DETERMINATION OF THE RIPPLE OF THE OUTPUT VOLTAGE OF BRUSHLESS DC GENERATORS

G. A. Sipailov, A. B. Tsukublin

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)

The replacement of commutator DC generators by systems with semiconductor devices requires solving the problem of obtaining a high-quality DC voltage with a limited number of rectifying elements. In this connection it is necessary to evaluate the influence of the rectification circuit and of the number of generator phases, under various loads at the rectifier output in magnitude and type (active, capacitive, inductive), on the magnitude of the alternating component of the rectified voltage.

Of the methods currently known in the theory of rectifier devices for evaluating the alternating component of the rectified voltage [1, 2, 3, 4], the authors used two methods for the analysis: evaluation of the ripple magnitude by the amplitude value [1] and by the effective value [4]. It is most expedient to carry out the analysis of the alternating component of the rectified voltage using the bridge rectification circuit shown in Fig. 1 as an example. It is known that the rectified voltage is the sum of the potentials of the positive and negative—

Fig. 1. Schematic diagram of the generator.

tive poles. However, in order to obtain an analytical expression for the ripple magnitude, it is necessary to align the pole potentials in time, taking the phase shift into account. In this case the ripple magnitude, as well as the mean value, depends on whether the number of phases is even or odd (Fig. 2a, b). If it is assumed that the generator provides a sinusoidal system of phase emf’s,

\[ e_k = E_m \sin \left[\gamma - (k - 1)\frac{2\pi}{m}\right], \qquad k = 1, 2, 3 \ldots m \]

Fig. 2. Rectified and phase voltages of the generator.
a) \(m\)—an odd number \((m=3)\)

b) \(m\)—an even number \((m=4)\)

(where the emf \(e_k\) should be taken as the emf in the air gap of the synchronous generator), then, taking into account the commutation angle and the natural firing angle [5], the rectified voltage can be determined over the period of variation by the following expressions:

for \(m\) equal to an even number

\[ U_d(\nu)=2E_m\cos\frac{\pi}{m}\,\sin\left(\nu+\frac{\pi}{m}\right)-I_d r', \]

\[ \nu=\frac{\pi}{2}-\frac{\pi}{m}-\alpha \div \frac{\pi}{2}-\frac{\pi}{m}-\alpha+\gamma \tag{1} \]

\[ U_d(\nu)=2E_m\sin\nu-2I_d r', \qquad \nu=\frac{\pi}{2}-\frac{\pi}{m}-\alpha+\gamma \div \frac{\pi}{2}+\frac{\pi}{m}-\alpha \]

for \(m\) equal to an odd number

\[ U_d(\nu)=2E_m\cos^2\frac{\pi}{2m}\cdot\sin\left(\nu+\frac{\pi}{m}\right)-\frac{3}{2}I_d r' \]

\[ \nu=\frac{\pi}{2}-\frac{\pi}{m}-\alpha \div \frac{\pi}{2}-\frac{\pi}{m}-\alpha+\gamma \tag{2} \]

\[ U_d(\nu)=2E_m\cos\frac{\pi}{2m}\,\sin\left(\nu+\frac{\pi}{2m}\right)-2I_d r' \]

\[ \nu=\frac{\pi}{2}-\frac{\pi}{m}-\alpha+\gamma \div \frac{\pi}{2}+\frac{\pi}{m}-\alpha, \]

where \(\nu=\omega t\) is time in synchronous radians,
\(r'=r_{\phi}+r_{\mathrm{v}}\) is the sum of the active resistances of the generator and the direct resistance of the valve.

Expressions (1) and (2) make it possible to estimate the number of phases from the standpoint of the ripple of the rectified voltage, both by amplitude values and by harmonic components.

