OPERATING STABILITY OF A BRUSHLESS DC ELECTRIC MACHINE AMPLIFIER UNDER LOAD CONDITIONS

E. N. Podborskii, A. I. Skorospeshkin, E. K. Dergobuzova

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

Analysis of stability by constructing stability regions is of great practical importance, since it makes it possible to determine what values the parameters adjusted during tuning must have in order for the amplifier to operate stably. Therefore, in the present article systems of differential equations are compiled and solved that describe the operation of the amplifier under active, active-inductive, and motor loads. The coefficients of the characteristic equations are calculated for a model of a brushless DC electric machine amplifier manufactured according to the scheme given in [1]. The stability regions are calculated by the Hurwitz criterion in the plane of two parameters varied during tuning: the degree of compensation \(\varepsilon_2\) and the angle of displacement of the commutation axis from the geometric neutral \(\beta_2\).

When operating with an active-inductive load, the dynamic properties of the amplifier are described by the following system of equations:

\[ \Phi_1 = \lambda_1\left(W_y i_y - W_{p1} i_2\right), \tag{1} \]

\[ \Phi_2 = \lambda_2\left(W_{p2} i_2 - \frac{W_3 I_3}{\pi}\beta_2\right), \tag{2} \]

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

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

\[ \sigma_k W_k p\Phi_2 + R_k i_k - r_{\mathrm{sh}} i_3 = 0, \tag{5} \]

\[ e_3 = C_2\Phi_2 = r_3 i_3 + r_k i_k + r_{\mathrm{n}} i_3 + pL_{\mathrm{n}} i_3 + \sigma_k W_k p\Phi_2 + \frac{C_2\lambda_2 W_3\beta_2}{\pi}\, i_3 - \tag{6} \]

\[ -\, b_3 W_3 p\Phi_2, \]

where

\[ i_3 \text{ is the load current,} \]

\[ r_{\mathrm{n}} \text{ is the load resistance,} \]

\[ L_{\mathrm{n}} \text{ is the inductance of the load circuit.} \]

The remaining notation is given in [1].

Solving system (1–6) with respect to the current \(i_3\), we obtain the dependence of the output current on the control signal

\[ \left[(1+pT_y)(1+pT_2+K''_p)+K_p(1+pT_y)-T_yK_p p\right] \left\{(1+pT_{\mathrm{n}}+ \right. \]

\[ +\frac{k_3\beta_2}{\pi}+\frac{k_3\beta_2}{\pi}\left[1+\frac{\sigma_3 W_3 p(1-\varepsilon_2)}{C_2}\right]\Biggr\}i_3- \left[\frac{T_2 W_3\beta_2}{W_{p2}\pi}(1+pT_y)p\right]\times \]
\[ \times\left\{\frac{k_2 r_2}{R_3}\left[1+\frac{\sigma_3 W_3 p(1-\varepsilon_2)}{C_2}\right]\right\}i_3 = \frac{K_2K_u}{R_3}\left[1+\frac{\sigma_3 W_3 p(1-\varepsilon_2)}{C_2}\right]U_y, \tag{7} \]

where

\[ R_3=r_3+\frac{r_k r_{\mathrm{sh}}}{R_k}+r_n, \]

\[ T_n=\frac{L_n}{R_3}\quad \text{is the time constant of the load circuit,} \]

\[ K_3=\frac{C_2\lambda_2 W_3}{R_3}. \]

Equation (7) corresponds to the characteristic equation

\[ a_0p^3+a_1p^2+a_2p+a_3=0, \tag{8} \]

where

\[ a_0=T_yT_2T_n; \]

\[ a_1=T_yT_2+\frac{T_yT_2k_3\beta_2}{\pi} +\left[T_y+T_2+T_yk_p''\right]T_n+ \]
\[ +\frac{(T_y+T_2+T_yk_p)K_3\beta_2\sigma_3W_3(1-\varepsilon_2)}{\pi C_2} -\frac{T_2K_3\beta_2\sigma_3W_3(1-\varepsilon_2)}{\pi C_2}; \]

\[ a_2=(T_y+T_2+T_yk_p'')\left(1+\frac{2k_3\beta_2}{\pi}\right) +\left(1+k_p''+k_p\right)\left[T_n+ \right. \]
\[ \left. +\frac{k_3\beta_2\sigma_3W_3(1-\varepsilon_2)}{\pi C_2}\right] -\frac{T_2k_3\beta_2}{\pi}; \]

\[ a_3=(1+k_p''+k_p)\left(1+\frac{2k_3\beta_2}{\pi}\right). \]

