CALCULATION OF THE INDUCTION FIELD OF A MULTIWIRE LINE WITH SPATIAL SYMMETRY

Yu. A. Sher

The article presents an approximate method, proposed by the author, for calculating the electromagnetic field of a multiwire line containing dissimilar (nonferrous and steel) conductors.

The calculation is divided into two stages:

a) calculation of currents in the conductors for any number of nonferrous and steel conductors and for arbitrary connection of the generator to the conductors;

b) calculation of the field in the induction zone.

The method can be used directly in two practically important problems:

1. In calculating the field strength of a bundle of conductors (inductive coupling with a moving train).

2. In determining the influence of high-frequency channels of power-transmission lines on communication lines. In this case, not only the currents in copper conductors but also the currents in steel cables may be taken into account.

Since frequencies of the order of hundreds of kc/s are used for h.f. channels, in formulas (9) it is necessary to take the expressions of the function \(J\) corresponding precisely to these frequencies.

The phenomenon of wave propagation in communication lines is more complex than in power-transmission lines; therefore, the application of the method is illustrated by examples of inductive coupling with a moving train.

1. Theoretical Part

Propagation of Waves in a Bundle of Conductors

The process of wave propagation along a multiwire line is described by the following system of equations:

\[ -\frac{d}{dz}\|U\|=\|Z\|\cdot\|I\|;\qquad -\frac{d}{dz}\|I\|=\|Y\|\cdot\|U\|; \tag{1} \]

where

\[ \|U\|= \begin{Vmatrix} U_1\\ U_2\\ \ldots\\ U_n \end{Vmatrix}; \qquad \|I\|= \begin{Vmatrix} I_1\\ I_2\\ \ldots\\ I_n \end{Vmatrix}; \qquad \|Z\|= \begin{Vmatrix} Z_{11}&Z_{12}&\ldots&Z_{1n}\\ Z_{21}&Z_{22}&\ldots&Z_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ Z_{n1}&Z_{n2}&\ldots&Z_{nn} \end{Vmatrix}; \]

\[ \|Y\|= \begin{Vmatrix} Y_{11}&(-Y_{12})&\ldots&(-Y_{1n})\\ (-Y_{21})&Y_{22}&\ldots&(-Y_{2n})\\ \ldots&\ldots&\ldots&\ldots\\ (-Y_{n1})&(-Y_{n2})&\ldots&Y_{nn} \end{Vmatrix}; \]

\(U_i\)—the voltage between the \(i\)-th conductor and the earth;

\(I_i\) is the current in the \(i\)-th conductor; \(Z_{ii}\) and \(Z_{ik}\) are the total self and mutual impedances, respectively; \(Y_{ii}\) and \(Y_{ik}\) are the total self and mutual admittances, respectively.

For a large number of conductors in a bundle, the quantitative dependences of the exact theory, based on equations (1), are practically unsuitable for engineering calculations.

Thus, for example, even in computing the partial capacitances of twenty conductors, determinants of the 20th order would have to be evaluated, and the total number of such computations would be 210 (190 computations of mutual and 20 of self partial capacitances). Therefore, for a large number of conductors, the self and mutual potential coefficients are usually averaged, which leads to simple calculation formulas for capacitance.

The assumption of equality of the self and mutual potential coefficients means replacing the real bundle of conductors of a multiconductor line by one with spatial symmetry. As experiments carried out by the author have shown, in determining the field strength of a bundle of conductors, wave attenuation, and other factors of inductive coupling, averaging the parameters is entirely permissible, and the theory of a multiconductor line with spatial symmetry can be used for an approximate description of the process of wave propagation in a bundle of conductors under inductive coupling.

The self impedances of copper and steel conductors differ sharply at high frequencies, which must be taken into account.

Averaging the parameters of the conductors simplifies system (1):

\[ \|Z\| = \left\| \begin{array}{cccc} Z_c & Z & \ldots & Z \\ Z & Z_c & \ldots & Z \\ \ldots & \ldots & \ldots & \ldots \\ Z & Z & \ldots & Z_M \\ Z & Z & \ldots & Z_M \end{array} \right\| \begin{array}{l} n\\[3.5em] m \end{array} \quad;\quad \|Y\| = \left\| \begin{array}{cccc} Y_S & (-Y) & \ldots & (-Y) \\ (-Y) & Y_S & \ldots & (-Y) \\ \ldots & \ldots & \ldots & \ldots \\ (-Y) & (-Y) & \ldots & Y_S & (-Y) \\ (-Y) & (-Y) & \ldots & (-Y) & Y_S \end{array} \right\| \quad;\tag{2} \]

\[ \underbrace{\hspace{4.5cm}}_{m+n} \qquad\qquad \underbrace{\hspace{5.5cm}}_{m+n} \]

where “\(n\)” is the number of steel conductors, and “\(m\)” the number of copper conductors in the bundle; \(Z_c\) and \(Z_M\) are the total self impedances of the steel and copper conductors, respectively; \(Z\) is the mutual impedance between any two conductors; \(Y_S\) and \(Y\) are the self and mutual admittances, respectively.

System (1), under the simplifications (2), has the solution (3) [1].

For steel conductors:

\[ U_i = P_{c1} e^{-k_1 z} + P'_{c1} e^{k_1 z} + P_{c2} e^{-k_2 z} + P'_{c2} e^{k_2 z} + P_{ic} e^{-k_c z} + P'_{ic} e^{k_c z}, \tag{3} \]

\[ I_i = Q_{c1} e^{-k_1 z} + Q'_{c1} e^{k_1 z} + Q_{c2} e^{-k_2 z} + Q'_{c2} e^{k_2 z} + Q_{ic} e^{-k_c z} + Q'_{ic} e^{k_c z}, \]

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

For copper conductors:

\[ U_i = P_{M1} e^{-k_1 z} + P'_{M1} e^{k_1 z} + P_{M2} e^{-k_2 z} + P'_{M2} e^{k_2 z} + P_{iM} e^{-k_M z} + P'_{iM} e^{k_M z}, \]

\[ I_i = Q_{M1} e^{-k_1 z} + Q'_{M1} e^{k_1 z} + Q_{M2} e^{-k_2 z} + Q'_{M2} e^{k_2 z} + Q_{iM} e^{-k_M z} + Q'_{iM} e^{k_M z}, \]

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

where \(U_i\) and \(I_i\) are the voltage and current in the \(i\)-th conductor; \(P\) and \(P'\) are the voltages of the incident and reflected waves at the beginning of the line; \(Q\) and \(Q'\) are the currents of the incident and reflected waves at the beginning of the line.

