Electrical Calculation of the Contact Network and Rail Circuit in Mine Electric Locomotive Haulage
Abstract Generated abstract
This paper develops calculation procedures for the contact network and rail return circuit in mine electric locomotive haulage. It discusses permissible voltage drops, arguing that short-term fluctuations of 35 to 40 percent may be acceptable, while average voltage drop should be set more strictly to preserve train schedules, and compares graph-cutting, moving-load, and probabilistic methods for sizing contact wires. The study further examines whether an economic cross-section is needed, concluding that voltage-drop criteria normally govern, and analyzes rail-circuit current distribution using existing theory, laboratory models, and corrections for finite mine rail circuits. It proposes a practical conservative check for voltage drop in rail sections most liable to cord burnout.
Full Text
Electrical Calculation of the Contact Network and Rail Circuit in Mine Electric Locomotive Haulage
PUKHOV E. M.
Associate Professor, Candidate of Technical Sciences
Calculation of the Cross Section of the Contact Network
1. Design standard for voltage drop in the contact network. The contact network of electric railways consists of the contact network proper, the rail circuit, suction, supply, and—when the cross section of the main contact wire is insufficient—reinforcing feeders suspended from it. In calculating the contact network one proceeds from the standards for the permissible voltage drop. Glavenergoprom1 recommends adopting as the standard the average permissible voltage drop over the whole network of 15–20% of the voltage at the buses of the traction substations; moreover, under maximum peak load the permissible voltage drop may be increased to 35–40%. Prof. F. N. Shklyarskii, as well as Prof. A. S. Begekhtin (Ugol’, No. 3, 1939), proceed, in calculating the contact network, from a permissible voltage drop of 20%, i.e., with a voltage at the buses of the traction substation of 275 V the design drop is \(\Delta u = 0.2 \cdot 275 = 55\) V. Assoc. Prof. S. A. Volotkovskii recommends a somewhat smaller value, namely only 15%, i.e. \(\Delta u = 0.15 \cdot 275 = 40\) V.
Fig. 1. Dependence of the change in tractive force and current during voltage fluctuations in the contact network.
Curve I—the dependence of speed on current at voltage \(U_1\); curve II—the same at voltage \(U_2 < U_1\).
First of all it must be noted that, from the standpoint of normal operation of the rolling stock, short-term voltage fluctuations of 35–40% are entirely permissible and present no danger. A short-term decrease of the voltage at the motor terminals even has a favorable effect on its operation, since in this case, at the first moment (Fig. 1), there will be a certain decrease in current. There will be no overloading of the motor in this case. Indeed, at the first moment, when the speed, owing to the inertia of the train, has not yet had time to decrease, the current will fall from the value \(I_1\) to the value \(I_2\)—the electric locomotive will operate, instead of according to characteristic I, according to characteristic II. The tractive force of the electric locomotive will in this case decrease to the value \(F_2\), which will entail a decrease in speed, and the process of slowing—
a change in speed will take place until the current \(I_2\) again reaches the value \(I_1\), after which the train will move at the new steady speed. An entirely different phenomenon will occur in the case of a sharp, sudden increase in voltage. In this case, when the train passes from characteristic II to characteristic I, the current \(I_1'\) will increase to the value \(I_3\), as a result of which the tractive force will also change from the value \(F_1'\) to \(F_3\)—the train will acquire a certain acceleration, after which the current will fall, and at the same time the tractive force of the electric locomotive will decrease. The current surge when the voltage is increased is especially significant and dangerous when, before the surge, the motor was operating on the saturated part of the characteristic \(E=f(I)\). A large current surge may entail a whole series of undesirable consequences—wheel slip, a circular fire on the commutator, etc.
Let us see how the value of the tractive force changes when the voltage in the contact network is sharply increased. Let, for the initial conditions, the speed of the electric locomotive be equal to \(v_0\) at voltage in the contact network \(U_0\) and current \(I_0\). Further, let at voltage \(U_1\) the speed and current of the electric locomotive be respectively equal to \(v_1\) and \(I_1\).
Obviously, we may write
\[ v_0=\frac{U_0}{k I_0}\quad \text{and}\quad v_1=\frac{U_1}{k I_1}; \tag{1} \]
where \(k\) is a constant coefficient depending on the parameters of the traction motor.
Since at the first moment after the voltage jump the speed \(v_0=v_1\), then
\[ \frac{U_0}{k I_0}=\frac{U_1}{k I_1}\quad \text{or}\quad I_1=I_0\,\frac{U_0+\Delta U}{U_0}=I_0\left(1+\frac{\Delta U}{U_0}\right) \tag{2} \]
\(\Delta U\) is the magnitude of the voltage surge.
Since the tractive force and the current in traction series motors operating according to the so-called “universal” characteristic are related to one another by an equation of the form:
\[ F_k=C_1 I-C_2, \tag{3} \]
where \(C_1\) and \(C_2\) are certain constant coefficients, different for different types of motors (in particular, for the TR-10 motor this relation may be expressed by the equation)
\[ F_k=12.5 I-150\ \text{kg}, \]
where \(F_k\) is the tractive force at the wheel rim,
then, substituting the values \(F_0\) and \(F_1\) in place of \(I_1\) and \(I_0\), we rewrite equation (2) in the following form:
\[ F_1=F_0\left(1+\frac{\Delta U}{U_0}\right)+150\,\frac{\Delta U}{U_0}; \tag{4} \]
Thus, with a sharp increase of the voltage in the contact network, for example by 40%, i.e. with \(\frac{\Delta U}{U}=0.4\), the tractive force \(F_1=1.4F_0+60\) kg, i.e. the surge of tractive force will be even somewhat greater than the current surge. However, it can be shown that, for an electric locomotive with TR-10 motors, the danger of wheel slip may arise when it is operating up to the surge with hourly current only when the voltage is increased by 29.5%.
Indeed, the danger of wheel slip will arise only when the coefficient of adhesion exceeds the value 0.25, i.e. when the force
…of traction will exceed the value
\[ F_1=\frac{6000\cdot 0.25}{2}=750\ \text{kg} \]
per motor (here 6000 kg is the weight of the electric locomotive).
Since the tractive force at the hourly current is \(F_0=545\ \text{kg}\) (from the motor characteristic), substituting the values \(F_0\) and \(F_1\) into equation (4), we obtain:
\[ 750=545\left(1+\frac{\Delta U}{U_0}\right)+150\frac{\Delta U}{U_0}, \]
whence we obtain
\[ \frac{\Delta U}{U_0}=0.295 \]
or 29.5% of the voltage in the contact network.
When the motor operates with a current equal to the continuous-power current, the danger of wheel slip does not arise at all. Since, on the one hand, the traction motor rarely operates under saturation conditions (especially in mine conditions), and since, on the other hand, surges of current and tractive force will be sharply smoothed out by the inertia of the train, which we have not taken into account up to now, it may be considered that short-term fluctuations of voltage in the contact network of the order of 35–40% are entirely permissible.
Something quite different must be said with regard to prolonged voltage fluctuations. The magnitude of the average permissible voltage fluctuation must be determined from other considerations. First of all, in solving this question we must ensure exact adherence to the train movement schedule. Since traction calculations are based on the motor characteristic constructed for a definite voltage, it is necessary that the average voltage at the pantographs of the electric locomotives be equal to the calculated voltage for which the motor characteristic has been constructed. If this condition is observed, normal fulfillment of the train movement schedule will be ensured. Since the average voltage drop up to the pantographs of the electric locomotives is always somewhat greater than the voltage drop in the contact network (Fig. 2), it is evident that with a voltage on the buses of the traction substation of 275 V, an average voltage at the pantographs of the electric locomotives of 250 V will be ensured only if the permissible voltage drop in the entire contact network does not exceed 20%, and in the contact wire proper approximately 15%, if it is taken into account that about 5% is lost in the rails, feeding and return feeders.
