DETERMINATION OF THE CURRENT AND VOLTAGE TRANSFORMATION RATIOS AND OF THE POWER FACTOR IN A MACHINE–VALVE FREQUENCY CONVERTER

A. A. Skripilev, B. E. Trofimenko, A. B. Tsunublin

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

At present, autonomous installations are increasingly using alternating-current power-supply systems. However, in the transition to alternating-current systems a whole series of problems arises, one of the most important of which is the problem of obtaining a constant frequency when the shaft rotational speed is variable.

A constant frequency at a variable rotational speed can be obtained by various methods; in this article we examine the operating features of one of the possible variants for stabilizing the frequency of an autonomous source by means of a machine–valve frequency converter. The power circuit of the converter is shown in Fig. 1. On the stator of the synchronous generator there are placed \(m\) three-phase windings, arranged at an angle of

\[ \frac{360}{m} \]

electrical degrees, which are connected in turn to the load at certain time intervals by closing the ends of the generator’s three-phase windings through a rectifier bridge.

Fig. 1. Schematic circuit diagram of a machine–valve frequency converter.

When consumers are supplied from a synchronous generator of comparable power—which is often the case in autonomous installations—of great importance in generator design is the determination of the coeffi-

... conversion ratios of phase current, voltage, and power factor.

In determining the indicated ratios, the following assumptions are adopted in this work:

1. The phase voltages of the synchronous generator are sinusoidal and symmetrical.
2. The valves are ideal switches.
3. Valve commutation occurs instantaneously, i.e., the valve commutation angle \(\gamma = 0\).

To determine the conversion ratios, it is necessary to determine the spectra of the phase currents and voltages of the generator and of the load. The spectrum of the output voltage can be found using the mathematical apparatus of switching functions. This method makes it possible to represent the output voltage as the product of the generator phase voltage and a certain switching function that depends on the operating mode of the converter.

The generator phase voltages are written in general form as:

\[ \begin{aligned} u_{Ae}(t) &= U_m \sin\left[\omega_1 t + \alpha_1 - \frac{e-1}{m}\cdot 2\pi\right],\\ u_{Be}(t) &= U_m \sin\left[\omega_1 t + \alpha_1 - \frac{3(e-1)+m}{3m}\cdot 2\pi\right],\\ u_{Ce}(t) &= U_m \sin\left[\omega_1 t + \alpha_1 - \frac{3(e-1)+2m}{3m}\cdot 2\pi\right], \end{aligned} \tag{1} \]

where \(U_m\) is the maximum value of the phase voltage of the synchronous generator;
\(\omega_1\) is the angular frequency of the generator voltage;
\(\alpha_1\) is the initial phase of the generator phase-voltage curve, taken as the reference with respect to the origin of coordinates;
\(m\) is the number of switched three-phase generator windings;
\(e\) is the ordinal number of the generator three-phase windings.

The expansion of the switching function into a harmonic series can be written in the following form [1]:

\[ f_e(t)=\frac{1}{m}+\frac{2}{k\pi}\sum_{k=1}^{\infty}\sin\frac{k\pi}{m}\cos k \times \]

\[ \times\left(\omega_k t+\alpha_k-\frac{e-1}{m}\cdot 2\pi\right), \tag{2} \]

where \(\omega_k\) is the angular frequency of the switching function;
\(\alpha_k\) is the initial phase of the switching function.

If the duration of the connected state of each three-phase generator winding to the load is equal to \(\tau\), and the period of the switching function is \(T_k\), then the zero terms in expansion (2) are equal to:

\[ n_0=\frac{T_k}{\tau}=m, \tag{3} \]

from which it is clear that, in the expansion of the switching function, terms with harmonics that are multiples of the number of three-phase generator windings are absent.

The output phase voltages of the converter are the sum of \(m\) modulated phase voltages:

\[ u_A(t)=\sum_{e=1}^{m} u_{Ae}(t)\cdot f_e(t); \]

\[ u_B(t)=\sum_{e=1}^{m} u_{B_e}(t)\cdot f_e(t); \]

\[ u_C(t)=\sum_{e=1}^{m} u_{C_e}(t)\cdot f_e(t); \tag{4} \]

Substituting into (4) the values of the phase voltages of the synchronous generator (1) and the commutation function (2), and carrying out the necessary calculations and transformations, we obtain:

\[ \begin{aligned} u_A(t)=\frac{mU_m}{\pi}\cdot \Bigg\{& \sin \frac{\pi \sin}{m}\,[(\omega_1-\omega_k)t+\alpha_1-\alpha_k]+ \\ &+\sum_{q=1}^{\infty}\frac{1}{qm-1}\cdot \sin \frac{qm-1}{m}\cdot \pi \cdot \sin[(\omega_1+qm\omega_k-\omega_k)t+\alpha_1+(qm-1)\alpha_k]+\\ &+\sum_{q=1}^{\infty}\frac{1}{qm+1}\cdot \sin \frac{qm+1}{m}\cdot \pi \cdot \sin[(\omega_1-qm\omega_k-\omega_k)t+\alpha_1-(qm+1)\alpha_k]\Bigg\}, \end{aligned} \tag{5} \]

where: \(q\) is any positive integer \((q=1, 2, 3 \ldots)\).

