Calculation of the No-Load Characteristic of a Combined-Type Phase-Sensitive Electric Machine Amplifier
Abstract Generated abstract
This paper develops a calculation procedure for the no-load characteristic of a two-stage phase-sensitive electric-machine amplifier with both stages combined in one magnetic core. Starting from first-harmonic magnetomotive-force expressions based on an approximate steel magnetization curve, the authors formulate an iterative method that accounts for the mutual influence of the superposed magnetic fields in teeth and yokes, then derives rotor and control-current quantities from the magnetic-circuit solution. Calculations for a model amplifier, performed on a Minsk-1 computer and compared with experimental characteristics, indicate practically satisfactory accuracy for normal induction levels, while also showing that a simpler calculation without field superposition can be adequate on the linear part of the characteristic.
Full Text
Calculation of the No-Load Characteristic of a Phase-Sensitive Combined-Type EMA
A. I. Skorospeshkin, M. L. Kostyrev
(Recommended by the seminar of the Departments of Electrical Machines and General Electrical Engineering).
The combination in one magnetic core of two stages of an alternating-current electric-machine amplifier (EMA) (Fig. 1) makes it possible to eliminate sliding electrical contacts and to make the machine compact [1].
Fig. 1. Schematic of a phase-sensitive EMA.
Here it is necessary to choose the ratio of the numbers of pole pairs, the shortening of pitches and the distribution of winding sections, as well as the order of connection of the coil groups in a phase, so that the forces of unilateral magnetic attraction and transformer coupling between the stages are absent.
The calculation of the magnetic circuit of combined machines is complicated by the fact that the specific ampere-turns in the steel sections of the magnetic core depend on the mutual displacement of the combined magnetic fields. There are various methods that take into account the mutual influence of the combined fields. One of them [2] makes it possible to obtain satisfactory accuracy, for a definite ratio of the numbers of pole pairs, by using curves of simultaneous magnetization of steel by currents of different frequency. Another, more labor-intensive method [3] is based on repeated graph-analytical calculation of the magnetic-core sections as the combined field is displaced relative to the one being calculated. An analysis of the magnetic circuit of combined machines [4], carried out on the basis of approximation of the steel magnetization curve by a hyperbolic ...
...with a sinusoid, leads to fairly simple formulas for calculating the m.m.f. in the steel sections of the magnetic circuit.
In the present work a calculation procedure is proposed, based on the results of the analysis in [4] and making it possible, in a comparatively simple way, to take into account the mutual influence of the combined fields. The procedure is intended for calculating the no-load characteristic of a two-stage phase-sensitive EMA of the combined type, but its principles may be used for calculating the magnetic circuits of two-stage electric machines of the combined type.
It is known [2, 4] that the effect of the higher harmonics in the curves of m.m.f. and currents on machine operation may be neglected, and the calculation may be carried out only for the first harmonic of the m.m.f.
The amplitudes of the first harmonics of the m.m.f. in the teeth [4] are
\[ F_{z1}=2\cdot L_z\cdot \alpha \cdot I_0(\beta B_{z m2})\cdot I_1(\beta B_{z m1}), \tag{1} \]
\[ F_{z2}=2\cdot L_z\cdot \alpha \cdot I_0(\beta B_{z m1})\cdot I_1(\beta B_{z m2}). \tag{2} \]
The indices 1 and 2 refer to the first and second combined machines. In the case of an EMA these are its 1st and 2nd stages.
\(I_0(\beta B)\) and \(I_1(\beta B)\) are Bessel functions of the first kind of purely imaginary argument, of zero and first order.
The coefficients \(\alpha\) and \(\beta\) depend on the grade of steel and on the portion of the magnetization curve.
A separate article will be devoted to the method for calculating the coefficients \(\alpha\) and \(\beta\), which ensure the best approximation of the magnetization curve, and also to allowance for the flattening of the magnetic field in the tooth zone and the nonuniformity of the field in the yokes.
The amplitudes of the first harmonics of the m.m.f. in the yokes are
\[ F_{a1}=\frac{4}{\pi}\cdot L_{a1}\alpha \cdot I_0(\beta B_{a m2})\cdot I_1(\beta B_{a m1}), \tag{3} \]
\[ F_{a2}=\frac{4}{\pi}\cdot L_{a2}\cdot \alpha \cdot I_0(\beta B_{a m1})\cdot I_1(\beta B_{a m2}). \tag{4} \]
\(B_{a m1}, B_{a m2}, B_{z m1}, B_{z m2}\) are the amplitudes of the first harmonics of the inductions in the yokes and teeth from the fields of stages I and II;
\(L_z, L_{a1}, L_{a2}\) are the corresponding lengths of the magnetic-circuit sections.
