CALCULATION OF METAL RHEOSTATS AND ADJUSTMENT OF CONTACTOR PANELS FOR MINE HOISTING INSTALLATIONS

I. A. Balashev

Methods for Calculating Rheostats

At various times, several methods have been proposed for calculating metal rheostats for three-phase asynchronous electric motors.

The principal ones are the following:

1. A method based on the use of a circle diagram. This method was developed by Prof. V. S. Kulebakin [1], as well as by other authors [4].

2. The method of mechanical characteristics, which subsequently became exceptionally widespread. This method has two variants: exact and simplified methods of calculation.

The exact method is based on the use of curvilinear mechanical characteristics of an asynchronous electric motor, both natural and rheostatic.

The simplified method is based on straightening the aforementioned characteristics in the form of rectilinear rays issuing from one pole—the intersection of the ordinate axis and the synchronous-speed line.

Each of the indicated methods has several modifications intended to make the calculation simple, visual, and universal, making extensive use of generalizations and graphical constructions.

For calculating rheostats of asynchronous electric motors used in hoisting installations, an exact method using curvilinear mechanical characteristics was developed by KhEMZ (Kharkov Electromechanical Plant), which manufactures rheostats, contactor panels, and electric motors of reinforced design for this purpose [5, 16, 17].

The use of the exact method of constructing curvilinear characteristics makes it possible to reduce the number of starting stages to five or six, chiefly at the cost of increasing the surges of dynamic torque, creating conditions for non-smooth starting and difficulty of control when stopping the machine.

Moreover, in all methods claiming accuracy of calculation and full utilization of the starting properties of the electric motor, the constancy of the voltage at the stator terminals is accepted without reservation. This assumption is in no way justified, since mine networks practically do not provide these conditions. Thus the exact method practically leads to jerky starting and to the possibility of passing through the overturning torque when the voltage is reduced, and a method that claims accuracy proves to be far from accurate.

The principal shortcoming of both the basic methods and the KhEMZ method should be considered the fact that the rheostat calculation is carried out without taking into account the operating duty, i.e., the diagrams of speed and forces that occur when the hoisting installation performs production operations. This last condition is specified as follows:

1. The starting and retarding characteristics of the electric motor must satisfy the technical requirements for hoisting and lowering loads.

2. A hoisting installation is a system with an elastic connection between the conveyance and the machine, and therefore the condition of smooth starting and stopping must be satisfied.

3. The condition of durability of the installation’s operation must be ensured; this is achieved to a considerable degree only when the excess (dynamic) torque has an increasing character.

Experience in the operation and adjustment of hoisting installations with contactor control shows that cases of mismatch between the power of the electric motor and the load regime are very frequent: the power of the former is excessive. In such cases, conditions are created not only for difficult control of the machine, but also for an emergency state of it. The surges of starting torque thereby become so great that the components of the mechanical part are subject to breakage and rapid wear of the ropes occurs.

Observations and analysis of experimental data ultimately lead to the conclusion that calculation of the rheostat for the asynchronous electric motor of a hoisting installation by the method of straightened characteristics, taking into account the diagrams of the installation operating regime (diagrams of speed and forces), gives much better results than other methods that claim accuracy but overlook more important circumstances.

On the basis of the considerations set forth, the author has proposed a graphical-analytical method for calculating the rheostat with allowance for diagrams of the operating regime. The method has been tested repeatedly in practice, where it has always given good results.

For both a designed and an existing installation, the diagrams of the operating regime, i.e., the diagrams of speed and forces, are known. The nominal voltage and rotor current and its nominal slip are likewise known, either from catalog data or from the motor data sheet. These data are sufficient for calculating a rheostat that provides the specified operating regime. Operating conditions—smooth starting, the possibility of lowering the load, emergency lifting of the cage above the cams, inspection of the ropes and shaft, etc.—must be taken into account additionally.

Graphical-Analytical Calculation

For the graphical-analytical calculation of the rheostat of the asynchronous electric drive of a hoisting machine, the operating regime of the hoisting installation and the operating regime for the given conditions must be specified, as well as the nominal values of the rotor voltage and current and the slip.

The main parameters determining the operating regime are:

1. The speed diagram of one complete hoisting cycle for the nominal load.
2. The diagram of torques on the drum shaft, or the diagram of forces at the radius of rope winding, if the drums are cylindrical.

The rheostat must be designed so as to provide, as accurately as possible, the specified operating regime during the unsteady states of starting and retardation, and also to allow for the features of the technological process of the hoisting operation.

In cage hoisting installations it is necessary to carry out overtravel lifting of the cage above the cams, and in skip installations—to provide the possibility of smooth departure from and entry into the unloading curves, smooth seating of the skip on the feeder carriage and closing of the latter, and also, in both cases, smooth tensioning of the rope, in order to avoid the occurrence in it of impermissibly large stresses from oscillations.

