Analysis of Errors and Performance of a Compensation Phase Magnetometer

L. M. Ananyev; S. F. Vasilevsky

(Presented by the scientific seminar of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI)

When tuning betatrons, a considerable part of the labor and time is spent on measuring and correcting the phase structure of the magnetic field. At present, certain advances have been made in the field of measuring phase inhomogeneities (PI) of the magnetic field of betatrons [1—5]. However, the existing methods of measurement do not sufficiently meet the increased requirements of betatron construction and require improvement.

In connection with the wide introduction of betatrons into industry and the organization of their small-series production, the problem of increasing the productivity of all adjustment work, including work connected with measuring and correcting the phase structure of the magnetic field—which takes up a considerable part of the time in tuning the accelerator electromagnet—has become urgent.

The need to determine the suitability of electromagnets before the start of adjustment work imposes increased requirements on the accuracy of the apparatus.

At present, differential and compensation methods for measuring PI are known.

It should be noted that, in measurements of PI, the most widespread is the compensation method, based on biasing the zero-field sensors with direct current. Therefore, it was precisely this method that we subjected to the most complete critical analysis.

The method is based on fixing the instant at which the magnetic field passes through the zero value and on compensating the shift between the signals of the zero-field sensors by biasing the reference sensor with direct current [1—2]. The essence of the compensation method is as follows.

In the betatron field, two zero-field sensors connected in opposition are installed: a reference (stationary) sensor and a measuring (movable) sensor (Fig. 1). The reference sensor is biased with a direct current regulated by resistance \(R\). An oscilloscope \(K.O.\) serves as the indicator. By changing the bias current, coincidence of the pulses of the reference and measuring zero-field sensors is achieved, and from the readings of the instrument at the moment of compensation the phase shift is determined.

In order to study the characteristics of the compensation method, its shortcomings, and ways of eliminating them, a [[unclear: continuation begins on next page]]

instrument—a compensation phase magnetometer, the complete schematic diagram of which is presented in Fig. 2.

The instrument includes two power supplies and a measuring circuit. The first voltage source (diodes \(D_1 \div D_4\)) is used for the main biasing of the zero-field sensors. The second rectifier (diodes \(D_5 \div D_8\)) is intended for the auxiliary bias, compensating the constant phase shift. Adjustment of the bias current is carried out coarsely by rheostats \(R_7, R_8\) and smoothly by potentiometers \(R_5, R_6\). To change the direction of the current through the sensor winding, polarity switches \(P_1, P_2\) are used.

Fig. 1. Compensation method for measuring phase inhomogeneity of a magnetic field.

The signal windings of the control and measuring sensors are connected through potentiometers \(R_9, R_{10}\), by means of which an optimal ratio of the signal levels on the oscilloscope screen can be obtained.

Changing the current-measurement ranges is carried out with the aid of the gang switch PP by switching the shunts of the M-24 microammeter. At the same time, in order to change the limits of adjustment of the bias current, in the different positions of switch PP the voltage on the secondary winding of the power transformer is changed by switching the number of turns of the secondary winding.

In position 1—1 of switches \(P_3, P_4\), the main-bias current is measured; in position 2—2 the same instrument measures the auxiliary-bias current.

The sensor leads are soldered to a six-pin ShR-6 connector. Permalloy peak-transformer sensors were used as zero-field sensors. The control sensor has two windings: one for the main bias, the second for the auxiliary bias, compensating the constant phase shift.

The experimentally determined principal technical characteristics of the instrument are as follows:

1. Sensitivity: 0.25 oersted/div.

2. Current-adjustment limits:

I range 0—0.3 mA,

II range 0—3 mA,

III range 0—30 mA,

IV range 0—300 mA.

1. Limits of measurement of phase inhomogeneity: from 1 to 50 oersteds.

For evaluation of the compensation method, its errors and performance are of greatest interest.

Investigation of the errors of the phase magnetometer showed that there is a considerable number of sources of error. Among them one may distinguish errors due to inaccuracy in setting the measuring disk, induced emf, inaccuracy in the calibration of the sensors and of the instrument measuring the bias current, etc.

The principal sources of errors, their influence on the accuracy of measurements, and ways of eliminating them are considered below.

