On the equation of the characteristic M=?(s) for a three-phase induction motor
Abstract Generated abstract
The paper addresses the difficulty of using the Kloss torque, slip characteristic in differential equations for three-phase induction motor drives under repeatedly short-time loading without starting regimes. It proposes replacing the working portion of the actual characteristic, up to about 0.9 of pull-out torque, by a straight line chosen so that its triangular area equals the area under the Kloss curve over the same slip interval. This derivation yields the approximate relation \(M_{dv}=0.9(M_n/s_n)s\), expressed through rated torque and rated slip, and therefore applicable to normal motors. Comparison with other linear approximations and constructed transient speed curves indicates that the proposed line more closely represents the working characteristic and is suitable for analytical study of transient operation within that range.
Full Text
On the Characteristic Equation \(M=\varphi(s)\) for a Three-Phase Induction Motor
M. F. Filippov
The characteristic \(M_{dv}=\varphi(s)\) for a three-phase induction motor is presented in Fig. 1 (curve \(oab\)). This characteristic can be expressed analytically by the well-known Kloss equation
\[ M_{dv}=M_{opr}\frac{2}{\dfrac{s_{opr}}{s}+\dfrac{s}{s_{opr}}}, \tag{1} \]
where \(s=\dfrac{\omega_s-\omega}{\omega_s}\) denotes the slip,
\[ \omega_s \]
is the synchronous angular velocity of rotor rotation,
\(M_{opr}\) is the maximum value of the torque on the curve \(oab\),
\(s_{opr}\) is the slip corresponding to \(M_{opr}\).
The complexity of the Kloss equation makes it difficult to solve differential equations of motion in a drive with a three-phase induction motor, for example, an equation of the form
\[ M_{dv}=M_{st}-J\omega_s\frac{ds}{dt}, \tag{2} \]
where \(J\) is the reduced moment of inertia of the drive,
\(M_{st}\) is the static moment of resistance to rotation of the drive.
In very many drives, an induction motor operates with a repeatedly short-time load continuously without starting regimes. In this case, under normal conditions the induction motor is never loaded up to the pull-out torque \(M_{opr}\), and for the working part of the characteristic \(M_{dv}=\varphi(s)\) one should take the segment \(oa\) of the curve \(oab\) (Fig. 1).
This working part of the characteristic can be replaced by a certain straight line, and then the solution of the differential equations of motion in a drive with an induction three-phase motor is considerably simplified.
With a corresponding choice of the parameters of the straight line replacing the actual characteristic \(M_{dv}=\varphi(s)\), it is possible, when solving the equations of motion, to obtain results that practically do not differ from the actual ones.
Instead of the working part \(oa\) of the characteristic \(oab\) (Fig. 1), let us take the straight line \(oc\), with the calculation made so that the area of the triangle \(ocd\) is equal to the area \(oad\).
The straight line \(oc\) can be expressed by the equation
\[ M=\frac{Mc}{s_a}s. \tag{3} \]
From the condition of equality of the areas \(ocd\) and \(oad\), we have
\[ \frac{M_c \cdot s_a}{2} = \int_{0}^{s_a} \frac{2M_{\mathrm{ovr}}\,ds}{\dfrac{s_{\mathrm{ovr}}}{s}+\dfrac{s}{s_{\mathrm{ovr}}}} = M_{\mathrm{ovr}}s_{\mathrm{ovr}}\ln \frac{s_{\mathrm{ovr}}^{2}+s_a^{2}}{s_{\mathrm{ovr}}^{2}}; \tag{4} \]
taking the value \(M_a=0.9M_{\mathrm{ovr}}\), we find from Kloss’s equation
\[ s_a=0.63\,s_{\mathrm{ovr}}, \tag{5} \]
substituting the obtained value of \(s_a\) into equation (4) and solving it with respect to \(M_c\), we find
\[ M_c=1.07\,M_{\mathrm{ovr}}; \tag{6} \]
substituting the values of \(s_a\) and \(M_c\) into equation (3), we obtain for the straight line \(oc\) the expression
\[ M=\frac{1.07\,M_{\mathrm{ovr}}}{0.63\,s_{\mathrm{ovr}}}\cdot S = 1.7\,\frac{\lambda M_n}{s_{\mathrm{ovr}}}\,s, \tag{7} \]
where
\[ \lambda=\frac{M_{\mathrm{ovr}}}{M_k}. \]
In turn, the pull-out slip \(s_{\mathrm{ovr}}\) can also be expressed through the rated slip, using Kloss’s equation, i.e.
