Issues in the Theory of a Brushless DC Electric Machine Amplifier under No-Load Conditions

E. N. PODBORSKY, A. I. SKOROSPESHKIN

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

This article considers the operating principle of the proposed brushless DC electric-machine amplifier and presents its theory under no-load conditions.

The brushless EMA is a two-stage single-armature amplifier, each stage of which is a DC machine with a semiconductor commutator. The armature winding of the first stage \(W_{p1}\) feeds, through a semiconductor commutator, the excitation winding of the second stage \(W_{p2}\). Both stages are made in a single magnetic core in such a way that there is no inductive coupling between them. The semiconductor commutator is a rectifier assembled in a bridge circuit using uncontrolled diodes. In the rotor slots located on the poles there is a compensating winding \(W_k\), whose axis is shifted by 90 electrical degrees relative to the winding \(W_{p2}\). The circuit of the amplifier is shown in Fig. 1. A semiconductor

Fig. 1

commutator with closed windings can be obtained by replacing each commutator plate with a pair of series-connected valves, between which the lead of the winding is connected [1]. By joining electrodes of the same name, we obtain a rectifier bridge. The voltages across all the valves are distributed in such a way that there is always an open pair of valves located dia-

metrically. The axis along which the open valves are located, called the commutation axis, is always located along the physical neutral. Thus, the brushes are, as it were, turned to the physical neutral.

Under no-load conditions let us set up the system of differential equations of the amplifier. In setting up the equations we shall operate with the resultant fluxes, acting independently of one another, since the use of mutual-inductance coefficients between windings greatly complicates the problem [2]. The magnetic circuit of the amplifier is assumed to be unsaturated, which is valid for operation at rated voltage.

Taking this into account, the system of equations has the following form:

Here

\[ \Phi_1=\lambda_1(W_y i_y-W_p i_2), \tag{1} \]

\[ \Phi_2=\lambda_2 W_{p2}i_2, \tag{2} \]

\[ U_y=r_y i_y+\sigma_y W_y p\Phi_1, \tag{3} \]

\[ e_2=C_1\Phi_1=r_2 i^2+\sigma_{p2}W_{p2}p\Phi_2+ \frac{C_1\lambda_1 W_{p1}}{\pi}\beta_1 i_2, \tag{4} \]

\[ \sigma_k W_k p\Phi_2+R_k i_k=0, \tag{5} \]

\[ e_3=C_2\Phi_2+\sigma_3 W_3 p\Phi_2-r_k i_k-\sigma_k W_k p\Phi_2. \tag{6} \]

Here

\(\Phi_1,\lambda_1\) are the flux and magnetic permeance of the first stage of the amplifier,

\(W_y, W_{p2}; W_k, W_3\) are the numbers of turns of the control winding, the excitation winding of the second stage, the compensating winding, and the output winding,

\[ W_p=\frac{W_{p1}\beta_1}{\pi}+W_z, \]

where

\(W_{p1}\) is the number of armature turns of the first stage;

\(\beta_1\) is the angle of displacement of the commutation axis of the first stage relative to the geometric neutral;

\[ W_z=\frac{F_{z1}}{i_2} \]

are fictitious turns corresponding to the m.m.f. due to the reaction of the commutating currents of the first stage;

\(i_y\) is the control current;

\(\Phi_2,\lambda_2\) are the flux and magnetic permeance of the second stage;

\(i_2\) is the excitation current of the second stage;

\(U_y\) is the control voltage;

\(\delta_y,\delta_{p2},\delta_k\) are the leakage coefficients of the control winding, the excitation winding of the second stage, and the compensating winding;

\(C_1, C_2\) are the coefficients of proportionality between the e.m.f.’s \(e_2\) and \(e_3\) and the fluxes \(\Phi_1\) and \(\Phi_2\);

\(r_2=r_{p1}+r_{p2}+2r_v\) are the active resistances of the armature windings of the first stage and the valve resistance;

\(R_k=r_k+r_{\sh}\) is the total resistance of the circuit compensating winding—shunt resistance.

System (1–6) characterizes the transient process of change of the e.m.f. for the case when the resistance \(r_2\) is taken as a constant quantity and the value of the output flux is linearly related to the current \(i_2\).

To derive the characteristic equation and transfer function of the amplifier under no-load conditions, we use system (1–6). We carry out the solution of this system assuming \(\beta_1=\mathrm{const}\). This assumption is quite acceptable, since the resistance \(r_2\) remains constant. Blա

Thanks to this assumption, the given system becomes linear and can be solved in general form [3].

