Determination of Parameters by Estimating Magnetic Fields in the Air Gap of a Synchronous Machine
Abstract Generated abstract
This paper addresses errors in conventional experimental methods for determining the direct and quadrature axis reactances of synchronous machines, especially small machines and machines excited by permanent magnets. It proposes determining armature reaction parameters from measured magnetic field distributions in the air gap under no load and load, using equivalent sinusoidal field curves, vector decomposition along the direct and quadrature axes, and corresponding electromotive forces measured with a conductor and angular position sensing arrangement. The method yields the armature reaction reactances and can also be extended to determine armature leakage reactance from measured voltages, currents, and phase angles. Experiments on a permanent magnet synchronous machine reportedly achieved an accuracy of about 2 to 5 percent, supporting the method’s use in experimental investigations.
Full Text
DETERMINATION OF PARAMETERS BY ESTIMATING MAGNETIC FIELDS IN THE AIR GAP OF A SYNCHRONOUS MACHINE
Yu. G. Bukgolec, V. V. Zhulovyan, Yu. P. Rybkin, G. A. Shanshurov
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
For calculating the characteristics of a synchronous machine it is necessary to know more accurately the values of the synchronous inductive reactances along the direct and quadrature axes. The results obtained analytically give a significant error owing to a whole series of assumptions made in deriving the equations.
Therefore, a whole series of methods has been developed for determining the parameters of synchronous machines experimentally, both for machines with electromagnetic excitation and for machines with excitation from permanent magnets. However, the known methods for determining the inductive reactances of synchronous machines have a number of shortcomings.
When determining the inductive reactance along the direct axis from the no-load and short-circuit characteristics (the unsaturated value \(x_d\)), a large error is introduced because of the active resistance of the armature windings. The same is also observed when determining the inductive reactance along the direct axis from the no-load and load characteristics for an inductive load (the saturated value \(x_d\)). The error due to the active resistance of the armature windings increases greatly as the size of the machine decreases.
When determining the inductive parameters by the slip method (along the direct axis \(x_d\) and the quadrature axis \(x_q\)), in addition to the error due to active resistance, an error is also introduced because of the influence of currents induced in the short-circuited rotor circuits, which distort the field in the air gap of the synchronous machine [2].
The methods of experimentally determining the inductive reactances of synchronous machines with electromagnetic excitation are completely inapplicable to machines with excitation from permanent magnets, owing to the impossibility of regulating the excitation flux.
For synchronous machines with excitation from permanent magnets, a number of methods have been developed that are modifications of the methods considered above [3]. These methods also have a whole series of specific shortcomings, consisting of the following:
1) it is necessary to have several machines of the same type and the same power;
2) it is necessary to have magnets with different degrees of magnetization;
3) considerable demagnetization of the magnet occurs during the experiment.
The known methods for determining the inductive parameters of synchronous machines make it possible to determine the synchronous inductances along the direct axis \(x_d\) and along the quadrature axis \(x_q\), which are composed, respectively,
from the armature reactance along the direct axis \(x_{ad}\) and along the quadrature axis \(x_{aq}\) and the armature leakage reactance \(x_s\).
\[ x_d = x_{ad} + x_s, \]
\[ x_q = x_{aq} + x_s. \]
It should be noted that the error in measuring synchronous reactances in machines excited by permanent magnets, due to the influence of active resistance, is considerably higher, since the value of \(x_{ad}\) (and in machines without pole shoes also \(x_{aq}\)) is significantly smaller than in machines with electromagnetic excitation, owing to the low magnetic permeability of the permanent-magnet material.
The most accurate determination of the parameters of a synchronous machine is possible on the basis of estimating the field pattern in the air gap at no load and under load. Knowing the field curves in the machine gap at no load and under load, one can isolate from the curve of the resultant field (under load) the curve of armature reaction. Then, by resolving the armature-reaction curve along the direct and quadrature axes, one can find the inductive reactances of armature reaction \(x_{ad}\) and \(x_{aq}\). This method of determining the indicated quantities follows directly from the classical theory of two reactions [1, 4, 5].