Fig. 2. Block diagram for calculating the no-load characteristic of a combined EMA.
Since the m.m.f.’s of stages I and II are coupled to each other through the inductions, it is proposed to calculate the characteristic by the method of successive approximations. The block diagram of such a procedure is presented in Fig. 2.
The total m.m.f. per pair of poles of the second cascade is
\[ \Sigma F_2 = F_{\delta 2}+F_{zp2}+F_{zc2}+F_{ap2}+F_{ac2}. \tag{5} \]
\(F_{\delta 2}, F_{zp2}, F_{zc2}, F_{ap2}, F_{ac2}\) are the components of the m.m.f. for the air gap, the teeth, and the stator and rotor yokes.
In Fig. 2 they, with the exception of \(F_{\delta 2}\), are denoted by \(F_{i2}\).
\(F_{\delta 2}\) is calculated by generally accepted formulas. The remaining m.m.f.’s are calculated by formulas (2,4).
In the first approximation, the values of the field flux densities of the first cascade may be taken as
\[ B_{z\mathrm{m}1}=\frac{C_B}{1-C_B}\cdot B_{z\mathrm{m}2}, \qquad C_B=0.4 \div 0.6. \tag{6} \]
\[ B_{a\mathrm{m}1}=\frac{C_B\cdot p_2}{(1-C_B)p_1}\cdot B_{a\mathrm{m}2}. \tag{7} \]
Recommendations for choosing \(C_B\) in design are given in [4]. In the second and subsequent approximations, the values of the field flux densities of the first cascade calculated in the preceding approximation are used.
The rms value of the current in the rotor is
\[ I_p=\frac{p_2}{0.9\cdot m_p\cdot \omega_{p2}}\cdot \Sigma F_2, \tag{8} \]
\(m_p\) is the number of phases of the rotor windings;
\(\omega_{p2}\) is the number of turns in a phase of winding \(W_{p2}\), taking into account the winding factor;
\(p_2\) is the number of pole pairs of the second cascade.
The rotor circuit equation is
\[ \dot E_{p1}+\dot E_{p2}=\dot I_p(r_p+jx_{sp}); \tag{9} \]
\(r_p=r_{p1}+r_{p2}\) is the active resistance of the rotor windings;
\(x_{sp}=x_{sp1}+x_{sp2}\) is the leakage reactance of the rotor windings;
\[ E_{p2}=\pi\sqrt{2}\cdot f_p\cdot \omega'_{p2}\cdot \Phi_2\cdot 10^{-4}. \tag{10} \]
\(f_p\) is the frequency of the current in the rotor;
\(\Phi_2\) is the magnetic flux of the field of the second cascade.
From equation (9), the rms value of the e.m.f. in the winding is
\[ E_{p1}=\sqrt{(E_{p2}+I_p\cdot x_{sp})^2+(I_p\cdot r_p)^2}. \tag{11} \]
From the calculated value of \(E_{p1}\), the flux densities in the air gap and in the steel sections of the first cascade are determined.
\(\Sigma F_1\) is calculated analogously to \(\Sigma F_2\). The values of the field flux densities of the second cascade in formulas (1,3) are taken from the same approximation as the flux densities of the first cascade field.
At the end of each approximation, the following conditions are checked:
\[ \left|(\Sigma F_1)_{n+1}-(\Sigma F_1)_n\right|\leq 0.05(F_1)_{n+1}; \tag{12} \]
\[ \left|(\Sigma F_2)_{n+1}-(\Sigma F_2)_n\right|\leq 0.05(\Sigma F_2)_{n+1} \tag{13} \]
\(n\) is the order of the approximation.
When conditions (13, 12) are satisfied, the m.m.f. of the control winding is calculated:
\[ F_y=\sqrt{(\Sigma F_1)^2+(F_{p1})^2+2(\Sigma F_1)\cdot F_{p1}\cdot \sin\varphi'_p}, \tag{14} \]
\[ F_{p1}=\frac{0.9\,m_p\cdot \omega'_{p1}\cdot I_p}{p_1}, \tag{15} \]
\[ \zeta'_p=\operatorname{arctg}\left(\frac{E_{p2}+I_p\cdot x_{sp}}{I_p\cdot r_p}\right), \tag{16} \]
\(\omega'_{p1}\) is the number of series turns in the winding phase, taking account of the winding factor;
\(p_1\) is the number of pole pairs of the first cascade.