Of the two indices attached to the coefficients \(P\) and \(Q\), the first indicates to which conductors, and the second—to which wave the given coefficient refers.

Further, \(k_1\), \(k_2\), \(k_c\), and \(k_M\) are the propagation constants of the independent waves.

\[ k_c=\sqrt{\gamma_{Sc}-\gamma_c}, \]

\[ k_M=\sqrt{\gamma_{SM}-\gamma_M}; \]

\[ k_{1,2}^{2}=\frac{k_c^{2}+k_M^{2}+n\gamma_c+m\gamma_M}{2}\ \pm \]

\[ \pm \sqrt{ \left(\frac{k_c^{2}-k_M^{2}}{2}\right)^{2} +\left(\frac{k_c^{2}-k_M^{2}}{2}\right)(n\gamma_c-m\gamma_M) +\left(\frac{n\gamma_c+m\gamma_M}{2}\right)^{2} }, \tag{4} \]

where

\[ \gamma_{Sc}=Z_cY_S-(m+n-1)ZY; \]

\[ \gamma_c=-Z_cY+ZY_S-(m+n-2)ZY; \]

\[ \gamma_{SM}=Z_MY_S-(m+n-1)ZY; \]

\[ \gamma_M=-Z_MY+ZY_S-(m+n-2)ZY. \]

Mathematical analysis shows [1] that in a multiwire line with spatial symmetry, containing two groups of conductors (steel and nonferrous), the following independent waves may exist:

a) \((m-1)\) waves with the same propagation constant \(k_M\). These waves propagate exclusively in the nonferrous conductors. The sum of the currents in all conductors is equal to zero; consequently, these waves are interconductor waves, and no currents flow through the ground;

b) \((n-1)\) waves with the same propagation constant \(k_c\). These waves propagate only in the steel conductors and are also interconductor waves;

c) two waves with different propagation constants \(k_1\) and \(k_2\). These waves propagate in all conductors of the bundle. The voltages and currents of each of them are mutually equal in all nonferrous and all steel conductors. For each of these waves the following relations hold:

\[ \frac{P_{M1}}{P_{c1}}=\eta_1;\qquad \frac{P_{M2}}{P_{c2}}=\eta_2;\qquad \frac{Q_{M1}}{Q_{c1}}=\eta'_1;\qquad \frac{Q_{M2}}{Q_{c2}}=\eta'_2, \tag{5} \]

where:

\(P_{M1}\) and \(P_{M2}\)—voltages on the nonferrous conductors of the first and second waves, respectively;

\(P_{c1}\) and \(P_{c2}\)—voltages on the steel conductors of the 1st and 2nd waves, respectively;

\(Q_{M1}\) and \(Q_{M2}\)—currents in the nonferrous conductors of the 1st and 2nd waves, respectively;
\(Q_{c1}\) and \(Q_{c2}\)—the same in the steel conductors.

\[ \eta_1=\frac{\gamma_M}{\gamma_c}\cdot \frac{(k_c^{2}-k_1^{2})}{(k_M^{2}-k_1^{2})}; \qquad \eta'_1=\frac{k_c^{2}-k_1^{2}}{k_M^{2}-k_1^{2}}; \]

\[ \eta_2=\frac{\gamma_M}{\gamma_c}\cdot \frac{(k_c^{2}-k_2^{2})}{(k_M^{2}-k_2^{2})}; \qquad \eta'_2=\frac{k_c^{2}-k_2^{2}}{k_M^{2}-k_2^{2}}. \tag{6} \]

The coefficients \(P\) and \(Q\) are determined from the conditions at the beginning of the line.

\[ \begin{aligned} P_{c1} &= \frac{ \dfrac{\eta_2}{n}\sum_{i=1}^{n} U_i(0) -\dfrac{1}{m}\sum_{i=n+1}^{n+m} U_i(0) }{ 2(\eta_2-\eta_1) } + W_{c1}\, \frac{ \dfrac{\eta'_2}{n}\sum_{i=1}^{n} I_i(0) -\dfrac{1}{m}\sum_{i=n+1}^{n+m} I_i(0) }{ 2(\eta'_2-\eta'_1) }; \\[1.2em] P_{c2} &= \frac{ \dfrac{1}{m}\sum_{i=n+1}^{n+m} U_i(0) -\dfrac{\eta_1}{n}\sum_{i=1}^{n} U_i(0) }{ 2(\eta_2-\eta_1) } + W_{c2}\, \frac{ \dfrac{1}{m}\sum_{i=n+1}^{n+m} I_i(0) -\dfrac{\eta'_1}{n}\sum_{i=1}^{n} I_i(0) }{ 2(\eta'_2-\eta'_1) }; \\[1.2em] P_{ic} &= \frac{1}{2} \left[ U_i(0)-\frac{1}{n}\sum_{i=1}^{n} U_i(0) \right] +\frac{1}{2}W_{cc} \left[ I_i(0)-\frac{1}{n}\sum_{i=1}^{n} I_i(0) \right]; \\[1.2em] P_{kM} &= \frac{1}{2} \left[ U_i(0)-\frac{1}{m}\sum_{i=n+1}^{n+m} U_i(0) \right] +\frac{1}{2}W_{MM} \left[ I_i(0)-\frac{1}{m}\sum_{i=n+1}^{n+m} I_i(0) \right]; \\[0.8em] &\qquad (i=1,2\ldots n.)\quad k=(n+1),(n+2)\ldots (n+m). \end{aligned} \tag{7} \]

