ELECTROMAGNETIC PHENOMENA IN THE AIR GAP OF AN INDUCTION MACHINE WITH AN OPEN MAGNETIC CIRCUIT

Yu. B. Volynsky

(Presented by the scientific seminar of the PTF of TPI)

The distribution of the magnetic field in the nonmagnetic gap of a flat linear induction machine (Fig. 1) in the general case depends on the width \(2a\) and thickness \(b\) of the layer of molten metal II with electrical conductivity \(\sigma\), moving with velocity \(v\) in a metallic channel I with thickness of the end walls \(\dfrac{d-b}{2}\) and electrical conductivity of the walls \(\sigma_{\text{st}}\), on the thickness \(\dfrac{\Delta-d}{2}\) of the thermal-insulation layer III, on the width of the inductor core \(2c\), and on the linear current loading \(A\) on the surface of each inductor.

Another filling of the nonmagnetic gap is also possible, either when the channel is made of ceramic \((\sigma_{\text{st}}=0)\), or when the thermal-insulation layer is absent \((\Delta=d)\).

Fig. 1. Diagram of an induction machine with an open magnetic circuit

At present there is a considerable number of works in which the distribution of the magnetic field in the gap of an induction machine with an open magnetic circuit is considered under various assumptions [1–8]. One of the first problems concerning the distribution of the field in a layer of liquid metal and in the electrically conducting walls of the channel was solved by I. A. Tyutin and E. K. Yankop [1]. In doing so, they pre-

It was assumed that the inductors and the channel have infinite width, and that the normal component of the resultant magnetic flux density is specified on the surface of the inductors; however, it was more convenient to assume a specified linear current loading, or an equivalent linear current density on the surfaces of the inductors [2].

In works carried out by Ya. Ya. Valdmanis, Ya. Ya. Liepeter, and Yu. Ya. Mikelson [3], electromagnetic phenomena in a conducting strip of infinite width were investigated with allowance for the higher spatial harmonics of the magnetic field. The assumption that the inductors have infinitely large width simplifies the study of the field distribution in the moving conducting layer, since in this case it is no longer necessary to take into account the transverse end effect; however, such an assumption is valid only for channels with longitudinal short-circuiting buses.

Extensive theoretical investigations carried out by A. I. Voldek and Kh. I. Yanes, under the assumption that the magnetic field in the gap is plane-parallel, made it possible to develop an engineering method for calculating induction machines with an open magnetic circuit [4–5], giving good agreement between calculated and experimental data for relatively small gaps and when using liquid metals with low electrical conductivity.

Solutions of the electromagnetic-field equations for a single-layer induction machine, taking into account secondary transverse and thickness end effects, were given by N. M. Okhremenko [6].

The most complete solution of the three-dimensional problem of the distribution of the electromagnetic field in the gap is given by N. M. Okhremenko in works [7–8]. In these works, the complex amplitudes of the resultant magnetic field in the layer of liquid metal, in the channel walls, and in the thermal-insulation layer are found, as well as the value of the magnetomotive force developed in the channel. All results were obtained under the assumption that the channel width is equal to the inductor width.

The case of a channel width greater than the inductor width is of undoubted interest; therefore, below we shall consider the problem of electromagnetic phenomena in such a channel with electrically conducting walls, under the assumption that the thermal-insulation layer is absent. The influence of a nonconducting thermal-insulation layer can be taken into account if the electrical conductivity of the channel walls is replaced by an equivalent conductivity that takes into account the thickness of the thermal-insulation layer.

We shall solve the problem under the following simplifying assumptions: a) the induction machine is infinitely long; b) the channel width \(2a\) is greater than the inductor width \(2c\); c) the magnetic permeability of the inductor steel is infinitely large; d) the surfaces of the channel walls perpendicular to \(oy\) are insulators; e) the liquid metal moves as a solid body with velocity \(v = v_x\); f) the inductor windings create only the fundamental harmonic of the magnetomotive force, and the linear current loading on the surface of the inductors varies according to the law \(A = A_m e^{j(\omega t-\alpha x)}\), where \(\alpha = \dfrac{\pi}{\tau}\) (\(\tau\) is the pole pitch). The amplitude value of the linear current loading on the surface of an inductor of width \(2c\) can be represented as a Fourier series with period \(2a\).