Since a DC BEMA is described by a system of equations whose characteristic equation is of no higher than fourth order, we shall use the Hurwitz criterion to construct the regions of stability.

In order to investigate the influence of the degree of compensation and of the displacement of the commutation axis from the geometric neutral, it is necessary to construct stability regions in the plane of two parameters, \(\varepsilon_2\) and \(\beta_2\).

For this purpose we represent the coefficients of the characteristic equation \(a_0, a_1, a_2, a_3\) in the form:

\[ \begin{aligned} a_0&=a_0'+x_2a_0'', &\qquad a_2&=a_2'+x_2a_2'',\\ a_1&=a_1'+x_2a_1'', &\qquad a_3&=a_3'+x_2a_3'', \end{aligned} \tag{9} \]

where \(x_2=\dfrac{2\beta_2}{\pi}\) is the relative displacement of the commutation axis,

\[ a_0'=T_yT_2T_n,\qquad a_0''=0, \]

\[ a_1'=T_yT_2+(T_y+T_2+T_yk_p'')T_n, \]

\[ a_1''=\frac{T_yT_2k_3}{2} +\frac{(T_y+T_2+T_y\kappa_p'')\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2} - \frac{T_2\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2}, \]

\[ a_2'=T_y+T_2+T_y\kappa_p''+(1+\kappa_p+\kappa_p'')T_n, \]

\[ a_2''=(T_y+T_2+T_y\kappa_p'')\kappa_3 +\frac{(1+\kappa_p+\kappa_p'')\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2} - \frac{T_2\kappa_3}{2}, \]

\[ a_3'=1+\kappa_p+\kappa_p'', \]

\[ a_3''=(1+\kappa_p+\kappa_p'')\kappa_3. \]

For a system whose third-order characteristic equation has positive coefficients, the stability boundary is determined by the condition

\[ a_1a_2-a_0a_3=0. \tag{10} \]

The magnitude of the shift of the commutation axis corresponding to stability was determined from expression (10), and also from the condition that each coefficient of the characteristic equation be equal to zero. Calculations showed that the minimum value of \(x_2\) at which the amplifier reaches the stability boundary is determined from the condition that the free term of equation (8) be zero and is equal to

\[ x_2=-\frac{a_3'}{a_3''}. \]

In this case the characteristic equation will have three roots:

\[ p_1=0,\qquad p_{2,3}=\frac{-a_1\pm\sqrt{a_1^2-4a_0a_2}}{2a_0}. \]

The roots \(p_{2,3}\) are complex with a negative real part. When the commutation axis is shifted from the stability boundary in the direction of rotation, the root \(p_1\) becomes negative and the transient process will proceed monotonically with damped oscillations.

When the commutation axis is shifted beyond the stability boundary against the direction of rotation, the coefficient \(a_3\) becomes negative. In this case at least one root will be positive, and a self-excitation process occurs in the amplifier.

In the case of an active load, characteristic equation (8) takes the form

\[ a_1p^2+a_2p+a_3=0. \tag{11} \]

For positive coefficients \(a_1\), \(a_2\), and \(a_3\), the amplifier will operate in the stability region; the roots of equation (11) will be either real negative when

\[ a_2^2>4a_1a_3, \]

or complex with a negative real part when

\[ a_2^2<4a_1a_3. \]

The boundary between periodic and aperiodic damped processes is the condition

\[ a_2^2=4a_1a_3. \]

For \(a_3=0\), the amplifier reaches the stability boundary; in this case the commutation axis is displaced opposite to the direction of armature rotation by the amount

\[ x_2=-\frac{a'_3}{a''_3}. \tag{12} \]

If the commutation axis is displaced opposite to the direction of rotation by an amount greater than that obtained from equality (12), a self-excitation process occurs in the amplifier.