The output e.m.f. of the amplifier under no-load conditions as a function of the control voltage will be equal to:

\[ e_{\mathrm{z}}\left[ p^{2}T_{\mathrm{y}}T_{2} +p\left(T_{\mathrm{y}}+T_{2}+T_{\mathrm{y}}K_{p}^{\prime\prime}\right) +1+K_{p}^{\prime\prime}+K_{p}^{\prime} \right] = \]

\[ = \left[ 1+\frac{\sigma_{3}W_{3}p(1-\varepsilon_{2})}{C_{2}} \right]K_{2}K_{\mathrm{u}}U_{\mathrm{y}}; \tag{7} \]

where

\[ K_{\mathrm{u}}=\frac{C_{1}\lambda_{2}W_{\mathrm{y}}}{r_{\mathrm{y}}} =\frac{C_{12}}{r_{\mathrm{y}}} \]

is the voltage transfer coefficient for the control winding.

\[ K_{p}^{\prime\prime} = \frac{\lambda_{1}W_{p1}C_{1}\beta_{1}}{r^{2}} \]

is the transfer coefficient caused by the armature reaction due to the current \(I_{2}\),

\[ K_{p}^{\prime}=\frac{\lambda_{1}C_{1}W_{z}}{r_{2}} \]

is the transfer coefficient caused by the reaction of the commutating currents of the first stage,

\[ K_{p}=K_{p}^{\prime\prime}+K_{p}^{\prime}, \]

\[ K_{2}=\frac{\lambda_{2}W_{p2}C_{2}}{r_{2}} \]

is the voltage transfer coefficient of the second amplifier stage,

\[ T_{\mathrm{y}}=\frac{\lambda_{\mathrm{y}}\sigma_{\mathrm{y}}W_{\mathrm{y}}^{2}}{r_{\mathrm{y}}} \]

is the time constant of the circuit of the single control winding,

\[ T_{2}=\frac{\lambda_{2}\sigma_{p2}W_{p2}^{2}}{r_{2}} \]

is the time constant of the excitation circuit of the second stage.

For the steady-state mode we obtain

\[ E_{\mathrm{z}}= \frac{K_{2}K_{\mathrm{u}}}{1+K_{p}^{\prime\prime}+K_{p}}U_{\mathrm{y}}. \tag{8} \]

In the first approximation, the term

\[ \frac{\sigma_{3}W_{3}p(1-\varepsilon_{2})}{C_{2}} \]

may be neglected as small in comparison with unity. In this case equation (7) takes the form

\[ e_{\mathrm{z}}= \frac{1+K_{p}^{\prime\prime}+K_{p}} {p^{2}T_{\mathrm{y}}T_{2} +p\left(T_{1}+T_{2}+T_{1}K_{p}^{\prime\prime}\right) +K_{p}+K_{p}^{\prime\prime}+1} \,E_{\mathrm{z}}. \tag{9} \]

The characteristic equation of the system will have the form

\[ D(p)=a_{0}p^{2}+a_{1}p+a_{2}, \]

where

\[ a_{0}=T_{\mathrm{y}}T_{2};\quad a_{1}=T_{\mathrm{y}}+T_{2}+T_{\mathrm{y}}K_{p}^{\prime\prime};\quad a_{2}=K_{p}+K_{p}^{\prime\prime}+1. \]

To analyze the transient process, we shall examine the characteristic equation (10), which has two roots.

\[ p_{1,2}= \frac{ -\left(T_y+T_2+T_yK_p''\right)\pm \sqrt{\left(T_y-T_2+T_yK_p''\right)^2-4T_yT_2K_p} }{ 2T_yT_2 } \tag{11} \]

It follows from (11) that, for a particular amplifier, the magnitude of the roots is determined by two time constants \(T_y\) and \(T_2\), and by the magnitude of the coefficients \(K_p\) and \(K_p''\).

In the process of adjusting the operation of the amplifier, the value of the time constant \(T_y\) may be varied by introducing an additional resistance.

The magnitude of the coefficients \(K_p\) and \(K_p''\) changes when the commutation axis of the first stage is shifted; moreover, it may take both positive and negative values depending on the direction of displacement of the commutation axis.

From (11) it is seen that the limiting case of an aperiodic process will occur when the amplifier parameters satisfy the relation

\[ \left(T_y-T_2+T_yK_p''\right)^2=4T_yT_2K_p'', \tag{12} \]

since, for \(4T_yT_2K_p>\left(T_y-T_2+T_yK_p''\right)^2\), the roots of the characteristic equation will be complex and an oscillatory process will be observed, whereas for \(4T_yT_2K_p<\left(T_y-T_2+T_yK_p''\right)^2\) the roots are real; consequently, the process will be aperiodic.

Let us introduce the coefficient

\[ \gamma=\frac{T_2}{T_y}. \]

Substituting \(\lambda\) into (12), we obtain

\[ \gamma^2-2\gamma\left(2K_p''+2K_p'+1\right)+\left(1+K_p''\right)^2=0. \tag{13} \]

Assigning various values to the coefficient \(K_p''\) and taking into account that it depends on the magnitude of the displacement of the commutation axis of the first stage, one can obtain the corresponding values of the parameter \(\lambda\) and, in the parameter plane, construct a curve that is the boundary between the regions with aperiodic and oscillatory modes (Fig. 2). The parameter plane is constructed for the model of the brushless EMU that is being studied in the laboratory.