The two other output phase voltages \(U_B(t)\) and \(U_C(t)\) are written similarly, with phase shifts of \(\dfrac{2\pi}{3}\) and \(\dfrac{4\pi}{3}\), respectively.

The spectrum of the phase load current, for a known phase voltage (5) and load parameters, can be obtained as follows:

\[ \begin{aligned} i_A(t)=\frac{mU_m}{\pi}\cdot \Bigg\{& \frac{\sin \dfrac{\pi}{m}}{Z_H}\cdot \sin[(\omega_1-\omega_k)t+\alpha_1-\alpha_k-\varphi_H]+\\ &+\sum_{q=1}^{\infty}\frac{1}{qm-1}\cdot \frac{\sin \dfrac{qm-1}{m}\pi}{Z'_q}\cdot \sin[(\omega_1+qm\omega_k-\omega_k)t+\alpha_1+(qm-1)\alpha_k-\\ &\qquad\qquad\qquad\qquad\qquad\qquad -\varphi'_q]+ \sum_{q=1}^{\infty}\frac{1}{qm+1}\cdot \frac{\sin \dfrac{qm+1}{m}\pi}{Z''_q}\cdot\\ &\qquad\qquad\qquad\qquad\qquad\qquad \cdot \sin[(\omega_1-qm\omega_k-\omega_k)t+\alpha_1-\\ &\qquad\qquad\qquad\qquad\qquad\qquad -(qm+1)\alpha_k-\varphi''_q]\Bigg\}, \end{aligned} \tag{6} \]

where \(Z_H\) and \(\varphi_H\) are the modulus of the complex load impedance and the angle of shift between the output current and voltage at the output frequency \(\omega_1-\omega_k\);

\[ Z_H=\sqrt{R_H^{2}+(\omega_1-\omega_k)^2 L_H^{2}}=\sqrt{R_H^{2}+X_{LH}^{2}}; \]

\[ \varphi_H=\operatorname{arc\,tg}\frac{(\omega_1-\omega_k)L_H}{R_H}; \tag{7} \]

\(Z'_q\), \(Z''_q\), \(\varphi'_q\), \(\varphi''_q\) are the moduli of the complex load impedances and the phase-shift angles between current and voltage for the higher harmonics.

\[ Z'_q=\sqrt{R_H^{2}+[\omega_1+(qm-1)\omega_k]^2 L_H^{2}}; \]

\[ \varphi'_q=\operatorname{arc\,tg}\frac{[\omega_1+(qm-1)\omega_k]L_H}{R_H}; \tag{8} \]

\[ Z_q''=\sqrt{R_{\mathrm{н}}^{\,2}+\left[\omega_1-(qm+1)\omega_k\right]^2L_{\mathrm{н}}^{\,2}}; \]

\[ \varphi_q''=\operatorname{arc\,tg}\frac{\left[\omega_1-(qm+1)\omega_k\right]L_{\mathrm{н}}}{R_{\mathrm{н}}}; \tag{9} \]

Given the known spectrum of the phase current of the load, the spectrum of the current of a generator phase is represented as the result of modulation of the output phase current by the commutation function [2]:

\[ \begin{aligned} i_{Ae}(t)&=i_A(t)\cdot f_e(t);\\ i_{Be}(t)&=i_B(t)\cdot f_e(t);\\ i_{Ce}(t)&=i_C(t)\cdot f_e(t). \end{aligned} \tag{10} \]

Since the spectra of the phase currents of all windings will be identical, differing only in phase angle, we shall determine the expression for the phase current of the generator winding taken as the reference, substituting into expression (10) the value of the phase current of the load and of the commutation function:

