CALCULATION OF THE EFFECT OF EDDY CURRENTS ON THE MAIN EXCITATION FIELD OF A DC MACHINE WITH A PRINTED ARMATURE

A. M. SHOR

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

To calculate the losses due to eddy currents in the printed armature winding of a DC machine, it is necessary to know the resulting magnetic field of the machine, obtained by superimposing on the main excitation field the field of reaction of the eddy currents.

In this work, a calculation is made of the resulting magnetic field in the region occupied by the conductors of the printed winding. The results obtained make it possible to carry out the loss calculation more accurately and to estimate the degree of influence of eddy currents on the main excitation field of the machine.

The calculation is performed under the following assumptions:

1. The nonuniformity of the distribution of the intensity of the external magnetic field of the machine over the width of the conductor, arising when the conductor moves in this field, is not taken into account. Thus, the conductor is considered not in a traveling magnetic field, as is the case in the machine, but in an alternating uniform magnetic field directed normal to the flat surface of the conductor.

2. The calculation is carried out with respect to the first harmonic of the external magnetic field. However, the results of the calculation make it possible to estimate the degree of influence of the reaction field on the higher harmonics of the main field.

3. The reaction of the eddy currents of an individual conductor is taken into account. Nevertheless, the results obtained can be used to determine the magnetic field produced by neighboring conductors.

4. A conductor of finite length and rectangular cross section is replaced by an infinite cylindrical conductor of elliptical cross section.

5. Displacement currents both inside and outside the conductor are not taken into account, which is quite permissible in the cases of low frequencies occurring in a DC machine.

The final formulation of the problem is as follows: an infinite cylinder of elliptical cross section, of width \(2b\), thickness \(2a\), with specific conductivity \(\gamma\) and magnetic permeability \(\mu_i\), is located in an alternating uniform magnetic field \(H = H_0 \sqrt{2}\sin \omega t\) (the corresponding complex \(\dot H_0\)) (Fig. 1). Determine the distribution of current density over the cross section of the conductor and the maximum magnitude of the distortion of the external field by the field of the eddy currents. From obvious considerations, the eddy currents will exert the maximum demagnetizing effect at the center of the conductor.

The calculation is carried out in an elliptic coordinate system \(\xi,\eta\); the coordinate \(\xi\) determines the size of confocal ellipses, and the coordinate \(\eta\) determines the position of confocal hyperbolas orthogonal to the ellipses.

Fig. 1. Conductor in an alternating magnetic field.

The coordinates \(\xi,\eta\) are related to the coordinates \(x,y\) by the relations:

\[ x + jy = h\,\operatorname{ch}(\xi + j\eta); \tag{1} \]

\[ x = h\,\operatorname{ch}\xi \cos\eta; \tag{2} \]

\[ y = h\,\operatorname{sh}\xi \sin\eta, \tag{3} \]

where \(2h\) is the distance between the foci of the ellipses. In addition, the width of the cylinder is \(2a = 2h\,\operatorname{ch}\xi_0\), and the thickness of the cylinder is \(2b = 2h\,\operatorname{sh}\xi_0\), where \(\xi_0\) is the coordinate determining the position of the contour of the elliptic conductor.

The calculation reduces to integrating Maxwell’s equations for the quasi-stationary regime

\[ \operatorname{rot}\dot{\vec H} = \gamma \dot{\vec E}, \tag{4} \]

\[ \operatorname{rot}\dot{\vec E} = -j\omega\mu\dot{\vec H} \tag{5} \]

in the exterior and interior regions of the conductor, taking into account the continuity of the tangential components of the magnetic-field intensity vector and the normal components of the magnetic-induction vector

\[ \dot H_{\eta e} = \dot H_{\eta i}, \tag{6} \]

\[ \mu_e \dot H_{\xi e} = \mu_i \dot H_{\xi i}. \tag{7} \]

