|Q\|= \left\| \begin{array}{c} Q_1\\ Q_2\\ \ldots\\ Q_{m+n} \end{array} \right\|. \tag{16} \]

Substitute the solutions (16) into system (7). The second derivatives of the solutions (16) will be:

\[ \frac{d^2}{dx^2}\|U\|=k^2\|P\|e^{-kx}+k^2\|P'\|e^{kx}= \]

\[ = \left\| \begin{array}{cccc} k^2&0&\ldots&0\\ 0&k^2&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&k^2 \end{array} \right\| \cdot \left\| \begin{array}{c} P_1\\ P_2\\ \ldots\\ P_{m+n} \end{array} \right\| e^{-kx} + \left\| \begin{array}{cccc} 0&k^2&\ldots&0\\ 0&k^2&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&k^2 \end{array} \right\| \cdot \left\| \begin{array}{c} P'_1\\ P'_2\\ \ldots\\ P'_{m+n} \end{array} \right\| e^{kx} \]

\[ \frac{d^2}{dx^2}\|I\|=k^2\|Q\|e^{-kx}+k^2\|Q'\|e^{kx}= \tag{17} \]

\[ = \left\| \begin{array}{cccc} k^2&0^2&\ldots&0\\ 0&k^2&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&k^2 \end{array} \right\| \cdot \left\| \begin{array}{c} Q_1\\ Q_2\\ \ldots\\ Q_{m+n} \end{array} \right\| e^{-kx} + \left\| \begin{array}{cccc} k^2&0&\ldots&0\\ 0&k^2&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&k^2 \end{array} \right\| \cdot \left\| \begin{array}{c} Q'_1\\ Q'_2\\ \ldots\\ Q'_{n+m} \end{array} \right\| \cdot e^{kx}. \]

In (17), instead of multiplying all elements of the matrices \(\|P\|\) and \(\|Q\|\) by \(k^2\), these matrices themselves are multiplied on the left by the diagonal matrices

\[ \left\| \begin{array}{cccc} k^2&0&\ldots&0\\ 0&k^2&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&k^2 \end{array} \right\| =\|k^2\|. \]

Further, system (7) will be rewritten in the form:

\[ \|k^2\|\cdot\|P\|\cdot e^{-kx} + \|k^2\|\cdot\|P'\|\cdot e^{kx} = \|\gamma zy\|\cdot\|P\|\cdot e^{-kx} + \|\gamma zy\|\cdot\|P'\|e^{kx}; \]

\[ \|k^2\|\cdot\|Q\|\cdot e^{-kx} + \|k^2\|\cdot\|Q'\|\cdot e^{kx} = \|\gamma yz\|\cdot\|Q\|e^{-kx} + \|\gamma yz\|\cdot\|Q'\|e^{kx}. \]

Equating the coefficients of identical exponentials, we obtain systems of equations from which all coefficients \(P\), \(P'\), \(Q\), and \(Q'\) must be determined.

\[ \bigl[\|\gamma zy\|-\|k^2\|\bigr]\cdot\|P'\|=0; \tag{18} \]

\[ \bigl[\|\gamma zy\|-\|k^2\|\bigr]\cdot\|P\|=0; \tag{19} \]

\[ \bigl[\|\gamma yz\|-\|k^2\|\bigr]\cdot\|Q'\|=0; \tag{20} \]

\[ \bigl[\|\gamma yz\|-\|k^2\|\bigr]\cdot\|Q'\|=0. \tag{21} \]

Expressions (18–21) are systems of homogeneous algebraic equations. Nonzero solutions for them are possible only under the condition that their principal determinants be equal to zero,

\[ \Delta_m^n= \left| \begin{array}{cccc} (\gamma_s^c-k^2)\gamma^c&\ldots&\gamma^c\gamma^c\\ \gamma^c(\gamma_s^c-k^2)&\ldots&\gamma^c\gamma^c\\ \ldots&\ldots&\ldots&\ldots\\ \gamma^M\gamma^M&\ldots&(\gamma_s^M-k^2)\gamma^M\\ \gamma^M\gamma^M&\ldots&\gamma^M(\gamma_s^M-k^2) \end{array} \right| =0; \qquad \Delta_m^n= \left| \begin{array}{cccc} (\gamma_s^c-k^2)\gamma^c&\ldots&\gamma^M\gamma^M\\ \gamma^c(\gamma_s^c-k^2)&\ldots&\gamma^M\gamma^M\\ \ldots&\ldots&\ldots&\ldots\\ \gamma^c\gamma^c&\ldots&(\gamma_s^M-k^2)\gamma^M\\ \gamma^c\gamma^c&\ldots&\gamma^M(\gamma_s^M-k^2) \end{array} \right| \tag{22} \]

Both determinants (22) are equal, since when all rows are replaced by columns, and the columns by rows, the value of the determinant does not change.

