Influence of the relative inductances of armature reaction and leakage on the operation of a synchronous reluctance machine at variable supply frequency
Abstract Generated abstract
This paper analyzes how the relative inductances of armature reaction and leakage affect a synchronous reluctance machine supplied at variable frequency, taking account of armature resistance. Using a modified equivalent circuit and relative machine characteristics, the authors derive expressions for current, electromagnetic parametric power, critical displacement angle, power factor, input power, and efficiency under a voltage law dependent on frequency. The analysis indicates that decreasing leakage and armature-reaction inductances can increase main-circuit current, that lower leakage inductance increases the critical displacement, and that the power factor at critical displacement generally falls as frequency rises. Efficiency at critical displacement is shown to decrease with lower frequency and greater magnetic asymmetry, while higher frequency and inductance values reduce the influence of asymmetry.
Full Text
EFFECT OF THE RELATIVE INDUCTANCES OF ARMATURE REACTION AND LEAKAGE ON THE OPERATION OF A SYNCHRONOUS RELUCTANCE MACHINE AT VARIABLE SUPPLY FREQUENCY
A. I. BERTINOV, Ya. A. GOLUBENKO
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
To analyze the effect of the relative inductances of armature reaction and leakage on the operation of a synchronous reluctance machine (SRM) under conditions of varying frequency, with allowance for the active resistance of the armature winding, we use the modified equivalent circuit proposed in [1] and the equations of the relative characteristics.
To obtain the results of the study in general form, let us consider the relative values of the SRM characteristics when the frequency and voltage are varied. The characteristics corresponding to the rated values \(f_n\) and \(U_n\) are taken as the base values.
Proceeding from the modified equivalent circuit, we shall also adopt the assumptions customary in these cases:
a) symmetry of the phases of the supply voltage, sinusoidal form of its curve and of the spatial distribution of the magnetic induction in the machine air gap;
b) the influence of saturation of the magnetic circuit and the phenomenon of current displacement in the windings are not taken into account;
c) the applied voltage, expressed in relative quantities, is a function of frequency, i.e., in general form:
\[ U^* = f^{*\,\nu}, \]
where
\[ \nu \]
characterizes the law of voltage variation with variation of frequency,
\[ U^*=\frac{U}{U_n}; \qquad f^*=\frac{f}{f_n}. \]
Here \(U\) and \(U_n\), \(f\) and \(f_n\) are, respectively, the current and rated values of voltage and frequency.
The nature of the variation of the relative inductances with variation of frequency is general, and at frequencies different from the rated one the parameters of the T-shaped modified equivalent circuit are expressed by the products:
\[ \beta_{1f}=f^*\beta_1 \quad \text{— for the armature winding,} \]
\[ \beta_{\mu f}=f^*\beta_\mu \quad \text{— for the magnetizing contour,} \]
\[ \delta_{\theta f}=f^*\delta_\theta; \qquad \beta_{\theta f}=f^*\beta_\theta \quad \text{— for the load branch.} \]
For the modified equivalent circuit with the magnetizing contour brought out to the terminals, respectively, we obtain:
\[ \beta_{0f}=f\beta_0^{*} \quad \text{— for the magnetizing circuit.} \]
\[ \delta_{\theta rf}=f\delta_{\theta r}^{*}; \qquad \beta_{\theta rf}=f\beta_{\theta r}^{*} \quad \text{— for the main circuit of the scheme.} \]
According to the adopted assumptions, the active resistances of the windings remain unchanged when the frequency changes.
The modified equivalent circuits of the SRM, in accordance with the parameters given for the fundamental harmonic of the voltage, are shown in Fig. 1. Using the circuit (Fig. 1b), we write the principal characteristics of the SRM, taking into account variation of frequency and voltage.
Fig. 1. Modified equivalent circuits of a synchronous reluctance machine at variable supply frequency.
- The expressions for the currents of the main and magnetizing circuits have the form
\[ I'_{\theta f}=I_i f^{*}\left(\sigma_1 A_{\theta f}\right)^{0.5}; \qquad I_{0f}=I_i\sigma_i f^{*}\left(\frac{A_{0f}}{\delta_0}\right)^{0.5}. \tag{1} \]
The maximum current of the main circuit is equal to
\[ I'_{omf}=I_i f^{*}\left(\sigma_1 A_{0mf}\right)^{0.5}. \tag{2} \]
Here
\[ I_i=\frac{U_{\mathrm{n}}}{r_1\sigma_1}; \qquad A_{0f}=\frac{1}{\sigma_1\left[(1+f^{*}\delta_{0r})^2+(f^{*}\beta_{0r})^2\right]}; \qquad A_{0i}=\frac{1}{\delta_0\left[1+(f^{*}i_0)^2\right]}; \]
\[ A_{0mf}=\frac{1}{\sigma_1\left[1+(f^{*}\beta_{0r})^2\right]}. \]
It follows from (1–2) that the effect of changes in the supply frequency on the machine currents, irrespective of the magnitude of the magnetic asymmetry \(k\), is determined to a considerable extent by the ratio of the relative inductances of armature reaction \(\beta'_{ad}\) and leakage \(\beta_1\), where
\[ \beta'_{ad}=\sigma_1\beta_0^{(k-1)} . \]
With a decrease in \(\beta^1_{ad}\) and \(\beta_1\), the maximum current of the main circuit increases for any values of magnetic asymmetry over the entire range of frequency variation. In this case we have
\[ \sigma_1=\sigma_1 e^{-j\gamma} =1+\frac{1+j\beta_{1f}}{\delta_M(1+j\beta_{Mf})} \]
the transformation coefficient of the equivalent circuit, expressed through relative parameters.
