Dependence of Current on Time during Braking of Standard-Design Squirrel-Cage Motors

I. G. Kuleev

Associate Professor, Candidate of Technical Sciences

Dependence of Current on Time during Braking by the Plugging Method for an Unloaded Motor

The equation of motion for the run-down process of an unloaded motor during braking by the plugging method may be written in the following form:

\[ -\frac{m_2 r_2 I_2^2}{9.81 \omega_1 S} = J \omega_1 \frac{dS}{dt}. \tag{1} \]

Equation (1) contains two variables: \(S\) and \(I_2\). The dependence of \(S\) on \(I_2\) may be found as follows.

At the initial instant of braking \(S = 2\). Consequently, the braking torque corresponding to this slip is

\[ M_{m2} = \frac{m_2 r_2 I_{2n}^{\,2}}{9.81 \omega_1 2} \tag{2} \]

Here \(I_{2n}\) is the rotor current at the initial instant of braking.

Dividing equality (2) by the expression for the braking torque, we obtain

\[ \frac{M_{m2}}{M_m} = \frac{I_{2n}^{\,2} S}{I_2^{\,2} 2}. \tag{3} \]

Dividing the numerator and denominator of equation (3) by \(M_{\max}\) gives

\[ \frac{M_{m2}}{M_m} = \frac{\dfrac{M_{m2}}{M_{\max}}}{\dfrac{M_m}{M_{\max}}}. \]

Substituting the values of \(\dfrac{M_{m2}}{M_{\max}}\) and \(\dfrac{M_m}{M_{\max}}\) from the Kloss equation, we obtain

\[ \frac{M_{m2}}{M_m} = \frac{2(S_m^2 + S^2)}{(S_m^2 + 4)S}. \tag{4} \]

Substituting the value of \(\dfrac{M_{m2}}{M_m}\) into equation (4), we shall have

\[ \frac{I_{2n}^{\,2} S}{I_2^{\,2} 2} = \frac{2(S_m^2 + S^2)}{(S_m^2 + 4)S}. \tag{5} \]

Equation (5) is a quadratic equation of the form

\[ S^2\left[(4+S_m^2)I_{2n}^2-4I_2^2\right]-4I_2^2S_m^2=0, \]

whence

\[ S=\pm \frac{2I_2S_m}{\sqrt{I_{2n}^2(4+S_m^2)-4I_2^2}} \tag{6} \]

and

\[ dS=\pm \frac{2I_{2n}^2(4+S_m^2)S_m\,dI_2} {\left[I_{2n}^2(4+S_m^2)-4I_2^2\right]^{3/2}} . \tag{7} \]

Substituting, in equation (1), the values of \(S\) and \(dS\) from equations (6, 7), we find an expression for the dependence of the current on time in the form

\[ t= \frac{39{,}24\,J\omega_1^2S_m^2}{m_2r_2} \int_{I_{2n}}^{I} \frac{I_{2n}^2(4+S_m^2)\,dI_2} {I_2\left[I_{2n}^2(4+S_m^2)-4I_2^2\right]} . \tag{8} \]

Integration gives

\[ t= -\frac{19{,}62\,J\omega_1^2S_m^2}{m_2r_2} \left[ \frac{1}{\varepsilon}\ln \frac{I_2^2S_m^2}{\varepsilon-4I_2^2} + \frac{4(I_2^2-I_{2n}^2)} {I_{2n}^2S_m^2(\varepsilon-4I_2^2)} \right], \tag{9} \]

where

\[ \varepsilon=I_{2n}^2(4+S_m^2). \]

Dependence of the current on time during braking by the counter-connection method of a loaded motor, \(M_c=\mathrm{Const}\).