The voltages and currents are related by the expressions:

\[ W_{c1}=\frac{P_{c1}}{Q_{c1}};\qquad W_{c2}=\frac{P_{c2}}{Q_{c2}};\qquad W_{cc}=\frac{P_{ic}}{Q_{ic}}; \]

\[ W_{M1}=\frac{P_{M1}}{Q_{M1}};\qquad W_{M2}=\frac{P_{M2}}{Q_{M2}};\qquad W_{MM}=\frac{P_{iM}}{Q_{iM}}; \]

\[ W_{cc}=\frac{Z_c-Z}{k_c};\qquad W_{MM}=\frac{Z_M-Z}{k_M}; \]

\[ W_{M1}=\left[Z_M+(m-1)Z+\frac{nZ}{\eta'_1}\right]\frac{1}{k_1}; \qquad W_{c1}=\frac{\eta'_1}{\eta_1}\cdot W_{M1}. \]

\[ W_{c2}=\left[Z_c+(n-1)Z+\eta'_2 m z\right]\cdot \frac{1}{k_2}; \qquad W_{m2}=\frac{\eta'_2}{\eta_2}\cdot W_{c2}. \tag{8} \]

In expressions (7) and (8) the following notation is used:

\(U_i(0)\) and \(I_i(0)\)—the voltage and current in the \(i\)-th wire at the beginning of the line (these quantities are assumed known).

\(P_{ic}\) and \(Q_{ic}\)—the voltage and current of the wave \(k_c\) in the \(i\)-th wire;
\(P_{kM}\) and \(Q_{kM}\)—the voltage and current of the wave \(k_M\) in the \(k\)-th wire;
\(P_{c1}\) and \(Q_{c1}\)—the voltage and current of the wave \(k_1\) in each of the steel wires (\(P_{c2}\) and \(Q_{c2}\)—the same for the wave \(k_2\));
\(P_{M1}\) and \(Q_{M1}\)—the voltage and current of the wave \(k_1\) in each of the nonferrous wires (\(P_{M2}\) and \(Q_{M2}\)—the same for the wave \(k_2\)).

Field of the “wire—earth” system

Experiments show that the nature of the variation of the electric field of the “wire—earth” system at frequencies of the order of \(2.5\) MHz is sufficiently

is accurately described by formulas (9) for the distances of interest—up to 70 m from the bundle of wires.

\[ \begin{aligned} E_x&=-j\,\frac{2kI}{\omega}\cdot 9\cdot 10^9\cdot \frac{\partial}{\partial x}\ln\frac{\rho''}{\rho'};\quad \left(\frac{\mathrm{V}}{\mathrm{m}}\right),\\ E_y&=-j\,\frac{2kI}{\omega}\cdot 9\cdot 10^9\cdot \frac{\partial}{\partial y}\ln\frac{\rho''}{\rho'};\quad \left(\frac{\mathrm{V}}{\mathrm{m}}\right),\\ E_z&=-j\,2\omega\cdot I\cdot 10^{-7} \left[\left(1+\frac{k^2c^2}{\omega^2}\right)\ln\frac{\rho''}{\rho'}+J\right];\quad \left(\frac{\mathrm{V}}{\mathrm{m}}\right). \end{aligned} \tag{9} \]

In these formulas:

\[ I=I_m(0)e^{j\omega t\pm kz};\qquad k=\sqrt{(R+j\omega L)(G+j\omega C)}; \]

\(E_x, E_y, E_z\) are the components of the electric-field intensity at the point \(M\);
\(I\) is the total current in the wire;
\(k\) is the propagation constant of the wave in the wire;
\(\omega\) is the angular frequency of the current;
\(c\) is the velocity of light in vacuum.
The meaning of \(\rho'\) and \(\rho''\) is clear from Fig. 1.
\(J\) is a function depending on the coordinates of the point and on the properties of the earth.

\[ J=2(A-jB); \]

\[ A=\frac{\cos\Theta}{\sqrt{2}\,p}-\frac{\cos3\Theta}{\sqrt{2}\,p^3}+\cdots \]

\[ B=\frac{\cos\Theta}{\sqrt{2}\,p}-\frac{\cos2\Theta}{p^2} +\frac{1}{2}\cdot\frac{\cos3\Theta}{p^3}+\cdots, \tag{10} \]

where

\[ p=\sqrt{4\pi\cdot\omega\cdot\sigma_{(\mathrm{cgs})}\cdot\rho''_{(\mathrm{cm})}}, \]

\(\sigma\) is the specific conductivity of the earth in units of the absolute electromagnetic system.

Fig. 1

The components of the magnetic-field intensity may be obtained directly from the equation:

\[ \operatorname{rot}\vec E=-j\omega\mu\vec H. \]

Expressions (9) may be used to calculate the field of a bundle of wires if the currents in the wires are known.

Using formulas (9), it is easy to show that the field of a multiwire line with currents in the wires equal in magnitude and phase reduces to the field of a single-wire line located at the mean height and carrying the total current.

2. Calculation formulas

Attenuation of waves

Using the assumptions customary for high frequencies:

1) \(G\ll j\omega C,\)
2) \(L_a\gg L_i\) and so on,

From the exact expressions of Section 1, as a result of a number of transformations, one can obtain the calculation formulas given below.

