Air-Gap Field Shape Factors of Alternating-Pulsating Machines with Magnetically Soft Pole Shoes

A. S. Zhibinov, Yu. N. Kroneberg, V. B. Gomzyakov

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

In machines of the alternating-pulsating type [1], in order to increase the depth of regulation, the poles excited by permanent magnets are, as a rule, made narrower than the poles of opposite polarity of electromagnetic excitation. In addition, in the process of regulation the ratio of the inductions of the narrow and wide poles changes. Although these features do not require the introduction of a fundamentally different system of field coefficients (as compared with normal synchronous machines), they also do not allow one to use their numerical values directly; therefore additional investigations are necessary.

In accordance with the principle of approximate superposition [2], the field of each pole may be considered separately, and then the harmonic components of the induction may be summed. In this case the amplitudes of the first harmonics of the excitation fields \((B_{1v})\), the longitudinal \((B_{1d})\) and transverse \((B_{1q})\) armature reactions are expressed as follows:

\[ B_{1v}=k_{fv}B_v+k_{f\sh}B_{\sh}; \tag{1} \]

\[ B_{1d}=(k_{dv}+k_{d\sh})B_{md}; \tag{2} \]

\[ B_{1q}=(k_{qv}+k_{q\sh})B_{mq}, \tag{3} \]

where \(k_{f\,v,\sh}, k_{d\,v,\sh}, k_{q\,v,\sh}\) are the field shape factors of the narrow and wide poles;

\(B_{v,\sh}\) is the induction of the excitation field along their axes;

\(B_{md,q}\) are the amplitudes of the inductions of the armature reactions along the longitudinal and transverse axes with a uniform gap.

In turn, the magnetic-field coefficients are expressed through the constant component \((\lambda_0)\), the first \((\lambda_1)\), and the second \((\lambda_2)\) harmonics of the relative specific magnetic permeance of the narrow and wide poles:

\[ k_{fy,\sh}=\lambda_{1y,\sh}; \tag{4} \]

\[ k_{dy,\sh}=0.5(\lambda_{0y,\sh}+\lambda_{2y,\sh}); \tag{5} \]

\[ k_{qy,\sh}=0.5(\lambda_{0y,\sh}-\lambda_{2y,\sh}). \tag{6} \]

The constant component of the permeance of the poles of an alternating-pulsating machine may be represented as the sum

\[ \lambda_{0y,\sh}=\lambda_0+\Delta\alpha, \tag{7} \]

where, according to [3, 4],

\[ \lambda_0=\alpha+\frac{2}{\pi}(1-\alpha)\left\{\frac{\delta}{\tau}\frac{1}{1-\alpha}\ln\left[1+\left(\frac{\delta}{\tau}\frac{1-\alpha}{2}\right)^2\right]+\operatorname{arctg}\left(\frac{\delta}{\tau}\frac{2}{1-\alpha}\right)\right\} \tag{8} \]

is the constant component of the specific magnetic permeance of the pole of a normal synchronous machine with pole arc

\[ \alpha=0.5(\alpha_y+\alpha_{\mathrm{ш}}) \tag{9} \]

and, in addition, it depends on the air gap \(\delta/\tau\), while the specificity of the alternating-pulsating machine is taken into account by the second term

\[ \Delta\alpha=\alpha_{y,\mathrm{ш}}-\alpha . \tag{10} \]

The harmonic components \(\lambda_n\) may be determined by using, for example, the expression for the specific magnetic permeance \(\lambda(x)\) given in [3], representing it (for simplicity of calculation) in three sections:

\[ \begin{array}{ll} \lambda=1, & 0\leq x\leq x_1;\\ \lambda=\lambda(x), & x_1\leq x\leq x_2;\\ \lambda=0, & x_2\leq x\leq \tau, \end{array} \tag{11} \]

where the coordinates \(x_1\) and \(x_2\) can be found from the condition of admissible errors.