Stepped metallic rheostats made of cast-iron elements, as adopted in hoisting installations, do not always fully satisfy the indicated requirements, especially in cases where the number of rheostat steps is small, as a result of which, during starting and stopping of the machine, the change in torque ...

of the drive occurs with impacts. The problem of designing a metal rheostat must be subordinated to the essential operating requirement—the reduction of the harmful effect of the impact-like application of torque on the elements of the hoisting installation.

Calculation plan

In Figs. 1 and 2 typical diagrams are given for the operating regime of a hoisting installation with skips unloaded through the bottom. For the specified regime

Fig. 1. Diagram of speed and acceleration of a skip hoist with unloading through the bottom.

it is necessary to calculate the metal rheostat, while striving to satisfy as fully as possible the requirement that the specified regime be carried out and that the starting characteristics be smooth in order to soften impacts. Fig. 3 shows a diagram of rheostat characteristics taking into account the above-mentioned requirements.

Fig. 2. Diagram of forces of a skip hoist.

Specifically, this is expressed in the fact that the values of the static and driving moments (in the present case, forces) are plotted on the diagram of Fig. 3, while the construction of the rheostat characteristics is subordinated to the above-mentioned requirement of reducing the initial starting jerks and increasing them as acceleration proceeds.

The construction of the characteristic diagram is therefore carried out in the following sequence:

1. In the coordinate axes \(S = f_1(M\%)\), for the given nominal slip, the straight line of the automatic characteristic of the electric motor is drawn.

Fig. 3. Rheostat characteristics of the electric motor.

1. On the abscissa axis, to scale, the nominal value of the torque is plotted, equal to \(100\%\), which is determined from the power of the electric motor.

The nominal torque of the electric motor on its shaft will be

\[ M_{\text{n}}=\frac{975\,P(\text{kW})}{n_c}\;(\text{kg—m}). \tag{1} \]

This moment, reduced to the drum shaft, will be

\[ M_{\text{nb}}=M_{\text{n}}\, i\, \eta_z \;(\text{kg—m}), \tag{2} \]

where \(i\) is the gear ratio of the reducer and \(\eta_z\) is the efficiency of the toothed gearing.

The nominal force at the winding radius will then be

\[ F_{\text{n}}=\frac{M_{\text{nb}}}{R_b}. \tag{2'} \]

To this nominal force in Fig. 3 there corresponds the force value of \(100\%\), plotted on the abscissa axis, whereby the scale of forces is determined.

1. In Fig. 3 we plot the values of the static and driving forces, taken from the diagram of Fig. 2 for the entire cycle of the hoisting operation. The construction points are indicated by the corresponding letters, as they are constructed on the regime diagram—Fig. 2.

2. Having a plan of the specified forces, we construct the obvious rheostat characteristics \(R_0\), \(R_1\), \(R_2\), and \(R_3\), necessary for carrying out the regime, subordinating them at the same time to the condition of a gradual increase in the driving force: smaller at the beginning, when the machine is motionless, and larger after the start of motion, when the driving force becomes greater than the static force. The basis for such a principle of constructing the characteristics is the proposition that substanti—

is governed by the laws of motion and is confirmed by experimental data and theoretical investigations, which show that with the skip stationary (as a result of elastic deformations in the rope) and with the drum beginning to move, an oscillatory process arises in the rope, considerably increasing the tension.

1. The total number of steps, apart from those indicated in item 4, is obtained by successive graphical construction in such a way that the oscillation of the dynamic component of the force remains within the lower limit equal to \(1.2 M_{\text{st}}\), called the switching moment or current, and the upper limit, determined by the overturning moment, equal to about twice the rated value.

In this construction the guiding factor will be the plotted line of driving forces, located within the above-indicated limits of the lines of switching moment (current) and overturning moment, as well as the same basic principle: the increase in starting-moment shocks (the increment of moment when passing from one step to the next) must increase as the speed increases, when the indicated shocks become less dangerous.

1. The first rheostat characteristics \(R_0\) and \(R_1\), besides starting purposes, must also be subject to the condition of fulfilling the specified tachogram in the third period of motion. The value of the forces in this period is plotted in Fig. 3 and consists of two steps. In addition, these rheostat steps are used for repair purposes in the shaft and for inspection of the ropes with empty skips, and in some cases—for braking against the current. In the latter case an additional condition is required: the presence of the necessary braking moment.

After the final placement of the rheostat characteristics, it is necessary to determine the numerical value of the resistance in order to fix the characteristic in its given position.

