SOME ISSUES IN THE OPERATION OF A LOW-POWER SYNCHRONOUS GENERATOR WITH A RECTIFIER LOAD

G. A. SIPAILOV, V. A. ZORIN, T. V. KUZNETSOVA, A. B. TSUKUBLIN

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

In creating autonomous direct-current installations of the low-power synchronous generator–rectifier type (SG–R), the researcher encounters unusual operating modes of the synchronous generator. The rectifier acts as a switch, switching the load current from one phase to another. The rectifier itself is a device possessing pronounced nonlinearity. It is therefore necessary to determine the influence of such a nonlinear load on the voltages and currents of the synchronous generator, on the armature reaction, and on the power factor.

The aim of the investigation is to establish the relationship between the values of the output direct current and voltage and the load of the synchronous generator itself. Determination of this equivalent load makes it possible to calculate, relatively simply, a synchronous generator operating into a rectifier unit.

As investigations have shown, the type of load (active, inductive, or capacitive) has a substantial influence on the operation of an SG–R unit, especially when the generator has a small number of phases and a sinusoidal emf. Owing to this phenomenon, there is a difference in determining the equivalent load of the generator for inductive and capacitive rectifier loads. A number of papers in the domestic and foreign technical literature [1, 2, 4] have been devoted to this question; however, the features inherent in low-power synchronous generators compel us to return once again to this topic. Such features include, first of all, the requirement of low weight and dimensions of the system, which does not permit the installation of an inductive-capacitive smoothing filter. Therefore the question of the quality of the dc voltage obtained, and the question of generator utilization, arise sharply here.

Another feature of low-power synchronous generator–rectifier systems is the relatively high value of the active resistance of the power source. In such installations a number of calculation coefficients cannot be computed without taking the active resistance into account, as is done in high-power systems, since it is commensurate with the reactive leakage resistance of the winding.

The present article gives an analysis of the operation of a low-power synchronous generator into a rectifier unit loaded with an active-induc-

tive load. The circuit for the case under study is shown in Fig. 1. In analyzing the operation of the system the following assumptions are adopted:

1. The valves have infinitely high resistance in the reverse direction.

2. The forward resistance of a valve is equal to the maximum forward voltage drop across the rectifying element divided by the maximum current of the element. This resistance is assumed to be constant in all operating modes of the rectifier.

Fig. 1. Actual circuit of the electrical network, a, and the calculated equivalent circuit, b, for operation of the rectifier with an active-inductive load.

1. The inductive load of the rectifier is so large that the rectified current is ideally smoothed.

2. The phase generated e.m.f. is sinusoidal.

3. The active resistance \(r_{\phi}\) and inductance \(L_{\phi}\) of the generator phases are taken as constant in all operating modes.

4. The number of generator phases is \(m \geqslant 3\).

In designing a synchronous generator, the calculated power, as well as all electromagnetic loadings, are chosen according to the fundamental harmonics of current and voltage; therefore it is necessary to establish the relation between the current at the rectifier output and the fundamental harmonics of the phase current and voltage of the generator itself. It is most convenient to establish this relation through certain coefficients:

1. \(\kappa_1 = \dfrac{I_{\phi}}{I_{d\mathrm{cp}}}\) — the ratio of the rms value of the phase current to the average value of the rectified current;

2. \(\kappa_{\phi 1} = \dfrac{I_{\phi 1}}{I_{\phi}}\) — the ratio of the rms value of the fundamental harmonic of the phase current to the rms value of the phase current.

With the aid of these coefficients one can establish the relation between the os-

of the fundamental harmonic of the phase current and the average value of the rectified current

\[ I_{\phi 1} = \kappa_1 \cdot \kappa_{\phi 1} \cdot I_{dcp}. \]

1. \(\kappa_u = \dfrac{U_\phi}{U_{dcp}}\) is the ratio of the rms value of the phase voltage to the average value of the rectified voltage.

2. \(\kappa_{\phi u} = \dfrac{U_{\phi 1}}{U_\phi}\) is the ratio of the rms value of the fundamental harmonic of the phase voltage to the rms value of the total phase-voltage curve.