In the exterior region (index \(e\)), taking into account assumption (5), equation (4) has the form

\[ \operatorname{rot}\dot{\vec H}_e = 0, \tag{8} \]

and, consequently, the field outside the conductor is potential and is described by Laplace’s equation for the scalar magnetic potential \(\varphi_m\)

\[ \nabla^2 \varphi_m = 0. \tag{9} \]

Equation (7) in elliptic coordinates has the form

\[ \frac{1}{l^{2}}\left(\frac{\partial^{2}\dot{\varphi}_{m}}{\partial \xi^{2}}+ \frac{\partial^{2}\dot{\varphi}_{m}}{\partial \eta^{2}}\right)=0 \tag{10} \]

\[ l=\frac{1}{\sqrt{2}}h(\operatorname{ch}2\xi-\cos 2\eta)^{\frac12} \]

is the Lamé coefficient.

Let us represent

\[ \dot{\varphi}_{m}=\dot{\varphi}_{m0}+\dot{\varphi}_{mb}, \]

where \(\dot{\varphi}_{m0}\) is the component of the potential due to the external homogeneous field;

\(\dot{\varphi}_{mb}\) is the component of the potential due to the influence of eddy currents;

\(\dot{\varphi}_{m0}, \dot{\varphi}_{mb}\) satisfy (9).

The expression for \(\dot{\varphi}_{m0}\) can be found by integrating the equation

\[ \dot{\overline{H}}_{0}=-\operatorname{grad}\dot{\varphi}_{m0}. \tag{11} \]

Taking (3) into account,

\[ \dot{\varphi}_{m0}=\dot{H}_{0}h\operatorname{sh}\xi\sin\eta. \tag{12} \]

Integrating (10) for \(\dot{\varphi}_{mb}\) by the Fourier method and choosing solutions that vanish at infinity, odd with respect to the \(x\)-axis and even with respect to the \(y\)-axis, we obtain

\[ \dot{\varphi}_{mb}=\sum_{k=0}^{\infty}B_{2k+1}e^{-(2k+1)\xi}\sin(2k+1)\eta. \tag{13} \]

Then

\[ \dot{\varphi}_{m}=\dot{H}_{0}h\operatorname{sh}\xi\sin\eta+ \sum_{k=0}^{\infty}B_{2k+1}e^{-(2k+1)\xi}\sin(2k+1)\eta. \tag{14} \]

The components of the magnetic-field intensity are

\[ \dot{H}_{\xi e}=-\frac{1}{l}\frac{\partial\dot{\varphi}_{m}}{\partial \xi} =-\frac{1}{l}\left[\dot{H}_{0}h\operatorname{ch}\xi\sin\eta- \sum_{k=0}^{\infty}(2k+1)B_{2k+1}e^{-(2k+1)\xi}\sin(2k+1)\eta\right], \tag{15} \]

\[ \dot{H}_{\eta e}=-\frac{1}{l}\frac{\partial\dot{\varphi}_{m}}{\partial \eta} =-\frac{1}{l}\left[\dot{H}_{0}h\operatorname{sh}\xi\cos\eta+ \sum_{k=0}^{\infty}(2k+1)B_{2k+1}e^{-(2k+1)\xi}\cos(2k+1)\eta\right]. \tag{16} \]

For the region inside the conductor (index \(i\)), the joint solution of (4), (5) with respect to \(\dot{\overline{E}}=\kappa\dot{\overline{E}}_{z}\) leads to the Helmholtz equation

\[ \nabla^{2}\dot{E}_{z}+\kappa_{1}^{2}\dot{E}_{z}=0, \tag{17} \]

\[ \kappa_{1}^{2}=-j\omega\mu_{i}\gamma. \tag{18} \]

In the elliptic coordinate system, (17) has the form

\[ \frac{\partial^2 \dot E_z}{\partial \xi^2} + \frac{\partial^2 \dot E_z}{\partial \eta^2} + 2q(\operatorname{ch}2\xi-\cos 2\eta)\dot E_z=0, \tag{19} \]

\[ q=\left(\frac{k_1 h}{2}\right)^2 . \tag{20} \]

The solution of (19) [1] is \(\dot E_z=\chi(\xi)\Phi(\eta)\), where \(\chi(\xi)\) is the solution of the modified Mathieu equation

\[ \frac{d^2\chi}{d\xi^2}-(a_1-2q\operatorname{ch}2\xi)\chi=0, \tag{21} \]

and \(\Phi(\eta)\) is the solution of the Mathieu equation

\[ \frac{d^2\Phi}{d\eta^2}+(a_1-2q\cos 2\eta)\Phi=0, \tag{22} \]

\(a_1\) being the separation constant.