Fig. 2. Schematic circuit diagram of the experimental compensation phase magnetometer.

Inaccuracy in Setting the Sensor in Azimuth

To move the measuring sensor in azimuth, it is mounted on a disk with azimuth marks in degrees. When setting the sensor to some azimuth $\Theta_i$ by rotating the disk, setting errors may arise because of inaccurate centering of the disk, inaccurate application of the degree marks, inaccurate positioning of the required mark against the pointer, etc. When the disk is set at points with an interval of $10^\circ$, the setting inaccuracy $\Delta \Theta_i$ is $(0.5—1)^\circ$, i.e., the relative setting error is $(5—10)\%$. In the graphical construction of the phase-inhomogeneity curve, the measurement result for the point $\Theta_i \pm \Delta \Theta$ is plotted for the point $\Theta_i$. As a result, the graph of the function $\Delta H = f(\Theta)$ passes above or below the true value. The error in the value of the inhomogeneity at a given point will depend on the steepness of the curve at this point. It can be shown that

\[ \delta(\Delta H)=\frac{\Delta(\Delta H)}{\Delta H_i} =\frac{(\Delta H_i-\Delta H_{i-1})}{\Delta H_i}\cdot\delta\Theta, \tag{1} \]

where $\Delta(\Delta H)$ is the absolute error of the PI due to the inaccuracy of sensor setting,

$\Delta H_i, \Delta H_{i-1}$ are the PI values at two neighboring points.

The difference $(\Delta H_i-\Delta H_{i-1})$ is equal to 0 when the PI values at two neighboring points are the same, and is equal to $\Delta H_i$ when $\Delta H_{i-1}=0$ and $\Delta H_i\ne0$. Therefore

\[ \delta(\Delta H)=\frac{\Delta(\Delta H)}{\Delta H_i} =(0\div1)\delta\Theta. \tag{2} \]

Thus, the limiting relative error of PI measurement due to the inaccuracy of setting the sensor in azimuth is

\[ \delta(\Delta H)=(5\div10)\%. \]

It is difficult to achieve more accurate setting of the sensor with the existing disk designs. It is more expedient to rotate the disk with a constant angular velocity $\omega_g$.

Error Due to the Presence of an emf Induced in the Sensor Winding

An emf with the frequency of the magnetic field is induced in the sensor winding. In addition to the direct current, an alternating current flows in the supply circuit; it creates an additional magnetic field $H'_{\sim}$ in the volume of the sensor.

As a result, equalization of the signal levels on the oscillograph screen is carried out with a compensating field $H_=+H'_{\sim}$, which differs from the field $H_=$ measured by the instrument by $H'_{\sim}$. A measurement error appears:

\[ \delta(\Delta H)=\frac{H'_{\sim}}{H_=+H'_{\sim}} =\frac{1}{1+\dfrac{i_gR_1(R_1+R_2)}{eR_2}}, \tag{3} \]

where $e$ is the emf induced in the sensor winding,

$i_g$ is the current through the divider (potentiometer $R$),

$R_1$ is the resistance of the lower arm of the divider (Fig. 1),

$R_2$ is the resistance of the rheostat in the supply circuit.

It follows from formula (3) that the error is especially large at small supply currents, when $R_1$ is small. To reduce the measurement error, the magnitude of the induced emf must be reduced. For this purpose it is advisable

to construct sensors with a small cross section and a small number of turns.

It is inadvisable to increase the divider current \(i_g\) beyond \(50 \div 100\) mA, since the power and the dimensions of the rectifier and of the divider resistor \(R\) increase.

To reduce the effect of the induced emf, it is necessary to include a choke in the bias circuit. In the presence of a choke \(L\), the relative error will be

\[ \delta(\Delta H)= \frac{1}{ 1+\dfrac{i_g R_1\sqrt{(R_1+R_2)+(\omega L)^2}}{eR_2} }. \tag{4} \]

When \(\omega L \gg R_2\), the error will be small. The influence of the inductance increases still further, since the phase of the alternating current is shifted and, in the limit, the zero of the main field may coincide with the zero of the additional field due to the induced emf. In this case it is recommended to carry out the measurement with the largest possible resistance \(R\) introduced in the potentiometer (increase \(R_1\)), and to adjust the bias current chiefly by means of the rheostat \(R_2\). Still better results can be obtained if the resistance of the divider \(R_1\) is kept constant, while the bias current \(i_p\) is adjusted either by regulating the voltage on the winding of the power transformer or by using a controlled rectifier.