Fig. 2. Variation of voltage up to the pantograph of an electric locomotive.
\(U_0\)—voltage on the buses of the substation; \(\Delta U_{\text{p.pr}}\)—voltage drop in the feeder wire; \(\Delta U_{\text{k.pr}}\)—the same in the contact wire; \(\Delta U_r\)—the same in the rail circuit; \(\Delta U_{\text{o.k}}\)—the same in the return cable; \(U_{\text{el}}\)—voltage at the pantograph of the electric locomotive.
To create a certain reserve for making up delays, adjusting the movement schedule, etc., for heavily loaded mine sections it should…
of sections, one should use the norm for the average voltage drop to the pantographs of electric locomotives of 10–15%, and for lightly loaded sections, the norm of 15–20%.
2. Methodology for calculating the contact network. When calculating the contact network by the average voltage drop in the line, the most accurate method is the method of cutting the train movement graph, the essence of which is as follows: from the graph, the most intensive period, lasting 1–2 hours, is selected. In this interval of time, the graph is cut every 1–2 min. For each cut, the current moment \(Il\) is found. Then, for the entire intensive period, the average calculated current moment is found
\[ Il_{\mathrm{av}}=\frac{\sum Il}{n}, \tag{5} \]
where \(n\) is the number of graph cuts.
From this average calculated current moment, the cross section of the contact wire is calculated by the usual formula
\[ q_{\mathrm{c.w.}}=\frac{17.5\,Il_{\mathrm{av}}}{\Delta U-r_p Il_{\mathrm{av}}}. \tag{6} \]
Here \(Il_{\mathrm{av}}\) is the average current moment in amp/km,
\(\Delta U\) is the permissible voltage drop in the contact network and rail circuit, in volts,
\(r_p\) is the per-kilometer resistance of the rail track in \(\Omega/\mathrm{km}\).
It is convenient to carry out the calculation itself in the form of the following table (Table 1).
Table 1
Table for calculating the cross section of the contact network
| No. of graph cut | Train Nos. | Distance from train to substation in km | Train current in amperes | Current moment | Total current moment for this cut |
|---|---|---|---|---|---|
| 1 | 1 | 0.5 | 30 | 15 | 75 |
| 1 | 2 | 1.5 | 40 | 60 | 75 |
Next, the obtained cross section is checked for the greatest permissible voltage drop by the same formula (6), for the maximum current moment obtained when cutting the graph. In this case the maximum permissible voltage drop must not exceed 35–40% of the voltage on the substation buses. The method of cutting the graph, although, despite a number of assumptions, it is the most accurate method, nevertheless requires a comparatively large expenditure of time, especially if the graph cuts are made every 2 min.
In view of this, in order to save time, instead of the graph-cutting method one may recommend one of the approximate methods—either Prof. A. V. Lebedev’s method, the so-called “moving loads method,” or the probability method, developed in detail by Prof. E. E. Rosenfeld. The first of these methods assumes movement according to a parallel graph, at constant speed, with a constant distance between trains, and the trains consume, throughout the entire time of movement, a current equal to the average train current. In the second method the trains consume
arbitrary current within the limits of the current consumed by the given type of trains; moreover, the distance between trains may be entirely arbitrary. The first of these methods gives somewhat understated voltage drops; the second, on the contrary, somewhat overstated ones. Thus, both these methods to a certain extent correct one another. Expressions for the principal quantities of interest to us are given in Table 21.
Table 2
| Method of moving loads | Method of probabilities | |
|---|---|---|
| 1. Voltage drop to the current collector during the entire time of motion under current with single-ended supply | $\dfrac{IRL(2N^2+1)}{6N}\,k_t$ | $\dfrac{IRL}{6}\,(2N+3\alpha-2)$ |
| 2. Power loss with single-ended supply | $\dfrac{I^2RL(2N^2+1)}{6}\,k_t$ | $\dfrac{I^2RLN}{6}\,(2N+1+3\beta)$ |
| 3. Average voltage drop to the end of the line | — | $\dfrac{iRLN}{2}$ |
In these formulas it is assumed that:
$R$ is the total resistance of the contact network in $\Omega/\mathrm{km}$, where
$R = r_p + r_{k.\mathrm{pr}}$, where $r_p$ and $r_{k.\mathrm{pr}}$ are the per-kilometer resistances of the rail track and the contact wire;
$I$ is the average train current, where
\[ I=\frac{I_1 n_1 + I_2 n_2}{n_1+n_2}; \]
here $I_1$ and $I_2$ are the average currents consumed by trains of different directions,
$n_1$ and $n_2$ are the number of such trains. In calculating $I$, one should take into account the additional expenditure of electrical energy for shunting, locomotive lighting, its controls, and losses in rheostats during starting periods. Usually the influence of these factors is allowed for by a coefficient of 1.1.
$N$2 $= n_1+n_2$ is the total number of trains of both directions simultaneously present on the section,
$k_t$3 is a correction coefficient,
$L$ is the haulage length in km,
$\alpha$ is the ratio of the entire operating time of the electric locomotive to its operating time under current.
Given the average voltage drop to the pantographs of the electric locomotives, it is easy first to find the total resistance of the network $R$, then the resistance of the contact wire $r_{k.\mathrm{pr}}$, and consequently also its cross-section.
3. Determination of the “economic” cross-section of the contact wire. It is easy to prove that there is no need to determine the so-called “economic” cross-section of the contact wire, since it will be several times greater than the cross-section required by the condition of the maximum voltage drop. Indeed, if we restrict ourselves
only by comparing two quantities that depend on the cross-section of the wire, namely the annual expenditures on the network and the cost of the energy lost in the line, and neglect all factors connected with changes in train speed due to changes in voltage, then one may write that
\[ A_{\mathrm{e}}=A_{\mathrm{k}}. \tag{7} \]
where \(A_{\mathrm{e}}\) is the cost of the electrical energy lost in the contact network over the course of a year,
\(A_{\mathrm{k}}\) is the depreciation charges and maintenance costs of the contact network.
It is obvious that
\[ A_{\mathrm{e}}=c\cdot \Delta W, \tag{8} \]
where \(c\) is the cost of a kilowatt-hour of energy in rubles.
The power losses in the network, according to Prof. V. E. Rosenfeld’s formula, are
\[ \Delta W=\frac{I^{2}RLN}{6}\,(2N+1+3\beta) \tag{9} \]
If we take the so-called annual number of loss hours1 for mining conditions to be about 6000 hours, then the cost of the electrical energy lost in the contact network over a year will be equal to
\[ A_{\mathrm{e}}=\frac{I^{2}RLN}{6}\,(2N+1+3\beta)\,\frac{6000}{1000}\,c\ \text{rubles}. \tag{10} \]
On the other hand, the annual network expenditures are
\[ A_{\mathrm{k}}=ab\gamma Lq_{\mathrm{ek}}, \tag{11} \]
here \(a\) is the depreciation charges and annual repair costs of the networks,
\(b\) is the cost of \(1\ \mathrm{kg}\) of copper wire,
\(\gamma\) is the specific weight of copper in \(\mathrm{kg}/\mathrm{dm}^{3}\).
