Methods for Measuring the Decay Index of the Magnetic Field in Accelerator Facilities

I. G. Leshchenko

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

In charged-particle accelerators, betatrons, and synchrotrons, a very important characteristic of the configuration of the controlling magnetic field is the magnetic-field decay index \(n\).

The value of the decay index, as is known, is determined by the relation

\[ n=-\frac{dH_z\cdot r}{H_z\cdot dr} \tag{1} \]

and in the working acceleration zone it must lie within the range from zero to unity.

All methods used up to the present time for determining the magnetic-field decay index—the graphical-analytic method, the analytic method, and two differential methods—may be called indirect methods. In these methods the experimentally obtained curve of the dependence \(H_z=f(r)\) is used.

To determine the magnetic-field decay index by the graphical-analytic method, the dependence of the voltage induced in the measuring probe coil on the radius, \(U_z=f(r)\), is recorded. From the curve (Fig. 1), for a given radius the values \(U_z\) and \(\Delta U_z\) are found, and the magnetic-field decay index is calculated from the formula

\[ n=\frac{\Delta U_z\cdot r}{U_z\cdot \Delta r}. \tag{2} \]

The sign before \(n\) is “+” if the field decreases, and “−” if the field increases with increasing radius.

The degree of accuracy in determining \(n\) in this case depends on the accuracy of measuring \(U_z=f(r)\) and on the accuracy of the graphical constructions. As the radius increases, the error in determining the field decay index increases, and at radii greater than 100 cm it practically becomes inadmissible (reaching 10 percent and more), especially when the field changes sharply.

In the analytic method, the voltage is measured at the terminals of a probe coil installed at equal intervals along the radius. Then the value of \(n\) at some point \(x\) can be determined from the expression:

\[ n_x=\frac{U_{x+1}-U_{x-1}}{U_x}\cdot\frac{r_x}{r_{x+1}-r_{x-1}}, \tag{3} \]

where \(x\) is the ordinal number of the point.

This method gives an even larger error in the determination of \(n\) than the grapho-analytical method, in which the errors of voltage measurement at individual points are smoothed when constructing the averaged curve \(U_z=f(r)\). The analytical method for determining \(n\) is sometimes used in accelerator installations with small radii, provided that the voltage on the magnetizing windings of the electromagnet is well stabilized.

Fig. 1

The accuracy of determining the decay index of the magnetic field is increased considerably by using a differential method of measuring the dependence \(U_z=f(r)\), instead of the direct one. In this case two oppositely connected measuring coils are used, one of which is installed immovably (\(U_{\mathrm{н}}=\mathrm{const}\)), while the other is moved along the radius at a given azimuth, and the difference of voltages \(\Delta U_x=U_x-U_{\mathrm{н}}\) is measured. The value of \(n\) is determined by the formula

\[ n_x=\frac{\Delta U_{x+1}-\Delta U_{x-1}}{\Delta U_x+U_{\mathrm{н}}}\cdot\frac{r_x}{r_{x+1}-r_{x-1}} \tag{4} \]

by grapho-analytical or analytical methods.

Despite the general increase in the accuracy of determining \(n\), this method is not without the shortcomings of grapho-analytical calculations and, in addition, has additional errors introduced by static and phase inhomogeneities of the magnetic field, especially when the stationary coil is placed at another azimuth and when \(\Delta U_x\) is small.

To eliminate the indicated errors in determining the decay index of the magnetic field by the differential method, two coils are used, as far as possible with identical effective cross sections and with the distance between their axes equal to \(\Delta r\). At the point at radius \(r\), where \(n\) is to be determined, the difference \(\Delta U_1\) and the coil voltages \(U_1\) and \(U_2\) are measured; then the coils are turned through \(180^\circ\) so that they exchange places, and the voltage difference \(\Delta U'_1\) and the voltages \(U'_1\) and \(U'_2\) are again measured. The required value of the magnetic-field decay index is determined from the formula

\[ n = 2\,\frac{\Delta U_1 + \Delta U'_1}{U_1 + U_2 + U'_1 + U'_2}\cdot \frac{r}{\Delta r}. \tag{5} \]

By the indicated method one can determine the magnetic-field decay index in accelerators with large radii with sufficient accuracy. However, a substantial shortcoming of this method should be considered to be that it requires analytical calculations and the manufacture of precise devices for rotating the paired coils during the measurement of \(\Delta U\) and \(U\).