Thus, with an active and active-inductive load, the amplifier will operate stably if the angle of displacement of the commutation axis opposite to the direction of rotation is less than the value determined by equality (12).

Let us consider the stability regions when the amplifier operates into a separately excited dc motor. The equations of the brushless EMU–motor system can be obtained by combining the equations of the amplifier and of the motor. For this purpose, the following must be added to equations (1–5):

\[ e_3=c_2\Phi_2=r_a i_3+r_k i_k+pL_a i_3+\sigma_k W_k p\Phi'_2-\sigma_3 W_3 p\Phi_2+e_{\partial}+\frac{K_3\beta_2}{\pi}i_3. \tag{13} \]

\[ j\frac{dn}{dt}=M_{\partial}-M_c, \tag{14} \]

\[ M_{\partial}=K_i i_3, \tag{15} \]

where

\[ \begin{aligned} e_{\partial}&=k_e n &&\text{— motor emf,}\\ n& &&\text{— rotational speed of the motor shaft,}\\ J& &&\text{— moment of inertia of the motor armature,}\\ M_{\partial}& &&\text{— driving torque of the motor,}\\ M_c& &&\text{— resisting torque,}\\ r_a& &&\text{— resistance of the motor armature,}\\ L_a& &&\text{— inductance of the armature circuit of the BEMU and the motor,}\\ k_i,\ k_e& &&\text{— constants.} \end{aligned} \]

This system of equations has been compiled under the following assumptions:

1) the magnetic system of the BEMU is not saturated,
2) the influence of hysteresis and eddy currents is not taken into account,
3) the motor flux is constant.

Solving this system with respect to speed and current, we obtain

\[ \bigl[(b-c)T_m p+1\bigr]n = \frac{R_a a}{K_e}U_y - \frac{M_c R_a(b-c)}{K_iK_e}, \tag{16} \]

\[ i_3= \frac{a}{b-c}U_y - \frac{K_e}{R_a(b-c)}, \tag{17} \]

where

\[ T_m=\frac{R_a J}{K_iK_e} \quad\text{— electromechanical time constant of the motor,} \]

\[ T_a=\frac{L_a}{R_a} \quad\text{— electromagnetic time constant of the armature circuit,} \]

\[ R_a=r_3+r_a+\frac{r_k r_{\mathrm{sh}}}{R_k}, \]

\[ a=\frac{K_u K_2}{R_a} \left[ 1+\frac{\sigma_3 W_3 p(1-\varepsilon_2)}{c_2} \right]; \tag{18} \]

\[ b=\left[(1+pT_y)(1+pT_2+\kappa_p^{\prime\prime})+\kappa_p(1+pT_y)-T_y\kappa_p p\right]\left\{\left(1+pT_a+\frac{\kappa_3\beta_2}{\pi}+\right.\right. \]
\[ \left.\left. +\frac{\kappa_3\beta_2}{\pi}\left[1+\frac{\sigma_3W_3p(1-\varepsilon_2)}{c_2}\right]\right)\right\}; \tag{19} \]

\[ C=\frac{T_2W_3\beta_2}{W_{p2}\pi}(1+pT_y)p\left\{\frac{\kappa_2 i_2}{R_a}\left[1+\frac{\sigma_3W_3p(1-\varepsilon_2)}{c_2}\right]\right\}. \tag{20} \]