Influence of a Large Constant Component of the FN

Because of the limited interpole space of the betatron electromagnet, it is not always possible to place the reference and measuring sensors at the same radius. When the zero-field sensors are located at different radii, a constant \(\Delta H_{=}\) and an alternating \(\Delta H_{\sim}\) component appear in the measured FN curve. The ratio \(\Delta H_{=}/\Delta H_{\sim}\) depends on the arrangement of the sensors and may be greater than 1. In this case, when measuring the FN, the error in determining the alternating component will be larger even when high-precision instruments are used,

\[ \delta(\Delta H)_{\sim}= \left[ \frac{\Delta H_{=}+\Delta H_{\sim}}{\Delta H_{\sim}} \right]\delta(\Delta H)_{\mathrm{pr}}, \tag{5} \]

where \(\delta(\Delta H)_{\mathrm{pr}}\) is the relative error of the instrument.

To reduce the influence of a large \(\Delta H_{=}\), we used an additional winding on the sensor, through which a current passed that compensated \(\Delta H_{=}\). In this case the accuracy of measuring \(\Delta H_{\sim}\) increased.

The influence of the constant component of the FN can also be eliminated by the method [7], which consists in placing both sensors (the reference and the measuring sensor) on a movable disk at the same radius and at an angle \(\Delta \Theta = \dfrac{360}{n}\) (\(n\) is the number of measurement points).

By successively moving the disk through the same angle \(\Delta \Theta\) and measuring the partial phase differences \(\varphi_i\), one can obtain the FN curve. The true phase shift at any \(i\)-th point of the field is equal to the sum of the measured partial phase differences:

\[ \varphi_i=\sum_{i=1}^{i}\bar{\varphi}_i. \tag{6} \]

Since the sensors are located at the same distance from the center and at a small azimuth from one another, the constant component of the FN is considerably reduced.

The limiting absolute measurement error at a certain point \(\Theta_i\) will be equal to

\[ \Delta \varphi_i=\sum_{i=1}^{i}\Delta \overline{\varphi}_i =\delta\varphi_{\mathrm{pr}}\cdot \sum_{i=1}^{i}|\overline{\varphi}_i|. \tag{7} \]

However, the limiting absolute error \(\Delta \varphi_i\) cannot characterize the accuracy of this method, since the value of the phase shift at a certain point is determined as the result of several measurements. With many measurements, errors with opposite signs are equally probable and, consequently, the possibility of their mutual cancellation increases. Therefore it is more correct to estimate the error from the deviation from the most probable value of the measured quantity. The most probable value of the measured quantity is the arithmetic mean of many measurements, and the absolute error of the arithmetic mean is

\[ \Delta \varphi_{\mathrm{sr}}= \frac{\delta\varphi_{\mathrm{pr}}\cdot \sum_{i=1}^{i}|\overline{\varphi}_i|}{\sqrt{n}}. \tag{8} \]

Comparison of expressions (8) and (5) shows that the accuracy of the proposed method is higher under the condition

\[ |\varphi_{=}+\varphi_{\sim}|>\frac{1}{\sqrt{n}}\cdot \sum_{i=1}^{i}|\overline{\varphi}_i|. \]

Fulfillment of this condition depends on the shape of the PhN curve. It is fulfilled for large constant phase shifts.

The results of the theoretical analysis and experimental verification of the errors are presented in Table 1.

Table 1

Errors of the compensation method for measuring PhN

As a result of the action of the listed sources of errors, the total limiting error reaches a value on the order of 50—78%.

In addition, conditions exist for the occurrence of subjective errors, which further reduces the accuracy and objectivity of the measurement results. Implementation of the proposed recommendations reduces the error of the method, but the accuracy of the measurements still remains insufficient (on the order of 25%).

The compensation method with adjustment of the zero-field sensors can be recommended for preliminary estimate measurements of the phase structure of accelerator electromagnets. When determining the quality of an electromagnet and the degree of its compliance with tolerances on deviations of the phase structure, a higher measurement accuracy is needed; therefore it is necessary to develop another, fundamentally new method for measuring FN.