To determine the complete topography of the magnetic-field decay in the gap of the interpolar space, such measurements must be carried out in different planes and at different azimuths, which requires a large expenditure of time. As a result, the complete topography of the magnetic field in accelerators has been studied insufficiently.

As a result of the investigations carried out, we have developed methods for the direct, rapid, and accurate measurement of the magnetic-field decay index by an instrument which we call an \(n\)-meter.

As a sensor for measuring the magnetic-field decay index, a magnetic probe is used, consisting of three measuring coils with identical effective cross sections. The two outer coils of the probe have a distance between their axes \(\Delta r\) and must give, in the decaying magnetic field, the value \(\Delta U_z\), while the third (middle) coil gives \(U_z\).

The circuits for measuring \(n\) using the indicated magnetic probes (\(n\)-meters) are based on the two methods described below.

Comparison Method

If, with the distance between the centers of the outer coils equal to unity, such a voltage \(U\) is obtained from the middle coil \(K_{\text{av}}\) at radius \(r\) that the numerical values of \(U\) and \(r\) are equal or multiples (with multiplicity 1, 10, 100, etc.), then the expression for determining the magnetic-field decay index may be written in the form:

\[ n = K \cdot \Delta U, \tag{6} \]

where \(K\) is a proportionality coefficient equal to \(\dfrac{r}{U\cdot \Delta r}\). In this case, a mandatory condition at the location of the magnetic probe is \(\Delta U/U = \text{const}\). This condition can be obtained when changing the voltage from the middle coil of the probe by varying the voltage at the terminals of the electromagnet windings, which is practically inconvenient and not always possible, or by means of paired potentiometers connected into the circuit of the middle and outer coils of the magnetic probe so that \(\Delta U/U = \text{const}\), and, consequently, it is possible to carry out numerical comparisons of voltage and radius. With a known constant \(K\), the scale of the voltmeter can be graduated in values of \(n\). However, the indicated circuit [1] requires the presence of two sensitive voltmeters with high internal resistance and two paired potentiometers, on which very stringent requirements are imposed with respect to simultaneous variation of the resistance of their arms.

The use, in the circuit of the middle coil, of a voltage divider with a division factor equal to the coefficient \(K\), and of a switch in the form of a toggle switch, made it possible to considerably simplify the circuit of the \(n\)-meter and to increase the accuracy of measuring the decay index of the magnetic field.

Fig. 2

Measurement of \(n\) according to this circuit (Fig. 2) is carried out in the following order:

1) with the toggle switch \(T\) set to the position “\(r\),” the magnetic probe of the \(n\)-meter is placed at the specified radius \(r\);

2) voltage divider II is set to the position corresponding to the proportionality coefficient \(K\);

3) by means of the potentiometer \(R_n\), a voltage value is set that is numerically equal to, or a multiple of, \(r\), for example, \(0.48\) volt \(\div\) \(48\) cm;

4) the toggle switch is switched to the position “\(n\),” and the voltmeter gives a direct reading \(\Delta U\), equal to the decay index of the magnetic field at the specified radius.

To take into account \(\Delta r\), which may be not equal to unity, an additional voltage divider I is used in the circuit of the middle coil, with a division factor numerically equal to \(\Delta r\) (Fig. 2).

In manufacturing the magnetic probe it is practically impossible to obtain all three coils with identical effective cross section, since the permissible error must be no more than \(0.001\%\). With a difference of effective cross sections amounting to \(0.01\%\), the value of \(n\) will change by the amount

\[ \Delta n = \frac{1}{10000}\cdot \frac{r}{\Delta r}. \]

Thus, for \(r = 100\) cm and \(\Delta r = 0.5\) cm, \(\Delta n = 0.02\).

In magnetic probes, an error in winding of one turn may cause a distortion of \(n\) by an inadmissible amount. Therefore, in order to obtain identical effective cross sections of the opposed coils with the required accuracy, it is necessary to connect in parallel with the coil having the larger effective cross section a “trimming” active resistance \(R_{\partial}\) (Fig. 2). The magnitude of the resistance is easily set by placing the probe in a strictly homogeneous magnetic field, in which \(\Delta U = 0\) is achieved. A solenoid may be used as the source of the homogeneous magnetic field; the magnetic probe is placed in its central part so that its center coincides with the axis of the solenoid.

