PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 160 1966

ON THE CHOICE OF A RECTIFICATION CIRCUIT

IN BRUSHLESS DIRECT-CURRENT GENERATORS

G. A. Sipailov, A. B. Tsukublin

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

The question of choosing a rectification circuit arises every time brushless direct-current generators of the synchronous generator–rectifier type are designed. This question becomes especially important in the design of exciters that are the first stage of brushless dc and ac electric-machine units [1, 2]. However, to date there are no specific recommendations on this matter in the technical literature.

The present article gives an analysis of the choice of a rectification circuit for a low-power brushless direct-current generator of the synchronous generator–rectifier type. A characteristic feature of such installations is that they must ensure maximum utilization of the generator over the entire range of load variation. At the same time it is necessary to take into account the simplicity and reliability of the installation, as well as minimum weight and dimensions. From this point of view, out of the entire variety of rectifier circuits one may single out only bridge rectification circuits and single-half-period circuits with the generator connected in star. As for other rectification circuits (circuits with an equalizing reactor, cascade circuits, etc.), because of their structural bulkiness they are not considered.

An important advantage of two-half-period rectification circuits over single-half-period ones is that, when supplied from a transformer, they give, for equal loads, a higher utilization factor and a lower ripple factor. When supplied from a synchronous generator, the operating picture of the rectifiers changes, which is connected with the presence of armature reaction in the synchronous generator. Therefore the choice of the rectification circuit must be made with allowance for the armature reaction of the synchronous generator, as well as the magnitude of the load.

In the analysis the following assumptions were made:

1. The valves are taken to be ideal, i.e., the forward resistance of the valve is not taken into account, and the reverse resistance is equal to infinity. It should be noted that this assumption is not entirely arbitrary, if one takes into account that low-power machines have relatively high values of the active resistance of the armature winding.

2. The generator gives a sinusoidal voltage both at no load and under load, i.e., the higher harmonics of the armature reaction caused by the nonsinusoidal nature of the armature current are completely eliminated by the damping circuits of the rotor.

As is known [1, 2], the operation of a low-power synchronous generator under load, i.e., with higher-harmonic armature reactions caused by nonsinusoidal armature current, is fully extinguished by the damping circuits of the rotor.

to the rectifier unit is characterized by a high generator \(\cos\varphi\), whose values lie within the range \(\cos\varphi = 1 \div 0.96\). Therefore, with a sufficient degree of accuracy one may assume that the valve load is active for the generator.

The vector diagram for this case is shown in Fig. 1. In accordance with the vector diagram, the phase voltage, at constant excitation current, is expressed as

\[ U_1 = \sqrt{ \frac{E_0^2 - 2 I_1^2 x_{0d} x_{0q}}{2} + E_0 \sqrt{ \frac{E_0^2}{4} - I_1^2 x_{aq}(x_{ad}-x_{aq}) } } - I_1 r . \tag{1} \]

\[ x_{1d}=x_d-x_d''\simeq x_{ad} \]

\[ x_{1q}=x_q-x_q''\simeq x_{aq} \]

Fig. 1. Vector diagram of the generator.

Fig. 2. External characteristics of the generator:
a — with a bridge rectification circuit,
b — with a single-half-period rectification circuit.

Expressing the rectified voltages and current through the known [3] coupling coefficients

\[ \frac{U_1}{U_d}=k_{1u} \quad\text{and}\quad \frac{I_1}{I_d}=k_{1i} \]

and substituting them into (1), we obtain the equation of the external characteristic of the generator operating into the rectifier:

\[ U_d = \frac{1}{k_{1u}} \sqrt{ \frac{E_0^2 - 2 I_d^2 x_{ad}x_{aq}k_{1i}^2}{2} + E_0 \sqrt{ \frac{E_0^2}{4} - I_d^2 x_{aq}(x_{ad}-x_{aq})k_{1i}^2 } } - I_d r \cdot k_{1i} \tag{2} \]

Assuming that the generator resistances remain constant when the rectification circuit is changed, we obtain that the slope of the characteristic depends mainly on the coefficients \(k_{1u}\) and \(k_{1i}\).

Figure 2 gives the external characteristics of the generator for single-half-period and two-half-period rectification circuits, obtained according to equation (2) for a three-phase generator. As can be seen from the curves, at some point (which we shall call the critical point) the characteristics intersect; i.e., up to this point the generator delivers greater power when operating with a bridge rectification circuit. However, under a load exceeding the critical value, owing to the armature reaction of the generator, the bridge rectification circuit gives generator power values considerably lower than the single-half-period circuit.