The periodic solutions of (21), (22) are the modified functions and the Mathieu functions of cosine and sine type [1]

\[ \chi(\xi)=\sum_{n=0}^{\infty}\left[D'_n Ce_n(\xi)+E'_n Se_n(\xi)\right], \tag{23} \]

\[ \Phi(\eta)=\sum_{n=0}^{\infty}\left[F'_n ce_n(\eta)+K'_n se_n(\eta)\right]. \tag{24} \]

From the aggregate of products of the functions (23), (24), only functions even with respect to the \(x\)-axis and odd with respect to the \(y\)-axis are selected,

\[ \dot E_z=\sum_{n=0}^{\infty}D_{2n+1}Ce_{2n+1}(\xi)\,ce_{2n+1}(\eta). \tag{25} \]

The components of the magnetic-field intensity from (5) are

\[ \dot H_{\xi i} = -\frac{1}{j\omega\mu_i l}\frac{\partial \dot E_z}{\partial\eta} = -\frac{1}{j\omega\mu_i l} \sum_{n=0}^{\infty} D_{2n+1}Ce_{2n+1}(\xi)\,ce'_{2n+1}(\eta), \tag{26} \]

\[ \dot H_{\eta i} = \frac{1}{j\omega\mu_i l}\frac{\partial \dot E_z}{\partial\xi} = \frac{1}{j\omega\mu_i l} \sum_{n=0}^{\infty} D_{2n+1}Ce'_{2n+1}(\xi)\,ce_{2n+1}(\eta). \tag{27} \]

Taking (6), (7) into account, we obtain

\[ \left. \begin{aligned} \dot H_0 h\,\operatorname{sh}\xi_0\cos\eta + \sum_{\kappa=1}^{\infty}\kappa B_\kappa e^{-\kappa\xi_0}\cos\kappa\eta &= -\frac{1}{j\omega\mu_i} \sum_{n=1}^{\infty} D_n Ce'_n(\xi_0)\,ce_n(\eta), \\ \dot H_0 h\,\operatorname{ch}\xi_0\sin\eta - \sum_{\kappa=1}^{\infty}\kappa B_\kappa e^{-\kappa\xi_0}\sin\kappa\eta &= \frac{1}{j\omega\mu_e} \sum_{n=1}^{\infty} D_n Ce_n(\xi_0)\,ce'_n(\eta). \end{aligned} \right\} \tag{28} \]

In (28), and in all subsequent cases, summation is performed only over odd \(\kappa, r\), and \(n\).

It can be shown that for imaginary \(q\)

\[ \begin{aligned} \cos \kappa \eta &= \sum_{n=1}^{\infty} A_{\kappa}^{\prime n}\, ce_n(\eta);\\ \sin \kappa \eta &= - \sum_{n=1}^{\infty} A_{\kappa}^{\prime n}\, ce_n'(\eta), \end{aligned} \tag{29} \]

where

\[ A_{\kappa}^{\prime n}=\frac{A_{\kappa}^{n}}{\Delta_n}, \qquad \Delta_n=\sum_{r=1}^{\infty} |A_r^n|^2 . \]