To obtain coils with a large effective cross section, three-frame rectangular magnetic probes were initially used, the design and dimensions of one of which are shown in Fig. 3. In addition,

probes with rectangular coils are more conveniently made monolithic, with a strictly fixed distance between the centers of the coils, for example, \(\Delta r = 1.00\) cm. Later we manufactured only two-frame probes (Fig. 3). In such a probe the turns of the middle coil are wound directly onto the two outer ones. Thus, with the same distance between the centers of the outer coils, a probe is obtained with a considerably larger effective cross section of each coil than in a three-frame probe. At present only two-frame magnetic probes have come into use, their effective cross section being \(100\)—\(500\ \text{cm}^2\) (PEL wire, \(0.05\) mm; number of turns \(1200\)—\(5000\)).

Fig. 3

To make the magnetic probes monolithic, since insignificant shifts of one coil relative to another can lead to serious distortions of the results of measuring \(n\), the probes are impregnated with bitumen or picein.

The use of \(n\)-meters based on the comparison method made it possible to increase significantly the speed and accuracy of measuring the decay index of the magnetic field in comparison with indirect methods of determining \(n\). In addition, the use of these instruments completely eliminated additional laborious graphical-analytical calculations.

The speed of measuring \(n\) by the comparison method increases tens of times and more; moreover, the larger the radius at which the measurements are made, the greater the relative speed of determining the magnetic-field decay index.

As an example, Fig. 4 shows the topography of the magnetic field in the working region of a 15 MeV betatron (\(r_o = 13\) cm), obtained with the aid of an \(n\)-meter over the course of one hour. In this case the decay index of the magnetic field was measured along the radius and in different planes every 5 mm.

The accuracy of measuring the decay index of the magnetic field with an \(n\)-meter is increased both by the simultaneous measurement of the voltage difference of the outer coils and by the strictly fixed distance between the centers of the outer coils of the magnetic probe. This is clearly seen from the character of the scatter of the points obtained when determining the field decay index \(n\) by some indirect method and with the aid of an \(n\)-meter.

The limits of possible measurement of the magnetic-field decay index with an \(n\)-meter depend on the magnetic-field strength, the distance \(\Delta r\) of the maximum radius at which the measurement of \(n\) is made, and on the sensitivity of the measuring instrument. Knowing the effective cross section of the end coils, the distance between their centers, and the sensitivity of the instrument, it is not difficult to determine the limits of possible measurement of \(n\) by the instrument (field strength and radius).

Fig. 4

In practice, if a sensitive voltmeter is used, for example of the LV-9 type, or, still better, of the “Vilnes” type (Czechoslovakia), the magnetic-field decay index can be measured in accelerators with radii greater than 10 meters.

Null Method

Null \(n\)-meters for measuring the magnetic-field decay index are based on the fact that the voltage \(U_2\) from the middle coil is reduced \(r\) times by voltage divider II, and \(R_r\)—by a potentiometer, while the \(n\)-potentiometer varies \(U_2/r\) so that

\[ \frac{U_2}{r}\cdot n = U_3 - U_1 = \Delta U. \tag{7} \]

The circuit for measuring \(n\) by the null method is shown in Fig. 5. Equality (7) is determined by a null instrument, which may be a voltmeter, a cathode-ray oscilloscope, etc. At \(\Delta r = 1.00\ \text{cm}\), the magnetic-field decay index can be measured only from 0 to \(+1\). For measuring \(n\) greater than unity, voltage divider III with a division factor of 10 is added to the circuit of the two oppositely connected coils. In this case measurements of \(n\) from 0 to 10 can be carried out. For measuring negative values of the magnetic-field decay index, a switch is provided for the direction of the current from the middle coil or from the two end coils.

The choice of the elements of the instrument circuit and the design of the magnetic probe must satisfy the same requirements as in the instrument circuit based on the comparison method. However, calibration of this instrument requires greater accuracy. The presence of several dividers makes a very careful calculation of the instrument circuit necessary.

Fig. 5

In view of the fact that the calibration of the potentiometer \(R_r\) must have a scale inversely proportional to \(r\), not one scale but several are used. This depends on the largest radius for which the \(n\)-meter is designed. For each range of radii an additional voltage divider is installed, calculated with allowance for the influence of \(R_r\) and of the \(n\)-potentiometer introduced into the elements of the divider.

Procedure for Measurement with the Instrument

1. The switch of divider II is set to the position corresponding to the range of radii at which the decay index of the magnetic field is measured.

2. The slider of the \(R_r\)-potentiometer is set to the position corresponding to the radius at which the magnetic probe is located.

