On the Calculation of the Toroidal Magnetic Core of a Stereobetatron
Abstract Generated abstract
This paper examines the design and calculation of a toroidal magnetic core for a two-chamber stereobetatron, motivated by field nonuniformities and inefficient use of cold-rolled electrical steel in conventional core geometries. It argues that a toroidal core made from strip steel can align magnetic flux with the rolling direction, improve use of the steel’s magnetic properties, and help produce a more homogeneous field, while requiring attention to unequal magnetic path lengths across the core. The authors derive calculation formulas for magnetic-field intensity, yoke and pole dimensions, winding window area, magnetic flux, leakage coefficients, and induction in the pole and yoke steel, including variants for cylindrical and distributed windings. The result is a practical procedure for selecting and refining the dimensions and magnetic characteristics of the stereobetatron magnetic circuit.
Full Text
ON THE CALCULATION OF THE TOROIDAL MAGNETIC CORE OF A STEREOBETATRON
V. I. Kudryavchenko, V. A. Moskalev
In the electromagnets of betatrons with an iron magnetic core, the induction is limited by the magnetic properties of the electrical steels used for their manufacture.
There are several types of magnetic-core designs [1]: Ш-shaped, multi-post, shell-type, and, finally, the magnetic core of the rectangular type of a two-chamber stereobetatron [2].
The magnetic field in the interpolar space of the electromagnet differs from an azimuthally symmetric one owing to the design features of the magnetic core:
\[ H = [H_0 + \Delta H(\Theta)] \cdot \sin(\omega t + \varphi(\Theta)). \tag{1} \]
The deviation of the magnetic field from an azimuthally symmetric one causes distortion of the orbits and plays the greatest role at the moment of injection:
\[ H_i = H_0(\omega t_i)\left[1 + \frac{h(\Theta)}{H_0} + \frac{\varphi(\Theta)}{\omega t_i}\right], \tag{2} \]
where \(H_0\) is the amplitude of the field in the gap;
\[ \frac{h(\Theta)}{H_0} \]
is the relative amplitude nonuniformity;
\[ \frac{\varphi(\Theta)}{\omega t_i} \]
is the relative phase nonuniformity, determined by the inequality of hysteresis losses and eddy currents in different sections of the magnetic core.
Cold-rolled electrical steels possessing high magnetic properties are used for manufacturing magnetic cores. The magnetic properties of these steels vary depending on the orientation of the magnetic-flux path relative to the rolling direction [3]. Thus, losses at an angle of \(45^\circ\) to the rolling direction for steel with a ribbed texture increase twofold, and with a cubic texture by a factor of 1.5. Any electromagnet design has sections with different rolling directions, which substantially reduces the effectiveness of using cold-rolled steels. Thus, for example, in a Ш-shaped magnetic core these sections amount to more than 40%.
The most successful design of a magnetic core would be one in which the lines of force of the magnetic flux were directed along the rolling direction throughout their entire length.
These requirements are met by the proposed design of the stereobetatron magnetic core, called toroidal (Fig. 1). It is easily made from thin strip steel with a yoke cross section approximating a circle, which makes it possible to obtain a uniform loading of the yoke steel by the magnetic flux and to provide a homogeneous magnetic field that does not require special correction [4, 5]. Since the lines of force of the magnetic field in this design always run in the direction of easy magnetization, the high magnetic properties of cold-rolled steels are used most effectively.
Fig. 1
In a magnetic core of this type, nonuniformity of magnetization over the cross section may arise, since the length of the magnetic path along the inner and outer diameters is not the same, and the field intensity created by the magnetizing windings in these regions is different; this may lead to saturation of the inner regions and, consequently, to deterioration of the magnetic properties and to the appearance of inhomogeneities.
Therefore the calculation of a toroidal magnetic core must be carried out with this circumstance taken into account. The magnetic-field intensity for a toroidal-type core is determined by the expression [6]:
\[ H = \frac{ F_{\text{н.о}} \ln \dfrac{R_{\text{н}}}{R_{\text{в}}} }{ 2\pi \left(R_{\text{н}} - R_{\text{в}}\right) }, \tag{3} \]
where \(F_{\text{н.о}}\) is the magnetomotive force,
\(R_{\text{в}}\) is the inner radius of the core,
\(R_{\text{н}}\) is the outer radius of the core.
