Calculation of Scattering Fields in the End Regions of Single-Layer Windings
Abstract Generated abstract
The paper addresses the calculation of leakage magnetic fields in the end regions of single-layer concentric windings, where three-dimensional geometry and nearby ferromagnetic boundaries make simplified force and inductance estimates unreliable. It formulates the problem in a bounded cylindrical region using vector potential components in cylindrical coordinates, expands current densities and potentials in spatial harmonics, and solves the resulting radial equations with modified Bessel functions under ideal ferromagnetic boundary conditions. The derived expressions give the vector potential and magnetic induction components in the relevant radial regions and are presented as a basis for computing conductor forces, inductances, losses, and local heating in end-region structures, with practical evaluation intended for digital computation.
Full Text
CALCULATION OF LEAKAGE FIELDS IN THE END REGIONS OF SINGLE-LAYER WINDINGS
V. I. Popov, G. A. Sipailov
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Determination of inductances, as well as of the forces acting on conductors in the end region of an electric machine, is associated with calculation of leakage fields. Since windings in the end region have complex spatial configurations, the leakage fields produced by them generally have a three-dimensional structure. Calculation of such fields is very difficult. In calculating leakage fields, many authors use simplified notions of the character of the magnetic field in the region of the end parts of electrical machines. For example, in [2], when calculating the field in the end space, the influence of the surrounding ferromagnetic surfaces is not taken into account, while in [1, 3, 15] the influence of the shields and the frame is disregarded. In addition, in [1, 3] flat windings are considered. The formulas for calculating the forces acting on the conductors, obtained in [1, 2, 3, 15], give results differing from one another by a factor of 1.5–3 and therefore prove to be unreliable [14].
The beginning of a rigorous analytical approach to the problem of calculating leakage in the region of the end parts by applying methods of electromagnetic-field theory was laid by A. I. Voldek [7, 8, 9]. A. I. Voldek’s method most fully takes into account the influence of the active steel, shields, rotor, and frame, but it was developed for an unrolled machine, in which the field is calculated for a region represented by an infinitely long rectangular parallelepiped [7]. Apparently, such an idealization is acceptable for multipole machines, which have a small curvature of the end space. For two-pole machines of the turbogenerator type, and especially for impulse generators with a single-layer concentric winding, which have large curvature, such an assumption is unacceptable.
The problem can be formulated in a more general form if the field is determined for a bounded cylindrical region, which is what the end space of an electric machine represents (Fig. 1).
The magnetic induction in the region of the end parts is determined through the vector potential A:
\[ \mathbf{B}=\operatorname{rot}\mathbf{A}. \tag{1} \]
Being the most general characteristic of the magnetic field, the vector potential makes it possible to determine practically all other field quantities: the flux distribution, the field energy, inductances, etc. Determining the vector potential A in a given region, however, is a rather difficult problem. Nevertheless, if the distribution of the current-density vector i, which excites the field in a space with magnetic pro-
Fig. 1. Schematic representation of the end region of an electric machine: 1 — rotor, 2 — winding, 3 — end shield, 4 — frame, 5 — active steel
with permeability \(\mu\), the vector potential is determined from Poisson’s equation
\[ \Delta A=-\mu i. \tag{2} \]
The vector \(A\) has the direction of the current-density vector. The concentric winding in the end zone has two spatial components of current density \(i_z\) and \(i_\varphi\) (Fig. 1); consequently, the problem reduces to finding the components of the vector potential \(A_z\) and \(A_\varphi\), which may be found from scalar equations written in cylindrical coordinates
\[ \frac{\partial^2 A_\varphi}{\partial r^2} +\frac{1}{r}\frac{\partial A_\varphi}{\partial r} +\frac{1}{r^2}\cdot\frac{\partial^2 A_\varphi}{\partial \varphi^2} +\frac{\partial^2 A_\varphi}{\partial z^2} =-\mu_0 i_\varphi, \tag{3} \]
\[ \frac{\partial^2 A_z}{\partial r^2} +\frac{1}{r}\frac{\partial A_z}{\partial r} +\frac{1}{r^2}\frac{\partial^2 A_z}{\partial \varphi^2} +\frac{\partial^2 A_z}{\partial z^2} =-\mu_0 i_z. \tag{4} \]
From the components of the vector potential \(A_z\) and \(A_\varphi\), using (1), the induction components \(B_z\), \(B_r\), and \(B_\varphi\) can be found at any point of the end region.
In integrating equations (3) and (4), the following assumptions are adopted:
1) displacement currents in the end-region domain and eddy currents in the winding conductors are absent;
2) the magnetic permeability of the surrounding ferromagnetic surfaces is \(\mu=\infty\);
3) the actual spatial configuration of the winding is replaced by a computational one. In Fig. 1 the computational configuration is shown by dashed lines.