Substituting (29) into (28), we obtain

\[ \left. \begin{aligned} &\sum_{\kappa=1}^{\infty}\sum_{n=1}^{\infty} \kappa B_{\kappa} A_{\kappa}^{\prime n} e^{-\kappa \xi_0} ce_n(\eta) +\frac{1}{j\omega\mu_i}\sum_{n=1}^{\infty} D_n Ce_n'(\xi_0) ce_n(\eta) = \\ &\hspace{3.5cm} = -\dot H_0 h\, \operatorname{sh}\xi_0 \sum_{n=1}^{\infty} A_1^{\prime n} ce_n(\eta),\\[1.0em] &\sum_{\kappa=1}^{\infty}\sum_{n=1}^{\infty} \kappa B_{\kappa} A_{\kappa}^{\prime n} e^{-\kappa \xi_0} ce_n(\eta) -\frac{1}{j\omega\mu_e}\sum_{n=1}^{\infty} D_n Ce_n'(\xi_0) ce_n(\eta) = \\ &\hspace{3.5cm} = \dot H_0 h\, \operatorname{ch}\xi_0 \sum_{n=1}^{\infty} A_1^{\prime n} ce_n(\eta). \end{aligned} \right\} \tag{30} \]

System (30) is satisfied if, for any \(n\),

\[ \left. \begin{aligned} &\sum_{\kappa=1}^{\infty} \kappa B_{\kappa} A_{\kappa}^{\prime n} e^{-\kappa \xi_0} +\frac{1}{j\omega\mu_i}D_n Ce_n'(\xi_0) = -\dot H_0 h\, \operatorname{sh}\xi_0 A_1^{\prime n};\\[0.8em] &\sum_{\kappa=1}^{\infty} \kappa B_{\kappa} A_{\kappa}^{\prime n} e^{-\kappa \xi_0} -\frac{1}{j\omega\mu_e}D_n Ce_n'(\xi_0) = \dot H_0 h\, \operatorname{ch}\xi_0 A_1^{\prime n}, \end{aligned} \right\} \tag{31} \]

whence

\[ D_n=-\dot H_0 \frac{j\omega\mu_e\mu_i h e^{\xi_0} A_1^{\prime n}} {\mu_e Ce_n'(\xi_0)+\mu_i Ce_n'(\xi_0)}, \tag{32} \]

\[ \sum_{\kappa=1}^{\infty} \kappa B_{\kappa} A_{\kappa}^{\prime n} e^{-\kappa \xi_0} = \frac{\dot H_0 h A_1^{\prime n} \left[\mu_e \operatorname{ch}\xi_0\, Ce_n'(\xi_0) -\mu_i \operatorname{sh}\xi_0\, Ce_n(\xi_0)\right]} {\mu_e Ce_n'(\xi_0)+\mu_i Ce_n'(\xi_0)} . \tag{33} \]

In investigating the exterior region, the coefficients \(B_{\kappa}\) can be determined, taking (33) into account, from an infinite system of equations for various \(n\).

For the case of a copper conductor one may take \(\mu_e=\mu_i=\mu_{0*}\); then

\[ D_n=-\dot H_0 \frac{j\omega\mu_0 h e^{\xi_0} A_1^{\prime n}} {Ce_n'(\xi_0)+Ce_n(\xi_0)}, \tag{33} \]

or

\[ D_n=-\dot H_0 \frac{j\omega\mu_0(a+b)A_1^{\prime n}} {Ce_n'(\xi_0)+Ce_n(\xi_0)} . \tag{34} \]

The current density in the conductor is

\[ \dot{\delta}=\dot{H}_0\varkappa_1^{\,2}(a+b)\sum_{n=1}^{\infty} A_1^{\prime n} \frac{Ce_n(\xi)\,ce_n(\eta)} {Ce_n'(\xi_0)+Ce_n(\xi)} . \tag{35} \]

The normal component of the magnetic-field intensity inside the conductor is

\[ \dot{H}_{\xi_i}=\dot{H}_0(a+b)\frac{1}{l}\sum_{n=1}^{\infty} A_1^{\prime n} \frac{Ce_n(\xi)\,ce_n'(\eta)} {Ce_n'(\xi_0)+Ce_n(\xi_0)} . \tag{36} \]

