ON THE CALCULATION OF THE MAGNETIC CIRCUIT OF COMBINED MACHINES

A. I. Skorospeshkin, M. L. Kostyrev

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The problems of calculating the magnetic circuit of combined machines have been treated in works [1, 2, and others]. In one of them [1], an analysis is given of two rotating fields combined in a single magnetic core, with the magnetization curve of steel approximated by the formula

\[ H=\alpha\cdot \operatorname{sh}^{\beta}\cdot B, \tag{1} \]

where \(\alpha\) and \(\beta\) depend on the grade of steel and on the portion of the magnetization curve. As a result of the analysis [1], it was established that the influence of higher harmonics in the m.m.f. may be neglected and the calculation may be carried out only for the first harmonics. The amplitudes of the first-harmonic m.m.f.’s on the teeth and yoke for the first and second combined machines are determined by the formulas [1]

\[ F_{z1}=2L_z\cdot \alpha \cdot I_0(\beta\cdot B_{zm2})\cdot I_1(\beta\cdot B_{zm1}), \tag{2} \]

\[ F_{z2}=2L_z\cdot \alpha \cdot I_0(\beta\cdot B_{zm1})\cdot I_1(\beta\cdot B_{zm2}), \tag{3} \]

\[ F_{a1}=\frac{4}{\pi}\cdot L_{a1}\cdot \alpha \cdot I_0(\beta\cdot B_{am2})\cdot I_1(\beta\cdot B_{am1}), \tag{4} \]

\[ F_{a2}=\frac{4}{\pi}\cdot L_{a2}\cdot \alpha \cdot I_0(\beta\cdot B_{am1})\cdot I_1(\beta\cdot B_{am2}), \tag{5} \]

\(I_0(\beta B)\), \(I_1(\beta B)\) are Bessel functions of a purely imaginary argument of the first kind, of zero and first order,

\(B_{zm1}, B_{zm2}, B_{am1}, B_{am2}\) are the amplitudes of the inductions in the teeth and yoke from the fields of the first and second machines,

\(L_z, L_{a1}, L_{a2}\) are the corresponding lengths of the sections.

The calculation method using formulas (2–5) is considerably simpler than another [2], which is associated with repeated grapho-analytical calculation of the magnetic circuit as the combined field is displaced relative to the field being calculated.

In addition, the method using approximation of the magnetization curve by an analytical dependence is convenient for programming on a digital computer.

The present article sets forth a method for calculating the coefficients \(\alpha\) and \(\beta\) with the aid of a digital computer, and gives calculated values of \(\alpha\) and \(\beta\) for steels E11 and E41.

For the calculation of the magnetic circuit with satisfactory accuracy, the phenomenon of flattening of the field curve in the zone of the air gap and teeth should be taken into account, as well as the nonuniformity of the field in the yoke.

One may take the coefficient of the calculated pole arc as \(a_i=\dfrac{2}{\pi}\) and use the magnetization curve of the steel in the teeth with allowance for field flattening, and the magnetization curve in the yoke with allowance for field nonuniformity [3]. Flattening in the gap may be allowed for in the usual way, through \(a_i\), determined from the resulting saturation in the air-gap zone.

The approximation of the magnetization curves with allowance for flattening and nonuniformity of the fields was carried out on the “Promin” digital computer by the method of quadratic interpolation. As is known, this method requires minimization of the quantity

\[ S=\sum_{i=1}^{k}(H_i-\alpha\cdot \operatorname{sh}\beta\cdot B_i)^2, \tag{6} \]

where \(B_i, H_i\) are the inductions and magnetic-field intensities at individual points of the magnetization curve.

Taking partial derivatives, we obtain

\[ \frac{\partial S}{\partial \alpha} = \sum H_i\cdot \operatorname{sh}\beta\cdot B_i - \alpha\cdot \sum \operatorname{sh}^2\beta\cdot B_i =0, \tag{7} \]

\[ \frac{\partial S}{\partial \beta} = \sum H_i\cdot \operatorname{ch}\beta\cdot B_i - \frac{1}{2}\cdot \alpha \sum B_i\cdot \operatorname{sh}2\beta\cdot B_i =0. \tag{8} \]

The transcendental equations (7, 8) were solved according to the scheme of Fig. 1. The initial value \(\beta_0\) is chosen deliberately small, but greater than zero. First \(\beta\) is computed with an accuracy up to 0.1; then the step \(\Delta\beta\) is set equal to 0.01 and \(\beta\) is computed with an accuracy up to 0.01, and so on.

With the number of points on the magnetization curve \(k\approx 10\), the machine time for computing \(\alpha,\beta\) to an accuracy of 5 digits does not exceed 15 min.

The results of calculating \(\alpha\) and \(\beta\) for steels Э11 and Э41 are given in Table 1. For the best approximation, the magnetization curve was divided into three sections. The maximum absolute error is not more than

\[ 1.8\ \frac{a}{\text{cm}}, \]

and the root-mean-square error is 6%.

For comparison of calculation procedures that do and do not take into account the flattening of the field in the tooth zone and the nonuniformity in the yoke, the no-load characteristics of a model two-stage EMU of the combined type [4] were calculated on the “Minsk-1” digital computer by both methods.

Fig. 1. Structural diagram for calculating the coefficients \(\alpha\) and \(\beta\)

Visible labels in the diagram:

• \(\beta_0\)
• \(\beta+\Delta\beta\)
• Calculation: \(\sum H_i\operatorname{sh}\beta B_i,\ \sum \operatorname{sh}^2\beta B_i,\ \sum B_iH_i\operatorname{ch}\beta B_i,\ \sum B_i\operatorname{sh}2\beta B_i\)
• Calculation: \(\displaystyle \alpha=\frac{\sum H_i\operatorname{sh}\beta B_i}{\sum \operatorname{sh}^2\beta B_i}\)
• Check of condition (8): \(\sum H_iB_i\operatorname{ch}\beta B_i-\alpha\beta\sum B_i\operatorname{sh}2\beta B_i<0\)
• If condition (8) is not satisfied
• Stop

From Fig. 2 it is evident that calculation without taking account of saturation and the nonuniformity of the fields (characteristic 3) leads to overestimated values of the magnetizing currents in comparison with the experimental characteristic 1. Taking these phenomena into account by the proposed method makes it possible to increase the accuracy of the calculation (characteristic 2).

Table 1

Fig. 2. No-load characteristics of a model combined-type phase-sensitive EMU

The calculation of the characteristics is carried out using average parameters. Owing to the scatter of parameters, the characteristics of each specimen will differ from the calculated ones. The problem of calculating tolerances for the parameters that guarantee agreement, with a specified accuracy, between the calculated and experimental characteristics for each specimen is very complicated for ordinary machines, all the more so for combined-type machines, and lies beyond the scope of the present article.

Conclusions

1. Approximating the magnetization curve of steel by a hyperbolic sine leads to a simple method for calculating the magnetic circuit of combined machines, convenient for programming on a digital computer.

2. Taking into account the flattening of the magnetic field in the teeth and the nonuniformity of the field in the yoke makes it possible to increase the accuracy of the calculation of combined-type machines.

References

1. V. S. Novokshenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk Polytechnic Institute, 1960.

2. M. S. Mikhailo-Mikulinskii. Calculation of the magnetic circuits of electrical machines with two rotating fields. Izvestiya vuzov, “Electromechanics,” 1953, No. 3.

3. I. M. Postnikov. Design of Electrical Machines. Kiev, 1960.

4. E. Mishkin. On Some Polyfield Amplifiers. “Power Apparatus and Systems,” April, 1956.