Attenuation of the interwire wave \(k_M\) in nonferrous wires

\[ b_M \cong \frac{R_M+R_S-R_m}{120(\xi_S-\xi_m)} . \tag{11} \]

Attenuation of the interwire wave \(k_c\) in steel wires

\[ b_c \cong \frac{R_c}{120(\xi_S-\xi_m)} . \tag{12} \]

Attenuation of the waves \(k_1\) and \(k_2\)

\[ b_{1,2} \cong \frac{1}{2} b_\psi \pm \sqrt{b_c \cdot b_{2\psi}+b_S^2}. \tag{13} \]

In these formulas:
\(R_M\) is the kilometer resistance of a nonferrous wire, taking account of the surface effect; \(R_c\) is the same for a steel wire; \(R_S\) is the mean additional resistance, depending on the properties of the earth. For \(f=2.5\) MHz, \(R_S\) is equal to

\[ R_S \cong \frac{4\omega \cdot 10^{-4}}{\sqrt{2\cdot p}}\left(\frac{\Omega}{\mathrm{km}}\right); \qquad p=2h_{(\mathrm{cm})}\cdot \sqrt{4\pi\sigma\omega}. \tag{14} \]

\(R_m\) is the mean additional mutual resistance. With a large number of wires, computation of this quantity requires a great expenditure of labor. As a result, \(R_m\) proves, for different profiles of communication lines, to be approximately \((7 \div 10\%)\) less than the value \(R_S\). One may take

\[ R_m \cong (0.9 \div 0.92)\cdot R_S, \tag{15} \]

and

\[ b_m \cong \frac{R_M+0.08\cdot R_S}{120(\xi_S-\xi_m)} . \]

Further,

\[ \xi_S=\ln \frac{2h_{\mathrm{avg}}}{a_{\mathrm{avg}}} \qquad \text{or} \qquad \xi_S=\frac{\alpha_S}{18\cdot 10^6}, \tag{16} \]

\[ \xi_m=\ln \frac{\rho'_{\mathrm{avg}}}{\rho'_{\mathrm{avg}}} \qquad \text{or} \qquad \xi_m=\frac{\alpha}{18\cdot 10^6}, \]

where \(h_{\mathrm{avg}}\) is the suspension height of the wires averaged over the entire bundle; \(a_{\mathrm{avg}}\) is the radius of the wires averaged over the entire bundle; \(\alpha_S\) is the self potential coefficient averaged over the entire bundle; \(\alpha\) is the mean value of the mutual potential coefficient.

In the formulas the following notation is also used:

\[ b_m \cong \frac{R_m}{120\xi_m}; \]

\[ \psi_n=1-nS \qquad b_\psi=b_c\cdot \psi_n+b_m'\cdot \psi_S, \]

\[ \psi_{2n}=1-2nS \qquad b_{2\psi}=b_c\cdot \psi_{2n}+b_m\cdot \psi_{2S}, \tag{17} \]

\[ \psi_S=(m+n)\cdot S \qquad b_S=b_m\cdot \psi_S-S\cdot n\cdot b_c; \]

\[ \psi_{2S}=(n-m)2\cdot S \]

\[ S=\frac{1}{\dfrac{\xi_S}{\xi_m}+m+n-1}, \]

where \(m\) and \(n\) are the numbers of nonferrous and steel wires, respectively.

Phase Coefficients

Neglecting the internal inductance of the wires leads to equality of the phase coefficients and, consequently, to the conclusion that the propagation velocity of all waves is the same. The interaction of waves having different propagation velocities is the cause of a wave-like variation of currents and voltages along the line; however, this circumstance plays a secondary role, and in a first approximation the assumption of equality of the phase coefficients may be accepted:

\[ \tau_1=\tau_2=\tau_M=\tau_c=\tau_0=\frac{\omega}{c}, \]

where \(c\) is the velocity of light in air.

Ratio of the currents of waves \(k_1\) and \(k_2\) in nonferrous and steel wires

In calculations one may take, approximately,

\[ k^2=(b+j\tau_0)^2 \simeq j\,2\tau_0 b-\tau_0^2 . \tag{18} \]

Then from expressions (6) we obtain the computational formulas:

\[ \eta'_1=\frac{Q_{M1}}{Q_{c1}}\simeq \frac{b_c-b_1}{b_M-b_1}; \tag{19} \]

\[ \eta'_2=-\frac{Q_{M2}}{Q_{c2}}\simeq \frac{b_c-b_2}{b_M-b_2}. \tag{20} \]

Computational formulas for the wave impedances

From expressions (8) we obtain:

\[ W_{MM}\simeq 60(\xi_s-\xi); \tag{21} \]

\[ W_{M1}\simeq 60\left[\xi_s+\left(m-1-\frac{n}{\eta'_1}\right)\xi\right]. \tag{22} \]

Currents of each wave at the beginning of the line

Expressions (7) and (8) can be simplified and reduced to computational formulas.

Since the attenuation of each of the waves on real sections of inductive communication exceeds \(1.5\) nepers, the multiwire line may be regarded as infinitely long. Then \(P'=0\) and \(Q'=0\). We obtain

\[ Q_{c1}=\frac{1}{\eta'_2-\eta'_1} \left[ \eta'_2\frac{1}{n}\sum_{i=1}^{n} I_i(0) - \frac{1}{m}\sum_{i=n+1}^{n+m} I_i(0) \right]; \tag{23} \]

\[ Q_{c2}=\frac{1}{\eta'_2-\eta'_1}\cdot \left[ \frac{1}{m}\sum_{i=n+1}^{n+m} I_i(0) - \eta'_1\frac{1}{n}\sum_{i=1}^{n} I_i(0) \right], \]

where \(I_i(0)\) is the total current in the steel and nonferrous wires at the beginning of the line.

