Features of Measuring Phase Inhomogeneity of Electromagnets under Mixed Power Supply of Betatrons
Abstract Generated abstract
This paper examines difficulties in measuring the phase inhomogeneity of betatron electromagnets operated with mixed power supply, consisting of a constant bias field and an alternating field. It analyzes how increasing premagnetization shifts the magnetic-field zero crossing into a region of reduced field-change rate and increased sinusoidal curvature, which lowers field-zero sensor pulse amplitude, increases pulse duration, and makes measurements more sensitive to field instability. The authors derive relations connecting phase inhomogeneity with and without bias, show that zero-level measurements may not represent injection-level conditions, and recommend measuring in the unbiased mode followed by recalculation. They also give a simple oscilloscope method for determining the degree of premagnetization needed for this recalculation.
Full Text
FEATURES OF MEASURING THE PHASE INHOMOGENEITY OF ELECTROMAGNETS UNDER MIXED POWER SUPPLY OF BETATRONS
L. M. Anan’ev, S. F. Vasilevskii, Yu. P. Yarushkin
(Presented by the scientific seminar of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI)
Mixed power supply of betatrons (a constant field plus an alternating field) is used to improve the characteristics of the accelerator. As shown in [1], biasing with a constant field facilitates the operating regime of the electromagnet, improves injection conditions, increases the economic efficiency of the installation, and reduces the phase inhomogeneity of the magnetic field.
Under mixed power supply of betatrons, the instant at which the magnetic field passes through zero is shifted to a region where the rate of change of the field strength \(\dot H^1_{\mathrm{om}}\) is small. It can be shown that \(\dot H^1_{\mathrm{om}}\) at the moment when the field passes through zero is related to the degree of biasing \(c_{\mathrm{n}}\) as follows:
\[ \dot H^1_{\mathrm{om}}=\omega H'_{\mathrm{om}}\sqrt{1-c_{\mathrm{n}}^2}, \tag{1} \]
where \(H^1_{\mathrm{om}}\) is the amplitude of the alternating-field strength under biasing,
\(\omega\) is the angular frequency of the field,
\(c_{\mathrm{n}}\leqslant 1\) is the degree of biasing.
It follows from this that, as \(c_{\mathrm{n}}\) increases, the rate \(\dot H^1_{\mathrm{om}}\) of change of the field strength at the moment when the field passes through zero decreases and tends to zero as \(c_{\mathrm{n}}\) tends to 1. In addition, for values of \(c_{\mathrm{n}}\) close to unity, the moment at which the magnetic field passes through zero lies in the region where the curvature \(k_0\) of the sinusoidal field (and, consequently, the nonlinearity as well) is greatest. We determine the curvature \(k_0\) of the sinusoidal field from the known formula [2]:
\[ k_0=\frac{\omega^2 H'_{\mathrm{om}}c_{\mathrm{n}}} {\left[1+(\omega H'_{\mathrm{om}})^2(1-c_{\mathrm{n}}^2)\right]^{3/2}}. \tag{2} \]
From expression (2) it is evident that, as \(c_{\mathrm{n}}\) tends to 1, the curvature \(K_0\) tends to the maximum value
\[ k_{0\max}=\omega^2 H'_{\mathrm{om}}. \tag{3} \]
These features of the magnetic field near the zero value under mixed power supply of betatrons (a decrease in the rate of change of the field strength \(\dot H^1_{\mathrm{om}}\) and an increase in the curvature \(k_0\)) appear when measuring the phase inhomogeneity of electromagnets. In the case when measurement of the phase structure of the magnetic field is carried out by a compensation ...
by the method with biasing of the peak-transformer sensors of the field zero [3], when \( \dot H^1_{\mathrm{om}} \) is reduced, the amplitude of the sensor pulses decreases.
In addition, a decrease in \( \dot H^1_{\mathrm{om}} \) causes an increase in the duration of the sensor pulses, which leads to an increase in the lower sensitivity threshold and in measurement errors.
The increase in nonlinearity near the field zero has an especially strong effect on the dependence of the results of measurements of phase inhomogeneity on field instability, which leads to nonconstancy of the degree of biasing \(c_{\mathrm{n}}\). This dependence is obtained by logarithmic differentiation of the formula known from [1], which relates the phase inhomogeneity in a system with biasing \(\Delta H_{\mathrm{n}}\) and without biasing \(\Delta H\),
\[ \Delta H_{\mathrm{n}}=\frac{\Delta H\cdot \sqrt{1-c_{\mathrm{n}}^{2}}}{1+c_{\mathrm{n}}}. \tag{4} \]
From (4) we obtain the relative change of the phase inhomogeneity in the mixed-power mode when the degree of biasing changes because of instability of the constant and alternating components of the magnetic-field intensity,
\[ \delta(\Delta H_{\mathrm{n}})=\frac{\Delta(\Delta H_{\mathrm{n}})}{\Delta H_{\mathrm{n}}} =\delta(\Delta H)+\frac{\Delta c_{\mathrm{n}}}{1-c_{\mathrm{n}}^{2}}, \tag{5} \]
where \(\delta(\Delta H)\) is the relative change of the PH in the mode without biasing, and \(\Delta c_{\mathrm{n}}\) is the absolute change in the degree of biasing.
