Determination of the Transient Process Time During Contactor Starting of a Hoisting Machine

V. D. Petunov

(Presented to the scientific-technical conference of the Mining Faculty)

If the overturning torque of the asynchronous motor \(M_k\) and the critical slip \(s_k\) are taken as the base units, and the active resistance of the stator winding is neglected, then we obtain the universal equation of the mechanical characteristic [1]

\[ \lambda = \frac{2\sigma}{1+\sigma^2}, \tag{1} \]

where

\[ \lambda=\frac{M}{M_k} \]

is the relative torque;

\[ \sigma=\frac{s}{s_k} \]

is the relative slip.

Equation (1) is suitable for all wound-rotor motors at any value of the active resistance in the rotor circuit.

To simplify calculations and graphical constructions, it is recommended [2, 3], instead of the curvilinear mechanical characteristic expressed by equation (1), to use rectified mechanical characteristics determined by the synchronous and nominal rotational speeds. The equation of the rectified characteristic is

\[ \lambda = k_1 \sigma, \tag{2} \]

where \(k_1\) is the angular coefficient characterizing the stiffness of the mechanical characteristic.

For the nominal torque of the motor,

\[ \frac{1}{\gamma_M}=k_1\sigma_n, \tag{3} \]

where \(\gamma_M\) is the overload capacity of the motor,
\(\sigma_n\) is the relative nominal slip.

The angular coefficient is

\[ k_1=\frac{2}{1+\sigma_n^2} =\frac{\gamma_M+\sqrt{\gamma_M^2-1}}{\gamma_M}. \tag{4} \]

From (4) it is evident that the angular coefficient depends only on the overload capacity of the motor. For motors used in mine hoisting, \(\lambda_m = 1.8 \div 2.8\) and \(k_1 = 1.83 \div 1.94\). Comparative calculations (Table 1) show that the curvilinear characteristic can be replaced by a straight line only in the interval of relative slips \(\sigma = 0 \div 0.3\).

Doctor of Technical Sciences L. B. Geiler [4] and engineer G. M. Kayalov [5], for an approximate replacement of the equation of the curvilinear characteristic, proposed the equation of a parabola

\[ \lambda = 2\sigma - \sigma^2 . \tag{5} \]

The relative error when using this equation is

\[ \frac{\Delta \lambda}{\lambda} = \frac{\sigma^3 - 2\sigma^2 - \sigma}{2}. \tag{6} \]

The maximum error is obtained at \(\sigma = 1/3\) and amounts to \(7.41\%\) of the motor pull-out torque. This error, converted to the rated torque, will amount to from \(13.3\) to \(20.7\%\), respectively, for motors with overload capacities of \(1.8\) and \(2.8\). Comparative calculations (Table 2) show that the parabolic equation (5) can be used at \(\sigma > 0.7\), i.e., in the region of slips close to critical.

The equation of the curvilinear mechanical characteristic (1) can be expressed in the following form

\[ \lambda = 2\sigma - \lambda \sigma^2 \tag{7} \]

and, for a more accurate replacement, the parabolic equation

\[ \lambda = 2\sigma - \beta \sigma^2, \tag{8} \]

where \(\beta\) is a correction coefficient determined by the starting conditions.

Obviously, equation (5) is a special case of equation (8), in which \(\beta = 1\). For contactor starting one may take

\[ \beta = \frac{\lambda_1 + \lambda_2}{2}, \tag{9} \]

where \(\lambda_1\) and \(\lambda_2\) are, respectively, the maximum starting torque and the switching torque.

From equations (7) and (8) it is evident that the smallest error is attained when the correction coefficient and the motor torque in relative units are equal. Calculations show (Table 2) that the equation of the curvilinear characteristic can be replaced by the parabolic equation (8) with the corresponding correction coefficient.

Good results are obtained by replacing equation (1) with the equation of a parabola that does not pass through the point of synchronous speed

\[ \lambda = 2\sigma - \beta \sigma^2 + \beta - 1, \tag{10} \]

where \(\beta = 1.04 \div 1.06\).

Table 1

Comparison of curvilinear and rectilinear mechanical characteristics

Comparison of the curvilinear mechanical characteristic and the parabolic form \(\lambda = 2\alpha - \beta\alpha\)

Comparison of the curvilinear mechanical characteristic and the parabolic characteristic of the form
\(\lambda = 2\zeta - \beta \zeta^{2} + \gamma\)

Table 3

Comparison of the curvilinear and mechanical characteristics with the parabolic form \(\lambda = 2s - 1.1s^2\)
Table 4

The calculations performed (Table 3) show that the parabola equation (10), for relative slips \(\sigma > 0.2\), replaces equation (1) with sufficient accuracy.

