On the Calculation of the Profile of the Pole Tips of a Betatron Electromagnet
Abstract Generated abstract
This paper addresses the design of betatron electromagnet pole tips needed to produce a radial magnetic-field distribution that provides stable axial and radial electron focusing while satisfying the betatron relation. It derives calculation methods based on treating the pole-tip surfaces as magnetic equipotentials and using the air-gap profile to approximate the magnetic line length, leading to formulas for the pole profile, straight-line approximations, correcting rims, and central inserts. By comparing exact and simplified calculations for betatrons of different geometries, the paper gives practical criteria for when each approximation is acceptable, recommending the more accurate gap formula for larger radius ratios and simpler straight profiles when deviations of the field index remain small. Tests of several manufactured accelerator electromagnets are reported as confirming the practical suitability of the proposed methods.
Full Text
On the Calculation of the Profile of the Pole Tips of a Betatron Electromagnet
M. F. Filippov
(Presented by the scientific seminar of the Faculty of Physics and Engineering)
Stable motion of electrons along a circular orbit in a betatron is possible only in the presence of the corresponding configuration of the magnetic field in the interpole space of the electromagnet, where the accelerating chamber is located.
The distribution of magnetic-flux density along the radius in the aperture region of the accelerating chamber, ensuring axial and radial focusing of the beam of accelerated electrons, must satisfy, as is known, the relation
\[ B = B_0 \frac{r_0^n}{r^n}, \tag{1} \]
where \(n\) is the exponent of the decrease of the magnetic field \((1 > n > 0)\),
\(B_0\) is the magnetic induction on the equilibrium orbit of radius \(r_0\),
\(B\) is the magnetic induction at radius \(r\).
In addition, for the equilibrium orbit the “betatron relation” must be satisfied during acceleration,
\[ \frac{B_{k0}}{B_0} = 2, \tag{1a} \]
where \(B_{k0}\) is the mean value of the magnetic induction in the circle of the equilibrium orbit of radius \(r_0\).
The profile of the pole tips of the betatron electromagnet is determined by the relations indicated above.
In published information on the design and manufacture of betatrons it is stated that the selection of the profile of the pole tips is carried out with the aid of an electrolytic bath, in which an electric field is created between electrodes whose shape corresponds to the profile of the pole tips of the betatron electromagnet. In some cases the selection of the profile of the pole tips was carried out on a reduced model of the electromagnet with solid poles, which are easier to manufacture than poles made of sheet steel with radially arranged plates.
The methods listed for selecting the profile of the pole tips are rather labor-consuming. Moreover, with the aid of an electrolytic bath it is not always possible to create an electric field completely analogous to the magnetic field in the interpole space, if it is necessary to take into account the influence of the leakage magnetic flux of the magnetizing windings.
The distribution of magnetic induction with radius in the gap between the poles of an electromagnet can be investigated with sufficient accuracy on a plane model (Fig. 1).
If the measuring coil—the “probe”—is placed on the axis \(a-a\) (at its various points), then the curve \(B=f(r)\) obtained on the plane model corresponds to the curve obtained on full poles of the same profile.
The plane model makes it possible to take into account the influence of the coils of the magnetizing winding and of the central inserts (cheeks) on the configuration of the magnetic field in the working zone of the interpole space. On the plane model it is possible to study the magnetic field with pole tips having slots for accommodating a compensating or bias winding.
Removable pole tips make it possible not only to replace one profile under investigation by another, but also to use the electromagnet of a single plane model for investigating the profiles of the pole tips of any betatrons.
Without excluding the selection of the pole-tip profile by the method of modeling, one should strive to determine the profile with sufficient accuracy by calculation.
In the electromagnet of a betatron, the magnetic reluctance of the air gap between the poles amounts to \((90 \div 95\%)\) of the total reluctance of the magnetic circuit. Taking this feature of the betatron magnetic system into account, and assuming that the magnetic flux is uniformly distributed over the cross section of the steel of the pole cores, the surfaces of the pole tips may be regarded as equipotential surfaces of the magnetic field. In this case the equality will hold
Fig. 1
\[ AW_z = 0.8\,\lambda B \simeq 0.8\,\lambda_0 B_0, \tag{2} \]
where \(AW_z\) is the magnetizing ampere-turns required to overcome the magnetic reluctance of the interpole space; \(\lambda_0\) is the length of the magnetic line of force in the air gap at the radius of the equilibrium orbit \((r_0)\); \(\lambda\) is the length of the line of force at radius \(r\).
