ESTIMATION OF ELECTRICAL LOSSES IN THE ELECTROMAGNET OF A PMB-6 BETATRON WITH PULSED POWER SUPPLY BY CURRENT OF INCREASED FREQUENCY

V. S. Loginov, A. A. Geizer, V. L. Chakhlov

(Submitted by the Scientific Research Institute of Nuclear Physics, Electronics, and Automation)

In [1] the possibility was shown of effectively using a pulsed power-supply mode for small betatrons. The transition from excitation of the electromagnet by alternating current at a frequency of 50 Hz to power supply by unipolar current pulses makes it possible to raise the pulse repetition frequency to 200 Hz per second, and consequently to increase by a factor of 4 the dose rate of bremsstrahlung generated by the betatron.

When pulsed power supply of betatrons is used, an important task is to solve the question of the distribution of electrical losses in the active parts of the electromagnet. The presence of additional losses, which are caused by significant leakage magnetic fluxes, leads to heating of the active parts and structural components of the electromagnet. A theoretical treatment of this question is associated with certain difficulties, since taking all factors into account leads to complication of the calculation formulas, making a general analysis difficult.

The purpose of the present work is an experimental estimate of the electrical losses in a betatron electromagnet on the basis of thermal measurements.

The experimental betatron installation was made on the basis of a PMB-6 betatron operating in continuous mode from a 50 Hz network [2]. The excitation-pulse parameters were as follows: current-pulse duration 2500 μsec, duty ratio equal to 2, current amplitude $I_m = 376$ A. The winding is made of a rectangular busbar

\[ \frac{2 \times 3.05 \times 10}{2 \times 3.8 \times 10.75} \]

with a cross section of 60 mm² and with the number of turns equal to 36. The magnetic core is assembled from sheets of electrical steel E330, 0.35 mm thick. Temperature measurements were carried out by the method of built-in temperature indicators. Copper-constantan thermocouples of diameter 0.2 mm, pre-calibrated from 0° to 150°C, served as temperature sensors (Fig. 1). The magnitude of the thermoelectric emf was measured with a PP-type dc potentiometer in the range from 0 to 5 mV. The measurement error did not exceed ±0.1 mV at an ambient-air temperature of $20 \pm 2^\circ$C and relative humidity from 30 to 80%. The relative error of measurement of the copper-constantan thermocouple due to thermal inertia is equal to [3]

\[ \delta=\frac{T_0t_0}{1+T_0t_0}, \tag{1} \]

where \(T_0\) is the heating constant of the betatron,

\[ t_0=\frac{mc}{aF} \]

is the heating constant of the thermocouple.

With pulsed excitation of the betatron electromagnet and in the absence of its cooling, the estimate of heat generation per unit volume was made according to the following relationship:

\[ q_{v0}=\left[\left(\frac{1}{k}-1\right)\frac{\gamma_{\mathrm{i}}}{\gamma_{\mathrm{m}}}C_{\mathrm{i}}+C_{\mathrm{m}}\right] \left[(\gamma_{\mathrm{m}}-\gamma_{\mathrm{i}})k+\gamma_{\mathrm{i}}\right]\frac{\partial \vartheta}{\partial \tau}, \tag{2} \]

where

\(k\) is the filling factor of copper (steel),

\(\gamma_{\mathrm{m}},\gamma_{\mathrm{i}}\) are the specific weights of copper and insulation,

\(C_{\mathrm{m}},C_{\mathrm{i}}\) are the specific heat capacities of copper and insulation,

\[ \frac{\partial \vartheta}{\partial \tau} \]

is the rate of change of temperature with time.

Fig. 1. Arrangement of thermocouples in the betatron electromagnet

Knowing the heat release and the volume of the body, the losses in it can be found:

\[ P=q_v\cdot V. \tag{2a} \]

On the other hand, the calculation of the main losses in the winding when the electromagnet is excited by a sinusoidal current is carried out by the formula

\[ P_{\mathrm{m}}=I_{\mathrm{eff}}^2\cdot R_0, \tag{3} \]

where

\[ R_0=\rho\frac{l}{s}, \]

\(\rho\) is the resistivity of the material of the winding conductor;

\(l\) is the length of the conductor, m;

\(s\) is the cross section of the conductor, mm.

A change in the temperature of the conductor is associated with a change in electrical conductivity; therefore, the resistance of a winding made of copper is found from the relation:

\[ r=r_0(1+\alpha' \upsilon), \tag{4} \]

where \(\alpha'\) is the temperature coefficient of resistance.

The effective value of the current is found from the known relation

\[ I_{\mathrm{eff}}=\sqrt{I_0^2+I_1^2+I_2^2+\cdots+I_n^2}, \tag{5} \]

where

\[ I_0 \]
is the dc component of the current;

\(I_1, I_2\ldots I_\ell\) are the rms values of the current of the individual harmonics, or, according to relation [4],

\[ I_{\mathrm{eff}}=\frac{I_m}{\sqrt{2S}}, \tag{6} \]

where

\(S\) is the duty factor of the pulsed process.

The magnitude of the additional losses in the winding can be estimated by the simple formula

\[ P_{\mathrm{add}}=|P|-P_m. \tag{7} \]

Table 1 gives the values of the losses in the betatron winding found from thermal measurements.

Table 1

Winding of the electromagnet of the PMB-6 betatron, made of rectangular busbar
(pulsed power supply \(f=200\ \text{Hz},\ t_c-t_0=25.4^\circ\text{C}\))

* The numbering of the rows is shown in Fig. 1.