For a concentric-type control winding,
\[ I_y=\frac{a_y\cdot F_y\cdot p_1}{\omega'_y}, \tag{17} \]
\[ \omega'_y=\omega_y\cdot \frac{y_{\mathrm{av}}}{\tau_1}, \tag{18} \]
\(\omega_y\) is the total number of turns of the control winding,
\(\tau_1\) is the pole pitch of the first cascade in slots,
\(y_{\mathrm{av}}\) is the average pitch of the control winding in slots,
\(a_y\) is the number of parallel branches of winding \(W_y\).
For a distributed control winding of the type of the armature winding of a direct-current machine,
\[ I_y=\frac{\pi\cdot a_y\cdot F_y\cdot p_1}{4\omega'_y}, \]
\[ \omega'_y=\omega_y\cdot K_{\mathrm{w}}. \tag{19} \]
Thus, the proposed method makes it possible, in a comparatively simple way, to take into account the mutual influence of the superposed fields.
Calculation by the ordinary method, without allowance for superposition, gives an underestimated value of the magnetizing current. But if the phenomenon of field flattening in the tooth zone and the nonuniformity of the field in the yoke are not taken into account, the calculated current will increase somewhat, which will lead to a certain compensation of the error caused by neglecting the superposition.
In the normal operating mode the electric machine amplifier operates with an unsaturated magnetic circuit \((B_{\max}<1.4\ \text{ml})\). The mmf on the steel sections of the magnetic circuit is small, and it becomes possible to calculate the no-load characteristic without allowing for superposition, by the simplified method.
In order to compare the simplified and proposed methods, calculations of the no-load characteristic of a model of a phase-sensitive electric machine amplifier of the combined type were carried out on the “Minsk-1” digital computer.
Figure 3 presents: experimental characteristic 1, characteristic 2, calculated with allowance for superposition by the proposed method, and characteristic 3, calculated by the simplified method. Comparison of the experimental and calculated characteristics shows that at maximum inductions \(B_{\max}\leq 1.6\ \text{ml}\) the simplified method gave errors in determining the rotor current and the control current:
\[ (\Delta I_p)_{\max}=10\%, \qquad (\Delta I_y)_{\max}=29\%, \]
\[ (\Delta I_p)_{\mathrm{av}}=-8\%, \qquad (\Delta I_y)_{\mathrm{av}}=+4\%. \]
The calculation error does not exceed the deviations determined by tolerances on the air gap, scatter in the parameters of the electrical steel, etc.
The method that takes superposition into account, at inductions \(B_{\max}\leq 1.6\ \text{ml}\), gave errors:
\[ (\Delta I_p)_{\max} \leq 11\%, \qquad (\Delta I_y)_{\max} \leq 6\%, \]
\[ (\Delta I_p)_{\mathrm{avg.}} = -8\%, \qquad (\Delta I_y)_{\mathrm{avg.}} = +2\%. \]
With inductions \(B_{\max} > 1.6\ \mathrm{T}\), curve 2 passes considerably below the experimental curve. This is explained by the fact that the model with the number of poles \(2p_1 = 2\) and \(2p_2 = 6\) was made on the basis of a 6-pole asynchronous motor. Therefore the yokes became saturated by the flux of cascade I earlier than the teeth did. The proposed calculation method is intended for machines with normal geometry and does not take into account phenomena associated with saturation of the yokes.
Fig. 3. No-load characteristics of the EMA model.
Conclusions
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The proposed method for calculating the no-load characteristic of a two-cascade combined-type EMA takes into account the mutual influence of the combined fields, is distinguished by its simplicity, and provides practically satisfactory calculation accuracy. The proposed approach can be used for calculating the magnetic circuit of other two-cascade combined-type machines (contactless converters, motors, etc.).
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The linear portion of the no-load characteristic of two-cascade combined-type EMAs may be calculated by a simplified method, without taking into account the combination of the fields, the flattening of the field in the tooth zone, and the nonuniformity of the field in the yokes.
References
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E. Mishkin. On Some Polyfield Amplifiers. “Power Apparatus and Systems,” April, 1956.
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V. M. Pavlinin. Calculation of the magnetic circuits of heteropolar electric machines combined in a single magnetic core. Proceedings of UPI, collection 138, 1964. Investigation of the electromagnetic and electromechanical processes of alternating-current machines.
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M. S. Mikhailov-Mikulinskii. Calculation of the magnetic circuits of electric machines with two rotating fields. Izv. vuzov, Elektromekhanika, No. 3, 1959.
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V. S. Novokshenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk Polytechnic Institute, 1960.
B. Order 934.