On the Requirements for Electromagnets of Betatron Installations and the Principles of Their Design

B. B. Gel'perin

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

Requirements Imposed on Electromagnets of Betatron Installations

An electromagnet is a magnetic circuit consisting of steel poles with pole pieces, a steel yoke, and an annular air gap. In the annular air gap and inside the ring, a controlling and accelerating magnetic field is created. A number of important technical conditions are imposed on this magnetic field; without their fulfillment, operation of a betatron installation is impossible.

As is known from betatron theory, the magnetic field of the electromagnet must satisfy the following conditions:

\[ r_o=\frac{1}{H_o}\sqrt{11.38E+1.113\cdot 10^{-5}E^2}\simeq \frac{E}{300H_o}, \tag{1} \]

\[ \Phi=2\pi r_o^2 H_o, \tag{2} \]

where \(E\) is the energy of the electrons at the end of acceleration,
\(r_o\) is the radius of the equilibrium (stationary) orbit,
\(H_o\) is the magnetic-field intensity in the air gap on the equilibrium orbit,
\(\Phi\) is the magnetic flux in the circle of radius \(r_o\).

Equation (1) determines \(r_o\) for the selected value of \(H_o\). Equation (2) provides an equilibrium orbit at the radius \(r_o\).

For stable motion of the electrons in the orbit, it is necessary that the variation of the magnetic field in the gap in the radial direction satisfy the equation

\[ H=H_o\frac{r_o^n}{r^n}, \tag{3} \]

where \(H\) is the magnetic-field intensity at an arbitrary radius \(r\), and \(n\) is the field fall-off index, which, for stable motion of the electron both in the axial and in the radial directions, must have the value

\[ 0.5<n<1, \tag{4} \]

usually \(n=0.67\) is adopted.

The value of \(n\) in the working region of the gap can be determined for any point by the following formula:

\[ n=-\frac{r\cdot dH}{H\cdot dr}. \tag{5} \]

The requirements for the magnetic field stated above were based on the assumption that the magnetic field in the annular gap is ideal. In reality it differs from a homogeneous one because of asymmetry of the design, technological tolerances (tolerances in machining, assembly, etc.), and other factors.

This nonuniformity of the magnetic field causes distortion of the orbit. The deviation of the electron trajectory from a circle may reach dimensions extending beyond the limits of the accelerating chamber. In this case the acceleration of the electrons will cease, since the bulk of the electrons will be deposited on the chamber walls.

If we denote the distance from the most remote point of the distorted orbit to the point on the ideal circle located on the same radius (the maximum distortion) by \(\Delta r_{\max}\), then, as is known from theory,

\[ \frac{\Delta r_{\max}}{r_0} = \sum_{l=1}^{l=\infty} {}_{\max} \frac{h_l\cos(l\Theta+\alpha_l)}{l^2+n-1}\,n . \tag{6} \]

On the right-hand side of equation (6) is given the sum of the harmonic components obtained as a result of expanding into a harmonic series the curve of distribution of magnetic-field intensities at the points of a circle of radius \(r_0\), where \(l\) is the number of the harmonic,

\(h_l\)—the amplitude of the \(l\)-th harmonic,

\(\alpha_l\)—the phase shift of the \(l\)-th harmonic.

In view of the fact that the betatron is excited by alternating current, the nonuniformity of the magnetic field at points of the orbit circle will be greatest at the moment when the magnetic-field intensity is close to zero. This means that the nonuniformity of the magnetic field has its sharpest effect at the moment of electron injection. For this case one may write

\[ h=h_c+h_t, \]

where \(h\) is the deviation of the magnetic-field intensity in the gap from the mean value,

\(h_c\) is the so-called static nonuniformity, caused by inequality of the amplitudes of the magnetic-field intensities,

\(h_t\) is the so-called temporal nonuniformity, caused by a phase shift between the magnetic-field intensities at different points of the circumference with respect to some one point taken as the initial point.