The equation of motion for the run-out process in this case may be written in the form

\[ -\frac{m_2r_2I_2^2}{\omega_1S\cdot 9{,}81}-M_c = J\omega_1\frac{dS}{dt}. \]

Substituting \(S\) and \(dS\) by their values from equations (6, 7), we obtain

\[ \frac{J\omega_1S_m\varepsilon\,dI_2} {dt(\varepsilon-4I_2^2)^{3/2}} = -\frac{m_2r_2I_2}{39{,}24\,\omega_1S_m} -M_c, \]

whence the dependence of the current on time is

\[ t= -J\omega_1^2S_m^2\varepsilon \int_{I_{2n}}^{I_2} \frac{dI_2} {(\varepsilon-4I_{2n}^2)^{3/2} \left( \frac{I_2m_2r_2\sqrt{\varepsilon-4I_2^2}}{39{,}24} \right)} . \tag{10} \]

Integration gives

\[ \begin{aligned} t= \frac{J\omega_1S_m}{M_c} \Bigg[ &\frac{4}{a_1a_2} \left( -\frac{I_2}{\sqrt{\varepsilon-4I_2^2}} + \frac{I_{2n}}{\sqrt{\varepsilon-4I_{2n}^2}} + \frac{4(a_1+a_2)}{a_1^2-a_2^2} \ln \frac{2\sqrt{\varepsilon-4I_2^2}} {\sqrt{\varepsilon-4I_{2n}^2}} \right) \\ &- \frac{a_1^2+4}{a_1(a_2-a_1)} \ln \frac{2(\sqrt{\varepsilon-4I_2^2}-a_1I_2)} {\sqrt{\varepsilon-4I_{2n}^2}-2a_1I_{2n}} + \frac{a_2^2+4}{a_2^2(a_2-a_1)} \ln \frac{2(\sqrt{\varepsilon-4I_2^2}-a_2I_2)} {\sqrt{\varepsilon-4I_{2n}^2}-2a_2I_{2n}} \Bigg], \end{aligned} \tag{11} \]

where

\[ a_{12}= -\frac{H\varepsilon}{2M_c} \pm \sqrt{\left(\frac{H\varepsilon}{2M_c}\right)^2-4}. \]

Dependence of Current on Time during Braking by the Counter-Connection Method for a Loaded Motor \(M_c = k'\omega_2^2\)

In this case the equation of motion for the running-down process may be written in the form

\[ -\frac{m_2 r_2 I_2^2}{\omega_1 S 9.81} - k(S-1)^2 = J\omega_1 \frac{dS}{dt}. \]

Here \(k(S-1)^2 = M_c = k'\omega_1^2(S-1)^2\).

Replacing \(S\) and \(dS\) by their values from equations (6, 7), we obtain

\[ -\frac{m_2 r_2 I_2 \sqrt{\varepsilon - 4 I_2^2}} {9.81 \cdot \omega_1 S_m \cdot 2} - k\left( -1 + \frac{2S_m I_2}{\sqrt{\varepsilon - 4I_2^2}} \right)^2 = J\omega_1 \frac{2\varepsilon S_m dI_2} {dt(\varepsilon - 4I_2^2)^{3/2}}, \]

whence the dependence of current on time is

\[ t = -4J\omega_1 S_m \varepsilon \int_{I_{2n}}^{I_2} \frac{dI_2} {(\varepsilon - 4I_2^2)^{1/2} \left\{ I_2 H(\varepsilon - 4I_2^2)^{3/2} + k\left[ \frac{\varepsilon - 4I_2^2}{2} - S_m I_2 \sqrt{\varepsilon - 4I_2^2} + 2S_m^2 I_2^2 \right] \right\}}, \tag{12} \]

where

\[ H = \frac{m_2 r_2}{9.81 \cdot \omega_1 S_m}. \]

The integral appearing on the right-hand side of equation (12) is evaluated by introducing a new variable.