Taking these coefficients into account, one can express the relation between the fundamental harmonic of the phase voltage and the average value of the rectified voltage as

\[ U_{\phi 1} = \kappa_u \cdot \kappa_{\phi u} \cdot U_{dcp}, \]

then the relation between the power of the synchronous generator and the output power can be represented in the following form:

\[ P_1 = m \cdot \kappa_1 \cdot \kappa_{\phi 1} \cdot \kappa_u \cdot \kappa_{\phi u} \cdot U_{dcp} \cdot I_{dcp} \cdot \cos \varphi_1 \]

or, denoting \(\kappa_p = m \cdot \kappa_1 \cdot \kappa_u \cdot \kappa_{\phi 1} \cdot \kappa_{\phi u} \cdot \cos \varphi_1;\ P_{dcp} = I_{dcp} U_{dcp}\),

\[ P_1 = \kappa_p \cdot P_{dcp}. \]

Let us consider how the indicated coefficients are related to the parameters of the generator and the rectifier circuit.

Ratio of Phase and Rectified Currents

As is known, during rectifier operation the presence of anode inductance (the inductance of the generator phase) makes an instantaneous transfer of current from phase to phase impossible. There is a certain interval of time during which both phases being switched operate in parallel (the commutation process). For a given load, the magnitude of the inductive reactance of the phases of the supply unit determines the number of simultaneously operating rectifiers. Here an analysis is given of the coefficients for the operating mode of the circuit when only two adjacent phases overlap simultaneously, i.e., for a bridge circuit this will be the process of alternation between two- and three-valve modes. Such operation of the SG–R system is observed when a three-phase low-power generator is used as the supply unit.

Indeed, for a three-phase bridge circuit the greatest commutation angle is \(60^\circ\). In this mode three rectifying elements operate simultaneously. If one takes into account that the commutation angle is determined mainly by the leakage inductive reactance and by the magnitude of the load current, while the load current, in turn, depends on the total inductive reactance of the generator, then continuous operation of three rectifying elements in this circuit is possible only with low load resistance (Fig. 1, b), or in modes close to short circuit. With a load not exceeding the critical point of maximum power of the synchronous generator, only alternating two- and three-valve modes are possible. Thus, the derived coefficients are suitable for three-phase systems consisting of a low-power synchronous generator and a bridge rectifier, and may also be used in multiphase systems when the commutation angle

\[ \gamma < \frac{\pi}{m}. \]

During commutation, the voltages on the phases being switched must be equal:

\[ U_a = U_b . \tag{1} \]

In accordance with the equivalent circuit for the instant of commutation (Fig. 1, b), the phase voltages can be determined as

\[ U_a = e_a - i_a r' - L_{\phi}\frac{di_a}{dt} \]

\[ U_b = e_b - i_b r' - L_{\phi}\frac{di_b}{dt}, \tag{2} \]

where \(e_a, e_b\) are the e.m.f.’s of the corresponding phases,
\(r' = r_{\phi} + r_b\) is the sum of the active resistances of the generator and the forward resistance of the rectifying element.

Starting from the assumption of an ideally smoothed rectified current, we obtain

\[ i_a + i_b = I_d,\qquad \frac{di_a}{dt} = -\frac{di_b}{dt}. \tag{3} \]

Consequently, for the instant of commutation,

\[ e_a - e_b = r'(I_d - 2i_b) - 2L_{\phi}\frac{di_b}{dt}. \]

At the initial instant of commutation, in fact the equality \(U_a = e_b\) takes place, i.e., commutation begins not at point \(K\), but at point \(K_1\), shifted relative to point \(K\) by the angle \(\alpha\) (Fig. 2, a). Expressing the values of the electromotive forces relative to point \(K_1\) (Fig. 2, a), for the commutation period one may write

Fig. 2. Curves of voltage a and current b of the generator phase during rectifier operation.

\[ e_a - e_b = 2E_m \sin\frac{\pi}{m}\cdot \sin(\omega t + \alpha), \tag{4} \]

and at the moment of switching on

\[ e_a-e_b=2E_m\cdot \sin \frac{\pi}{m}\cdot \sin a. \tag{5} \]

Since an instantaneous increase of the current in phase \(b\) is impossible, and also since \(\gamma < \frac{\pi}{m}\), at the moment of switching on \(i_b=0\), \(\frac{di_b}{dt}=0\), and the shift angle is determined as

\[ a=\arcsin\left(-\frac{I_d r'}{2E_m\sin \frac{\pi}{m}}\right). \tag{6} \]

