On the Calculation-Based Determination of the Starting Current of Electric Motors of Face Machines and the Short-Circuit Current in Low-Voltage Section Networks

Ya. V. Petrov

(Presented by the scientific seminar of the Mining-Mechanical Faculty of TPI)

As is known, when relatively powerful electric motors of face machines and mechanisms are started, considerable voltage losses arise in the section networks of mines. As a result, the starting currents decrease, while the currents of normally operating consumers, on the contrary, increase, provided that the torques of static resistances remain constant.

The magnitudes of the starting currents of electric motors, or the voltage losses caused by them, must be taken into account when calculating the networks of mine sections, when considering the operating conditions of relatively powerful electric motors—for example, cutting machines and combines—under real conditions, and when selecting the settings of maximum-current and thermal relays of automatic switches and starters [1].

However, up to the present time there have been no simple and, at the same time, sufficiently accurate methods for the calculation-based determination of the starting currents of electric motors of face machines and mechanisms; for this reason, calculations connected with determining the starting current or the voltage at the terminals of the motor being started are, as a rule, cumbersome or considerably inaccurate.

In the literature known to us [2, 3, 4, 5], the following methods are indicated for determining the starting currents of electric motors.

According to [2], the starting current of an electric motor \(I_{pf}\) can be determined by the formula

\[ I_{pf} = \frac{I_{pn}} {1 + \dfrac{I_{pn}(I_{nd}+I_{nt})}{20 I_{nd}\cdot I_{nt}}}, \]

where: \(I_{pn}\) is the starting current of the electric motor at rated voltage at its terminals at the moment of starting;

\(I_{nd}\) is the rated current of the motor;

\(I_{nt}\) is the rated secondary current of the transformer to which the motor is connected.

This expression is convenient for practical calculations, but is not sufficiently accurate, since it does not take proper account of the parameters of the section network. Here the starting current remains constant regardless of the length and cross section of the cables, as well as the magnitude of the load of normally operating consumers.

Usually the magnitude of the starting current of an electric motor is found from the following simple expression [3, 4]

\[ I_{pf}=I_{pn}\frac{U_p}{U_n}, \]

where \(U_p\) is the voltage at the terminals of the motor being started, taken from the condition of creating the required initial starting torque \((U_p < U_n)\);

\(U_n\) is the rated voltage of the electric motor.

Although fundamentally correct, the indicated expression does not make it possible to determine the actual magnitude of the starting current and starting voltage, since both the starting current and the starting voltage depend on the parameters of the section network, which are not taken into account by this expression. Using the generally accepted method of checking a section network for the starting of an electric motor, by successive approximations one can, of course, find the actual values of \(I_{pf}\) and \(U_p\), but the excessive laboriousness of such work is obvious.

Determination of the starting current by the method of the resultant resistance of the section network during the starting of the electric motor under consideration and the normal operation of the other consumers of the section is generally carried out as follows [4]

\[ I_{pf}=\frac{U_{nt}-\Delta U_{tk}}{\sqrt{3}Z_{kd}}, \]

where \(U_{nt}\) is the rated secondary voltage of the transformer;

\(\Delta U_{tk}\) is the voltage loss in the transformer and cables connected in series between the transformer and the point of the network to which the cable feeding only the electric motor being started is connected;

\(Z_{kd}\) is the geometric sum of the resistances of the electric-motor cable and the starting resistance of the latter.

In view of the considerable laboriousness of determining \(\Delta U_{tk}\), especially in the presence of the load of normally operating consumers, the method of resultant resistance, although it gives the most reliable results, is practically inapplicable in its existing form.

A simplified variety of the method of resultant resistance is a procedure in which the resultant resistance of the network is found by scalar addition of the resistances of the transformer \(Z_t\), the cables \(Z_k\) connected in series between the transformer and the motor being started, and the starting resistance of the motor \(Z_{pd}\). In this case [5]

\[ I_{pf}=\frac{U_{nt}}{\sqrt{3}(Z_t+Z_k+Z_{pd})}. \]

This procedure is fairly simple; however, it is inferior to the method of resultant resistance in accuracy, since scalar resistance values are added and the influence of the load of normally operating consumers on the magnitude of \(I_{pf}\) is neglected.

