ELECTROMAGNETIC TRANSIENT PROCESS IN A RELAY AUTOMATIC CONTROL SYSTEM OF TRACTION-MOTOR CURRENT DURING A STEP CHANGE IN SUPPLY VOLTAGE

L. V. VOLYNETS, A. P. ZAITSEV

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

Let us consider the case of a sudden disappearance of the voltage of the supply network, with its subsequent restoration, when regulating the starting current of a traction motor by shunting a stage of the starting resistance with a thyristor switch. Let us assume that such a disturbance occurs at the beginning of regulation on the next stage. The equations characterizing this process must be represented in full coordinates, which vary within considerable limits.

Fig. 1

The magnetization characteristic of a series-excitation motor is nonlinear; this gives rise to a nonlinear dependence of the inductance $L_\mu$, and consequently also of the time constant $T_\mu$ of the magnetizing circuit, on the current $i_\mu$. The phase-plane method makes it possible to study the character of variation of the regulated quantity also in the case where the nonlinearities $\Phi_\mu(i_\mu)$ and $T_\mu(i_\mu)$ are specified graphically. Constructing the phase trajectory in the coordinates $\left(i, \dfrac{di}{dt}\right)$ is difficult, since, in order to take into account the above-mentioned nonlinearities, it is necessary to know the values of the magnetizing current $i_\mu$ corresponding to the instantaneous values of the armature current $i$. It is advisable to use a special phase plane with coordinates $(i_\mu, \dot{i}_\mu)$, for which we shall write the systems of equations describing this mode in the form:

\[ \begin{aligned} T_s \frac{di}{dt} + i + k_s(i - i_\mu) + c_e \omega_\partial \Phi_\mu(i_\mu) &= U_c,\\ T_\mu(i_\mu)\frac{di_\mu}{dt} &= k_s(i i_\mu), \end{aligned} \tag{1} \]

where

\[ k_s=\frac{r_k}{r_i}; \]

\(r_k\) is the resistance of the equivalent eddy-current circuit;

\[ r_i = r - c_e \omega_\partial k_{\mathrm{ra}}; \]

\(k_{\mathrm{ra}}\) is the armature-reaction coefficient;
\(T_s\) is the armature-circuit time constant.

The system of equations (1) is reduced to the form:

\[ \begin{aligned} \frac{di}{dt} &= \frac{1}{T_s}\left[U_c - c_e\omega_\partial\Phi_\mu(i_\mu) - (1+k_s)i + k_s i_\mu\right],\\ \frac{di_\mu}{dt} &= \frac{k_s}{T_\mu(i_\mu)}(i-i_\mu). \end{aligned} \tag{2} \]

Let us use the phase plane with a rectangular coordinate system. We are interested in the single phase trajectory beginning at the point \((i_0, i_{\mu0})\), at which the automatic control system is located at the moment when the supply voltage is restored. Therefore we shall apply the method of canonical polygons. This method has rather high accuracy and makes it possible to construct a narrow “corridor,” bounded by polygons, between which the desired phase trajectory is located. Suppose that the motion of the representing point begins at the origin \((i=0,\ i_\mu=0)\). Until the inequality ceases to hold,

\[ i < I_y + i_g, \]

where

\(I_y\) is the setting current,
\(i_g\) is half the width of the hysteresis loop of the relay element of the automatic control system,

the comparison circuit of the automatic control system issues a command to shunt the controlled stage \(r_p\) by the thyristor switch, and in the expression for \(r_i\) the value \(r\) without \(r_p\) should be substituted.

The presence of the regulator’s total delay \(\tau_0\) leads to the fact that motion with the stage removed continues even after condition (3) ceases to be satisfied, and this circumstance causes a significant “overshoot” of the armature current relative to the setting current.

Let us denote:

\[ \frac{di_\mu}{dt}=P(i,i_\mu);\qquad \frac{di}{dt}=Q(i,i_\mu). \tag{4} \]

Then the coordinates \(i\) and \(i_\mu\) can be represented in the form of a certain vector field of instantaneous phase velocities

\[ P(i,i_\mu)\,\overline{a}+Q(i,i_\mu)\,\overline{b}, \tag{5} \]

where \(\overline{a}\) and \(\overline{b}\) are unit vectors along the axes \(i_\mu\) and \(i\), respectively.

The steady-state point on the phase plane is found by jointly solving the equations:

\[ \left. \begin{aligned} P(i,i_\mu)&=0\\ Q(i,i_\mu)&=0 \end{aligned} \right\} \tag{6} \]

This solution is determined by constructing the so-called “zero” and

“infinite” isoclines and by finding their points of intersection. For the zero isocline we have:

\[ \operatorname{tg}\varphi_0=\frac{Q(i,i_\mu)}{P(i,i_\mu)}=0. \]

The equation of the zero isocline is

\[ Q(i,i_\mu)=\frac{1}{T_s}\left[U_c-c_e\omega_{\partial}\Phi_\mu(i_\mu)-(1+k_s)i+k_s i_\mu\right]. \tag{7} \]

Its graph for the numerical data

\[ U_c=750\ \text{V},\qquad c_e\omega_{\partial}=7\ \text{s}^{-1},\qquad r_i=0.5\ \Omega,\qquad r_k=0.6\ \Omega \]

and for the dependence \(\Phi_\mu(i_\mu)\) shown in Fig. 1 is presented in Fig. 2.