For this we use the equation

\[ R = R_{\text{n}} \frac{S^0\!/_{0}}{M^0\!/_{0}} . \tag{3} \]

Here \(R_{\text{n}}\) is the rated rotor resistance, determined from the rated values of the rotor voltage and current known from the motor data sheet, namely:

\[ R_{\text{n}} = \frac{U_{\text{nr}}}{\sqrt{3}\, I_{\text{nr}}}. \tag{4} \]

The values of the slip \(S^0\!/_{0}\) and moment \(M^0\!/_{0}\) are taken from the constructed diagram of Fig. 3 for each given characteristic. The resistance value of each step of one rotor phase is obtained as the difference between the preceding and following total resistance of the rotor-phase circuit, i.e.,

\[ \begin{aligned} r_1 &= R_0 - R_1 \\ r_2 &= R_1 - R_2 \\ &\ldots \\ r_n &= R_{n-1} - R_n \end{aligned} \tag{5} \]

Acceleration Time

In Fig. 1 the acceleration time of the machine is determined on the basis of the constancy of the dynamic force. With a stepwise change of this force, the acceleration time is determined differently.

The dynamic component in the equation of motion with cylindrical drums has the form

\[ F = mj = m \frac{dv}{dt}. \tag{6} \]

As can be seen from Fig. 3, the change in the dynamic force as a function of speed is expressed by segments of straight lines, i.e., it has a linear dependence, which makes it possible to determine analytically the duration of the time for the speed to increase from the value \(V_1\) to \(V_2\) under the action of the dynamic force, also varying linearly from \(F_1\) to \(F_2\), as shown in Fig. 4.

From equation (6) we have

\[ dt = \frac{m}{F}\cdot dv. \]

\[ t_{1-2} = m \int_{V}^{V_2} \frac{dv}{F}. \tag{7} \]

If the dynamic force is constant, then

\[ t_{1-2} = \frac{m}{F}(V_2 - V_1). \tag{8} \]

Fig. 4. Change of speed under the action of a varying dynamic force.

With a linear change in \(F\), from the similarity of triangles 2, 3, 4 and 1, 2, 5—Fig. 4, we have

\[ \frac{F - F_2}{F_1 - F_2} = \frac{V_2 - V}{V_2 - V_1} \]

or

\[ F = F_2 + \frac{F_1 - F_2}{V_2 - V_1}(V_2 - V) = F_2 + \frac{F_1 - F_2}{V_2 - V_1}\cdot V_2 - \frac{F_1 - F_2}{V_2 - V_1}\cdot V. \tag{9} \]

Introducing the abbreviated designations

\[ F_2 + \frac{F_1 - F_2}{V_2 - V_1}\cdot V_2 = m \quad \text{and} \quad \frac{F_1 - F_2}{V_2 - V_1} = n, \]

we obtain

\[ F = m - nv. \tag{10} \]

Substituting the value of \(F\) from (10) into equation (7), we obtain

\[ t_{1-2} = m \int_{V_1}^{V_2} \frac{dv}{m - nv} = \frac{m}{n}\ln \frac{m - nV_1}{m - nV_2}. \tag{11} \]

Substituting again the values of \(m\) and \(n\) into the solution obtained, after transformation we obtain

\[ t_{1-2} = \frac{m(V_2 - V_1)}{F_1 - F_2}\cdot \ln \frac{F_1}{F_2}. \tag{12} \]

The indices 1 and 2 refer to the change in speed and dynamic force (counted from the static force) on the given rheostat step, from the moment of closing of the given contactor to the closing of the next contactor. This equation is used to calculate the acceleration time of the machine.

In the case when the dwell time on a given rheostat step is regulated not by a single time relay, but in combination by means of a current relay (type RE-191) and a time relay (type RE-100 or RE-180), the value of the dynamic component \(F_1\) in formula (12) is reckoned not from the points of intersection of the horizontal line with the lines of the rheostat characteristics (points \(4h, 5h, 6h\), and \(7h\) in Fig. 3), but from the intersection of the release line of the current relay—\(I_{\mathrm{tr}}\)—with the lines of the rheostat characteristics, i.e., from points 4, 5, 6, and 7 to points \(4k, 5k, 6k, 7k\) in Fig. 3. The latter points are located on the switching-current line \((F_{\mathrm{sw}})\).

In the above-described general form of the grapho-analytical method for calculating the rheostat for a mine hoisting installation, it was assumed that:

a) the natural and all rheostat characteristics of the electric motor are rays of straight lines issuing from one center of synchronous speed;

b) the magnitude of the dynamic moment (dynamic force) during the periods of acceleration and deceleration varies according to a linear law;

c) transient electromagnetic processes in the electric motor do not noticeably affect the dynamics of the accelerated and retarded motion of the installation.

The assumptions made render the calculation clear and allow the specific features of the hoisting installation to be taken into account; at the same time, however, they introduce unavoidable errors, as in any other method of calculation, if only because the magnitude of the resistances changes with heating depending on the degree of heating, deviations are inevitable when selecting resistance boxes, etc. Experience shows that the assumptions made, while simplifying the calculation, do not introduce a significant error, and the result of the calculation fully satisfies practical requirements.

EXAMPLE 1. A hoisting installation equipped with 4-ton skips raises coal from a depth of 250 m. Drum type \(2 \times 5 \times 1.7\), dead weight of the skips 4.0 t, mass of the moving parts reduced to a radius of 6000, gear ratio coefficient 1:10.2, maximum speed 6 m/sec, length of unloading path 3 m.