The amplitude value of the ripple of the rectified voltage can be represented as

\[ \varepsilon_M= \frac{U_d(\nu)_{\max}-U_d(\nu)_{\min}} {U_d(\nu)_{\max}+U_d(\nu)_{\min}} . \tag{3} \]

Substituting into (3) the expressions for the maximum and minimum voltages in accordance with (1) and (2), we obtain that for \(m\) equal to an even number

\[ \varepsilon_M= \frac{ 1-\cos\frac{\pi}{m}\cos(\gamma-\alpha)-\sin\alpha\sin\frac{\pi}{m} }{ 1+\cos\frac{\pi}{m}\cos(\gamma-\alpha)-3\sin\alpha\sin\frac{\pi}{m} }, \tag{4} \]

and for \(m\) equal to an odd number, if

\[ \gamma<\frac{\pi}{2m}+\alpha, \]

\[ \varepsilon_M= \frac{ 1-\cos\frac{\pi}{2m}\cos(\gamma-\alpha)-\sin\alpha\sin\frac{\pi}{2m} }{ 1+\cos\frac{\pi}{2m}\cos(\gamma-\alpha)-7\sin\alpha\cdot\sin\frac{\pi}{2m} } \tag{5} \]

and if \(\gamma>\frac{\pi}{2m}+\alpha\), then

\[ \xi_{\mathrm{m}} = \frac{\sin(\gamma-a)-\sin\alpha} {\sin(\gamma-a)+2\operatorname{ctg}\frac{\pi}{2m}\cos(\gamma-a)-z\sin\alpha}. \tag{6} \]

Analysis of expressions (4)–(6) convincingly shows that, with an even number of phases, the amplitude value of the ripple of the rectified voltage is considerably greater than with an odd number, even under relatively large generator loads. Thus, in terms of the absolute value of the alternating component, at small commutation angles a six-phase bridge circuit is equivalent to a three-phase one, while a five-phase system gives ripples smaller than an eight-phase one even at appreciable relative commutation angles (Fig. 3). If it is also taken into account that, as the number of phases increases, the utilization factor of the generator decreases [5], then, when designing a brushless dc generator, the choice of an odd number of phases is preferable.

Fig. 3. Dependence of \(\varepsilon_{\mathrm{m}}\) on the number of phases and the commutation angle

In assessing the influence of generator parameters on the amplitude value of the ripple, it must be noted that for low-power sources, in which the ratio of the active resistance of a phase to the leakage inductive reactance is close to unity, the ripple magnitude is practically independent of the voltage drop in the active resistance, especially for \(m>5\), and depends to a large extent on the valve commutation angle (leakage inductance).

Therefore, in calculations one may neglect the active voltage drop in expressions (1) and (2), which gives a considerable simplification of the calculation formulas.

The value of the ripple coefficient in terms of amplitude value, although convenient for analyzing the applicability of one or another rectification circuit, does not make it possible to calculate the current or voltage ripple when the type of load at the generator output is changed.

One of the methods that makes it possible to take the type of load into account is to determine the harmonic components of the ripple emf. The latter can be found by expanding the function defined by equations (1) or (2) into a harmonic series. Calculations show that, in the analysis, it is sufficient to confine oneself only to the fundamental component of the ripple, varying with frequency \(f_{\mathrm{p}}=m'f_{\mathrm{g}}\), where \(m'=m\) when \(m\) is an even number

and \(m' = 2m\) when \(m\) is an odd number. Neglecting the higher harmonics, as calculations show, gives only a \((3 \div 5)\%\) error in determining the ripple emf. Taking (1) and (2) into account, the ripple emf may be represented by the following expression:

\[ e_m = \frac{m'}{\pi} E_m \frac{\sin \dfrac{\pi}{m}}{(m')^2 - 1} \sqrt{ (m')^2 \left[ \sin^2(\gamma-\alpha) - 2\cos m'\gamma \sin(\gamma-\alpha)\sin\alpha + \sin^2\alpha \right] + 2m'\sin m'\gamma \sin\gamma + \cos^2(\gamma-\alpha) + 2\cos(\gamma-\alpha)\cos m'\gamma\cos\alpha + \cos^2\alpha }. \tag{7} \]

As the generator load increases, both the commutation angle \((\gamma)\) and the natural firing angle \((\alpha)\) increase. In this case the commutation angle increases considerably faster [5], which leads to an increase in the ripple emf as the load current rises.

