ANALYSIS OF QUASI-STATIC PROCESSES IN A RELAY SYSTEM OF REGENERATIVE BRAKING WITH A SEPARATELY EXCITED TRACTION MOTOR BY THE ROOT-LOCUS METHOD

V. V. Bolotov

(Presented by the Scientific and Technical Seminar of the Research Institute of AEM)

It is known that a harmonically linearized nonlinear system must be considered as a linear system lying on the stability boundary [1]. This follows from the characteristic equation of the system (Fig. 1), provided the linear part has good filtering properties,

\[ W_\ell \cdot K(A) = -1, \tag{1} \]

where

\[ K(A) \]

is the harmonic transfer coefficient of the nonlinear element.

Fig. 1. Block diagram of a nonlinear system.

Expression (1) shows that, in the complex plane, the amplitude–phase characteristic of the open-loop system will pass through the point \((-1, j0)\), which determines the stability boundary.

Therefore, when self-oscillations are present in a system described by a high-order differential equation (not lower than the third), its investigation is reduced mainly to determining the parameters of the self-oscillations, which is most simply carried out by the root-locus method.

The advantage of this method is that it is not necessary to construct the locus itself or to determine the roots of the characteristic equation of the closed-loop system, since the system is on the stability boundary.

The braking mode is described by the differential equations:

\[ \begin{gathered} U_{\mathrm{в}} = i_{\mathrm{в}} r_{\mathrm{в}} + L_s \frac{d i_{\mathrm{в}}}{dt} + L_\mu \frac{d i_\mu}{dt},\\[6pt] L_\mu \frac{d i_\mu}{dt} = i_{\mathrm{k}} r_{\mathrm{k}},\\[6pt] i_{\mathrm{в}} + i_\mu + i_{\mathrm{k}} \end{gathered} \tag{2} \]

The block diagram of the system is shown in Figures 2 and 3 (the latter presents the diagram after transformation), where

Fig. 2. Structural diagram of the relay system of regenerative braking with independent excitation.

Fig. 3. Structural diagram after transformation.

\(U_{\mathrm{в}}, i_{\mathrm{в}}, r_{\mathrm{в}}\) — voltage, current, and resistance of the field winding;

\(L_{\mu}, i_{\mu}\) — inductance and current of the magnetizing circuit;

\(i_{\mathrm{к}}, r_{\mathrm{к}}\) — current and equivalent resistance of the eddy-current circuit;

\(T_{\mathrm{к}}, T_s, T_{\mu}, T_{\mathrm{я}}\) — time constants: of the eddy-current circuit, leakage of the main poles, the magnetizing circuit, and the armature circuit.

\[ W_1 = \frac{(1/r_{\mathrm{в}})\,(T_{\mathrm{к}}\cdot p + 1)} {T_s \cdot T_{\mathrm{к}} p^2 + (T_s + T_{\mathrm{к}} + T_{\mu})p + 1} \]
— transfer function of the field winding;

\[ W_2 = \frac{k_1}{T_{\mathrm{к}}p + 1} \]
— transfer function of the magnetic circuit of the motor;

\[ K_1 = \frac{\Delta \Phi}{\Delta i_{\mu}} \]
— tangent of the angle of inclination of the tangent to the magnetization curve at the point determined by the mean value \(i_{\mu}\);

\[ K_2 = \frac{E}{\Phi} \]
— transmission coefficient relating the rotational emf to the useful flux of the motor;

\[ W_3 = \frac{1/r_{\mathrm{я}}}{T_{\mathrm{я}}p + K} \]
— transfer function of the armature circuit, taking into account the demagnetizing effect of the armature reaction by the coefficient \(K_p\);

\[ K = 1 + \frac{K_p \cdot K_2}{r_{\mathrm{я}}} \]
— coefficient;

\(K_3\) — transmission coefficient of the armature-current sensor;

\(\Phi_0\) — excitation flux of the motor without taking armature reaction into account;

\[ \Phi=\Phi_0-\Phi_p \]

\[ \begin{aligned} \Phi_p&\text{ — armature-reaction flux;}\\ \Phi&\text{ — useful flux;}\\ U_c&\text{ — supply-line voltage;}\\ U_d&\text{ — voltage at the output of the current sensor;}\\ U_e&\text{ — reference voltage proportional to the prescribed value of the armature current;}\\ U_y&\text{ — voltage at the output of the relay element.} \end{aligned} \]

If the output coordinate (the armature current) is compared with the specified one at discrete instants of time, then the hysteresis of the system becomes negligibly small and may be disregarded. Then, according to [1],

\[ K(A)=\frac{4B}{\pi\cdot A}, \tag{3} \]

where \(B\) is determined according to the form of the relay characteristic (Fig. 4);
\(A\) is the amplitude of the signal at the input of the nonlinear element.

Fig. 4. Characteristic of the nonlinear element.

According to Fig. 3, the transfer function of the linearized open-loop system is written in the form

\[ W_p=\frac{K_p\cdot K(A)} {\left[T_s\cdot T_k p^2+\left(T_s+T_k+T_{\text{я}}\right)p+1\right]\left(T_{\text{в}}p+K\right)}, \tag{4} \]

where

\[ K_p=\frac{K_1\cdot K_2\cdot K_3}{r_{\text{я}}\cdot r_{\text{в}}\cdot K}. \]

Having determined the roots of the characteristic equation for the open-loop system (4), one can write the equation of the root loci

\[ -\Theta_1-\Theta_2-\Theta_3-\omega_0\tau=-180^\circ, \tag{5} \]

where \(\Theta_i\) are the phase angles of the vectors drawn from the poles to the point \(\omega_0\) on the imaginary axis of the complex plane (Fig. 5);
\(\omega_0\) is the frequency of self-oscillations in the system;
\(\tau\) is a fixed delay.

If the value of \(\tau\) has a noticeable influence, then equation (5) is transcendental and is solved by trial-and-error. However, within practically encountered limits the value \(\omega_0\tau\) is small and may be neglected. Then, for a sufficiently small value of \(P_1\), the frequency of self-oscillations can be determined analytically:

\[ \omega_0=\sqrt{p_2\cdot p_3}. \tag{6} \]

The value of \(\omega_0\) found must be plotted on the imaginary axis (Fig. 5) and equation (5) checked. The harmonic transfer coefficient according to (3)

Fig. 5. Zero-pole portrait.

\[ K(A_0)=\frac{K_{\mathrm{gr}}}{K_l}=\frac{4B}{\pi \cdot A_0}, \tag{7} \]

where \(K_{\mathrm{gr}}\) is the transfer coefficient of the linearized system at the point \(\omega_0\).

The other parameter of the self-oscillations, \(A_0\), at the input of the nonlinear element is determined as

\[ A_0=\frac{4B \cdot K_l}{\pi \cdot K_{\mathrm{gr}}}. \tag{8} \]

After conversion to armature current, the amplitude of the current self-oscillations is

\[ \Delta I_{\mathrm{a}}=\frac{A_0}{K_3}. \tag{9} \]

The filtering properties of the linear part of the system are checked on the basis of the obtained self-oscillation frequency \(\omega_0\), according to [2]:

\[ \left|\frac{W_l(iK\omega)}{W_l(i\omega)}\right| \ll 1. \tag{10} \]

References

1. E. G. Uderman. Approximate Investigation of Self-Oscillations by the Root-Hodograph Method. Moscow, “Energiya,” 1967.

2. Theory of Automatic Control. Edited by A. V. Netushil. Moscow, “Vysshaya shkola,” 1972.