Further, calculations and the physical meaning show that the coefficients \(\eta_1'\) and \(\eta_2'\) differ sharply in magnitude. If wave \(k_1\) has small attenuation, then the main part of the energy is concentrated in the non-steel wires and the ratio of currents, expressed by the coefficient \(\eta_1'\), is large. Conversely, for wave \(k_2\), which has large attenuation, this ratio \((\eta_2')\) is less than unity. Consequently,

\[ \eta_1' \gg 1;\qquad \eta_2' < 1. \tag{24} \]

If the transmitter is connected to the non-steel wires, then the average current in the non-steel wires is much greater than the average value of the current in the steel wires multiplied by the small coefficient \(\eta_2' < 1\).

\[ \frac{\eta_2'}{n}\sum_{i=1}^{n} I_i(0) \ll \frac{1}{m}\sum_{n+1}^{n+m} I_i(0). \tag{25} \]

Taking into account expressions (24), (25), and also neglecting the value \(Q_{m2}\), we obtain the calculation formulas:

\[ Q_{m1} \cong \frac{1}{m}\sum_{n+1}^{n+m} I_i(0),\qquad Q_{c1}=\frac{Q_{m1}}{\eta_1'}, \tag{26} \]

\[ Q_{km}\cong I_k(0)-Q_{m1}. \]

Waves \(k_c\) and \(k_2\) propagate with attenuation exceeding \(1.0\ \dfrac{\mathrm{nep}}{\mathrm{km}}\), and already at distances greater than \(5\text{–}6\ \mathrm{km}\) the field is determined only by the currents of waves \(k_1\) and \(k_m\). Therefore, one may restrict oneself to calculating the latter.

Thus, for calculating the field at distances exceeding \(5\text{–}6\ \mathrm{km}\) from the transmitter, in the first approximation it is sufficient to take into account the data of current measurements only in the non-steel wires.

Formulas for field strength

Formulas (9) are simplified if one takes into account that

\[ k=b+j\tau_0 \cong j\tau_0;\qquad \frac{k}{\omega}\cong \frac{j\tau_0}{\omega}=\frac{j}{c};\qquad \left(\frac{kc}{\omega}\right)^2\cong -1. \]

Of the three components—\(E_x\), \(E_y\), and \(E_z\)—the vertical component \(E_y\) plays the principal role in the receiving process. It is expedient to calculate the communication range from its magnitude.

For distances greater than \(10\text{–}15\ \mathrm{m}\) from the bundle we have:

\[ \frac{\partial}{\partial y}\ln\frac{\rho''}{\rho'} = \frac{2h(x^2+h^2-y^2)} {[x^2+(h+y)^2]\,[x^2+(h-y)^2]}. \]

For \((h+y)^2 \ll x^2\), we obtain the approximate expression:

\[ \frac{\partial}{\partial y}\ln\frac{\rho''}{\rho'} \cong \frac{2h}{x^2}. \]

Finally,

\[ E_y \cong 60\cdot I\,\frac{2h}{\lambda^2}\left(\frac{v}{m}\right);\qquad E_z \cong j\,2\omega I\cdot 10^{-7}\cdot J\left(\frac{v}{m}\right), \tag{27} \]

where \(x,y\) are the coordinates of the point in meters; \(h\) is the height of suspension of the wire in meters; \(I\) is the current in amperes.

If the field \(\Delta E_y\) is produced by a pair of equal currents, equal in magnitude and oppositely directed, the following approximate formulas may be used:

Fig. 2

a) for the vertical arrangement of the currents (wires),

\[ \Delta E_y \simeq 120 \cdot I \cdot \frac{2d_{\text{vert}}}{x^2} \cdot \left(\frac{v}{m}\right), \tag{27'} \]

b) for the horizontal arrangement of the currents,

\[ \Delta E_y \simeq 120 \cdot I \cdot \frac{2h \cdot 2d_{\text{horiz}}}{x^3} \left(\frac{v}{m}\right), \tag{27''} \]

c) an inclined arrangement of the currents is replaced by two pairs: a horizontal and a vertical one.

3. Measurement of Currents at the Beginning of Lines

For measuring currents in wires without opening the circuit, it is expedient to use methods applied in measuring current in antennas and feeders. As an example of a possible design, an instrument manufactured at the Department of Transport Communications of TEMIIT under the supervision of Assoc. Prof. R. M. Shevchuk is shown (Fig. 3).

The main loop, 30 cm long and 3–4 cm wide, is closed through a variable capacitor—for tuning to resonance at the frequency of the current being measured. A sensitive microammeter connected through a coupling loop, with a high-frequency cuprous rectifier, is calibrated

Fig. 3

in milliamperes against thermomilliammeters. If the instrument is suspended on a wire, then at a fixed frequency the e.m.f. and the currents in the loops are proportional to the current being measured in the wire.

The measurement ranges are changed by switching the cuprous rectifier. The parts are mounted on a textolite plate.

Examples of Calculations

Example 1. There is a multiwire line (Fig. 4), consisting of 14 steel wires of diameter 5 mm and two copper wires of diameter 4 mm, arranged on brackets. The mean suspension height for the entire bundle is 5 m. The conductivity of the soil is \(50 \cdot 10^{-14}\) electromagnetic units. The frequency of the generator is 2.39 MHz.

Calculation of Wave Attenuation

1. The total internal resistance, taking the surface effect into account, is found from the formula:

\[ R + j\omega L_i = R_0 \frac{x_0(1+j)}{2\sqrt{2}}, \tag{28} \]

where

\[ x_0 = a\sqrt{\omega\mu\sigma}, \qquad j=\sqrt{-1}, \]

\(\mu=\mu_0\mu'\) is the magnetic permeability of the wire metal;
\(a\) is its radius; \(\sigma\) is the conductivity of the metal; \(R_0\) is the resistance to direct current; \(L_i\) is the internal inductance.

For a copper wire we obtain

\[ R_m + j\omega L_{im} \simeq 34(1+j)\,\frac{\text{ohm}}{\text{km}}, \]

and for a steel wire

\[ R_c + j\omega L_{ic} \simeq 875(1+j)\,\frac{\text{ohm}}{\text{km}}. \]

The relative magnetic permeability of steel is taken as \(\mu'=100\).

