Transient Processes in the System “Shunt Exciter—Generator Field Winding”

S. S. Kropanin

(Presented by the scientific seminar of the Electromechanical Faculty)

The study of transient processes in direct-current machines, taking nonlinearities into account, is of great practical importance. Taking account of the nonlinear parameters of machines makes it possible to reveal substantially new properties and phenomena which often cannot be detected if these parameters are treated as linear. The solution of nonlinear problems is often associated with considerable difficulties of a mathematical nature. For a number of cases there are no unified methods for the analytical expression of the laws of variation of individual nonlinear parameters.

In practice, methods of analytical approximation are used, since they provide sufficient accuracy in expressing the approximated quantity itself, as well as sufficient accuracy and simplicity in expressing derivatives, with a comparatively simple form of solution suitable for investigation [1].

Fig. 1.

For the system whose circuit is shown in Fig. 1, after the closing of switch \(P\) and for a certain relationship among the parameters, the following types of motion, shown in Fig. 2 [2, 3, 4], are possible:
a) establishment of a new equilibrium state; b) undamped periodic oscillations; c) aperiodic damped oscillations.

In view of the complexity of the phenomena occurring in the system under consideration, and of the influence of a large number of factors on the character of the transient regimes, it is necessary to make the following series of assumptions, which would make it possible to investigate the operation of the system by a method sufficiently simple and suitable for practice, with accuracy sufficient for practical purposes [5].

1. We assume that the rotational speed of the exciter armature is constant.
2. In the calculations we do not take into account the armature reaction of the exciter.
3. We do not take hysteresis phenomena into account.
4. We neglect the influence of eddy currents in the massive parts of electrical machines.
5. We neglect the mutual induction between the armature winding and the field winding of the exciter.

Fig. 2 a, b.

1. We neglect the inductance of the armature winding \(L_y\) of the exciter.
2. We take as constant the inductances of the exciter field winding \(L_v\) and of the generator field winding \(L_g\).
3. We take as constant the resistance of the exciter armature winding.
4. We take the function \(e_v(i_v)\) to be substantially nonlinear.

It must be noted that the assumptions made will affect the accuracy of the result; however, these assumptions are unavoidable, since our aim is to investigate transient regimes by methods sufficiently simple for practice, allowing satisfactory results to be obtained with the least expenditure of labor and time.

The expression indicated above, \(e_v(i_v)\), is a nonlinear function of the excitation current of the exciter, \(i_v\), and can be represented in the form of a no-load characteristic. As is known, in the literature [1, 6] there are many methods for approximating the magnetization curve of electrical machines. Of all the formulas known to us that give an analytical expression for the magnetization curve, we chose the formula

Fig. 2c.

\[ e_v(i_v)=A e_{vn}\arctan B i_v, \tag{1} \]

which gives the best approximation both of the magnetization curve and of its derivative with respect to the excitation current [6]. In formula (1), \(e_{vn}\) is the nominal value of the no-load emf of the exciter, and \(A\) and \(B\) are constant coefficients determined from the form of the experimental no-load characteristic.

In Fig. 3 the experimental no-load characteristic of the exciter (curve 1) and the characteristic calculated from equation (1) (curve 2) are presented. Comparison of the actual and calculated characteristics gives good agreement. The greatest error is \(2.5\%\). The use of the average static no-load characteristic of the exciter in calculating transient conditions introduces an inaccuracy into the results obtained, but this must be accepted, since taking account of hysteresis and of the dynamic no-load characteristic presents considerable difficulties.

Fig. 3.

Let us write the equations characterizing the transient processes in the system under consideration, taking into account the assumptions adopted above and the expression for the no-load emf of the exciter according to formula (1), after closing the switch \(P\).

\[ A e_{vn}\operatorname{arc\,tg} B i_v = i_v r_v+i_{ya}r_{ya}+L_v\frac{di_v}{dt}, \tag{2} \]

\[ A e_{vn}\operatorname{arc\,tg} B i_v = i_2 r_g+i_{ya}r_{ya}+L_g\frac{di_2}{dt}, \tag{3} \]

\[ i_{ya}=i_v+i_2, \tag{4} \]

where \(r_v\) is the resistance of the field winding of the exciter,
\(r_g\) is the resistance of the field winding of the generator,
\(i_{ya}\) is the current in the armature winding of the exciter,
\(i_2\) is the current in the field winding of the generator.