Knowing the angle \(a\), one can determine the law of variation of the current during commutation. If the origin of time is made to coincide with point \(k_1\), then: \(\omega t=0\div \gamma\),

\[ i_b=0.5I_d\left(1-e^{-\frac{t}{T}}\right)+ \kappa\left[\sin(\omega t-\Theta+a)+\sin(\Theta-a)e^{-\frac{t}{T}}\right], \tag{7} \]

where

\[ \Theta=\operatorname{arctg}\frac{x_\phi}{r'};\qquad \kappa=\frac{E_m\cdot \sin \frac{\pi}{m}}{\sqrt{(r')^2+x_\phi^2}};\qquad T=\frac{L_\phi}{r'};\qquad x_\phi=\omega L_\phi. \]

Thus, the phase-current curve (Fig. 2, b) may be represented by the following equations:

\[ \left. \begin{aligned} &\omega t=0\div a, && i_b=0,\\[0.4em] &\omega t=a\div b,\quad && i_b=0.5I_d\left(1-e^{-\frac{\omega t-a}{\omega T}}\right) +\kappa\left[\sin(\omega t-\Theta+a-a)+\sin(\Theta-a)e^{-\frac{\omega t-a}{\omega T}}\right],\\[0.4em] &\omega t=b\div c, && i_b=I_d,\\[0.4em] &\omega t=c\div d,\quad && i_b=0.5I_d\left(1+e^{-\frac{\omega t-c}{\omega T}}\right) -\kappa\left[\sin(\omega t-\Theta+a-c)+\sin(\Theta-a)e^{-\frac{\omega t-c}{\omega T}}\right],\\[0.4em] &\omega t=d\div \pi, && i_b=0. \end{aligned} \right\} \tag{8} \]

The duration of commutation \(\gamma\) can be calculated from equation (7), since by the end of phase commutation, i.e., at \(\omega t-a=\gamma\), \(i_b=I_d\),

\[ I_d=0.5\left(1-e^{-\frac{\gamma}{\omega T}}\right)I_d +\kappa\left[\sin(\gamma-\Theta+a)+\sin(\Theta-a)e^{-\frac{\gamma}{\omega T}}\right]. \]

Solving this equation for \(\gamma\) in general form presents great difficulties; therefore the solution is found either graphically or by approximate methods for some particular case. As studies [1, 3] have shown, when calculating the commutation angle one may neglect the active resistance of the phases of the feeding unit. The solution obtained in this case has a degree of accuracy sufficient for practice:

\[ \gamma=\arccos\left(1-\frac{I_d x_\phi}{E_m\cdot \sin \frac{\pi}{m}}\right). \]

With the determination of the commutation angle it becomes possible to determine the rms value of the phase current

\[ I_{\phi} = \left[ \frac{1}{\pi}\int_a^b \left\{ 0.5 I_d\left(1-e^{-\frac{\omega t-a}{\omega T}}\right) + \kappa\left[ \sin(\omega t-\Theta+\alpha-a) + \sin(\Theta-\alpha)e^{-\frac{\omega t-a}{\omega T}} \right] \right\}^2 d\omega t + \frac{1}{\pi}\int_b^c I_d^2\,d\omega t + \frac{1}{\pi}\int_c^d \left\{ 0.5 I_d\left(1+e^{-\frac{\omega t-c}{\omega T}}\right) - \kappa\left[ \sin(\omega t-\Theta+\alpha-c) + \sin(\Theta-\alpha)e^{-\frac{\omega t-c}{\omega T}} \right] \right\}^2 d\omega t \right]^{\frac{1}{2}}, \tag{9} \]

and, consequently, also the coefficient \(\kappa_1=\dfrac{I_{\phi}}{I_{d\text{av}}}\). However, calculation of the coefficient \(\kappa_1\) using the phase-current values from (9) is very laborious because of the cumbersomeness of expression (9). This calculation can be considerably simplified if it is assumed that the current curve has the form of the trapezoid shown in Fig. 2,b by the dashed line. Replacing the actual curve by a trapezoid gives sufficient practical accuracy in determining the coupling coefficient \(\kappa_1\) for low-power synchronous generator—rectifier systems, for which the active resistance of the phase is comparable with the leakage inductive resistance. As a result of this replacement we obtain a simplified formula for calculating the coupling coefficient, analogously to [1],

\[ \kappa_1=\sqrt{\frac{2}{m}-\frac{\gamma}{3\pi}}. \tag{10} \]

Thus, as the commutation angle increases, i.e., as the generator load increases, the rms value of the phase current and, consequently, the coefficient \(\kappa_1\) decrease. For a three-phase bridge circuit this variation lies within the limits \(0.82 \div 0.74\).

