Some Features of Synchronous Generator Operation with a Frequency Converter
Abstract Generated abstract
This paper analyzes the operation of a synchronous generator without, or with only a weak, damper system when supplying a directly coupled thyristor frequency converter. Using simplified harmonic and magnetomotive force representations, the authors show that converter control produces modulation of the generator armature reaction, flux, emf, and output voltage at twice the converter output frequency, with the effect depending on winding displacement and load symmetry. The study also examines how pulse width excitation regulation responds to this low frequency voltage modulation, tending to compensate it while self excitation may reinforce oscillations. Experimental tests on a 3 kW inductor generator with three armature windings are reported to support the analysis, and the paper recommends placing three phase winding groups in the same phase zones to suppress modulation under symmetrical converter loading.
Full Text
SOME FEATURES OF OPERATION OF A SYNCHRONOUS GENERATOR FEEDING A FREQUENCY CONVERTER
B. V. LUKUTIN, A. B. TSUKUBLIN
Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering
To create autonomous sources of electric power with a stable output-voltage frequency under variable generator rotational speed, thyristor frequency converters are at present widely used [1, 2]. It is obvious that a load such as a frequency converter has a characteristic effect on the operation of the generator. The present article examines certain features that arise when a generator without a damper system operates feeding a directly coupled frequency converter, in particular, modulation of the generator voltage by the converter output frequency and its influence on the voltage-stabilization system.
One possible variant of the system—a synchronous generator–frequency converter (SG–FC) with a three-phase output—is shown in Fig. 1. Such a system consists of a synchronous generator SG with three armature windings, three controlled rectifiers CR, and three in—
Fig. 1. Schematic circuit diagram of a three-phase SG–FC system
vectors I. In the steady-state regime the generator delivers an alternating voltage with frequency \(f\); the output-voltage curve of the controlled rectifier is formed from sections corresponding to the instantaneous values of the generator voltage during the operating intervals of the individual valves. By varying the firing angles of the thyristors according to a certain law with a specified frequency of the secondary network, it is possible to obtain, at the rectifier output, a voltage whose value varies periodically according to any law. Inverter I serves to obtain an alternating voltage from the pulsating voltage of the rectifier. The nature of the formation of the generator phase-current curve for an FC with sinusoidal output voltage is shown in Fig. 2. The curve is shown for the case
Fig. 2. Oscillogram of the generator phase current
of operation of the SG–FC system with an active load at \(f/f_0 = 8/1\). Separating the first harmonic from the generator phase-current curve, we note that it is amplitude-modulated by the known output frequency of the converter \(f_0\). Neglecting the higher harmonics and the commutation processes in the rectifier, we replace the actual generator-current curve by an equivalent curve, which is a sinusoid with frequency \(f\), amplitude-modulated at frequency \(2f_0\). Then the currents in the generator phases for an SG–FC system with a single-phase output are determined as:
\[ i_1 = \sqrt{2}\, I (k + \sin 2\omega_0 t)\sin \omega t ; \]
\[ i_2 = \sqrt{2}\, I (k + \sin 2\omega_0 t)\sin (\omega t - 120^\circ) ; \tag{1} \]
\[ i_3 = \sqrt{2}\, I (k + \sin 2\omega_0 t)\sin (\omega t - 240^\circ), \]
where \(k\) is a constant coefficient determining the modulation depth of the voltage of the controlled rectifier.