The kinematic regime of hoisting is presented in Fig. 5. The cycle duration is \(T_{\mathrm{c}} = 67\) sec, the duration of motion is 61 sec. On the basis of the kinematic elements obtained, a diagram of the acting forces during hoisting is constructed, the construction of which is shown in Fig. 6. The basis for the construction is the equation of dynamics for cylindrical drums without tail rope

Fig. 5. Kinematics of hoisting with 4-ton skips.

\[ F = kQ + p_i(H - 2x) + mj \tag{13} \]

Here \(k = 1.15,\ Q = 4000\ \text{kg},\ H = 250\ \text{m},\ m = 6000\), and the accelerations \(j\) are shown in the diagram.

On the basis of the force diagram we determine the effective force and the power at the winding radius

\[ P_{\mathrm{eff}}=\frac{F_{\mathrm{eff}}\cdot V}{102\,\eta_{3}}, \tag{14} \]

Fig. 6. Force diagram for one hoisting cycle with 4-ton skips.

where

\[ F_{\mathrm{eff}}= \sqrt{ \frac{\sum F^{2}t} {t_{2}+0.5(t_{1}+t_{3})+\dfrac{\theta}{4}} }. \]

The values of the forces are shown in Fig. 6.

As a result we obtain \(F_{\mathrm{eff}}=5900\ \mathrm{kg}\),

and

\[ P_{\mathrm{eff}}=380\ \mathrm{kW}. \]

The constructional rated power of the motor is taken as \(400\ \mathrm{kW}\), \(250\ \mathrm{rpm}\), with rotor data

\[ \begin{aligned} I_{\mathrm{nr}}&=500\ \mathrm{A},\\ U_{\mathrm{nr}}&=520\ \mathrm{V},\\ S_{\mathrm{n}}&=5\%. \end{aligned} \]

These data are sufficient for calculating the rheostat.

Construction of Rheostatic Characteristics

The construction of the rheostatic characteristics for the given specific case is shown in Fig. 7. Points 1–10 show the values of the acting forces over the entire hoisting cycle, taken from the diagram of the specified operating duty—Fig. 6.

The position of the characteristics \(R_{0}\), \(R_{1}\), and \(R_{2}\) is determined by the need to ensure realization of the designed force diagram, as shown in Fig. 7.

To construct the subsequent characteristics \(R_{3}\), \(R_{4}\), etc., it is necessary to determine the limits of variation of the accelerating dynamic force. Usually the variation of the dynamic force (or torque) is determined as follows:

a) the lower limit, called the switching current (on the scale of stator or rotor currents), must be 15–20% higher than the current required to overcome the static force. The latter is shown on the diagram by the value \(5.4\ \mathrm{t}\);

b) the upper limit, or peak, of the moments is determined by the value of the overturning moment of the electric motor, taken for large electric motors as about 200% of the rated moment.

Fig. 7. Rheostat characteristics of a hoisting motor with 4-ton skips.

If one proceeds from these limits of fluctuation of the moments, then it is possible to obtain a rheostat consisting of 5 steps, but then the points of dynamic moment during transition from one step to another will rise above 8 t.

It is appropriate to mention that, in this hoisting installation, the rheostat was designed by the design organization with large increments of dynamic force, reaching up to 10 t, as a result of which, during trial tests after installation, the crosspieces of the split device of the loose drum were broken, causing a prolonged shutdown of the mine. In order to reduce the jolts during starting, the increment of dynamic moment was reduced to 5 t, which was achieved with 7 rheostat steps. With the seven-step rheostat the machine has been operating for more than 5 years, without significant failures attributable to the rheostat.

This part in the calculation of a metal rheostat is very important, and an incautious assignment of the limits of variation of the dynamic moment leads to an emergency condition of the equipment, although in this case a small number of rheostat steps is obtained.

A more detailed analysis of the operating experience of installations with metal rheostats gives grounds to assert that the number of failures of the mechanical part and the rapid wear of ropes are explained precisely by the small number of steps of the starting rheostat. In confirmation of the validity of this assertion, one could cite a large amount of factual material speaking against stepwise starting of large reversible hoisting machines.

In the resolutions of the technical conference on the restoration of mines of the Donets Basin (Stalino), published in 1945 [21], we also find: “With respect to improving stationary equipment, item 4 states that ‘it is necessary to revise the number of steps at the magnetic stations of large hoisting machines in the direction of increasing them, in connection with the observed non-smooth starting of motors at five-step stations.’”

For our part, we consider it expedient, for the purpose of accident-free and durable operation of large hoisting installations having a mass reduced to the winding radius of more than 4–5 thousand units, to provide smooth starting,

which can be achieved with 7 or 8 rheostat steps, depending on the nominal rotor slip. The latter must be on the order of 4–5%, which is not always the case in large low-speed electric motors with a wound rotor.