The end portion of the winding on one pole division is replaced by one curvilinear section with current \(i_\varphi\) and two rectilinear sections with currents \(i_z\). Comparison of the real and computational configurations shows that they are close, and therefore the error in determining the field is small.
Expanding the current densities \(i_\varphi\) and \(i_z\) in series in eigenfunctions—
...along the coordinate \(z\) and taking into account that the current is periodic along the bore of the stator, we may write
\[ i_\varphi=\sum_{\nu=1}^{\infty}\sum_{n=0}^{\infty} i_{n\varphi}(r)\cdot \cos mz \cdot \sin p\nu\varphi, \tag{5} \]
\[ i_z=\sum_{\nu=1}^{\infty}\sum_{n=1}^{\infty} i_{nz}(r)\sin mz \cdot \cos p\nu\varphi, \tag{6} \]
where \(\nu\) is the order of the spatial harmonic of the current density with respect to the coordinate \(\varphi\),
\[ n=0,1,2,3,\ldots \]
\(i_{n\varphi}(r)\) and \(i_{nz}(r)\) are the coefficients of the series, which in the general case are functions of the coordinate \(r\) (the amplitudes of the spatial harmonics of the current density),
\[ m=\frac{\pi n}{C}, \]
where \(C\) is the period of the spatial harmonic with respect to the coordinate \(z\) (Fig. 1),
\(p\) is the number of pole pairs.
For practical calculations it is sufficient to consider only the fundamental harmonic \(\nu=1\), since the influence of higher harmonics on the magnitude of the magnetic field in the end regions is small [8]. Then the expressions for the components of the vector potential \(A_\varphi\) and \(A_z\) can be represented in the form
\[ A_\varphi=\sum_{n=0}^{\infty} V_n(r)\cos mz\sin p\varphi, \tag{7} \]
\[ A_z=\sum_{n=1}^{\infty} W_n(r)\sin mz\cos p\varphi. \tag{8} \]
Here \(V_n(r)\) and \(W_n(r)\) are the coefficients of the series, which are functions of only one variable \(r\).
Substituting the series (5), (7) into (3) and (6), (8) into (4), we obtain equations for determining the coefficients \(V_n(r)\) and \(W_n(r)\):
\[ \frac{\partial^2 V_n(r)}{\partial r^2} +\frac{1}{r}\frac{\partial V_n(r)}{\partial r} -\left(m^2+\frac{p^2}{r^2}\right)V_n(r) =-\mu_0 i_{n\varphi}, \tag{9} \]
\[ \frac{\partial^2 V_0(r)}{\partial r^2} +\frac{1}{r}\frac{\partial V_0(r)}{\partial r} -\frac{p^2}{r^2}V_0(r) =-\mu_0 i_{0\varphi}, \tag{9a} \]
\[ \frac{\partial^2 W_n(r)}{\partial r^2} +\frac{1}{r}\frac{\partial W_n(r)}{\partial r} -\left[m^2+\frac{p^2}{r^2}\right]W_n(r) =-\mu_0 i_{nz}. \tag{10} \]
The solution by the method of variation of arbitrary constants for the Bessel equations of the same type (9) and (10) is:
\[ V_n(r)= \left( N_v-\mu_0\int_{r_1}^{r} i_{n\varphi}(\rho)K_p(m\rho)\rho\,d\rho \right)\cdot I_p(mr)+ \]
\[ +\left( M_v+\mu_0\int_{r_1}^{r} i_{n\varphi}(\rho)I_p(m\rho)\rho\,d\rho \right)\cdot K_p(mr), \]
\[ W_n(r)= \left( N_w-\mu_0\int_{r_1}^{r} i_{nz}(\rho)K_p(m\rho)\rho\,d\rho \right)I_p(mr)+ \]
$$ +\left(M_w+\mu_0\int_{r_1}^{r} i_{nz}(\rho) I_p(m\rho)\rho\,d\rho\right)K_p(mr), $$
where
\(I_p\) and \(K_p\) are modified Bessel functions of the first and second kind, respectively,
\(N_v, N_w, M_v, M_w\) are constants of integration,
\(r_1\) is some fixed value of \(r\).
Expression (9a), corresponding to \(n=0\), represents Euler’s equation, whose general integral can be written as
$$ V_0=\left(N_0-\frac{\mu_0}{2p}\int_{r_1}^{r} i_{0\varphi}(\rho)\rho^{1-p}\,d\rho\right)r^p+ $$
$$ +\left(M_0+\frac{\mu_0}{2p}\int_{r_1}^{r} i_{0\varphi}(\rho)\rho^{1+p}\,d\rho\right)r^{-p}, $$
where \(N_0\) and \(M_0\) are constants of integration.
The validity of these solutions can be verified by substituting them into the corresponding equations, taking into account that the derivative of an integral with respect to a variable upper limit is equal to the integrand at this limit.