Substituting the values of \(A_{\mathrm{e}}\) and \(A_{\mathrm{k}}\) into equation (7), we find the “economic” cross-section of the contact wire
\[ q_{\mathrm{ek}}\simeq 4.22I\sqrt{\frac{N(2N+1+3\beta)c}{ab\gamma}}. \tag{12} \]
The average voltage drop in the contact network (Table 2) is
\[ \Delta U=\frac{IRLN}{2}. \]
Hence the wire cross-section, calculated by voltage drop, is
\[ q=\frac{1000ILN}{2k\Delta U}=0.22\,ILN. \]
If the haulage length is expressed in kilometers and the voltage drop is taken to be \(\simeq 40\ \mathrm{V}\), \(c=0.05\) rubles/kWh, \(a=0.15\), \(\gamma=8.9\ \mathrm{kg}/\mathrm{dm}^{3}\), and \(b=2\) rubles/kg, then
\[ \frac{q_{\mathrm{ek}}}{q}=\frac{2.63}{L}\sqrt{\frac{2N+1+3\beta}{N}}. \tag{13} \]
Assigning various values to \(L\), \(N\), and \(\beta\), we see that for ordinary lengths of electric-locomotive haulage the “economic” cross-section of the contact wire considerably exceeds the cross-section calculated according to the permissible drop—
… voltage. Under conditions of a shortage of non-ferrous materials, there is no sense in calculating the cross section of the contact wire on the basis of minimum annual expenditures.
Conclusions
Summarizing what has been said, one should conclude that the calculation of the cross section of the contact network must be carried out on the basis of the average operating conditions of the electric locomotives on the section, i.e., on the basis of the average voltage drop to the pantographs of the electric locomotives. In addition, the adopted cross section should be checked for the most severe case of the train-movement schedule.
II. Calculation of the Rail Circuit
1. Distribution of currents in the rail circuit for “infinite” and “finite” circuits.
The methodology for calculating the rail circuit in electric-locomotive haulage has, up to the present time, hardly been developed, although its correct calculation is extremely necessary from the standpoint of developing measures for protection against stray currents.
Since two-sided feeding of haulage tracks has not yet become widespread under mine conditions, chiefly because of the insignificant length of haulage, we shall confine ourselves to considering the case of one-sided feeding.
Forwald, in his article “Streckenimpedanz und Schienenstrom bei Einphasenbahnen,” gave the current-distribution scheme in this case in the following form (Fig. 3).
Fig. 3. Distribution of currents in the rail circuit according to Forwald.
However, in this article Forwald did not give the relation between the train current and the currents flowing off toward the working face and also in the direction of the traction substation. In addition, the value he gave for the coefficient \(k\) is valid only for single-phase current.
Let us present this scheme in a somewhat different form (Fig. 4).
Fig. 4. Diagram of current distribution in the rail circuit.
At the point of application of the load \(B\), the full train current \(I_0\) branches into currents \(I_1\) and \(I_2\). Part of the currents \(i_1\) and \(i_2\) flows along the rail, while another part, owing to incomplete insulation of the rail from the ground, branches off into the latter. For mine conditions it is important to know not only the law of variation of the currents on the section \(AB\) between the substation and the train, but also on the section \(BC\), especially in the case when electric blasting of shot-firing cables is used in the mine.
Let us introduce the following notation:
\(r_p\) — the resistance per kilometer of the rail track, taking into account the resistance of the joints, in \(\Omega/\text{km}\),
\(r_z\) — the transition resistance from rail to ground, also in \(\Omega/\text{km}\),
\[ p=\sqrt{\frac{r_p}{r_z}}, \]
\(I_0\) — total train current,
\(i_1,i_2,i_3\) — currents in the rails on sections \(AB\), \(BC\), and \(AD\),
\(U_1,U_2,U_3\) — rail potentials with respect to the ground on the same sections,
\(I_1,I_2,I_3,I_4\) — components of the total train current at points \(A\) and \(B\).
Although the specific resistance of the ground is fairly large, nevertheless, owing to its large volume, the resistance of the ground may be freely neglected.
In order to find the pattern of current variation along the rail track, let us find, in general form, the dependence between \(i\), \(p\), and the lengths. At any point of the rail track, the magnitude of the current passing into the ground over the section \(dx\) will be
\[ di_z=\frac{U_p-U_z}{r_z}\,dx. \tag{14} \]
Here \(U_p\) and \(U_z\) are the potentials of the rail and the ground at the point under consideration. From this equation we find
\[ \frac{di_z}{dx}=\frac{U_p-U_z}{r_z}. \]
Let us take the second derivative of \(i\) with respect to \(x\), taking the ground potential \(U_z\) as a constant value,
\[ \frac{d^2 i_z}{dx^2}=\frac{dU_p}{dx}\,\frac{1}{r_z}. \tag{15} \]
Since the train current \(I_0=i_z+i_p\) does not depend on \(x\), then
\[ di_p=-di_z, \]
i.e., each change of current in the rail corresponds to an equal change, but with the opposite sign, of the current branching into the ground. The elementary change in the rail potential, evidently, may be expressed in the following form:
\[ dU_p=-i_p r_p\,dx \tag{16} \]
or
\[ \frac{dU_p}{dx}=-i_p r_p. \]
Substituting this expression into equation (15),
\[ \frac{d^2 i_z}{dx^2}=-\frac{d^2 i_p}{dx^2}=-i_p\frac{r_p}{r_z}=-i_p p^2 \]
or
\[ \frac{d^2 i_p}{dx^2}-i_p p^2=0.^1) \tag{17} \]
Thus, we obtain the desired dependence in the form of a differential equation.
The integral of this equation has the form
\[ i=C_1 ch\,px+P_2 sh\,px, \tag{18} \]
where \(C_1\) and \(C_2\) are constants of integration, different for the different sections of the rail circuit considered by us.
\(^{1)}\) Prof. Lebedev. Fundamentals of Electric Traction. ONTI, 1937.
Let us consider separately each of the sections \(AB\), \(BC\), and \(CD\).
I. Section \(BC\). We shall take point \(B\) as the origin of coordinates. Obviously, when \(x_2=0\), \(i_2=I_2=C_1\), and when \(x_2=l_2\), \(i_2=0\), since at an infinite distance from the train the current in the rail becomes equal to zero.
Thus, \(I_2 ch\,pl_2+C_2 sh\,pl_2=0\), or \(C_2=-I_2 cth\,pl_2\); the variation of the current \(i_2\) in section \(BC\) can be expressed by the following equation:
\[ i_2=I_2 ch\,px_2-I_2 cth\,pl_2\,sh\,px_2 . \tag{19} \]
II. Section \(AD\). The origin of coordinates is point \(A\). When \(x_3=0\), \(i_3=I_3=C'_1\), and when \(x_3=l_3\), \(i_3=0\). Analogously to the preceding case, we obtain
\[ i_3=-I_3 ch\,px_3+I_3 cth\,pl_3\,sh\,px_3 . \tag{20} \]
III section \(AB\). The origin of coordinates is point \(B\).