For the infinite isocline we have:

\[ \operatorname{tg}\varphi(\infty)=\frac{Q(i,i_\mu)}{P(i,i_\mu)}=\infty. \tag{8} \]

Fig. 2

Its equation is

\[ P(i,i_\mu)=-\frac{k_s}{T_\mu i_\mu}(i-i_\mu)=0, \tag{9} \]

or

\[ i=i_\mu . \tag{10} \]

The straight line (10) is also shown in Fig. 2. The equilibrium point \(M\) has coordinates, in relative units: \(I^*_{\infty}/I^*_{\infty \mu}=0.31\).

The construction of the polygons is carried out as follows. For the starting point of motion \(O\), the values are determined:

\[ \left(\frac{di_\mu}{dt}\right)_0=P(i_0,i_{\mu0});\qquad \left(\frac{di}{dt}\right)_0=Q(i_0,i_{\mu0}). \tag{11} \]

The direction of the field vector at point \(O\) is determined from the signs of the derivatives (11), and the tangent of the angle of inclination of the field vector \(\varphi_0\) from their ratio. In our particular case \(\varphi_0=90^\circ\). In the direction of the field vector, an arbitrary step is taken to point \(A\) (the smaller it is, the higher the desired accuracy of construction). At point \(A\), the inclination of the field vector is again determined according to (11), and \(\operatorname{tg}\varphi_A\) is found. In our case \(\varphi_A\approx90^\circ\). We take a second step to point \(B\). The angle of inclination of the field vector at point \(B\): \(\varphi_B<90^\circ\). From

from point \(A\) parallel to the field vector at point \(B\), i.e., at an angle \(\varphi_B\) to the abscissa axis, a straight-line segment of arbitrary length is drawn. A segment of the isocline \(1\), passing through point \(B\) and intersecting the drawn straight-line segment at point \(B\), is constructed. This construction is carried out according to the equation:

\[ \tan \varphi_B=-\frac{Q(i,i_\mu)}{P(i,i_\mu)} , \tag{12} \]

into which arbitrary values of the variable \(i_\mu\) (or \(i\)) are substituted, and the values of the second variable \(i\) (or \(i_\mu\)) are found. The next step is made from point \(B\) to point \(C\), and point \(C\) is found by construction. As a result, left-canonical \((OABC\ldots)\) and right-canonical \((OABC\ldots)\) polygons are obtained, between which the desired phase trajectory is located symmetrically. In Fig. 2 this trajectory is constructed for \(T_s=0.02\ \text{s}\) and for the dependence \(T_\mu(v)\), \(i_\mu(v)\), shown in Fig. 1.

The construction of the transient process in time \(i' f_1(t')\) can be performed by approximate integration of one of equations (2). Let us write the first of them in increments:

\[ \frac{\Delta i}{\Delta t} = \frac{1}{T_s} \left[ U_c-c_e\omega_\partial \Phi_\mu(i_\mu) -(1+k_s)+k_s i_\mu \right]. \tag{13} \]

Here the values \(i\) and \(i_\mu\) are related to each other by the phase trajectory found. To determine the time increments \(\Delta t_n\), it is necessary to substitute the armature-current increments \(\Delta i_n\) and the corresponding average values of the currents \(i_{\mathrm{av}}\) and \(i_{\mu,\mathrm{av}}\) into equation (13). The sum of the armature-current increments gives the values \(i_n\), corresponding to the values of time \(t_n\), also obtained by summing all preceding time increments.

Fig. 3

Figure 3 shows the curve of the transient process \(i=f(t)\), constructed from equation (13), with the time count begun from point \(t=t_1\). This curve makes it possible quite simply to construct the dependence \(i_\mu=t_2(t)\), using the interrelation of \(i\) and \(i_\mu\) through the phase trajectory. The curve \(i_\mu=t_2(t)\), on a certain scale, reflects the character of the change in flux during the transient process. At the instant \(t_2\) inequality (3) ceases to be satisfied (the hysteresis zone is not shown in Fig. 3), the comparison circuit issues a command to close the thyristor switch, and the delay time \(\tau_0\) begins to be counted. At the instant of its end, \(t_3\), the thyristor switch closes, the regulated step \(r_p\) is introduced, and the current \(i\), having reached the maximum value \(i_{\max}\), decreases.

A study of the transient-process curves shows that, when supply is restored to a rotating series-excitation motor, the rate of rise of the armature current exceeds the rate of rise of the magnetic flux by tens of times. Therefore, exceptionally high requirements must be imposed on the speed of response of the regulator.