The constants of integration are found from the boundary conditions. The field is determined for the region of space bounded by ferromagnetic surfaces with magnetic permeability \(\mu=\infty\). This means that the components of induction tangential to the surfaces are equal to zero for any value of the angle \(\varphi\).
The components of induction for the vector potential \(A_\varphi\) are determined from equation (1)
$$ B_r=-\frac{\partial A_\varphi}{\partial z}, $$
$$ B_\varphi=0, $$
$$ B_z=\frac{A_\varphi}{r}+\frac{\partial A_\varphi}{\partial r}. $$
For the tangential components of induction (boundary conditions):
a) on the end surfaces
$$ 1)\quad \left(\frac{\partial A_\varphi}{\partial z}\right)_{z=0}=0, \qquad 2)\quad \left(\frac{\partial A_\varphi}{\partial z}\right)_{z=C}=0; $$
b) on the cylindrical surface
$$ 3)\quad \left(\frac{A_\varphi}{r}+\frac{\partial A_\varphi}{\partial r}\right)_{r=r_1}=0, \qquad 4)\quad \left(\frac{A_\varphi}{r}+\frac{\partial A_\varphi}{\partial r}\right)_{r=r_2}=0. $$
Conditions 1 and 2 are satisfied by specifying solution (7). Consequently, the constants of integration \(N_v\) and \(M_v\) must be found from conditions (3) and (4), for which one can obtain
$$ \left(\frac{V_n(r)}{r}+\frac{\partial V_n(r)}{\partial r}\right)_{r=r_1}=0; \qquad \left(\frac{V_n(r)}{r}+\frac{\partial V_n(r)}{\partial r}\right)_{r=r_2}=0. $$
By analogy with the preceding, the constants \(N_w\) and \(M_w\) can be found from the conditions
\[ \left(\frac{\partial W_n(r)}{\partial r}\right)_{r=r_1}=0;\qquad \left(\frac{\partial W_n(r)}{\partial r}\right)_{r=r_2}=0, \]
and the constants \(N_0\) and \(M_0\) from
\[ \left(\frac{V_0(r)}{r}+\frac{\partial V_0(r)}{\partial r}\right)_{r=r_1}=0;\qquad \left(\frac{V_0(r)}{r}+\frac{\partial V_0(r)}{\partial r}\right)_{r=r_2}=0 . \]
In determining the constants of integration it is necessary to calculate integrals of the form
\[ \int_{r_1}^{r_2} i_{n\varphi}(\rho)K_p(m\rho)\rho\,d\rho;\quad \int_{r_1}^{r_2} i_{0\varphi}(\rho)\rho^{1-p}\,d\rho\ldots \quad \text{etc.} \]
Since the current is distributed over region II, and in regions I and III is equal to zero, the indicated integrals exist only in the interval \(R_1<r<R_2\). In addition, region II forms the surfaces \(r=\text{const}\), and, consequently, the current density in it does not depend on the variable \(r\) (see Fig. 1).
The constants of integration determined from the solutions of equations (9), (9a), and (10), taking into account the boundary conditions considered, take the form
\[ N_v=\mu_0 i_{n\varphi} b R_k \beta_v \frac{f_vK_p(mR_k)+\varepsilon_v I_p(mR_k)} {\beta_v f_v-\varepsilon_v\lambda_v}, \]
\[ M_v=\mu_0 i_{n\varphi} b R_k \lambda_v \frac{f_vK_p(mR_k)+\varepsilon_v I_p(mR_k)} {\beta_v f_v-\varepsilon_v\lambda_v}, \]
\[ N_w=\mu_0 i_{nz} b R_k \beta_w \frac{f_wK_p(mR_k)+\varepsilon_w I_p(mR_k)} {\beta_w f_w-\varepsilon_w\lambda_w}, \]
\[ M_w=\mu_0 i_{nz} b R_k \lambda_w \frac{f_wK_p(mR_k)+\varepsilon_w I_p(mR_k)} {\beta_w f_w-\varepsilon_w\lambda_w}, \]
\[ N_0=\mu_0 i_{0\varphi} \frac{S\cdot(p+1)r_2^p+T(p-1)r_2^{-p}} {(p+1)\cdot(r_2^p-r_1^{2p}r_2^{-p})}, \]
\[ M_0=\mu_0 i_{0\varphi} \frac{S\cdot(p+1)r_2^p+T(p-1)r_2^{-p}} {(p-1)(r_2^p-r_1^{2p}r_2^{-p})}\cdot r_1^{2p}, \]
where
\[ \beta_v=(p+1)K_{p-1}(mr_1)+(p-1)K_{p+1}(mr_1), \]
\[ f_v=(p+1)I_{p-1}(mr_2)+(p-1)I_{p+1}(mr_2), \]
\[ \varepsilon_v=(p+1)K_{p-1}(mr_2)+(p-1)K_{p+1}(mr_2), \]
\[ \lambda_v=(p+1)I_{p-1}(mr_1)+(p-1)I_{p+1}(mr_1), \]
\[ \beta_w=K_{p-1}(mr_1)+K_{p+1}(mr_1), \]
\[ f_w=I_{p-1}(mr_2)+I_{p+1}(mr_2), \]
\[ \varepsilon_w=K_{p-1}(mr_2)+K_{p+1}(mr_2), \]
\[ \lambda_w=I_{p-1}(mr_1)+I_{p+1}(mr_1), \]
\[ S=\frac{R_2^{2-p}-R_1^{2-p}}{2p(2-p)}, \]
\[ T=\frac{R_2^{2+p}-R_1^{2+p}}{2p(2+p)}, \]
\[
b=R_2-R_1
\]
is the radial width of the winding,
\[
R_k=\frac{R_2+R_1}{2}
\]
is the mean radius of the winding.