Higher harmonics in the phase-current curve

As equations (8) show, the phase-current curve of a synchronous generator feeding a rectifier is nonsinusoidal and contains a certain percentage of higher harmonics.

The need to determine and take account of the higher harmonics is due to the fact that these harmonics produce additional losses and distort the waveform of the e.m.f. curve. However, expansion into a harmonic series of the curve described by equations (8), even in the simplest case of a three-phase bridge circuit, leads to cumbersome expressions [3] that are difficult to analyze and calculate. If the same simplifications are adopted as in calculating the rms value of the phase current, i.e., if the curve of the commutation process is considered rectilinear (dashed lines in Fig. 2,b), then the phase-current curve is expressed by relatively simple equations (11)

\[ \omega t = 0 \div \frac{\pi}{2}-\frac{\pi}{m}-\frac{\gamma}{2}, \qquad i_b=0 \]

\[ \omega t = \frac{\pi}{2}-\frac{\pi}{m}-\frac{\gamma}{2} \div \frac{\pi}{2}-\frac{\pi}{m}+\frac{\gamma}{2}, \qquad i_b=\frac{I_d}{\gamma} \left( \omega t-\frac{\pi}{2}+\frac{\pi}{m}+\frac{\gamma}{2} \right), \]

\[ \omega t = \frac{\pi}{2}-\frac{\pi}{m}+\frac{\gamma}{2} \div \frac{\pi}{2}+\frac{\pi}{m}-\frac{\gamma}{2}, \qquad i_b=I_d \]

\[ \omega t=\frac{\pi}{2}+\frac{\pi}{m}-\frac{\gamma}{2}\div \frac{\pi}{2}+\frac{\pi}{m}+\frac{\gamma}{2}, \qquad i_b=-\frac{I_d}{\gamma}\left(\omega t-\frac{\pi}{2}-\frac{\pi}{m}+\frac{\gamma}{2}\right), \]

\[ \omega t=\frac{\pi}{2}+\frac{\pi}{m}+\frac{\gamma}{2}\div \pi, \qquad i_b=0. \tag{11} \]

Expanding such a function into a harmonic series no longer presents any special difficulties, and the amplitude value of any odd harmonic can be expressed as

\[ I_{\phi mk}=\frac{8I_d}{\pi\cdot \gamma\cdot k^2}\cdot \sin k\cdot \frac{\gamma}{2}\cdot \cos k\left(\frac{\pi}{2}-\frac{\pi}{m}\right), \tag{12} \]

where \(k=1, 3, 5,\ldots\).

Even harmonics, as well as harmonics that are multiples of the number of phases of the supply unit, are absent from the expansion.

Thus, passing to the form factor of the phase-current curve, we obtain

\[ k_{\phi I}= \frac{ 2\sqrt{3}\sin \frac{\gamma}{2}\sin \frac{\pi}{m} }{ \sqrt{\pi\cdot \frac{\gamma}{2}\cdot \sqrt{\frac{3\pi}{m}-\frac{\gamma}{2}}} }, \tag{13} \]

In particular, for a three-phase bridge rectifier circuit,

\[ k_{\phi I}=1.69\cdot \frac{ \sin \frac{\gamma}{2} }{ \frac{\gamma}{2}\cdot \sqrt{\pi-\frac{\gamma}{2}} }. \]

From the formula for the expansion of the function it is evident that, as the commutation angle increases, the amplitude of the higher harmonics decreases; at the same time the value of the current-curve form factor tends to unity. The changes in \(k_{\phi I}\) for the system “three-phase low-power synchronous generator—bridge rectifier” are insignificant and lie within the limits \(k_{\phi}=0.95\div 1\).

3. Relation between Phase and Rectified Voltages

Having determined the phase current for the entire period, one can find the dependence of the rectified voltage on the emf magnitude, the parameters of the supply unit, and the load current. Since during commutation

\[ U_a=U_b, \]

\[ U_a=e_a-i_a r'-L_{\phi}\frac{di_a}{dt} =e_b-i_b r'-L_{\phi}\frac{di_b}{dt}, \]

and also

\[ -\frac{di_a}{dt}=\frac{di_b}{dt} \quad \text{and} \quad I_d=i_a+i_b, \]

then

\[ U_b=\frac{e_a+e_b}{2}-\frac{I_d r'}{2}. \tag{14} \]

In Fig. 2, \(a\), the positive half-wave of the phase voltage of a synchronous generator is shown. It is evident from the figure that the potential of the positive pole \(U_d^+\) during commutation relative to the zero-

of the generator point is equal to

\[ U_d^+ = U_b = \frac{e_a+e_b}{2}-\frac{I_d r'}{2}, \]

and without commutation \(U_d^+ = e_b - I_d r'\).