The magnitude of the losses in the winding, calculated by (3), was \(P_m=254.82\), whereas in actuality it was \(795.10\ \text{W}\). In this case the added losses are equal to \(540.28\ \text{W}\). Thus, in a winding made of rectangular busbar with pulsed power supply by current of increased frequency (\(f=200\ \text{Hz}\)), the losses found from relation (3) are 3.12 times smaller than those observed in actuality.

To reduce the additional losses in the winding, the rectangular busbar was replaced by a stranded cable consisting of 90 parallel conductors with a cross section of \(0.442\ \text{mm}^2\). The number of turns in the coil-

was 36. Table 2 gives data on the electrical losses in a winding made of multiconductor cable.

The losses calculated from relation (3) are equal to 613.8 W, while those from the thermal experiment are 907.2 W. In this case the additional losses amount to 293.4 W.

Table 2

Winding of the PMB-6 betatron made of multiconductor cable

* The calculations for busbar rows Nos. 3 and 4 were performed assuming the same temperature distribution in the lower (rows Nos. 1, 2) and upper parts of the electromagnet. This assumption is valid for multistory betatrons.

As can be seen from Table 2, the value of the additional-loss coefficient

\[ \left(K_{\text{add}}=\frac{P_{\text{test}}}{P_m}\right) \]

in the winding made of multiconductor cable ($K_{\text{add}}=1.48$) is 2.1 times smaller than in the winding made of rectangular conductor ($K_{\text{add}}=3.12$) (Table 1). In addition, these tables show that the losses are distributed nonuniformly. A comparison of the losses under pulsed supply (Tables 1, 2) with the losses under continuous supply of the electromagnet with current of frequency 50 Hz (Table 3) shows that

Table 3

Winding of an industrial PMB-6 betatron, made of rectangular busbar
($f=50$ Hz, $t_{\text{avg}}=21.5^\circ$C, $I=25$ A)

* The rows belong to the primary (excitation) winding. For these rows the copper filling factor is $k_m=0.804$, and in the remaining rows it is $k_m=0.751$.

In the first case, the additional losses are of more decisive importance than in the second case. This makes it possible to conclude that the greatest leakage fluxes are concentrated at the poles, and that they exert a greater influence on a winding made of rectangular busbar than when supplied at a frequency \(f=50\) Hz. Such an approach to estimating heat generation in the heat-releasing elements of betatrons makes it possible, when the geometrical dimensions are known, to estimate the magnitude of the leakage fluxes.

The electrical losses in the steel under sinusoidal variation of the induction can be calculated by the Steinmetz formula:

\[ P=\left[K_1 f B_m^n+K_2 f^2 B_m^2\right]G, \tag{8} \]

where the first term in brackets represents the specific hysteresis losses, and the second the eddy-current losses. The exponent \(n\) for electrical steels lies within the range \(2\text{–}2.5\). It is known that the total losses in steel, determined experimentally, are greater than the hysteresis and eddy-current losses. Then the total losses are determined from the formula

\[ P_{\mathrm{st}}=P_{\mathrm{h}}+P_{\mathrm{e}}+P_{\mathrm{add}}. \tag{9} \]

Table 4 gives the values of losses in the magnetic circuit obtained from thermal tests and calculated by (8).

Table 4

Magnetic circuit of the industrial PMB-6 betatron \((f=50\) Hz)

* The magnetic flux is directed at an angle of \(90^\circ\) to the rolling direction.

For pulsed supply, the losses in the magnetic circuit are determined from the relation

\[ P=\left[K_1 B_m^n+K_2 B_m^2 f\right]\frac{N}{2}\,G. \tag{10} \]

Table 5 gives the values of losses in the magnetic circuit found experimentally and calculated by (9) at \(f=200\) Hz and \(N=200\) pulses per second.

An analysis of the results obtained (Table 5) shows that with pulsed supply there is an increase in the additional losses in the pole core and pole tip owing to the flux crossing the laminations. An increase in the additional losses caused by this phenomenon in a betatron with continuous supply (Table 4) is observed in the yokes of the magnetic circuit; however, this increase is considerably smaller.

Table 5

* The magnetic flux is directed at an angle of 90° to the rolling direction.

The total losses in the betatron electromagnet are determined by the formula

\[ P_{\mathrm{em}} = P_{\mathrm{m}} + P_{\mathrm{st}}. \tag{11} \]

The calculation results are given in Table 6.

Table 6

Total losses in betatron electromagnets

Conclusions

1. On the basis of thermal measurements, an estimate has been made of the electrical losses in the individual heat-generating elements of the betatron electromagnets.

2. At a supply-current frequency of $f = 50$ Hz, the additional losses are small. This makes it possible to calculate the electrical losses by the usual method adopted in transformer engineering.

3. With pulsed powering of betatrons by a current of frequency $f = 200$ Hz and with a pulse duty factor equal to two, the value of the actual losses in a winding made of a rectangular-section copper conductor is 2–3 times higher than the losses calculated from relation (3).

4. Replacing the copper busbar with an equivalent-cross-section multistrand cable leads to a reduction in additional losses and, consequently, to a reduction in the maximum temperature difference in the winding under identical cooling conditions.

References

1. L. M. Ananyev, Yu. A. Otrubyannikov. Some problems of pulsed power supply for electromagnets of induction accelerators. Pribory i tekhnika eksperimenta, 1967, No. 3.

2. L. M. Ananyev et al. Small-size betatrons and their application in flaw detection. Defektoskopiya, 1968, No. 5.

3. Ya.-V. I. Kostyrko. Accounting for the principal errors of copper–constantan microthermocouples in studies of the heating of electrical machines. Izv. vuzov, series Electromechanics, 1971, No. 6.

4. Yu. A. Otrubyannikov. Candidate’s dissertation. Tomsk, 1967.