Let

\[ \varepsilon - 4I_2^2 = Z^2 I_2^2, \tag{13} \]

then

\[ -dI_2 = \frac{I_2 Z\,dZ}{4+Z^2}; \qquad I_2 = \frac{\varepsilon}{4+Z^2}. \]

Substituting the values of \(I_2\) and \(dI_2\) into equation (13), we obtain

\[ t = \frac{4J\omega_1 S_m}{k} \int_{I_{2n}}^{I_2} \frac{(4+Z^2)dZ} {Z^4 + Z^3\left(\frac{H\varepsilon}{k}-2S_m\right) + Z^2(1+S_m^2)2 - Z8S_m + 8S_m^2}. \]

Denoting \(\dfrac{H\varepsilon}{k} - 2S_m\) by \(h\), and \(2(1+S_m^2)\) by \(g\), we shall have

\[ t = \frac{4J\omega_1 S_m}{k} \int_{I_{2n}}^{I_2} \frac{(4+Z^2)dZ} {Z^4 + Z^3 h + Z^2 g - Z8S_m + 8S_m^2}. \tag{14} \]

Integration gives:
(See the author’s paper, “Running-Up of Standard-Design Squirrel-Cage Motors,” Izvestiya TPI, vol. 63, 1944.)

\[ t = \frac{4J\omega_1 S_m}{k} \left[ A \ln \frac{ \sqrt{\dfrac{\varepsilon}{I_2^2} - 1 - \dfrac{f}{2} + d} }{ \sqrt{\dfrac{\varepsilon}{I_{2n}^2} - 1 - \dfrac{f}{2} + d} } + \right. \]

\[ \begin{aligned} &+ B \ln \frac{ \sqrt{\dfrac{\varepsilon}{I_2^2}-1-\dfrac{f}{2}-d} }{ \sqrt{\dfrac{\varepsilon}{I_3^2}-1-\dfrac{f}{2}-d} } \\ &+ \frac{C}{2}\ln \frac{ \dfrac{\varepsilon}{I_{2n}^2}-1 + f\sqrt{\dfrac{\varepsilon}{I_2^2}-1+\dfrac{f^2}{4}+e^2} }{ \dfrac{\varepsilon}{4I_{2n}^2}-1 + f\sqrt{\dfrac{\varepsilon}{4I_{2n}^2}-1+\dfrac{f^2}{4}+e^2} } \\ &+ \frac{2D-cf}{2e} \left( \operatorname{arctg} \frac{ 2\sqrt{\dfrac{\varepsilon}{I_2^2}-1+f} }{2e} - \operatorname{arctg} \frac{ 2\sqrt{\dfrac{\varepsilon}{4I_{2n}^2}-1+f} }{2e} \right) \Bigg], \end{aligned} \tag{15} \]

where

\[ A= \frac{ 4+\dfrac{f^2}{4}+fd+d^2 }{ 2d\left(f^2+2fd+d^2+e^2\right) }, \qquad B= \frac{ 4+\dfrac{f^2}{4}-fd+d^2 }{ 2d\left(f^2-2fd+d^2+e^2\right) }; \]

\[ C=-A-B; \qquad D=1-A\left(d+\frac{f}{2}\right)+B\left(d-\frac{f}{2}\right)+cf. \]

Fig. 1

Fig. 2

Fig. 3

The values of \(I\) during plugging braking, obtained from formulas and taken from oscillograms (1, 2) for the same instants of time, are given in Table 1 below.

Table 1

The calculations and oscillographic recordings were made for a motor having the following data: type T-750, \(U_1 = 220\) V, \(E_2 = 145\) V, \(I_1 = 38\) A, \(I_2 = 45.5\) A, \(n = 715\) rpm, \(\cos \varphi = 0.8\), \(r_1 = 0.4\) ohm, \(r_2 = 0.08\) ohm, \(r_k = 1.1\) ohm, \(X_1 = 1.35\) ohm, \(X_2 = X_1\), \(Z_k = 2.92\) ohm, \(S_m = 0.213\), \(J = 0.05\) kgm sec\(^2\), \(M_n = 11\) kgm, \(M_{\max} = 30\) kgm.

Dependence of current on time during dynamic braking of an unloaded motor.