The instantaneous value of the voltage of a rectifier connected in a bridge circuit is equal to the difference of the potentials of the positive and negative poles. The average value of the rectified voltage of such a rectifier is determined as the difference of the average values of the pole potentials. Since the potential of the negative pole, with a phase shift, varies according to the same law as the potential of the positive pole, but has the opposite sign, the average value of the rectified voltage is numerically equal to twice the average value of the potential of the positive pole.

Taking into account that

\[ e_a = E_m \sin\left(\omega t+\frac{\pi}{2}-\frac{\pi}{m}\right) \quad \text{and} \quad e_b = E_m \sin\left(\omega t+\frac{\pi}{2}+\frac{\pi}{m}\right), \]

we obtain the average value of the rectified voltage

\[ U_{d\mathrm{av}}= \frac{m}{\pi}\int_{\alpha}^{\alpha+\gamma} \left[\frac{e_a+e_b}{2}-\frac{I_d r'}{2}\right]\,d\omega t + \frac{m}{\pi}\int_{\alpha+\gamma}^{\gamma+\frac{2\pi}{m}} (e_b-I_d r')\,d\omega t . \]

Carrying out the integration and simplifications, we finally obtain

\[ U_{d\mathrm{av}}= \frac{m}{\pi}E_m\sin\frac{\pi}{m} \,[\cos\alpha+\cos(\alpha+\gamma)] - \frac{m}{2\pi}\,I_d r'\,\gamma - 2I_d r'. \tag{15} \]

During one half-period the phase voltage undergoes several changes as a result of rectifier commutation. If point \(O\) (Fig. 2, a) is taken as zero, then the phase voltage is expressed as:

\[ \begin{aligned} \omega t &= 0 \div \frac{\pi}{2}-\frac{\pi}{m}-\alpha, & U_b &= e_b, \\[4pt] \omega t &= \frac{\pi}{2}-\frac{\pi}{m}-\alpha \div \frac{\pi}{2}-\frac{\pi}{m}-\alpha+\gamma, & U_b &= \frac{e_a+e_b}{2}-\frac{I_d r'}{2}, \\[4pt] \omega t &= \frac{\pi}{2}-\frac{\pi}{m}-\alpha+\gamma \div \frac{\pi}{2}+\frac{\pi}{m}-\alpha, & U_b &= e_b-I_d r', \\[4pt] \omega t &= \frac{\pi}{2}+\frac{\pi}{m}-\alpha \div \frac{\pi}{2}+\frac{\pi}{m}-\alpha+\gamma, & U_b &= \frac{e_b-e_c}{2}-\frac{I_d r'}{2}, \tag{16} \\[4pt] \omega t &= \frac{\pi}{2}+\frac{\pi}{m}-\alpha+\gamma \div \pi, & U_b &= e_b . \end{aligned} \]

The rms value of such a function is determined by the well-known formula

\[ U_b=\sqrt{\frac{1}{\pi}\int_0^\pi [f(x)]^2\,dx}. \]

Substituting the values of the function, carrying out the integration and simplifications, we finally obtain

\[ U_{\phi}=E_m\left[\frac{1}{\pi}\left\{\left(\frac{\pi}{2}-\gamma\right)+\cos\frac{2\pi}{m}\,[\gamma+\cos(\gamma-2\alpha)\sin\gamma]\right\}+\right. \]
\[ \left. +\frac{I_d^2(r')^2}{E_m^2}\left(\frac{2\pi}{m}-\frac{\gamma}{2}\right) -4\frac{I_dr'}{E_m}\cos\left(\alpha-\frac{\gamma}{2}\right)\cos\frac{\gamma}{2}\sin\frac{\pi}{m} \right]^{1/2}. \tag{17} \]

The ratio of the rms value of the phase voltage to the average rectified voltage gives the coefficient already adopted by us,

\[ k_u=\frac{U_{\phi}}{U_{d\text{av}}}. \]

Thus, we see that the coefficient \(k_u\) depends on the commutation angle and on the parameters of the supplying unit. As the commutation angle increases, i.e., as the load increases, both the rms value of the phase voltage and the average rectified voltage decrease. The value of the coefficient \(k_u\) depends on which of the voltages decreases faster. However, as studies have shown, for three-phase generators the changes in the coefficient \(k_u\) are insignificant.