In the general case, the field curves in the air gap may be nonsinusoidal, despite all the measures (shortening of windings, skew of pole tips, skew of slots, etc.) provided in the machine in order to obtain a sinusoidal field shape in the gap.
In this case the nonsinusoidal field curve in the gap must be replaced by a sinusoid whose area is equal to the area of the nonsinusoidal curve. The sinusoid obtained in this way will be called the equivalent sinusoid.
This method, therefore, makes it possible to determine the inductive reactances of armature reaction in pure form. If it is necessary to obtain the values of the synchronous inductive reactances \(x_d\) and \(x_q\), it is necessary to know the leakage reactance of the armature winding \(x_s\), which is determined by one of the known methods.
The essence of the method for determining the parameters of a synchronous machine on the basis of estimating the field pattern in the air gap is explained in Figs. 1, 2, 3, and 4.
Figure 1 shows the no-load field of a synchronous machine, \(B_0\), and the resultant field in the machine gap under load, \(B_\delta\). From the field in the gap under load \(B_\delta\), the armature-reaction field \(B_a\) is graphically isolated and resolved into components along the direct axis of the machine (axis \(d\)), \(B_{ad}\), and along the quadrature axis of the machine (axis \(q\)), \(B_{aq}\).
\[ \overline{B}_a = \overline{B}_{ad} + \overline{B}_{aq}. \tag{1} \]
Figure 2 gives a spatial diagram for a synchronous machine, where the magnitude of the armature-reaction field is determined vectorially.
The determination of the parameters of a synchronous machine is somewhat simplified if one passes from the fields in the air gap to the corresponding electromotive forces produced by these fields.
Figure 3 gives a simplified vector diagram of a synchronous machine. The vector \(\dot{E}_0\) represents the no-load e.m.f.; the vector \(\dot{E}_\delta\), the e.m.f. in the air gap. The magnitudes of these e.m.f.’s are determined by the corresponding no-load fields and the resultant field under load. The diagram also shows the resolution of the armature current \(I_a\) and of the armature-reaction e.m.f. into components along the direct and quadrature axes.
Fig. 1
Fig. 2 Fig. 3
The indicated vector diagram, in order to provide an analytical determination of the armature-reaction emf \(E_a\) and its components along the direct and quadrature axes (\(E_{ad}\) and \(E_{aq}\)), has been transformed to the form shown in Fig. 4. In the diagram the vector \(\dot E_0\) is aligned with the positive direction of the real axis, i.e., the real axis is aligned with the \(q\) axis, and the imaginary axis with the \(d\) axis.
On the basis of the vector diagram one may write
\[ \dot E_a = \dot E_\delta - \dot E_0 \tag{2} \]
or
\[ E_a e^\alpha = E_\delta e^{\theta'} - E_0 . \tag{3} \]
Fig. 4
Thus, determining the components of the armature-reaction emf along the direct and quadrature axes reduces to determining the imaginary and real parts of the vector \(E_{a1}^{\alpha}\).
\[ E_a e^\alpha = E_{aq} + jE_{ad}. \tag{4} \]
The problem of determining the parameters of a synchronous machine, as shown above, reduces to separating out the armature-reaction emf and determining the displacement angles \(\psi\) between \(E_0\) and \(I_0\), and \(\Theta'\)—between the no-load field and the resultant field.
Figure 5 presents a circuit for determining the above-mentioned quantities and, consequently, the inductive reactances \(x_{ad}\) and \(x_{aq}\). The driving motor 1 is connected to the power source 2 and rotates at rated speed. The rotor of the machine under investigation 3 and sensor 7 are rigidly connected to the driving motor and to each other.
