On a Method for Calculating the Starting Characteristics of Synchronous Salient-Pole Machines
Abstract Generated abstract
The article examines methods for calculating the static starting characteristics of salient-pole synchronous machines operating asynchronously, focusing on the dependence of average torque and stator current on slip under steady rotor speed. It classifies available approaches into exact, approximate, and combined methods, deriving formulas from the Park-Gorev equations for the exact case and from separate direct- and quadrature-axis equivalent circuits for simplified calculation. The authors indicate when the more complex exact method is warranted, particularly for new machine designs or cases with significant axis asymmetry and stator resistance, while noting that approximate methods are limited because they cannot adequately represent the influence of rotor asymmetry on starting characteristics.
Full Text
ON METHODS FOR CALCULATING THE STARTING CHARACTERISTICS OF SALIENT-POLE SYNCHRONOUS MACHINES
E. V. Kononenko, T. V. Chesheva
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
The principal task in investigating the asynchronous mode of a salient-pole synchronous machine is the analysis of the laws governing the variation of currents, magnetomotive forces, magnetic fields, and torques. In calculating the asynchronous mode, one usually confines oneself to determining only the starting characteristics, understood as the dependences of the average torque and the stator current on slip. Since the most widespread and reliable means of studying the starting processes of synchronous and asynchronous motors is the consideration of steady-state characteristics, this article presents formulas for calculating only the static starting characteristics of the asynchronous mode. This means that in all calculations the rotor speed at any instant of time is assumed to be constant.
The methods used in studying and calculating the starting characteristics of synchronous motors are divided into two principal categories: exact and approximate. At the same time, exact calculation is considered rather complicated. This circumstance gave rise to the appearance of many simplified methods. However, the variety of methods currently available does not at all provide a method that is both simple in calculation and accurate in results; on the contrary, it leads to difficulties in choosing a method for practical calculations and for studying the asynchronous mode of new types of synchronous machines [1—15].
As a result of a critical analysis of existing methods for studying and calculating the characteristics of the asynchronous mode of salient-pole synchronous motors, the need was found for a strict classification, subdividing these methods into exact and approximate ones, with recommendations regarding their use. This classification, together with the calculation formulas, may be presented as follows.
Exact method
The basic condition for an exact calculation of the starting characteristics of salient-pole synchronous machines is the determination of expressions for all currents and torques from the complete system of Park–Gorev equations. For a steady-state asynchronous operating mode, these equations in complex form may be represented as follows (with the excitation winding short-circuited):
\[
\overline{U}_d=\overline{I}_d\left[r+jsX_d(js)\right]-\overline{I}_qX_q(js)(1-s),
\]
\[
\overline{U}_q=\overline{I}_dX_d(js)(1-s)+\overline{I}_q\left[r+jsX_q(js)\right],
\tag{1}
\]
where
\[ X_d(js)=x_d\frac{(1+jsT'_d)(1+jsT''_d)} {(1+jsT'_{do})(1+jsT''_{do})} \tag{2} \]
\[ X_q(js)=\frac{1+jsT''_q}{1+jsT''_{qo}}\cdot x_q . \]
If a symmetrical system of voltages is applied to the stator winding, then the voltage components \(U_d\) and \(U_q\), at a constant rotor speed of rotation, are determined by
\[
U_d=U\cos(st+\delta_0),
\]
\[
U_q=U\sin(st+\delta_0),
\tag{3}
\]
where \(\delta_0=\alpha_0-\gamma_0\) is the angle formed by the resultant voltage vector with the rotor \(d\)-axis at the initial instant of time. Assuming that the actual quantities are determined as the real parts of the complexes, one may write
\[ \overline{U}_d=Ue^{j(st+\delta_0)}, \tag{4} \]
\[ \overline{U}_q=jUe^{j(st+\delta_0)} . \tag{5} \]
Solving the system of equations (1) with respect to the currents \(\overline{I}_d\) and \(\overline{I}_q\), one can determine the components of the space vector of the stator current
\[ \overline{i}_s=\overline{i}_{s1}+\overline{i}_{s2}, \tag{6} \]
where
\[ \overline{i}_{s1}=\frac{1}{2}\left(\overline{I}_d+j\overline{I}_q\right) \]
is the space vector of the positive-sequence current;
\[ \overline{i}_{s2}=\frac{1}{2}\left(\overline{I}_d^{*}+j\overline{I}_q^{*}\right) \]
is the space vector of the negative-sequence current.