3. The slider of the \(n\)-potentiometer is brought to such a position that the zero instrument should indicate zero—practically, a minimum—owing to the presence of a phase shift between the voltages of the outer coils.

When measuring the decay index of the magnetic field with unidirectional pulsed supply of the electromagnet, a cathode oscilloscope must be used as the voltage null instrument.

It should be noted, incidentally, that the principal advantage of the null \(n\)-meter is the possibility of measuring the decay index of the magnetic field in the pulsed operating mode of the installation over a wide frequency range, whereas all indirect methods for determining \(n\) and the comparison method, when the field decay index is measured directly with an \(n\)-meter in this operating mode of accelerators, have an inadmissible error.

As a result of checking and testing \(n\)-meter instruments that have found application in the laboratory of the Tomsk Polytechnic Institute, the following is recommended in order to avoid complicated adjustment of \(n\)-meters and to reduce errors in measuring the decay index of the magnetic field:

1) take account of \(\Delta r\), which is not equal to unity, not by means of an additional voltage divider, but by changing the effective cross section of the middle coil when manufacturing the magnetic probe;

2) strive to make the errors introduced in measuring part of the voltage from the middle coil and the voltage difference from the outer coils of the magnetic probe differ little from one another;

3) calculate the \(n\)-meter for measuring the decay index of the magnetic field in pulsed mode for a definite range of radii, so as not to install the additional voltage divider II.

When measuring the decay index of the magnetic field with the indicated instruments, the influence of radial phase nonuniformity of the magnetic field and pole saturation is practically not taken into account, although, as is known [1, 2], these influences can be very serious, especially in large accelerators.

Thus, taking into account the phase shift, the expression for the instantaneous value of the magnetic-field decay index can be written in the form:

\[ n_t=\frac{\Delta U_{t\varphi}}{U_t}\cdot \frac{r}{\Delta r}, \tag{8} \]

where \(\Delta U_{t\varphi}\) is the instantaneous value of the voltage difference, taking into account the phase shift of the fluxes between the centers of the outer coils,
\(U_t\) is the instantaneous value of the voltage of the middle coil. Assuming that the phase shift \(\Delta\varphi\) over the interval \(\Delta r\) is a constant quantity, independent of the amplitude value of the magnetic-field induction, for small phase-shift angles the value \(\Delta U_{t\varphi}\) can be represented as:

\[ \Delta U_{t\varphi}=U_1\cos\omega t-U_3\cdot \cos(\omega t-\Delta\varphi) =(U_1-U_3)\cos\omega t-U_3\sin\omega t\cdot \Delta\varphi . \tag{9} \]

The mean value of the voltage over a quarter period is

\[ \Delta U_{cp}=\frac{2}{\pi}\int_{0}^{\pi/2}\Delta U_{t\varphi}\,d\omega t =-\frac{2}{\pi}\left(U_1-U_3\right)+\frac{2}{\pi}U_3\cdot\Delta\varphi \]

\[ =\frac{2}{\pi}\left(U_3\cdot\Delta\varphi-\Delta U\right). \tag{10} \]

Since the tube voltmeter is calibrated for the effective value, then

\[ \Delta U_{\text{eff}}=\frac{\Delta U_{cp}\cdot\pi}{2\sqrt{2}} =\frac{-\Delta U+U_3\cdot\Delta\varphi}{\sqrt{2}}, \tag{11} \]

and, consequently,

\[ n_{cp}=\frac{\sqrt{2}\cdot\Delta U_{\text{eff}}}{U_{2max}}\cdot\frac{r}{\Delta r} =\frac{-\Delta U+U_3\cdot\Delta\varphi}{U_{2max}}\cdot\frac{r}{\Delta r}. \tag{12} \]

Thus, the reading of the \(n\)-meter changes, owing to the radial phase inhomogeneity over the interval \(\Delta r\), by the amount

\[ n_{\varphi}=\frac{U_3\cdot\Delta\varphi}{\Delta r}\cdot\frac{r}{U_2}. \tag{13} \]

To estimate the error introduced, let us find the ratio

\[ \frac{n_{\varphi}}{n_c} =\frac{U_3\cdot\Delta\varphi\cdot r}{U_2\cdot\Delta r}\cdot\frac{U_2\cdot\Delta r}{\Delta U\cdot r} =\frac{U_3\cdot\Delta\varphi}{\Delta U}. \tag{14} \]