The solution of the posed problem reduces to finding the magnetic-flux-density vectors \(\dot{B} = \dot{B}_m e^{j(\omega t-\alpha x)}\) and the current-density vectors

\[ \dot I = I_m e^{j(\omega t-\alpha x)}, \]

which characterize the resultant electromagnetic field in the region of the liquid metal and in the channel walls, and to establish expressions for the power losses.

For slowly moving conducting media, the system of Maxwell equations can be reduced to the following inhomogeneous equations:

\[ \nabla^2 \dot B_{\mathrm{I}} = j\mu_0 \sigma_{\mathrm{st}}\omega \dot B_{\mathrm{I}}, \tag{1} \]

\[ \nabla^2 \dot I_{\mathrm{I}} = j\mu_0 \sigma_{\mathrm{st}}\omega \dot I_{\mathrm{I}}, \tag{2} \]

\[ \nabla^2 \dot B_{\mathrm{II}} = j\mu_0 \sigma \omega S \dot B_{\mathrm{II}}, \tag{3} \]

\[ \nabla^2 \dot I_{\mathrm{II}} = j\mu_0 \sigma \omega S \dot I_{\mathrm{II}} . \tag{4} \]

The subscript I denotes quantities relating to the channel walls, and the subscript II to the layer of liquid metal.

The simultaneous solution of equations (1–4) made it possible to obtain expressions for the amplitude values of the components \(\dot B\) and \(\dot I\)

\[ \dot B_{m\mathrm{I}x} = A \sum_{\nu=0}^{\infty} B_\nu \frac{\dot D_\nu \operatorname{sh}\dot\gamma_\nu z+\dot E_\nu \operatorname{ch}\dot\gamma_\nu z}{\dot c_\nu} \cos \beta_\nu y, \tag{5} \]

\[ \dot B_{m\mathrm{I}y} = -\,j\frac{A}{a}\sum_{\nu=0}^{\infty} B_\nu \frac{\dot D_\nu \operatorname{sh}\dot\gamma_\nu z+\dot E_\nu \operatorname{ch}\dot\gamma_\nu z}{\dot c_\nu} \sin \beta_\nu y, \tag{6} \]

\[ \dot B_{m\mathrm{I}z} = j\frac{A}{a}\sum_{\nu=0}^{\infty} B_\nu \frac{\dot D_\nu \operatorname{ch}\dot\gamma_\nu z+\dot E_\nu \operatorname{sh}\dot\gamma_\nu z}{\dot c_\nu} \cos \beta_\nu y. \tag{7} \]

\[ \dot B_{m\mathrm{II}x} = A \sum_{\nu=0}^{\infty} B_\nu \frac{\dot\delta_\alpha}{\dot C_\nu}\cdot \operatorname{sh}\dot\delta_\nu z \cos \beta_\nu y, \tag{8} \]

\[ \dot B_{m\mathrm{II}y} = -\,j\frac{A}{a}\sum_{\nu=0}^{\infty} B_\nu \frac{\dot\delta_\nu \beta_\nu}{\dot c_\nu}\cdot \operatorname{sh}\dot\delta_\nu z \sin \beta_\nu y, \tag{9} \]

\[ \dot B_{m\mathrm{II}z} = j\frac{A}{a}\sum_{\nu=0}^{\infty} B_\nu \frac{\left(\alpha^2+\beta_\nu^2\right)}{\dot c_\nu}\cdot \operatorname{ch}\dot\delta_\nu z \cos \beta_\nu y, \tag{10} \]

\[ \dot I_{m\mathrm{I}x} = -\frac{\sigma_{\mathrm{st}}\omega}{a}A \sum_{\nu=0}^{\infty} B_\nu \frac{\beta_\nu}{\dot\gamma_\nu} \frac{\dot D_\nu \operatorname{ch}\dot\gamma_\nu z+\dot E_\nu \operatorname{sh}\dot\gamma_\nu z}{\dot c_\nu} \sin \beta_\nu y, \tag{11} \]

\[ \dot I_{m\mathrm{I}y} = j\sigma_{\mathrm{st}}\omega A \sum_{\nu=0}^{\infty} B_\nu \frac{\dot D_\nu \operatorname{ch}\dot\gamma_\nu z+\dot E_\nu \operatorname{sh}\dot\gamma_\nu z}{\dot\gamma_\nu \dot c_\nu} \cos \beta_\nu y, \tag{12} \]

\[ \dot I_{m\mathrm{II}x} = -\frac{\sigma\omega S A}{a} \sum_{\nu=0}^{\infty} B_\nu \frac{\beta_\nu}{\dot c_\nu} \operatorname{ch}\dot\delta_\nu z \sin \beta_\nu y, \tag{13} \]