Expression (22) is the characteristic equation of degree \((m+n)\) with respect to \((k^2)\). The further task consists in determining all the roots of the characteristic equation (22). For this purpose the determinant (22) can be represented in the form of factors (see the Appendix):

\[ \Delta_m^n=(k_c^2-k^2)^{n-1}(k_m^2-k^2)^{m-1} \left[(k_c^2-k^2)\cdot(k_m^2-k^2)+n\gamma^c(k_m^2-k^2)+m\gamma^m(k_c^2-k^2)\right], \tag{23} \]

where

\[ k_c^2=\gamma_s^c-\gamma^c,\qquad k_m^2=\gamma_s^m-\gamma^m. \]

Equating each of the three factors to zero, we find all \(m+n\) roots of the equation.

The first factor

\[ (k_c^2-k^2)^{n-1}=0 \]

gives \((n-1)\) multiple roots

\[ k^2=k_c^2;\qquad k=\pm k_c=\pm\sqrt{\gamma_s^c-\gamma^c}. \tag{24} \]

The second factor

\[ (k_m^2-k^2)^{m-1}=0 \]

gives \((m-1)\) multiple roots

\[ k^2=k_m^2;\qquad k=\pm k_m=\pm\sqrt{\gamma_s^m-\gamma^m}. \tag{25} \]

Equating the 3rd factor to zero, we find the remaining two roots from the quadratic equation with respect to \(k^2\)

\[ (k_c^2-k^2)\cdot(k_m^2-k^2)+n\gamma^c(k_m^2-k^2)+m\gamma^m(k_c^2-k^2)=0. \tag{26} \]

Its solution is:

\[ k_{1,2}^2=\frac{b}{2}\pm\sqrt{\left(\frac{b}{2}\right)^2-a}, \tag{27} \]

where

\[ b=k_m^2+k_c^2+m\gamma^m+n\gamma^c;\qquad a=k_m^2k_c^2+m\gamma^m k_c^2+n\gamma^c k_m^2. \tag{28} \]

The solution of the characteristic equation gives grounds for drawing the following conclusion.

In the system under consideration, consisting of \((m+n)\) conductors, there may exist \((n-1)\) waves with the same propagation constant \(k_c\); \((m-1)\) waves with the same propagation constant \(k_m\), and two waves with different propagation constants \(k_1\) and \(k_2\).

3. Relations between the vectors of each of the waves

After determining all the roots of the characteristic equation (22), the systems of equations (18–21) can be used to determine all the coefficients \(P, P'\) and \(Q, Q'\).

As is seen from expressions (18–21), systems (18) and (19) are exactly the same, as are systems (20) and (21). Let us consider one of them, for example (18). Substituting one of the roots \(k_p\) into the equations, dividing the first \(n\) equations by \(\gamma^c\ne0\), and the last \(m\) equations by \(\gamma^m\ne0\), denoting:

\[ C_p=\frac{\gamma_s^c-k_p^2}{\gamma^c},\qquad M_p=\frac{\gamma_s^m-k_p^2}{\gamma^m}, \]

we obtain the following system:

\[ \begin{gathered} C_p P_{1p}+P_{2p}+\ldots+P_{(n+1)p}+\ldots+P_{(n+m)p}=0,\\ P_{1p}+C_p P_{2p}+\ldots+P_{(n+1)p}+\ldots+P_{(n+m)p}=0,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ P_{1p}+P_{2p}+\ldots+M_p(P_{n+1})_p+\ldots+P_{(n+m)p}=0,\\ P_{1p}+P_{2p}+\ldots+P_{(n+1)p}+\ldots+M_p P_{(n+m)p}=0. \end{gathered} \tag{29} \]

For the unknown coefficients \(P\), two indices have been introduced: the first is the index of the coefficient in the given particular solution, and the second index refers to the root of the characteristic equation substituted into the given system of equations (18).