The equation of motion during dynamic braking of an unloaded motor may be written in the following form:

\[ -\frac{m_2 r_2 I_2^2}{\omega_1 s \cdot 9.81} = J \omega_1 \frac{ds}{dt}. \]

Substituting for \(s\) and \(ds\) their values (see the author’s paper “Dependence of Current on Time during Starting of Standard-Design Squirrel-Cage Motors,” Izvestiya TPI, vol. 63, 1944), we obtain

\[ t = \frac{9.81 J \omega_1^2 S_{m1}^2}{m_2 r_2} \int_{I_{2n}}^{0.051\,2n} \frac{I_{2n}^2(1+S_{m1}^2)\,dI_2} {I_2\left[I_{2n}^2(1+S_{m1}^2)-I_2^2\right]^2}, \tag{16} \]

where \(I_{2n}\) is the rotor current at the first instant after connecting the stator winding to the direct-current network. Since the rotor current differs little from the starting current, it may, with accuracy sufficient for practice, be taken equal to the latter.

Integration gives

\[ t = -\frac{9.81 J \omega_1^2 S_{m1}^2}{m_2 r_2} \left\{ \frac{1}{2\varepsilon} \ln \frac{I_2^2 S_{m1}^2} {I_{2n}^2(1+S_{m1}^2)-I_2^2} + \right. \]

\[ \left. + \frac{1}{2\left[I_{2n}^2(1+S_{m1}^2)-I_2^2\right]} - \frac{1}{2S_{m1}^2 I_{2n}^2} \right\}. \tag{17} \]

Dependence of current on time during braking of a loaded motor \(M_c = \mathrm{Const}\).

The equation of motion in this case may be written in the form

\[ -\frac{m_2 r_2 I_2^2}{9.81 \omega_1 S} - M_c = J \omega_1 \frac{dS}{dt}. \]

Replacing \(S\) and \(dS\) by their values (see the preceding section), we obtain

\[ \frac{J\omega_1 S_{m1}\varepsilon\, dI_2}{\left(\varepsilon-I_2^2\right)^3\,dt} = -\frac{m_2 r_2 I_2\sqrt{\varepsilon-I_2^2}}{9.81\,\omega_1 S_{m1}} -M_c, \]

whence the dependence of the current on time is

\[ t=-J\omega_1^2 S_{m1}\varepsilon \int_{I_{2n}}^{0.05 I_{2n}} \frac{dI_2}{ \left(\varepsilon-I_2^2\right)^{3/2} \left( \frac{I_2 m_2 r_2}{9.81} \sqrt{\varepsilon-I_2^2} +M_c\omega_1 S_{m1} \right) }. \]

Here \(I_{2n}\) is the starting current of the rotor. Here, as before, the upper limit of integration is taken equal to \(0.05 I_{2n}\), in order to obtain a finite value for \(t\).

Integration gives

\[ t=-\frac{J\omega_1 S_{m1}}{M_c} \left[ \frac{1}{a_1a_2} \left( -\frac{I_2}{\sqrt{\varepsilon-I_2^2}} + \frac{I_{2n}}{\sqrt{\varepsilon-I_{2n}^2}} \right) + \right. \]

\[ \left. +\frac{a_1+a_2}{a_1^2a_2^2} \ln \frac{I_2\sqrt{\varepsilon-I_2^2}} {I_2\sqrt{\varepsilon-I_{2n}^2}} - \right. \]

\[ \left. -\frac{a_1^2+1}{a_1^2(a_2-a_1)} \ln\left( \frac{I_{2n}\sqrt{\varepsilon-I_2^2}} {I_2\sqrt{\varepsilon-I_{2n}^2}} -a_1 \right) + \frac{a_2^2+1}{a_2^2(a_2-a_1)} \ln\left( \frac{I_{2n}\sqrt{\varepsilon-I_2^2}} {\sqrt{\varepsilon-I_{2n}^2}} -a_2 \right) \right]. \]

The values of \(I\) obtained from formulas (17) and taken from the oscillogram (curve 4) for the same instants of time for the previously mentioned motor are given in the following Table 2.

Fig. 4

Table 2

Noticed Misprints