Approximate Determination of the Magnetic Flux Leakage Coefficient of the Poles of a Betatron Electromagnet

M. F. Filippov

(Presented by the scientific seminar of the Physics and Technology Faculty)

In calculating the magnetic circuit of the electromagnet of a betatron, synchrotron, cyclotron, etc., in order correctly to determine the dimensions of the magnetic core and the ampere-turns of the magnetizing winding, it is necessary to take into account the leakage coefficient of the magnetic flux of the poles (Fig. 1)

\[ \sigma_{n}=\frac{\Phi_{n}}{\Phi_{H}}= \frac{\Phi_{H}+\Phi_{p}}{\Phi_{H}}= 1+\frac{\Phi_{p}}{\Phi_{H}}, \tag{1} \]

where \(\Phi_{H}\) is the magnetic flux in the median plane of the interpole space within a circle of radius \(r_{H}\),
\(r_{H}\) is the outer radius of the pole tips,
\(\Phi_{n}\) is the magnetic flux in the pole cores at their junction with the yoke,
\(\Phi_{p}\) is the magnetic flux leakage of the poles.

The leakage coefficient of the pole flux, depending on the shape and dimensions of the interpole space, varies within rather wide limits and has a very substantial effect on the results of the calculation of the magnetic circuit.

Determination of the coefficient \(\sigma_{n}\) by analytical or graphical methods is possible when the dimensions of the magnetic core and the total ampere-turns of the magnetizing winding are already known, i.e., only after the calculation of the magnetic circuit. In order correctly to calculate the dimensions of the magnetic core and the magnetizing ampere-turns, it is necessary in the first version of the calculation to assume the coefficient \(\sigma_{n}\) approximately, refining its value subsequently. Graphical methods for calculating the leakage coefficient are rather complicated and laborious. Therefore, the closer the approximately adopted value of the leakage coefficient is to the true one, the less time will be required to perform the calculation of the magnetic circuit.

Taking into account that geometrically similar electromagnets of different dimensions will have the same leakage coefficient, one can, using measurement data for the coefficient \(\sigma_{n}\) obtained on electromagnets, find a fairly simple and practically sufficiently accurate equation for determining this coefficient.

As is known, in electromagnets of circular accelerators the presence of a large air gap between the pole faces is inevitable—

...by the pole tips occupied by the accelerating chamber. In this case the magnetic resistance of the steel of the magnetic circuit is very small in comparison with the magnetic resistance of the air gap, and the surface of the pole tips may be regarded as an equipotential surface of the magnetic field of the interpole space (Fig. 1). For the ampere-turns overcoming the magnetic resistance of the interpole space, one may write the equality

\[ A W_{\delta}=0.8 B_{0}\delta_{0}=0.8 B_{p}\lambda_{p}, \tag{2} \]

where

\[ \begin{aligned} B_{0}&\text{ is the magnetic induction in the interpole space at the radius of the}\\ &\text{ equilibrium orbit } r_{0},\\ \delta_{0}&\text{ is the air gap at the radius } r_{0},\\ B_{p}\ \text{and}\ \lambda_{p}&\text{ are the mean calculated induction and the calculated length of the mean}\\ &\text{ line of force of the leakage flux of the poles.} \end{aligned} \]

Fig. 1

From equation (2) we have:

\[ B_{p}=\frac{\delta_{0}}{\lambda_{p}}B_{0}=k_{p}B_{0}. \tag{3} \]

It is known that the leakage flux of the poles \(\Phi_{p}\) is concentrated mainly on the lateral surface of the poles near the air gap.

For simplicity, we assume that the leakage flux is uniformly distributed over the height \(h_{p}=\delta_{0}\) on the lateral surface of the poles (Fig. 1).

In this case the magnetic leakage flux is

\[ \Phi_{p}=2\pi r_{n}h_{p}B_{p}=2\pi r_{n}\delta_{0}k_{p}B_{0}. \tag{4} \]

The magnetic flux in the midplane of the interpole space in a circle of radius \(r_{n}\), when the betatron condition \((B_{k}:B_{0}=2)\) [1] is satisfied, is equal to

\[ \Phi_{n}=2\pi r_{0}^{2}B_{0}+\frac{2\pi r_{0}^{n}B_{0}}{2-n}\left(r_{n}^{\,2-n}-r_{0}^{\,2-n}\right), \]

where \(n\) is the exponent of the radial decrease of the field,

\(B_{k}\) is the mean induction in a circle of radius \(r_{0}\).

