INVESTIGATION OF THE OPERATION OF AN AUTONOMOUS SYNCHRONOUS GENERATOR FEEDING A CONTROLLED RECTIFIER

B. L. Lukutin, A. B. Tsukublin

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The wide use of thyristor frequency converters as autonomous electric-power sources poses new problems in the investigation of a synchronous generator that is the primary source of energy in such systems. Of particular interest is the armature reaction of a machine loaded by a frequency converter when the powers of the load and the generator are commensurate.

Let us consider the processes in a synchronous generator operating into a controlled rectifier over a control period. The rectifier control angles vary according to the arccosinusoidal law:

\[ \alpha = \arccos \nu \sin \omega_2 t, \]

where \(\nu\) is the modulation-depth coefficient;

\(\omega_2\) is the angular frequency of the output voltage.

With variation of the control angle, the magnitude of the phase current, the power factor, and the duration of the commutation angles \(\gamma\) will change. As a result, the armature flux over the rectifier control period will vary both in magnitude and in its position in space.

Let us conventionally divide the transient process in the generator caused by variation of the control angle \(\alpha\) into two components: a high-frequency component, due to the commutating action of the rectifier, and a low-frequency component, due to the discrete variation of the valve control angle.

The physical picture of the low-frequency transient process in the machine is determined mainly by the variation of the first harmonic of the generator phase current over the rectifier control period. Therefore, the low-frequency transient process may be considered without taking into account the commutation angle \(\gamma\), whose influence is accounted for by the high-frequency transient process. To determine the law of variation of the first harmonic of the phase current, we divide the actual current curve into sections equal in duration to the period of the machine voltage. The division points are chosen at the instants when the curve of the corresponding phase voltage crosses zero. As a result, we obtain \(k = \dfrac{\omega_1}{2\omega_2}\) functions for each phase:

\[ i_{1A}(\omega_1 t),\quad i_{2A}(\omega_1 t),\ \ldots,\ i_{kA}(\omega_1 t); \]

\[ i_{1B}(\omega_1 t),\quad i_{2B}(\omega_1 t),\ \ldots,\ i_{kB}(\omega_1 t); \]

\[ i_{1C}(\omega_1 t),\quad i_{2C}(\omega_1 t),\ \ldots,\ i_{kC}(\omega_1 t); \]

where \(\omega_1\) is the angular frequency of the generator.

Expansion of each of these functions in a Fourier series makes it possible to determine the amplitude and phase of the first harmonic:

\[ A_{1ij}=\sqrt{a_{1ij}^{2}+b_{1ij}^{2}}; \]

\[ \varphi_{1ij}=\operatorname{arc\,tg}\frac{a_{1ij}}{b_{1ij}}; \]

where \(a_{1ij}\) and \(b_{1ij}\) are the coefficients of the cosine and sine components of the harmonic series.

Since the rectifier control angles vary according to an arcsine law, the operating time of the bridge valves is different. Consequently, the amplitude and phase of the first harmonic of the current differ both from interval to interval and among the generator phases at each instant of time. Extracting from this system the positive phase sequence,

\[ \dot A_{11}=\frac{1}{3}\left(\dot A_{1iA}+a\dot A_{1iB}+a^{2}\dot A_{1iC}\right), \]

we find that the positive sequence of the first harmonics of the phase currents varies in amplitude from 0 at \(\alpha=90^\circ\) to a maximum value at \(\alpha=0^\circ\), and in phase from a purely inductive current at \(\alpha=90^\circ\) to a purely active current at \(\alpha=0^\circ\). The laws governing the change of phase and amplitude, for sufficiently large ratios

\[ \frac{\omega_1}{\omega_2} \]

(for example, \(\omega_1\) and \(\omega_2\) differ by an order of magnitude), approach sinusoidal laws.

Knowing the law of variation of the positive sequence of currents and of the generator voltage, its actual rectifier load can be replaced by a certain equivalent phase load. This load will be symmetrical with respect to the phases, while its magnitude and character depend on the control angle \(\alpha\): at \(\alpha=90^\circ\) it is an infinitely large inductive load, and at \(\alpha=0\) a finite purely active one. In doing so, we neglect the higher current harmonics, as well as the negative- and zero-sequence components. The admissibility of such an idealization is justified by the fact that only the positive sequence of the phases of the first harmonic rotates synchronously with the inductor and, consequently, has the determining influence on the flux of the machine. The magnitude of the resistance of the equivalent load of each generator phase is determined by the expression

\[ Z_{\mathrm{e}}(\omega_2 t)=\frac{R}{\gamma \sin \omega_2 t}, \]

where \(R\) is the equivalent phase resistance of an uncontrolled rectifier.