\[ \dot I_{m\mathrm{II}y} = j\sigma\omega S A \sum_{\nu=0}^{\infty} B_\nu \frac{1}{\dot c_\nu} \operatorname{ch}\dot\delta_\nu z \cos \beta_\nu y, \tag{14} \]

\[ \dot I_{m\mathrm{I}z}=\dot I_{m\mathrm{II}z}=0, \]

where

\[ A=\frac{4}{\pi}\mu_0 A_m,\qquad \beta_\nu=\frac{2\nu+1}{2a}\pi,\qquad B_\nu=\frac{\sin \beta_\nu c}{(2\nu+1)} . \]

\[ \dot{\gamma}_{\nu}^{\,2}=\alpha^2+\beta_{\nu}^{2}+j\mu_0\sigma_{\mathrm{st}}\omega,\qquad \dot{\delta}_{\nu}^{\,2}=\alpha^2+\beta_{\nu}^{2}+j\mu_0\sigma\omega s, \tag{15} \]

\[ \dot{C}_{\nu}=\dot{\gamma}_{\nu}\operatorname{ch}\dot{\delta}_{\nu}\frac{b}{2}\operatorname{sh}\dot{\gamma}_{\nu}\frac{(d-b)}{2} +\dot{\delta}_{\nu}\operatorname{sh}\dot{\delta}_{\nu}\frac{b}{2}\operatorname{ch}\dot{\gamma}_{\nu}\frac{(d-b)}{2}, \tag{16} \]

\[ \dot{D}_{\nu}=\dot{\gamma}_{\nu}\operatorname{ch}\dot{\gamma}_{\nu}\frac{b}{2}\operatorname{ch}\dot{\delta}_{\nu}\frac{b}{2} -\dot{\delta}_{\nu}\operatorname{sh}\dot{\gamma}_{\nu}\frac{b}{2}\operatorname{sh}\dot{\delta}_{\nu}\frac{b}{2}, \tag{17} \]

\[ \dot{E}_{\nu}=\dot{\delta}_{\nu}\operatorname{sh}\dot{\delta}_{\nu}\frac{b}{2}\operatorname{ch}\dot{\gamma}_{\nu}\frac{b}{2} -\dot{\gamma}_{\nu}\operatorname{sh}\dot{\gamma}_{\nu}\frac{b}{2}\operatorname{ch}\dot{\delta}_{\nu}\frac{b}{2}. \tag{18} \]

The powers transmitted to the layer of molten metal and to the channel walls were found using the Umov--Poynting theorem:

\[ P_{\mathrm{II}}=\frac{1}{2}\frac{A^2}{\mu_0\alpha^2}\cdot\omega la \sum_{\nu=0}^{\infty}(\alpha^2+\beta_{\nu}^{2})B_{\nu}^{2} \cdot \frac{\psi_{\nu}\operatorname{sh}\varphi_{\nu}b+\varphi_{\nu}\sin\psi_{\nu}b}{|\dot{C}_{\nu}|^2}, \tag{19} \]

\[ Q_{\mathrm{II}}=\frac{1}{2}\frac{A^2}{\mu_0\alpha^2}\omega la \sum_{\nu=0}^{\infty}(\alpha^2+\beta_{\nu}^{2})B_{\nu}^{2} \cdot \frac{\varphi_{\nu}\operatorname{sh}\varphi_{\nu}b+\psi_{\nu}\sin\psi_{\nu}b}{|\dot{C}_{\nu}|^2}, \tag{20} \]

\[ P_{\mathrm{I}}=\frac{1}{2}\frac{A^2}{\mu_0\alpha^2}\omega la \sum_{\nu=0}^{\infty} \frac{(\alpha^2+\beta_{\nu}^{2})B_{\nu}^{2}} {(\xi_{\nu}^{2}+\eta_{\nu}^{2})|\dot{C}_{\nu}|^2} \times \]

\[ \times\Bigl\{ \eta_{\nu}\bigl[(|\dot{D}_{\nu}|^2+|\dot{E}_{\nu}|^2)\cdot(\operatorname{sh}\xi_{\nu}d-\operatorname{sh}\xi_{\nu}b) +(\dot{D}_{\nu}\dot{E}_{\nu}^{*}+\dot{E}_{\nu}\dot{D}_{\nu}^{*})\cdot(\operatorname{ch}\xi_{\nu}d-\operatorname{ch}\xi_{\nu}b)\bigr]+ \]

