Calculation of the Characteristics of a Brushless Single-Phase Synchronous Generator of Increased Frequency
Abstract Generated abstract
This paper examines a cascade brushless single-phase synchronous generator intended for increased-frequency autonomous power systems and combining magnetic and electrical circuits. It describes the operating principle in which the rotor winding, connected through a half-wave rectifier, serves both as the first-stage output winding and as the second-stage excitation winding, while the reverse-synchronous field is used as current feedback. A calculation method is developed for the rotor current, rectifier operating mode, no-load characteristic, and external characteristic, including the effect of transformer coupling between windings. Comparison with tests on a 150 W, 150 Hz model indicates close agreement between calculated and experimental characteristics, and shows that use of the reverse-synchronous field increases the stiffness of the external characteristic and the generator power.
Full Text
CALCULATION OF THE CHARACTERISTICS OF A BRUSHLESS SINGLE-PHASE HIGH-FREQUENCY SYNCHRONOUS GENERATOR
A. P. Felzing, M. L. Kostyrev, A. I. Skorospeleshkin
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
Research conducted in the USSR on the problem of optimizing the frequency of industrial alternating current indicates the unconditional expediency of using an increased frequency of 100–400 Hz [1]. As sources of increased frequency for autonomous systems, brushless generators are being used ever more widely, including cascade brushless generators with combined magnetic and, sometimes, electrical circuits. A cascade generator has electromagnetic excitation and permits regulation of the output voltage over a wide range. The combination of two or more machines in one magnetic circuit makes it possible to construct the generator in a single-machine design, to reduce overall dimensions and weight by decreasing the consumption of structural materials. Electrical combination makes it possible additionally to reduce the consumption of active materials, decrease the number of windings, and thereby simplify the design of the machine.
The proposed article describes the operating principle and a method for calculating the principal characteristics of a cascade brushless single-phase high-frequency synchronous generator with combined magnetic and electrical circuits, whose schematic diagram is shown in Fig. 1.
The generator is a cascade connection of two synchronous machines, the winding circuits and numbers of pole pairs of which are selected and
Fig. 1. Diagram of a brushless single-phase generator with combined magnetic and electrical circuits
are arranged in such a way that there is no mutual-inductance coupling between them. A feature of the circuit in Fig. 1 is that the rotor winding \(W_R\), closed through a half-wave rectifier, simultaneously performs two functions:
a) the output winding of the first stage;
b) the excitation winding of the second stage.
The generator operates as follows. When voltage is applied to the excitation winding \(W_f\) and the rotor rotates, an emf is induced in the winding \(W_R\) with frequency
\[ f_R = P_1 \cdot n, \tag{1} \]
where
\(P_1\) is the number of pole pairs of the first stage,
\(n\) is the rotor speed in rev/sec.
This emf, through the half-wave rectifier, supplies the rotor winding with direct current. The rectified current, flowing through the rotor winding, creates a magnetic flux stationary with respect to the rotor, with the number of pole pairs \(P_2\). In this case the first harmonics of the m.m.f. of the individual winding phases, produced by the dc component of the current, coincide in time phase, are displaced in space by the angle \(\dfrac{2\pi}{m}\), and give a geometrical sum of zero [2]. The harmonic components of the m.m.f. that are multiples of \(m\) coincide in time phase and are displaced in space by the angle
\[ \frac{2\pi}{m}\cdot \nu_m, \]
where \(\nu_m = k \cdot m,\ k = 1, 3, 5\).
\(m\) is the number of phases of the rotor winding; i.e., in space they coincide and are superposed on one another.
Consequently, the resultant constant m.m.f. of all phases of the rotor winding due to the rectified current is the result of the superposition of harmonics that are multiples of the number of phases of this winding. All other harmonics of the constant m.m.f. of the phases sum to zero.
The number of pole pairs of the steady magnetic field created by the rotor winding is
\[ P_2 = \nu_m \cdot P_1. \tag{2} \]
The amplitude value of the constant m.m.f. of the combined winding per pole pair is
\[ F_R = \frac{4}{\pi}\cdot \frac{m \cdot W_R \cdot k_{\mathrm{ob}\nu m}}{P_2} \cdot I_R, \tag{3} \]
where \(W_R\) is the number of turns in a phase,
\(K_{\mathrm{ob}\nu m}\) is the winding factor for the harmonic \(\nu_m\),
\(I_R\) is the dc component of the current in a phase of the rotor winding.
Under the action of this m.m.f., an emf is induced in the output winding \(W_S\) with frequency
\[ f_S = p_2 \cdot n. \tag{4} \]
When an alternating load current flows through it, the stator winding \(W_S\) creates a pulsating field, which may be represented as consisting of two magnetic fields rotating in opposite directions, with amplitudes equal to one half of the amplitude of the pulsating field.
In conventional generators the reverse-synchronous field (RSF) leads to a number of undesirable phenomena: additional losses arise in the excitation winding, the waveform of the output voltage is distorted, etc. In the proposed circuit of the brushless generator, the RSF is used as current feedback. The emf induced in the rotor winding by it ...
OSP through a half-wave rectifier feeds this same winding with a direct current proportional to the load current.
The appearance of a reverse-synchronous field when the generator is switched on to a load and the operation of the rectifier in the short-circuit mode determine the specific nature of the method for calculating the characteristics of such a generator.