With certain assumptions, it appears possible to simplify the method of resultant resistance, practically without reducing its accuracy, if the concept of the equivalent length of cables is used [5].

The starting current of an electric motor can be determined as follows:

\[ I_{pf}=\frac{U_{nt}-\Delta U_t}{\sqrt{3}Z_n}, \tag{1} \]

here

$Z_n$ is the resistance of the starting circuit¹ (of one of its phases), taking into account the effect on the value of $Z_n$ of the load of normally operating consumers;

$\Delta U_m$ is the voltage loss in the transformer when the current of normally operating consumers flows.

The voltage loss in the transformer can easily be calculated by one of the known formulas, for example,

\[ \Delta U_m=\sqrt{3}\,I_n(r_m\cos\varphi_n+x_m\sin\varphi_n), \]

where $I_n$ is the total current of the normally operating consumers, flowing through the transformer at the moment the motor is started; it is assumed that $I_n$ does not depend on $I_{n\phi}$;

$\cos\varphi_n$ is the power factor of the normally operating consumers;

$r_m$ and $x_m$ are the active and inductive resistances of the transformer (Table 2).

The greatest difficulties arise in determining the resistance of the starting circuit.

If no load of normally operating consumers is connected to the starting circuit, then, neglecting the resistance of the network up to the transformer (system), we have

\[ Z_n=\sqrt{(r_m+r_k+r_{n\partial})^2+(x_m+x_k+x_{n\partial})^2}, \tag{2} \]

where $r_k$ and $x_k$ are the active and inductive resistances of the cables from the transformer to the motor being started;

$r_{n\partial}$ and $x_{n\partial}$ are the active and inductive resistances of the motor during starting.

Replacing the cables of different cross section, connected in series between the transformer and the motor being started, by a cable of one cross section, for example $50\ \text{mm}^2$, for $r_k$ and $x_k$ we may write

\[ r_k=0.0188\sum_{i=1}^{n}\frac{l_i}{s_i} =0.0188\left(\frac{l_1}{s_1}+\frac{l_2}{s_2}+\ldots+\frac{l_n}{s_n}\right)= \]

\[ =0.0188\left(\frac{l_1}{50}K_{e1}+\frac{l_2}{50}K_{e2}+\ldots+\frac{l_n}{50}K_{en}\right)= \]

\[ =0.0188\cdot\frac{1}{50}\,l_e, \]

\[ x_k=\frac{0.07}{10^3}\sum_{i=1}^{n}l_i. \]

Here

\[ 0.0188\ \frac{\Omega\cdot \text{mm}^2}{\text{m}} \]

is the resistivity of conductor copper at a temperature of $+20^\circ\text{C}$; $l_1,l_2,\ldots,l_n,\text{m}$ and $S_1,S_2,\ldots,S_n,\text{mm}^2$ are the length and cross section of the co—

of the corresponding cable; \(K_{e1}, K_{e2}, \ldots K_{en}\) are equivalence coefficients for the corresponding cable cross sections; the values of the coefficients are given in Table 1.

\(l_e, m\) is the length of cable with a cross section of \(50\ \mathrm{mm}^2\), equivalent to the total length of cables of various cross sections laid between the transformer and the motor being started; in the general case

\[ l_e=\sum_{i=1}^{n} l_i K_{ei}; \]

\[ \frac{0.07}{10^3}\ \Omega/m \]

is the inductive resistance of one meter of cable (taken to be the same for all cross sections and types of low-voltage power cables used in mine districts).