Analysis of system (29) for the case of simple roots \(k_1^2\) and \(k_2^2\) presents no difficulties. It is known [4] that, when simple roots of the characteristic equation

\[ \Delta_k=0 \]

are substituted into the determinant \(\Delta_k\), at least one of the minors of order one less is nonzero (for the determinant of system (29), all minors of order \((m+n-1)\) are nonzero). Thus, the rank of the principal matrix is \(r=(m+n-1)\), and consequently there exist \(r\) independent relations among the coefficients \(P\), so that \((m+n-1)\) unknowns can be eliminated and expressed through arbitrary constants.

The number of arbitrary constants is

\[ (m+n)-r=1. \]

Deleting one of the equations, for example the first, and moving the column \(P_{1p}\) as free unknowns to the right-hand side, we determine all the remaining \(m+n-1\) coefficients by Cramer’s rule.

Having done this, we establish the following:

a) for each of the waves (\(k_1\) and \(k_2\)), the voltages on all steel wires, except the first, taken as the basic one, are equal to

\[ P_{ip}=-P_{1p}\, \frac{M_p-1}{(C_p-1)(M_p-1)+(n-1)(M_p-1)+m(C_p-1)} \tag{30} \]

\[ i=2,3\ldots n, \]

\[ p=1,2; \]

b) for each of the waves (\(k_1\) and \(k_2\)), the voltages on all copper wires are equal to

\[ P_{ip}=-P_{1p}\, \frac{C_p-1}{(C_p-1)(M_p-1)+(n-1)(M_p-1)+m(C_p-1)}; \tag{31} \]

\[ i=n+1,\ n+2\ldots n+m, \]

\[ p=1,2. \]

It remains to determine the dependence between the voltage \(P_{1p}\) and the voltages on the steel wires. For this it is necessary to substitute into expression (30) the values \((k_1\) and \(k_2)\) determined in (27).

Carrying out the transformations, we obtain

\[ \text{for wave } k_1:\quad P_{11}=P_{i1};\quad i=2,3\ldots n, \tag{32} \]

\[ \text{for wave } k_2:\quad P_{12}=P_{i2};\quad i=2,3\ldots n. \]

Now the final conclusion concerning the voltage ratios can be formulated:

1. For each of the waves \(k_1\) and \(k_2\), the voltages on all steel and all copper conductors are equal.

2. The ratio of the voltage on the copper conductors to the voltage on the steel conductors is, for the wave \(k_1\),

\[ \eta_1=\frac{C_1-1}{M_1-1} = -\frac{\gamma^{\mathrm{м}}\left(k_c^2-k_1^2\right)} {\gamma^{\mathrm{с}}\left(k_{\mathrm{м}}^2-k_1^2\right)}; \tag{33} \]

for the wave \(k_2\),

\[ \eta_2=\frac{C_2-1}{M_2-1} = -\frac{\gamma^{\mathrm{м}}\left(k_c^2-k_2^2\right)} {\gamma^{\mathrm{с}}\left(k_{\mathrm{м}}^2-k_2^2\right)}. \]

The solution of system (29), upon substitution into it of \(k_1^2\) and \(k_2^2\), gave two arbitrary constants \(P_{11}\) and \(P_{12}\). If the copper conductor is taken as the reference conductor, then the arbitrary constants will be \(P_{i1}\) and \(P_{i2}\) \((i>n)\), and the voltages on the steel conductors will be related to \(P_{i1}\) and \(P_{i2}\) by the coefficients \(\eta_1\) and \(\eta_2\), which in magnitude are reciprocal to the coefficients in (33). Thus, we have obtained particular solutions in the form:

\[ n\left\{ \begin{vmatrix} P_{c1}\\ P_{c1}\\ \ldots\\ P_{c1}\eta_1\\ P_{c1}\eta_1 \end{vmatrix} e^{-k_1x} + \begin{vmatrix} P'_{c1}\\ P'_{c1}\\ \ldots\\ P'_{c1}\eta_1\\ P'_{c1}\eta_1 \end{vmatrix} e^{k_1x}; \qquad n\left\{ \begin{vmatrix} P_{c2}\\ P_{c2}\\ \ldots\\ P_{c2}\eta_2\\ P_{c2}\eta_2 \end{vmatrix} e^{-k_2x} + \begin{vmatrix} P'_{c2}\\ P'_{c2}\\ \ldots\\ P'_{c2}\eta_2\\ P'_{c2}\eta_2 \end{vmatrix} e^{k_2x}, \tag{34} \]

where \(P_{c1}\) is the voltage on the steel conductors for the 1st wave,
\(P_{c2}\) for the 2nd wave.