The diagram of rheostat characteristics, Fig. 7, makes it possible to determine the resistance of all rheostat steps and to select a rheostat from the catalog range of resistance boxes.

Calculation of Resistance Values

For calculating the resistances we use equation (3)

\[ R=\frac{U_{\mathrm{nr}}}{\sqrt{3}\,I_{\mathrm{nr}}}\cdot \frac{S\%}{M\%} =\frac{520}{\sqrt{3}\cdot 500}\cdot \frac{S\%}{M\%} =0.6\,\frac{S\%}{M\%}. \tag{15} \]

Substituting the corresponding values of slip and torque from Fig. 7, we obtain

\[ R_0=0.6\,\frac{100}{30}=2\ \text{ohms} \]

\[ R_1=0.6\,\frac{100}{70}=0.86\ \text{ohms} \]

\[ R_2=0.6\,\frac{100}{120}=0.50\ \text{ohms} \]

\[ R_3=0.6\,\frac{84}{184}=0.274\ \text{ohms} \]

\[ R_4=0.6\,\frac{55}{193}=0.171\ \text{ohms} \]

\[ R_5=0.6\,\frac{34}{206}=0.0794\ \text{ohms} \]

\[ R_6=0.6\,\frac{20}{215}=0.0558\ \text{ohms} \]

\[ R_7=0.6\,\frac{11}{220}=0.030\ \text{ohms}. \]

Now let us determine the resistance of each step by formula (5):

\[ r_1=R_0-R_1=1.14\ \text{ohms} \]

\[ r_2=R_1-R_2=0.36\ \text{ohms} \]

\[ r_3=R_2-R_3=0.226\ \text{ohms} \]

\[ r_4=R_3-R_4=0.103\ \text{ohms} \]

\[ r_5=R_4-R_5=0.0916\ \text{ohms} \]

\[ r_6=R_5-R_6=0.0236\ \text{ohms} \]

\[ r_7=R_6-R_7=0.0258\ \text{ohms}. \]

\[ \text{Sum}\quad \Sigma_r=1.970\ \text{ohms} \]

Time delay of the time relays of the contactors

The time during which a rheostat step remains under current is determined by the time delay of the contactors’ time relays.

This time is determined by formula (12) and from the diagram of characteristics in Fig. 7. There is no current relay, and the time delay at this step is regulated only by the time relay.

For the acceleration relays 1RU and 2RU, as preliminary steps ensuring a smooth increase of torque, the time delay is determined by the relay’s own time, about 0.3 sec. For the subsequent relays we have—for relay 3RU and 4RU

\[ t_{a-b}=\frac{6000\cdot 1}{1800-600}\cdot \ln \frac{1800}{600} =\frac{6}{1.2}\cdot 1.1=5.5\ \text{sec}, \]

\[ t_{c-d}=\frac{6000(2.85-1)}{5650-1850}\cdot \ln \frac{5650}{1850} =3.2\ \text{sec}, \text{ etc.} \]

The time delays calculated in this way are shown on the corresponding characteristics of Fig. 7.

The total acceleration time up to entry onto the automatic characteristic is equal to 13.6 sec, instead of the calculated 12.7 sec according to Fig. 5.

Thermal calculation of the resistances

On the abscissa axis of Fig. 7 we plot, on the scale of rotor currents, the rotor current, taking into account that 100% of the scale corresponds to the nominal value of the current in the rotor, equal to

\[ I_{\text{nr}}=\frac{400\cdot 1000}{\sqrt{3}\cdot 520\cdot 0.9}=500\ a. \]

During the starting of the electric motor, the current at the steps varies in magnitude, and the time during which the given step remains under current is also different. These changes are shown in Fig. 8, constructed from the data of Fig. 7.

Fig. 8. Change of current at the rheostat steps over time.

First we determine the effective values of the currents at the individual steps by the formula for root-mean-square values

\[ I_{\text{eff}}=\sqrt{\frac{I_1^2+I_2^2+I_1 I_2}{3}}. \tag{17} \]

Calculation shows that in this case one may use the arithmetic-mean value

\[ I_{\mathrm{eff}} \cong \frac{I_1 + I_2}{2}. \tag{18} \]

The effective currents calculated in this way are shown in Fig. 8 by a dotted line.

As can be seen from the diagram, the magnitude of the effective currents changes during starting. To estimate the thermal effect of these currents, it is necessary to determine the equivalent current on the given step.