Table 1

Values of the equivalence coefficients for reducing various cable cross sections to a cross section of \(50\ \mathrm{mm}^2\)

In view of the relatively small magnitude of the inductive resistance of cables in comparison with the resistance of the other elements of the starting circuit, and also the fact that the average value of the coefficients \(K_e\) for the cable cross sections most commonly used in mine districts approaches 1 (Table 1), it is permissible to express \(X_k\) as follows:

\[ X_k \approx \frac{0.07}{10^3}\left(l_1 K_{e1}+l_2 K_{e2}+\cdots+l_n K_{en}\right) = \frac{0.07}{10^3}l_e . \]

The power factors of TMS transformers under short-circuit conditions and of mine induction squirrel-cage motors during starting differ little from one another. Therefore, with a small error, the vector addition of the resistances of the transformer and the motor being started may be replaced by a scalar one, assuming that the power factors of the transformer and of the motor being started are identical and equal to the motor power factor during starting.

Then expression (2), after the simplest transformations, may be rewritten as:

\[ Z_p \approx \frac{1}{100} \sqrt{ 10^4(Z_m+Z_{nd})^2+ (7.5\cos\varphi_{nd}+1.4\sin\varphi_{nd}) } \times \sqrt{ (Z_m+Z_{nd})l_e+0.00147l_e^2 }. \tag{3} \]

The starting current of the electric motor is

\[ I_{pd}= \frac{100U_{nm}} {\sqrt{3}\sqrt{ 10^4(Z_m+Z_{nd})^2+ (7.5\cos\varphi_{nd}+1.4\sin\varphi_{nd})(Z_m+Z_{nd})l_e+ 0.00147l_e^2 }} . \tag{4} \]

Voltage at the terminals of the electric motor at the moment of starting

\[ U_{n0}=\sqrt{3}\,I_{n\phi}Z_{n0}. \tag{5} \]

In expressions (3, 4, 5):

\(Z_m\) is the total impedance of the transformer, ohms (Table 2);

\[ Z_{n0}=\frac{U_n}{\sqrt{3}\,I_{nn}} \]

is the starting impedance of the motor, ohms;

\(\cos\varphi_{n0}\) is the power factor of the motor at starting.

The data needed in calculations for the transformers and coal-cutter electric motors most commonly encountered in practice are given in Tables 2 and 3. The impedances of other transformers or electric motors are determined from generally known formulas.

Table 2

Impedances of power transformers

Table 3

Starting parameters of coal-cutter electric motors

From expression (4) a family of curves \(I_{n\phi}=f(l_e)\) can readily be constructed for various transformers and the most frequently used electric motors whose starting conditions are of interest. Such curves in a rectangular coordinate system for the electric motors MA—191/3M, MA—191/10, and MAD—191/11, for various transformer-capacity variants, are shown in Figs. 1, 2, and 3. In constructing the curves, \(U_{nm}=400\) V was assumed. If the voltage at the transformer \(U_m\) differs from 400 V owing to the use of the transformer tap terminals, or because of the presence of loads of normally operating consumers, or for any other reason, then in all such cases the current values found from the curves should be multiplied by

\[ \frac{U_m}{400}. \]

The accuracy of calculations by formula (4) or from the curves is practically the same as the accuracy of calculations by the resultant-impedance method, while surpassing the latter in simplicity. Calculations from the curves are especially convenient, since in essence they amount only to determining the equivalent cable length.

If a load of normally operating consumers is connected to the starting circuit, determination of the starting current becomes considerably more complicated, since the magnitude of the latter is determined not only by the resistance of the starting circuit, but also by the load of the normally operating consumers, which, in turn, depends on the starting current.

Fig. 1. Curves for determining the starting current of the electric motor
MA—191/3M

To solve analytically, with exactness, the problem of determining the starting current under these conditions is difficult or even impossible, especially for the rather branched sectional networks encountered in practice.

Therefore, when there is a load of normally operating consumers, this problem has to be solved with a known approximation in comparison with the results of calculations by the method of the resulting resistance, which, like the proposed method, assumes that the load of normally operating consumers does not depend on the starting current.

The reduction of the motor starting current due to the action of the current of the normal load flowing through the starting circuit can evidently be taken into account by a decrease of the voltage at the transformer (1) and by a certain increase in the length of the equivalent cable in those sections through which the current of the normal load passes. Then it proves possible to use, as the calculation circuit, the same simple circuit as in the absence of the load of normally operating consumers, which leads to a solution analogous to (4).