\[ +\xi_{\nu}\bigl[(|\dot{D}_{\nu}|^2-|\dot{E}_{\nu}|^2)\cdot(\sin\eta_{\nu}d-\sin\eta_{\nu}b) +j(\dot{D}_{\nu}\dot{E}_{\nu}^{*}-\dot{E}_{\nu}\dot{D}_{\nu}^{*})\cdot (\cos\eta_{\nu}d-\cos\eta_{\nu}b)\bigr]\Bigr\} \tag{21} \]

\[ Q_{\mathrm{I}}=\frac{1}{2}\frac{A^2}{\mu_0\alpha^2}\omega la \sum_{\nu=0}^{\infty} \frac{(\alpha^2+\beta_{\nu}^{2})B_{\nu}^{2}} {(\xi_{\nu}^{2}+\eta_{\nu}^{2})|\dot{C}_{\nu}|^2} \times \]

\[ \times\Bigl\{ \xi_{\nu}\bigl[(|\dot{D}_{\nu}|^2-|\dot{E}_{\nu}|^2)\cdot(\operatorname{sh}\xi_{\nu}d-\operatorname{sh}\xi_{\nu}b) +(\dot{D}_{\nu}\dot{E}_{\nu}^{*}+\dot{E}_{\nu}\dot{D}_{\nu}^{*})\cdot(\operatorname{ch}\xi_{\nu}d-\operatorname{ch}\xi_{\nu}b)\bigr]- \]

\[ -\eta_{\nu}\bigl[(|\dot{D}_{\nu}|^2-|\dot{E}_{\nu}|^2)\cdot(\sin\eta_{\nu}d-\sin\eta_{\nu}b) +j(\dot{D}_{\nu}\dot{E}_{\nu}^{*}-\dot{D}_{\nu}^{*}\dot{E}_{\nu})\times \]

\[ \times(\cos\eta_{\nu}d-\cos\eta_{\nu}b)\bigr]\Bigr\}. \tag{22} \]

Here

\[ \dot{\delta}_{\nu}=\varphi_{\nu}+j\psi_{\nu},\qquad \dot{\gamma}_{\nu}=\xi_{\nu}+j\eta_{\nu}, \]

\[ |\dot{C}_{\nu}|^2=\frac{1}{4}\Bigl\{ (\xi_{\nu}^{2}+\eta_{\nu}^{2})(\operatorname{ch}\varphi_{\nu}b+\cos\psi_{\nu}b) [\operatorname{ch}\xi_{\nu}(d-b)-\cos\eta_{\nu}(d-b)]+ \]

\[ +(\varphi_{\nu}^{2}+\psi_{\nu}^{2})(\operatorname{ch}\varphi_{\nu}b-\cos\psi_{\nu}b) [\operatorname{ch}\xi_{\nu}(d-b)+\cos\eta_{\nu}(d-b)]+ \]

\[ +2(\varphi_{\nu}\xi_{\nu}+\eta_{\nu}\psi_{\nu}) [\operatorname{sh}\varphi_{\nu}b\,\operatorname{sh}\xi_{\nu}(d-b) +\sin\psi_{\nu}b\,\sin\eta_{\nu}(d-b)]+ \]

\[ +2(\xi_{\nu}\psi_{\nu}-\eta_{\nu}\varphi_{\nu}) [\sin\psi_{\nu}b\,\operatorname{sh}\xi_{\nu}(d-b) -\operatorname{sh}\varphi_{\nu}b\,\sin\eta_{\nu}(d-b)]\Bigr\}, \tag{23} \]

\[ |\dot{D}_{\nu}|^2+|\dot{E}_{\nu}|^2= \frac{1}{2}\Bigl\{ \operatorname{ch}\xi_{\nu}b\bigl[(\varphi_{\nu}^{2}+\psi_{\nu}^{2})(\operatorname{ch}\varphi_{\nu}b-\cos\psi_{\nu}b)+ \]

\[ +(\xi_{\nu}^{2}+\eta_{\nu}^{2})(\operatorname{ch}\varphi_{\nu}b+\cos\psi_{\nu}b)\bigr] -2\operatorname{sh}\xi_{\nu}b\bigl[(\xi_{\nu}\varphi_{\nu}+\eta_{\nu}\psi_{\nu})\times \]