The magnitude of the equivalent current is determined by the general formula

\[ I_{\mathrm{eq}}=\sqrt{\frac{\sum I_{\mathrm{eff}}^{2}t}{\sum t}}. \tag{19} \]

For our case the formula will have the form

\[ I_{\mathrm{eq}}=\sqrt{\frac{I_{1\mathrm{e}}^{2}t_1+I_{2\mathrm{e}}^{2}t_2}{t_1+t_2}}. \tag{20} \]

In addition, it is necessary to determine the relative duration of the stay of the given step under current, which is determined by the formula

\[ PV\%=\frac{t_c\cdot100}{T}. \tag{21} \]

Since the catalog data indicate the permissible current value for a definite duration of switching-on, for example, 12.5 or 20%, the actual durations of switching-on obtained from formula (21) must be recalculated to standard values. The recalculation is made according to the heat-balance formula

\[ I_1^2 PV_1\% = I_2^2 PV_2\% \tag{22} \]

i.e., when reducing to a 12.5% duration of switching-on, we shall have

\[ I=I_c\sqrt{\frac{PV_c\%}{12.5}}. \tag{23} \]

The currents on the steps calculated in this way are given in Table 1.

Table 1. Reduced currents on the steps

Selection of Resistance Boxes

The selection of resistance boxes by current and by the magnitude of ohmic resistance is shown in Fig. 9.

To facilitate selection, the electrical characteristics are given in Table 2 and the permissible operating currents in Table 3 for cast-iron elements of boxes; moreover, the data are given for standard types that have found wide application in mine hoisting machines.

Fig. 9. Diagram for assembling resistance boxes.

Table 2. Electrical data of standard resistance boxes made of 20 elements

Table 3. Permissible operating currents in amperes for cast-iron elements of resistance boxes (at a heating temperature of 300°C)

For calculating the rheostat it is convenient to use a calculation form. The calculation form for the example considered above is given in Tables 4 and 5.

Calculation form.

Table 4. A. Principal parameters of the hoisting installation

Table 5. B. Rheostat calculation

\[ \Sigma_r = 1\,970 \]

\[ \Sigma = 1\,995 \]

Explanations for the tables of the form—Figs. 4 and 5.

1. The calculated value of the resistance per rotor phase is determined by the formula

\[ R = \frac{U_{nr}}{\sqrt{3} J_{nr}} \cdot \frac{S\%}{M\%} \tag{24} \]

1. The time during which a stage is under current is found from the time delay of the acceleration relay. The latter is calculated by the formula:

\[ t=\frac{m(V_2-V_1)}{F_1-F_2}\ln\frac{F_1}{F_2}. \tag{25} \]

1. The effective current is determined by the formula

\[ J_{\mathrm{eff}}=\sqrt{\frac{J_1^2+J_2^2+J_1J_2}{3}} \quad \text{or} \quad J_{\mathrm{eff}}\simeq \frac{J_1+J_2}{2}. \tag{26} \]

1. The equivalent current at the rheostat stage is determined by the formula

\[ J_{\mathrm{eq}}=\sqrt{\frac{\sum F^2t}{\sum t}}. \tag{27} \]

1. The relative duration for which a stage remains under current is found by the formula

\[ PV\%=\frac{tc100}{T_{\mathrm{c}}}. \tag{28} \]

1. The current reduced to \(PV\%=12.5\%\) is determined by the formula

\[ J=J_{\mathrm{eq}}\sqrt{\frac{PV\%}{12.5}}. \tag{29} \]

1. When selecting factory resistance boxes, it is necessary to assemble them from one or two types for convenience of replacement in operation.

EXAMPLE 2. Calculation of a rheostat for a cage freight-hoisting installation (PM-16 winch)

Characteristics of the installation

The hoisting installation consists of two single-deck cages with a useful hoisted load of 1000 kg and the weight of one empty mine car of 700 kg. At the level of the pithead landing and at the upper receiving platform the cages come to rest on catches.

Fig. 10. Speed diagram of the PM-16 hoisting winch, \(2\times 2.5\times 0.75\).

In addition to the main freight operations, auxiliary operations are also performed: lowering and hoisting people, equipment, and materials, shaft repairs, etc.

The operating regime of the hoisting installation is given in Figs. 10 and 11; on the basis of these, the diagram of rheostat characteristics shown in Fig. 12 has been constructed.

The calculated quantities are given in Tables 6, 7, 8, 9, and 10.

Fig. 11. Force diagram of the PM-16 hoisting winch.

Fig. 12. Rheostat characteristics of the hoisting motor of the PM-16 winch.

Table 6. Main parameters of the installation

Table 7. Calculation of resistance values

Table 8. Time delays of the contactor time relays

Instruction: 1) The speed increment at a given stage is taken from the diagram.
2) Calculation formula for the time

\[ t=\frac{2400\,\Delta V}{F_1-F_2}\cdot \ln\frac{F_1}{F_2} \]

3) If an RT-type RE-191 current relay is used, the time delay on the time relay is determined by the formula of item 2, but the dynamic component \(F_1\) is counted from the release line of the current relay to the line of the torque or switching current, as indicated above.