The increase in the length of the equivalent cable laid between the transformer and the motor being started can be calculated on the basis of the following considerations.

The total voltage loss in the cable, through which the current of the normally operating consumers \(I_n\) and the starting current of the motor under consideration \(I_{n\delta}\) flow simultaneously:

\[ \begin{aligned} \Delta U_{nn} &= \Delta U_p + \Delta U_n = I_{n\delta}(r_k \cos \varphi_{n0} + x_k \sin \varphi_{n0}) +{}\\ &\quad + I_n(r_k \cos \varphi_n + x_k \sin \varphi_n) = I_{n\delta}l_3 \times{}\\ &\quad \times \left(\frac{0{,}0188}{50}\cos \varphi_{n0} + \frac{0{,}07}{10^3}\sin \varphi_{n0}\right) + I_n l_3 \times{}\\ &\quad \times \left(\frac{0{,}0188}{50}\cos \varphi_n + \frac{0{,}07}{10^3}\sin \varphi_n\right). \end{aligned} \]

Fig. 2. Curves for determining the starting current of the electric motor MA-19/10

Divide both sides of the last equality by

\[ \Delta U_p = I_{n\delta}l_3 \left(\frac{0{,}0188}{50}\cos \varphi_{n0} + \frac{0{,}07}{10^3}\sin \varphi_{n0}\right), \]

we obtain

\[ \frac{\Delta U_{np}}{\Delta U_n} = 1+ \frac{ I_n\left(\frac{0.0188}{50}\cos\varphi_n+\frac{0.07}{10^3}\sin\varphi_n\right) }{ I_{n\phi}\left(\frac{0.0188}{50}\cos\varphi_{nd}+\frac{0.07}{10^3}\sin\varphi_{nd}\right) }. \]

Solving this equation with respect to \(I_{n\phi}\) is possible, but somewhat cumbersome, and therefore, for the sake of considerable simplification, it is permissible here to take \(I_{n\phi}\approx I_{pn}\), which somewhat reduces the resistance of the starting circuit as compared with the actual value. But since the scalar addition

Fig. 3. Curves for determining the starting current of the electric motor MAD-191/11

of the resistances \(Z_t\) and \(Z_{nd}\) gives an error of the opposite sign, the total error in calculating the value of the starting current is small. It is also possible,

as studies have shown, the second terms standing in the parentheses of the last expression may be neglected. Then

\[ \frac{\Delta U_{np}}{\Delta U_p} \simeq 1+\frac{I_n \cos \varphi_n}{I_{pn}\cos \varphi_{nd}} . \]

Since in both cases the cross section of the equivalent cable is the same, and the voltage losses are proportional to the cable lengths, the ratio of voltage losses can be replaced by the ratio of the corresponding lengths of the equivalent cable:

\[ \frac{l_{en}}{l_e}=1+\frac{I_n \cos \varphi_n}{I_{pn}\cos \varphi_{nd}}=K_n \]

or

\[ l_{en}=l_e K_n=lK_eK_n, \tag{7} \]

where
\(l_e\) is the equivalent length of the cable, determined without taking into account the load of consumers operating normally;
\(l_{en}\) is the same, taking into account the load of consumers operating normally;
\(l\) is the actual length of the cable of the given cross section;

\[ K_n=1+\frac{I_n\cos \varphi_n}{I_{pn}\cos \varphi_{nd}}>1.0 \]

is the coefficient of increase in the equivalent length of the cable due to the influence of the load of consumers operating normally.

