Rotor Slot Leakage in Advanced Synchronous Reluctance Motors
Abstract Generated abstract
This paper examines the calculation of rotor slot leakage reactance in advanced synchronous reluctance motors whose cast aluminum squirrel-cage rotors include slots of complex configuration. It analyzes leakage flux behavior separately for transverse and longitudinal pulsating magnetic fields, using flux-linkage based permeance calculations after dividing the slot into simple geometric regions. Experimental frequency dependences of rotor parameters are used to support the assumption that current displacement into the upper part of such slots is absent under the studied geometry. The derived formulas indicate that the magnetic permeance of slot leakage in a transverse pulsating field is nearly twice that in a longitudinal pulsating field, a distinction that should be included in two-reaction analytical treatments of these machines.
Full Text
Rotor Slot Leakage in Advanced Synchronous Reluctance Motors
E. V. Kononenko, T. V. Chesheva
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
In the rotor design of advanced synchronous reluctance motors (SRM), in addition to the usual slots of oval cross-section and the slots of the interpolar space, there are slots of complex configuration. They consist of two slots of oval cross-section, arranged symmetrically with respect to the transverse axis of the rotor, and internal layers between them (Fig. 1). This design feature considerably complicates the calculation of the rotor parameters. In the present article only questions associated with the calculation of the inductive
Fig. 1. Transverse section of the rotor
reactance of the slot leakage of these machines will be considered. This is due to the fact that at present there are no ready-made calculation formulas for determining the magnetic permeance coefficient of the leakage flux between the walls of a slot of such configuration, \(\lambda_p\). In the general case, the rotor winding of an SRM is an aluminum cast squirrel cage, intended for the purposes of asynchronous starting and for increasing operating stability.
The transformer couplings between the stator winding and the circuits of the rotor winding are usually effected through rotating mag-
...the magnetic fields. We replace these fields by two pulsating fields along the axes $d$ and $q$, which induce the corresponding current components in the bars of the rotor winding. With a sinusoidal distribution of the components of these currents over the bars, their maximum values along the axes $d$ and $q$ will occur in the bars displaced from the axis of the corresponding field by a distance
\[ \frac{\tau}{2} \]
(Fig. 2). A detailed consideration of the physical phenomena makes it possible to note the following:
- A magnetic field pulsating along the $d$ axis induces in bars symmetrically located with respect to the $q$ axis currents that are equal in magnitude and direction.
Fig. 2. Currents in the bars of the rotor winding of an SRM:
$a$ — in a longitudinal pulsating magnetic field; $b$ — in a transverse pulsating magnetic field.
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A magnetic field pulsating along the $q$ axis induces in bars symmetrically located with respect to the $d$ axis currents that are equal in magnitude and direction.
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A magnetic field pulsating along the $d$ axis induces in bars of complex shape a current of one direction; whereas a field pulsating along the $q$ axis induces currents of different directions in these same bars.
In addition, it is necessary to take into account that the bars of one pole, arranged symmetrically with respect to the \(d\)-axis, belong to one and the same longitudinal circuit, but at the same time belong to the like-named transverse circuits of adjacent poles, formed by bars arranged symmetrically with respect to the \(q\)-axis. Consequently, the leakage field of a slot of complex shape will differ depending on what field induces currents in the bars. For this reason, the subsequent analysis must be carried out separately for these two cases.
