Toward the Calculation of the Parameters of the Rotor Winding of Synchronous Reluctance Motors

E. V. Kononenko, T. V. Chesheva

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

In calculating the parameters of synchronous reluctance motors (SRM), the rotor winding may be considered in the following two ways. According to the first, the real multi-contour rotor windings along the $d$ and $q$ axes are replaced by two equivalent windings whose width is equal to the pole pitch. The magnetic axis of one of them coincides with the longitudinal $d$ axis, and the magnetic axis of the other coincides with the transverse $q$ axis [4, 6, 10, etc.]. The other method consists in the fact that every two bars of the rotor winding, located symmetrically with respect to the $d$ and $q$ axes, form a separate circuit of the rotor winding, respectively along the $d$ and $q$ axes [1, 2, 3, 4, etc.]. The first method is used to simplify analytical studies and, in comparison with the second, will henceforth be called the simplified method. When real rotor windings are replaced by equivalent ones, it is assumed that the distribution of the components of the currents along the $d$ and $q$ axes in the bars of the rotor winding is sinusoidal.

In those cases where it becomes necessary to study processes in the rotor winding itself, the simplified representation of it by two equivalent circuits proves insufficient [6, 7]. In the rotor of improved SRMs there is a distributed short-circuited winding, which is located in slots of different cross section: oval slots, interpole spaces, and slots of complex shape obtained with the aid of additional internal slots connecting two slots symmetrically located with respect to the $q$ axis. This circumstance casts doubt on the assumption adopted in the simplified method of a sinusoidal spatial distribution of the current components in the rotor bars along the $d$ and $q$ axes. This means that, in order to obtain a calculation method providing sufficient accuracy for practical use, it is necessary to carry out analytical investigations not only by the simplified method but also by the exact method. Naturally, the exact method is very cumbersome and laborious, but it is necessary for assessing the degree of error of the simplified method.

Reduction of Circuits

When transformer couplings arise between the stator and rotor windings, currents will be induced in the latter. Since, in an improved SRM, the bars of the rotor winding are arranged symmetrically with respect to the $d$ and $q$ axes, the bars located at the same ...

at equal distances from the axes, will be traversed by currents equal in magnitude and opposite in direction (the processes along the \(d\) and \(q\) axes are considered separately) [12]. Therefore it may be assumed that they form separate circuits.

In order not to choose different systems of base quantities for the stator and rotor windings in calculations, and also when compiling exact equivalent circuits, it is necessary to refer the rotor windings to the stator windings [5]. To refer individual rotor circuits to the stator winding it is necessary to determine the transformation coefficients for currents \(k_i\), voltages \(k_u\), and resistances \(k_z=\dfrac{k_u}{k_i}\) for each rotor circuit formed in the rotor axes \(d\) and \(q\). We adopt such a system of transformation coefficients as would ensure reciprocity of the mutual inductances between the phases of the transformed circuits and the rotor-winding circuits coupled with them, that is,

\[ k_u k_i=\frac{2}{m}, \tag{1} \]

where \(m\) is the number of phases of the stator winding.

We determine the current transformation coefficient \(k_i\) from the condition of equality of the fundamental harmonics of the m.m.f. of the referred and real circuits [4, 7]. We shall denote the maximum values of the currents in the separately taken circuits of the rotor winding by

\[ I_{rd1},\ I_{rd2},\ldots I_{rdk}\quad \text{along the longitudinal axis,} \]

\[ I_{rq1},\ I_{rq2},\ldots I_{rqk}\quad \text{along the transverse axis,} \]

and the referred values will be denoted with a prime.

The amplitudes of the first harmonics of the m.m.f. created by the stator winding when the current of the \(k\)-th rotor circuit flows through it, along the \(d\) and \(q\) axes respectively, are

\[ F_{ad1(k)}=\frac{m}{2}\cdot \frac{4}{\pi}\cdot \frac{W_1 K_{\mathrm{ob}1}}{2p} I'_{rdk}, \tag{2} \]

\[ F_{aq1(k)}=\frac{m}{2}\cdot \frac{4}{\pi}\cdot \frac{W_1 K_{\mathrm{ob}1}}{2p} I'_{rqk}, \tag{3} \]

where \(m\), \(W_1\), \(K_{\mathrm{ob}1}\) are respectively the number of phases, the number of turns, and the winding factor of the stator winding;
\(p\) is the number of pole pairs.