For \(Q\) and \(Q'\) the expressions are identical. It is only necessary to replace \(P_1\) by \(Q_1\), and \(P'_1\) by \(Q'_1\), where

\[ \eta'_1= -\frac{\gamma^{\mathrm{с}}(C_1-1)} {\gamma^{\mathrm{м}}(M_1-1)} = \frac{k_c^2-k_1^2}{k_{\mathrm{м}}^2-k_1^2}; \qquad \eta'_2= -\frac{\gamma^{\mathrm{с}}(C_2-1)} {\gamma^{\mathrm{м}}(M_2-1)} = \frac{k_c^2-k_2^2}{k_{\mathrm{м}}^2-k_2^2}. \tag{34a} \]

Next consider system (29) for the case of multiple roots. Substitution into (29) of the root \(k^2=k_c^2=\gamma_s^{\mathrm{с}}-\gamma^{\mathrm{с}}\) reduces the first \(n\) equations to a single equation, since

\[ C=\frac{\gamma_s^{\mathrm{с}}-k_c^2}{\gamma^{\mathrm{с}}}=1. \]

In this case the rank of the coefficient matrix proves to be equal to \(m+1\). Consequently, the number of arbitrary constants of the system is equal to

\[ (m+n)-(m+1)=n-1. \]

We rewrite system (29), moving to the right-hand side, as free unknowns, \(P_1, P_2,\ldots,P_{n-1}\), and denoting:

\[ P_1+P_2+\ldots P_{n-1}=\sum_{i=1}^{n-1}P_i;\qquad M_c=M_p\quad \text{for } k_p=k_c. \]

\[ \left\{ \begin{aligned} P_n+P_{n+1}+\ldots+P_{m+n}&=-\sum_{i=1}^{n-1}P_i\\ P_n+M_cP_{n+1}+\ldots P_{m+n}&=-\sum_{i=1}^{n-1}P_i\\ &\cdots\\ P_n+P_{n+1}+\ldots M_cP_{m+n}&=-\sum_{i=1}^{n-1}P_i . \end{aligned} \right. \tag{35} \]

Since the determinant of this system is nonzero, the unknowns \(P_n\ldots P_{n+m}\) can be determined by Cramer’s rule.

We obtain
\[ P_n=-(P_1+P_2+\ldots+P_{n-1}), \]
\[ P_{n+1}=P_{n+2}=\ldots=P_{n+m}=0 \tag{36} \]
(the determinants in the numerator turn out to have proportional columns for \(P_{n+1}\div P_{n+m}\)).

Thus, for a wave with propagation constant \(k_c\), the voltages on all the colored conductors are equal to zero, while the voltages on the steel conductors form a balanced system of vectors, since their sum is zero (an interconductor wave).

Substitution of the root \(k=k_m\) gives the same results with respect to the colored conductors, i.e.
\[ P_1=P_2=\ldots P_n=0, \]
\[ P_{n+m}=-(P_{n+1}+P_{n+2}+\ldots P_{m+n-1}). \tag{37} \]

For a wave with propagation constant \(k_m\), the voltages on all the steel conductors are equal to zero, while the voltages on the colored conductors form a balanced system of vectors, since their sum is zero.

We obtain analogous results for the currents (it is sufficient to replace \(P\) by \(Q\) and \(P'\) by \(Q'\)).

Having determined all the coefficients from system (29), we shall form particular solutions in the following way: for the first wave with propagation constant \(k_c\) we leave the arbitrary constant \(P_{1c}\), setting the remaining arbitrary constants equal to zero; for the second wave \(k_c\) we leave \(P_{2c}\), and so on.

Then for the \(p\)-th wave \(k_c\) the particular solution is written in the form:
\[ \begin{array}{c} p\text{-th row}\left\{ \begin{Vmatrix} 0\\ P_{pc}\\ 0\\ \ldots\\ -P_{pc} \end{Vmatrix} \right. \\[-0.5em] n\text{-th row} \end{array} e^{-k_cx} + \begin{array}{c} \left. \begin{Vmatrix} 0\\ P'_{pc}\\ 0\\ \ldots\\ -P'_{pc} \end{Vmatrix} \right\} \end{array} e^{k_cx}, \tag{38} \]
where
\(P_{pc}\) is the voltage of the wave \(k_c\) on conductor \(p\).