From equations (1) and (2) it follows that
\[ \lambda = \lambda_0 \frac{r^n}{r_0^n}. \tag{3} \]
In the middle part of the working zone of the interpole space, where the “edge effect” is not manifested, for example at radius \(r\) (Fig. 2), the magnetic line of force may be represented as an arc of a circle with radius \(R\) (arc \(ab\), Fig. 2). Owing to the small curvature of the line \(ab\), it may be assumed that the length of the line of force \(\lambda\) is equal to the length of the air gap \(\delta\) corresponding to the radius \(r\) (line \(cd\), Fig. 2).
As is seen from Fig. 2,
\[ \lambda = 2\alpha R \quad \text{and} \quad \frac{\delta}{2} = R \tan \alpha, \]
i.e., the relation
\[ \frac{\delta}{\lambda}=\frac{\tg\alpha}{\alpha}. \]
The angle \(\alpha\) for betatrons of 5 MeV is about \(10^\circ\); for more powerful betatrons it is considerably smaller. Taking \(\alpha = 10^\circ\), we find
\[ \frac{\delta}{\lambda}=\frac{\tg\alpha}{\alpha}=\frac{0.1763}{0.1745}=1.010 . \]
Consequently, the difference between \(\lambda\) and \(\delta\) in this case amounts to only 1%. In engineering calculations it is quite permissible to take \(\lambda=\delta\). Then, in accordance with equation (3), we obtain
\[ \delta=\delta_0\frac{r^n}{r_0^n}. \tag{4} \]
The length of the air gap \(\delta\) at the radius of the equilibrium stable orbit \(r_0\) depends on the axial dimension of the cross section of the accelerating chamber. The choice of the dimensions of the chamber cross section is not considered in this article; i.e., the dimensions \(\delta_0\), \(r_0\), \(r_c\), and \(r_n\) are assumed to be known.
Fig. 2
Using equation (4), one can, for various values (in the interval from \(r_c\) to \(r_n\)), determine the corresponding values of \(\delta\), i.e., determine the profile of the pole tips. But equation (4) is valid only in the case when the length of a magnetic line of force differs little from the corresponding length of the air gap. This is true over a considerable part of the working region of the interpole space. As one approaches the edge of the pole tips, the curvature of the magnetic lines of force increases and their length differs noticeably from the magnitude of the air gap at the same radius. The change in the curvature of the lines of force is due to the influence of the “edge effect.”
To reduce the influence of the edge effect on the configuration of the magnetic field in the working region, the pole tips are provided with a correcting rim \(K\) (Fig. 3).
Distortion of the magnetic field in the working region is also caused by the edge effect of the central projections used in the first betatron designs. Replacing the projections with two or three cylindrical inserts (Fig. 3), with a proper choice of the gaps between them, almost completely eliminates the distortion of the magnetic field near the inner wall of the accelerating chamber.
In the first betatron designs, an attempt was made to ensure a constant value of \(n\) in the median plane throughout the entire working region. Later it was established that the conditions for capturing electrons into acceleration are improved if, at the injection radius, \(n\) is close to unity.
Fig. 3
Studies of the magnetic field in various betatrons manufactured at the Tomsk Polytechnic Institute showed that quite good results are obtained by a correcting rim with dimensions
\[ x_{\mathrm{k}} = 0.26\delta_{\mathrm{n}} \quad \text{and} \quad y_{\mathrm{k}} = 0.1\delta_{\mathrm{n}} \tag{5} \]
and that, when calculating the profile of the pole tips, \(n\) may be regarded as \(\mathrm{const}\) and equal to its value at the radius \(r_0\).
The calculated curve of the pole-tip profile (the line \(pq\), Fig. 3), for a constant value of \(n\), is close to a straight line; and, in order to simplify manufacture of the die, it may be taken as a straight line. In this case, when specifying the profile, it is sufficient to indicate the angle of inclination of the straight line \(pq\) to the median plane of the interpolar space (the angle \(\alpha\), Fig. 3)
\[ \alpha = \operatorname{arc\,tg}\frac{\delta_{\mathrm{n}}-\delta_{\mathrm{v}}}{2(r_{\mathrm{n}}-r_{\mathrm{c}})}, \tag{6} \]
where \(\delta_{\mathrm{v}}\) and \(\delta_{\mathrm{n}}\) are computed from equation (4)
\[ \delta_{\mathrm{n}} = \delta_0\frac{r_{\mathrm{n}}^{\,n}}{r_0^{\,n}} \quad \text{and} \quad \delta_{\mathrm{v}} = \delta_0\frac{r_{\mathrm{c}}^{\,n}}{r_0^{\,n}} . \tag{7} \]
In small betatrons, where \(z = \dfrac{r_0}{r_{\mathrm{c}}} > 1.5\), when point \(u\) does not noticeably coincide with the straight line \(pq\) and the value of \(n\) may differ from the specified one by 10% or more, one should use equation (4) to determine \(\delta\) also for two intermediate radii and construct the profile from five points.