\[ \begin{aligned} i_{A_1}(t)=& \frac{U_m\sin\frac{\pi}{m}}{\pi\cdot Z_{\mathrm{н}}}\, \sin\left[(\omega_1-\omega_k)t+\alpha_1-\alpha_k-\varphi_{\mathrm{н}}\right]+\\ &+\frac{U_m}{\pi}\sum_{q=1}^{\infty}\frac{1}{qm-1}\, \frac{\sin\frac{qm-1}{m}\pi}{Z_q'}\, \sin\left[(\omega_1+qm\omega_k-\omega_k)t+\alpha_1+\right.\\ &\left.\qquad +(qm-1)\alpha_k-\varphi_q'\right] +\frac{U_m}{\pi}\sum_{q=1}^{\infty}\frac{1}{qm+1}\times\\ &\times \frac{\sin\frac{qm+1}{m}\pi}{Z_q''}\, \sin\left[(\omega_1-qm\omega_k-\omega_k)t+\alpha_1-(qm+1)\alpha_k-\varphi_q''\right]+\\ &+\frac{2mU_m}{\pi^2}\sum_{\kappa=1}^{\infty} \frac{\sin\frac{\pi}{m}\cdot\sin\frac{\kappa\pi}{m}}{\kappa Z_{\mathrm{н}}}\, \sin\left[\omega_1 t+\alpha_1-\varphi_{\mathrm{н}}-(\omega_k t+\alpha_k)\right]\times\\ &\times \cos\kappa(\omega_k t+\alpha_k) +\frac{2mU_m}{\pi^2}\sum_{q=1}^{\infty}\sum_{\kappa=1}^{\infty} \frac{1}{(qm-1)\kappa}\times\\ &\times \frac{\sin\frac{qm-1}{m}\pi\cdot\sin\frac{\kappa\pi}{m}}{Z_q'}\, \sin\left[\omega_1t+\alpha_1+\varphi_q'+(qm-1)(\omega_k t+\alpha_k)\right]\times\\ &\times \cos\kappa(\omega_k t+\alpha_k) +\frac{2mU_m}{\pi^2}\sum_{q=1}^{\infty}\sum_{\kappa=1}^{\infty} \frac{1}{(qm+1)\kappa}\times\\ &\times \frac{\sin\frac{qm+1}{m}\pi\cdot\sin\frac{\kappa\pi}{m}}{Z_q''}\, \sin\left[\omega_1t+\alpha_1-\varphi_q''-(qm+1)(\omega_k t+\alpha_k)\right]\times\\ &\times \cos\kappa(\omega_k t+\alpha_k). \end{aligned} \tag{11} \]

The expressions obtained, (6) and (11), make it possible to determine the rms value of the phase currents of the load and of the generator, from which we find the phase-current transformation factor for various ratios of the frequencies \(\omega_1\) and \(\omega\).

\[ k_i = \frac{I_A}{I_{A_1}}, \tag{12} \]

where \(I_A\) is the rms value of the load phase current;
\(I_{A_1}\) is the rms value of the generator phase current.

It has been found that \(k_i\) depends neither on the type of load nor on the ratio \(\omega_1\) and \(\omega\). When the load \(\cos \varphi\) decreases, the rms value of the load phase current decreases; at the same time the generator phase current decreases, while \(k_i\) remains constant. The phase-current transformation ratio is determined only by the relationship between the phases of the synchronous generator and the load. The greater the number of three-phase windings placed in the generator, the higher \(k_i\); the dependence of the phase-current transformation ratio of a machine-valve frequency converter on the number of three-phase generator windings is shown in Fig. 2.

Fig. 2. Dependence of the phase-current transformation ratio of a machine-valve frequency converter on the number of three-phase generator windings.

The voltage transformation ratio \(k_u\) is determined by the ratio of the rms value of the fundamental harmonic of the output voltage \(U_A\) to the rms value of the fundamental harmonic of the generator voltage \(U_{A_1}\).

The voltage transformation ratio characterizes the degree of utilization of the primary generator voltage.

From expression (5)

\[ k_U = \frac{m}{\pi} \cdot \sin \frac{\pi}{m}. \tag{13} \]

With sinusoidal phase voltage of the generator, \(k_U\) depends on the number of three-phase generator windings. The law of variation of the voltage transformation ratio with the number of three-phase generator windings is given in Fig. 3.

Fig. 3. Dependence of the voltage transformation ratio on the number of three-phase generator windings.

When determining the armature reaction of a synchronous generator—which is especially important if the powers of the generator and of the load are commensurate—it is necessary to determine the transformation ratio with respect to the fundamental harmonics of the load phase current \(I_A'\) and the generator current \(I_{A_1}'\),

\[ k_I'=\frac{I_A'}{I_{A_1}'}, \tag{14} \]

where: \(I_A'\) is the rms value of the fundamental harmonic of the load phase current;

\(I_{A_1}'\) is the rms value of the fundamental harmonic of the generator phase.