Assuming that \(U_3\) differs little from \(U_2\), from expression (2), which does not take into account the phase shift between the voltages in the outer coils, we find

\[ \frac{\Delta U}{U_3}\simeq \frac{\Delta U}{U_2}=\frac{n_c\cdot\Delta r}{r}. \tag{15} \]

From equations (14) and (15)

\[ \frac{n_{\varphi}}{n_c}=\frac{r}{n_c}\cdot\frac{\Delta\varphi}{\Delta r}\cdot 100\%. \tag{16} \]

On the basis of experimental data on measuring the radial phase inhomogeneity of the magnetic field, it is known that \(\Delta\varphi/\Delta r\) in the working acceleration zone of betatrons and synchrotrons does not exceed \(0.2\text{--}0.3\ \mu\text{sec}/\text{cm}\), or \((0.2\text{--}0.3)\pi\cdot10^{-4}\ \text{cm}^{-1}\). Consequently,

\[ \frac{n^{\varphi}}{n_c}<0.3\pi\,\frac{r}{n_c}\cdot10^{-4}. \tag{17} \]

For example, for a betatron with \(r_o = 50\) cm, where \(n = 0.6\) and \(r = 52\) cm, \(n_\varphi\) amounts to less than \(0.8\%\) of \(n_c\).

The greater the width of the working acceleration zone, the smaller \(\Delta \varphi/\Delta r\), and therefore, even in accelerators with radii of several meters, one may practically assume that the phase shift of the fluxes in the end coils of the magnetic probe does not affect the reading of the \(n\)-meter (within the sensitivity of the instrument, up to \(1\)—\(2\%\)).

Thus, if a tube voltmeter is used as the measuring instrument in \(n\)-meters, then a certain “average” value of the magnetic-field decay index is determined, close to the static value, \(n_c\).

When a cathode-ray oscilloscope is used, strictly speaking, it is not \(n\) that is measured, but some other quantity (a fictitious \(n\), which we shall denote \(n'\)), since

\[ n' = -\,\frac{\Delta U_z}{U_z}\cdot \frac{r}{\Delta r} = -\,\frac{\Delta \dot H_z}{\dot H_z \Delta r}\cdot \frac{r}{\Delta r}. \tag{18} \]

Since at the beginning of the field rise, under continuous electromagnet supply (and practically also under pulsed supply), the equality

\[ \frac{\Delta H_z}{H_z} = \frac{\Delta \dot H_z}{\dot H_z}, \tag{19} \]

is sufficiently valid, the measurement of the magnetic-field decay index can therefore be carried out without great error also with the aid of a cathode-ray oscilloscope.

Fig. 6

Fig. 7

For measuring the magnetic-field decay index while taking into account the influence of saturation of the pole correcting teeth and the radial phase nonuniformity, we have developed special \(n\)-meter circuits.

Figure 6 shows the circuit of an \(n\)-meter based on the comparison method. This circuit makes it possible to measure \(n\) at the beginning of the rise of the magnetic field without the influence of radial phase nonuniformity, and at the end of it, but already with full allowance for the influence of pole saturation. The amplifier indicated by a dashed line in Fig. 6 is not obligatory, but is used when the voltage signals at the integrator output are weak. The time marker may be used for the accurate determination of the instant of time corresponding to a magnetic-field phase of \(90^\circ\) [1].

A modified circuit of a zero \(n\)-meter (Fig. 7), i.e., with the addition of an integrator with an amplifier and a time marker, makes it possible to measure the indica-

the magnetic-field decay index, both with continuous and with pulsed powering of the electromagnet, at any moment of time, taking into account the influence of radial phase nonuniformity and pole saturation.

The time-marker circuit is one variety of the ordinary reactive trigger circuits with cathode coupling. The circuit sensor is a permalloy pickup placed in the magnetic field of the installation at the radius of the equilibrium orbit.

In conclusion, we note that the \(n\)-meters with small-volume measuring probes discussed above make it possible, quite simply and with high accuracy, to carry out a number of experiments connected with studying the influence, on the configuration of the magnetic field, of the arrangement of various current-carrying windings, magnetic shunts, bellows, steel chambers, etc.

REFERENCES

1. Leshchenko I. G., Calculation, construction, and experimental study of the electromagnet of a 100 MeV betatron. Candidate’s dissertation, TPI, 1955.

2. Blewett M. H., Eddy-Current Phenomena in the Cosmotron, Rev. Sci. Instr., 24, 9, 1953.