Without detriment to the accuracy of the calculations, one may take \(n=1\); then

\[ \Phi_{n}\simeq 2\pi r_{0}r_{n}B_{0}. \tag{5} \]

Substituting the values of \(\Phi_p\) and \(\Phi_n\) from equations (4 and 5) into equation (1), we obtain

\[ \sigma_p = 1 + k_p \frac{\delta_0}{r_0}. \tag{6} \]

Equation (6) applies to the simplest design of a betatron electromagnet, when the guiding and accelerating magnetic fields are produced by common pole cores (Fig. 1). If the poles of the electromagnet have annular slots for accommodating compensating windings (Fig. 2a, b), then, in addition to the general magnetic-flux leakage coefficient of the poles \(\sigma_p\), calculated from equation (6), it is necessary to know the leakage coefficient of the guiding annular pole \(\sigma_y\).

Fig. 2

The leakage coefficient of the guiding annular pole, taking into account the leakage flux only on the outside, is

\[ \sigma_{y n} = \frac{\Phi_{n v} + \Phi_p}{\Phi_{n v}} = 1 + \frac{\Phi_p}{\Phi_{n v}}, \tag{7} \]

where \(\Phi_{n v}\) is the magnetic flux in the ring bounded by the radii \(r_n\) and \(r_v\) in the median plane between the poles (Fig. 2a, b)

\[ \Phi_{n v} = \frac{2\pi B_0 r_0^n}{2-n} \left(r_n^{\,2-n} - r_v^{\,2-n}\right) \simeq 2\pi B_0 r_0 (r_n-r_v). \]

Substituting the obtained value of \(\Phi_{n v}\) and the value of \(\Phi_p\) from equation (4) into equation (7), we find

\[ \sigma_{y n} = 1 + k_p \frac{r_n \delta_0}{r_0(r_n-r_v)}. \tag{8} \]

In equation (8), leakage of the pole flux is taken into account only from the outside of the guiding pole, when on the inside of the slot the winding closes the air gap (Fig. 2b).

If the air gap on the inside of the guiding pole is open (Fig. 2a), then the leakage coefficient of the guiding pole will be equal to

\[ \sigma_y = 1 + (k_p' r_n + k_p r_v) \frac{\delta_0}{r_0(r_n-r_v)}. \tag{9} \]

The coefficient \(k_p^{1}\), which takes into account the leakage flux from the inner side of the guiding pole, differs somewhat from the coefficient \(k_p\), which takes into account leakage from the outer side. But if the difference between \(k_p\) and \(k_p^{1}\) is neglected, taking \(k_p^{1}=k_p\), and assuming \((r_n+r_v)=2r_0\), we obtain a less accurate but simpler equation for determining the total leakage-flux coefficient of the guiding poles of a betatron with annular grooves on the poles:

\[ \sigma_y = 2 k_p \frac{\delta_0}{b_y}, \tag{10} \]

where \(b_y\) is the width of the ring of the guiding pole.

In a synchrotron, where the central core of the magnetic circuit is absent, the electromagnet has the form of a ring with a C-shaped or Ш-shaped cross section of the magnetic circuit (Fig. 2в, г).

With a C-shaped magnetic circuit, the pole leakage-flux coefficient is calculated from equation (9 or 10); with a Ш-shaped magnetic circuit, from equation (10).

The magnetic flux in the interpole space of a cyclotron at the radius of the pole pieces \(r_n\) is

\[ \Phi_{nц} \cong \pi r_n^{2} B\delta . \]

Using equations (1) and (4), we find for the cyclotron

\[ \sigma_n = 1 + \frac{\Phi_p}{\Phi_{nц}} = 1 + 2 k_p \frac{\delta}{r_n}. \tag{11} \]

Determination of the pole leakage-flux coefficient of an electromagnet by the equations given above presents no difficulty if the value of the coefficient \(k_p\) is known, since in the process of calculating the electromagnet, by the time the need arises to determine \(\sigma_n\), all the necessary dimensions of the interpole space are already known.

