Determination of Additional Losses in the Bars of a Stator Winding Placed in the Machine Air Gap
Abstract Generated abstract
This paper analyzes additional eddy-current losses in stator-winding bars placed in the air gap of synchronous machines with a smooth armature, a design proposed to reduce stator mass and leakage reactance in pulse-power generators. Using a vector-potential formulation of Maxwell’s equations under simplifying assumptions about field geometry and conductor dimensions, the authors derive expressions for eddy-current density, loss per unit bar length, and an additional-resistance coefficient as functions of conductor geometry, conductivity, permeability, frequency, and field penetration depth. Calculations for impact generators indicate that, for uninsulated elementary conductors, additional copper losses can exceed principal no-load losses by about 1.5 to 2 times, while subdivision into insulated elementary conductors reduces them substantially. Comparison with model-generator experiments shows good agreement, supporting the conclusion that these losses need not limit slotless stator designs for pulse-power generators.
Full Text
Proceedings of the Tomsk Polytechnic Institute
named after S. M. Kirov
Vol. 200 1974
DETERMINATION OF ADDITIONAL LOSSES IN THE BARS OF THE STATOR WINDING PLACED IN THE AIR GAP OF THE MACHINE
L. O. OGANYAN, K. A. KHORKOV
Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering
The use in synchronous machines of an armature winding placed in the air gap of the machine makes it possible to design the toothed zone of the machine more compactly, which leads to a reduction in the weight of the stator steel and insulation by approximately a factor of two. It becomes possible to increase significantly the induction in the air gap of the machine. The magnitude of the inductive leakage reactance of the stator winding is reduced. The latter is especially important for impact-power generators. However, in a winding arranged on a smooth armature, additional electrical losses arise, the magnitude of which depends on the main flux of the machine. It is of interest to estimate the magnitude of these additional losses relative to the principal losses in the copper of the stator winding.
We determine the additional losses under the assumption that the stator winding is developed in the tangential direction, the transverse dimensions of the bars are small in comparison with their length, the external field is plane-parallel, and its distribution along the sides of each bar is similar and known. We do not take into account the influence of eddy currents on the external field. The calculation scheme of the problem is given in Fig. 1. We seek the solution with the aid of the vector potential $\overline{A}$, related to the induction in the volume of the field under consideration by the relation
\[ \overline{B} = \operatorname{rot}\overline{A}. \tag{1} \]
Fig. 1. Calculation scheme
If the bars of the stator winding have specific conductivity $\gamma$ and magnetic permeability $\mu$, and the induction vector varies in time with angular frequency $\omega$, then Maxwell’s equation [1], describing the field inside the bar, can be reduced to the form
\[ \frac{\partial^2 \dot{A}}{\partial x^2} + \frac{\partial^2 \dot{A}}{\partial y^2} = j\omega\gamma\mu \dot{A}, \tag{2} \]
where
\[ \dot A=\sum_{n=1}^{s}\left(D_1 e^{\lambda y}+D_2 e^{-\lambda y}\right)\cos kx; \tag{3} \]
\[ \lambda=\sqrt{k^2+p^2}; \tag{4} \]
\[ p^2=j\omega\gamma\mu; \tag{5} \]
\[ k^2=\left(\frac{n\pi}{T}\right)^2, \tag{6} \]
\(D_1;\ D_2\) are constants of integration,
\(\lambda\) is the spatial frequency of the field in the body of the shield,
\(T\) is the period of expansion. For the surfaces \(x=0\) and \(x=b\), \(T=a\); for the surface \(y=a\), \(T=b\).
The constants of integration are determined from the condition of continuity of the tangential components of the field on the surfaces \(x=0,\ x=b\) and \(y=a\), and from the absence of the indicated component on the surface \(y=0\), owing to the location of the bar in the immediate vicinity of a ferromagnetic surface with \(\mu=\infty\).
As a result of the transformations we have
\[ \dot A_1=\frac{2\dot B_1}{\pi}\sum_{n=1}^{s}(-1)^{n+1}\frac{1}{(2n-1)} \cdot \frac{ \operatorname{sh}\frac{\lambda_1}{2}\left(x-\frac{b}{2}\right) }{ \lambda_1\operatorname{ch}\frac{\lambda_1}{2}b } \cos\frac{n\pi}{a}y; \tag{7} \]
\[ \dot A_2=\frac{2\dot B_2}{\pi}\sum_{n=1}^{s}(-1)^{n+1}\frac{1}{(2n-1)} \cdot \frac{\operatorname{ch}\lambda_2 y}{\lambda_2\operatorname{sh}\lambda_2 a} \cdot \cos\frac{n\pi}{b}x; \tag{8} \]
where
\[ \lambda_1=\sqrt{\left(\frac{n\pi}{a}\right)^2+j\omega\gamma\mu}; \]
\[ \lambda_2=\sqrt{\left(\frac{n\pi}{b}\right)^2+j\omega\gamma\mu}; \]
\[ \dot B_1=\sum_{n=0}^{s}\dot B_{1T}\cos\frac{n\pi}{a}y; \tag{9} \]
\[ \dot B_2=\sum_{n=0}^{s}\dot B_{2T}\cos\frac{n\pi}{b}x; \tag{10} \]
\(\dot B_{1T};\ \dot B_{2T}\) are the Fourier-series coefficients in the expansion of the induction curve, respectively, over the intervals \(0\le y\le a\) and \(0\le x\le b\).
