Graphical-Analytical Calculation of Rail Circuit Parameters
Abstract Generated abstract
This article presents a graphical-analytical method for determining the electrical parameters of direct-current and alternating-current rail circuits from open-circuit and short-circuit measurements. It reformulates standard calculations for rail and ballast specific resistance, and for alternating-current circuits also the rail impedance angle, by introducing correction coefficients expressed as functions of measured impedance ratios and phase data. These coefficients are obtained from graphs rather than logarithmic, hyperbolic, or complex-function tables, reducing the computation to simple arithmetic after voltage, current, and phase measurements. The method is intended to simplify repeated practical calculations while retaining accuracy sufficient for engineering use.
Full Text
Graphical-Analytical Calculation of Rail Circuit Parameters
V. M. Razin
The article proposes, for practical use, a graphical-analytical method for determining the electrical parameters of DC and AC rail circuits from measurements by the open-circuit and short-circuit method.
1. DC rail circuit
When measuring the electrical parameters of DC rail circuits by the open-circuit and short-circuit method, at the feeding end of the rail circuit the voltage and current are measured both with the relay end open \((U_x \text{ and } I_x)\) and with it short-circuited \((U_k \text{ and } I_k)\). From these measurement data, the values of the input resistances of the rail circuit under open-circuit conditions and under short circuit at the receiving end are determined:
\[ R_x = \frac{U_x}{I_x}, \tag{1} \]
\[ R_k = \frac{U_k}{I_k}. \tag{2} \]
If the ballast resistance is large in comparison with the resistance of the rails, then for short rail circuits, with accuracy sufficient for practical purposes, one takes:
\[ r_p = \frac{R_k}{l}, \tag{3} \]
\[ r_b = R_x l, \tag{4} \]
where \(r_p\) is the specific resistance of the rails in \(\Omega/\text{km}\),
\(r_b\) is the specific resistance of the ballast in \(\Omega/\text{km}\),
\(l\) is the length of the rail circuit in km.
If the specific ballast resistance is small, or if the rail circuit has a great length, then the calculation is usually carried out according to the following scheme [1; 2].
The surge impedance of the rail circuit is determined by the expression:
\[ R_v = \sqrt{R_k R_x}. \tag{5} \]
The value of the attenuation constant is found from the formula:
\[ \gamma = \frac{\operatorname{arth}\sqrt{\dfrac{R_k}{R_x}}}{l} = \frac{1}{l}\cdot \frac{1}{2}\ln \frac{1+\sqrt{\dfrac{R_k}{R_x}}} {1-\sqrt{\dfrac{R_k}{R_x}}}. \tag{6} \]
Then the sought specific resistances of the rails and ballast are determined by simple relations:
\[ r_p = R_{\mathrm{в}}\gamma \tag{7} \]
\[ r_{\mathrm{б}}=\frac{R_{\mathrm{з}}}{\gamma}. \tag{8} \]
To compute the parameters in this case, tables of logarithms or hyperbolic functions are required, which sometimes presents a certain inconvenience. By means of simple transformations and by replacing tabular data with a graphical dependence, the calculation formulas can be reduced to a form more convenient for calculations. Using relations (5) and (6), equalities (7) and (8) can be represented in the form:
\[ r_p=\sqrt{R_k R_x}\, \frac{\operatorname{arth}\sqrt{\dfrac{R_k}{R_x}}}{l} = \frac{R_k}{l}\, \frac{1}{\sqrt{\dfrac{R_k}{R_x}}}\, \operatorname{arth}\sqrt{\dfrac{R_k}{R_x}} . \tag{9} \]
\[ r_{\mathrm{б}}= \frac{l\sqrt{R_k R_x}}{\operatorname{arth}\sqrt{\dfrac{R_k}{R_x}}} = \frac{R_x l}{\dfrac{1}{\sqrt{\dfrac{R_k}{R_x}}}\operatorname{arth}\sqrt{\dfrac{R_k}{R_x}}}. \tag{10} \]
Denote
\[ \frac{R_k}{R_x}=n,\quad \text{where } n\leq 1. \tag{11} \]
Then
\[ r_p=\frac{R_k}{l}\,\frac{1}{\sqrt{n}}\operatorname{arth}\sqrt{n} =\frac{R_k}{kl}, \tag{12} \]
\[ r_{\mathrm{б}}= \frac{R_x l}{\dfrac{1}{\sqrt{n}}\operatorname{arth}\sqrt{n}} = R_x\cdot l\cdot k. \tag{13} \]
Here it is assumed that
\[ k=\frac{\sqrt{n}}{\operatorname{arth}\sqrt{n}}. \tag{14} \]
The coefficient \(k=f(n)\) in formulas (11) and (12) is, in a certain sense, a correction coefficient to formulas (3) and (4), taking into account the influence of current leakage through the ballast. The dependence \(k=f(n)\) is presented graphically in Fig. 1. Thus, determination of the parameters of a direct-current rail circuit reduces to the following simple computational operations. From the measurement data obtained by the open-circuit and short-circuit method, we compute \(R_x\) and \(R_k\) from formulas (1) and (2). Then, from formula (11), we find the value \(n\). After that, from the graph in Fig. 1, we determine the value of the correction coefficient \(k\) for the found value of the parameter \(n\).
