Conditions for the Computational Equivalence of an Induction Motor with an Asymmetrical Wound Rotor and a Squirrel-Cage Rotor
Abstract Generated abstract
This paper addresses how an induction motor with an asymmetrical squirrel-cage rotor, a case arising from casting defects or broken bars, can be represented computationally by an equivalent motor with an asymmetrical wound rotor. Using symmetrical-component decomposition of the multiphase rotor currents, it argues that only positive- and negative-sequence components produce resultant magnetomotive forces interacting with the stator, while other sequences contribute to heating losses. The paper derives conditions for equivalence based on preservation of the positive and negative resultant currents, magnetomotive forces, and inversion coefficients, and formulates equations for determining the impedances of the three phases of the equivalent wound rotor. It concludes with a calculation procedure for obtaining these impedances and notes that operating characteristics may then be found by established methods, with additional allowance for losses from all rotor current sequences.
Full Text
CONDITIONS FOR COMPUTATIONAL EQUIVALENCE OF AN INDUCTION MOTOR WITH AN ASYMMETRIC WOUND AND SQUIRREL-CAGE ROTOR
M. P. TABINSKII
(Recommended by the scientific seminar of the Electromechanical Faculty)
At the present time there is a well-developed theory of the operation of an induction motor with asymmetry in the wound-rotor circuit \([1, 2, 3, 4]\), both in steady-state and in transient regimes. In the literature known to us, the operation of an induction motor with an asymmetric squirrel-cage rotor with a simple cage—not to mention a deep-bar cage or a double squirrel cage—has not been treated at all. Meanwhile, rotor asymmetry in these motors occurs frequently: it arises either during manufacture of the motor because of casting defects when the rotor is filled, or during operation as a result of broken bars.
The operating conditions of an induction motor with an asymmetric squirrel-cage rotor can be investigated if the real motor is replaced by an equivalent motor with a wound rotor having analogous asymmetry. The justification of this replacement is given below.
The rotor of an induction motor may be represented in the form of a multiphase generator whose phases are connected by short-circuiting rings into a polygon (Fig. 1).
In Fig. 1 the following are denoted:
\(z_k\) — the total resistance of a sector of the short-circuiting ring;
\(z_c\) — the total resistance of a rotor bar (phase);
\(E_{p1}, E_{p2}, \ldots\) — the e.m.f. induced by the magnetic field of the air gap in the rotor phases.
Fig. 1. Equivalent circuit of the squirrel-cage rotor of an induction motor.
Asymmetry of the rotor resistances leads to an asymmetric system of currents, the vector diagram of which forms an asymmetric star. The asymmetric system of rotor currents can be decomposed into
the symmetrical components of the zero, first, second, etc. sequences, respectively, and there will be as many sequences as there are rotor phases. The method of decomposing a multiphase asymmetrical system is set forth in [5]. It is also proved there that the first sequence gives a resultant m.m.f. equal to:
\[ \dot F_{p1}=\dot F_{1/1}+\dot F_{1/2}+\dot F_{1/3}+\ldots+\dot F_{1/n} =\dot F_{1/1}\cdot \frac{n}{2} =\frac{n}{2}F_{1/1}e^{j(\omega t+\alpha)} \tag{1} \]
where
\[
\dot F_{1/1}=\sqrt{2}\,|I_{1/1}|\,w_1
\]
is the m.m.f. produced by one phase when a current of the first sequence flows;
\[
\dot I_{1/1}
\]
is the rms value of the current of the first (positive) sequence.
The sequence \((n-1)\) produces a resultant m.m.f. rotating in the opposite direction. Its magnitude is equal to
\[ \dot F_{p(n-1)}=\dot F_{(n-1)/1}+\dot F_{(n-1)/2}+\ldots+\dot F_{(n-1)/n} =\frac{n}{2}\dot F_{(n-1)/1}= \]
\[ =\frac{n}{2}F_{(n-1)/1}e^{j(\omega t+\alpha)}, \tag{2} \]
where
\[
\dot F_{(n-1)/1}=\sqrt{2}\,|I_{(n-1)/1}|\,w_1
\]
is the m.m.f. produced by one phase when a current of the last (negative) sequence flows.
The currents of the zero, second, third, etc., up to the \((n-2)\)-th sequence, flowing through the rotor winding, do not produce a resultant m.m.f.
It follows from this that, in an induction motor with an asymmetrical squirrel-cage rotor, only the symmetrical components of the rotor currents of the positive and negative sequences interact with the stator.