1. We compute the mean additional resistance \(R_s\)

\[ p=2h\sqrt{4\pi\omega\sigma} =2\cdot 500\sqrt{4\pi\cdot 1.5\cdot 10^7 \cdot 50\cdot 10^{-14}} \simeq 9.7 \]

for a wire situated at the mean height \(h=5\) m.

\[ R_s \simeq \frac{4\omega\cdot 10^{-4}}{\sqrt{2}\cdot p} = \frac{4\cdot 1.5\cdot 10^7\cdot 10^{-4}}{\sqrt{2}\cdot 9.7} \simeq 440\,\frac{\text{ohm}}{\text{km}}. \]

1. Instead of computing the mean, for the entire bundle, of the quantity \(\xi_s=\ln \dfrac{2h}{a}\), we find it for the mean wire, considering it to be steel:

\[ \xi_s=\ln \frac{10^4}{2.5}=8.3. \]

1. To compute the mean value \(\xi_m=\ln \dfrac{\rho''}{\rho'}\), the simplest procedure is as follows: divide the bundle into two equal parts, replacing each of them by a wire averaged over height. In the present case the imaginary wires are at heights of six and four meters.

\[ \xi_m=\ln \frac{6+4}{6-4}=1.6. \]

1. We find the value of the kilometric attenuation of interwire waves in the copper wires:

\[ b_m \simeq \frac{R_m+0.08R_s}{120(\xi_s-\xi_m)} = \frac{34+35}{120(8.3-1.6)} = \frac{69}{805} \simeq 86\,\frac{\text{m nep}}{\text{km}}. \]

Fig. 4

1. The value of the kilometric attenuation of interwire waves in steel wires:

\[ b_c \cong \frac{R_c}{120(\xi_s-\xi_m)}=\frac{875}{805}\cong 1.1\ \frac{\text{nep}}{\text{km}} . \]

1. Next we compute the auxiliary quantities:

\[ R_m=0.92\cdot 440 \cong 410\ \frac{\text{ohm}}{\text{km}};\qquad b_m=\frac{R_m}{120\xi_m}=\frac{440}{120\cdot 1.6}\cong 2.14\ \frac{\text{nep}}{\text{km}} . \]

\[ s=\frac{1}{\dfrac{\xi_s}{\xi_m}+m+n-1} =\frac{1}{\dfrac{8.3}{1.6}+2+14-1}\cong 0.05. \]

\[ \psi_n=1-ns=1-14\cdot 0.05=0.3, \]

\[ \psi_{2n}=1-2n\cdot s=-0.4, \]

\[ \psi_s=(n+m)\cdot s=(14+2)\cdot 0.05=0.8, \]

\[ \psi_{2s}=(n-m)\cdot 2s=12\cdot 0.1=1.2, \]

\[ b_y=b_c\psi_n+b_m\psi_s=1.1\cdot 0.3+2.14\cdot 0.8=2.05, \]

\[ b_{2y}=b_c\psi_{2n}+b_m\cdot \psi_{2s}=1.1(-0.4)+2.14\cdot 1.2=2.15, \]

\[ b_s=b_m\psi_s-sn\cdot b_c=2.14\cdot 0.8-0.05\cdot 14\cdot 1.1=1.68-0.77=0.95. \]

1. We determine the attenuation of waves \(k_1\) and \(k_2\)

\[ b_{1,2}\cong \frac{1}{2}b_y \mp \frac{1}{2}\sqrt{b_c b_{2y}+b_s^2} =\frac{1}{2}\cdot 2.05 \mp \sqrt{1.1\cdot 2.12+0.95^2}; \]

\[ b_1=1.025-0.89=0.135\ \frac{\text{nep}}{\text{km}};\qquad b_2=1.02+0.89\cong 1.9\ \frac{\text{nep}}{\text{km}} . \]

1. Ratio of currents in the nonferrous and steel wires:

\[ \eta_1'=\frac{b_c-b_1}{b_m-b_1} =\frac{1.1-0.135}{0.085-0.135} =-\frac{0.965}{0.05}=-19 \]

1. Suppose the transmitter is connected into the phantom circuit “two nonferrous wires—ground.” Measurement of the currents in the nonferrous wires (outside the station territory) gave:

\[ I_1(0)=30\ \text{ma},\qquad I_2(0)=30\ \text{ma}. \]

The current of wave \(k_1\) in each of the nonferrous wires:

\[ Q_{m1}=\frac{1}{m}\sum_{n+1}^{n+m} I_k(0)=\frac{30+30}{2}=30\ \text{ma}, \]

and in each of the steel wires:

\[ Q_{c1}=\frac{Q_{m1}}{\eta_1'}=\frac{30}{-19}=-1.5\ \text{ma}. \]

Wave currents \(k_m\) in the nonferrous wires:

\[ Q_{1m}=I_1(0)-Q_{m1}=30-30=0, \]

\[ Q_{2m}=I_2(0)-Q_{m1}=30-30=0. \]

The equivalent current of the nonferrous wires \(I_{\mathrm{eq}\,m}=2\cdot 30=60\ \text{mA}\) is located at the mean height \(6.7\ \text{m}\). The equivalent current of the steel wires

\[ I_{\mathrm{eq}\,s}=\frac{-30.14}{19}\simeq -22\ \text{mA} \]

is located at the mean height for the steel wires, \(5\ \text{m}\).