It is evident from expression (5) that, when \(c_{\mathrm{n}}\) is close to 1, small changes in the degree of biasing lead to a sharp change in the phase inhomogeneity and, consequently, to sharp changes in the results of repeated measurements.
In the mode without biasing, owing to the considerable linearity of the initial portion of the sinusoidal field, the difference between the phase inhomogeneity \(\Delta H_{0}\) measured at the zero level and that at the injection level \(\Delta H_{i}\) is so small that it is neglected. In the mixed-power mode, the increase in the nonlinearity of the field region near the zero value leads to the fact that the results of measuring the phase structure at the zero-field level cannot characterize the phase structure at the injection level \(H_{i}\).
By simple mathematical calculations one can obtain the following expression, relating the phase inhomogeneity at the injection level \(\Delta H_i\) and at the field-zero level \(\Delta H_0\),
\[ \frac{\Delta H_i}{\Delta H_0} = \frac{\sin(\Theta_{\mathrm{n}}\pm \varphi-\varphi_i)-\sin(\Theta_{\mathrm{n}}-\varphi_i)} {\sin(\Theta_{\mathrm{n}}\pm \varphi)-\sin\Theta_{\mathrm{n}}}, \tag{6} \]
where \(\Theta_{\mathrm{n}}\) is the biasing angle,
\(\varphi_i\) is the injection phase,
\(\varphi\) is the phase shift due to the field inhomogeneity.
It follows from this that, in order to obtain values of the phase structure of the magnetic field at the moment of injection, it is necessary to bring the measuring field-zero sensor to the injection level by means of additional biasing.
The features considered make it difficult to measure the phase structure of electromagnets under mixed power supply. It is advisable to measure the phase inhomogeneity in the mode of electromagnet supply without biasing by a constant field and to recalculate the results obtained using the formulas.
In this case, if the amplitude of the purely alternating field is
\[ H_{\sim}=H_{\mathrm{om}}=H'_{\mathrm{om}}(1+c_{\mathrm{n}}), \]
then the recalculation is carried out according to the expression
\[ \Delta H_{\text{p}}=\Delta H'\frac{\sqrt{1-c_{\text{p}}^{2}}}{1+c_{\text{p}}}. \tag{7} \]
If \(H_{\sim}=H'_{\text{om}}\) (the dc bias is simply switched off), then
\[ \Delta H_{\text{p}}=\Delta H'\frac{1}{\sqrt{1+c_{\text{p}}}}. \tag{8} \]
As is evident from expressions (7, 8), to recalculate the measurement results in the mode without bias into values of the phase inhomogeneity under mixed power supply, it is necessary to know the degree of premagnetization,
\[ c_{\text{p}}=\frac{H_{0}}{H'_{\text{om}}}. \]
The easiest way to measure \(c_{\text{p}}\) is with an oscilloscope, without measuring the constant field \(H_{0}\) and the alternating field \(H'_{\text{om}}\).
If the signals of a peak-transformer field-zero sensor placed in an electromagnet with mixed power supply are applied to the oscilloscope input, then on the oscilloscope screen a pattern similar to that shown in Fig. 1 will be seen. The sinusoidal curve corresponds to the emf induced in the turns of the sensor winding, and the sharp spikes correspond to remagnetization of the permalloy core at the instant when the field passes through zero. The distance \(L\) represents, on a certain scale, the period of variation of the magnetic field, while the length \(l\) depends on the degree of premagnetization \(c_{\text{p}}\). Having measured these lengths \(L\) and \(l\), the degree of premagnetization is easily determined from the formula
Fig. 1.
\[ c_{\text{p}}=\cos \pi\frac{l}{L}. \tag{9} \]
Thus, under mixed power supply of betatrons, it is advisable to measure the phase inhomogeneity (PH) in the mode without dc bias, using the same apparatus as for measuring PH in a purely sinusoidal field, with the corresponding recalculation according to the measured value.
REFERENCES
- Yu. P. Yarushkin. Candidate dissertation. Tomsk, TPI, 1965.
- M. Ya. Vygodsky. Handbook of Higher Mathematics. Gostekhteorizdat. Moscow, 1956.
- L. M. Anan’ev et al. Induction Electron Accelerator—Betatron. Gosatomizdat, 1961.