On the Calculation of the Stray Magnetic Field in Betatrons
Abstract Generated abstract
This paper addresses the calculation of magnetic stray flux in betatron magnetic circuits, a factor needed for sizing yokes and columns and avoiding local saturation. It argues that the field outside the annular working gap can, to practical accuracy, be modeled as the field of a finite-height current busbar, with corrections for finite iron permeability and for nearby excitation windings. Formulae are derived for the field components and for the resulting stray flux along the pole perimeter, including a simplified expression when the winding influence is negligible. The work provides a practical computational scheme for incorporating extra-gap magnetic flux into betatron core design.
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On the Calculation of the Magnetic Stray Field in Betatrons
B. B. Gel’perin
(Presented by the scientific seminar of the Physico-Technical Faculty)
For the correct calculation of the cross sections of the yoke and columns of the magnetic circuit of a betatron, it is necessary to take into account the magnetic stray flux of the betatron, which closes outside the annular working gap onto the cylindrical parts of the core. This flux is channeled through the core of the yoke and the columns and, if it is not taken into account, may cause supersaturation of individual elements of the magnetic circuit.
The author’s theoretical investigations have shown that the field of the extra-annular working gap can, with sufficient accuracy for practice, be represented as the field of a busbar of height equal to the gap \(2\delta\), with current
\[ J = 0.8\,B\,2\delta \cdot 10^{-1} = 0.16\,B\delta, \tag{1} \]
where \(B\) is the induction in the gap \(2\delta\) (Fig. 1a and b).
Fig. 1
Taking into account that the pole radius is sufficiently large, one may consider a plane-parallel field, and then the field of the busbar \(2\delta\) (Fig. 1b) can be calculated as follows:
\[ H_x = \frac{2J}{\delta}\ln \frac{\rho_2}{\rho_1}, \tag{2} \]
\[ H_y = -\frac{2J}{\delta}\alpha. \tag{3} \]
In (2) and (3) the coefficient 2 takes into account the reflection of the bus from iron with permeability \(\mu=\infty\). For finite \(\mu\), the two must be replaced by the coefficient
\[ \frac{2\mu-1}{\mu}. \tag{4} \]
If the exciting windings are located sufficiently close to the gap and to the core, then their influence must be taken into account (Fig. 2a, 2b, and 2c).
Fig. 2
From the figures shown it is seen that Fig. 2a is transformed into Fig. 2c, and the latter, in turn, is transformed into Fig. 3.
The field strength at the point \(M(xy)\) from each coil of Fig. 2c is
\[ H_y = -\frac{J\cdot 10^{-1}}{2ab} \left\{ (x' + a)(\beta'_2-\alpha'_2) - (x' - a)(\beta'_1-\alpha'_1) + (y' + b)\ln\frac{\rho'_2}{\rho'_1} - (y' - b)\ln\frac{r'_2}{r'_1} \right\}, \tag{5} \]
\[ H_x = -\frac{J\cdot 10^{-1}}{2ab} \left\{ (y' + b)(\beta'_1-\beta'_2) - (y' - b)(\alpha'_1-\alpha'_2) + (x' + a)\ln\frac{\rho'_2}{r'_2} - (x' - a)\ln\frac{\rho'_1}{r'_1} \right\}, \tag{6} \]
— ampere-turns of the exciting winding.
For the reflected (left) coil, if the permeability of the iron is finite, it is necessary to multiply equations (5 and 6) by the coefficient
\[ \frac{\mu-1}{\mu}. \tag{7} \]
Fig. 3
For the case when the influence of the excitation winding is small, it may be assumed that it is removed to infinity; then \(H_x\) and \(H_y\), according to equations 5 and 6, become zero.
Fig. 4
In this case the magnetic stray flux \((\Phi_p)\) is readily calculated along the \(y\)-axis as follows (Eq. 2):
\[ \Phi_p = p \int_{\delta}^{y} H_x\,dy = \frac{2J}{\delta}\,p \int_{\delta}^{n\delta} \ln \frac{\rho_2}{\rho_1}\,dy, \]
where \(p\) is the perimeter of the pole, and \(y = n\delta\).
Taking into account that \(\rho_2 \simeq y + \delta\) and \(\rho_1 = y - \delta\), we obtain
\[ \Phi_p = 2Jp\left[(n+1)\ln(n+1) - (n-1)\ln(n-1) - 1.4\right] = 2JpA, \tag{6} \]
where
\[ A = (n+1)\ln(n+1) - (n-1)\ln(n-1) - 1.4. \tag{7} \]
Figure 4 gives the curve \(A = f(n)\).