1. We find the currents at any point of the line. For example, a section of \(30\ \text{km}\) introduces an attenuation \(30\cdot 0.135=4.05\ \text{nepers}\), which corresponds to a reduction of the currents by \(e^{4.05}=58\simeq 60\) times,

\[ I'_{\mathrm{eq}\,m}\simeq \frac{60\ \text{mA}}{60}=1\ \text{mA},\qquad I'_{\mathrm{eq}\,s}=\frac{-22\ \text{mA}}{60}=-0.37\ \text{mA}. \]

1. We compute the vertical component of the field at distances from the bundle greater than \(15\ \text{m}\). For example, at a distance of \(30\ \text{m}\) we have, from the currents of the nonferrous wires:

\[ E_{y m}=\frac{60\cdot 1\cdot 10^{-3}\cdot 2\cdot 6.7}{30^2}\simeq 890\ \frac{\mu\text{V}}{\text{m}}, \]

from the currents of the steel wires:

\[ E_{y s}=-\frac{60\cdot 0.37\cdot 10^{-3}\cdot 2\cdot 5}{30^2}\simeq -250\ \frac{\mu\text{V}}{\text{m}}. \]

The resultant field is

\[ 890-250=640\ \frac{\mu\text{V}}{\text{m}}. \]

1. The voltage on each of the nonferrous wires is

\[ W_{m1}\simeq 60\left[\xi_s+\left(m-1+\frac{n}{\eta_1}\right)\xi\right]= \]

\[ =60\left[8.3+\left(1+\frac{14}{(-19)}\right)1.6\right]\simeq 60\cdot 8.7=523\ \Omega, \]

\[ P_{m1}=W_{m1}\cdot Q_{m1}=523\ \Omega\cdot 0.5\ \text{mA}\simeq 260\ \text{mV}. \]

1. If we introduce the concept of the transition attenuation “bundle–antenna,” defining it as the logarithm of the ratio of the voltage on the excited wires to the emf in the antenna, then for an antenna with an effective height of \(1\ \text{m}\) we find:

\[ g=\ln \frac{U_{\text{line}}}{E_{\text{ant}}}\simeq \ln \frac{260}{0.64}=6\ \text{nepers}. \]

If the influence of the mass of the locomotive is taken into account by an approximate coefficient equal to \(0.5\), then we obtain:

\[ g=\ln \frac{260}{0.64\cdot 0.5}=6.7\ \text{nepers}. \]

Example 2. A multiwire line with the wires arranged on brackets (see Fig. 5) contains 22 steel wires of diameter 5 mm and 4 bimetallic wires of diameter 4 mm. The transmitter frequency is 2100 kHz. The soil conductivity is \(50 \cdot 10^{-14}\) electromagnetic units.

1. The internal resistance (for \(\mu' = 100\)) of the steel wires is

\[ R_c + j \omega L_{ic} = 850(1+j)\,\frac{\Omega}{\text{km}}; \]

of the nonferrous wires,

\[ R_m + j \omega L_{im} \simeq 32(1+j)\,\frac{\Omega}{\text{km}}. \]

1. The auxiliary parameter \(p\) for the mean height \(h = 5\) m is

\[ p = 2h\sqrt{4\pi\omega\sigma} = 10^3 \cdot 2\pi \sqrt{2 f \sigma} = 2\pi \cdot 10^3 \sqrt{2 \cdot 2{,}1 \cdot 10^6 \cdot 5 \cdot 10^{-13}} \simeq 9. \]

Fig. 5

The additional resistance \(R_s\) is

\[ R_s \simeq \frac{4\omega \cdot 10^{-4}}{\sqrt{2}\,p} = \frac{5{,}3 \cdot 10^{-4}}{\sqrt{2}\cdot 9} = 420\ \Omega. \]

1. For the middle wire at height 5 m we find \(\xi_s\):

\[ \xi_s = \ln \frac{2h}{a} = \ln \frac{10^4}{2{,}5} = 8{,}3. \]

1. The quantity \(\xi_m\) for each support is calculated in the same way as in the preceding example: \(\xi'_m = 1{,}65\).

We compute the mean value \(\xi''_m\) for wires located on different supports approximately for the two central wires of each support:

\[ \xi''_m \simeq \ln \frac{\sqrt{10^2 + 2^2}}{2} = \ln \frac{10{,}2}{2} = 1{,}63. \]

The mean value for the whole bundle is

\[ \xi_m = \frac{\xi'_m + \xi''_m}{2} = 1{,}64. \]

1. Attenuation of interwire waves in the nonferrous wires:

\[ b_m \simeq \frac{R_m + 0{,}08 R_s}{120 \cdot (\xi_s - \xi_m)} = \frac{32 + 34}{120 \cdot (8{,}3 - 1{,}64)} = \frac{66}{790} \simeq 84\,\frac{\text{m nep}}{\text{km}}. \]

1. Attenuation of interwire waves in the steel wires:

\[ b_c = \frac{R_c}{120(\xi_s - \xi_m)} = \frac{850}{790} = 1{,}08 \simeq 1{,}1\,\frac{\text{nep}}{\text{km}}. \]

1. We compute the auxiliary quantities

\[ R_m = 0{,}92 \cdot R_s = 0{,}92 \cdot 420 = 386\,\frac{\Omega}{\text{km}}, \]

\[ b_m = \frac{R_m}{120 \xi_m} = \frac{386}{120 \cdot 1{,}64} = 1{,}95\,\frac{\text{nep}}{\text{km}}. \]

\[ s=\frac{1}{\frac{8.3}{1.64}+22+4-1}=0.033,\qquad ns=0.725,\qquad ms=0.132. \]

\[ \psi_n=1-ns=1-0.725=0.275;\qquad \psi_{2n}=1-2ns=1-1.45=-0.45, \]

\[ \psi_s=(m+n)\cdot s=0.86;\qquad \psi_{2s}=(n-m)2s=(22-4)\cdot 0.066=1.19. \]

\[ b_\psi=b_c\psi_n+b_m\psi_s=1.1\cdot 0.275+1.95\cdot 0.86=1.98. \]

\[ b_{2\psi}=b_c\psi_{2n}+b_m\psi_{2s}=1.1(-0.45)+1.95\cdot 1.19=1.825. \]

\[ b_s=1.95\cdot 0.86-0.725\cdot 1.1=0.88. \]

1. Attenuation of waves \(k_1\) and \(k_2\).

\[ b_{1,2}=\frac{1}{2}\cdot 1.98 \mp \sqrt{1.1\cdot 1.82+0.88^2}=0.99\mp 0.835, \]

\[ b_1=0.155\ \frac{\text{nep}}{\text{km}}.\qquad b_2=1.825\ \frac{\text{nep}}{\text{km}}. \]