With an inductive load at the rectifier output, \(L_n=\infty\), the ripple coefficient can be determined as the ratio of the ripple emf to the average value of the rectified voltage, obtained in accordance with expressions (1) or (2):

\[ \varepsilon = \frac{2}{(m')^2 - 1}\, \frac{ \sqrt{ (m')^2 \left[ \sin^2(\gamma-\alpha) - 2\cos m'\gamma \sin(\gamma-\alpha)\sin\alpha + \sin^2\alpha \right] } }{ \cos\alpha + \cos(\gamma-\alpha) + \dfrac{ 2m'\sin m'\gamma \sin\gamma + \cos^2(\gamma-\alpha) + 2\cos(\gamma-\alpha)\cos m'\gamma\cos\alpha + \cos^2\alpha }{ \gamma\sin\alpha - \dfrac{4\pi}{m}\sin\alpha } }. \tag{8} \]

Since, as the load increases, the average value of the rectified voltage decreases while the ripple emf increases, this leads to a considerable increase in the ripple coefficient. Fig. 4 illustrates this dependence for a three-phase bridge rectifier operating from

Fig. 4. Change in the ripple coefficient of the three-phase circuit at \(L_n=\infty\); 1 — calculation, 2 — experiment.

…of a low-power synchronous generator. The increase in ripple with increasing load must be taken into account when selecting the filtering system.

For any other loads at the generator output, both in type and in magnitude, the alternating component of the rectified voltage can be determined from the equivalent circuit shown in Fig. 5. The equivalent parameters of the generator for the direct-current circuit are expressed, according to [4], as average values over the period of variation of the rectified voltage and for the frequency corresponding to the ripple frequency.

Fig. 5. Circuit for calculating current and voltage ripple

For a bridge circuit, the equivalent resistances can be determined as

\[ x_{\mathrm{э}} = x_{\phi} m'\left(2-\frac{3\gamma}{2\pi}\right), \qquad r_{\mathrm{э}} = r'\left(2-\frac{3\gamma}{2\pi}\right). \tag{9} \]

With known parameters of the generator and the load, the current and voltage ripples of the generator can be determined as

\[ \dot{i}_{\mathrm{п}}=\frac{\dot{E}_{\mathrm{п}}}{z_{\mathrm{г}}+z_{\mathrm{н}}}, \qquad \dot{U}_{\mathrm{п}}=\dot{i}_{\mathrm{п}} z_{\mathrm{н}}, \tag{10} \]

where \(z_{\mathrm{г}}=r_{\mathrm{э}}+i x_{\mathrm{э}}\).

Experimental investigations carried out by the authors (Fig. 4) showed that the proposed method for calculating the ripple of the rectified voltage gives an accuracy satisfactory for engineering calculations.

Thus, the investigations performed make it possible to draw the following conclusions:

1. To reduce the alternating component of the rectified voltage in brushless direct-current generators, it is necessary to use an odd number of phases. In this case, for low-power generators the number of phases \(m=5, 7\) is the maximum possible from the standpoint of technological feasibility and the utilization factor of the power source.

2. The ripple factor of the rectified voltage can be determined with accuracy sufficient for practice by calculating the ripple emf under ideally smoothed current, with subsequent recalculation for the actual load.

3. In all cases, to reduce the ripple of the rectified voltage it is necessary to design power sources with low inductive leakage reactance.

References

1. M. P. Kostenko, L. R. Neiman, G. N. Blavdzevich. Electromagnetic Processes in Systems with Powerful Rectifier Installations. Publishing House of the Academy of Sciences of the USSR, 1964.

2. V. N. Aksenov. Rectifiers and Transformer Substations. Svyazizdat, 1962.

3. G. G. Fedoseev. Rectifiers and Stabilizers. Gosizdat, “Iskusstvo,” 1960.

4. A. M. Pintsov. Calculation of the harmonics of rectified current and voltage. Elektrichestvo, No. 12, 1956.

5. 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, 1966.