The amplitudes of the spatial harmonics of the current density \(i_{0\varphi}\), \(i_{n\varphi}\), and \(i_{nz}\) are determined from (5) and (6):
\[ i_{0\varphi}=\frac{2Iq}{C\pi b}\,\sin\frac{\pi}{2}\left(1-\frac{\gamma}{2}\right), \]
\[ i_{n\varphi}=\frac{8Iq}{\pi^2 b(C_2-C_1)}\cdot \frac{1}{n}\cos\frac{m(C_2+C_1)}{2}\times \]
\[ \times \sin\frac{m(C_2-C_1)}{2}\,\sin\frac{\pi}{2}\left(1-\frac{\gamma}{2}\right), \]
\[ i_{nz}=-\frac{16I q p k_p}{\pi^2 b(R_2+R_1)}\,\frac{1}{n}\,\sin^2\frac{mC_0}{2}, \]
where
\[
\gamma
\]
is the ratio of the wound part of the stator to the full circumference;
\[
k_p
\]
is the winding distribution coefficient;
\[
I
\]
is the current flowing through the winding;
\[
C_1,C_2,C_0
\]
are geometrical dimensions (Fig. 1).
Expressions for the coefficients of series (7)
for region I, \(r_1<r<R_1\),
\[ V_{n\mathrm{I}}(r)=N_v\left[\beta_v I_p(mr)+\lambda_v K_p(mr)\right]; \]
for region II, \(R_1<r<R_2\),
\[ V_{n\mathrm{II}}(r)=N_v\left[\beta_v I_p(mr)+\lambda_v K_p(mr)\right] -\mu_0 I_p(mr)\int_{R_1}^{r} i_{n\varphi}(\rho)K_p(m\rho)\rho\,d\rho\;+ \]
\[ +\mu_0 K_p(mr)\int_{R_1}^{r} i_{n\varphi}(\rho)I_p(m\rho)\rho\,d\rho; \]
for region III, \(R_2<r<r_2\),
\[ V_{n\mathrm{III}}(r)=N_v\left[\varepsilon_v I_p(mr)+f_v K_p(mr)\right]. \]
Similarly, for the coefficients of series (8),
\[ W_{n\mathrm{I}}(r)=N_w\left[\beta_w I_p(mr)+\lambda_w K_p(mr)\right], \]
\[ W_{n\mathrm{II}}(r)=N_w\left[\beta_w I_p(mr)+\lambda_w K_p(mr)\right]- \]
\[ -\mu_0 I_p(mr)\int_{R_1}^{r} i_{nz}(\rho)K_p(m\rho)\rho\,d\rho +\mu_0 K_p(mr)\int_{R_1}^{r} i_{nz}(\rho)I_p(m\rho)\rho\,d\rho; \]
\[ W_{n\mathrm{III}}(r)=N_w\left[\varepsilon_w I_p(mr)+f_w K_p(mr)\right]. \]
The components of the induction at any point of the end-winding space are determined as
\[ B_{r\delta}=\frac{1}{r}\frac{\partial A_{z\delta}}{\partial \varphi} -\frac{\partial A_{\varphi\delta}}{\partial z}, \]
\[ B_{\varphi\delta}=-\frac{\partial A_{z\delta}}{\partial r}, \]
\[ B_{z\delta}=\frac{1}{r}A_{\varphi\delta} +\frac{\partial A_{\varphi\delta}}{\partial r}, \]
where \(\delta\) is the region in which the induction is determined.
The method considered for determining the field in the end region of single-layer windings is the first stage in studying the forces acting on the conductors of the end part of a surge generator. It is also applicable to the calculation of inductances, the study of losses, and local heating in the structural elements of the end region. The difficulties associated with the use of special functions and with the cumbersomeness of the calculations can be successfully overcome by using electronic digital computers.
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