Under actual operating conditions of hoisting machines, a reduction of voltage during starting by \(5\%\) from the nominal value often occurs. With such a voltage reduction, with sufficient accuracy—as is confirmed by the calculations given in Table 4—one may use the parabola equation

\[ \lambda = 2\sigma - 1.1\sigma^2 . \tag{11} \]

The graphical construction of all the indicated characteristics is given in Fig. 1.

Let us write the equation of motion during the starting of a hoisting machine with a constant winding radius in the following form:

\[ \lambda - \lambda_{st} = -T \frac{d\sigma}{dt}, \tag{12} \]

where \(\lambda_{st}\) is the relative moment of static resistance, which is considered constant over the entire starting time, since the load due to the difference in rope weights during starting amounts in practice to less than \(3\%\) of the weight of the useful load;

\(T\) is the mechanical time constant, sec.

Fig. 1. Universal mechanical characteristics of an asynchronous motor.

Curves constructed:

1—according to the equation

\[ \lambda=\frac{2\sigma}{1+\sigma^2}; \]

2—according to the equation \(\lambda=K_1\sigma\), with \(K_1=1.90\);

3—according to the equation \(\lambda=2\sigma-\sigma^2\);

4—according to the equation \(\lambda=2\sigma-\beta\sigma^2+\beta-1\), with \(\beta=1.06\);

5—according to the equation

\[ \lambda=2\sigma-1.1\sigma^2 . \]

The mechanical time constant is

\[ T=\frac{\mu v_0 S_k}{\gamma_M}, \tag{13} \]

where \(\mu\) is the ratio of the reduced mass to the nominal motor force on the drum shaft;

\(v_0\) is the hoisting speed corresponding to the synchronous speed of the motor, m/sec.

The duration of the transient process is

\[ t = T \int_{\sigma_1}^{\sigma_2} \frac{d\sigma}{\lambda-\lambda_{st}}, \tag{14} \]

where \(\sigma_1\) and \(\sigma_2\) are, respectively, the initial and final slip in relative units.

The simultaneous solution of (1) and (14) gives

\[ t = T \int_{\sigma_1}^{\sigma_2} \frac{(\sigma^2+1)\,d\sigma} {\lambda_{cm}\sigma^2 - 2\sigma + \gamma_{cm}} . \tag{15} \]

Equation (15) is split into two tabular integrals. After substituting the limits of integration and transformations, we obtain

\[ t=T\alpha^2\left[ C\ln\frac{(A-\sigma^2)(\sigma_1-B)}{(A-\sigma_2)(\sigma_2-B)} -\ln\frac{2\alpha\sigma_1-\sigma_1^2-1}{2\alpha\sigma_2-\sigma_2^2-1} -\frac{\sigma_1-\sigma_2}{\alpha} \right], \tag{16} \]

where

\[ \alpha=\frac{1}{s_{\mathrm{st}}}; \]

\[ B=\alpha-\sqrt{\alpha^2-1}; \]

\[ A=\alpha+\sqrt{\alpha^2-1}; \]

\[ C=-\frac{\alpha}{\sqrt{\alpha^2-1}}. \]

The expression obtained is considered the most accurate for determining the acceleration time and is given in approximately the same form in the literature [6]. When the final slip differs only slightly from the steady-state slip—for example, when the difference between them is 15–20%—we have \(B \approx \sigma_2\). In this case the first term of equation (16) gives a substantial error, because of which the overall result may have an error of up to 30–50%. Therefore formulas that include the difference of close quantities [7] should be avoided, and instead of (16) the following equation should be used:

\[ t=T\alpha^2\left[ C\ln\frac{A\sigma_1+B\sigma_2-\sigma_1\sigma_2-1} {A\sigma_2+B\sigma_1-\sigma_1\sigma_2-1} -\ln\frac{2\alpha\sigma_1-\sigma_1^2-1}{2\alpha\sigma_2-\sigma_2^2-1} -\frac{\sigma_1-\sigma_2}{\alpha} \right]. \tag{17} \]

However, equations (16) and (17) are complicated for practical use. The problem posed is solved more simply by using equations (2), (5), (8), and (10).

The simultaneous solution of (2) and (14) gives

\[ t=-\frac{T}{k_1}\ln\frac{k_1\sigma_1-k_{\mathrm{st}}}{k_1\sigma_2-k_{\mathrm{st}}}. \tag{18} \]

The resulting formula (18) is the simplest, but it can be used only for relative slips up to 0.33.