According to [4], its magnitude is related to the rectifier load resistance as

\[ R=\frac{r_{\mathrm{n}}\cdot k_{1U}}{k_{1I}}; \]

where

\[ k_{1I}=\frac{I_{\phi 1}}{I_{\mathrm{n}}};\qquad k_{1U}=\frac{U_{\phi 1}}{U_{\mathrm{n}}}; \]

\(U_{\phi 1}\), \(I_{\phi 1}\) are the rms values of the first harmonics of the generator phase voltages and currents;

\(I_{\mathrm{n}}\), \(U_{\mathrm{n}}\) are the mean values of the rectified current and voltage.

The relation between the active and inductive components of the load equivalent to the controlled rectifier is determined as

\[ r_{\mathrm{e}}=R; \]

\[ x_{\mathrm{e}}=R\sqrt{\frac{1}{\nu^{2}}\sec^{2}\omega_{2}t-1}. \]

Accordingly, the armature flux will vary in time both in its magnitude and in its position in space. As \(\alpha\) decreases from \(90^\circ\) to \(0^\circ\), it will rotate in space, while at the same time increasing in amplitude from a purely longitudinal, demagnetizing flux to a purely transverse one.

It is natural to assume that its maximum demagnetizing action will appear at some intermediate value of the control angle \(\alpha\). As a result, modulation of the total magnetic flux of the machine occurs at the control frequency of the rectifier. This modulation causes modulation, at the same frequency, of the generator voltage. The degree of modulation, under the condition that the no-load emf remains constant, will depend on the rotational speed of the generator, increasing as the latter increases, on the magnitude of the rectifier load, and on the depth of regulation \(\nu\).

The high-frequency transient process is determined by the presence of source inductances and by the commutating action of the rectifier. Under normal-load conditions, operation of the rectifier is accompanied by periodic two-phase short circuits through the valves. The direct sequence of the overlap current appears as a short-circuit current behind an additional resistance equal to the resistance of the negative-sequence system with respect to the overlapping anodes. The armature reaction over the short interval of the short circuit practically does not manifest itself. However, such a picture is valid only for a single short circuit. With a rectifier load, short circuits occur cyclically, and the armature reaction will ultimately begin to manifest itself the more strongly, the greater the commutation angle \(\gamma\).

It is known that the transient process in the synchronous-generator–rectifier system is approximately described by a first-order differential equation [1, 3]. Consequently, when \(\alpha\) is decreased discretely, the generator current will increase exponentially, and the commutation angle \(\gamma\) will increase.

During short circuits in the generator circuit, currents arise in the stator winding, creating the armature-reaction flux \(\Phi_{\mathrm{ad}}\). Under the action of the current of the rotor circuits a flux \(\Delta\Phi_{\mathrm{r}}\) is produced, consisting of two components, one of which is due to the free current of the field winding, while the other is a consequence of the free currents of the damper circuits. If the short circuit lasted sufficiently long, the flux of the machine would decrease to some definite value corresponding to the established short-circuit current.

However, after a certain interval of time the short circuit ceases; as a result, the armature-reaction flux decreases and the resultant flux becomes smaller than its forced value. In steady state, for a definite ratio of the commutation and non-commutation intervals, by the time of the next overlap the rotor flux increases to its previous value.

As the commutation angle \(\gamma\) increases, the ratio between the duration of the commutation and non-commutation intervals changes, and the rotor flux, by the beginning of the next overlap, cannot reach its initial value. The duration of the transient process caused by the change in the angle \(\gamma\) will be determined by the corresponding subtransient and transient time constants, depending on the magnitude of the angle \(\gamma\) and on the parameters of the machine. The change in the commutation angle affects the magnitude of the rectified current, and the power factor \(\varphi\) determines the high-frequency component of the transient process.

In the steady-state regime, for analysis of the high-frequency transient processes of the system one can use the method proposed by M. G. Shekhtman, I. A. Glebov, A. F. Krogerius et al., according to which the main magnetic flux of the machine depends little on the commutating action of the rectifier [1]. The admissibility of such a simplification is possible both for machines having damper windings on the inductor and for machines without damper windings [2]. Thus, in the simplified method of analysis the synchronous generator is represented, for the rectifier, as a system consisting of symmetrical undistorted e.m.f.’s behind the commutation inductive reactance. For machines with damper windings the value of this reactance is equal to

\[ x_k=\frac{x_d''+x_2}{2}. \tag{1} \]

Calculation of the synchronous-generator—controlled-rectifier system, taking into account both the high-frequency and low-frequency components of the transient process, as well as the commensurability of the load and generator powers, can be performed by the method proposed below.