Since the induction motor is connected to a symmetrical system and the stator winding is symmetrical, only positive-sequence power can be transmitted from the stator to the rotor. In the rotor of the motor, owing to the asymmetry of the resistances, the positive-sequence power is partially converted into power of the negative, zero, second, etc. sequences. The electromagnetic power of the positive and negative sequences is converted into mechanical power; the electromagnetic power of the remaining sequences is converted only into heat, increasing the heating of the motor.
Fig. 2. Power-balance diagram of an induction motor with an asymmetrical squirrel-cage rotor for \(S>0.5\):
\(P_1\)—power supplied from the network,
\(P_{c1}\)—losses in the stator steel,
\(P_{m1}\)—losses in the stator copper,
\(P_{m2}\)—losses in the rotor copper,
\(P_2\)—mechanical power.
The conversion of the network power flow in an induction motor with an asymmetrical squirrel-cage rotor can be illustrated by the power-balance diagram (Fig. 2).
The foregoing makes it possible to conclude that the analogy in the operation of an induction motor with an asymmetrical phase-wound rotor and with an asymmetrical squirrel-cage rotor will hold when the positive resultant m.m.f. of the windings of both rotors and the negative resultant ...
of the m.m.f. of the rotor windings. The fundamental harmonic of the resultant m.m.f. of the windings is calculated by the formula [6]
\[ F_{\mathrm{res}}=\frac{m}{2}F_{\phi 1}=0.45\frac{m\cdot w\cdot k_{\mathrm{w}}}{p}I, \tag{3} \]
where: \(m\) is the number of rotor phases;
\(\quad w\) is the number of turns per phase;
\(\quad k_{\mathrm{w}}\) is the winding factor;
\(\quad p\) is the number of pole pairs.
The power on the motor shaft in the case of a squirrel-cage rotor will be smaller, since part of the positive-sequence power is converted into loss power due to currents of the second, third, etc., sequence.
Thus, for an equivalent replacement of a squirrel-cage rotor by a phase-wound rotor it is necessary to find the resistances in the phase-rotor circuit \(z_a, z_b, z_c\), for which the same positive and negative resultant currents and m.m.f.’s are obtained as in the asymmetric squirrel-cage rotor.
The degree of asymmetry in the rotor of an induction motor is characterized by the inversion coefficient [1].
\[ \dot{\gamma}=\frac{\dot{i}_{(n-1)/1}}{\dot{i}_{1/1}}. \tag{4} \]
The inversion coefficients for positive and negative phase sequence of the applied voltage are, in the general case, different. They are equal only under mirror asymmetry, when an axis can be drawn in the rotor with respect to which the asymmetric sections are arranged symmetrically.
For an induction motor with a three-phase rotor, for positive phase sequence the inversion coefficient is calculated by the formula [1]
\[ \dot{\gamma}_1=-\left(\frac{\dot{i}_{2/1}}{\dot{i}_{1/1}}\right)_{\mathrm{pos}} =-\frac{z_a+az_b+a^2z_c}{z_a+z_b+z_c}. \tag{5} \]
For negative phase sequence
\[ \dot{\gamma}_2=-\left(\frac{\dot{i}_{2/1}}{\dot{i}_{1/1}}\right)_{\mathrm{neg}} =-\frac{z_a+a^2z_b+az_c}{z_a+z_b+z_c}, \tag{6} \]
where
\[ a=e^{j120^\circ} \]
\(z_a; z_b; z_c\) are the total impedances of the rotor phases.
It is obvious that, when replacing an induction motor with a squirrel-cage rotor by an equivalent motor with a phase-wound rotor, the inversion coefficients \(\dot{\gamma}_1\) and \(\dot{\gamma}_2\) must be preserved.