Analogous particular solutions are obtained for all waves \(k_m\).

4. Wave impedances

As is known, the number of arbitrary constants of an ordinary system of differential equations is equal to the degree of the characteristic equation with respect to \(k\) [4].

In the present case this number is equal to

\[ 2(m+n). \]

Thus there must also be \(2(m+n)\) arbitrary constants, determined from the boundary conditions. However, all the particular solutions obtained contain altogether \(4(m+n)\) constants: \(2(m+n)\) coefficients \(P\) and \(2(m+n)\) coefficients \(Q\). Consequently, the coefficients \(Q\) can be expressed in terms of \(P\), or conversely. For this purpose it is convenient to use system (1).

Substituting into it the particular solutions for the waves \(k_1\) and \(k_2\), and equating the coefficients at identical exponentials, we find the following relations between the coefficients \(P\) and \(Q\):

for the wave \(k_1\):

\[ W_{M1}=\frac{P_{M1}}{Q_{M1}} =\left[Z_s^{M}+(m-1)Z+\frac{nZ}{\eta'_1}\right]\cdot\frac{1}{k_1}; \tag{39} \]

\[ W_{c1}=\frac{P_{c1}}{Q_{c1}} =\frac{\eta'_1}{\eta_1} \left[Z_s^{M}+(m-1)Z-\frac{nZ}{\eta'_1}\right]\cdot\frac{1}{k_1}; \]

for the wave \(k_2\)

\[ W_{c2}=\frac{P_{c2}}{Q_{c2}} =\left[Z_s^{c}+(n+1)Z+\eta'_2 mZ\right]\cdot\frac{1}{k_2}; \]

\[ W_{M2}=\frac{P_{M2}}{Q_{M2}} =\frac{\eta'_2}{\eta_2} \left[Z_s^{c}+(n-1)Z+\eta'_2 mZ\right]\cdot\frac{1}{k_2^2}, \tag{40} \]

where the index \(c\) refers to steel conductors, and \(M\) to copper conductors.

For the primed coefficients we obtain:

\[ \frac{P'}{Q'}=-\frac{P}{Q}=-W. \tag{41} \]

Since \(P\) and \(P'\) have the dimension of voltage, while \(Q\) and \(Q'\) have the dimension of currents, their ratios \(W\) may be regarded as wave impedance. Its value relates the voltages of the incident and reflected waves to the currents of these waves in each conductor.

The wave impedances for the waves \(k_c\) and \(k_M\) are found in the same way, by substituting all the particular solutions (38) into equations (1) and equating the coefficients at identical exponentials:

\[ W_{cc}=\frac{Z_s^{c}-Z}{k_c}; \]

\[ W_{MM}=\frac{Z_s^{M}-Z}{k_M}, \tag{42} \]

where \(W_{cc}\) is the wave impedance of all waves with propagation constant \(k_c\), and \(W_{MM}\) of all waves with constant \(k_M\). As was noted, from expressions (36 and 37), the \(k_c\)-waves exist only in the steel conductors, and the \(k_M\)-waves only in the copper conductors. Therefore \(W_{cc}\), the wave impedance of the wave \(k_c\), is exactly equal to the wave impedance of the interconductor wave of the symmetrical bundle of steel conductors, while \(W_{MM}\) is the expression for the wave impedance of the interconductor wave of the symmetrical bundle of copper conductors.

5. General solution of the original system of equations

After determining the relations between the coefficients \(P, P'\) and \(Q, Q'\), there remain \(2(n+m)\) independent constants: all \(2(m+n)\) constants must be determined from the boundary conditions and, consequently, the relations between them are determined not by the parameters of the line, but by its operating regime.