Experience in working with a compensation phase magnetometer shows that the process of measuring FN requires large expenditures of labor and time, and the measurement operations are performed by highly qualified operators.

In order to determine the causes of the low productivity and ways of eliminating them, an analysis was made of the productivity of the FN measurement process by the compensation method.

It was established that the low productivity is connected primarily with the large number of manual and computational operations performed by the experimenter when carrying out the measurements. These include:

1) setting the sensor at the required azimuth,
2) adjusting the compensation current to align the signal levels on the oscilloscope screen;
3) reading the instrument indications (microammeter),
4) recording the instrument indications,
5) constructing the graph of the FN curve,
6) harmonic analysis of the FN curve.

The results of the experimental determination of the time spent on measuring FN (items 1, 2, 3, 4 of the listed operations) are presented in Table 2. The experimentally determined average time for a measurement cycle (36 points) was 20 min, with 2 people engaged in the measurements.

Up to 5 min are spent on constructing the graph of the FN curve. The time expenditures are especially large for the process of harmonic analysis of the FN curve. Timing of the process of determining the amplitudes and phases of the first three harmonics by the method of K. P. Yakovlev [6] showed that up to 30 min are spent on the process of harmonic analysis.

Increasing the productivity of the measurement process is possible only by automating the manual and computational operations. However, the possibilities for automating the compensation method are limited. Thus, to automate operations 2, 3, and 6, devices for reading images are needed, which is still technically difficult.

At present there are numerous techniques for measurement by converting the measured nonelectrical quantity into an electrical signal and using electronic means. Recording and analysis of the electrical signal at the present level of technology do not present any special difficulties. To implement the achievements of electronics, the method of measuring FN itself should be changed. It is advisable to measure FN by converting the time interval (or phase), set by the zero-field sensors, into a voltage without using an oscilloscope or other means of visual indication of the compensation instant.

To automate the measurement process, movement of the field sensor in azimuth must also be carried out automatically. This

can be carried out with the aid of a mechanical system driven by an electric motor.

Table 2

Productivity of the process of measuring FN by the compensation method.

The angular velocity of rotation of the moving zero-field sensor must be constant and much less than the frequency of the voltage supplying the electromagnet. Since the period of the supply voltage is $T_{\mathrm{s}}=0.02$ sec, the time for one revolution of the moving sensor may be taken as 1 sec (two orders of magnitude greater than $T_{\mathrm{s}}$). The electronic measuring device must have sufficient speed to manage, during one revolution, to register the incoming signals, which is entirely possible.

Taking into account the time required for the establishment of transient processes, the total time for measurement and analysis with such a system will be no more than 1 min. Consequently, the productivity of the process of measuring and analyzing the FN of the magnetic field can be increased almost 100-fold. At the same time, the use of the sensor makes it possible to considerably reduce the error in measuring FN due to the inaccuracy of setting the sensor in azimuth.

Conclusion

Theoretical and experimental studies of a compensation phase magnetometer with tuning of the zero-field sensors make it possible to draw the following conclusions.

The most important technical characteristics of the instrument—the accuracy and productivity of the measurement process—no longer meet the increased requirements of betatron construction, even with substantial improvement of the method for measuring FN. To improve these characteristics, it is advisable to develop fundamentally new methods of measu-

of the FN without visual indication of the moment of compensation, by using electronic automation means.

The instrument constructed and the methodology developed can be used for rough estimation measurements of the phase structure of a magnetic field.

REFERENCES

1. L. M. Anan'ev et al. Induction Electron Betatron Accelerator. Gosatomizdat, 1961.

2. L. M. Anan'ev. Candidate dissertation. Tomsk, 1956.

3. Yu. K. Petrov. Phase magnetometer. Scientific report of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI, 1961.

4. G. M. Tumaikin. Permalloy magnetometer. Scientific report of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI, 1961.

5. G. M. Tumaikin et al. Proceedings of the Fourth Interuniversity Conference on Electron Accelerators. Izd. “Vysshaya shkola,” Moscow, 1964.

6. K. P. Yakovlev. Mathematical Processing of Measurement Results. ITL, 1950.

7. Registration certificate No. 32-58120. Compensation phase magnetometer, 1966.