Table 9. Thermal calculation of the rheostat

4) Calculation formula for the effective current

\[ J_{\mathrm{eff}}=\sqrt{\frac{J_1^2+J_2^2+J_1J_2}{3}}\cong \frac{J_1+J_2}{2}. \tag{31} \]

For the equivalent current

\[ J_{\mathrm{eq}}=\sqrt{\frac{J_{\mathrm{e}1}^{2}t_1+J_{\mathrm{e}2}^{2}t_2}{t_1+t_2}}. \tag{32} \]

Table 10. Selection of resistance boxes

Table 11. Contactors of the Kharkov Electromechanical Plant

Pull-in voltage with heated coil

\[ U_{\min}\cong 0.61U_{\mathrm{n}} \]

Adjustment of the current relay

A three-phase current relay of type RE-191 is connected in parallel to each phase of the last rheostat stage. The purpose of the current relay is to create a time delay according to current during starting, while in this case the time relay serves as an additional time delay for switching on the next contactor. The connection diagram of the current relay—RT—in the control circuit is shown in Fig. 13.

The contacts of the current relay RT are included in the circuit of all time relays (1RU—7RU in Fig. 13). As long as the RT contacts are closed, no successive time relay will operate. When the starting-current force drops to the value of the rotor current, called the relay release current and set within the limits

\[ J_{\mathrm{rel}}=(1.4\div 1.6)\,J_{\mathrm{nom}}, \]

the RT contacts will open and the next time relay will come into operation.

Upon expiration of the setting time, the time relay, by means of its block contacts 1RU2--7RU2, Fig. 13, will close the circuit of the next rotor contactor, thereby shunting this stage of the rheostat.

At the moment the current relay (CR) operates, we have the following relation between the rotor currents \(J_{\mathrm{rel}}\) and the current relay current \(i_p\):

\[ U_p = J_{\mathrm{rel}} r_c = i_p Z_p . \tag{33} \]

In these equalities, \(U_p\) is the voltage drop across the resistance \(r_c\) of the last stage of the rheostat phase, and \(Z_p\) is the total resistance of the current-relay circuit.

The total resistance of the relay circuit consists of the additional active resistance of the setting rheostat, \(r_d\), connected in series

Fig. 13. Circuit diagram of contactor control for the KhEMZ hoisting machine.

with the winding of the current-relay coil and of the relay itself, \(r_p\), and the inductive resistance of the relay coil, \(x_p\).

From equation (33) we obtain the relay release current

\[ J_{\mathrm{rel}}=\frac{i_p}{r_c}\cdot Z_p =\frac{i_p}{r_c}\sqrt{(r_d+r_p)^2+x_p^2}. \tag{33'} \]

During the period of machine acceleration, the rotor-current frequency changes from 50 to 10 cycles/sec, which causes a decrease in the inductive resistance, \(x_p\). In addition, the magnitude of the inductive resistance depends to a great extent on

—gap, \(\delta\), between the pole and the relay armature. This gap can be adjusted, thereby accomplishing the setting of the relay.

In Fig. 14 are shown the relay circuit and the dependence \(Z_p=f_1(c)\) and \(Z_p=f_2(\delta)\), i.e., the dependence of the total relay resistance

\[ Z_p=\sqrt{r_p^2+x_p^2} \tag{34} \]

as a function of the supply frequency, \(c\), and of the size of the gap, \(\delta\).

The current relay is adjusted according to the following plan:

1. On the diagram of rheostat characteristics, Fig. 3, the relay release-current line, \(J_{\text{rel}}\), is plotted for two slip values—zero and unity.

2. The multiple of the relay release current at zero slip is chosen within the limits \((1.4 \div 1.6)J_n\). This multiple is obtained on the basis that

Fig. 14. Circuit and characteristics of a current relay of type RE-191.

the lower limiting switching current is greater than the current corresponding to the static force by 15–20%. Increasing this current by another 10–20%, in order to make possible operation of the time relay with a certain delay, gives the upper point of the release current.

1. The multiple of the release current at slip equal to unity is obtained from equation (33)

\[ k_1=\frac{J_{\text{rel}1}}{J_{\text{rel}0}} = \frac{\sqrt{(r_d+r_p)^2+x_{p1}^2}} {\sqrt{(r_d+r_p)^2+x_{p0}^2}} \simeq \frac{\sqrt{r_d^2+x_{p1}^2}} {\sqrt{r_d+x_{p0}^2}} \simeq \frac{Z_{p1}}{Z_{p0}} \tag{35} \]

The values of the resistances at slip equal to unity, \(Z_{p1}\), and at slip equal to zero, \(Z_{p0}\), are taken from the relay characteristics—Fig. 14.

1. The parameters of the current relay are selected so as to be most favorable for its operation; they lie within the following limits: \(U_p=13 \div 15\) volts, \(r_d=2Z_p=10\text{–}12\) ohms, \(\delta=2 \div 4\) mm, and the release current of the relay itself is about \(1.5a\). The correctness of the selected operating parameters is checked by formula (33) for the release current at slip equal to zero.

From the two obtained values of the release current, the release line of relay—RT is plotted on the diagram of rheostat characteristics.

1. Adjustment of the setting of the current relay is carried out by changing the spring tension, i.e., by changing the mechanical characteristic of relay—RT.