The general solution of the original system is written in the form:

\[ U_i=P_{c1}e^{-k_1x}+P'_{c1}e^{k_1x}+P_{c2}e^{-k_2x}+P'_{c2}e^{k_2x}+P_{ic}e^{-k_cx}+P'_{ic}e^{k_cx}, \]

\[ I_i=\frac{P_{c1}e^{-k_1x}-P'_{c1}e^{k_1x}}{W_{c1}} +\frac{P_{c2}e^{-k_2x}-P'_{c2}e^{k_2x}}{W_{c2}} +\frac{P_{ic}e^{-k_cx}-P'_{ic}e^{k_cx}}{W_{cc}}, \tag{43} \]

for all steel conductors \(i=1,2\ldots n\), and

\[ U_i=P_{m1}e^{-k_1x}+P'_{m1}e^{k_1x}+P_{m2}e^{-k_2x}+P'_{m2}e^{k_2x}+P_{i.m}e^{-k_mx}+P'_{i.m}e^{k_mx}; \]

\[ I_i=\frac{P_{m1}e^{-k_1x}-P'_{m1}e^{k_1x}}{W_{m1}} +\frac{P_{m2}e^{-k_2x}-P'_{m2}e^{k_2x}}{W_{m2}} +\frac{P_{i.m}e^{-k_mx}-P'_{i.m}e^{k_mx}}{W_{m.m}}; \tag{44} \]

for all copper conductors \(i=(n+1),(n+2)\ldots(n+m)\).

Here, as before, two indices have been introduced for the voltages, currents, and wave impedances of the individual waves. The first index indicates to which conductors, and the second—to which wave—the given quantity pertains.

6. Determination of the arbitrary constants from the conditions at the beginning of the line

At the beginning of the line all voltages and currents are given. Putting \(x=0\), we obtain \(2(n+m)\) equations for determining the \(2(n+m)\) arbitrary constants. Solving them with respect to \(P\) and \(Q\) and denoting

\[ U_{x=0}=U(0);\qquad I_{x=0}=I(0), \]

we obtain

\[ P_{c1}= \frac{ \dfrac{\eta_2}{n}\sum_{i=1}^{n}U_i(0) -\dfrac{1}{m}\sum_{i=n+1}^{n+m}U_i(0) }{ 2(\eta_2-\eta_1) } + W_{c1}\, \frac{ \dfrac{\eta'_2}{n}\sum_{i=1}^{n}I_i(0) -\dfrac{1}{m}\sum_{i=n+1}^{n+m}I_i(0) }{ 2(\eta'_2-\eta'_1) }; \tag{45} \]

\[ P_{c2}= \frac{ \dfrac{1}{m}\sum_{i=n+1}^{n+m}U_i(0) -\dfrac{\eta_1}{n}\sum_{i=1}^{n}U_i(0) }{ 2(\eta_2-\eta_1) } + W_{c2}\, \frac{ \dfrac{1}{m}\sum_{i=n+1}^{n+m}I_i(0) -\dfrac{\eta'_1}{n}\sum_{i=1}^{n}I_i(0) }{ 2(\eta'_2-\eta'_1) }; \]

\[ P_{ic}= \frac{1}{2} \left[ U_i(0)-\frac{1}{n}\sum_{i=1}^{n}U_i(0) \right] + \frac{1}{2}W_{cc} \left[ I_i(0)-\frac{1}{n}\sum_{i=1}^{n}I_i(0) \right]; \]

\[ i=1,2\ldots n. \]

\[ P_{i.m}= \frac{1}{2} \left[ U_i(0)-\frac{1}{m}\sum_{i=n+1}^{n+m}U_i(0) \right] + \frac{1}{2}W_{m.m} \left[ I_i(0)-\frac{1}{m}\sum_{i=n+1}^{n+m}I_i(0) \right]; \]

\[ i=(n+1),(n+2)\ldots(n+m). \]

The expressions for \(P'_1, P'_2, P'_{1c}\), and \(P'_{1m}\) differ only in the signs before \(W_{c1}, W_{c2}, W_{cc}\), and \(W_{mm}\).

Suppose that the waves \(k_c\) and \(k_1\) have greater attenuation than \(k_m\) and \(k_2\). By eliminating the waves \(k_c\) and \(k_1\), we thereby achieve concentration of the energy in the waves \(k_m\) and \(k_2\), which have smaller attenuation. Let us determine under what terminal conditions the coefficients \(P_1, P'_1, P_{ic}\), and \(P'_{ic}\) become zero.