In this case, the transition from the field of the pole of a normal synchronous machine satisfying condition (9) to the field of the wide (narrow) pole of an alternating-pulsating machine consists in the fact that the first section will increase (decrease) by the amount \(\frac{\tau}{2}\Delta\alpha\), the second will be displaced along the \(x\)-axis by the same amount, and the third, correspondingly, will decrease (increase).

Then the amplitude of the \(n\)-th harmonic of the permeance of the alternating-pulsating machine will be expressed in the form

\[ \lambda_{ny,\mathrm{ш}}=A_n\sin n\frac{\pi}{2}\Delta\alpha+B_n\cos n\frac{\pi}{2}\Delta\alpha; \tag{12} \]

\[ A_n=-\frac{2}{n\pi}\cos n\frac{\pi}{\tau}x_1-\frac{2}{\tau}\int_{x_1}^{x_2}\lambda(\cdot)\sin n\frac{\pi}{\tau}x\,dx; \tag{13} \]

\[ B_n=-\frac{2}{n\pi}\sin n\frac{\pi}{\tau}x_1+\frac{2}{\tau}\int_{x_1}^{x_2}\lambda(x)\cos n\frac{\pi}{\tau}x\,dx, \tag{14} \]

In calculating the coefficients \(A_n\) and \(B_n\), it is expedient to use the coordinate transformation [3], after which we obtain

\[ A_n=\frac{2}{n\pi}\cos nf(t_1)-\frac{4}{\pi}\frac{\delta}{\tau}\int_{t_1}^{t_2} f_1(t)\sin nf(t)\,dt; \tag{15} \]

\[ B_n=\frac{2}{n\pi}\sin nf(t_1)+\frac{4}{\pi}\frac{\delta}{\tau}\int_{t_1}^{t_2} f_1(t)\cos nf(t)\,dt, \tag{16} \]

where

\[ f(t)=\frac{\pi}{2}\alpha-(1-\alpha)\left[\frac{\delta}{\tau}\frac{1}{1-\alpha}\ln\frac{(t+1)(q^2-t)}{(t-1)(q^2+t)}-2\operatorname{arctg}\frac{t}{q}\right], \tag{17} \]

\[ f_1(t)=\frac{t(q^2+1)}{(t^2-1)(q^2-t^2)}; \tag{18} \]

\[ q=\frac{\tau}{\delta}\,\frac{1-\alpha}{2} +\sqrt{\left(\frac{\tau}{\delta}\,\frac{1-\alpha}{2}\right)^2+1}. \tag{19} \]

The coefficients \(\lambda_0, A_1, B_1, A_2, B_2\) for practically used values of the pole arcs \(\alpha\) and air gaps \(\delta/\tau\) were calculated on the BESM-4 computer center and are given, respectively, in Tables 1–5. The error in computing the harmonics (numerical integration, simplifications on the intervals \(0\le x\le x_1\) and \(x_2\le x\le\tau\)) was taken as \(\Delta\lambda_{1,2}<5\cdot10^{-4}\), which is quite satisfactory for practical calculations.

Table 1

Table 2

Table 3

Table 4

Table 5

References

1. Yu. N. Kroneberg, V. B. Gomzyakov, A. S. Zhibinov. Designs of alternating-pulsating machines. In: Design of devices for electrical power supply of electric drives. Vol. 2. Electromechanical devices and elements of technology. Moscow, “Energiya,” 1973, pp. 22–27.

2. V. V. Apsit. A system of assumptions and simplified methods for investigating the magnetic field in the air gap of electrical machines with claw-shaped poles. In: Contactless Electrical Machines. Riga, Publishing House of the Academy of Sciences of the Latvian SSR, 1963, issue 3, pp. 5–28.

3. Z. K. Sika. Magnetic permeance of the air gap of a machine with claw-shaped poles. In: The Magnetic Field in Electrical Machines. Riga, “Zinatne” Publishing House, 1965, pp. 125–149.

4. Yu. N. Kroneberg, A. S. Zhibinov, V. B. Gomzyakov. Coefficients of the calculated pole arc of alternating-pulsating machines with magnetically soft pole shoes. Izvestiya TPI, vol. 228 (in press).