(Conjugate quantities are denoted by an asterisk.)
The rms value of the stator current for any values of slip, except \(s=0\) and \(s=1.0\), is determined by the formula
\[ I=\sqrt{I_1^2+I_2^2}, \tag{7} \]
where \(I_1\) and \(I_2\) are the amplitudes of the currents \(\overline{i}_{s1}\) and \(\overline{i}_{s2}\).
At \(s=1.0\), the magnitude of the current depends on the position of the rotor in space and varies within the limits from \(I_{\max}=I_1+I_2\) to \(I_{\min}=I_1-I_2\). Knowing the expressions for the current components, one can calculate the average torque as the sum of the positive- and negative-sequence torques.
To calculate the starting characteristics of salient-pole synchronous motors by the above method on a digital computer, all formulas (1)–(7) can be transformed to the following form:
\[ a'=r+\left(\frac{1}{2}-s\right)(B+D), \]
\[ b'=\left(\frac{1}{2}-s\right)(A+C), \]
\[ c'=r^2-rs(B+D)+(AC-BD)(1-2s), \]
\[ d'=rs(A+C)+(BC+AD)(1-2s), \]
\[ a''=\left(\frac{1}{2}-s\right)(D-B), \]
\[ b''=\left(\frac{1}{2}-s\right)(C-A), \]
\[ A=\frac{a_1}{b_1},\qquad B=\frac{a_2}{b_1},\qquad C=\frac{c_1}{b_2},\qquad D=\frac{c_2}{b_2}, \]
\[
a_1=x_d-s^2x_d\left[T'_dT''_d+T'_{do}T''_{do}-(T'_d+T''_d)(T'_{do}+T''_{do})\right]+
\]
\[
{}+s^4x_dT'_dT''_dT'_{do}T''_{do},
\]
\[
a_2=s^3x_d\left[T'_dT''_d(T'_{do}+T''_{do})-T'_{do}T''_{do}(T'_d+T''_d)\right]-
\]
\[
{}-sx_d\left[(T'_{do}+T''_{do})-(T'_d+T''_d)\right],
\]
\[ b_1=1-2s^2T'_{do}T''_{do}+s^4T_{do}^{\prime 2}T_{do}^{\prime\prime 2}+s^2(T'_{do}+T''_{do})^2, \]
\[ b_2=1+s^2T_{qo}^{\prime\prime 2}, \]
\[ c_1=x_q(1+s^2T''_qT''_{qo}), \]
\[ c_2=x_qs(T''_q-T_{qo}), \]
\[ I_{1a}=\frac{a'c'-b'd'}{c'^2+d'^2},\qquad I_{1p}=\frac{-b'c'-a'd'}{c'^2+d'^2}, \]
\[ I_1=\sqrt{I_{1a}^{2}+I_{1p}^{2}}, \]
\[ I_{2a}=\frac{a''c'+b''d'}{c'^2+d'^2},\qquad I_{2p}=\frac{b''c'-a''d'}{c'^2+d'^2}, \]
\[ I_2=\sqrt{I_{2a}^{2}+I_{2p}^{2}},\qquad I=\sqrt{I_1^2+I_2^2}, \]
\[ M_1=UI_{1a}-rI_1^2,\qquad M_2=-\frac{r}{1-2s}I_2^2, \]
\[ M_c=M_1+M_2. \]
Amplitude of the pulsating torque
\[ M_{\mathrm{puls}}=I_2\sqrt{\,U^2+\left[\frac{2r(1-s)}{1-2s}I_1\right]^2-4UI_{1a}\frac{r(1-s)}{1-2s}\,}. \]
Despite the fairly high degree of complexity of this method, it should be recommended in those cases when:
1) the asynchronous mode of a synchronous machine of a new design is being investigated, i.e., for machines for which a detailed study of machine operation in the asynchronous mode is required;
2) the active resistance of the stator winding is sufficiently large and the difference in the steady-state values of the operational resistances along the direct \(d\) and quadrature \(q\) axes of the machine is very significant.
