DEVICE FOR MEASURING THE MAGNETIC FIELD DECAY INDEX IN BETATRONS

A. E. KAZAK, E. G. KADLUBOVICH, G. A. KUNITSYN, I. G. LESHCHENKO

(Presented by the scientific seminar of the Research Institute of EI and the Department of IIT AVTF)

In setting up the electromagnets of betatrons and synchrotrons, great importance is attached to measuring the magnetic-field decay index \(n\), which is determined by the expression

\[ n = \frac{\Delta H}{\Delta r}\cdot \frac{r}{H}, \]

where \(\dfrac{\Delta H}{\Delta r}\) is the rate of decrease of the magnetic field along the radius from the center of the accelerator,

\(H\) is the magnetic-field strength at the same radius.

There are various methods for indirect and direct measurement of the average and instantaneous values of the magnetic-field decay index [1]. It is known that indirect methods for determining \(n\) from the experimentally obtained dependence \(H=f(r)\) are labor-intensive and have a comparatively large error in accelerators with large orbit radii.

Methods for direct measurement of the magnetic-field decay index, based on the measurement by field sensors of the voltage difference proportional to \(\Delta H\) over the interval \(\Delta r\), and of the voltage at radius \(r\), proportional to \(H\), with subsequent circuit transformations, are more accurate and much more productive.

Below, a variant of an \(n\)-meter device is described for direct measurement of both average and instantaneous values of the magnetic-field decay index.

The block diagram of the device is shown in Fig. 1. As the sensor, a probe is used consisting of three measuring coils, analogous to those described previously in [1].

The measurement is based on the use of a compensation circuit that makes it possible to perform the division operation

\[ \frac{\Delta U_i / \Delta r}{U}, \]

and to take into account the radius of the probe-center position \(r\) by means of a potentiometer, the slider of which is connected to the probe and placed on a special disk in the gap of the accelerator electromagnet.

The division operation is carried out according to the circuit shown in Fig. 2. The output quantities of the circuit, that is, the dividend and the divisor, are the increments of the resistance values \(\Delta r\) and \(\Delta R\) of the arms \(r_3\) and \(R_3\) of the unbalanced bridges \(M_1\) and \(M_2\).

Thermoresistors with indirect heating, TKP-300, having high sensitivity to the power dissipated in the heater [3], are used as the resistors \(r_3\) and \(R_3\). The output quantity of the circuit is the current \(I = \dfrac{\Delta U_i \Delta r}{U}\). Taking into account the radius \(r\), set by the potentiometer \(h_0\), the readings of the measuring instrument are directly proportional to the average value of the decay index of the magnetic field over the period. The heaters are supplied with voltages proportional to \(\Delta H\) and \(H\), taken from the probe.

Fig. 1

Fig. 2

When bridge \(M_1\) is in balance, the output voltage \(U_1\), equal to the input voltage of the amplifier, will be zero, and the instrument will indicate a zero current value. As a result of the unbalance of bridge \(M_1\), a voltage will appear at its output

\[ U_1 = U_0 \cdot \frac{\Delta r}{4 \cdot r}, \tag{2} \]

where \(U_0\) is the supply voltage of bridge \(M_1\).

The output voltage is amplified by an amplifier and fed to the input of the second bridge. The unbalance of the second bridge is compensated with the unbalance of the first bridge up to complete compensation

\[ U_2=\frac{I\cdot \Delta R}{4}, \tag{3} \]

in the ideal case \(U_1=U_2\),

\[ U_0=\frac{\Delta r}{4\cdot r}=I\cdot \frac{\Delta R}{4}, \]

whence

\[ I_{\mathrm{out}.1}=\frac{1}{r}\cdot U_0\cdot \frac{\Delta r}{\Delta R}. \tag{4} \]

For \(U_1\ne U_2\), the relative undercompensation is

\[ a=\frac{U_1-U_2}{U_1}=\frac{1}{1+K\cdot \beta}. \tag{5} \]