Taking into account that the sum of the voltages of the wave \(k_c\) over all wires is equal to zero, we find from (43) that \(P_{ic}=P'_{ic}=0\) in the case when the voltages on all steel wires and the currents are equal at the beginning of the line:

\[ U_1(0)=U_2(0)=\ldots=U_n(0), \]

\[ I_1(0)=I_2(0)=\ldots=I_n(0). \tag{47} \]

From (45) we obtain, for the condition

\[ P_{c1}=P'_{c1}=0; \]

\[ -\frac{\eta_2}{n}\sum_{i=1}^{n} U_i(0) = \frac{1}{m}\sum_{i=n+1}^{n+m} U_i(0); \]

\[ -\frac{\eta'_2}{n}\sum_{i=1}^{n} I_i(0) = \frac{1}{m}\sum_{i=n+1}^{n+m} I_i(0). \tag{48} \]

Conditions (47) and (48) together give:

\[ \eta_2 U_{cm}(0) = \frac{1}{m}\sum_{i=n+1}^{n+m} U_i(0); \]

\[ \eta_2 I_{cm}(0) = \frac{1}{m}\sum_{i=n+1}^{m+n} I_i(0). \tag{49} \]

Condition (49) shows that the voltages on the steel wires must be \(\eta_2\) times smaller than the average voltage of the copper wires.

Further, from expressions (37) and (44) it is clear that, when the terminal voltages of the copper wires are equal, the interwire wave \(k_m\) in the copper wires also disappears.

In conclusion, let us consider the conditions under which groups of equipotential wires arise.\(^{1)}\) Expressions (43–46) make it possible to draw the following conclusions:

1. The generator is connected to all copper wires. Then the interwire wave \(k_m\) is absent. If, in addition, the terminal resistances of all steel wires are equal, i.e., all terminal voltages on these wires are equal, then the interwire wave in the steel wires is also absent. In the bundle there exist 2 groups of wires with identical voltages: the group of steel wires and the group of copper wires.

The same result, of course, is obtained when the generator is connected to all steel wires, with equality of the terminal conditions of the copper wires.

1. The generator is connected only to several copper wires. The remaining copper wires are not excited. Since in this case the terminal voltages of the copper wires are not all equal, there exists an interwire wave \(k_m\) in addition to the waves \(k_1\) and \(k_2\). If, in addition, all terminal

\(^{1)}\) A group of equipotential wires may be replaced by one equivalent wire.

the resistances of the steel wires are equal, and if the terminal conditions of all unexcited copper wires are also equal, then three groups of wires are formed, with voltages equal for all wires of each group:

a) the group of steel wires,
b) the group of unexcited copper wires,
c) the group of excited copper wires.

1. In the general case, the number of groups of wires with identical voltages depends on the number of groups of wires with identical terminal conditions—separately for copper and steel.

Examples of practical use of the results obtained will be presented in a separate paper.

Appendix

Transform the determinant (23)

\[ \Delta_m^n = \left| \begin{array}{ccccc} (\gamma_s^c-k^2)&\gamma^c&\ldots&\gamma^c&\gamma^c\\ \gamma^c&(\gamma_s^c-k^2)&\ldots&\gamma^c&\gamma^c\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \gamma^\mu&\gamma^\mu&\ldots&(\gamma_s^\mu-k^2)&\gamma^\mu\\ \gamma^\mu&\gamma^\mu&\ldots&\gamma^\mu&(\gamma_s^\mu-k^2) \end{array} \right| = (\gamma^c)^n(\gamma^\mu)^m \left| \begin{array}{ccccc} C&1&\ldots&1&1\\ 1&C&\ldots&1&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&M&1\\ 1&1&\ldots&1&M \end{array} \right| \tag{50} \]

where

\[ C=\frac{\gamma_s^c-k^2}{\gamma^c}, \qquad M=\frac{\gamma_s^\mu-k^2}{\gamma^\mu}. \]

Adding all the remaining columns to the last column, we obtain:

\[ \left| \begin{array}{ccccc} C&1&\ldots&1&C+m+n-1\\ 1&C&\ldots&1&C+m+n-1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&M&M+n+m-1\\ 1&1&\ldots&1&M+n+m-1 \end{array} \right| = \left| \begin{array}{ccccc} C&1&\ldots&1&C+m+n-1\\ 1&C&\ldots&1&C+m+n-1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&M&C+m+n-1\\ 1&1&\ldots&1&C+m+n-1 \end{array} \right| + \left| \begin{array}{ccccc} C&1&\ldots&1&0\\ 1&C&\ldots&1&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&M&(M-C)\\ 1&1&\ldots&1&(M-C) \end{array} \right|. \]