The ratio of \(\dot{H}_{\xi_i}\) to \(-\dot{H}_0\) at the point \(\xi=0,\ \eta=\dfrac{\pi}{2}\), which determines the maximum weakening of the external field by the field of the eddy currents, is

\[ \left[ \frac{\dot{H}_{\xi_i}}{\dot{H}_0} \right]_{\substack{\xi=0\\ \eta=\frac{\pi}{2}}} = -\,e^{\xi_0}\sum_{n=1}^{\infty} A_1^{\prime n}(q) \frac{Ce_n(0,q)\,ce_n'\!\left(\dfrac{\pi}{2},q\right)} {Ce_n'(\xi_0,q)+Ce_n(\xi_0,q)} . \tag{37} \]

The Mathieu functions entering into (37) depend on \(q\), which makes it possible to analyze

\[ \left[ \frac{\dot{H}_{\xi_i}}{\dot{H}_0} \right]_{\substack{\xi=0\\ \eta=\frac{\pi}{2}}} \]

as a function of the frequency and the dimensions of the conductor.

Let us determine the limiting values of \(q\) for a copper printed conductor.

Taking the magnetic permeability of copper as \(\mu_{\mathrm{M}}=\mu_0=4\pi\cdot10^{-9}\ \dfrac{\mathrm{H}}{\mathrm{cm}}\), the specific electrical conductivity as \(\gamma_{\mathrm{M}}=5.6\cdot10^5\ \dfrac{1}{\Omega\cdot\mathrm{cm}}\), the angular frequency as \(\omega=670\ \dfrac{1}{\mathrm{s}}\) (which corresponds to \(p=4\cdot10^4\ \mathrm{rev/min}\)), and \(h\simeq a=0.1\ \mathrm{cm}\), we obtain for \(q\) the value

\[ q=-\,j\,\frac{\omega\mu\gamma h^2}{4}=-\,j\,0.08, \]

i.e., a very small quantity.

For small \(q\), formula (37) can be substantially simplified. Using expansions of the Mathieu functions in circular and hyperbolic functions [1] (indices only odd),

\[ \left. \begin{aligned} Ce_n(\xi,q)&=\sum_{k=1}^{\infty} A_k^{n}\operatorname{ch} k\xi,\\ ce_n(\eta,q)&=\sum_{k=1}^{\infty} A_k^{n}\cos k\eta, \end{aligned} \right\} \tag{38} \]

and the expressions for the coefficients \(A_k^n\) for small \(q\) \((r\geqslant0,\ m>0)\),

\[ \left. \begin{aligned} A_{m+2r}^{m}&\simeq(-1)^r \frac{m!}{r!(m+r)!}\left(\frac{q}{4}\right)^r,\\ A_{m-2r}^{m}&\simeq \frac{(m-r-1)!}{r!(m-1)!}\left(\frac{q}{4}\right)^r \end{aligned} \right\} \tag{39} \]

and, retaining terms with powers of \(\left(\dfrac{q}{4}\right)\) no higher than the first in (37), we obtain

\[ \left[ \frac{\dot H_{\xi_i}}{\dot H_0} \right]_{\substack{\xi=0\\ \eta=-\frac{\pi}{2}}} \simeq \frac{1-\dfrac{q}{4}}{1-\dfrac{q}{4} e^{2\xi_0}} -\frac{3}{4} e^{-2\xi_0}\left(\frac{q}{4}\right), \tag{40} \]

\[ \xi_0 = \operatorname{Arth}\frac{b}{a}. \tag{41} \]

It follows from (40) that even in the case of the limiting value \(q=-j0.08\) the ratio

\[ \left[ \frac{\dot H_{\xi_i}}{\dot H_0} \right]_{\substack{\xi=0\\ \eta=-\frac{\pi}{2}}} \]

practically does not differ from 1, i.e., there is no distortion of the external field. This makes it possible, when calculating eddy-current losses in the printed conductors of DC machines in the absence of steel in the armature, to disregard the influence of the eddy-current reaction on the main excitation field of the machine.

REFERENCES

1. N. W. McLachlan. Theory and Applications of Mathieu Functions. IIL, Moscow, 1953.