The characteristic equation for the given case will have the form:

\[ A_0p^4+A_1p^3+A_2p^2+A_3p+A_4=0, \tag{21} \]

where

\[ A_0=T_yT_2T_aT_m, \]

\[ A_1=T_yT_2T_m+(T_y+T_2+T_y\kappa_p^{\prime\prime})T_aT_m+ \]

\[ +\frac{(T_y+T_2+T_y\kappa_p^{\prime\prime})T_m\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2} -x_2\frac{T_2T_m\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2}+ \]

\[ +-\frac{T_yT_2T_m\kappa_3}{2}\,x_2, \]

\[ A_2=T_m(T_y+T_2+T_y\kappa_p^{\prime\prime})+(1+\kappa_p+\kappa_p^{\prime\prime})T_aT_m+x_2T_m\left[(T_y+\right. \]

\[ \left.+T_2+T_y\kappa_p^{\prime\prime})\kappa_3+ \frac{(1+\kappa_p+\kappa_p^{\prime\prime})\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2} -\frac{T_2\kappa_3}{2}\right], \]

\[ A_3=T_m(1+\kappa_p+\kappa_p^{\prime\prime})+x_2T_m(1+\kappa_p+\kappa_p^{\prime\prime})\kappa_3, \qquad A_4=1. \]

Usually the electromagnetic time constant of the armature circuits may be neglected. Therefore, for \(T_a=0\) we obtain \(A_0=0\),

\[ A_1^{\prime}=T_yT_2T_m, \]

\[ A_1^{\prime\prime}\frac{T_yT_2T_m\kappa_3}{2} +\frac{(T_y+T_2+T_y\kappa_p^{\prime\prime})T_m\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2} - \]

\[ -\frac{T_2T_m\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2}, \]

\[ A_2^{\prime}=T_m(T_y+T_2+T_y\kappa_p^{\prime\prime}), \]

\[ A_2^{\prime\prime}=T_m\left[(T_y+T_2+T_y\kappa_p^{\prime\prime})\kappa_3+ \frac{(1+\kappa_p+\kappa_p^{\prime\prime})\kappa_3\sigma_3W_3(1-\varepsilon_2)}{2c_2} -\right. \]

\[ \left.-\frac{T_2\kappa_3}{2}\right], \]

\[ A_3^{\prime}=T_m(1+\kappa_p+\kappa_p^{\prime\prime}), \]

\[ A_3^{\prime\prime}=T_m(1+\kappa_p+\kappa_p^{\prime\prime})\kappa_3, \qquad A_4=1. \]

Here

\[ A_0=A_0^{\prime}+x_2A_0^{\prime\prime}, \qquad A_2=A_2^{\prime}+x_2A_2^{\prime\prime}, \]

\[ A_1=A_1^{\prime}+x_2A_1^{\prime\prime}, \qquad A_3=A_3^{\prime}+x_2A_3^{\prime\prime}. \]

To construct the stability regions we shall use the Hurwitz criterion for a fourth-order system, which is written in the form

\[ A_3(A_1A_2 - A_0A_3) - A_1^2A_4 = 0. \tag{22} \]

Specifying the degree of compensation \(\varepsilon_2\), from condition (22) we find the values of the shift of the commutation axis corresponding to the stability boundary. Figure 1 shows the regions of stable and unstable operation of the amplifier with the MI-11F motor.

Fig. 1

From consideration of the stability region it follows that, when the commutation axis is shifted in the direction of rotation, the amplifier operates stably; in this case the roots of the characteristic equation may be either all real and negative, or one root is real and negative and two are complex with a negative real part. In the first case an aperiodic transient process will occur; in the second, the transient process proceeds monotonically, but damped oscillations are superimposed on the transient curve.

It should be noted that the boundary of the stability region is constructed from the condition that the angle of shift of the commutation axis is independent of the degree of compensation. In fact, at the same load the angle of shift of the commutation axis is positive and is the larger the smaller the undercompensation; with overcompensation the shift is negative. Therefore, for a degree of compensation \(\varepsilon_2 < 1\), the amplifier will operate stably at any load, since in this case \(x_2 > 0\). An unstable mode is possible only with overcompensation.

Thus, the analytical investigation carried out has shown that a brushless DC EMA, when operating under load conditions, will be stable for \(\varepsilon_2 < 1\), independently of the type of load.

REFERENCES

1. E. N. Podborskii, A. I. Skorospeshkin. Questions of the theory of a brushless DC EMA in no-load operation. Izvestiya TPI, vol. 190, 1968.

2. D. V. Vasil’ev, V. G. Chuich. Automatic Control Systems. Vysshaya Shkola Publishing House, Moscow, 1967.