ACCOUNTING FOR THE INFLUENCE OF SATURATION ON THE PARAMETERS OF SYNCHRONOUS REACTIVE MOTORS

E. V. KONONENKO, A. N. AIFERT, B. P. GARGANEEV

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

In calculating magnetic fields in the air gap of synchronous salient-pole machines, including synchronous reactive motors [1], the two-reaction theory is used, which necessitates the application of the principle of superposition of magnetic fields. The latter is strictly valid only for unsaturated magnetic circuits. In order to make better use of active materials, synchronous reactive motors are made with a saturated magnetic circuit; therefore, this circumstance must be taken into account in calculations.

In [2, 3] the influence of saturation is taken into account on the basis of an approximately determined picture of the magnetic field in the air gap at various load angles of a synchronous reactive motor. With a more accurate construction of the field pattern, and for improved synchronous reactive motors, it is practically impossible to use this method. When accounting for the influence of saturation of the magnetic circuit on the parameters of synchronous reactive motors, it is advisable to use the method of “approximate superposition,” which has found wide application in the investigation of electrical machines [4, 5]. This method assumes that saturation affects only the magnitude of the magnetic flux, while the form of the induction-distribution curves in the air gap does not depend on the degree of saturation. The decrease in the magnitude of the magnetic flux in this case can be taken into account in the same way as the slotting of the stator and rotor surfaces—by increasing the air gap by means of a saturation coefficient.

Guided by this method, one can calculate with sufficient accuracy the synchronous inductive reactances of a synchronous reactive motor in the limiting positions of the rotor: when the longitudinal axis of the rotor coincides with the axis of the magnetomotive force (m.m.f.) of the stator winding \((x_d)\), and in the case when the transverse axis of the rotor coincides with the axis of the m.m.f. of the stator winding \((x_q)\). In investigating operating modes and calculating the operating characteristics of reactive motors, it is usually assumed that the reactances \(x_d\) and \(x_q\) do not depend on the magnitude of the load. This can explain the circumstance that, when calculated by existing methods, the operating characteristics differ significantly from those obtained experimentally. When taking into account the influence of saturation on the parameters of synchronous reactive motors under changing load, it is necessary to consider the following. Under ideal no-load conditions the main magnetic flux closes along the longitudinal axis of the rotor. As the load increases, the magnetic flux along the longitudinal axis of the rotor decreases, since

as part of the main magnetic flux passes along the transverse axis of the rotor, and the magnetic flux along the transverse axis increases. This leads to the fact that, with increasing load, saturation along the longitudinal axis decreases, while saturation along the transverse axis increases. The change in saturation of the magnetic circuit along the axes \(d\) and \(q\) causes the corresponding change in the reactances \(x_d\) and \(x_q\). As the load increases, the synchronous inductive reactance along the longitudinal axis \(x_d\) increases, while the synchronous inductive reactance along the transverse axis \(x_q\) decreases.

Applying the method of approximate superposition, it is natural to assume that the change in saturation along the axes \(d\) and \(q\) does not exert a mutual influence. This makes it possible to calculate or determine experimentally the dependences of the synchronous reactive resistances \(x_d\) and \(x_q\) as functions of the voltage \(U\) applied to the stator winding. Figure 1 presents the dependences \(x_d=f(u)\) and \(x_q=f(u)\), determined experimentally for an improved synchronous reluctance motor.

Fig. 1. Dependences of the parameters \(x_d\), \(x_q\) on the voltage applied to the stator winding.

When calculating operating characteristics (for \(U=\mathrm{const}\)), the influence of saturation on the parameters of a synchronous reluctance motor can be determined as follows. For any specified load, determined by the angle \(\Theta\), the voltage components along the longitudinal \(U_d\) and transverse \((U_q)\) axes of the rotor are found:

\[ \begin{aligned} U_d &= U \sin \theta,\\ U_q &= U \cos \theta \end{aligned} \tag{1} \]

The longitudinal component of the voltage creates a magnetic flux along the transverse axis; the transverse component of the voltage creates a magnetic flux along the longitudinal axis. Consequently, from the known dependence \(x_d=f(u)\) and the voltage component \(U_q\), for any specified load angle \((\Theta)\), the value of \(x_d\) can be determined with allowance for saturation. The resistance \(x_q\) is determined analogously when \(x_q=f(U)\) and \(U_d\) are known.

Figure 2 gives the curves of the variation of the parameters \(x_d\) and \(x_q\) and of the ratio

\[ \frac{x_d}{x_q} \]

as functions of the load angle \((\Theta)\), constructed by the method described above with allowance for the data of Fig. 1. As follows from these curves, with increasing load \(x_d\) increases, while \(x_q\) decreases. Moreover, the change in \(x_d\) occurs over a considerably wider range than that in \(x_q\). The latter is explained by the fact that, in improved synchronous reluctance motors, saturation of the magnetic circuit along the transverse axis has less influence on the operation of the motor, since the magnetic flux

Fig. 2. Dependences of the parameters \(X_d\), \(X_q\), and the ratio \(\frac{x_d}{x_q}\) on the load angle.

along the transverse axis encounters on its path significant nonmagnetic sections (air gap, additional internal slots, interpolar space). The change in the parameters \(x_d\) and \(x_q\) leads to the fact that the ratio \(\frac{x_d}{x_q}\), with increasing load, rises from a value equal to 3.1 (at \(\Theta = 0^\circ\)) to 6.7 (at \(\Theta = 45^\circ\)). It is known that the ratio \(\frac{x_d}{x_q}\), in the main, also determines the magnitude of the maximum values of \(\cos(\varphi_{\min})\) and the power \(P_{2\max}\) of the motor. Consequently, when saturation is taken into account, with increasing load \(\cos(\varphi_{\min})\) and \(P_{2\max}\) increase.

