COMPENSATION OF ARMATURE REACTION

IN BRUSHLESS EMA

A. I. SKOROSPECHKIN, E. N. PODBORSKII

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

With the development of semiconductor technology, it has become possible to create brushless EMAs that have no sliding contacts; however, the method of compensating armature reaction in the proposed circuits [1] of these amplifiers has the substantial disadvantage that it makes it possible to compensate only the longitudinal component of the m.m.f. of armature reaction.

Figure 1 shows the circuit of a brushless EMA in which the armature reaction is compensated along both the longitudinal and the transverse axes. Winding 1, with the number of pole pairs \(p_d\), serves for starting the asynchronous motor. When control winding 2 is connected to a constant voltage, a control flux with the number of pole pairs \(P_1\) arises in the stator of the amplifier, and an e.m.f. is induced in winding 3. Rectifier 4, assembled using semiconductor diodes, is intended for

Fig. 1.

of the supply of winding 5, which creates a flux with the number of pole pairs \(P_2\), rotating relative to the stator. In winding 6 this flux induces an e.m.f. When rectifier 7 is connected to the load, an alternating current flows through winding 6 and creates the m.m.f. of armature reaction. Compensation of the longitudinal component of the m.m.f. of armature reaction is achieved by increasing the current in winding 5 as the load increases, by means of feedback winding 8.

The degree of longitudinal compensation is regulated by resistance 9. To compensate the transverse component of the m.m.f. of armature reaction, winding 10 is used, shifted by 90 electrical degrees relative to winding 5. Proportional to the load current there will increase: the m.m.f. of winding 11, the voltage at the output of rectifier 12, supplied from winding 13, and, consequently, the current in winding 10. The degree of transverse compensation is regulated by resistance 14.

Winding 13 serves as the secondary winding for the driving motor and at the same time is the generator winding for supplying rectifier 12, while winding 11, with the number of pole pairs \(P_1\), serves as the excitation winding.

Fig. 2.

Figure 2 presents the vector diagram of the output stage of an implicit-pole amplifier under the assumption that the current, voltage, and e.m.f. vary with time according to a sinusoidal law and that the machine parameters are constant. To simplify the derivations, and also to obtain the form of the equation of the external characteristic most convenient for practical calculations, the voltage vector is directed along the real axis \(\alpha\), which leads the \(\beta\) axis by \(90^\circ\) when the vectors rotate counterclockwise. In the figure the following are denoted:

\[ E_{s0} \]
— the phase value of the e.m.f. of the output winding at no load,

\[ E_k \]
— the compensation e.m.f.,

\[ E_{kd},\ E_{kq} \]
— the compensating e.m.f.’s along the longitudinal and transverse axes, respectively,

\[ j I_s X_{cx}\sin\varphi \]
— the voltage drop in the synchronous inductive reactance, projected onto the voltage vector,

$j I_s X_{cx}\cos\varphi$ — the voltage drop in the synchronous inductive reactance, projected onto the axis,

$I_s r_s$ — the voltage drop in the active resistance of the output winding.

From the diagram in Fig. 2 we find the phase voltage:

\[ U_n = \sqrt{(E_{so}+E_{kd})^2+E_{kq}^{\,2}-(I_s X_{cx}\cos\varphi-I_s r_s\sin\varphi)^2} - I_s X_{cx}\sin\varphi - I_s r_s\cos\varphi, \tag{1} \]

where

\[ E_{kd}=R_{\alpha d}\frac{W_{ocd}}{W_y}\cdot I_{hd}K_{ocd}, \]

\[ R_{\alpha d}=\frac{E_{so}}{I_y} \]

is the magnetizing resistance,

$W_{ocd}$ — number of turns of the longitudinal-axis feedback winding,

$W_y$ — number of turns of the control winding,

\[ I_{ocd}=K_{ocd}I_n \]

is the current of the longitudinal-axis feedback winding,

$K_{ocd}$ — longitudinal-axis feedback coefficient,

\[ E_{kq}=R_{\alpha q}K_{ocq}I_n; \]

$R_{dq}$ — magnetizing resistance along the transverse axis,

\[ I_{ocq}=K_{ocq}I_n \]

is the current of the transverse-axis feedback winding.

The voltage at the output of the amplifier, for a constant control signal, in the general case will be equal to

\[ U_n=\frac{1}{K_u}\left[ \sqrt{ \left(E_{so}+R_{\alpha d}\frac{W_{ocd}}{W_y}K_{ocd}I_n\right)^2 + \left(R_{\alpha q}K_{ocq}I_n\right)^2 - K_I^{\,2}I_n^{\,2}(X_{cx}\cos\varphi-r_s\sin\varphi)^2 } - K_I X_{cx} I_n\sin\varphi - K_I I_n r_s\cos\varphi \right], \tag{2} \]

where

\[ K_u=\frac{U_s}{U_n} \]

is the ratio of the rms value of the phase voltage to the rectified voltage,

\[ K_I=\frac{I_s}{I_n} \]

is the ratio of the rms value of the phase current to the rectified current.