Fig. 2

The self-excitation mode can arise only in the case when one of the coefficients of the characteristic equation becomes negative. Analysis of the coefficients of this equation shows that only the constant term of the equation \(a_2\) can become negative. Thus, the second limiting case of an aperiodic process will occur when the system parameters satisfy the relation

\[ a_2 = K_p + K_p'' + 1 = 0, \]

i.e., for all values

\[ K_p + K_p'' < -1 \]

we shall have self-excitation of the amplifier.

Let us find the value of the relative displacement of the commutation axis of the first stage,

\[ X_1 = -\frac{2\beta_1}{\pi}, \]

at which the self-excitation process begins. To this end, we represent the coefficient \(K_p\) as the sum of two terms

\[ K_p = K_p'' + K_p' = \pm \frac{C_1 \gamma_1 W_{p1}}{2r_2} X + \frac{C_1 \gamma_1 W_z}{r_2}. \tag{15} \]

Substituting (15) into (14),

\[ -\frac{C_1 \gamma_1 W_{p1}}{2r_2} X + \frac{C_1 \gamma_1 W_z}{r_2} + 1 = 0. \tag{16} \]

The minus sign before the first term indicates that this equality is possible only when the commutation axis is displaced opposite to the direction of rotation of the armature. From the last equality it follows that

\[ X_1 = \frac{W_z}{W_{p1}} + \frac{r_2 W_y}{C_{12} W_{p1}}. \tag{17} \]

Thus, for values

\[ X_1 > \frac{W_{z1}}{W_{p1}} + \frac{r_2 W_y}{C_{12} W_p} \]

the amplifier operates in the self-excitation mode.

From the physical point of view, the self-excitation mode can be explained by the fact that displacement of the commutation axis of the first stage from the geometrical neutral against the direction of rotation of the armature produces a longitudinal magnetizing armature reaction which, for values \(X_1\) determined by expression (17), compensates the action of all demagnetizing factors, and, for a larger displacement, may exceed them; this brings the amplifier into the self-excitation mode.

The magnitude and direction of displacement of the commutation axis of the first stage from the geometrical neutral can change only when the degree of compensation of the transverse armature reaction of the first stage is changed by means of a special winding placed on the stator and supplied from the output voltage. If this winding is absent, then displacement of the commutation axis is possible only in the direction of rotation of the armature. The oscillatory process, as indicated above, is possible when

\[ 4T_y T_2 K_p > (T_y - T_2 + T_y K_p'')^2. \tag{18} \]

The roots of the characteristic equation then become complex, namely

\[ P_{1,2} = -\frac{1}{T_e} \pm j\omega, \tag{19} \]

where

\[ T_{\mathrm{e}}=\frac{2T_yT_2}{T_y+T_2+T_yK_p^{\prime\prime}} \]

is the equivalent time constant,

\[ \omega=\sqrt{\frac{K_p}{T_yT_2}-\left(\frac{T_y-T_2+T_yK_p^{\prime\prime}}{2T_yT_2}\right)^2} \]

is the natural frequency of oscillations of the amplifier.

In the general case, the equation for the build-up of the emf of the amplifier under an abrupt application of control at constant voltage can be written in the following form [3]:

\[ e_3=E_3\left[1-e^{-\frac{t}{T_{\mathrm{e}}}}\left(\cos\omega t+\frac{1}{T_{\mathrm{e}}\omega}\sin\omega t\right)\right]. \tag{20} \]

Thus, the value of the equivalent time constant is determined by the values of the time constants \(T_y\) and \(T_2\) and by the magnitude of the displacement of the commutation axis from the geometrical neutral. Increasing the displacement of the commutation axis leads to a decrease in the equivalent time constant.

The second factor characterizing the transient process is the oscillation frequency \(\omega\), which depends to a considerable degree on the coefficient \(K_p^{\prime\prime}\). Displacement of the commutation axis in the direction of armature rotation leads to an increase in the oscillation frequency. At \(\omega=0\), the oscillatory process becomes aperiodic.

From an analysis of the no-load operating modes of a brushless EMA, the following conclusions may be drawn:

1. The character of the transient process is influenced mainly by the coefficients \(K_p^{\prime\prime}\) and \(K_p\). In the manufactured amplifier, the coefficient \(K_p^{\prime\prime}\) is determined by the magnitude and direction of displacement of the commutation axis of the first stage. Depending on it, three types of transient processes may exist in the amplifier: (a) aperiodic, (b) oscillatory, and (c) a mode of mild self-excitation.

2. The value of the equivalent time constant depends on \(T_y\) and \(T_2\), and also on the magnitude of the displacement of the commutation axis.

3. The oscillation frequency is determined chiefly by the coefficient \(K_p\); in other words, in the manufactured amplifier it depends on the displacement of the commutation axis.

References

1. V. L. Lototskii. Investigation of brushless direct-current machines with a semiconductor commutator. Proceedings of the Moscow Aviation Institute, 1966.

2. D. P. Morozov. On a method for compiling differential equations for complex circuits of electric drives of mechanisms. Proceedings of MEI, issue XXII, 1956.

3. V. P. Lagunov. On the problem of investigating the stable operation of a transverse-field electric-machine amplifier, taking account of the nonlinearity of its parameters. Dissertation, Tomsk, 1964.