The presence of leakage reactance in the phases of the rotor winding leads to the fact that the diodes, operating in such a deep mode, conduct current practically throughout the entire period. Therefore the diode commutation angle, neglecting the active resistance of the phase, is \(\gamma \simeq 180^\circ\). The current in each phase exists throughout the entire period, and its average value is [3, 4]
\[ I_{R1}=\frac{E_{01}}{X_{d1}}, \tag{5} \]
where \(E_{01}\) is the e.m.f. induced by the excitation flux in the rotor winding, and \(X_{d1}\) is the synchronous reactance of the first cascade. Thus, the constant component of the current flowing through the rotor winding is equal to the three-phase short-circuit current of the first cascade. From the condition of equality of the actual and referred m.m.f. created by the winding \(W_f\), the coefficient for referring the excitation winding to the rotor is
\[ K_{i1}=\frac{I_f}{I_f'}=\frac{0.9 \cdot m \cdot W_R \cdot K_{\mathrm{ob}}}{W_f \cdot K_f}, \tag{6} \]
where \(W_f\), \(K_f\) are the number of turns and the field-form factor of the excitation winding,
\(K_{\mathrm{ob}}\) is the winding factor of the rotor winding with respect to the field of the first cascade.
The constant component of the current in the rotor from the excitation field is
\[ I_{R1}=I_{\mu1}-\frac{I_f}{K_{i1}}, \tag{7} \]
where \(I_{\mu1}\) is the magnetizing current of the first cascade.
The windings \(W_S\) and \(W_R\), with respect to the OSP, may be regarded as windings of a certain short-circuited transformer [5]. From the condition of equality of the actual and referred m.m.f. created by the stator winding, the coefficient for referring the stator winding to the rotor is
\[ K_{i2}=2\frac{m \cdot W_R \cdot K_{\mathrm{ob}m}}{W_S \cdot K_{\mathrm{ob}S}}, \tag{8} \]
where \(K_{\mathrm{ob}S}\) is the winding factor of the stator winding. The constant component of the current in the rotor from the reverse-synchronous field is
\[ I_{R2}=I_{\mu2}-\frac{I_S}{K_{i2}}, \tag{9} \]
where \(I_{\mu2}\) is the magnetizing current of the second cascade with respect to the OSP.
The resultant direct current in the phase of the rotor winding is
\[ I_R=I_{R1}+I_{R2}. \tag{10} \]
The presence of transformer coupling between the windings \(W_S\) and \(W_R\) leads to an additional voltage drop at the generator output. Taking this into account, the voltage at the generator output is
\[ U_S=\dot{E}_{02}-j(X_a+X)\cdot \dot{I}-R\cdot \dot{I}, \tag{11} \]
\[ X=X_{\sigma S}+X_2,\qquad R=r_S+r_2; \]
where
$r_s$ and $r_2$ are the active resistances of the positive- and negative-sequence components of winding $W_s$,
$X_{\sigma s}$, $X_a$, $X_2$ are the leakage reactance and the positive- and negative-sequence reactances of winding $W_s$.
Fig. 2. Vector diagram of the emf of a brushless single-phase generator
In Fig. 2 a vector diagram of the emf has been constructed, from which we obtain the following expression for the external characteristic of the single-phase combined generator:
\[ U_s=\sqrt{E_{02}^{\,2}-\left[\left(X_d\cos\varphi-R\sin\varphi\right)\cdot I_s\right]^2} -\left(X_d\sin\varphi+R\cos\varphi\right)\cdot I_s, \tag{12} \]
where
\[ X_d=X_a+X. \]
Using the proposed method for determining the current in the rotor and accounting for the influence of the OSP, the no-load characteristic and the external characteristic of the combined single-phase generator were calculated; these are shown in Figs. 3 and 4. For comparison, the experimentally obtained characteristics are also shown in the same figures. In addition, Fig. 4 presents the experimentally obtained external characteristic without using the OSP. The experiments were carried out on a generator model with combined magnetic electric circuits, manufactured on the basis of an AO—32—6 motor. Data of the model:
\[ \begin{aligned} P&=150\ \text{W},\\ f_s&=150\ \text{Hz},\\ n&=3000\ \text{rpm}. \end{aligned} \]
As can be seen from the figures presented, the calculated and experimental curves agree with one another quite closely. Some discrepancy is explained by the fact that, in calculating the no-load characteristic, iron losses were not taken into account, and in calculating the external characteristic, the magnetizing currents were not taken into account.
From expression (10) and from comparison of the curves in Fig. 4 it is evident that the use of the OSP substantially increases the stiffness of the external characteristic and, consequently, also the power of the cascade single-phase brushless synchronous generator.
Fig. 3. No-load characteristic: 1) calculated, 2) experimental
Fig. 4. External characteristics: 1) calculated, 2) experimental, 3) experimental without using the OSS
Conclusions
- A single-phase synchronous generator with combined magnetic and electric circuits has a simple design and rigid external characteristics owing to the use of an inverse-synchronous field.
The proposed method for taking into account the influence of the combined winding and the operating mode of the rectifier makes it possible, with sufficient accuracy for practice, to calculate all the characteristics of a single-phase combined generator.
References
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