The greatest influence is exerted by the temporal nonuniformity of the magnetic field. Usually, in betatron design practice, temporal nonuniformity is measured by the angle \(\gamma\)—the phase shift of the magnetic-field intensity

\[ \gamma_{\max} = \sum_{l=1}^{l=\infty} {}_{\max} \frac{\gamma_l\cos(l\Theta+\alpha_l)}{l^2+n-1}. \]

To simplify the calculations and with a certain margin, the value of \(\gamma_{\max}\) may be determined from the equation

\[ \gamma_{\max} = \sum_{l=1}^{l=\infty} \frac{\gamma_l\cdot n}{l^2+n-1}. \tag{7} \]

In exactly the same way,

\[ h_{c\max}=\sum_{l=1}^{l=\infty}\frac{h_{lc} n}{l^2+n-1}. \tag{8} \]

The greatest relative deviation of the magnetic-field intensity is

\[ h_{\max}\%=\gamma_{\max}\frac{29E}{\sqrt{U}}\frac{P_z}{P_z+P_c}(1+m)\cdot 10^{-6}, \tag{9} \]

where \(U\) is the injection voltage,
\(P_z\) is the magnetic conductance of the air annular gap,
\(P_c\) is the magnetic conductance of the central part of the magnetic circuit (through the inserts),

\[ m=\frac{\Phi_c}{\Phi_z} \]

is the ratio of the magnetic flux in the central part to the flux of the annular gap.

It follows from equation (9) that the greater the injection voltage, the higher the permissible value of \(\gamma_{\max}\) may be. However, increasing the injection voltage makes the injector and the accelerating chamber more expensive. It is obvious that the wider the annular gap in the electromagnet, and consequently the wider the accelerating chamber, the larger the “working region” of the betatron and therefore the greater the permissible value of

\[ \frac{\Delta r_{\max}}{r_0}; \]

consequently, higher tolerances on the nonuniformity of the magnetic field, \(\gamma_{\max}\) and \(h_{c\max}\), can be obtained. It should be taken into account that widening the annular gap also causes an increase of the gap in height, which entails an increase in the weight of the installation and in its cost. In order to build an electromagnet with a smaller nonuniformity of the magnetic field, it is necessary to understand the nature of the origin of the static and temporal nonuniformity.

Static nonuniformity arises mainly from asymmetry of the design, inaccuracy in the manufacture and assembly of the pole tips and of the entire electromagnet as a whole, nonuniformity of the gaps, and, finally, edge effects when the magnetic field passes through structural gaps, i.e., mainly from the unequal resistance of individual field tubes. Temporal asymmetry arises from different active components of the current in the individual closed tubes of the magnetic field. This difference in active current is caused by the nonuniform distribution of the flux over the yoke packs, by the passage of magnetic flux from one pack to another across the steel sheets, by different work-hardening of the steel during its processing, etc.

Therefore, in order to obtain the greatest uniformity of the magnetic field, it is necessary to strive, on the one hand, to create a symmetrical electromagnet design and, on the other hand, to use some compensating devices for equalizing the temporal nonuniformity of the magnetic field in the gap.

The principles of designing symmetrical electromagnets and compensating devices will be discussed below.

In betatrons of the design of the Moscow Transformer Plant, the following dimensional ratios are used:

\[ \frac{v}{r_0}=0.4\div 0.7 \qquad \text{and} \qquad \frac{h_z}{r_0}=0.2\div 0.4, \]

where \(v\) is the width of the annular gap,
\(h_z\) is the height of the annular gap at the equilibrium orbit.

In this case \(\gamma_{\max}\) varies up to 1 minute, and \(h_{c\max}\) up to \(0.01H\).

In the radial direction, as indicated above, the magnetic-field intensity must decrease according to the law expressed by equation (3). But in the radial direction there is also a time shift between the field intensities at points along the radius. This occurs because of the different active-current components for individual parallel tubes taken along the radius at some one azimuth.

The tolerance for radial time nonuniformity ranges up to \(1'\) without any appreciable narrowing of the working region of the magnet.