Let us calculate a system consisting of a synchronous generator operating into a rectifier controlled according to an arccosinusoidal law. The rectifier is loaded by an active-inductive load; the frequency ratio \(\omega_1/\omega_2\) is large.

Because of the inductive character of the load, the curve of the rectifier output current has a sinusoidal form of frequency \(\omega_2\). Divide the output sinusoid of the rectifier into \(N\) equal, sufficiently small intervals of duration \(\Delta t\), and assume that the rectifier current does not vary over the discretization interval. Then, on each interval, we have a steady-state operating regime of the synchronous generator through the rectifier into an infinitely large inductance. The transition from one interval to another is effected by a jump, due to a change in the control angle \(\alpha\).

For each time segment \(\Delta t\) we shall use the simplified calculation method.

The rms value of the calculated undistorted e.m.f. \(E\) is most simply determined from the vector diagram of voltages, which for an explicitly-pole low-power synchronous generator will have the form shown in Fig. 1.

We assume that the parameters of the generator and load are known: \(x_d, x_q, x_d'\), \(x_d''\), \(x_2, r, E_0, x_{\text{н}}, r_{\text{н}}\).

The equivalent circuit, taking into account the assumptions made, is shown in Fig. 2. The phase e.m.f.’s of the circuit are described by the equations

\[ e_1=\sqrt{2}\,E\sin\omega_1 t; \]

\[ e_2=\sqrt{2}\,E\sin(\omega_1 t+120^\circ); \]

\[ e_3=\sqrt{2}\,E\sin(\omega_1 t+240^\circ), \]

where \(E\) is the rms value of the undistorted e.m.f. over the discretization interval.

The transient process of the current in the system is approximately described by the equation

\[ \left(L_{\text{н}}+L_{\phi}\right)\frac{d I_{\text{н}}}{dt} +\left(r_{\text{н}}+r+\frac{3x_{\phi}}{\pi}\right)I_{\text{н}} = \frac{3\sqrt{6}}{\pi}E\cos\alpha, \tag{2} \]

from which, for the \(i\)-th interval, knowing the values of \(I_{\text{н}}\) and \(E\) on the preceding interval, one can determine the current increment \(\Delta I_{\text{н}i}\):

\[ \Delta I_{\text{н}i} = \frac{ \frac{3\sqrt{6}}{\pi}E_{i-1}\cos\alpha_i - \left(r_{\text{н}}+r+\frac{3x_{\phi}}{\pi}\right)I_{\text{н}i-1} }{ L_{\text{н}}+L_{\phi} }\Delta t . \tag{3} \]

![Vector diagram]

Fig. 1. Vector diagram

![Calculation circuit]

Fig. 2. Calculation circuit

In accordance with the reasoning given above, the quantity \(x_{\phi}\) depends on the commutation angle \(\gamma\), and in our case it should be taken as the mean value between \(x_d''\) and \(x_d\). However, the inductance that determines the value in equation (3) is the load inductance \(L_n\). As a rule, the value \(L_n\) is one or two orders of magnitude greater than the value of the phase inductance of the generator.

From the found increment \(\Delta I_{ni}\), we determine the total current in the \(i\)-th interval:

\[ I_{ni}=I_{ni-1}+\Delta I_{ni}. \tag{4} \]

Assuming, in a first approximation, that the magnitude of the undistorted emf did not change when the rectifier control angle changed from the value

from $\alpha_{i-1}$ to $\alpha_i$, we determine all quantities of interest to us on the $i$-th interval.

\[ \gamma_i=\arccos\left[\cos\alpha_i-\frac{2I_{\mathrm{n}i}x_k}{\sqrt{6}E_{i-1}}\right]-\alpha_i; \tag{5} \]

\[ I_{1\phi i}=\frac{\sqrt{6}}{\pi}I_{\mathrm{n}i}\cos\frac{\gamma_i}{2}; \tag{6} \]

\[ \varphi_i=\alpha_i+\frac{\gamma_i}{2}, \tag{7} \]

where $x_k$ is the inductive commutation reactance, determined by formula (1);

$\gamma_i$ is the commutation angle on the $i$-th interval;

$I_{1\phi i}$ is the rms value of the first harmonic of the generator phase current on the $i$-th interval;

$\varphi_i$ is the angle of displacement between the first harmonic of the generator phase current and the calculated e.m.f. $E_{i-1}$.