Approximate method
The use of simplified methods has become possible because the exact methods for calculating the starting characteristics of synchronous salient-pole machines prove to be rather complex.
In calculating starting characteristics by the approximate method, the possibility is admitted of considering separately the phenomena along the two axes
rotor of a synchronous salient-pole machine. The expressions for the current complexes \(\overline{I}_d\) and \(\overline{I}_q\) in this case can be determined from equivalent circuits analogous to the well-known equivalent circuits of asynchronous motors, i.e.,
\[ \overline{I}_d=\frac{1}{r+jX_d(js)}\,\overline{U}, \qquad \overline{I}_q=\frac{1}{r+jX_q(js)}\,\overline{U}. \tag{8} \]
The calculation of the steady-state values of the operational resistances \(X_d(js)\) and \(X_q(js)\) must always be sufficiently accurate; i.e., simplifications in calculating the steady-state values of the operational resistances and simplifications in the method of calculating starting characteristics are not one and the same thing.
Determination of the average torque consists in finding the torque twice: first, under the assumption that the machine has a symmetrical rotor and that its parameters along both axes are equal to the parameters along the direct axis; second, under the assumption that the machine has a symmetrical rotor and that its parameters along both axes are equal to the parameters along the quadrature axis. In this case the average torque is simply the mean value of the losses in the rotor circuits of the equivalent circuits along the direct and quadrature axes, if a voltage equal to \(U\) is applied to each of them.
The steady-state values of the operational resistances along the two rotor axes are represented in the form
\[ jX_d(js)=R_{ds}+jX_{ds}, \qquad jX_q(js)=R_{qs}+jX_{qs}. \]
The effective value of the current in the stator winding will be equal to
\[ I=\sqrt{\frac{1}{2}I_d^2+I_q^2} \tag{10} \]
The average torque is determined as follows:
\[ M_c=\frac{1}{2}(M_d+M_q)=\frac{1}{2}\left(I_d^2R_{ds}+I_q^2R_{qs}\right). \tag{11} \]
After transformation, the real and imaginary parts of \(jX_d(js)\) and \(jX_q(js)\) can be written as follows:
\[ R_{ds}=\frac{a_2}{b_1},\qquad X_{ds}=\frac{a_1}{b_1},\qquad R_{qs}=\frac{c_2}{b_2},\qquad X_{qs}=\frac{c_1}{b_2}. \tag{12} \]
It follows from the equations given that the dependences \(M_d=f(s)\) and \(M_q=f(s)\) are no different from the mechanical characteristics of an asynchronous motor with a symmetrical rotor. Consequently, when these calculation formulas are applied in the calculation, it is impossible to analyze the influence of rotor asymmetry on the starting characteristics of a synchronous machine.
Thus, approximate calculation methods based on separate consideration of the phenomena along the direct and quadrature axes of the rotor can be used only for a definite ratio of the parameters of the synchronous machine.
Combined Method
The combined method for calculating the static starting characteristics of synchronous salient-pole machines assumes an approximate calculation of the current complexes \(I_d\) and \(I_q\), while the torques are determined from formulas
of the exact method. In other words, the currents are determined from the equivalent circuits of the machine along the two axes, considered separately, as in the case of asynchronous motors with a symmetrical rotor. The average torque is determined by the formula
\[ M_c = M_1 + M_2 . \]
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