We determine the distribution of eddy-current density in the winding bar as
\[ \dot\delta=-j\omega\gamma\dot A, \tag{11} \]
and the losses in the bar as
\[ P=\frac{1}{2\gamma}\int_V |\dot\delta|^2\,dV. \tag{12} \]
Proceeding from (7, 8, 11 and 12), after transformations we obtain an expression for calculating the losses per unit length of the bar:
\[ P_1=\frac{1}{8\pi^2}\omega^2 c^3\gamma a \sum_{n=1}^{s}\frac{B_1^2}{(2n-1)^2}\eta_1; \tag{13} \]
\[ P_2=\frac{1}{16\pi^2}\omega^2 c^3\gamma b \sum_{n=1}^{\infty}\frac{B_2^2}{(2n-1)^2}\eta_2, \tag{14} \]
where \(c\) is the depth of penetration of the field into the body of the conductor;
\[ c=\sqrt{\frac{2}{\omega\gamma\mu}}; \tag{15} \]
\[ \eta_1= \frac{\operatorname{sh}\dfrac{b}{c}-\sin\dfrac{b}{c}} {\operatorname{ch}\dfrac{b}{c}+\cos\dfrac{b}{c}}; \tag{16} \]
\[ \eta_2= \frac{\operatorname{sh}2\dfrac{a}{c}+\sin 2\dfrac{a}{c}} {\operatorname{ch}2\dfrac{a}{c}-\cos 2\dfrac{a}{c}}. \tag{17} \]
If the winding consists of \(m\) bars, then the losses due to eddy currents in the copper of the stator winding per unit length will be equal to
\[ P_{\text{add}}=m(P_1+P_2). \tag{18} \]
We determine the resistance of the bars to the eddy current as
\[ r_{\text{eddy}}=\frac{P_{\text{add}}}{I_{\text{eddy}}^2}, \tag{19} \]
where
\[ i_{\text{eddy}}=\int_S \dot{\delta}\,dS. \tag{20} \]
Taking into account (13, 14, 12, and 11), we obtain
\[ r_{\text{eddy}}=\frac{\pi^2}{16}\frac{1}{\gamma c} \left[\frac{2}{a}\eta_1+\frac{1}{b}\eta_2\right]. \tag{21} \]
If the ohmic resistance of the bar is
\[ r_0=\frac{1}{\gamma ab},\quad \text{then } r_{\text{eddy}}=k_b r_0, \tag{22} \]
whence
\[ k_b=\frac{\pi^2}{16c}\left[a\eta_2+2b\eta_1\right]. \tag{23} \]
Using the method described, losses per unit length of the bars were calculated for impact generators with a smooth armature, made with \(2p=2, 4,\) and \(6\) within the limiting dimensions of an eight-pole generator [2]. The calculation results are presented in Table 1.
Table 1
| \(2p\) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| \(f\) (Hz) | 12.5 | 25 | 37.5 | 50 |
| \(P_{\text{add}}\) (MW) | 2.18 | 3.1 | 3.8 | 4.4 |
| \(P_{xx}\) (MW) | 2.2 | 2.3 | 2.5 | 2.7 |
The additional losses in the copper of the bars of the stator winding were calculated for the worst-case variant of using bars made of uninsulated
elementary conductors. Calculations show that the level of additional losses exceeds the principal no-load losses of the machine by a factor of 1.5–2.0. The magnitude of the additional losses is significantly reduced when the bars are made of insulated elementary conductors. Figure 2 presents the dependences of the coefficient \(k_{\mathrm{v}}\) on the relative geometric dimensions of the stator-winding bars \(\left(\dfrac{b}{c}; \dfrac{a}{c}\ \text{and}\ \dfrac{a}{b}\right)\).
Analysis of the curves in Fig. 2 shows that a reduction in the additional resistance of the bar, and consequently in the additional losses due to eddy currents, is possible by dividing the bar cross-section into elementary conductors.
Fig. 2. Dependence of the coefficient \(k_{\mathrm{v}}\) on the relative geometric dimensions of the bar:
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\(k_{\mathrm{v}} = f\left(\dfrac{a}{b}\right)\) at \(\dfrac{b}{c} = 0.2\);
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\(k_{\mathrm{v}} = f\left(\dfrac{a}{b}\right)\) at \(\dfrac{b}{c} = 2\)
Fig. 3. Dependence of additional losses in the stator-winding bars of a model impulse generator with a smooth armature on the induction in the machine air gap. \(*\) — experimental points
The described method for calculating additional losses was verified by experimental studies on a model impulse generator with a smooth armature, made in the dimensions of an AR-7 asynchronous motor [3]. The calculated curve of additional losses as a function of induc-
winding in the air gap of the machine is presented in Fig. 3. In the same figure, the symbol (*) denotes experimental data kindly provided to us by Associate Professor V. F. Kulakov. The good agreement between the results of the calculations and the experiments is evident.
Thus, the results of the investigations show that additional losses from eddy currents in the bars of the stator winding are not a limiting factor in the application of a slotless stator design in pulse-power generators.
REFERENCES
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S. D. Kupalyan. Theoretical Foundations of Electrical Engineering. “Energiya,” Moscow, 1970, Part 3.
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G. A. Sipailov, K. A. Khor’kov, L. O. Oganyan, V. Z. Khor’kova. On the limiting energies of a pulse generator with a smooth armature. Proceedings of the All-Union Conference “Power Pulse Systems.” VNIIITI Publishing House, 1973.
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G. A. Sipailov, K. A. Khor’kov, V. F. Kulakov, V. Z. Khor’kova, V. I. Andreev, V. S. Baklin. A model of a synchronous generator with a smooth armature. Izvestiya TPI, vol. 265, Tomsk State University Publishing House, Tomsk, 1973.