The obtained values $I_{1\phi i}$, $\gamma_i$, $\varphi_i$ would be valid for the case when the generator power is much greater than the load power. In the case of commensurability of the load power and the generator power, the value $E$ will vary with the change in the angle $\alpha$. Therefore, it is necessary in the calculation to take into account the change in the undistorted e.m.f. of the generator over the period of control by the rectifier.

From the vector diagram of voltages we have, for the $i$-th interval:

\[ E_i'\sin\Theta_i=I_{1\phi i}\cos(\varphi_i+\Theta_i)(x_q-x_k); \tag{8} \]

\[ E_i'\cos\Theta_i=E_0-I_{1\phi i}\sin(\varphi_i+\Theta_i)(x_d-x^k). \tag{9} \]

Solving the system of equations (8), (9) with respect to the unknowns $\Theta_i$ and $E_i'$, we find a new value $E_i'$, differing from $E_i=E_{i-1}$. Averaging the obtained values $E_i'$ and $E_i$, we substitute their mean value into equation (3). Next we compute new values of $I_{1\phi i}$, $\gamma_i$, $\varphi_i$; from them we again find $E_i$, etc. By organizing an iterative cycle in this manner, we calculate, with the prescribed accuracy, the values of all parameters of interest to us.

Solving in this way the system of equations (3)—(9) on each interval $\Delta t$, we obtain the law of variation of all parameters of the system over the control period of the rectifier.

According to this procedure, a system was investigated consisting of a low-power synchronous generator operating into a rectifier controlled according to an arcsinusoidal law. The rectifier load is active-inductive, with $\cos\varphi=0.8$. The machine parameters in relative units are: $x_d=1.1$; $x_q=0.68$; $r=0.04$; $x_d'=0.33$; $x_d''=0.15$; $E_0=1.18$; the rectifier load parameters: $r_{\mathrm{n}}=2.88$; $x_{\mathrm{n}}=0.202$.

The calculations were carried out on a digital computer; the results are presented in Fig. 3a, b, where curve 1 is plotted for the ratio $\omega_1/\omega_2=48$; 2—for $\omega_1/\omega_2=32$; 3—for $\omega_1/\omega_2=16$.

The results of the calculations fully agree with the description given above of the physical picture of the transient processes in the synchronous generator over the control period of the rectifier. Thus, in Fig. 3a one can see the modulation of the rms value of the calculated e.m.f. $E$, and its form confirms our assumption that the greatest demagnetizing action of the armature flux appears at a certain intermediate value of the rectifier control angle. The modulation depth increases with increasing generator rotational speed.

Fig. 3a, b. Calculated curves: a) rms value of the emf \(E\); b) rms value of the current \(I_{1\varphi}\) and angle \(\varphi\); 1 — for the ratio \(\omega_1/\omega_2 = 48\); 2 — for \(\omega_1/\omega_2 = 32\); 3 — for \(\omega_1/\omega_2 = 16\).

Experimental investigations of the synchronous-generator—controlled-rectifier system attest to the correctness of the theoretical propositions and of the calculation procedure developed on their basis. Figure 4 shows oscillograms of the voltage at the rectifier load and of the generator phase emf. A high-frequency inductor generator was used as the model generator.

Fig. 4. Oscillograms of the voltage at the rectifier load (1) and of the phase emf (2)

Conclusions

1. When a synchronous generator operates into a rectifier controlled according to some periodic law, the armature flux changes in its magnitude and spatial position in accordance with the law of control of the rectifier.

2. In the case where the powers of the generator and the rectifier are commensurate, changes in the armature flux cause modulation of the machine emf. The depth of modulation depends on the load magnitude and the parameters of the generator, while its form is determined by the law of control of the rectifier.

3. Calculation of the synchronous-generator—controlled-rectifier system can be carried out by the proposed procedure, based on the method of successive intervals.

References

1. I. A. Glebov. Excitation Systems of Synchronous Generators with Controlled Converters. Publishing House of the USSR Academy of Sciences, 1960.

2. A. F. Kroteris. Contactless DC Generators. Publishing House of the Academy of Sciences of the Latvian SSR, Riga, 1961.

3. A. A. Bulgakov. A New Theory of Controlled Rectifiers. Moscow, Nauka, 1970.

4. G. A. Sipailov, V. A. Zorin, T. V. Kuznetsova, A. B. Tsukublin. Some questions concerning the operation of a low-power synchronous generator with a rectifier load. Izvestiya TPI, vol. 145, Tomsk, TSU Publishing House, 1966.