As a result, the sought values of the specific resistances of the rails and ballast are determined from relations (12) and (13).
2. Alternating-Current Rail Circuit
For an alternating-current rail circuit, in addition to determining the specific resistances of the rails and ballast under alternating current, it is also necessary to determine the electrical angle of the rail resistance, caused by
the presence of inductive resistance of the rails to alternating current. Suppose that the measurement of an alternating-current rail circuit is carried out by the open-circuit and short-circuit method with the use of a phase meter. In this case, with the aid of a voltmeter, ammeter, and phase meter, the voltage, current, and phase-shift angle between current and voltage at the feeding end are measured both with a short circuit at the receiving end \((U_k, I_k\) and \(\varphi_k)\), and with the relay circuit at it open \((U_x, I_x\) and \(\varphi_x)\). The subsequent calculations are carried out according to the following scheme [1; 2; 3]. We determine the input impedance of the rail circuit under open-circuit and short-circuit conditions:
Fig. 1
\[ Z_k=\frac{U_k}{I_k}, \tag{15} \]
\[ Z_x=\frac{U_x}{I_x}. \tag{16} \]
Specific resistance of the rails
\[ Z_p=\sqrt{Z_k Z_x(\beta^2+\alpha^2)}. \tag{17} \]
Specific resistance of the ballast
\[ Z_b=\sqrt{\frac{Z_k Z_x}{\beta^2+\alpha^2}}=\frac{Z_k Z_x}{Z_p}. \tag{18} \]
Angle of the rail resistance
\[ \varphi_p=\varphi_x+\varphi_k. \]
The values of the coefficients \(\alpha\) and \(\beta\) are determined from the relations:
\[ \operatorname{th} 2\beta l= \frac{2\sqrt{Z_k Z_x}}{Z_k+Z_x} \cos\left(\frac{\varphi_k-\varphi_x}{2}\right), \tag{19} \]
\[ \operatorname{tg} 2\alpha l= \frac{2\sqrt{Z_k Z_x}}{Z_x-Z_k} \sin\left(\frac{\varphi_k-\varphi_x}{2}\right). \tag{20} \]
Here \(l\) is the length of the rail circuit.
If the value \(\alpha\) has been found, then the value \(\beta\) is more simply determined from the equation:
\[ \beta=\frac{\alpha}{\operatorname{tg}\dfrac{\varphi_k+\varphi_x}{2}}. \tag{21} \]
As is evident from the expressions given above, the greatest difficulty here is the computation of the coefficients \(\alpha\) and \(\beta\). When nomograms of the hyperbolic tangent of a complex argument are used, the quantities \(\alpha\) and \(\beta\) are determined as follows [1]. The modulus of the quantity
\[ \operatorname{th}\gamma l=\operatorname{th}(\beta l+j\alpha l) \]
is equal to:
\[ T=\sqrt{\frac{Z_k}{Z_x}}, \tag{22} \]
and the argument
\[ \varphi_t=\frac{\varphi_k-\varphi_x}{2}. \tag{23} \]
Hence we have:
\[ \operatorname{th}\gamma l=\operatorname{th}(\beta l+j\alpha l)=T e^{j\varphi_t}. \tag{24} \]
Using the data of this equation, on nomograms of the hyperbolic tangent of a complex argument, available in the literature [1], we find \(\alpha l\) and \(\beta l\). After this we compute the unit resistance of the rails
\[ Z_p=\frac{\sqrt{Z_k Z_x\left[(\alpha l)^2+(\beta l)^2\right]}}{l}. \tag{25} \]
The unit resistance of the ballast is
\[ Z_b=\frac{Z_k Z_x}{Z_p}. \]
The angle of the rail resistance is
\[ \varphi_p=\varphi_x+\varphi_k. \tag{26} \]
The computation process is somewhat simplified; however, computation by formula (25), when there is a large number of calculations, introduces a certain difficulty. In addition, a certain skill in using nomograms is required in order to obtain sufficient accuracy of the computations.