1. External characteristic under “complete transverse compensation”

This mode occurs when

\[ R_{\alpha q}K_{ocq}=K_I(X_{cx}\cos\varphi-r_s\sin\varphi) \]

and is practically realized by setting the slider of resistance 14 to such a position that the phase-voltage curve has the least distortion over the entire range of load variation.

In this case the equation of the external characteristic is significantly simplified and takes the form:

\[ U_n=\frac{1}{K_u}\left(E_{so}+R_{\alpha d}\frac{W_{ocd}}{W_y}K_{ocd}I_n\right) - K_I X_{cx} I_n\sin\varphi - K_I r_s I_n\cos\varphi. \tag{3} \]

It is evident from equation (3) that, by changing the longitudinal-axis feedback coefficient with resistance 9, one can regulate the stiffness of the external characteristics.

It should be noted that equation (3) is valid only for constant amplifier parameters; when the load changes, the inductance of the output winding and the mutual inductance of the output winding and winding 5 change. But taking into account that the ratio of these inductances practically does not depend on the load current, the equation of the external characteristic of a brushless EMA with full transverse compensation and an unsaturated first stage takes the form:

\[ U_{\mathrm{н}} = \frac{1}{K_U} \left(E_{S0}+K_{\mathrm{ocd}}R_{ad}I_{\mathrm{н}}\right) \frac{X_{cx}}{X_{cx0}} - \frac{K_I}{K_U}X_{cx}I_{\mathrm{н}}\sin\varphi - \frac{K_I}{K_U}r_s I_{\mathrm{н}}\cos\varphi . \]

In this equation the angle

\[ \varphi = \frac{\gamma}{2}+\alpha, \]

where \(\gamma\) is the commutation angle,
\(\alpha\) is the firing angle.

The angles \(\gamma\) and \(\alpha\) may be calculated by the formulas of [2].

\(X_{cx0}\) is the unsaturated synchronous inductive reactance of the output stage.

The coefficients \(K_I\) and \(K_U\) are determined respectively from curves 1 and 2 of Fig. 3, while the dependence of \(X_{cx}\) for several values of \(K_{\mathrm{ocd}}\) is determined from curves 3 and 4 of Fig. 3.

Fig. 3.

External characteristics with full transverse compensation were obtained experimentally with different stiffness, for which \(K_{\mathrm{ocq}}\approx \mathrm{const}\), and \(K_{\mathrm{ocd}}=\mathrm{var}\).

In Fig. 4 these characteristics are shown by solid lines, and, for comparison, characteristics calculated from equation (4) are shown by dashed lines. As can be seen from the figure, the calculations give quite satisfactory results. Important advantages of this regime are the straight-line nature of the external characteristics and the minimum level of output-voltage pulsations.

Fig. 4.

However, these characteristics are straight-line only up to certain values of the load currents. With a further increase in the load, as was established experimentally, saturation of the first and additional amplifier stages begins. This leads to the fact that the increase of the currents in windings 5 and 10 slows down, the armature reaction along the longitudinal and transverse axes proves to be undercompensated, and the linearity of the external characteristics is disturbed. The output voltage then begins to decrease more sharply.

II. External characteristic of the amplifier without transverse compensation, with an emf of longitudinal compensation linearly dependent on the load current

In the existing circuits [1] of brushless EMAs, only this operating mode is possible; in the present amplifier it is obtained with winding 11 short-circuited.

Figure 5 presents experimentally obtained external characteristics for various feedback coefficients \(K_{\mathrm{ocd}}\) (curves 1, 2, 3). The nonlinearity of these characteristics makes practical use difficult and necessitates the application of additional devices for obtaining linear characteristics.

III. External characteristic of the amplifier without longitudinal compensation, with compensation of the transverse armature reaction proportional to the load current

This operating mode is realized with winding 8 short-circuited. In Fig. 5, curves 4, 5, 6 show the external characteristics obtained experimentally at various degrees of transverse compensation. It is evident from the curves that the external characteristic has the greater curvature, the more the degree of transverse compensation differs from full transverse compensation, which corresponds to curve 5.

Fig. 5.

Conclusions

1. Of greatest interest for practical purposes is the mode of full transverse compensation, which provides linear external characteristics.

2. The entire family of external characteristics, including the linear ones, is realized by means of only a single linear current feedback; therefore, no additional devices are required to obtain linear characteristics.

3. The magnitude of the steady-state short-circuit current is determined by the saturation of the first and additional stages of the amplifier and can be selected in advance from the condition of permissible overload.