Let us denote the ratio \(\dfrac{R_{\mathrm{n}}}{R_{\mathrm{v}}}\) by \(\alpha\), and expression (3) is written as:
\[ H=\frac{F_{\mathrm{n.o}}\ln\alpha}{2\pi\left(R_{\mathrm{n}}-R_{\mathrm{v}}\right)}, \tag{4} \]
whence
\[ l=\frac{2\pi R_{\mathrm{v}}(\alpha-1)}{\ln\alpha}. \tag{5} \]
Usually the field intensity is determined at the mean radius:
\[ H_{\mathrm{av}}=\frac{F_{\mathrm{n.o}}}{\pi\left(R_{\mathrm{n}}+R_{\mathrm{v}}\right)} =\frac{F_{\mathrm{n.o}}}{\pi R_{\mathrm{v}}(\alpha+1)}. \tag{6} \]
Figure 2 gives the dependence \(\dfrac{H}{H_{\mathrm{av}}}=f(\alpha)\), calculated from the formula
\[ \frac{H}{H_{\mathrm{av}}}=\frac{\alpha+1}{2(\alpha-1)}\ln\alpha . \]
As is not difficult to see from Fig. 2, for the ratio \(\dfrac{R_{\mathrm{n}}}{R_{\mathrm{v}}}>2\) it is necessary to calculate the core according to expression (4), since calculation according to expression (6) may lead to saturation of the inner part of the core.
Fig. 2
The method for calculating the dimensions of the poles and of the interpole space is set forth in works [7, 8]. The pole height and the dimensions of the yoke of the magnetic circuit depend on the cross-sectional area of the coils of the magnetizing windings. The magnetizing force necessary to determine the dimensions of the coils is determined from the expression
\[ F_{\mathrm{n.o}}=0.8\,H_{\mathrm{om}}\cdot 2\delta_{0}+4\pi h_{\mathrm{p}}H_{\mathrm{st.p}}+l_{\mathrm{core}}\cdot H_{\mathrm{st.core}}. \tag{7} \]
We assume that the magnetic-field intensity in the steel of the yoke is twice as high as in the steel of the pole; then:
\[ r_{\mathrm{y}}=\frac{r_{\mathrm{p}}}{\sqrt{2}}. \]
The cross-sectional area of the window in a toroidal magnetic core is found from the relation
\[ S_{\mathrm{w}}=\frac{\pi R_{\mathrm{in}}^{2}}{2}+2h_{\mathrm{p}}R_{\mathrm{in}} =\frac{F_{\mathrm{w}}}{2\sqrt{2}\,j_{\mathrm{w}}}, \tag{8} \]
\[ j_{\mathrm{w}}=K_{0}j_{2}, \tag{9} \]
where \(j_{\mathrm{w}}\) is the current density in the window,
\(j_{2}\) is the current density in the copper of the winding,
\(K_{0}\) is the coefficient of filling the window with copper.
Express \(R_{\mathrm{out}}\) through \(r_{\mathrm{p}}\):
\[ R_{\mathrm{out}}=R_{\mathrm{in}}+2\frac{r_{\mathrm{p}}}{\sqrt{2}}. \tag{10} \]
From expression (3)
\[ l_{\mathrm{avg}}= \frac{2\pi\left(R_{\mathrm{out}}-R_{\mathrm{in}}\right)} {\ln \dfrac{R_{\mathrm{out}}}{R_{\mathrm{in}}}}. \tag{11} \]
Hence, substituting (10) into (11), we obtain:
\[ l_{\mathrm{avg}}=\frac{4\pi r_{\mathrm{p}}}{\sqrt{2}\ln\alpha}. \tag{12} \]
Thus, from (8)
\[ \frac{\pi R_{\mathrm{in}}^{2}}{2}+2h_{\mathrm{p}}R_{\mathrm{in}} = \frac{ 0.8H_{\mathrm{om}}\cdot 2\delta_{0} +4h_{\mathrm{p}}H_{\mathrm{st.p}} +\dfrac{4\pi r_{\mathrm{p}}}{\ln\alpha\cdot\sqrt{2}}\cdot H_{\mathrm{st.y}} } {2\sqrt{2}\,j_{\mathrm{w}}}. \tag{13} \]
Solving (13) with respect to \(R_{\mathrm{in}}\), we find:
\[ R_{\mathrm{in}}= \frac{-K_{2}+\sqrt{K_{2}^{2}+4K_{1}K_{3}}}{2K_{1}}, \tag{14} \]
where
\[ K_{1}=2\pi\ln\alpha\,j_{\mathrm{w}}, \]
\[ K_{2}=8\ln\alpha\,j_{\mathrm{w}}h_{\mathrm{p}}, \]
\[ K_{3}=1.6\sqrt{2}\ln\alpha\cdot\delta_{0}\cdot H_{\mathrm{om}} +4\sqrt{2}\ln\alpha\,h_{\mathrm{p}}H_{\mathrm{st.p}} +4\pi r_{\mathrm{p}}H_{\mathrm{st.y}}. \]
Expression (14) has been obtained under the condition that the magnetizing winding is made of the cylindrical type, and the magnetic core has the form of a toroid with rectilinear sections formed by the poles.