The solution of equations (5) and (14) gives

\[ t=\frac{T}{2a}\ln\frac{(A-\sigma_2)(\sigma_1-B)} {(A-\sigma_1)(\sigma_2-B)}, \tag{19} \]

where

\[ a=\sqrt{1-k_{\mathrm{st}}};\qquad A=1+a;\qquad B=1-a. \]

Expanding the parentheses in equation (19), we obtain

\[ t=-\frac{T}{2a}\ln \frac{A\sigma_1+B\sigma_2-\sigma_1\sigma_2-\lambda_{cm}} {A\sigma_2+B\sigma_1-\sigma_1\sigma_2-\lambda_{cm}}. \tag{20} \]

The equations obtained, (19–20), are considerably simpler than (16–17). Their use is recommended by Doctor of Technical Sciences L. B. Geiler [4] and Candidate of Technical Sciences L. G. Zhivov [6]. However, L. B. Geiler considers the case of a sudden increase of the load on the motor shaft and its operation near the pull-out point; in this case, replacement of the curvilinear characteristic by a parabolic one gives almost no discrepancies in torque and slip, and consequently also in the time of the transient process. It should only be remembered that, for these formulas, the critical slip is determined from the equation of the parabolic characteristic

\[ s_k=s_n\gamma_M\left(1+\sqrt{1-\frac{1}{\gamma_M}}\right), \tag{21} \]

where \(s_n\) is the nominal slip.

The simultaneous solution of equations (8) and (14) gives the transient-process time

\[ t=-\frac{T}{2a}\ln \frac{(A-\beta\sigma_2)(\beta\sigma_1-B)} {(A-\beta\sigma_1)(\beta\sigma_2-B)} \tag{22} \]

or

\[ t=-\frac{T}{2a}\ln \frac{A\sigma_1+B\sigma_2-\beta\sigma_1\sigma_2-\lambda_{cm}} {A\sigma_2+B\sigma_1-\beta\sigma_1\sigma_2-\lambda_{cm}}, \tag{23} \]

where

\[ a=\sqrt{1-\beta\lambda_{cm}};\qquad A=1+a;\qquad B=1-a. \]

The torque limits for determining the correction factor are found from the equation of the curvilinear mechanical characteristic by substituting the corresponding slips.

The simultaneous solution of equations (10) and (14) gives the transient-process time

\[ t=-\frac{T}{2a}\ln \frac{(A-\beta\sigma_2)(\beta\sigma_1-B)} {(A-\beta\sigma_1)(\beta\sigma_2-B)} \tag{24} \]

or

\[ t=-\frac{T}{2a}\ln \frac{A\sigma_1+B\sigma_2-\beta\sigma_1\sigma_2+\beta-1-\lambda_{cm}} {A\sigma_2+B\sigma_1-\beta\sigma_1\sigma_2+\beta-1-\lambda_{cm}}, \tag{25} \]

where

\[ a=\sqrt{1-\beta(1+\lambda_{cm}-\beta)};\qquad A=1+a;\qquad B=1-a. \]

If \(\beta=1.04\div1.06\), then one may take

\[ a=1.03\sqrt{1-\lambda_{cm}}. \tag{26} \]

It is often necessary to determine the acceleration time on the natural characteristic of the motor from the instant of switching to the instant of static resistance. This section of the mechanical characteris-

characteristic may be replaced by a straight-line segment. Let us take the acceleration corresponding to the initial torque of static resistance, amounting to \(1\%\) of the acceleration at switching [6]. Then the required time is

\[ t \simeq 2.5\,T . \tag{27} \]

The total release time is composed of the acceleration time on the individual steps, taking into account the time of the preliminary steps and the time required to attain the steady speed on the natural characteristic.

References

1. Bychkov V. P. Graphical methods for constructing curves of an asynchronous motor. Elektrichestvo, No. 9, 1945.

2. Balashev I. A. Calculation of metallic rheostats and adjustment of contactor panels of mine hoisting installations. Izvestiya TPI, vol. 67, 1949.

3. Shklyarskii F. N. Physico-mechanical foundations of electric mine hoisting. Ugletekhizdat, 1953.

4. Geiler L. B. On the transient regime of an asynchronous motor under a stepwise change of load on the shaft. Elektrichestvo, No. 8, 1937.

5. Kayalov G. M. Operation of an asynchronous motor at a rotating torque close to the pull-out torque. Vestnik elektropromyshlennosti, No. 7, 1937.

6. Zhivov I. G. Drive and automation of mine hoisting machines. Mashgiz, 1952.

7. Bezikovich Ya. S. Approximate calculations. GITTL, 1941.