Principles of Designing Electromagnets

In accordance with the technical requirements imposed on the electromagnet of a betatron, a number of principles have been developed that form the basis for designing betatron electromagnets.

a) Shape of the pole tips

First of all, the shape of the pole tips must be selected. Since the magnetic-field intensity along the radius must decrease in the direction of increasing radius according to the law of equation (3), then, neglecting the drop of the magnetic potential in the steel, one may roughly assume that

\[ h_{3x} \simeq C r_x^n, \tag{10} \]

where \(h_{3x}\) is the height of the air gap at the radius vector \(r_x\), and \(C\) is a constant.

The pole tip is a conical surface (Fig. 1). The tooth used at the edge of the pole tip is intended to reduce the influence of the spreading of the magnetic field from the axis of the poles toward the periphery.

Fig. 1

The dimensions of the central cylindrical part of the magnet are chosen so that equation (2) is satisfied; this part may be made removable, in the form of an insert, or as an extension of the pole. If this part is removable, it can be used to adjust the equilibrium orbit during manufacture of the betatron. For this purpose it is deliberately made somewhat greater in height, and is then ground down to the required size according to test data.

The calculation of the reduction in the height of the inserts is carried out on the basis that the change in the central flux through the insert, as a function of the displacement of the radius of the equilibrium orbit \(\Delta r\) required by the test, is determined by the following formula:

\[ \Delta \Phi = 2 \pi r \Delta r \cdot H(1 - n). \]

Knowing the value of \(\Delta \Phi\), one can determine the change in the gap between the inserts, and consequently the change in the height of the insert.

If the central cylindrical part is a direct continuation of the pole, then adjustment of the equilibrium orbit is carried out by changing the gaps in the magnet (main and auxiliary), thereby changing the ratio between the mean magnetic-field intensity inside the orbit and the intensity on the orbit.

Since the electromagnet is supplied with alternating current and the pole, as a result, must be assembled from radial sheets of transformer steel (to avoid strong overheating of the steel), structurally the pole shoe is not made separately, but forms a single whole with the pole. At MTZ, each sheet of transformer steel of the pole is stamped already with the specified shape of the pole shoe (Figs. 4c and 6b).

As studies have shown, the presence of inserts disrupts the law of magnetic-field decrease in the air gap near the inserts if the height of the latter is calculated from equation (10). In the region closest to the inserts, the quantity $n$ becomes greater than 1, and in connection with this part of the so-called “working” region of the betatron is lost. To compensate for the influence of the cylindrical central part, a small recess is made in the pole (Fig. 1). The dimensions of the recess are difficult to calculate and are usually selected experimentally. This recess, as we shall see below, is used for placing a special winding in it.

b) Adjustment of the position of the equilibrium orbit during operation of the betatron

The optimum intensity of the gamma radiation of the betatron is affected by the position of the equilibrium orbit relative to the injector. Therefore, it seems to us that, in order to obtain greater intensity, “fine adjustment” of the radius of the equilibrium orbit during operation of the betatron will be of great importance. For this purpose, a winding located in the recesses of the pole may be used. By connecting this winding to an adjustable voltage from the source feeding the main winding of the electromagnet, it is possible to change the magnitude of the flux inside the orbit and, consequently, its radius. The same winding can at the same time be used also for a “contractor” circuit, making it possible to contract the orbit during injection, and for an “extractor” circuit, making it possible to expand or contract the orbit at the end of acceleration, for bringing the accelerated electrons out onto an external or internal target. The basic circuit for the simultaneous use of one and the same winding for the purposes of supply by 50-cycle and pulsed current is shown in Fig. 2.

Fig. 2

The principle of operation of this circuit is as follows. When supplied by a 50-cycle voltage, the electrical resistance of the inductance $L$ is small and, thus, it as it were “short-circuits” the pulse generator $U_i$. The electrical resistance of the capacitance $C$ is large, and the current it consumes at 50 hertz is small; therefore it is practically not any appreciable load for the generator $U_{50}$.

When, at the required moment, the pulse generator $U_i$ is connected, then for it, on the contrary, the inductance $L$ presents a large electrical resistance, and the capacitance $C$ a small one. In this connection there is no danger to the insulation of the 50-hertz generator.