Substituting here the values of \(U_1\) and \(U_2\),

\[ \frac{1}{1+K\cdot \beta}=1-\frac{I\cdot r\cdot \Delta R}{U_0\cdot \Delta r}, \]

whence

\[ I_{\mathrm{out}.2}=\frac{K\cdot \beta\cdot U_0\cdot \Delta r}{r\cdot \Delta R\cdot (1+K\cdot \beta)}. \tag{6} \]

The output voltage of the second bridge is

\[ U_2=\frac{1}{4}\cdot \frac{\Delta R}{R}\cdot U_{\mathrm{out.amp}}; \]

\[ \beta=\frac{U_2}{U_{\mathrm{out.amp}}}=\frac{1}{4}\cdot \frac{\Delta R}{R}. \tag{7} \]

Substitute the value of \(\beta\) into the output current

\[ I_{\mathrm{out}.2}=\frac{K\cdot \Delta r\cdot U_0}{r\cdot (4R+K\cdot \Delta R)}. \tag{8} \]

By specifying the value of the relative undercompensation, one can determine the gain coefficient of the amplifier \(K\):

\[ a=\frac{I_{\mathrm{out}.1}-I_{\mathrm{out}.2}}{I_{\mathrm{out}.1}}=\frac{4R}{4R+K\cdot \Delta R}, \]

whence

\[ K=\frac{4\cdot R(1-a)}{\Delta R}. \tag{9} \]

For the selected parameters of the TKP-300 thermistors and a relative undercompensation of \(0.001\), the amplifier gain coefficient is equal to 5000. The amplifier is made as a three-stage amplifier, using transistors.

The voltage at the potentiometer output is equal to

\[ U_{\mathrm{out}}=K\cdot \frac{\Delta U/\Delta r}{U}\cdot r, \tag{10} \]

i.e., the output voltage is proportional to \(n=n=\dfrac{\Delta U\cdot r}{U\cdot \Delta r}\) (the magnetic-field decay index). The potentiometer slider is rigidly connected

with a slide into which the sensor is inserted during measurements. The potentiometer itself is located on the measuring disk along the radius of the electromagnet air gap.

As the recording device of the instrument, providing the greatest measurement accuracy, a V3-7 tube millivoltmeter is used; during measurements it is connected to special terminals on the front panel of the instrument. In this case the scale of the millivoltmeter is first calibrated directly in \(n\).

To measure the instantaneous values of the magnetic-field decay index, a device is used whose block diagram is shown in Fig. 1. By means of a phase shifter the sinusoid is shifted from 0 to 90°. The filter serves to preserve the shape of the sinusoid. To increase the stability of operation and reduce the noise level at the input of the instrument, an emitter follower EP is used. The Schmitt trigger TSh is used to form rectangular pulses.

The differentiating circuit DTs converts these pulses into trigger pulses, which start the one-shot. From the output of the triggered multivibrator ZhM, negative pulses with a duration on the order of microseconds are fed to the four-diode sample-and-hold circuits ChSOP\(_1\) and ChSOP\(_2\) and open the diodes for the duration of these microseconds. During the action time of the pulses the diodes are open and the capacitors at the output (Fig. 3) are charged to the corresponding instantaneous values of the measured voltages, which are supplied from converter D [4].

Fig. 3

In order to use, for determining \(n\), the circuit that was employed for measuring average values, these pulses, proportional to the instantaneous values of the measured voltages, must be applied to the arms of bridges \(M_1\) and \(M_2\).

But since the time constants of the heated thermoresistors are of the order of 13.5 sec, it is necessary to convert the short pulses from the one-shot into broad ones. This can be achieved by increasing the discharge time constants of the capacitors. For this purpose emitter followers EP\(_1\) are installed on compound triodes.

If the sinusoid is shifted from 0° to 90°, then the pulse is thereby also shifted; for a time equal to its duration (on the order of microseconds) it opens the sample-and-hold circuits. At the output, voltages are obtained that are proportional to the instantaneous values \(dU\) and \(U\) coming from the sensor. In the “check” position of switch \(P_3\) and in positions I of switches \(P_1\) and \(P_2\), the voltages \(\Delta U\) and \(U\) of the sensor are checked in succession. In the “measurement” position of switch \(P_3\), measurements of average and instantaneous values are made. During measurement

for average values, switches \(P_1\) and \(P_2\) are simultaneously set to position 2; for instantaneous values, to position 3.