Transverse Field
The leakage field of the winding induces in it a self-induction emf, or the so-called leakage emf, which is taken into account by means of the leakage inductive reactance. On its path the leakage field encounters mainly the reluctance of nonmagnetic gaps; therefore, when deriving the calculation formulas, the reluctance of the steel may be neglected and the leakage inductance may be regarded as
\[ L_{\sigma} = \mathrm{const}. \]
Figure 3 shows a slot of complex shape, most often used in the rotor design of improved SynRM machines. The total depth of the slot may well be such that problems arise associated with the need to take account of the skin-effect phenomenon. In any case, this question should be considered separately. To this end, let us analyze the possible distribution of the magnetic lines of force of the leakage flux closing across the slot. Since we are considering
Fig. 3. Rotor slot of complex shape
slot located in a transverse pulsating magnetic field; then, when currents of different directions flow through bars 1 and 2, the entire slot may be regarded as two identical slots separated by a nonconducting partition along the q-axis. The smallest value of the nonmagnetic interlayer occurs in the region of the slot opening. This means that in this part of the slot the leakage flux will be greater than in any other. In addition, the complex configuration of the slot leads one to assume a nonuniform distribution of the magnetic lines of force of the leakage flux over the entire height of the slot along the line O—O. It may be considered that
Fig. 4. Distribution of the magnetic lines of force of a slot in a transverse pulsating magnetic field
each magnetic line of force of the slot leakage flux crosses the slot twice: in the region of the slot opening and in any other part. Consequently, the smallest leakage e.m.f.’s are induced in the lower sections of the bar, and the largest in the upper part. It is natural to suppose that sections with a smaller slot width will have greater leakage; that is, if it is at all meaningful to speak of the phenomenon of current displacement in this slot, then its existence is explained only by the nonuniform distribution of the magnetic lines of force over the sections of the slot. In any case, it may be considered that displacement of the current into the upper part of the slot is absent here. Figure 4 shows the distribution of the magnetic lines of force of the slot. Figure 5 shows the experimental dependences of the parameters of the SynRM on frequency. The parameters were determined for an advanced SynRM, the rotor of which had the design shown in Fig. 1. In accordance with the designations of Fig. 3, slot 1 of complex shape of the experimental rotor had the following dimensions: \(h_{\text{sh}}=1.45\) mm; \(h_1=2.25\) mm; \(h_2=4.9\) mm; \(h_3=5\) mm; \(h_4=22\) mm; \(b_{\text{sh}}=1.5\) mm; \(d_1=3.8\) mm; \(d_2=4.5\) mm; \(b_1=4\) mm; \(b_2=4\) mm; total slot height 15.6 mm.
The slot located on the axial line of the pole has the dimensions: \(h_{\text{sh}}=1.45\) mm; \(h_1=2.25\) mm; \(h_2=4.9\) mm; \(h_3=1.2\) mm; \(h_4=30\) mm; \(b_{\text{sh}}=1.5\) mm; \(d_2=4.5\) mm; \(d_1=3.8\) mm; \(b_1=5.0\) mm; \(b_2=5.0\) mm; total slot height 23.1 mm.
The parameters of the rotor winding were determined by the static method [4]. The experimental dependences confirm the assumption made above regarding the absence of current displacement if the rotor construction contains slots of the configuration described above, while the height of the remaining slots is sufficiently small.
Fig. 5. Dependences of the parameters on frequency for the rotor (Fig. 1); \(x_d''\) and \(x_q''\) are referred to a frequency of 50 Hz
To obtain an expression for the magnetic permeance coefficient of the leakage of such a slot, we shall use the generally known method [1, 2, 3, etc.]. The coefficient of conductance of slot leakage \(\lambda_{\text{п}}\) depends only on the geometric dimensions of the slot. Its value is determined from the flux linkage of the induction lines passing across the slot with the part of the bar in the slot. The complicated configuration of a single slot bar can always be divided into simple geometric figures. Thus, the slot shown in Fig. 3 can be divided into five regions. The slot is filled with aluminum, i.e., there is one bar in the slot. The magnetic permeance coefficient of the slot can be represented as the sum of the magnetic permeances of the individual regions
\[ \lambda_{\text{п}} = \lambda_{\text{пII}} + \lambda_{\text{пIII}} + \lambda_{\text{IV}} + \lambda_{\text{пV}}. \tag{1} \]
The method for determining the individual components of the permeances is as follows [3]. As an example, let us consider region III of the slot. The leakage flux per unit height \(h_x\) of each region is determined. To do this, we isolate an elementary magnetic flux tube of width \(dx\) at a distance \(x\) from the upper boundary of the region and determine the magnitude of the elementary flux for this tube per unit length of the rotor