Substituting the value of the exciter armature current \(i_{ya}\) from equation (4) into equation (2), we express the generator field current as

\[ i_2= \frac{ A e_{vn}\operatorname{arc\,tg} B i_v -i_v(r_v+r_{ya})-L_v\frac{di_v}{dt} }{r_{ya}} . \tag{5} \]

Differentiating (5) with respect to time, we shall have

\[ \frac{di_2}{dt} = \frac{ \left[ \frac{AB e_{vn}}{1+B^2 i_v^2} -(r_v+r_{ya}) \right]\frac{di_v}{dt} - L_v\frac{d^2 i_v}{dt^2} }{r_{ya}} . \tag{6} \]

Substituting (4), (5), and (6) into (3) and solving with respect to the exciter field current, we obtain the equation of motion in the system

\[ \frac{d^2 i_v}{dt^2} + \left[ \frac{r_v+r_{ya}}{L_v} + \frac{r_2+r_{ya}}{L_g} - \frac{AB e_{vn}}{L_v(1+B^2 i_v^2)} \right] \frac{di_v}{dt} + \frac{ (r_v r_{ya}+r_2 r_{ya}+r_v r_2)i_v - r_2 A e_{vn}\operatorname{arc\,tg} B i_v }{L_v L_g} =0 . \tag{7} \]

Equation (7) is a nonlinear differential equation of the second order, the analytical solution of which presents considerable difficulty.

To solve equation (7), it is necessary to choose a method that would be uniform for all possible types of motion in the system (Fig. 2), would possess sufficient accuracy and clarity, and would make it possible to evaluate the qualitative picture of the motion without an analytical solution of the nonlinear differential equation. All the requirements indicated above are met by the grapho-analytical method of isoclines, which gives a solution in the phase plane with subsequent graphical construction of transient regimes in the current–time coordinate axes [7, 8]. It should be noted that:

1. For the solution it is necessary to know the numerical value of all parameters of the system.

2. Directly from the phase portrait one can judge only the magnitude of the amplitude of the desired variable. The duration of the transient process can be determined only after additional constructions in the current–time coordinate axes.

Let us denote in equation (7)

\[ \frac{di_v}{dt}=y \quad\text{and}\quad \frac{dy}{di_v}=\Theta, \tag{8} \]

where \(\Theta=\tan\alpha\) is the tangent of the angle of inclination of the tangent to the integral curve drawn at the point of intersection of the required integral curve with the isocline.

Substituting (8) into (7) and solving the latter with respect to \(V\), we obtain the equation of the isoclines

\[ y= -\frac{ A e_{\mathrm{vn}}\arctg B i_{\mathrm{v}} -\left(r_{\mathrm{v}}r_{\mathrm{ya}}+r_{\mathrm{g}}r_{\mathrm{ya}}+r_{\mathrm{v}}r_{\mathrm{g}}\right)i_{\mathrm{v}} }{ L_{\mathrm{v}}L_{\mathrm{g}} \left[ \frac{r_{\mathrm{v}}+r_{\mathrm{ya}}}{L_{\mathrm{v}}} + \frac{r_{\mathrm{g}}+r_{\mathrm{ya}}}{L_{\mathrm{g}}} - \frac{AB e_{\mathrm{vn}}}{L_{\mathrm{v}}\left(1+B^{2}i_{\mathrm{v}}^{2}\right)} +\Theta \right] }. \tag{9} \]

Assigning various values to \(\Theta\) from \(+\infty\) to \(0\) and to \(-\infty\), and varying the values of the exciter field current within real limits, we construct a family of isoclines in the phase plane derivative—current, and, by the method set forth in [7, 8], construct the integral curve. The initial conditions for constructing the integral curve are:

\[ t=0;\qquad \frac{d i_{\mathrm{v\,init.}}}{dt}=0;\qquad i_{\mathrm{v}}=i_{\mathrm{v\,init.}}, \]

where \(i_{\mathrm{v\,init.}}\) is the initial value of the exciter field current before the switch is closed.