Thus, in all possible cases of distribution of electric power over the section, the starting current of the electric motor may be determined from the expression

\[ I_{pf}= \frac{100\left(U_{nm}-\Delta U_m\right)} {\sqrt{3}\sqrt{10^4\left(Z_m+Z_{nd}\right)^2+\left(7.5\cos \varphi_{nd}+1.4\sin \varphi_{nd}\right)\left(Z_m+Z_{nd}\right)l_{en}+0.00147\,l_{en}^2}} \tag{8} \]

The voltage at some point of the equivalent cable is

\[ U=U_{nm}-\Delta U'_m-\frac{l'_{en}}{l_{en}}\left(U_{nm}-\Delta U'_m-U_{pd}\right), \tag{9} \]

here
\(l'_{en}\) is the equivalent length of the cable from the transformer to the point under consideration;
\(l_{en}\) is the same, from the transformer to the motor being started;
\(\Delta U'_m\) is the voltage loss in the transformer when currents \(I_n\) and \(I_{pf}\) flow.

\[ \Delta U'_m = \sqrt{3}I_n\left(r_m\cos \varphi_n+x_m\sin \varphi_n\right) +\sqrt{3}I_{pf}\times \]

\[ \times\left(r_m\cos \varphi_p+x_m\sin \varphi_p\right) = \Delta U_m+\sqrt{3}I_{pf}\times \]

\[ \times\left(r_m\cos \varphi_p+x_m\sin \varphi_p\right), \]

here

\[ \cos \varphi_p=\frac{R_p}{Z_p} = \frac{R_m+R_{nd}+R_k}{Z_p} \simeq \frac{\left(Z_m+Z_{nd}\right)\cos \varphi_{nd}+0.375\cdot 10^{-3}l_{en}}{Z_p} \]

is the power factor of the starting circuit.

A number of calculations have shown that determination of the starting currents of electric motors by expression (8) is performed with errors (mainly positive) not exceeding 1.0–2.5%, if, as the control values of the starting currents, one takes the values found by the method of the resultant network resistance. At the same time, the proposed method compares favorably with the method of the resultant network resistance in its simplicity,

which is especially felt with a branched power-supply scheme for a section. The curves presented in Figs. 1, 2, 3 make it possible, very simply and with the least expenditure of time, to find the starting currents of electric motors, no matter how complex the electric-power distribution scheme in the section may be.

According to the safety regulations in coal and shale mines, the selection of the settings of the maximum-current relays of circuit breakers and starters must be carried out taking into account the two-pole short-circuit currents in the section low-voltage network. These currents are determined from the curves \(I_{\infty}^{(2)} = f(l_z)\), constructed for individual transformers operating alone [5].

It is easy to show that the steady-state currents of a two-pole short circuit in the section low-voltage network can also be found analytically just as simply, from an expression analogous to (4), i.e.,

\[ I_{\infty}^{(2)}=\frac{100U_{nt}}{2\sqrt{m+n l_z+p l_z^2}}\,a, \tag{10} \]

where the quantities \(m\) and \(n\) do not depend on the length and cross section of the cables and, for a given transformer, are constant; the values of \(m\) and \(n\) are given in Table 4, and are also presented as curves in Fig. 4, which make it possible to determine \(m\) and \(n\) also when transformers operate in parallel; \(p=0.00147\) in all cases, as above.

Table 4

The dependences \(I_{\infty}^{(2)} = f(l_z)\) for individual transformers are represented by curves similar to the curves [5], but located somewhat higher, since the determination of short-circuit currents by expression (10) is carried out more accurately.

Example 1. Determine the starting current of the electric motor MA-191/10 for the circuit of Fig. 5.

Schematic labels in Fig. 5: TMSh-180/6; cable sections of \(200\,\text{m}\) with cross sections \(95\,\text{mm}^2\), \(70\,\text{mm}^2\), and \(35\,\text{mm}^2\); points \(a\) and \(b\); \(I_{n1}=100\,a\); \(I_{n2}=60\,a\); \(\cos\varphi_{n1}=\cos\varphi_{n2}=0.75\); MA-191/10.

Fig. 5. Calculation scheme.