\[ d\Phi_x = B_x dx. \tag{2} \]
Since the flux is linked only with the part of the bar in the slot \(\dfrac{x}{h_x}\), the flux linkage is
\[ d\psi_x = d\Phi_x \cdot \frac{x}{h_x}. \tag{3} \]
The field intensity \(H_x\) is determined from the law of total current for the given elementary tube
\[ \int H_x dl = i \frac{x}{h_x}. \tag{4} \]
Since we neglect the magnetic resistance of the steel, then
\[ \begin{aligned} H_x \cdot (b_{\sh} + b_x) &= i \frac{x}{h_x} \\ H_x &= \frac{ix}{h_x \cdot (b_{\sh} + b_x)} \\ B_x &= \mu_0 H_x \end{aligned} \tag{5} \]
The elementary flux linkage in the element is
\[ d\psi_x = \mu_0 \frac{ix^2 dx}{h_x^2 (b_{\sh} + bx)} . \tag{6} \]
The total flux linkage of the section is
\[ \Psi = \mu_0 i \lambda = \int_0^{h_x} d\psi_x = \mu_0 i \frac{h_x}{3(b_{\sh} + b_x)} . \tag{7} \]
Whence
\[ \lambda = \frac{h_x}{3(b_{\sh} + b_x)} . \tag{8} \]
For the slot (Fig. 3), the coefficients of magnetic permeance of the leakage flux, in accordance with (8), will have the following form:
\[ \lambda_{\text{pII}} = \frac{0.623}{2} + \frac{2h_1}{3(3b_{\sh} + d_2)} , \tag{9} \]
\[ \lambda_{\text{pIII}} = \frac{2h_2}{3(2b_{\sh} + d_2 + d_1)} , \tag{10} \]
\[ \lambda_{\text{pIV}} = \frac{2h_3}{3(2b_{\sh} + d_1 + b_1)} , \tag{11} \]
\[ \lambda_{\text{pV}} = \frac{h_4}{3(b_{\sh} + b_2)} . \tag{12} \]
The coefficient of magnetic permeance of leakage for the entire slot will be equal to
\[ \lambda_{\text{pq}} = 2 \left[ \frac{0.623}{2} + \frac{2h_1}{3(3b_{\sh} + d_2)} + \frac{2h_2}{3(2b_{\sh} + d_2 + d_1)} + \frac{2h_3}{3(2b_{\sh} + d_1 + b_1)} + \frac{h_4}{3(b_{\sh} + b_2)} \right] . \tag{13} \]
Longitudinal Field
In the longitudinal pulsating magnetic field, a slot of complex shape in the rotor of an improved SRM is traversed by current of one direction; consequently, in this case this slot cannot be regarded as consisting of two separate identical slots with insulation between them. A picture of the possible distribution of magnetic lines of force for the conditions under consideration is presented in Fig. 6. From the fig-
Fig. 6. Distribution of the magnetic lines of force of a slot in a longitudinally magnetically pulsating field
From the figure it is evident that the induction lines close through the steel and cross the slot twice. The magnetic permeance coefficient of the slot, as in the preceding case, is determined from the flux linkage of the induction lines passing across the slot with the part of the bar in the slot. Let us consider the regions of the open slot. Here it may be assumed that the magnetic flux is constant, since it embraces the full slot current, consisting of two bars. Using the previous notation and method, we obtain expressions for the magnetic permeance coefficient of this part of the slot:
\[ \Phi = Bh_{\text{sh}};\qquad \psi = Bh_{\text{sh}};\qquad B=\mu_0 H_x; \]
\[ H_x\cdot(2b_{\text{sh}})=i;\qquad H_x=\frac{i}{2b_{\text{sh}}};\qquad \psi=\mu_0 i\,\frac{h_{\text{sh}}}{2b_{\text{sh}}}; \]
\[ \lambda_I=\frac{h_{\text{sh}}}{2b_{\text{sh}}}. \tag{14} \]
For the second section,
\[ d_1=2\cdot\frac{d_2+b_{\text{sh}}}{2}=d_2+b_{\text{sh}}. \tag{15} \]
\[ \lambda_{II}=0.623+\frac{h_1}{3(d_2+b_{\text{sh}})}. \tag{16} \]
Similarly, for the remaining ones,
\[ \lambda_{III}=\frac{h_2}{3(d_2+d_1)}; \tag{17} \]
\[ \lambda_{IV}=\frac{h_3}{3(d_1+b_1)}; \tag{18} \]
\[ \lambda_V=\frac{h_4}{6b_2}. \tag{19} \]
The magnetic permeance coefficient of leakage of the entire slot will be equal to
\[ \lambda_{nd}=\frac{h_{\text{sh}}}{2b_{\text{sh}}}+0.623+ \frac{h_1}{3(d_2+b_{\text{sh}})}+ \frac{h_2}{3(d_2+b_1)}+ \]
\[ + \frac{\bar h_3}{3(d_1+b_1)} + \frac{\bar h_4}{6b_2}. \tag{20} \]
Calculation by formulas (13) and (20) shows that, in a transverse pulsating magnetic field, the leakage of a slot of complex shape will be greater than the leakage of this slot in a longitudinal pulsating magnetic field by almost a factor of 2.
Conclusions
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In slots of complex shape, present on the rotors of advanced synchronous reluctance motors, current displacement into the upper part of the slot is absent even in the case when the total slot height exceeds 20 mm.
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When using the theory of two reactions in analytical studies of the above-mentioned electrical machines, it is necessary to take into account the fact that the leakage of a slot of complex shape differs in longitudinal and transverse pulsating magnetic fields.
References
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I. M. Postnikov. Generalized Theory and Transient Processes of Electrical Machines, Tekhnika Publishing House, Kiev, 1966.
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M. P. Kostenko. Electrical Machines. Special Part, GEI, 1949.