When \(x_1=0\), \(i_1=I_1=C''_1\); when \(x_1=l_1\),
\[ i_1=I_4=C''_1 ch\,pl_1+C''_2 sh\,pl_1, \]
or
\[ C''_2=\frac{I_4-I_1 ch\,pl_1}{sh\,pl_1} \]
and, finally,
\[ i_1=I_1 ch\,px_1+\frac{I_4-I_1 ch\,pl_1}{sh\,pl_1}\,sh\,pl_1 . \tag{21} \]
For convenience of calculation, we express the currents \(I_1\), \(I_2\), \(I_3\), and \(I_4\) in terms of the train current \(I_0\):
\[ \begin{aligned} I_1+I_2&=I_0,\\ I_3+I_4&=I_0 \end{aligned} \tag{22} \]
For points \(A\) and \(B\), moreover, we can write the following equalities:
\[ \left|\frac{di_1}{dx_1}\right|_{\substack{\text{when}\\ x_1=0}} = \left|\frac{di_2}{dx_2}\right|_{\substack{\text{when}\\ x_2=0}} \quad\text{and}\quad \left|\frac{di_3}{dx_3}\right|_{\substack{\text{when}\\ x_3=0}} = \left|\frac{di_1}{dx_1}\right|_{\substack{\text{when}\\ x_1=l_1}}; \tag{23} \]
the first of these equalities gives the following relation between the currents
\[ I_1 sh\,pl_1+\frac{I_4-I_1 ch\,pl_1}{sh\,pl_1}\,ch\,pl_1 = I_3 cth\,pl_3, \]
the second equality gives
\[ I_1 ch\,pl_1-I_2 cth\,pl_2\,sh\,pl_1=I_4. \]
From these equalities, and also from equation (22), we first find the current \(I_1\), and then the currents \(I_2\) and \(I_3\). Without giving the successive course of the solution, because of its cumbersomeness, we present the final dependences:
\[ \left. \begin{aligned} I_1&=\frac{ch\,pl_2\,sh\,p(l_1+l_2)+ch\,pl_3\,sh\,pl_2} {sh\,p(l_1+l_2+l_3)}\,I_0,\\[4pt] I_2&=\frac{sh\,pl_2\,sh\,p(l_1+l_2)-ch\,pl_3\,sh\,pl_2} {sh\,p(l_1+l_2+l_3)}\,I_0,\\[4pt] I_3&=\frac{sh\,pl_3\,ch\,p(l_1+l_2)-ch\,pl_2\,sh\,pl_3} {sh\,p(l_1+l_2+l_3)}\,I_0,\\[4pt] I_4&=\frac{ch\,pl_2\,sh\,pl_3+ch\,pl_3\,sh\,p(l_1+l_2)} {ch\,p(l_1+l_2+l_3)}\,I_0. \end{aligned} \right\} \tag{24} \]
Since, in deriving these formulas, the sections \(AD\) and \(BC\) were taken to be of infinite length, in order to obtain expressions for the currents \(i_2\), \(i_1\), and \(i_3\), it is necessary to get rid of the expressions containing the lengths \(l_2\) and \(l_3\), since in
In this case indeterminacies of the form \(\frac{\infty}{\infty}\) are obtained. For this purpose we find the values \(\frac{I_1}{I_2}\) and \(\frac{I_3}{I_4}\) for \(l_2=\infty\) and \(l_3=\infty\):
\[ \frac{I_1}{I_2} = \frac{ ch\,pl_2\,sh\,pl_1\,ch\,pl_3 + ch\,pl_2\,sh\,pl_3\,ch\,pl_1 + ch\,pl_3\,sh\,pl_2 }{ sh\,pl_2\,ch\,pl_1\,ch\,pl_3 + sh\,pl_2\,sh\,pl_3\,sh\,pl_1 - ch\,pl_3\,sh\,pl_2 } = \]
\[ = \frac{ ch\,pl_3\,sh\,pl_2 \left[ cth\,pl_2\,sh\,pl_1 + ch\,pl_2\,th\,pl_3\,ch\,pl_1 + 1 \right] }{ ch\,pl_3\,sh\,pl_2 \left[ ch\,pl_1 + sh\,pl_1\,th\,pl_3 - 1 \right] }; \]
since \(cth\,\infty=1\) and \(th\,\infty=1\), then
\[ \frac{I_1}{I_2} = \frac{sh\,pl_1+ch\,pl_1+1}{sh\,pl_1+ch\,pl_1-1} = \frac{ 2sh\,\frac{pl_1}{2}\,ch\,\frac{pl_1}{2} + 2ch^2\frac{pl_1}{2} }{ 2sh\,\frac{pl_1}{2}\,ch\,\frac{pl_1}{2} + 2sh^2\frac{pl_1}{2} } = cth\,\frac{pl_1}{2}. \tag{25} \]
Thus we obtain the known Dell’Arciprete formula.
Analogously to the preceding, we also find \(\frac{I_3}{I_4}\) for \(l_2=\infty\)
\[ \frac{I_3}{I_4}=th\,\frac{pl_1}{2} \tag{26} \]
Using the derived expressions for \(\frac{I_1}{I_2}\) and \(\frac{I_3}{I_4}\), we find \(I_1, I_2, I_3\), and \(I_4\) through the total train current \(I_0\)
\[ \frac{I_1+I_2}{I_2}=cth\,\frac{pl_1}{2}+1, \]
or
\[ I_2= \frac{I_0}{1+cth\,\frac{pl_1}{2}} = \frac{I_0\,th\,\frac{pl_1}{2}}{1+th\,\frac{pl_1}{2}}. \]
In exactly the same way we find
\[ \left. \begin{aligned} I_1&=\frac{I_0}{1+th\,\frac{pl_1}{2}},\\[6pt] I_3&=\frac{I_0\,th\,\frac{pl_1}{2}}{1+th\,\frac{pl_1}{2}},\\[6pt] I_4&=\frac{I_0}{1+th\,\frac{pl_1}{2}}. \end{aligned} \right\} \tag{27} \]
Substituting the values \(I_1, I_2, I_3\), and \(I_4\) into equations 19, 20, 21, we find:
\[ i_1=I_1\,ch\,px_1+\frac{I_4-I_1\,ch\,pl_1}{sh\,pl_1}\,sh\,px_1 \]
or, after substitution and simplification,
\[ i_1 = I_2 \frac{\operatorname{sh} p(l_1-x_1)+\operatorname{sh} px_1} {\operatorname{sh} pl_1\left(1+\operatorname{th}\frac{pl_1}{2}\right)}\, I_0 . \tag{28a} \]
Similarly,
\[ i_2 = I_2(\operatorname{ch} px_2-\operatorname{cth} pl_2\, \operatorname{sh} px_2) \]
or
\[ i_2 = \frac{ \operatorname{th}\frac{pl_1}{2}\operatorname{ch} px_2 - \operatorname{cth} pl_2\, \operatorname{sh} px_2 } {1+\operatorname{th}\frac{pl_1}{2}}\, I_0 . \tag{28b} \]
Finally,
\[ i_3 = I_3(\operatorname{cth} pl_3\, \operatorname{sh} px_3-\operatorname{ch} px_3)= \]
\[ = \frac{ \operatorname{th}\frac{pl_1}{2} } { 1+\operatorname{th}\frac{pl_1}{2} } (\operatorname{cth} pl_3\, \operatorname{sh} px_3-\operatorname{cth} px_3) I_0 . \tag{28c} \]
Let us find the potential difference on the sections \(AB\) and \(BC\), which are the most important from the point of view of protection against stray currents.