The fundamental harmonic of the output current has the frequency \(f=f_1-f_k\) and is described by the following expression:

\[ i_A'(t)=\frac{mU_m}{\pi}\cdot \frac{\sin \frac{\pi}{m}}{Z_H}\cdot \sin \bigl[(\omega_1-\omega_k)t+\alpha_1-\alpha_k+\varphi_H\bigr], \tag{15} \]

The expression for the fundamental harmonic of the generator phase is found from (11). It is easy to verify that harmonics with \(k=1,\ k=qm\pm1\) have the frequency \(\omega_1\). The fundamental harmonic of the generator phase is thus determined as the sum of an infinite number of sinusoids with frequency \(\omega_1\) and increasing phase angle:

\[ i_{A_1}'(t)= \frac{mU_m}{\pi^2}\cdot \frac{\sin^2 \frac{\pi}{m}}{Z_H} \sin(\omega_1 t+\alpha_1-\varphi_H)+ \]

\[ +\frac{mU_m}{\pi^2}\sum_{q=1}^{\infty} \frac{\sin^2 \frac{qm-1}{m}\cdot \pi}{(qm-1)^2 Z_q'} \sin(\omega_1 t+\alpha_1-\varphi_q')+ \]

\[ +\frac{mU_m}{\pi^2}\sum_{q=1}^{\infty} \frac{\sin^2 \frac{qm+1}{m}\cdot \pi}{(qm+1)^2 Z_q''} \cdot \sin(\omega_1 t+\alpha_1-\varphi_q''). \tag{16} \]

From relations (15) and (16), the rms values of the fundamental harmonics of the load phase current and the generator phase current are determined, and from them the transformation ratio with respect to the fundamental harmonics of the phase current. Figure 4 shows the dependence of the variation of \(k_I'\) on the ratio of the generator frequency to the commutation frequency \(\omega_1/\omega_k\) for the number of three-phase generator windings \(m=3\). The nature of the variation of \(k_I'\) for \(m=4\) and \(m=6\) is analogous to that shown in Fig. 4. For engineering calculations, when determining \(k_I'\), the second and third terms in expression (16) may be taken equal to zero; then the transformation ratio with respect to the fundamental harmonics of the phase current is determined as:

\[ k_I'=\frac{\pi}{\sin \frac{\pi}{m}}. \tag{17} \]

The calculation error in this case does not exceed 3%.

Another important energy-performance indicator in calculating a synchronous generator operating in a machine-valve frequency converter is \(\cos\varphi_g\), which in the general case depends on the ratio of the generator frequency to the output frequency and on the character of the load. The generator power factor with respect to the fundamental harmonics of current and voltage, \(\cos\varphi_g\), is found from expression (16), for a given \(\cos\varphi_H\), by geometrically summing the individual components of the phase current

Fig. 4. Dependence of the variation of the phase-current transformation coefficient with respect to the fundamental harmonics for different frequency ratios \(\omega_1/\omega_k\).

of the generator. Figure 5 presents the range of variation of \(\cos \varphi_g\) when \(\cos \varphi_n\) is varied for the range of variation of \(\omega_1/\omega_k\) from 0.5 to 3 at \(m=3\). The dependences \(\cos \varphi_g=f(\cos \varphi_n)\) for \(m=4\), \(m=6\) are analogous to the one presented and are not given in the article. As a result of the calculations it was found that, for \(\omega_1/\omega_k<1\) and an inductive load, the power factor of the generator is higher than the power factor of the load; conversely, for \(1<\omega_1/\omega_k<2\), \(\cos \varphi_g\) is less than \(\cos \varphi_n\); further, for \(\omega_1/\omega_k>2\), \(\cos \varphi_g\) becomes

Fig. 5. Range of variation of the generator power factor for different load power factors.

greater than \(\cos \varphi_n\). However, the greatest difference of \(\cos \varphi_g\) from \(\cos \varphi_n\) for \(0.5<\omega_1/\omega_k<1\) does not exceed 6% for \(m=3\); therefore, with accuracy acceptable for engineering calculations, one may set \(\cos \varphi_g=\cos \varphi_n\). But at a low load power factor (0.2–0.4) and a frequency ratio \(\omega_1/\omega_k\) close to 3, \(\cos \varphi_g\) increases considerably compared with \(\cos \varphi_n\) (up to 20%).

Thus, as a result of the study of the machine-valve frequency converter it has been established:

1. The phase-current transformation coefficient \(k_I\) does not depend on the type

load and on the frequency ratio $\omega_1/\omega$ and is determined by the phase relationship of the synchronous generator and the load.

1. In the operating mode of the machine-valve frequency converter with frequency ratio $\omega_1/\omega_k < 1$, the power factor of the generator is always higher than the power factor of the load $(\cos\varphi_g > \cos\varphi_n)$.

2. The voltage conversion coefficient for a given machine-valve converter is determined by the expression:

\[ k_U = \frac{m}{\pi}\sin\frac{\pi}{m}. \]

REFERENCES

1. I. S. Gonorovsky. Fundamentals of Radio Engineering. Svyazizdat, 1957.

2. N. Z. Masteev, G. S. Myshyk. Criteria for evaluating methods of frequency conversion for converters with direct coupling. Elektrotekhnika, No. 1, 1972.