Using equations (1), (2), (3), (4), (5), one can compose equations for finding the unknowns \(z_a, z_b, z_c\).
\[ z_a(1+\dot{\gamma}_1)+z_b(a+\dot{\gamma}_1)+z_c(a^2+\dot{\gamma}_1)=0, \tag{7} \]
\[ z_a(1+\dot{\gamma}_2)+z_b(a^2+\dot{\gamma}_2)+z_c(a+\dot{\gamma}_2)=0. \tag{8} \]
The missing equation for finding the three unknowns can be found by Kirchhoff’s law for one section of the three-phase rotor circuit (Fig. 3):
\[ (\dot{i}_{1/a}+\dot{i}_{2/a})z_a-(a^2\dot{i}_{1/a}+a\dot{i}_{2/a})z_b=\dot{U}_{AB}. \tag{9} \]
The following quantities appear in equation (9):
\[ \dot I_{1/a}=\frac{n}{6}\,\frac{\dot I_{1/1}}{w_2\cdot k_2}, \tag{10a} \]
\[ \dot I_{2/a}=\frac{n}{6}\,\frac{\dot I_{(n-1)/1}}{w_2\cdot k_2} \tag{10b} \]
—the positive- and negative-sequence components of the currents of the phase rotor, found from the equality of the resultant e.m.f.’s of the rotor windings (3);
\[ \dot I_{1/1};\ \dot I_{(n-1)/1} \]
—the positive- and negative-sequence components of the currents of the squirrel-cage rotor;
\(n\)—the number of rotor bars (phases);
\[ \dot U_{\mathrm{AB}}=\dot E_{\mathrm A}-\dot E_{\mathrm B} \tag{11} \]
—the line voltage on the winding of the phase rotor;
\(k_2\)—the winding factor;
\[ \dot E_{\mathrm A}=E_{\mathrm p1}\cdot 2\cdot w_2\cdot k_2\cdot \sin\omega t, \tag{12a} \]
\[ \dot E_{\mathrm B}=E_{\mathrm p1}\cdot 2\cdot w_2\cdot k_2\cdot \sin(\omega t-120^\circ) \tag{12b} \]
—e.m.f.’s in the phases of the phase rotor;
\(E_{\mathrm p1}\)—e.m.f. in a phase of the squirrel-cage rotor (Fig. 1).
Fig. 3. Equivalent circuit of the phase rotor.
The positive- and negative-sequence currents of the squirrel-cage rotor \(\dot i_{1/1}\) and \(\dot i_{(n-1)/1}\) are found from the formula [5]:
\[ \dot i_{1/1}=\frac{1}{n}\left(i_{c1}+r i_{c2}+r^2 i_{c3}+\cdots+r^{(n-1)}i_{cn}\right), \tag{13} \]
\[ \dot i_{(n-1)/1}=\frac{1}{n}\left(i_{c1}+r^{(n-1)}i_{c2}+r^{2(n-1)}i_{c3}+\cdots+r^{(n-1)(n-1)}i_{cn}\right), \tag{14} \]
where
\[ r=e^{j\theta}=e^{j\left(\frac{360^\circ\cdot P}{n}\right)} \]
is the rotation factor.
The currents in the rotor bars \(i_{c1}, i_{c2}, i_{c3}\), etc., entering formulas (13) and (14), are determined by solving the electric circuit (Fig. 1) by the mesh-current method, or with the aid of the theorem on current changes in an electric circuit, or with the aid of directed graphs. It is necessary to note that the replacement of a polygon by an equivalent star, used to simplify the calculations, is permissible only under symmetrical conditions.
The procedure for calculating the complete resistances of a phase rotor equivalent to an asymmetric squirrel-cage rotor is as follows:
1) The equivalent circuit of the squirrel-cage asymmetric rotor is found (Fig. 1).
2) Having specified the magnitude of the e.m.f. in the rotor bars:
\[ E_{\mathrm p1}=E_m\sin\omega t, \]
\[ E_{\mathrm p2}=E_m\sin(\omega t-\varphi), \]
\[ E_{\mathrm p3}=E_m\sin(\omega t-2\varphi), \]
\[ \cdots\cdots\cdots\cdots \]
where
\[ \varphi=\frac{360^\circ\cdot P}{n}. \]
find the current distribution in the rotor bars for the direct and reverse alternation of the phases of the rotor e.m.f.;
3) The currents of the positive- and negative-sequence components of the short-circuited rotor are found for the direct and reverse alternation of the phases of the rotor e.m.f.;
4) The inversion coefficients \(\gamma_1\) and \(\gamma_2\) are determined for the direct and reverse alternation of the phases of the e.m.f. by formula (4);
5) Using equations (7)–(12), the total impedances of the phases of the equivalent rotor \(z_a, z_b, z_c\) are determined.
After determining the parameters of the equivalent-rotor equivalent circuit, the operating characteristics of the induction motor can be calculated by known methods. In determining \(\eta\) and \(P_2\), it is necessary to take into account the power of the losses in the short-circuited rotor due to currents of all sequences.
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