1. Ratio of currents in nonferrous and steel wires

\[ \eta'_1=\frac{b_c-b_1}{b_m-b_1} =\frac{1.1-0.155}{0.084-0.155} =\frac{0.945}{-0.071} =-13.3. \]

1. Let the transmitter be connected to wire No. 24. The measured data for the currents in the nonferrous wires are:

\[ I_{24}(0)=100\ \text{mA}, \]

\[ I_{25}(0)=40\ \text{mA}, \]

\[ I_{20}(0)=20\ \text{mA}, \]

\[ I_{21}(0)=20\ \text{mA}. \]

The current of wave \(k_1\) in each nonferrous wire is

\[ Q_{m1}=\frac{1}{m}\sum I_x(0)=\frac{1}{4}(100-40-20-20)=5\ \text{mA}. \]

The current of the same wave in each steel wire is

\[ Q_{c1}=\frac{Q_{m1}}{\eta'_1}=\frac{5}{-13.3}\simeq -0.38\ \text{mA}. \]

The equivalent current of wave \(k_1\) in the nonferrous wires, \(4\cdot 5=20\ \text{mA}\), is located at the mean height for these wires,

\[ \frac{1}{4}(7.0+6.4+4.0+4.6)=5.5\ \text{m}. \]

The equivalent current of wave \(k_1\) in the steel wires, \(-22\cdot 0.38\simeq -8.4\ \text{mA}\), is located approximately at the same height.

The currents of the interwire waves \(k_m\) in the nonferrous wires are:

\[ Q_{25}=I_{25}(0)-Q_{m1}=-40-5=-45\ \text{mA}, \]

\[ Q_{20}=I_{20}(0)-Q_{m1}=-20-5=-25\ \text{mA}, \]

\[ Q_{21}=I_{21}(0)-Q_{m1}=-20-5=-25\ \text{mA}, \]

\[ Q_{24}=I_{24}(0)-Q_{m1}=100-5=95\ \text{mA}. \]

1. Let us find the field of the currents of wave \(k_1\) at the beginning of the line. The equivalent currents \(+20\ \mathrm{ma}\) and \(-8.4\ \mathrm{ma}\) are located horizontally (\(h \cong 5.5\ \mathrm{m}\)). A pair of currents \(\pm 8.4\ \mathrm{ma}\), equal in magnitude and oppositely directed, creates a field decreasing, in accordance with formula \((27^{\mathrm{II}})\), inversely proportional to the cube of the distance. Therefore, at distances greater than \(10\)–\(15\ \mathrm{m}\) this field may be neglected. The field of the earth wave will be, for a current \(20-8.4=11.6\ \mathrm{ma}\) at a distance of \(20\ \mathrm{m}\) from the bundle:

\[ E_{y_k} \cong \frac{60\cdot I\cdot 2h}{x^2} = \frac{60\cdot 11.6\cdot 10^{-3}\cdot 2\cdot 5.5}{400} = 19\ \frac{\mathrm{mv}}{\mathrm{m}}. \]

1. Next we compute the field of the currents of the interwire waves \(k_m\) at the beginning of the line. An interwire wave \(\pm 45\ \mathrm{ma}\), with distance between currents \(2d=0.6\ \mathrm{m}\), creates the field:

\[ \Delta E'_{y_m} \cong \frac{120\cdot I\cdot 2d}{x^2} = \frac{120\cdot 45\cdot 10^{-3}\cdot 0.6}{400} = 8.1\ \frac{\mathrm{mv}}{\mathrm{m}}. \]

Two equal currents in wires No. 20 and 21 may be replaced by the equivalent

\[ 2\cdot(-25)=-50\ \mathrm{ma}, \]

located at a height

\[ \frac{1}{2}(4.0+4.6)=4.3\ \mathrm{m}, \]

after which the field of the interwire wave \(\pm 50\ \mathrm{ma}\), with distance between currents \(2d=2.7\ \mathrm{m}\), is found from formula \((27')\):

\[ \Delta E''_{y_m} \cong \frac{120\cdot 50\cdot 10^{-3}(-2.7)}{400} = -40.5\ \frac{\mathrm{mv}}{\mathrm{m}}. \]

The current \((-50)\ \mathrm{ma}\), flowing in wires No. 20 and 21, is located above the current \(+50\ \mathrm{ma}\), flowing in wire No. 24. Since, at the distances under consideration (more than \(10\)–\(15\ \mathrm{m}\) from the bundle), the field of the upper wire is stronger than that of the lower one, the resulting value \(\Delta E''_y\) turns out to be negative.

The total field of wave \(k_m\) is

\[ -40.5+8.1=-32.4\ \frac{\mathrm{mv}}{\mathrm{m}}, \]

and the field of wave \(k_1\) is

\[ 19\ \frac{\mathrm{mv}}{\mathrm{m}}. \]

1. At a distance of \(30\ \mathrm{km}\) from the point where the transmitter is connected, the field of wave \(k_m\) will have the value:

\[ -32.4 e^{-0.084\cdot 30} \cong -2.6\ \frac{\mathrm{mv}}{\mathrm{m}}, \]

and the field of wave \(k_1\)

\[ 19\cdot e^{-0.155\cdot 30} \cong 0.19\ \frac{\mathrm{mv}}{\mathrm{m}}; \]

in all,

\[ -2.6+0.2=-2.4\ \frac{\mathrm{mv}}{\mathrm{m}}. \]

The method of calculation set forth above gives somewhat overestimated data, since it does not take into account the influence of a number of factors: a) reflection of waves at points of nonuniformity; b) branches and the influence of extraneous multiwire lines; c) radiation at frequencies of \(2\)–\(3\ \mathrm{MHz}\).

It is important to take into account the influence of the mass of the locomotive, which considerably distorts the electric field at the locations of the antennas. This may be done by introducing a correction coefficient determined experimentally.

REFERENCES

1. Sher, Yu. A. Theory of a multiwire line with spatial symmetry. Izv. TPI, vol. 72, 1952.