Knowing how the parameters \(x_d\) and \(x_q\) vary as a function of the load angle, one can calculate the operating characteristics with allowance for saturation. The simplest way to calculate the operating characteristics is by means of circle diagrams. For a given value of the load, the values of \(x_d\), \(x_q\) are determined with allowance for saturation, and a circle diagram is constructed, from which all quantities characterizing the given operating mode of the motor are determined. By constructing in this way the required number of points for different load angles, the operating characteristics are determined.

Fig. 3 presents the operating characteristics of an improved synchronous reluctance motor, calculated with allowance for the influence of saturation by the method set forth above and determined experimentally. The curves of the dependence of the reactances \(x_d\), \(x_q\) on voltage and load for this motor are presented in Figs. 1 and 2.

As follows from Fig. 3, the calculated data of the operating characteristics practically coincide with the experimental data. This makes it possible to conclude that the proposed method for calculating operating characteristics with allowance for the influence of saturation of the magnetic circuit reflects with sufficient accuracy the processes occurring in synchronous reluctance motors and can be recommended for practice.

In calculating the operating characteristics considered above, the dependences \(x_d = f(U)\) and \(x_q = f(U)\) were determined experimentally. In order to calculate these dependences, it is necessary to construct the magnetization characteristics along the longitudinal and transverse axes of the synchronous reluctance motor. For this it is necessary to calculate the magnetic circuit along the \(d\) and \(q\) axes at various voltage values. To calculate the magnetic circuit and the reactances \(x_d\) and \(x_q\), it is necessary to know the coefficients characterizing the magnetic field in the air gap [1].

Fig. 3. Operating characteristics of a synchronous reactive motor.
o — calculated points, x — experimentally determined.

The calculation of the magnetic circuit along the \(d\)-axis is carried out under the assumption that the entire magnetic flux closes along the longitudinal axis. The calculation procedure is as follows.

For the specified voltage, the first harmonic of the emf is determined:

\[ E_{ad1}=k_{ed}\cdot U, \tag{2} \]

where \(k_{ed}\) is a coefficient that takes into account the voltage drop in the active resistance and leakage resistance of the stator winding due to the current \(I_d\).

The first harmonic of the magnetic flux \(\Phi_{d1}\) is determined by the formula

\[ \Phi_{d1}=\frac{E_{ad1}}{4.44 W_1 k_{ob1} f}. \tag{3} \]

When calculating the magnetic circuit, knowledge of the value \(\Phi_{d1}\) is insufficient, since the field curve is not sinusoidal; and in order to calculate the induction and the mmf in each section of the magnetic circuit, it is necessary to know the real value of the magnetic flux.

The total magnetic flux along the longitudinal axis is determined as

\[ \Phi_d=k_{\phi d}\cdot \Phi_{d1}. \tag{4} \]

The mmf in the teeth and in the air gap is determined from the maximum induction in the air gap, equal to

\[ B_{md}=\frac{\Phi_d}{a_d\tau\cdot l}. \tag{5} \]

Calculation of the magnetic circuit for the rated voltage makes it possible to determine the magnetizing component of the no-load current. By assigning different values of the emf, one can calculate the magnetization characteristic, using which, for any specified voltage value, one can determine the saturation coefficient along the longitudinal axis \(k_{\mu d}\), equal to the ratio of the entire mmf of the magnetic circuit to the mmf of the air gap, and take into account the effect of saturation on the inductive reactance \(x_d\).

The calculation of the magnetic circuit along the \(q\)-axis is carried out under the assumption that the entire magnetic flux closes along the transverse axis. In the calculation

equations (2)--(5) are used, in which the indices \(d\) are replaced by \(q\). Having calculated the magnetic circuit for various mmf's, one constructs the magnetization characteristic along the transverse axis, by means of which one can determine the saturation coefficient along the transverse axis \(k_{\mu q}\) and thus take into account the effect of saturation on the resistance \(x_q\).

The synchronous inductive reactances along the direct and transverse axes are determined from the expression:

\[ x_d = x_s + x_{ad}; \tag{6} \]

\[ x_q = x_s + x_{aq}; \tag{7} \]

\[ x_{ad} = x_m \frac{k_d}{k_{\mu d}}; \tag{8} \]

where

\[ x_{aq} = x_m \frac{k_q}{k_{\mu q}}; \tag{9} \]

\[ x_m = \frac{4}{\pi} f_1 \mu_0 \frac{m w_1^2 k_{\mathrm{w}1}^{\,2} l_{\delta 1}}{p} \frac{\tau \cdot 1}{k_\delta \cdot \delta}. \tag{10} \]

\(x_m\) is the inductive reactance under the assumption that the rotor is cylindrical, expressed through the design data of the motor.

References

1. E. V. Kononenko, A. N. Aifert. On the calculation of magnetic fields of synchronous reluctance machines, Proceedings of TPI, vol. 145, 1966.

2. W. Schuisky, Beitrag zur Theorie und Berechnung des Reaktionsmotors, AFE, vol. 39, 1950.

3. A. M. Langen. Features of calculating the synchronous mode of a reluctance motor, Electricity, No. 9, 1964.

4. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines, Part II, GEI, 1958.

5. V. V. Apsit. A system of assumptions and simplified methods for investigating the magnetic field in the air gap of electric machines with claw-shaped poles, in: Contactless Electrical Machines, III. Riga, 1963.