In the case where the manufacture of a distributed winding is envisaged, it is advisable to make the magnetic core in such a way that the yokes and poles form the shape of a regular toroid, but to take the ratio
\[ \frac{R_{\mathrm{out}}}{R_{\mathrm{in}}}=\alpha \]
smaller, since for large values of \(\alpha\) an asymmetric construction is obtained. The leakage fluxes in such a construction increase, and the nonuniformities of the magnetic field increase.
For the case of a distributed winding, expression (8) is written as:
\[ \frac{\pi R_{\mathrm{in}}^{2}}{2} = \frac{ 0.8H_{\mathrm{om}}2\delta_{0} +\dfrac{4\pi r_{\mathrm{p}}}{\ln\alpha\sqrt{2}}\cdot H_{\mathrm{st.y}} } {2\sqrt{2}\,j_{\mathrm{w}}}. \tag{15} \]
Whence
\[ R_{\mathrm{in}}= \sqrt{ \frac{ 1.6\sqrt{2}\ln\alpha H_{\mathrm{om}}\delta_{0} +4\pi r_{\mathrm{p}} } {2\pi\ln\alpha\,j_{\mathrm{w}}} }. \tag{16} \]
The magnetic flux between the poles in a circle of radius \(r_{\mathrm{n}}\) is determined by [1]:
\[ \Phi_{\mathrm{n}}=2\pi r_0^2\cdot H_{\mathrm{om}}+ \frac{2\pi r_0^n H_{\mathrm{om}}}{2-n}\left(r_{\mathrm{n}}^{2-n}-r_0^{2-n}\right). \tag{17} \]
The leakage coefficient of the magnetic flux of the poles is
\[ \sigma_{\mathrm{n}}=1+0.56\,\frac{\delta_0}{r_0}. \tag{18} \]
The total magnetic flux in the pole core is
\[ \Phi_{\mathrm{p}}=\sigma_{\mathrm{n}}\cdot\Phi_{\mathrm{n}}. \tag{19} \]
The magnetic flux in the mean section of the yoke is
\[ \Phi_{\mathrm{y}}=\sigma_0\cdot\Phi_{\mathrm{p}}, \tag{20} \]
where \(\sigma_0\) is a coefficient taking into account the leakage magnetic flux of the magnetizing windings.
The magnetic induction in the steel of the poles is:
\[ B_{\mathrm{p}}=\frac{\Phi_{\mathrm{p}}}{\pi r_{\mathrm{n}}^2 K_{\mathrm{p}}}, \tag{21} \]
where \(K_{\mathrm{p}}\) is the coefficient of filling of the pole with steel.
The magnetic induction in the yoke steel is
\[ B_{\mathrm{y}}=\frac{4\Phi_{\mathrm{y}}}{\pi\,[R_{\mathrm{v}}(\alpha-1)]^2 K_{\mathrm{y}}}, \tag{22} \]
where \(K_{\mathrm{y}}\) is the coefficient of filling of the yoke with steel.
Thus, all dimensions of the magnetic circuit and its magnetic characteristics have been determined. Since assumptions were made in the calculations, it is accepted that
\[ B_{\mathrm{y}}=2B_{\mathrm{p}}, \tag{23} \]
and, after an arbitrary choice of \(\alpha\) has been made, a refined calculation of the magnetic circuit is carried out.
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