According to equation (52), we write
\[ -U_{AB}=\int_{0}^{l_1} i_1 r_p\, dx_1 = -\frac{2(\operatorname{ch} pl_1-1)} {\operatorname{sh} pl_1\left(1+\operatorname{th}\frac{pl_1}{2}\right)} \,\frac{r_p}{p}\, I_0 = -\frac{2\operatorname{th}\frac{pl_1}{2}} {1+\operatorname{th}\frac{pl_1}{2}} \,\frac{r_p}{p}\, I_0 ; \tag{29} \]
further,
\[ -U_{BC}=\int_{0}^{l_2} i_2 r_p\, dx_2 = -\frac{\operatorname{th}\frac{pl_1}{2}} {1+\operatorname{th}\frac{pl_1}{2}} \,\frac{r_p}{p}\, I_0 \left[ \operatorname{sh} pl_2-\operatorname{cth} pl_2(\operatorname{ch} pl_2-1) \right] = \]
\[ = -\frac{ \operatorname{th}\frac{pl_1}{2}\operatorname{th}\frac{pl_1}{2} } { 1-\operatorname{th}\frac{pl_1}{2} } \,\frac{r_p}{p}\, I_0 . \tag{30} \]
To construct the picture of current exchange between the rails and the ground, it is necessary to find the potential of the rail with respect to the ground for any point of the rail track.
On section \(AB\), the potential of the rails with respect to the ground will vary according to the following law (equation 29):
\[ U_{A-B}= -\frac{I_0} {\operatorname{sh} pl_1\left(1+\operatorname{th}\frac{pl_1}{2}\right)} \,\frac{r_p}{p} \left[ \operatorname{ch} px_1-\operatorname{ch} p(l_1-x_1) \right] \]
and on section \(BC\),
\[ U_{B-C}= -\frac{ I_0\operatorname{th}\frac{pl_1}{2}\operatorname{th}\frac{pl_2}{2} } { 1+\operatorname{th}\frac{pl_1}{2} } \,\frac{r_p}{p} \left[ \operatorname{sh} px_2-\operatorname{cth} pl_2\, \operatorname{ch} px_2 \right]. \tag{31} \]
Dell’Arciprete’s formulas were derived, as was indicated above, for infinitely long sections \(AD\) and \(BC\); meanwhile, there are indications in the literature that the current \(i_2\) in section \(BC\), at a comparatively short distance from the train, almost completely disappears. Thus, according to data of Assoc. Prof. Ozerny (“Ugol’,” 1938, No. 150), the current \(i_2\) disappears in some cases already at a distance of 60 m from the train, amounting at a distance of 50 m from the train to only 1.3% of the initial value of the train current. True, in other cases this percentage is considerably larger. Thus, for example, according to the same experimental data of Assoc. Prof. Ozerny, on grade No. 6, substation No. 3, the current at a distance of 100 m from the train amounted to 16.1%, decreasing to 3.2% at a distance of 200 m.
Let us derive formulas for the values of the currents \(I_1\), \(I_2\), \(I_3\), and \(I_4\), and also the currents \(i_1\), \(i_2\), and \(i_3\), for the case when the lengths \(l_2\) and \(l_3\) have a finite value. Let us suppose that in sections \(AD\) and \(BC\), at a distance of \(a\) meters—in the first case from the substation and in the second from the train—the current disappears. In this case, for section \(BC\) we may write:
\[ \left. \begin{aligned} i_1 &= I_1 \operatorname{ch} px_1 + \frac{I_4 - I_1 \operatorname{ch} pa}{\operatorname{sh} pa}\operatorname{sh} px_1,\\ i_2 &= I_2 \operatorname{ch} px_2 - I_2 \operatorname{cth} pa\, \operatorname{sh} px_2,\\ i_3 &= I_3\left(\operatorname{cth} pa\, \operatorname{sh} px_3 - \operatorname{ch} px_3\right). \end{aligned} \right\} \tag{32} \]
Moreover, as we saw earlier,
\[ I_1 = I_4 = \frac{\operatorname{ch} pa\, \operatorname{sh} p(l_1+a) + \operatorname{ch} pa\, \operatorname{sh} pa} {\operatorname{sh} p(l_1+2a)} I_0 = \]
\[ = \frac{\operatorname{ch} pa\,[\operatorname{sh} p(l_1+a)+\operatorname{sh} pa]} {\operatorname{sh} p(l_1+2a)} I_0 = \frac{2\operatorname{ch} pa\, \operatorname{ch}\frac{pl_1}{2}\, \operatorname{sh} p\left(\frac{l_1}{2}+a\right)} {\operatorname{sh} p(l_1+2a)} \]
and
\[ I_2 = I_3 = \frac{2\operatorname{sh} pa\, \operatorname{sh}\frac{pl_1}{2}\, \operatorname{sh} p\left(\frac{l_1}{2}+a\right)} {\operatorname{sh} p(l_1+2a)} I_0 . \tag{33} \]
Thus, in contrast to the case considered by Dell’Arciprete, for our case, when \(l_2\) and \(l_3\) have finite length, we may write
\[ \frac{I_1}{I_2} = \frac{ 2\operatorname{ch} pa\, \operatorname{ch}\frac{pl_1}{2}\, \operatorname{sh} p(l_1+2a) }{ 2\operatorname{sh} pa\, \operatorname{sh}\frac{pl_1}{2}\, \operatorname{sh} p(l_1+2a) } = \operatorname{cth} pa\, \operatorname{cth}\frac{pl_1}{2} \]
and
\[ \frac{I_3}{I_4} = \operatorname{th} pa\, \operatorname{th}\frac{pl_1}{2}. \tag{34} \]
The difference in this case is expressed in the appearance of the additional factors \(\operatorname{cth} pa\) and \(\operatorname{th} pa\).
The currents \(I_1\) and \(I_2\) through the total train current may be expressed by the following formulas:
\[ \left. \begin{aligned} I_1 &= \frac{I_0}{1+\operatorname{th} pa\, \operatorname{th}\frac{pl_1}{2}},\\[6pt] I_2 &= \frac{I_0\, \operatorname{th} pa\, \operatorname{th}\frac{pl_1}{2}} {1+\operatorname{th} pa\, \operatorname{th}\frac{pl_1}{2}}. \end{aligned} \right\} \tag{35} \]
we obtain, further, the following expressions for the currents \(i_2, i_1\), and \(i_3\):
\[ \left. \begin{aligned} i_2&=-I_2\,ch\,px_2-I_2\,cth\,pa\,sh\,px_2 \\ &=-(cth\,px_2-cth\,pa\,sh\,px_2)\, \frac{th\,pa\,th\,\dfrac{pl_1}{2}}{1+th\,pa\,th\,\dfrac{pl_1}{2}}\,I_0,\\[1.2em] i_3&=(cth\,pa\,sh\,px_3-ch\,px_3)\, \frac{th\,pa\,th\,\dfrac{pl_1}{2}}{1+th\,pa\,th\,\dfrac{pl_1}{2}}\,I_0,\\[1.2em] i_1&=\frac{\left(ch\,px_1-sh\,px_1\,th\,\dfrac{pa}{2}\right)I_0} {1+th\,pa\,th\,\dfrac{pl_1}{2}}. \end{aligned} \right\} \tag{36} \]
Finally, let us determine, for the case under consideration, the potential differences on the sections \(AB\) and \(BC\).
\[ -\,U_{AB}=\int_{0}^{l_1} i_1 r_p\,dx_1 = \frac{ch\,pl-sh\,pl_1\,th\,\dfrac{pa}{2}-1} {1+th\,pa\,th\,\dfrac{pl_1}{2}}\, \frac{r_p}{p}\,I_0, \]
\[ -\,U_{BC}=\int_{0}^{l_1} i_2 r_p\,dx_2 = \frac{(cth\,pl_2\,th\,pa-th\,pa-sh\,pl_2)\,th\,\dfrac{pl_1}{2}} {1+th\,pa\,th\,\dfrac{pl_1}{2}}\, \frac{r_p}{p}\,I_0. \tag{37} \]
Thus we have obtained all the data of interest to us also for the case when the lengths of the sections \(AD\) and \(BC\) have a finite magnitude.