The m.m.f. of a separately taken rotor circuit, created when current \(I_{rdk}\) flows through it, varies along the air gap in the form of a rectangular wave (Fig. 1) with height \(F_{rdk}\) or \(F_{rqk}\). The amplitudes of the first harmonic of the m.m.f. created by the \(k\)-th circuit when current \(I_{rdk}\) or \(I_{rqk}\) flows through it will respectively have the form:

\[ F_{rd1(k)}=\frac{4}{\pi} I_{rdk}\cdot \sin \frac{\pi}{\tau}\frac{\tau_k}{2}, \tag{4} \]

\[ F_{rq1(k)}=\frac{4}{\pi} I_{rqk}\cdot \cos \frac{\pi}{\tau}\frac{\tau_k}{2}, \tag{5} \]

where \(\tau_k\) is the pitch of the \(k\)-th circuit of the rotor winding,
\(\tau\) is the pole pitch.

Equating the left-hand sides of (2) and (4), (3) and (5), we obtain the current transformation coefficients for any rotor circuit, respectively along the \(d\) and \(q\) axes:

Figure 1. M.m.f. of a rotor-winding circuit

\[ k_{idk}=\frac{I'_{rdk}}{I_{rdk}}=\frac{2}{m}\cdot\frac{2p}{W_1K_{ob1}}\cdot\sin\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}, \tag{6} \]

\[ k_{iqk}=\frac{I'_{rqk}}{I_{rqk}}=\frac{2}{m}\cdot\frac{2p}{W_1K_{ob1}}\cdot\cos\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}. \tag{7} \]

From equations (6) and (7) it is evident that each separately taken circuit of the rotor winding along the \(d\) and \(q\) axes has its own reduction coefficients, the values of which vary depending on the pitch of the separately taken circuit. The reduction coefficients of voltages and resistances are determined from expression (1):

\[ k_{udk}=\frac{2}{m}\cdot\frac{1}{k_{idk}} =\frac{W_1K_{ob1}}{2p}\cdot \frac{1}{\sin\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}} \tag{8} \]

\[ k_{uqk}=\frac{2}{m}\cdot\frac{1}{k_{iqk}} =\frac{W_1K_{ob1}}{2p}\cdot \frac{1}{\cos\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}}, \tag{9} \]

\[ k_{rdk}=\frac{k_{udk}}{k_{idk}} =\frac{m}{2}\cdot\left(\frac{W_1K_{ob1}}{2p}\right)^2 \frac{1}{\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}}, \tag{10} \]

\[ k_{rqk}=\frac{k_{uqk}}{k_{iqk}} =\frac{m}{2}\cdot\left(\frac{W_1K_{ob1}}{2p}\right)^2 \cdot\frac{1}{\cos^2\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}}. \tag{11} \]

Basic Equations

The basic equations for an improved SRD having \(K\)-circuits per one pole pitch are usually written in relative units. To express all quantities in relative units, the base units of the primary circuit are used \([7, 9]\). The choice of base units for the quantities of the secondary, rotor circuits is made implicitly and is determined by the reduction coefficients.

We shall assume that the acceleration of the rotor is negligible, that is, we shall confine ourselves to the investigation of the steady-state regime at asynchronous rotational speed. Then the system of equations for the flux linkages of the wind—

of the stator and the voltage-balance equations for the short-circuited circuits can be written in complex form:

for the longitudinal axis

\[ \dot{\psi}_d = X_{a1d}\dot{i}_{1d} + X_{a2d}\dot{i}_{2d} +\ldots+ X_{akd}\dot{i}_{kd} + X_d\dot{i}_d, \tag{12} \]

\[ \left. \begin{aligned} 0&=Z_{11d}\dot{i}_{1d}+Z_{12d}\dot{i}_{2d}+\ldots+Z_{1kd}\dot{i}_{kd}+X_{1ad}\dot{i}_d\\ 0&=Z_{21d}\dot{i}_{1d}+Z_{22d}\dot{i}_{2d}+\ldots+Z_{2kd}\dot{i}_{kd}+X_{2ad}\dot{i}_d\\ &\ .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\\ &\ .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\\ 0&=Z_{k1d}\dot{i}_{1d}+Z_{k2d}\dot{i}_{2d}+\ldots+Z_{kkd}\dot{i}_{kd}+X_{kad}\dot{i}_d \end{aligned} \right\} \tag{13} \]

for the transverse axis

\[ \dot{\psi}_q = X_{a1q}\dot{i}_{1q} + X_{a2q}\dot{i}_{2q} +\ldots+ X_{akq}\dot{i}_{kq} + X_q\dot{i}_q \tag{14} \]