2. After plotting the release line of the current relay, the time relay must be adjusted. The method of this adjustment remains the same as in the circuit with only a time relay (see formulas 7—12 and the chapter—Run-up time), with the only difference that the dynamic component of the forces is measured within the limits from the release line of the current relay to the line of switching currents, as indicated earlier.

After calculating the time delay of each time relay, the operation of the automatic equipment for combined starting with relay—RT and time relay—RV must be checked for the case of lowering a normal load. For this check it is necessary to determine the run-up time of the hoisting installation under the influence of the lowering load. For this purpose, diagrams of the operating mode when lowering the load must be constructed.

In this case the total time delay of the time relay—RV must be equal to the run-up time under the action of the lowering load, which ensures proper operation of the automatic equipment when lowering the load, i.e., the entry of the asynchronous electric motor into generator mode after reaching synchronous speed.

Without this check, the meaning and advantages of combined starting automation with a current relay and time relays are lost.

If the total time delay on the time relays proves to be greater than the acceleration time when lowering the load, it is necessary to shift the line of the release currents of the current relay to the left and perform the check again.

Selection of Contactors

The contactors and rheostat are switched in according to a two-pole (Fig. 3) or three-pole circuit, as shown in Fig. 13.

In the first case, the line current of the rheostat passes through the contactor, and in the second—the phase current.

Table 11 gives the technical data of KhEMZ contactors, according to which they are selected for the given conditions.

For example 1, with a nominal rotor current of 500 A, three-pole contactors of type KT-4 will be sufficient, passing a phase current of 300 A, or a line current of \(300 \cdot \sqrt{3} = 520\) A.

For example 2, where the nominal rotor current is 120 A, two-pole contactors of type KT-3 for 150 A will be sufficient; and with a three-pole circuit, when the current through the contactors will be \(120 : \sqrt{3} = 70\) A, contactors of type KT-2 for a continuous current of 75 A will be sufficient.

All contactors of the panel are installed of the same type in order to simplify maintenance.

REFERENCES

1. Kulebakin V. S.—Electrical Apparatus, Vol. 1. Starting and Regulating Rheostats, Gosenergoizdat, Moscow—Leningrad, 1932.

2. En’ko V. V.—On the Question of Calculating Starting Rheostats, Vestn. elektroprom. No. 4, 1936.

3. Popov V. K.—Fundamentals of Electric Drive, Gosenergoizdat, Moscow—Leningrad, 1945 (Bibliography).

4. Fateev A. V. and Norenkovskii B. I.—Collection of Examples and Problems in the Theory of Electric Drive, GONTI, 1939.

5. Rossoshanskii I. A.—Calculation of Universal Characteristics and Starting Resistance of Three-Phase Asynchronous Motors, Vestn. elektroprom. No. 3, 1941.

6. Shtein I. M. Construction of the Characteristic of an Asynchronous Electric Motor with Contact Rings, Vestnik elektroprom. No. 8, 1940.

7. Frankfurt Ya. L.—Construction of the Mechanical Characteristic of an Asynchronous Motor, Vestn. elektroprom. No. 4, 1947.

1. Soroker, T. G.—On the formula for the torque of an asynchronous motor, Vestn. elektroprom., No. 9, 1939.

2. Geiler, L. B.—Formula for the torque of an asynchronous motor, Vestn. elektroprom., No. 12, 1938.

3. Bychkov, V. P.—Graphical methods for constructing curves of an asynchronous motor, El-vo, No. 9, 1945.

4. Konovalov, N. I.—Characteristics of asynchronous motors in a system of relative units, El-vo, No. 9, 1945.

5. Bulgakov, V. A.—Electrical control apparatus, Gosenergoizdat, Moscow–Leningrad, 1947.

6. Shenfer, K. I., and Sviridenko, P. A.—Energy recuperation in industry, Energoizdat, Moscow–Leningrad, 1946.

7. Khalizev, G. P.—Starting and regulating apparatus, Moscow Energy Institute, 1937.

8. Veshenevskii, S. N.—Calculation of resistances for electric motors, GONTI, 1938.

9. Shklyarskii, F. N.—Calculation of starting resistances and settings of acceleration relays for an asynchronous hoisting motor, Zapiski Leningradskogo gornogo instituta, vol. 14.

10. Rossoshanskii, A. I.—Asynchronous drive of a mine hoisting machine, Vestnik KhEMZ, No. 4 (23), 1939.

11. Kazovskii, E. Ya.—Transient regimes in asynchronous machines, El-vo, No. 6, 1947 (Bibliography on the theory of asynchronous machines).

12. Horwood, P. B.—Control of electric motors, New York, 1944.

13. Bakharev, V. M.—On the fatigue of steel under repeated overloads, Trudy TsIAM, No. 91, NKAP, 1945.

14. Resolutions of the technical conference on the restoration of mines of the Donets Basin, Stalino, 1945.