2. Methodology for the experimental determination of the parameters of the rail circuit. First of all, let us dwell briefly on the methodology for determining the transition resistance between the rails and the soil. To determine this quantity there are a number of methods. Most often in practice one has to use two of them, namely the Ohio Brass method and the method of Prof. R. Podoskov. The first of these methods makes it possible, simultaneously with determining the transition resistance between the rail and the soil, also to determine the resistance of the rail track itself. Measurements by this method1 are carried out according to the scheme in Fig. 5.
The measurement scheme according to the method of Prof. R. Podoskov is shown in Fig. 6.
The resistance of the rail track \(R_p\) and the transition resistance between the rail and the soil \(R_g\) by the Ohio Brass method may be found from the following formulas:
\[ \begin{gathered} R_p=\frac{U_1-U_2}{2(I_1+I_2)}\\[0.6em] \text{and}\\[0.6em] R_g=\frac{U_1+U_2}{8(I_1-I_2)} \end{gathered} \left\} \tag{38} \]
We shall present the derivation of these formulas, since this derivation is absent in the literature, while it is extremely necessary for estimating the degree of accuracy obtained in measurements.
Fig. 5. Measurement of the transition resistance from rail to soil by the “Ohio Brass” method.
As is known, for a circuit with distributed constants one may write the following equations:
\[ \left. \begin{aligned} U_1 &= U_2 \operatorname{ch}\sqrt{rg}\,l + I_2 \sqrt{\frac{r}{g}}\, \operatorname{sh}\sqrt{rg}\,l,\\ I_1 &= U_2 \sqrt{\frac{g}{r}}\, \operatorname{sh}\sqrt{rg}\,l + I_2 \operatorname{ch}\sqrt{rg}\,l \end{aligned} \right\} \tag{39} \]
where \(r\) is the resistance of one strand of the rail track,
\(g\) is the conductance between the rail and the soil,
\(l\) is the length of the measured section of the rail track in km.
Fig. 6. Measurement of the transition resistance from rail to soil by the method of Prof. R. Podolskii.
Let us reduce equations (39) to a somewhat different form.
Denote
\[ \sqrt{rg}=p, \]
then
\[ \sqrt{\frac{r}{g}}=\frac{r}{p}=\frac{p}{g}, \]
and equations (39) will take the following form:
\[ \left. \begin{aligned} U_1 &= U_2 \operatorname{ch} pl + \frac{I_2 p}{g}\operatorname{sh} pl,\\ \text{or}\qquad \frac{I_2 p}{g}\operatorname{sh} pl &= U_1 - U_2 \operatorname{ch} pl,\\ \text{and}\qquad I_1 &= U_2 \frac{g}{p}\operatorname{sh} pl + I_2 \operatorname{ch} pl. \end{aligned} \right\} \tag{40} \]
Multiplying both sides of the last equation by \(\dfrac{p}{g}\operatorname{sh} pl\), it then assumes the following form:
\[ \frac{I_1p}{g}\operatorname{sh} pl = U_2\operatorname{sh}^2 pl + \frac{I_2p}{g}\operatorname{sh} pl\,\operatorname{ch} pl. \]
Substituting into this equation, instead of \(\dfrac{I_2p}{g}\operatorname{sh} pl\), its value from the first equation (40),
\[ -\frac{I_1p}{g}\operatorname{sh} pl = U_2\operatorname{sh}^2 pl + U_1\operatorname{ch} pl - U_2\operatorname{ch}^2 pl = U_1\operatorname{ch} pl-U_2, \]
since
\[ \operatorname{ch}^2 pl-\operatorname{sh}^2 pl=1. \]
Thus, as a result of the transformations, we obtain the following system of equations:
\[ \left. \begin{aligned} -\frac{I_2p}{g}\operatorname{sh} pl &= U_1-U_2\operatorname{ch} pl,\\ -\frac{I_1p}{g}\operatorname{sh} pl &= U_1\operatorname{ch} pl-U_2 \end{aligned} \right\} \tag{41} \]
To obtain the value of \(R_g\), subtract the first equation from the second:
\[ \frac{(I_1-I_2)p}{g}\operatorname{sh} pl=(U_1+U_2)(\operatorname{ch} pl-1), \]
whence
\[ g=\frac{(I_1-I_2)p\operatorname{sh} pl}{(U_1+U_2)(\operatorname{ch} pl-1)} \]
or
\[ R_g=\frac{(U_1+U_2)(\operatorname{ch} pl-1)}{(I_1-I_2)p\operatorname{sh} pl}. \]
Since the basic equations (39) were set up by us for one rail strand, whereas in fact in the Ohio Brass circuit there are not one but two of them, and since, further, in the circuit the leakage current passes through two resistances \(R_g\) connected in series, it is obvious that, for the circuit considered by us, the result obtained for \(R_g\) must be divided by 4, i.e.,
\[ R_g=\frac{(U_1+U_2)(\operatorname{ch} pl-1)}{4(I_1-I_2)p\operatorname{sh} pl}. \tag{42} \]
For an approximate determination of \(R_g\), expand \(\operatorname{ch} pl\) and \(\operatorname{sh} pl\) in series, and in expanding \(\operatorname{ch} pl\) neglect all terms of the series beginning with the third, and in expanding \(\operatorname{sh} pl\), beginning with the second; then
\[ \operatorname{ch} pl \simeq 1+\frac{p^2l^2}{2} \]
and
\[ \operatorname{sh} pl \simeq pl. \]
If the values obtained for \(\operatorname{ch} pl\) and \(\operatorname{sh} pl\) are substituted into equation (42), the transition resistance from rail to ground may approximately be expressed as
\[ R_g \simeq \frac{U_1+U_2}{8(I_1-I_2)}\, l\ \Omega\,\text{km}. \tag{42a} \]
Since neglecting terms in the expansion of \(\operatorname{ch} pl\) and \(\operatorname{sh} pl\) into a series—in the first case beginning with the third term, and in the second beginning with the second—entails a certain error, which increases as the value of \(pl\) increases, Fig. 7 gives a curve of the corrections that must be introduced in calculating \(R_g\).
Correcting coefficient
| \(pl\) | correction coeff. |
|---|---|
| 0.5 | 1.02 |
| 1.0 | 1.08 |
| 1.5 | 1.18 |
| 2.0 | 1.32 |
| 3.0 | 1.62 |
| 4.0 | 2.00 |
| 5.0 | 2.57 |
Fig. 7. Curve of corrections in determining the transition resistance and the resistance of the rail track by the “Ohio Brass” method.
When determining the transition resistance, the result obtained by the “Ohio Brass” formula should be divided by the correction coefficient corresponding to the value of \(pl\). When determining the resistance of the rail track, the result must be multiplied by the correction coefficient.
The computation of \(R_g\) must be performed as follows. First of all we determine the value of \(pl\), for which we divide equations (41) by one another. After reduction we obtain
\[ \frac{I_2}{I_1}=\frac{U_2-U_1 \operatorname{ch} pl}{U_1 \operatorname{ch} pl-U_2}, \]
whence we obtain
\[ \operatorname{ch} pl=\frac{U_1 I_1+U_2 I_2}{U_1 I_2+U_2 I_1}. \tag{43} \]
Knowing from the experiment all the quantities entering this expression and using tables of hyperbolic functions, we find \(pl\) and, consequently, the corresponding correction factor from the graph in Fig. 7.