It is easy to show that the calculation equations for an alternating-current rail circuit can be reduced to the form of equations (12) and (13), where the value of the correction coefficient \(k\) can be determined graphically.
Indeed, determining the values \(\alpha\) and \(\beta\) from equations (19) and (20), substituting these values into equations (17) and (18), and carrying out simple transformations, we obtain:
\[ Z_p=\frac{Z_k}{l}\times \frac{ \sqrt{ \operatorname{ar^2 th}\left[ \frac{2\sqrt{\dfrac{Z_k}{Z_x}}}{1+\dfrac{Z_k}{Z_x}} \cos\left(\frac{\varphi_k-\varphi_x}{2}\right) \right] + \operatorname{ar^2 tg}\left[ \frac{2\sqrt{\dfrac{Z_k}{Z_x}}}{1-\dfrac{Z_k}{Z_x}} \sin\left(\frac{\varphi_k-\varphi_x}{2}\right) \right] } }{ 2\sqrt{\dfrac{Z_k}{Z_x}} }. \tag{27} \]
\[ Z_b=Z_x l\times \frac{ 2\sqrt{\dfrac{Z_k}{Z_x}} }{ \sqrt{ \operatorname{ar^2 th}\left[ \frac{2\sqrt{\dfrac{Z_k}{Z_x}}}{1+\dfrac{Z_k}{Z_x}} \cos\left(\frac{\varphi_k-\varphi_x}{2}\right) \right] + \operatorname{ar^2 tg}\left[ \frac{2\sqrt{\dfrac{Z_k}{Z_x}}}{1-\dfrac{Z_k}{Z_x}} \sin\left(\frac{\varphi_k-\varphi_x}{2}\right) \right] } }. \tag{28} \]
Introducing the notation:
\[ n=\frac{Z_k}{Z_x}, \tag{29} \]
\[ \varphi=\frac{\varphi_k-\varphi_x}{2}, \tag{30} \]
\[ k= \frac{2\sqrt{n}}{ \sqrt{ \operatorname{ar^2 th}\left(\frac{2\sqrt{n}}{1+n}\cos\varphi\right) + \operatorname{ar^2 tg}\left(\frac{2\sqrt{n}}{1-n}\sin\varphi\right) } }, \tag{31} \]
we obtain:
\[ Z_p = \frac{Z_k}{kl}, \tag{32} \]
\[ Z_b = Z_x \cdot kl. \tag{33} \]
The dependence \(k=f(n_1\varphi)\) is presented graphically in Fig. 2.
Thus, in the case of an alternating-current rail circuit as well, the calculation of the specific resistances of the rails and ballast is reduced to the simplest mathematical operations. On the basis of the measurement data we calculate \(Z_k\) and \(Z_x\) by formulas (15) and (16), then by formulas (29) and (30) we find the values of the auxiliary parameters \(n\) and \(\varphi\). For the computed values of the parameters \(n\) and \(\varphi\), from the graphs in Fig. 2 we find the value of the correction coefficient \(k\), and by formulas (32) and (33) we calculate the desired values of the specific resistances of the rails and ballast. The impedance angle of the rails is calculated by formula (26).
Fig. 2
As is clear from the foregoing, the proposed method for calculating the parameters of direct- and alternating-current rail circuits considerably simplifies the computational work, is distinguished by simplicity and clarity with accuracy quite sufficient for practical purposes, and does not require the use of mathematical tables for the calculations.
All this taken together can yield a substantial saving of time when carrying out a large number of calculations of this kind.
LITERATURE
- Bryleev A. M. and Ryazantsev B. S. Rail Circuits. Transzheldorizdat, 1952.
- Vakhnin M. I. Line Devices of Centralized Traffic Control. Transzheldorizdat, 1947.
- Ponomarenko L. P. Alternating-Current Rail Circuits, 27th collection, NIISS NKPS, 1933.