The current of each phase creates a magnetomotive force (m.m.f.):
\[ F_1 = \frac{2\sqrt{2}\, I W k_w}{\pi p} (k + \sin 2\omega_0 t)\cos \alpha \sin \omega t ; \]
\[ F_2 = \frac{2\sqrt{2}\, I W k_w}{\pi p} (k + \sin 2\omega_0 t)\cos (\alpha - 120^\circ)\sin (\omega t - 120^\circ) ; \tag{2} \]
\[ F_3 = \frac{2\sqrt{2}\, I W k_w}{\pi p} (k + \sin 2\omega_0 t)\cos (\alpha - 240^\circ)\sin (\omega t - 240^\circ). \]
The magnetomotive force of the three-phase winding will be equal to
\[ F_\Sigma = \frac{3\sqrt{2}\, I W k_w}{\pi p} (k + \sin 2\omega_0 t)\sin \omega t - \alpha . \tag{3} \]
Thus, the armature-reaction flux is modulated at frequency \(2\omega_0\). Consequently, the total flux of the machine, determined by the flux of the field winding and the armature-reaction flux, proves to be modulated at the same frequency. This leads to modulation, at twice the output frequency of the converter, of the generator emf. In the field winding, by analogy
...with a transformer whose primary winding is the armature winding, an emf with frequency \(2\omega_0\) is also induced.
Let us next consider the SG–FC system with a three-phase output, shown in Fig. 1. Using the reasoning given above, let us assume that each winding of the SG is loaded with a symmetrical load, the magnitude of which varies sinusoidally with frequency \(2\omega_0\) and is shifted in time by \(120^\circ\).
Then the magnetomotive force of each three-phase winding is determined according to expression (3):
\[ F_{\Sigma}^{1} = \frac{3\sqrt{2}\, I W k_w}{\pi p} \left(k+\sin 2\omega_0 t\right)\sin(\omega t-\alpha); \]
\[ F_{\Sigma}^{2} = \frac{3\sqrt{2}\, I W k_w}{\pi p} \left[k+\sin(2\omega_0 t-120^\circ)\right]\sin(\omega t-\alpha-\gamma_1); \tag{4} \]
\[ F_{\Sigma}^{3} = \frac{3\sqrt{2}\, I W k_w}{\pi p} \left[k+\sin(2\omega_0 t-240^\circ)\right]\sin(\omega t-\alpha-\gamma_2), \]
where \(\gamma_1\) and \(\gamma_2\) are the angles of displacement between the three-phase windings.
The resultant mmf of the machine is determined by summing the mmf of each three-phase winding:
\[ F_{\Sigma} = \frac{3\sqrt{2}\, I W k_w}{\pi p} \left\{ 4k\cos\frac{\omega t-\alpha}{2} \cos\frac{\omega t-\alpha-\gamma_1}{2} \cos\frac{\omega t-\alpha-\gamma_2}{2} +\right. \]
\[ \left. +\sin(\omega t-\alpha) \left[ \sin 2\omega_0 t +\sin(2\omega_0 t-120^\circ)\cos\gamma_1 +\sin(2\omega_0 t-240^\circ)\cos\gamma_2 \right] +\right. \]
\[ \left. +\cos(\omega t-\alpha) \left[ \sin(2\omega_0 t-120^\circ)\sin\gamma_1 +\sin(2\omega_0 t-240^\circ)\sin\gamma_2 \right] \right\}. \tag{5} \]
Analyzing expression (5), we conclude that, with a symmetrical load of the machine, the resultant armature mmf is modulated at frequency \(2\omega_0\), except in the case when \(\gamma_1=\gamma_2=0\), i.e., when all three armature windings lie in the same slots. In this case the resultant mmf of the machine is equal to
\[ F_{\Sigma} = \frac{3\sqrt{2}\, I W k_w}{\pi p}\sin(\omega t-\alpha)\,3k . \tag{6} \]
In the case of an asymmetrical load of the individual three-phase windings of the generator, with asymmetry of the FC output phases, the resultant mmf is determined as
\[ F_{\Sigma} = \frac{3\sqrt{2}\, I W k_w}{\pi p}\sin(\omega t-\alpha) \left[(I_m+2I)k+(I_m-I)\sin 2\omega_0 t\right]. \tag{7} \]
It follows from this that the magnetic flux of the machine is also modulated at frequency \(2\omega_0\), and, consequently, the emf of the generator will be modulated at the same frequency. It is interesting to note that the emf modulation, both in the case of an asymmetrical load and in the case of a spatial displacement of the windings in all three-phase windings, will be in phase. In the field winding, as in the case of a machine with a single armature winding, a transformer emf will appear.