The choice of profile should also include the determination of the calculated air gap \(\delta_c\) in the central part of the interpolar space (Fig. 3)
\[ \delta_c=\delta_1+\delta_2+\delta_3 . \]
Assuming, as before, that the surfaces of the pole tips are equipotential surfaces of the magnetic field, one may write that
\[ 0.8 B_{oa}\delta_0=0.8 B_{ca}\delta_c+aw_c(\delta_v-\delta_c), \]
whence
\[ \delta_c=\frac{0.8 B_{oa}\delta_0-aw_c\delta_v}{0.8B_{ca}-aw_c}, \tag{8} \]
where \(B_{oa}\) is the calculated induction on the equilibrium orbit of radius \(r_0\),
\(B_{ca}\) is the calculated mean induction in the circle of radius \(r_c\), when the betatron relation is satisfied,
\(aw_c\) is the specific ampere-turns corresponding to the induction in the steel of the cylindrical inserts \((B_{cm}=B_{ca}\cdot k_c)\),
\(k_c\) is the coefficient of filling of the inserts with steel.
The thickness (height) of each insert (Fig. 3) is
\[ h_c=\frac{\delta_v-\delta_c}{2}. \tag{9} \]
Assuming that in practice the line \(pg\), for betatrons with \(\gamma<1.5\), differs little from a straight line, one may recommend an even simpler method for determining the angle of inclination of this straight line to the median plane of the interpolar space.
Differentiating equation (4), we obtain
\[ d\delta=\frac{\delta_0}{r_0}\,n\left(\frac{r}{r_0}\right)^{n-1}dr. \tag{10} \]
Dividing equation (10) by equation (4) and solving for \(n\), we find
\[ n=\frac{r}{\delta}\cdot\frac{d\delta}{dr}. \tag{11} \]
If \(\alpha\) is the angle between the median plane of the interpolar space and the tangent to the profile curve at the point corresponding to the gap \(\delta\) and radius \(r\) (Fig. 2), then
\[ \frac{1}{2}\frac{d\delta}{dr}=\operatorname{tg}\alpha . \]
On the other hand, in Fig. 2,
\[ \frac{\delta}{2}=R\operatorname{tg}\alpha . \]
Using the relations given above and equation (11), we find
\[ n=\frac{r}{R}. \tag{12} \]
Equation (12) shows that if the profile of the pole tips in the working zone is represented by the straight line \(pg\), then \(n\) will not be constant at different radii. Only when \(R=r\) will \(n\) be unchanged and equal to unity. Nevertheless, as indicated above, for betatrons with \(\gamma<1.5\) the profile of the pole tips may in practice be made in the form of a straight line, since deviations of the field index from the mean value do not substantially affect the configuration of the magnetic field. In this case it is necessary to provide the specified (calculated) value \(n_0\) near the equilibrium orbit.
with radius \(r_0\), where the air gap is \(\delta_0\). In this case the radius \(R_0\) indicated in Fig. 3 will be equal to
\[ R_0=\frac{r_0}{n_0}. \tag{13} \]
and, correspondingly, the angle of inclination of the straight line \(pq\) to the plane of the orbit is
\[ \alpha_0=\operatorname{arc tg}\frac{\delta_0}{2r_0}\,n_0 . \tag{14} \]
Table 1 gives, for comparison, the results of calculating the profile by the three methods described above, i.e., from equations (4, 6—7, and 14), for a small betatron \((A=5\ \text{MeV})\), where the inaccuracies of the calculation method are most appreciable. In the calculations, the influence of the edge effect was not taken into account.
Fig. 4
In Fig. 4 are shown the curves of the dependence \(n=f(r)\) for the profile of the pole tips, calculated from equation (14), for three different betatrons:
- \(A=5\ \text{MeV}\) \((\gamma=1.74)\)
- \(A=15\ \text{MeV}\) \((\gamma=1.44)\)
- \(A=25\ \text{MeV}\) \((\gamma=1.25)\)
the calculated field index being \(n=1/\gamma\).
The results of the calculations presented in Table 1 show that, for the profile of the pole tips calculated approximately (from equations 6—7 and 14), the curve \(n=f(r)\) deviates the less from \(n_0=\text{const}\), the closer the calculated value of the field index is to unity.