According to expression (7) and the data of Tables 1 and 3, we find

\[ l_{en}=200\cdot0.55\left(1+\frac{160\cdot0.75}{510\cdot0.54}\right)+200\cdot0.73\left(1+\frac{60\cdot0.75}{510\cdot0.54}\right)+ \]

\[ +200\cdot1.42=158+170+284=612\ \text{m}. \]

Voltage loss in the transformer from the current of normally operating consumers

\[ \Delta U_t=\sqrt{3}\,160(0.020\cdot0.75+0.045\cdot0.66)\simeq 12\ \text{V}. \]

Starting current of the electric motor according to expression (8)

\[ I_{pf}= \frac{100(400-12)} {\sqrt{3}\cdot\sqrt{10^4(0.049+0.431)^2+(1.5\cdot0.54+}} \]

\[ \overline{+\,1.4\cdot0.84)(0.049+0.431)612-0.00147\cdot612^2} \]

\[ =\frac{100\cdot388}{\sqrt{3}\cdot0.661}\simeq339\ \text{A}. \]

Legend in the figure:

1 — \(m\) at 400 V
2 — \(n\) at 420 V
3 — \(n\) at 400 V
4 — [[unclear: same designation]] 420 V

Fig. 4. Values of the coefficients \(m\) and \(n\) for determining the current of a two-pole short circuit.

Power factor of the starting circuit

\[ \cos\varphi_n= \frac{(0.049+0.431)\cdot0.54+0.375\cdot10^{-3}\cdot612}{0.661} =0.737. \]

Voltage at the terminals of the motor being started

\[ U_{n\partial}=\sqrt{3\cdot 339\cdot 0.431}=253\ \text{V}. \]

Voltage loss in the transformer due to the normal-load current and the starting current of the electric motor

\[ \Delta U'_m=12+\sqrt{3}\cdot 339(0.020\cdot 0.737+0.045\cdot 0.676)=38\ \text{V}. \]

Voltage at point \(v\)

\[ U_v=400-38-\frac{158+170}{612}(400-38-253)=304\ \text{V}. \]

The starting current, determined by the method of the resultant resistance, is \(I_{n\phi}=342\ \text{A}\). The error is

\[ \frac{342-339}{342}\cdot 100=0.9\%. \]

From the curve in Fig. 2, for \(l_{\text{eq}}=612\ \text{m}\), \(I'_{n\phi}=351\ \text{A}\). Taking into account the voltage loss in the transformer \(\Delta U_m=12\ \text{V}\), we obtain

\[ I_{n\phi}=351\cdot \frac{400-12}{400}\simeq 340\ \text{A}. \]

Example 2. Find the current \(I_{\infty}^{(2)}\) at point \(b\) of the circuit of the preceding example.

\(l_{\text{e}}=200\cdot 0.55+200\cdot 0.73=256\ \text{m}\); according to (10) and Table 4, for \(U_{nm}=400\ \text{V}\),

\[ I_{\infty}^{(2)}= \frac{100\cdot 400}{2\sqrt{32+0.293\cdot 256+0.00147\cdot 256^2}} \simeq 1403\ \text{A}. \]

Conclusions

1. Existing methods for determining the starting currents of electric motors of face machines do not possess sufficient accuracy or simplicity.

2. The proposed calculation method makes it possible to determine the starting currents of electric motors of face machines and the currents of a two-pole short circuit, taking into account the operation of other motors connected to the network under calculation, much more simply and requiring less time than existing methods.

LITERATURE

1. Leibov R. M. Settings of thermal relays of PMV-1355 and PMV-1365 starters. Collection of articles “Mining Mechanics and Electrical Engineering” of the Donetsk Industrial Institute, Ugletekhizdat, 1954.

2. Berstel V. N. Basic technical requirements for electric motors of new face machines. Ugol, No. 1–2, 1945.

3. Ozernoi M. I. Mining Electrical Engineering. Ugletekhizdat, 1951.

4. Mikheev Yu. A., Faibisovich I. L. Calculation of the cable network of a mine section. Ugletekhizdat, 1951.

5. Efremov V. K. and Khaesh M. M. Protection against short-circuit currents in underground low-voltage electrical networks. Promyshlennaya energetika, No. 12, 1952.

1. By the starting circuit is meant the circuit composed of the transformer windings, the cable conductors successively connected between the transformer and the motor being started, and the windings of the motor being started. ↩