\[ \left. \begin{aligned} 0&=Z_{11q}\dot{i}_{1q}+Z_{12q}\dot{i}_{2q}+\ldots+Z_{1kq}\dot{i}_{kq}+X_{1aq}\dot{i}_q\\ 0&=Z_{21q}\dot{i}_{1q}+Z_{22q}\dot{i}_{2q}+\ldots+Z_{2kq}\dot{i}_{kq}+X_{2aq}\dot{i}_q\\ &\ .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\\ &\ .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad .\\ 0&=Z_{k1q}\dot{i}_{1q}+Z_{k2q}\dot{i}_{2q}+\ldots+Z_{kkq}\dot{i}_{kq}+X_{kaq}\dot{i}_q, \end{aligned} \right\} \tag{15} \]

where \(K\) is the number of rotor circuits per pole pitch;

\[ Z_{km}=\frac{R_{km}}{S}+jX_{km} \]

is the mutual-induction impedance between the rotor circuits;

\[ Z_{kk}=\frac{R_{kk}}{S}+jX_{kk} \]

is the total impedance of the \(K\)-th rotor circuit.

The transformed stator voltage-balance equations for the steady asynchronous regime can be written in the form:

\[ \left. \begin{aligned} \dot{U}_d&=js\dot{\psi}_d+r_a i_q-(1-s)\dot{\psi}_d\\ \dot{U}_q&=js\dot{\psi}_d+r_a i_d+(1-s)\dot{\psi}_d \end{aligned} \right\} \tag{16} \]

where

\[ \left. \begin{aligned} \dot{\psi}_d&=x_d(js)i_d\\ \dot{\psi}_q&=x_q(js)i_q \end{aligned} \right\} \tag{17} \]

\(x_d(js)\) and \(x_q(js)\) are the steady values of the operational impedances along the \(d\) and \(q\) axes.

In all cases \(x_d(js)\) and \(x_q(js)\) can be expressed as ratios of two determinants, each of which, in expanded form, is a polynomial. Expressions for the steady values of the operational impedances can be obtained as follows: solving system (12) and (13) with respect to the flux linkages \(\psi_d\), and system (14) and (15) with respect to the flux linkages \(\psi_q\), we obtain:

\[ \dot{\psi}_d=\frac{\Delta D}{\Delta d}i_d; \tag{18} \]

\[ \dot{\psi}_q=\frac{\Delta Q}{\Delta q}i_q, \tag{19} \]

where

\[ \Delta_D = \begin{vmatrix} x_d & x_{a1d} & x_{a2d} & \ldots & x_{akd}\\ X_{1ad} & Z_{11d} & Z_{12d} & \ldots & Z_{1kd}\\ X_{2ad} & Z_{21d} & Z_{22d} & \ldots & Z_{2kd}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ X_{kad} & Z_{k1d} & Z_{k2d} & \ldots & Z_{kkd} \end{vmatrix}, \tag{20} \]

\[ \Delta_d = \begin{vmatrix} Z_{11d} & Z_{12d} & \ldots & Z_{1kd}\\ Z_{21d} & Z_{22d} & \ldots & Z_{2kd}\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ Z_{k1d} & Z_{k2d} & \ldots & Z_{kkd} \end{vmatrix}, \tag{21} \]

\[ \Delta_Q = \begin{vmatrix} x_q & x_{a1q} & x_{a2q} & \ldots & x_{akq}\\ X_{1aq} & Z_{11q} & Z_{12q} & \ldots & Z_{1kq}\\ X_{2aq} & Z_{21q} & Z_{22q} & \ldots & Z_{2kq}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ X_{kaq} & Z_{k1q} & Z_{k2q} & \ldots & Z_{kkq} \end{vmatrix}, \tag{22} \]

\[ \Delta_q = \begin{vmatrix} Z_{11q} & Z_{12q} & \ldots & Z_{1kq}\\ Z_{21q} & Z_{22q} & \ldots & Z_{2kq}\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ Z_{k1q} & Z_{k2q} & \ldots & Z_{kkq} \end{vmatrix}. \tag{23} \]

Then from equations (18) and (19), together with equations (17), we obtain expressions for the established values of the operator impedances:

\[ x_d(js)=\frac{\Delta_D}{\Delta_d}; \tag{24} \]

\[ x_q(js)=\frac{\Delta_Q}{\Delta_q}. \tag{25} \]

The expressions \(x_d(js)\) and \(x_q(js)\), obtained in complex form, make it possible immediately to determine the real and imaginary parts of these impedances.