Near the inner wall of the accelerating chamber, at a distance
\[ \Delta r=\frac{2}{3}(r_0-r_c)=2\ \text{cm} \]
from the radius of the equilibrium orbit, for the profile of the pole tips calculated from equations (6—7), the deviation of the field index from the calculated value \((n_0=\text{const})\) will be, for \(n_0=0.5\), \(11.6\%\), and for \(n_0=0.75\), \(6.0\%\).
Table 1
| \(r\), cm | By equation 4: \(n=0.5\), \(\delta\), cm | By equation 4: \(n=0.5\), \(\tan\alpha\) | By equation 4: \(n=0.75\), \(\delta\), cm | By equation 4: \(n=0.75\), \(\tan\alpha\) | By equations 6–7: \(\tan\alpha=0.180\), \(\delta\), cm | By equations 6–7: \(\tan\alpha=0.180\), \(n\) | By equations 6–7: \(\tan\alpha=0.266\), \(\delta\), cm | By equations 6–7: \(\tan\alpha=0.266\), \(n\) | By equation 14: \(\tan\alpha=0.176\), \(\delta\), cm | By equation 14: \(\tan\alpha=0.176\), \(n\) | By equation 14: \(\tan\alpha=0.263\), \(\delta\), cm | By equation 14: \(\tan\alpha=0.263\), \(n\) | Note |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4 | 3,72 | 0,232 | 3,24 | 0,304 | 3,72 | 0,388 | 3,24 | 0,656 | 3,87 | 0,374 | 3,32 | 0,633 | |
| 5 | 4,16 | 0,208 | 3,82 | 0,286 | 4,07 | 0,442 | 3,77 | 0,705 | 4,22 | 0,416 | 3,85 | 0,682 | \(A = 5\) MeV |
| 6 | 4,56 | 0,19 | 4,38 | 0,274 | 4,43 | 0,488 | 4,31 | 0,741 | 4,57 | 0,461 | 4,38 | 0,72 | \(r_0 = 7\) cm |
| 7 | 4,92 | 0,175 | 4,92 | 0,264 | 4,79 | 0,525 | 4,84 | 0,77 | 4,92 | 0,500 | 4,92 | 0,75 | \(r_c = 4\) cm |
| 8 | 5,26 | 0,161 | 5,44 | 0,255 | 5,15 | 0,559 | 5,38 | 0,79 | 5,27 | 0,532 | 5,45 | 0,773 | \(r_n = 10\) cm |
| 9 | 5,58 | 0,155 | 5,94 | 0,247 | 5,51 | 0,587 | 5,91 | 0,81 | 5,62 | 0,562 | 5,97 | 0,794 | \(\delta_0 = 4,92\) |
| 10 | 5,88 | 0,147 | 6,43 | 0,241 | 5,88 | 0,613 | 6,43 | 0,83 | 5,98 | 0,588 | 6,5 | 0,81 | \(\gamma=\dfrac{r_0}{r_c}=1,74\) |
For a profile calculated from equation 14, we shall have: for \(n_0=0.5\), \(16.8\%\); for \(n_0=0.75\), \(9.1\%\).
Deviations of this order substantially disturb the correspondence between the configuration of the magnetic field of the working zone and the aperture of the accelerating chamber. In such cases, equation (4), as the more accurate one, should be used for calculating the profile.
The curves in Fig. 4 show the influence of \(\gamma\) on the magnitude of the deviation of the field index from the calculated value in an approximate calculation of the pole-tip profile.
At a distance \(\Delta r=-\dfrac{2}{3}(r_0-r_c)\) from the radius \(r_0\), the deviation of \(n\) from the calculated \(n_0=\dfrac{2}{3}\) will be: for \(\gamma=1.74\), \(11.4\%\); for \(\gamma=1.44\), \(7.8\%\); for \(\gamma=1.25\), \(4.5\%\).
Allowing the deviation of the field index from the calculated value near the inner wall of the accelerating chamber to be no more than \(5\%\), it should be recommended that, when calculating the profile of the pole tips of a betatron electromagnet, only equation (4) be used when \(\gamma>1.5\).
If \(\gamma=1.5\div1.3\), equations (6–7) may be used. For \(\gamma<1.3\), equation (14) is also suitable.
It would seem that, for reliability, the more accurate equation (4) should be used in calculating the profile; however, for a simplified profile with a straight line \((pq)\), it is considerably easier—and therefore also cheaper—to manufacture the die for the pole plates. Therefore, whenever possible, one should strive to use a simplified profile for the pole tips.
The described methods for calculating the pole-tip profile were used by us in designing several different constructions of electromagnets for induction accelerating installations. The results of tests of the manufactured installations demonstrated the complete suitability of these methods for practical calculations.