The coefficient \(k_p\) for geometrically similar electromagnets of any size is a constant quantity if the magnetic induction in the steel of the magnetic circuit has a value at which the phenomenon of saturation is not yet appreciably manifested. In the electromagnets under consideration, owing to the presence of a large air gap, saturation under ordinary conditions has an insignificant effect on the value of \(k_p\).

Measurements of leakage-flux coefficients in electromagnets differing in size and construction have shown that deviations from similarity within the limits occurring in ordinary constructions of the same type have practically no effect on the magnitude of the coefficient \(k_p\).

Thus, the coefficient \(k_p\) can be determined experimentally on an electromagnet of any size of a type close to that being designed.

In certain cases, chiefly in large synchrotrons and synchrophasotrons, particularly high accuracy is required in determining the pole leakage-flux coefficient. The most accurate results can be obtained by measuring the leakage coefficient on a reduced model of the electromagnet being designed, but the fabrication of a model entails a considerable expenditure of funds and time.

With sufficient accuracy, the coefficient \(k_p\), and consequently also the flux leakage coefficient, can be determined with the aid of a small flat model. The flat model represents an electromagnet with a C-shaped magnetic circuit made of sheets of transformer steel. The pole pieces of the model are removable, which makes it possible, on one and the same electromagnet, to select any specified ratio \(\frac{\delta}{b}\) (Fig. 3) and to investigate poles of different profiles.

To determine \(k_p\), it is necessary to have the curve of the distribution of magnetic induction along the midline of the interpole space, i.e., the curve \(B = f(b)\).

As indicated above, the leakage flux of the poles may be considered concentrated at the height \(h_p = \delta\) of the lateral surface of the poles (Fig. 1). In this case, the calculated leakage flux per linear centimeter of the width of the lateral surface of the model pole on the outer side (Fig. 3) is

\[ F_p = F_2 - F_1 . \]

The mean calculated induction of the leakage flux from the outer side of the model pole is

\[ B_p = \frac{F_p}{\delta}. \]

The coefficient \(k_p\) is determined from equation (3)

\[ k_p = \frac{B_p}{B_0}. \]

Similarly, the coefficient \(k_p^1\) is determined for the lateral surface of the pole on the inner side.

As a result of measurements of the leakage coefficients on a flat model and on actual electromagnets, the value of the coefficient \(k_p\) was found to be close to

\[ k_p = 0.56. \]

Fig. 3

Thus, for an approximate determination of the leakage coefficient of the magnetic flux of the poles, the following equations may be used.

1. The total leakage coefficient of the flux of betatron poles

\[ \sigma_n = 1 + 0.56 \frac{\delta_0}{r_0}. \]

1. The leakage coefficient of the flux of the annular control poles of a betatron, taking into account leakage only from the outer side,

\[ \sigma_{yn} = 1 + 0.56 \frac{r_n \cdot \delta_0}{r_0 \cdot b_y}. \]

1. The leakage coefficient of the flux of the annular control poles, taking into account leakage from the outer and inner sides, and the leakage coefficient of the pole flux in a synchrotron

\[ \sigma_y = 1 + 1.12 \frac{\delta_0}{b_y}. \]

1. The leakage coefficient of the magnetic flux of cyclotron poles

\[ \sigma_n = 1 + 1.12 \frac{\delta}{r_n}. \]

The coefficient \(k_p = 0.56\) was obtained with the coils of the magnetizing winding located on the pole cores.

In large accelerators with a pole radius greater than \(100\ \text{cm}\), the values of the coefficients \(\sigma_p\) and \(\sigma_y\) are obtained more accurately if, in the equations given above, the gap between the correcting teeth \(\delta_k\) is substituted for \(\delta_0\).

In some, admittedly very rare, cases the coils of the magnetizing winding may be located on the yokes of the magnetic circuit. With such an arrangement of the coils, the leakage coefficient of the pole flux will be greater than with coils located on the poles.

REFERENCES

1. Filippov, M. F. On the calculation of the radius of the equilibrium orbit of a betatron. Izvestiya TPI, vol. 87, 1957.