We proceed in exactly the same way in determining the resistance of the rail track. Add both equations (41), first replacing \(P/g\) by \(r/p\).
We obtain
\[ \frac{(I_1+I_2)r}{p}\operatorname{sh} pl=(U_1-U_2)(\operatorname{ch} pl+1) \]
or
\[ r=\frac{(U_1-U_2)p(\operatorname{ch} pl+1)}{(I_1+I_2)\operatorname{sh} pl}; \]
taking into account that in the Ohio Brass circuit the rail strands are connected in series, the actual resistance of a single-track line consisting of 2 strands will obviously be 4 times smaller, i.e.,
\[ R_p=\frac{1}{4}\cdot\frac{U_1-U_2}{I_1+I_2}\cdot\frac{p(\operatorname{ch} pl+1)}{\operatorname{sh} pl}. \tag{44} \]
For an approximate determination of \(R_g\), we again expand \(\operatorname{ch} pl\) and \(\operatorname{sh} pl\) into a series; in expanding \(\operatorname{ch} pl\), we also neglect the second term of the series. Then
\[ R_p \approx \frac{U_1-U_2}{2(I_1+I_2)l}\ \Omega/\text{km}. \]
The procedure for calculating \(R_p\) is the same as for calculating \(R_g\). By Prof. Podoskiy’s method, as was indicated above, only the transition resistance from rail to ground can be determined, and the value of this resistance is determined by the formula:
\[ R_g=\frac{U_1+U_2}{2(I_1-I_2)}\,l\ \Omega\cdot\text{km}. \tag{45} \]
To derive this formula, let us represent Prof. Podosky’s circuit in the following form (Fig. 8). For this circuit, obviously, the following equations may be written:
\[ \begin{aligned} I_1 &= I + I_{g1} + I_{g2},\\ I_{g1} &= U_1 \frac{g}{2},\\ I_{g2} &= U_2 \frac{g}{2}; \end{aligned} \tag{46} \]
Fig. 8. Equivalent circuit for measuring the resistance from rail to earth by the method of Prof. R. Podosky.
adding the second and third equations, we obtain
\[ I_{g1} + I_{g2} = (U_1 + U_2)\frac{g}{2} \]
or, replacing
\[ I_{g1} + I_{g2} \text{ by } I_1 - I_2, \]
we obtain
\[ g = \frac{2(I_1 - I_2)}{U_1 + U_2}. \]
The sought resistance \(R_g\), obviously, will be
\[ R_g = \frac{U_1 + U_2}{2(I_1 - I_2)}. \]
3. Characterization of the Ohio Brass and Prof. Podosky methods. Experimental determination of the parameters of the rail circuit, carried out at the mines in Temir-Tau and Salaire, Novosibirsk Region, made it possible to draw the following conclusions regarding the merits and shortcomings of both methods.
Prof. Podosky’s method, although it does not require the joints to be bonded, also—as it were—does not require the presence of an external current source, since, it would seem, the currents flowing in the rail track can be used; however, this only appears so at first glance. Even when an artificial condition is created in the rail circuit by means of a rheostat, the measurement results prove to be completely unacceptable, because the traction currents, being superposed on the currents created under the artificial condition, completely distort the results. To eliminate this defect, two measurements could be made: the first with the rheostat disconnected and the second with the rheostat connected, but these measurements must be made over a very short interval of time and simultaneously. The second, still more substantial defect of the method under consideration lies in the specific nature of mine conditions. To measure the potential of two points of the rail track relative to “earth,” it is necessary to drive a pin at a distance of at least 20–30 m from the rail, which cannot be done under mine conditions. Nor is it possible to eliminate the error that arises when the pin is driven at a distance of only 1 m from the rail, since the formula for determining \(R_g\) contains the sum \(U_1 + U_2\), and not their difference. The third short—
Fig. 8-a. Circuit for measuring rail currents.
The drawback of Prof. Podosky’s method is the necessity of having sensitive voltmeters for measuring \(U_1\) and \(U_2\), and with a large internal resistance, since the leakage resistance of the electrode is an added resistance to the voltmeter and thus can strongly affect the accuracy of the voltmeter readings if the latter is taken with a small internal resistance. Finally, a large influence on the accuracy of the measurements is exerted by the contact transition resistances when measuring the voltage difference over a comparatively small section of the rail circuit (1–2 meters)—these resistances are, in magnitude, commensurable with the resistance of the rail section.
This defect, however, is easily eliminated either by using special clamps with low transition resistance1, or by using one of the compensation circuits. In particular, under laboratory conditions I used the circuit shown in Fig. 8-a. Incidentally, in studying the distribution of currents along the rail track this circuit, although it somewhat complicates the measurement process, can nevertheless be of substantial benefit. In view of all the above-mentioned shortcomings, Prof. Podosky’s method must be recognized as unacceptable for mine conditions. In any case, it has not been possible to obtain any sufficiently reliable results with the aid of this method. Prof. Podosky’s method can be applied only on surface mine tracks.
The Ohio Brass method is not especially convenient under operating conditions, since it requires unbolting the joints on both sides of the measured section. By this method it is quite impossible to determine the resistance of the rail track unless special clamps having a small transition resistance are used. This is evident not only from the experiments which I carried out at the Temir-Tau and Salair mines, but also from the experiments of Doc. Ozernoi (see “Coal,” No. 150), who obtained for the resistance of the rail track values of the order of 0.84 to 320 \(\Omega/\text{km}\) for rail type N-91 (the resistance without joints of a single-track two-rail track is \(0.047 \Omega/\text{km}\)). Doc. Ozernoi explained the enormous discrepancy in the results by insufficiently good insulation of the section being measured. Incidentally, with another method of measuring the resistance of the rail track on these same sections he obtained more probable results—the joint resistance in the very worst case was equal to the resistance of 20 m of continuous rail. The discrepancy in the results lies in the measurement method itself. When measuring such small quantities as the resistance of a rail track, one must not neglect the resistance of the lead wires and the transition resistance of the contacts. Under mine conditions these resistances may prove greater than the resistance of the rail track itself.
By virtue of these considerations, the resistance of a rail track should reliably be measured either by the bridge method (for a description see the article by Eng. Livshits, “Stray Currents in Mines during Haulage by Contact Electric Locomotives,” Mining Journal, No. 1, 1930), or by one of the approximate methods recommended by C. Michaike in his article “Streustrommessungen,” ETZ, 1926, p. 5.
Measurement of the transition resistance from rail to soil by the Ohio Brass method gives comparatively satisfactory results.
It should, however, be pointed out that in the practice of everyday observation of stray currents under mine conditions it is nevertheless more expedient to proceed to the so-called indirect methods of determining both the transition—
…of the resistance from rail to earth, as well as of the resistance of the track circuit itself.
The main shortcoming of indirect methods is that these methods do not give the desired parameters of the rail circuit at each, even relatively small, section, which is sometimes very important (for example, for sections near dynamite depots). In addition, indirect methods require knowledge of the laws of current distribution along a track circuit of “finite” length; these laws, however, cannot yet be formulated with sufficient accuracy.
4. Current distribution in the rail circuit and approximate determination of the voltage drop in it in design. E. Dell’Arciprete, in his article “Il binario come conduttore di ritorno nella trazione elettrica”1, gave the following picture of the current distribution along the track circuit (Fig. 9) for two cases, namely:
Fig. 9. Current distribution in the rail circuit according to Dell’Arciprete. The full train current (100%) is plotted along the ordinate axis.