- The electromagnetic parametric power developed by the machine at variable supply frequency is determined by the expression
\[ P_{nf}=P_i\sigma_1\sigma_{0r}A_{0f}f^{*\,2\nu+1}, \]
where
\[ P_i=m_1 I_i U_{\mathrm{n}}. \]
The critical displacement angle \(\Theta_{nfk}\), determining the static overload capacity, and the maximum parametric power \(P_{nfk}\) are determined by the expressions
\[ \theta_{nfk}=\pm \operatorname{arctg} f^{*}\beta'_0 \left[1+(f^{*}\beta_{0r})^2\right]^{-0.5}; \qquad P_{nfm}=\pm \frac{0.5P_i f^{*\,2\nu}} {1+\left[1+(f^{*}\beta_{0r})^2\right]^{0.5}} . \tag{4} \]
The plus sign refers to the motor mode, the minus sign to the generator mode.
Studies have shown that over the entire range of frequency variation the following holds:
a) irrespective of the magnetic-asymmetry coefficient \((k=1\div 5)\), with a decrease in the relative leakage inductance the critical displacement increases;
b) the effect of magnetic asymmetry on the magnitude of the critical displacement increases with increasing \(\beta_1\).
- The general expression for the power factor corresponding to the angle \(\varphi_1\) [1] has, under load, the form
\[ \cos\varphi_{1f}=(1+f^{*}\delta_{0r})\sqrt{\sigma_1 A_{\vartheta f}}, \tag{5} \]
and at critical displacement—
\[ \cos\varphi_{1fk}=\left[0.5\left(1+\sqrt{\sigma_1 A_{\vartheta mf}}\right)\right]^{0.5}. \tag{6} \]
The relative inductances of armature reaction and leakage at variable supply frequency also have a substantial effect on the power factor of the machine.
From analysis of expression (6) and the curves in Fig. 2, the following follows:
Fig. 2. Influence of magnetic saliency and the relative inductances of armature reaction and leakage on the power factor at critical displacement and variable supply frequency.
\[
\cos \varphi_{1fk} = 1(k) \ \text{as a function of } f^{*} \text{ and } \beta'_{ad}.
\quad \rule{1.5cm}{0.4pt}\ \beta_1 = 10;
\]
\[
--\ \beta_1 = 0.5;
\]
a) over the entire range of frequency variation \((f^{*}=0.5 \div 3)\), with increasing frequency the power factor corresponding to the critical displacement \((\cos \varphi_{1fk})\) decreases for all values of the relative inductances of armature reaction \(\beta'_{ad}\), leakage \(\beta_1\), and magnetic saliency.
b) the increase in \(\cos \varphi_{1fk}\) as the frequency decreases occurs more rapidly at smaller relative inductances of armature reaction and leakage; moreover, the influence of the relative leakage inductance is manifested to a greater degree than that of the relative armature-reaction inductance. With increasing magnetic saliency this increase slows down.
- The power supplied from the network is determined by the expression
\[ P_{1f}=P_1 \sigma_{1f}^{*\,2}\left[A_{\theta f}\left(1+\delta_{\theta fr}^{*}\right)+A_{01}\right]. \tag{7} \]
Assuming, in the operating mode of the machine, the shaft power to be equal to the parametric (electromagnetic) power \(P_{\mathrm{nf}}=P_{2f}\), for the efficiency without taking into account the active component of the current of the magnetizing circuit we obtain
\[ \eta=\frac{\delta_{\theta f}}{1+\delta_{\theta f}}. \tag{8} \]
The efficiency corresponding to the critical displacement at variable supply frequency, by analogy with (8), is determined by the following expression:
\[ \eta_{fk}=\frac{f\delta_{0rfk}^{*}}{1+f\delta_{0rfk}^{*}}, \tag{9} \]
where
\[ \delta_{0rfk}=\frac{\beta'_0}{\operatorname{tg}\theta_{nrk}} \]
is the relative active resistance of the main circuit of the modified equivalent circuit at critical slip under conditions of varying frequency.
Analysis of (9) shows that:
-
With increasing magnetic asymmetry, the efficiency corresponding to the critical slip at variable supply frequency, \(\eta_{fk}\), decreases.
-
As the frequency decreases, \(\eta_{fk}\) decreases for all values of the relative inductances of armature reaction \(\beta'_{ad}\) and leakage \(\beta_1\).
-
With increasing supply frequency and relative inductances of armature reaction and leakage, the influence of the magnetic-asymmetry coefficient on \(\eta_{fk}\) is smoothed out to a considerable extent.
Conclusions
The influence of frequency variation on the current, electromagnetic power (torque), power factor, and efficiency of synchronous reluctance machines depends to a considerable degree on the relative inductances of armature reaction and leakage.
References
- A. I. Bertinov. Electrical Machines of Aviation Automation. Oborongiz, 1961.