The operation of this circuit was tested, and positive results were obtained.

c) Extraction of the beam of accelerated electrons onto the target

Extraction of the beam of accelerated electrons onto the target may be carried out in two ways: by means of an “extractor” circuit and by means of saturating inserts.

The operation of the “extractor” circuit consists in the fact that, at the end of acceleration, at the required moment, a current pulse of several tens of microseconds’ duration is fed to a special winding. Depending on the sign of the current pulse applied to the winding, the flux inside the orbit will either increase or decrease and, consequently, the orbit radius will either increase (displacement of the electrons onto an external target, usually placed on the rear side of the injector) or decrease (displacement of the electrons onto an internal target). The advantage of the “extractor” circuit is that, first, the current pulse can be applied at any moment during acceleration and, consequently, the energy of the accelerated electrons can be varied very easily; and, second, the displacement of the accelerated electrons can be carried out onto an external target, which can be placed closer to the object being irradiated and thereby provide a greater irradiation intensity of the object, since the irradiation intensity, as is known, varies inversely with the square of the distance of the object from the target.

Displacement of the accelerated electrons by means of saturating inserts is based on the following principle: at the end of acceleration, when the magnetic-field strength approaches its amplitude, the central cylindrical part of the magnet begins to saturate. The increase of the central magnetic flux begins to decrease, and the radius of the electron orbit also begins to decrease, until the electrons reach the internal target.

The advantage of this method of displacement is that there is no need to use special windings and, chiefly, a special pulse generator. However, a major drawback of this method is the impossibility of regulating the gamma-radiation energy of the installation and of displacing the electrons onto an external target.

г) Compensation of magnetic-field inhomogeneity

As indicated above, magnetic-field inhomogeneity manifests itself most sharply during the injection of electrons into the chamber. In this case the temporal inhomogeneity of the magnetic field is of greatest importance. Therefore, the main attention must be devoted to compensating the temporal inhomogeneity. Since the source of the temporal inhomogeneity is the different active component of the current in the individual tubes of the field, it is obvious that the field should be divided into separate sections (tubes) and these should be encircled by windings fed by an active current (shifted by \(90^\circ\) with respect to the main current). The division of the electromagnet into sections is shown in Figs. 3b, 4b, and 5b.

As follows from equation (6), there is no need to compensate the active component of the current in every section. It is sufficient only to reduce the equivalent inhomogeneity, according to equation (6), to an admissible value. For this purpose, after decomposing the curve of the “natural” distribution of the magnetic-field strengths around the circumference into a harmonic series, those sections are determined in which compensation gives the greatest effect. The windings are fed with active current from the same current source that supplies the main windings (or by transformation of this current source), through a relatively large resistance. A schematic power-supply circuit is presented in Fig. 7.

If special windings for compensation are not provided in the electromagnet, compensation can be carried out from the side of the yoke or the column, by encircling with turns one or another part of the yoke or column. In a two-yoke construction of the electromagnet this method is less effective, since the compensating flux of the pole “spreads out” over a fairly large angle and does not create a “sharp” compensation.

Fig. 4.

a) Detail A

b) Detail A

c)

Fig. 3.

a)

b) Detail A; Detail A

![Diagram labels: “Section winding”; “Transformer”; “Power supply for the main windings.”]

Fig. 7.

Fig. 6.

a)

b)

Fig. 5.

a)

b)

d) Basic designs of electromagnets

Figures 3 and 4 show four-yoke, and Figs. 5 and 6 two-yoke, designs of an electromagnet. As can be seen, the four-yoke design is a symmetrical system. All parallel sections of the field have identical geometrical dimensions. In this connection all sections must have the same magnetic reluctance. The exception to this is the outer sections adjacent to the edges of the columns. Because the magnetic flux passes across the sheets of the magnetic core, the saturation of these stacks differs from the saturation in other sections and, consequently, they have a different magnetic reluctance, which causes a certain inhomogeneity of the magnetic field. In addition, bending of the sheets in this design creates work-hardening of the steel, which causes additional losses and, consequently, an increase in the active component of the current in these sections. Nevertheless, the inhomogeneity of the magnetic field in the four-yoke design is considerably smaller than in the two-yoke design, since in the latter the inhomogeneity of the magnetic field is caused by the asymmetry of the design.