The error in measuring the average value of the magnetic-field decay index depends entirely on the error of the vacuum-tube millivoltmeter used as the output instrument, and on the errors of the electronic circuit and the sensor.

\[ \delta = \delta_{\text{v.m}} + \delta_{\text{el.circ}} + \delta_{\text{s}} . \]

The measurement error of the vacuum-tube millivoltmeter is \(1.5\%\).

Errors due to changes in ambient temperature do not affect the measurement, since changes in the resistances of thermistors \(\Delta r\) and \(\Delta R\) depend on temperature to the same degree and, as a result of division, the dependence of the output quantity on temperature is eliminated.

The error of the electronic circuit is specified when determining the amplifier gain through the magnitude of the relative noncompensation. For this instrument \(\delta_{\text{el.circ}} = 0.2\%\). The sensor error may be divided into the following groups:

a) error due to possible incorrect installation of the coils along the radius;

b) error due to possible skewing of the coils;

c) temperature error.

The first error is determined by the formula [1]

\[ \frac{\Delta n}{n} = 1 - \left( \frac{r + \Delta r}{r} \right)^{1-n}, \]

where \(\Delta r\) is the displacement of the coil along the radius.

When the sensor is displaced by \(0.5\ \text{mm}\) at the radius of the equilibrium orbit \((r = 24.5\ \text{cm})\), where the actual \(n = 0.062\), the measurement gives a relative error of \(0.2\%\).

The error arising due to skewing of the coils is equal to

\[ \frac{\Delta n}{n} = 1 - \frac{1}{\cos \alpha}, \]

where \(\alpha\) is the angle of deviation.

At \(\alpha = 2^\circ\) the error is \(0.5\%\).

Temperature error occurs when the measuring probe is heated in the electromagnet gap and is calculated by the formula

\[ \frac{\Delta n}{n} = \frac{\Delta R}{R_{\text{p}} + R_{\text{k}} + \Delta R_{\text{k}}}. \]

Here \(\Delta R_{\text{k}} = R_{\text{k}20^\circ}\cdot \alpha' \cdot \Delta T\) is the increase in coil resistance with increasing temperature; \(\alpha'\) is the mean temperature coefficient (for copper, \(0.00393\));

\(R_{\text{k}20^\circ}\) is the resistance of the measuring coil at \(20^\circ\text{C}\).

For the sensor used in this instrument, when the temperature is increased by \(1^\circ\text{C}\), the error is \(0.13\%\). If, when moving the sensor from one point to another, the line voltage changes by \(1\%\), this will also lead to a reading error equal to \(1\%\) [1]. Therefore, during measurements it is necessary to observe the condition of line-voltage stability \((0.05 \div 0.01\%)\). In this case the total maximum error will be of the order of \(2\%\).

The error of the electronic circuit used for measuring instantaneous values of the magnetic-field decay index may be neglected. The pulses charging the capacitors in \(\mathrm{ChSOP}_1\) and \(\mathrm{ChSOP}_2\) follow at an interval of \(0.02\ \text{s}\). Owing to emitter followers with high input resistances, the discharge constants are equal—

is \(\tau = 0.082\) sec, i.e., the capacitors do not have time to charge during the pulse repetition interval.

The resistance of the thermistor will be kept constant at the selected instantaneous voltage value if, even allowing for an error of the electronic circuit of up to 5%, that is, \(\tau = 0.082 \pm 0.0041\) sec.

REFERENCES

1. I. G. Leshchenko, Development of a Method for Measuring Magnetic Fields in Accelerators. Proceedings of TPI, 1956.

2. P. P. Ornatskii, Automatic Measuring Instruments. Tekhnika Publishing House, 1965.

3. V. Yu. Lavrinenko, Handbook of Semiconductor Devices. Tekhnika Publishing House, 1964.

4. G. Partridge, Electronic Measuring Instruments. Gosenergoizdat, 1961.