\[ \frac{M_n}{M_{\mathrm{ovr}}} = \frac{2}{\dfrac{s_{\mathrm{ovr}}}{s_n}+\dfrac{s_n}{s_{\mathrm{ovr}}}} = \frac{1}{\lambda}, \]
whence
\[ s_{\mathrm{ovr}}=s_n\left(\lambda+\sqrt{\lambda^2-1}\right). \tag{8} \]
After substituting this value into equation (7), we obtain
\[ M= \frac{1.7\lambda M_n}{s_n\left(\lambda+\sqrt{\lambda^2-1}\right)}\,s, \tag{9} \]
or
\[ M=\frac{K M_n}{s_n}\,s, \tag{10} \]
where
\[ K=\frac{1.7\lambda}{\lambda+\sqrt{\lambda^2-1}}. \tag{11} \]
The value of \(\lambda\) in normal three-phase induction motors varies within the limits
\[ \lambda=1.8-3. \]
For these values of \(\lambda\), the coefficient \(K\) changes very little—
\[ K=0.925-0.875. \]
Consequently, with sufficient accuracy, the average value \(k=0.9\) may be substituted into equation (11); then equation (11) is written in the form
\[ M=0.9\,\frac{M_n}{s_k}\,s. \tag{12} \]
Fig. 1
Fig. 2
Thus, instead of the working part of the actual characteristic
$M_{dv}=\varphi(s)$ of a three-phase induction motor, one may use the equation of a straight line
\[ \boxed{M_{dv}=As}, \tag{13} \]
where
\[ A=0.9\,\frac{M_n}{s_n}=0.9\,\frac{M_n n_s}{n_s-n_n}. \]
The obtained equation is valid for all normal three-phase induction motors, since it is expressed through the basic parameters of the motor.
It should be noted that, when solving the differential equations of a motor in drives with an induction motor, instead of the actual characteristic one sometimes uses the straight line $ol$ (Fig. 1), whose equation is obtained from Kloss’s formula (equation 1) at small slips, i.e.,
\[ M_{dv}=\frac{2M_{opr}}{s_{opr}}\,s. \tag{15} \]
In addition, Fig. 1 shows the straight line $of$, which, instead of the actual characteristic, is used by engineer A. T. Golovan in studying the operation of the electric drive of forging-and-pressing machines. The equation of this straight line has the form:
\[ M_{dv}=\frac{M_n s}{s_n 1.37}. \tag{16} \]
As can be seen from Fig. 1, the straight line $ol$ and the straight line $of$ diverge very strongly from the actual characteristic of the induction motor, while at the same time the straight line $oc$, proposed by the author, is closest to the working part of the characteristic.
Figure 2 shows the curves of the transient process $n=f(t)$ of an induction motor when a constant load is applied
\[ M_{st}=\text{const}. \]
The curves are constructed by the method of proportions for the four characteristics under consideration:
\[ M_{dv}=\frac{2M_{opr}}{\dfrac{s_{opr}}{s}+\dfrac{s}{s_{opr}}}; \tag{1} \]
\[ M_{dv}=\frac{M_n}{1.37\cdot s_n}\,s; \tag{16} \]
\[ M_{dv}=\frac{2M_{opr}}{s_{opr}}\,s; \tag{15} \]
\[ \boxed{M_{dv}=0.9\,\frac{M_n}{s_n}\,s} \tag{13} \]
As the results of the construction show, the characteristic expressed by equation 13 gives the most accurate results and, consequently, is entirely suitable for the analytical investigation of transient conditions of an induction motor when the load of the latter varies within the limits of the working part of the characteristic $M_{dv}=\varphi(s)$.