After constructing the integral curve, using the method set forth in [7, 8], we construct the transient process in the current—time coordinate axes.

Let us carry out the above calculations for a specific system.

a) Transition from one established equilibrium state to another after closing switch \(P\).

Assigning the corresponding parameters of the system (Fig. 2a), we construct the isoclines and the integral curve in Fig. 4a. Using the method indicated in [7, 8], in Fig. 5a we construct the transient processes for the exciter field current in the current—time coordinate axes.

Fig. 4a.

b) Undamped periodic oscillations.

Increasing the active resistance of the exciter field-winding circuit and leaving the remaining parameters of the system unchanged, we obtain a regime of undamped periodic oscillations. The system parameters adopted for this case are indicated in Fig. 2b. Using the method indicated above, in Fig. 4b we construct a family of isoclines, by which we carry out the construction of the integral curve of the transient process for the exciter field current. In Fig. 5b we construct the transient processes in the current—time coordinate axes.

c) Aperiodic damped oscillations.

With a further increase in the active resistance of the exciter field-winding circuit, we can obtain a regime of aperiodic—

… of aperiodically damped oscillations. If the investigation of the above transient regimes is of considerable interest for practical purposes (since motors operating in these regimes may be used to actuate the working elements of the corresponding mechanisms), then the use of an electric motor operating in the regime of aperiodically damped oscillations is hardly expedient. We shall omit the exposition of the method for constructing the transient processes for this case (c), since it has a certain special feature.

Fig. 4 b.

In the laboratory of the Department of Electrification of Industrial Enterprises of the Tomsk Polytechnic Institute named after S. M. Kirov, we

Fig. 5 a.

the system under consideration is subjected when applied to the main drive of a longitudinal-planing machine [3, 9].

The construction of transient regimes for the generator field current, the exciter armature current, and the exciter voltage can be carried out by the method set forth above. However, using the already known dependences \(\frac{di_{\mathrm{в}}}{dt}(i_{\mathrm{в}})\) and \(i_{\mathrm{в}}(t)\) (Figs. 4, 5) and equation (5), it is possible, with less labor, to construct the transient regimes for the generator field current. Using equation (4) and the graphs \(i_{\mathrm{в}}(t)\) and \(i_2(t)\) obtained by us, one can, by simple geometrical summation, construct the graph of the transient regime for the exciter armature current. All calculations are shown in Fig. 5b.

Fig. 5b.

A comparison of the actual and the calculated curves of the transient processes in Fig. 2b and Fig. 5b makes it possible to assess positively the method we used, since the results obtained satisfactorily reflect both the qualitative and the quantitative picture of the transient regimes. For the exciter field current the error in amplitude is \(4.3\%\), and in period \(5.4\%\). For the generator field current, respectively, \(21.8\) and \(5.4\%\). For the armature current, \(16.0\) and \(5.4\%\).

The practical interest of the questions considered lies in the fact that the method presented makes it possible to calculate transient processes in electrical machines quite simply and with sufficient accuracy for practice, taking into account only the principal nonlinearity \(e_{\mathrm{в}}(i_{\mathrm{в}})\).

LITERATURE

1. Feldbaum A. A. Introduction to the Theory of Nonlinear Systems, GEI, 1948.
2. Shuleikin M. V. “On Electrical Oscillations,” Journal of the Russian Physico-Chemical Society, vol. XLII, physics section, issue 1, 1910.

1. Govorkov V. A. Works of Academician M. V. Shuleikin on electrical machines, “Elektrichestvo,” No. 5, 1952.

2. Gandzha L. I. and Kropanin S. S. On oscillations in the exciter of the “generator—motor” system, Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, vol. 82, 1956.

3. Ermolin N. P. Transient processes in direct-current machines. GEI, 1951.

4. Arkhangel’skii B. I. Analytical expression for the magnetization curve of electrical machines, “Elektrichestvo,” No. 3, 1950.

5. Andronov A. A. and Khaikin S. E. Theory of Oscillations, ONTI, part I, 1937.

6. Under the general editorship of Polivanov K. M. Physical foundations of electrical engineering. GEI, 1950.

7. Kropanin S. S. Author’s certificate for the invention “Electromascope,” No. 91805, 1951.