Fig. 5
At no load we align the axis of the signal winding 8 and one of the phases of the machine under investigation 3. The signal from conductor 4 is fed to oscillograph II and to a voltmeter with high internal resistance. We oscillograph the no-load emf curve and measure the value of the emf \(E^0\). We switch on the load 5 of the machine under investigation 3,
of the synchronous machine 3 under investigation, we oscillograph the e.m.f. curve under load and measure the value of the e.m.f. \(E_\delta\). From the signal winding 8 of the angular-position sensor 7, the signal is fed to the angle sensor “\(\Theta'\)”—9 and “\(\psi\)”—10. From conductor 4 the signal is fed to the angle sensor “\(\Theta'\)”—9, and from the additional resistance 6 to the angle sensor “\(\psi\)”—10. The sensors give the values of the angles \(\Theta'\) and \(\psi\), proportional to the time shifts of the e.m.f. \(E_\delta\) and the load current \(I_a\) relative to the e.m.f. of the signal winding, fixed in time.
The no-load e.m.f. curves of the conductor and the e.m.f. curves due to the resultant field under load completely reproduce the field curves in the air gap of the synchronous machine.
Using formula (4), we determine the e.m.f. of the armature reaction along the \(d\) and \(q\) axes, respectively.
\[ E_{an} e^{\alpha} = E_{aqn} + jE_{adn}, \tag{5} \]
where
\(E_{an}\) is the armature-reaction e.m.f. induced in conductor 4 by the armature-reaction field;
\(E_{aqn}\) is the e.m.f. of conductor 4 due to the transverse component of the armature reaction in the air gap of the synchronous machine;
\(E_{adn}\) is the e.m.f. of conductor 4 due to the longitudinal component of the armature reaction in the air gap of the synchronous machine.
Using the known winding data of the machine (the number of turns \(w\) and the winding factor \(k_{\mathrm{ob}}\)), we reduce the armature-reaction e.m.f. of the conductor along the \(d\) and \(q\) axes to the armature-reaction e.m.f. of the winding along the \(d\) and \(q\) axes.
\[ E_{ad} = E_{adn}\cdot k_{\mathrm{ob}} w \]
\[ E_{aq} = E_{aqn}\cdot k_{\mathrm{ob}} w . \]
Knowing the magnitude of the load current and the displacement angle \(\psi\) between the vectors \(E_\delta\) and \(I_a\) (Figs. 2 and 3), we determine the projections of the phase current along the \(d\) and \(q\) axes, respectively.
\[ I_d = I_a \cdot \sin \psi \]
\[ I_q = I_a \cdot \cos \psi \]
From the known values of the armature-reaction e.m.f. and the phase current along the corresponding \(d\) and \(q\) axes, we determine, respectively, the magnitudes of the inductive reactances of the armature reaction along the longitudinal and transverse axes.
\[ x'_{ad} = \frac{E_{ad}}{I_d} \]
\[ x_{aq} = \frac{E_{aq}}{I_q}. \tag{8} \]
The displacement angles \(\psi\) and \(\Theta'\) are measured by one of the methods described in [6]. The choice of the measurement method is determined by the required degree of accuracy in measuring the inductive parameters of the synchronous machine.
This method also makes it possible to determine the leakage reactance of the armature winding. From the diagram in Fig. 2 we have that
\[ \dot E_\delta + \dot U = \dot E_i . \tag{6} \]
Thus, by measuring the phase-shift angles between the resultant e.m.f. in the air gap \(E_\delta\) and the voltage \(\dot U\), and between \(E_\delta\) and the load current \(I_a\), as well as the magnitudes \(E_\delta\), \(U\), and \(I_a\), one can determine the total impedance of the stator winding and, consequently, the leakage reactance of the stator winding \(x_s\).
Experimental determination of the parameters by the method considered, carried out for a synchronous machine with excitation from permanent magnets, provided sufficiently high accuracy (on the order of 2–5%) and can be recommended for experimental investigations.
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