Case I: resistance of the track circuit \(r_p = 0.03\ \Omega/\text{km}\), transition resistance from rail to earth \(r_z = 0.8\ \Omega/\text{km}\), hence \(p = 0.194\).
Case II: \(r_p = 1.0\ \Omega/\text{km}\), \(r_z = 0.8\ \Omega/\text{km}\), \(p = 1.12\). In both cases the distance from the train to the substation is \(l = 10\ \text{km}\).
As can be seen from Fig. 9, the pattern of currents in the rail circuit changes sharply depending on
\[ \frac{pl}{2}. \]
For the first case considered by Dell’Arciprete,
\[ \frac{pl}{2} = .97 \]
the current, both on section \(AB\) and on section \(BC\), decreases slowly, whereas in the second case
\[ \left(\frac{pl}{2} = 5.6\right) \]
on both sections the current decreases steeply.
Incidentally, exactly the same picture was also obtained on the models that I built when studying current distribution in the rail circuit. The model was constructed as follows: instead of a rail, \(3\ \text{m}\) of copper wire with \(d = 1.6\ \text{mm}\) were taken; 3 meters of the wire have a resistance of \(0.026\ \Omega\). If it is assumed that the resistance of \(1\ \text{km}\) of double-track line, made with a rail of the heavy mine type, is \(0.028\ \Omega/\text{km}\), then the scale for lengths will be approximately equal to \(1/3000\), i.e., \(3\ \text{m}\) of the model correspond to \(1\ \text{km}\) of track
\[ \left(\frac{0.084}{0.026} = 3\right). \]
As transitional resistances, 60 coils of manganin wire, each 43 cm long \((d = 0.4\ \text{mm})\), were adopted. The resistance of one coil is \(1.56\ \Omega\); consequently, for all 60 coils,
\[ \frac{1.56}{60} = 0.026\ \Omega. \]
Thus, for the constructed model \(p = 1\).
The distribution of currents on the model was as follows (Table 4).
Table 4
Table of current distribution in the rail circuit
on the model \(p = 1\)
| Distance from the train, in meters | Current as % of train current | Distance from the train, in meters | Current as % of train current |
|---|---|---|---|
| 25 | 84.5 | 25 | 5.0 |
| 167 | 81.5 | 50 | 4.3 |
| 642 | 99.0 | 133 | 2.9 |
| 675 | 94.5 | ||
| at the substation 742 | 100.0 |
For the model \(p = 15\), the following data were obtained (Table 5).
Table 5
Table of current distribution in the rail circuit
on the model \(p = 15\)
| Distance from the train, in meters | Current as % of train current | Distance from the train, in meters | Current as % of train current |
|---|---|---|---|
| 25 | 50.0 | 25 | 40 |
| 100 | 20.2 | 100 | 16.5 |
| 200 | 10.1 | 175 | 2.6 |
| 300 | 5.0 | ||
| 850 | 5.0 | ||
| 950 | 10.1 | ||
| 1050 | 15.2 | ||
| 1150 | 37.9 |
The pattern of current distribution obtained on the models is shown in Figs. 10 and 11.
The graphical relationship between the ratio of currents \(\frac{I_2}{I_1}\) and \(pl\), both for the case of an “infinite” rail circuit and for a “finite” one, is given in Fig. 12; moreover, for the latter case the distance \(a\) from the train at which the current in the rail circuit almost disappears was taken as 200 m.
The laws derived by Dell’ Arciprete have been checked repeatedly on our mainline railways and everywhere have been confirmed with accuracy sufficient for practice. For mine conditions these laws are not fully confirm—
![Figure 10 graph: current distribution in the rail circuit. Axis labels include “current in % of train current,” “substation,” “train,” and distance \(s, m\).]
Fig. 10. Distribution of currents in the rail circuit
(model: \(\rho = 1,\ l = 0.75\ \mathrm{km}\)).
![Figure 11 graph: current distribution in the rail circuit. Axis labels include “current in % of train current,” “substation,” and “train.”]
Fig. 11. Distribution of currents in the rail circuit
(model: \(\rho = 15,\ e = 1.25\ \mathrm{mm}\)).
![Figure 12 graph: dependence of \(j_2/j_1\) on \(\rho l\), with curves labeled “infinite rail circuit,” \(\rho a = 1.0\), \(\rho a = 0.8\), \(\rho a = 0.6\), \(\rho a = 0.4\), and \(\rho a = 0.2\).]
Fig. 12. Dependence of \(\dfrac{j_2}{j_1}\) on \(\rho l\) for various \(\rho a\) and for an infinite rail circuit.
Note: \(a = 0.2\ \mathrm{km}\).
are given, it is necessary to introduce corrections for the “finite” length of the rail circuit. The formulas derived for this case in the present work are not yet entirely accurately confirmed by experiment, but they undoubtedly reflect reality more closely than the formulas given by Dell’ Arciprete.
For the section \(BC\), which is the most dangerous from the standpoint of premature cord burnout, the following approximate method of checking the rail circuit may be proposed.
As is evident from the preceding, the current \(I_2\) under mine conditions is always considerably less than half the train current, even with the poorest condition of the rail track. Indeed, we have seen that the current
\[ I_2 = \frac{\operatorname{th} pa \operatorname{th} \dfrac{p l_1}{2}} {1+\operatorname{th} pa \operatorname{th} \dfrac{p l_1}{2}} I_0 . \]
Assigning the value \(a = 200\ \text{m}\) and various values of \(p\) and \(l\), we obtain that only when \(p = 20\) will the current \(I_2\) be equal to half the train current. If a mine rail of type \(H\)—91 is taken and it is assumed that, owing to the poor condition of the joints, the resistance of the rail track will be 3 times greater than normal, i.e. \(r_p = 0.047 \cdot 3 = 0.14\ \Omega/\text{km}\), then for this case the transition resistance from rail to earth is
\[ r_z = \frac{r_p}{p^2} = \frac{0.14}{20^2} = 0.0007\ \Omega\ \text{km}. \]
Meanwhile, even in the worst case, the transition resistance is considerably higher. According to the experiments of Assoc. Prof. Ozerny (“Coal,” 1938, No. 150), in the worst case \(r_z = 0.0047\ \Omega/\text{km}\).
Thus, taking for \(I_2\) a value equal to half the train starting current, we introduce a certain margin in determining the voltage drop in section \(BC\); the latter may be taken as
\[ U_{BC}=\frac{1}{2} I_{\text{start}} r_p l_{BC}; \tag{47} \]
here \(r_p\) is understood to be the resistance of the rail track without taking into account the shunting influence of the ground, which again gives a certain margin in determining \(U_{BC}\). According to the “norms,” this voltage drop must not exceed \(2.5\ \text{V}/\text{km}\). It is nevertheless necessary to note that the validity of the latter norm is disputed—quite rightly—by some authors, in particular Assoc. Prof. Ozerny (see “Coal,” 1938, No. 150).
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The number of trains
\[ N=\frac{2A_{\mathrm{cm}}}{k\cdot n\cdot q}, \]
where $A_{\mathrm{cm}}$ is the shift output of the haulage tracks,
$k=T_{\mathrm{cm}}/T_{\mathrm{reis}}$ is the number of trips of the electric locomotive per shift,
$n$ is the number of mine cars in the train,
$q$ is the useful capacity of a mine car. ↩ -
V. E. Rozenfeld, Yu. E. Ryvkin, I. A. Lakshtovsky.—Electric Traction of Trains, p. 691, 1940. ↩