The supertransient reactances along the \(d\) and \(q\) axes are determined from the expressions for \(x_d(js)\) and \(x_q(js)\) under the condition that the active resistance of the rotor circuits is equal to zero. That is,

\[ x_d''=\frac{\Delta_D''}{\Delta_d''}; \tag{26} \]

\[ x_q''=\frac{\Delta_Q''}{\Delta_q''}, \tag{27} \]

where \(\Delta_D''\), \(\Delta_d''\), \(\Delta_Q''\), and \(\Delta_q''\) are obtained from equations (12)–(15) for \(R_{km}\) and \(R_{kk}=0\).

The reactive and active resistance of the rotor winding along the \(d\) and \(q\) axes can be determined from the formulas for the operator impedances \(x_d(js)\) and \(x_q(js)\), obtained by the simplified method [6, 7 et al.]:

\[ x_d(js)=x_d-\frac{X_{ad}^{2}}{X_{rd}+R_{zd}}, \tag{28} \]

\[ x_q(js)=x_q-\frac{X_{aq}^{2}}{X_{rq}+R_{rq}}. \tag{29} \]

If the total impedance of the rotor winding is denoted by

\[ Z_{rd}=jX_{rd}+\frac{R_{rd}}{S}\ \text{along the } d \text{ axis}, \tag{30} \]

\[ z_{rq}=jX_{rq}+\frac{R_{rq}}{S}\quad \text{along the } q \text{ axis}, \tag{31} \]

then from formulas (28), (29) one may write

\[ Z_{rd}=\frac{X_{ad}^{2}}{x_d-x_d(js)}, \tag{32} \]

\[ Z_{rq}=\frac{X_{aq}^{2}}{x_q-x_q(js)}. \tag{33} \]

The complex calculation method makes it possible to reduce the amount of calculation almost by half, since it eliminates the need for separate calculations of the active and reactive resistances of the rotor winding, and of the active and reactive components of the steady-state values of the machine operator impedances. Using the formulas presented, the parameters of the rotor winding were calculated for two improved salient-pole synchronous machines in asynchronous operation, more precisely for \(S=1\), by the refined and by the simplified methods. In addition, the parameters of these machines were also determined experimentally, by the static method. In the first machine the rotor was a design in which there were 5 rods per pole, while on the rotor pole of the second machine there were 6 rods. In the first machine the air gap was \(\delta=0.75\) mm, and in the second \(\delta=0.35\) mm.

Below is a table of comparative values of the parameters of these machines, calculated by the two methods and determined experimentally.

Table 1

Longitudinal axis (\(\delta=0.75\) mm)

Table 2

Longitudinal axis (\(\delta=0.35\) mm)

Conclusions

1. Calculations performed by the refined method are characterized by considerable complexity, the degree of which depends largely on the construction of the rotor sheet of iron. This method cannot be recommended for practical use.

2. A comparative analysis of calculations by the refined and simplified methods with the experimental results makes it possible to recommend, for practical calculations, the method of a simplified representation of a multiloop rotor winding by two equivalent ones.

3. The results of calculations of the steady asynchronous operating mode by the refined method differ hardly at all from the results of calculating these same parameters by the simplified method.

4. The refined method for calculating the parameters makes it possible to calculate the currents in the individual rods of the rotor winding along the $d$ and $q$ axes, and thereby opens the way for further analytical and experimental investigation of the operation of these machines in transient modes and, in particular, in the asynchronous mode.

References

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2. L. A. Kilgor. Calculation of Synchronous Machine Constants Reactances and Time Constants Affecting Transient Characteristics, Trans. AIEE, 1931, v. 50.

3. A. W. Rankin. The Direct and Quadraturexes Equivalent Circuits of the Synchronous Machine, Trans. AIEE, 1945, v. 64.

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6. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes of a Synchronous Machine. GEI, 1960.

7. Ya. B. Danilevich, V. V. Dombrovskii, E. Ya. Kazovskii. Parameters of AC Electrical Machines. Nauka Publishing House, 1965.

8. Ch. Concordia. Synchronous Machines. Transient and Steady-State Processes, translated from English, GEI, 1959.

9. M. I. Alyab’ev. General Theory of Ship Electrical Machines. Sudostroenie Publishing House, L., 1965.

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12. E. V. Kononenko, T. V. Chesheva. Slot Leakage of the Rotor of Improved Synchronous Reluctance Motors. Present collection of works, Izv. TPI.