It should be noted that the value of \(E_m\), which in turn depends strongly on the armature reaction, has a substantial influence on the value of \(k_u\).

Fundamental Harmonic of the Phase Voltage

Analysis of equations (16) shows that the phase-voltage curve is nonsinusoidal even with a sinusoidal emf. As the load increases it is distorted: the relative value of the fundamental harmonic decreases and the higher harmonics increase markedly. Therefore, for the correct selection of the power of a synchronous generator it is necessary to estimate the magnitude of the fundamental harmonic of the phase voltage. The magnitude of the fundamental harmonic can be determined by expanding the curve described by equations (16) into a harmonic series, i.e.,

\[ U_{\phi_1}=\frac{\sqrt{a_1^2+b_1^2}}{\sqrt{2}}, \]

where \(a_1\) and \(b_1\) are the amplitude values of the first harmonics of the Fourier series. As a general rule, the values \(a_1\) and \(b_1\) can be found as

\[ a_1=\frac{2}{\pi}\int_0^\pi f(x)\cos x\,dx;\qquad b_1=\frac{2}{\pi}\int_0^\pi f(x)\sin x\,dx. \tag{17′} \]

In particular, for a three-phase generator these coefficients will be equal to

\[ a_1=\frac{E_m}{\pi}\sin\gamma\sin(2\alpha-\gamma) -\frac{2I_dr'}{\pi}\cos\left(\alpha-\frac{\gamma}{2}\right)\sin\frac{\pi}{3}\cos\frac{\gamma}{2}, \tag{18} \]

\[ b_1=E_m-\frac{3E_m}{2\pi}\left[\gamma-\sin\gamma\cdot\cos(\gamma-2\alpha) -\frac{4I_dr'}{\pi}\sin\left(\alpha-\frac{\gamma}{2}\right)\sin\frac{\pi}{3}\cos\frac{\gamma}{2}\right]. \tag{19} \]

Thus, the relative magnitude of the fundamental harmonic is expressed through the coefficient

\[ k_{\phi u}=\frac{\sqrt{a_1^2+b_1^2}}{\sqrt{2}\cdot U_{\phi}}. \]

Substituting the values \(U_{\varphi}\), \(a_1\), and \(b_1\), one can establish that the coefficient \(k_{\varphi u}\) depends on the commutation angle, the generator parameters, and the load current. As the commutation angle increases, i.e., as the load increases, the form factor of the phase voltage decreases.

Generator Power Factor

When calculating a synchronous generator, a necessary quantity is also the time phase-shift angle between the first harmonics of the phase current and voltage, \(\varphi_1\), which determines the generator power factor.

Figure 3 shows the actual curves of the phase current and voltage and their first harmonics. The time shift between the fundamental harmonics of current and voltage is determined by

\[ \varphi_1 = \frac{\gamma}{2} - \alpha - \psi_u, \]

where the angle \(\psi_u = \operatorname{arctg}\left(-\frac{a_1}{b_1}\right)\); \(a_1\) and \(b_1\) are calculated according to equations (17, 18, 19).

Fig. 3. Current and voltage curves for determining \(\cos \varphi_1\).

As studies have shown, the value of the angle \(\varphi_1\) is small, so that \(\cos \varphi_1 \simeq 0.97 \div 0.95\). This indicates that the operation of a synchronous generator feeding a rectifier unit may, as a first approximation, be regarded as operation into a purely active load. As the rectifier load increases, the angle \(\varphi_1\) increases somewhat.

Thus, having determined the coefficients relating the input and output of the rectifier, one can specify the equivalent AC load—

current for which the synchronous generator is also designed. The relations derived will subsequently serve as the basis for a method of calculating a low-power synchronous generator operating into a rectifier unit.

REFERENCES

1. V. I. Aksenov. Rectifiers and transformer substations. Svyazizdat, 1961.

2. M. G. Shekhtman. Operation of a generator with a rectifier load. Proceedings of the Leningrad Industrial Institute, No. 3, 1940.

3. A. F. Krogeris. Higher harmonics of current and voltage in the power sources of three-phase bridge rectifier circuits. In: Problems of Power Engineering, Academy of Sciences of the Latvian SSR, Riga, 1957.

4. W. L. Shiling. Armature reaction of the exciter and excitation conditions of a brushless generator with rotating rectifiers. Appl. and Industry, No. 51, November, 1960.