In the first determinant on the right-hand side of the equality we take \((C+m+n-1)\) outside the determinant sign and subtract the last column from all the remaining ones. Expanding, further, the determinant by the elements of the last row, we obtain

\[ \Delta_1 = \left| \begin{array}{ccccc} C&1&\ldots&1&1\\ 1&C&\ldots&1&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&M&1\\ 1&1&\ldots&1&1 \end{array} \right| \cdot (C+m+n-1) = \left| \begin{array}{ccccc} (C-1)&0&\ldots&0&1\\ 0&(C-1)&\ldots&0&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\ldots&(M-1)&1\\ 0&0&\ldots&0&1 \end{array} \right| \,(C+m+n-1) \]

\[ = [C+m+n-1]\,(C-1)^n(M-1)^{m-1}. \]

In the second determinant we take \((M-C)\) outside the determinant sign and subtract the last column from all the others:

\[ \Delta_{II}=(M-C)\cdot \begin{vmatrix} C&1&\ldots&1&0\\ 1&C&\ldots&1&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&M&1\\ 1&1&\ldots&1&1 \end{vmatrix} = (M-C)\cdot \begin{vmatrix} C&1&\ldots&1&0\\ 1&C&\ldots&1&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\ldots&(M-1)&1\\ 0&0&\ldots&0&1 \end{vmatrix}. \]

In the lower left corner there is a minor consisting entirely of zeros. We can therefore represent the second determinant as the product of the minor

\[ \left. \begin{vmatrix} C&1&\ldots&1\\ 1&C&\ldots&1\\ \cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&C \end{vmatrix} \right\} n \]

by its algebraic complement [3]

\[ \left. \begin{vmatrix} (M-1)&0&\ldots&1\\ 0&(M-1)&\ldots&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\ldots&1 \end{vmatrix} \right\} m \]

Both minors are readily transformed:

\[ \begin{vmatrix} (M-1)&0&\ldots&1\\ 0&(M-1)&\ldots&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\ldots&1 \end{vmatrix} =(M-1)^{m-1}; \quad \begin{vmatrix} C&1&\ldots&1\\ 1&C&\ldots&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&C \end{vmatrix} =(C+n-1) \begin{vmatrix} 1&1&\ldots&1\\ 1&C&\ldots&1\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&1&\ldots&C \end{vmatrix} = \]

\[ =(C+n-1)\cdot \begin{vmatrix} 1&0&\ldots&0\\ 1&(C-1)&\ldots&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 1&0&\ldots&(C-1) \end{vmatrix} =(C+n-1)\cdot(C-1)^{\,n-1}. \]

Finally, the determinant \(\Delta_m^n\) is written as follows:

\[ \begin{aligned} \Delta_m^n &=\{(C+m+n-1)(C-1)^n(M-1)^{m-1} +(M-C)(M-1)^{m-1}(C-1)^{n-1} \\ &\qquad\cdot(C+n-1)\}(\gamma^c)^n(\gamma^u)^m \\ &=(C-1)^{n-1}\cdot(M-1)^{m-1} \{(C-1)(C+m+n-1) \\ &\qquad +(M-C)(C+n-1)\}(\gamma^c)^n(\gamma^u)^m \\ &=(C-1)^{n-1}\cdot(M-1)^{m-1} \{(C-1)(M-1)\cdot n(M-1)+m(C-1)\} (\gamma^c)^n(\gamma^u)^m . \end{aligned} \tag{52} \]

Substitution of \(C\) and \(M\) from (55) leads to expression (23).

References

1. Kovalenkov, V. I. “Decomposition of the Equations Expressing Electromagnetic Processes in Linear Circuits.” Avtomatika i Telemekhanika, vol. VIII, no. 4, 1947.

2. Neiman, L. R., and Kalantarov, P. L. Theoretical Foundations of Electrical Engineering. Part III. Energoizdat, 1948.

3. Kurosh, A. G. A Course in Higher Algebra. GITTL, 1949.

4. Stepanov, V. V. A Course in Differential Equations. OGIZ, Gostekhizdat, 1950.

5. Nelepets, V. “Inductive Radio Communication.” Radio, no. 1, 1951.

6. Rylov, A. P., Chernetsov, V., and Khachevsky, B. “Train Radio Communication on the Omsk Railway.” Zheleznodorozhnyi Transport, no. 2, 1950.