The presence of low-frequency modulation of the generator output voltage affects the generator excitation control system. Let us consider the widely used relay regulator with pulse-width modulation [3]. Regulation will be carried out according to voltage deviation. Modulation of the width of the pulses applied to the generator field winding is accomplished by comparing a sawtooth voltage, whose dc component is proportional to the generator output voltage, with a reference voltage.
In the case of operation of a generator with a weak damper system on
the frequency converter, the generator voltage is modulated by twice the output frequency of the converter. Graphs explaining the operation of the stabilization system are shown in Fig. 3. The current of the generator field winding is determined by the expression:
\[ I_{\mathrm{ov,\,av}}=\frac{U_{\mathrm{pit}}}{R_{\mathrm{ov}}}k_{\mathrm{z.i}};\quad k_{\mathrm{z.i}}=\frac{t_{\mathrm{i}}}{T_{\mathrm{v}}}, \]
where \(U_{\mathrm{pit}}\) is the voltage of the output stage of the amplifier;
\(R_{\mathrm{ov}}\) is the resistance of the field winding;
\(k_{\mathrm{z.i}}\) is the pulse duty factor;
\(t_{\mathrm{i}}\) is the duration of the current pulse;
\(T_{\mathrm{v}}\) is the pulse repetition period.
Fig. 3. Graphs illustrating the operation of the relay element of the regulator
From Fig. 3 we determine the pulse duration:
\[ t_{\mathrm{ni}}=\frac{U_{\mathrm{st}}-U_i(\omega_0)}{\operatorname{tg}\alpha},\quad t_{\mathrm{ni}}\leq T_{\mathrm{v}}, \tag{8} \]
where \(t_{\mathrm{ni}}\) is the duration of the \(i\)-th pulse;
\(U_i(\omega_0)\) is the instantaneous value of the component of the generator voltage varying at frequency \(2\omega_0\).
Expression (8) shows that the duration of the current pulses in the field winding changes in accordance with the change in \(U(\omega_0)\), i.e., the average current in the field winding will vary with frequency \(2\omega_0\) (provided that the time constant of the winding is less than \(1/2f_0\)). Thus, the excitation regulator tends to stabilize the total flux in the machine, reducing the modulation of the generator voltage. If the generator operates with self-excitation, the voltage \(U_{\mathrm{pit}}\) will also be modulated. In this case, modulation of the voltage of the output stage \(U_{\mathrm{pit}}\) will tend to amplify the oscillations of the generator voltage, while the regulator, by means of pulse-width modulation, will stabilize it.
A SG–FC system was built and tested according to the circuit considered. As the generator, an inductor generator with a power of 3 kW and three armature windings was used. The test results confirm the correctness of the assumptions and conclusions we made. An oscillogram of the generator phase current is shown in Fig. 2.
Conclusions
- When operating into a rectifier controlled according to some periodic law, in generators with a weak damper system there arises
there occurs modulation of the output voltage with a period equal to the period of the rectifier control law.
- When the generator operates over a wide range of rotational speeds, the modulation increases at the upper speed.
As a result of modulation of the generator voltage, the voltage-stabilization system operates in a periodic mode, striving to compensate for the modulation of the machine’s total flux. Consequently, in addition to maintaining the generator voltage constant when the magnitude of the load and the rotational speed change, the regulator must also eliminate its modulation.
- To eliminate voltage modulation in SG–FC systems with a three-phase generator-winding output, the phases should be arranged in the same phase zones. In this case, with a symmetrical load of the FC, modulation of the generator voltage is absent.
REFERENCES
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I. Ya. Bernshtein. Thyristor frequency converters without a dc link. “Energiya,” 1968.
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P. A. Rovinskii, V. A. Tikan. Valve frequency converters without a dc link. L., “Nauka,” 1965.
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V. L. Bunakov and R. G. Gasparov. Semiconductor voltage and frequency regulators for electric machines. “Energiya,” 1966.