The value of the load at the critical point and its relation to the machine parameters can be obtained by the simultaneous solution of the equations

(2), for the half-wave and bridge rectification circuits. Denoting by \(k_{1U}^{\prime\prime}\), \(k_{1i}^{\prime\prime}\) the values of the coefficients for the bridge circuit, and by \(k_{1U}^{\prime}\), \(k_{1i}^{\prime}\) for the half-wave circuit, we obtain

\[ \left[ \frac{ \left(R_{\mathrm{nkr}}\frac{k_{1U}^{\prime}}{k_{1i}^{\prime}}+r\right)^2+x_{aq}^2 }{ \left(R_{\mathrm{nkr}}\frac{K_{1U}^{\prime\prime}}{K_{1i}^{\prime\prime}}+r\right)^2+x_{aq}^2 } \right]^{\frac{1}{2}} = \frac{K_{1i}^{\prime}}{K_{1i}^{\prime\prime}} \cdot \frac{ \left(R_{\mathrm{nkr}}\frac{k_{1U}^{\prime}}{k_{1i}^{\prime\prime}}+r\right)^2+x_{ad}x_{aq} }{ \left(R_{\mathrm{nkr}}\frac{K_{1U}^{\prime\prime}}{K_{1i}^{\prime\prime}}+r\right)^2+x_{ad}x_{aq} }. \tag{3} \]

Solving this equation with respect to \(R_{\mathrm{nkr}}\) gives the relation between the critical load and the generator parameters. For machines with a uniform air gap, equation (3) is simplified by taking \(x_{ad}=x_{aq}\) and \(x_{1d}=x_{1q}=x_d\), and takes the form

\[ \frac{k_{1i}^{\prime}}{k_{1i}^{\prime\prime}} = \left[ \frac{ \left(R_{\mathrm{nkr}}\frac{k_{1U}^{\prime\prime}}{k_{1i}^{\prime\prime}}+r\right)^2+x_d^2 }{ \left(R_{\mathrm{nkr}}\frac{k_{1U}^{\prime}}{k_{1i}^{\prime}}+r\right)^2+x_d^2 } \right]^{\frac{1}{2}} . \tag{4} \]

Solving this equation leads to simple relations between the load resistance and the generator parameters

\[ R_{\mathrm{nkr}}= \frac{-r\,\Delta k_p+\sqrt{r^2\Delta k_p^2-z_r\Delta k_{1i}\Delta k_{1U}}} {\Delta K_{1U}} \tag{5} \]

where

\[ \Delta k_p=k_{1U}^{\prime}k_{1i}^{\prime}-k_{1U}^{\prime\prime}\cdot k_{1i}^{\prime\prime}, \]

\[ \Delta k_{1U}=(k_{1U}^{\prime})^2-(k_{1U}^{\prime\prime})^2, \]

\[ \Delta k_{1i}=(k_{1i}^{\prime})^2-(k_{1i}^{\prime\prime})^2, \]

\[ z_r^2=x_d^2+r^2. \]

Taking into account that in low-power synchronous generators \(z_r \gg r\), we simplify expression (5) to the form:

\[ R_{\mathrm{nkr}}\simeq z_r\cdot \sqrt{ \frac{(k_{1i}^{\prime\prime})^2-(k_{1i}^{\prime})^2} {(k_{1U}^{\prime})^2-(k_{1U}^{\prime\prime})^2} }. \tag{6} \]

Thus, when the load resistance is above \(R_{\mathrm{nkr}}\), in all cases the bridge rectification circuit is the most advantageous. At large loads it is necessary to choose the half-wave rectification circuit, which in this region ensures the greatest utilization of the generator.

Since the coefficients \(k_{1U}\) and \(k_{1i}\) are complex functions of the load, the number of phases, etc., a preliminary choice of the rectification circuit can be made on the basis of consideration of an idealized rectifier. Taking this into account, the critical load resistance

\[ R_{\mathrm{nkr}}=z_r\cdot \frac{4}{\pi}\cdot \frac{\sin^2\frac{\pi}{m}}{\frac{\pi}{m}} . \tag{7} \]

As is evident from (7), the critical load resistance depends on the number of phases of the generator. Fig. 3, illustrating this dependence, clearly shows that with an increase in the number of phases the limits of application of bridge circuits expand, and for a sufficiently large number of phases the circuits

Fig. 3. Dependence of \(\dfrac{R_{\mathrm{nkr}}}{z_r}\) on the number of phases.

are practically identical. Therefore, in designing brushless direct-current generators, as well as various kinds of brushless exciters, the question of applying one or another rectifier circuit must be decided simultaneously with the question of selecting the number of phases, the generator parameters, and also the required level of ripple of the rectified voltage or current.

From this point of view, for exciters one may recommend multiphase systems with a single-half-period rectification circuit, and for generators operating into a high-ohmic load, bridge rectifier circuits with a relatively small number of phases.

REFERENCES

1. A. I. Skorospeshkin, E. F. Obergan, E. N. Podborsky. Brushless electromagnetic amplifier (BEMA). Izvestiya TPI, vol. 145, 1966.

2. A. F. Krogeris. Operating conditions of power sources for three-phase bridge rectifier circuits. Dissertation, Riga, 1954.

3. 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.