\[ \times\operatorname{sh}\varphi_{\nu}b+(\eta_{\nu}\varphi_{\nu}-\xi_{\nu}\psi_{\nu})\cdot\sin\psi_{\nu}b\bigr]\Bigr\} \tag{24} \]

\[ |\dot D_\nu|^2-|\dot E_\nu|^2= \frac{1}{2}\{\cos \eta_\nu b[(\xi_\nu^2+\eta_\nu^2)(\operatorname{ch}\varphi_\nu b+\cos \psi_\nu b) -(\varphi_\nu^2+\psi_\nu^2)\times \]
\[ \times(\operatorname{ch}\varphi_\nu b-\cos \psi_\nu b)] -2\sin \eta_\nu b[(\eta_\nu\varphi_\nu-\xi_\nu\psi_\nu)\operatorname{sh}\varphi_\nu b -(\xi_\nu\varphi_\nu+\eta_\nu\psi_\nu)\sin\psi_\nu b]\}, \tag{25} \]

\[ \dot D_\nu \dot E_\nu^{*}+\dot D_\nu^{*}\dot E_\nu= \frac{1}{2}\{2\operatorname{ch}\xi_\nu b[(\xi_\nu\varphi_\nu+\eta_\nu\psi_\nu)\operatorname{sh}\varphi_\nu b +(\eta_\nu\varphi_\nu-\xi_\nu\psi_\nu)\sin\psi_\nu b]- \]
\[ -\operatorname{sh}\xi_\nu b[(\xi_\nu^2+\eta_\nu^2)\cdot(\operatorname{ch}\varphi_\nu b+\cos\psi_\nu b) +(\varphi_\nu^2+\psi_\nu^2)\cdot(\operatorname{ch}\varphi_\nu b-\cos\psi_\nu b)]\} \tag{26} \]

\[ \dot D_\nu \dot E_\nu^{*}-\dot D_\nu^{*}\dot E_\nu= \frac{j}{2}\{\sin \eta_\nu b[(\varphi_\nu^2+\psi_\nu^2)(\cos\psi_\nu b-\operatorname{ch}\varphi_\nu b) +(\xi_\nu^2+\eta_\nu^2)\times \]
\[ \times(\operatorname{ch}\varphi_\nu b+\cos\psi_\nu b)] +2\cos\eta_\nu b[\operatorname{sh}\varphi_\nu b(\eta_\nu\varphi_\nu-\xi_\nu\psi_\nu)- \]
\[ -\sin\psi_\nu b(\eta_\nu\psi_\nu+\xi_\nu\psi_\nu)]\} \tag{27} \]

The relations obtained, (19)—(22), can be used in constructing a refined equivalent circuit of an induction machine with an open magnetic circuit.

LITERATURE

1. I. A. Tyutin, E. K. Yankop. Electromagnetic processes in induction pumps for liquid metals. Applied Magnetohydrodynamics, Proceedings of the Institute of Physics, Academy of Sciences of the Latvian SSR, issue VIII, 1956.

2. Ya. Ya. Liepeter, R. A. Petrovich. On the theory of flat induction pumps. Izvestiya of the Academy of Sciences of the Latvian SSR, Series of Physical and Technical Sciences, 1964, No. 1.

3. Ya. Ya. Valdmanis, Ya. Ya. Liepeter, Yu. Ya. Mikelson. Influence of higher spatial harmonics of the field on electrodynamic forces and Joule losses in a conducting strip moving in a traveling magnetic field. Izvestiya of the Academy of Sciences of the Latvian SSR, Series of Physical and Technical Sciences, 1965, No. 6.

4. A. I. Voldek. Currents and forces in the liquid-metal layer of flat induction pumps. Proceedings of Higher Educational Institutions, “Electrical Engineering,” 1959, No. 1.

5. A. I. Voldek, Kh. I. Yanes. Transverse end effect in a flat induction pump with an electrically conducting channel. Proceedings of the Tallinn Polytechnic Institute, Series A, 1962, No. 197.

6. N. M. Okremenko. Electromagnetic phenomena in flat induction pumps. “Electricity,” 1960, No. 3.

7. N. M. Okremenko. Investigation of the spatial distribution of magnetic fields and electromagnetic phenomena in induction pumps. “Magnetohydrodynamics,” 1965, No. 1.

8. N. M. Okremenko. Transverse end effect in flat linear induction pumps. “Magnetohydrodynamics,” 1965, No. 3.