Thus, inhomogeneity of the magnetic field is inherent in the very design of a two-yoke electromagnet. If one also takes into account that the field inhomogeneity is aggravated by manufacturing and assembly tolerances, it is quite obvious that the two-yoke design gives a significantly greater field inhomogeneity than the four-yoke design and requires more adjustment and compensation.

An advantage of the four-yoke MTZ design is also that the pole of the electromagnet forms a single whole with the yoke and the column, so that the pole does not require additional fastening.

A disadvantage of the four-yoke design is that it requires large dimensions of transformer steel for manufacture, since the pole, yoke, and column are cut from the sheet all at once. The use of cold-rolled steel for such a design is difficult, since cold-rolled steel has different magnetic reluctance along and across the rolling direction, is more sensitive to work-hardening, and to other stresses in the steel. Since the dimensions of cold-rolled steel sheets are at present small, the latest designs of MTZ electromagnets have been developed from cold-rolled steel and in a two-yoke version.

e) Cooling of electromagnets

Only the smallest betatron installations, supplied by alternating current with a frequency of 50 cycles, can be made with natural air cooling. More powerful installations, beginning with energies of 25 MeV and higher, when supplied by alternating current with a frequency above 50 cycles, must have forced cooling. As a rule, cooling with forced circulation of air is used. Taking into account that during operation of the betatron it is necessary to remove its upper half, the fan is connected to the electromagnet from below, on the stationary half. In order to render the air harmless in the betatron room, the radioactive air from the betatron must be removed; therefore the fan operates in air-extraction mode, as is shown in Figs. 4a and 6a.

The cross sections of the ducts are calculated so that the air velocities are of the order of 10–15 m/sec. After assembly of the magnet, the duct cross sections can be throttled to obtain the calculated air velocities. The powers of the fan motors were 3–5 kW. The fans were of medium pressure, 200 mm water column, with a capacity of 3000–5000 m³/hour.

Power-Supply Circuit for the Electromagnets

Betatron installations consume a large amount of reactive power. In order to relieve the supply network of the burden of carrying a large reactive power, a bank of static capacitors is connected to the electromagnet winding. The capacitance of the capacitor banks is adjusted close to resonance with the inductance of the electromagnet, so that the network carries mainly only a small active load.

To avoid surges when switching on the network, as a rule, a regulator is included in the supply circuit, allowing the voltage to be raised gradually from a low value or from zero.

For magnetizing the electromagnet, a winding is used whose number of turns is chosen so as to make the fullest possible use, in terms of voltage, of the available capacitor banks.

In cases where the voltage of the magnetizing winding of the electromagnet differs greatly from the line voltage, an auxiliary winding is used, called the supply winding. The number of turns of this winding is selected according to the supply voltage, and the current in accordance with the active current consumed by the supply circuit (for reserve, a power factor of 0.5 is adopted).

Fig. 8

Supplying the electromagnet with alternating current at a frequency of 50 hertz can be carried out directly from the network (Fig. 8). It is desirable that a separate feeder be assigned for supplying the betatron installation, in order to avoid large voltage fluctuations during operation.

Judging by the condition of the networks in industrial districts, it may be expected that the frequency and the magnitude of the voltage in them are more or less stable, so that there is no need to introduce into the circuit a device for stabilizing voltage and frequency. In an extreme case, automatic voltage stabilization developed by MTZ may be used.

When alternating current is used to supply windings at frequencies above 50 hertz (150, 200, 300, 500), either a four-machine unit or a tube generator may be used.

The four-machine unit consists of an asynchronous motor supplied from the network at a frequency of 50 hertz. On the same shaft with this motor there is a direct-current generator. The direct-current generator supplies a direct-current motor, on the same shaft with which there is a synchronous generator supplying the electromagnet winding.

The presence of an intermediate direct-current link makes it easy to regulate the frequency of the synchronous generator and to stabilize the frequency and, if necessary, the voltage.

It is also possible to use a static frequency-tripler circuit. This circuit, however, still requires further development with regard to greater voltage stabilization.