MATHEMATICAL MODELING OF THE OPERATION OF AN IMPACT GENERATOR TAKING INTO ACCOUNT CHANGES IN ROTATIONAL SPEED AND SATURATION OF THE MAGNETIC CIRCUIT

A. V. Loos, G. A. Sipailov

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

During the operation of an impact generator—when it is short-circuited or connected to an inductive load at the moment when the e.m.f. passes through zero—during the first or part of the first half-period of the e.m.f., the kinetic energy of the rotor is converted into the electromagnetic energy of the stator and the load, owing to which, by the moment of maximum current, the rotor speed decreases by 5–10 percent. During the second or part of the second half-period of the e.m.f., as the current decreases to zero, the electromagnetic energy of the stator and the load is again converted into the kinetic energy of the rotating rotor, causing a corresponding change in rotational speed. Therefore, a complete analysis of the operation of an impact generator in the general case must be carried out by solving a system of differential equations for the machine, including the voltage-balance equations of the generator and load circuits and the equation of motion of the rotor.

The system of differential equations of the electromechanical transient process of an impact generator is nonlinear. The variable coefficients in the voltage equations of the machine circuits are complex functions of time, whose character of variation is determined from the solution of the torque equation. The need to consider the mechanical and electromagnetic processes jointly makes their investigation rather laborious.

As experience shows, variable saturation of the magnetic circuit has a great influence on processes during sudden short circuits of synchronous machines [1].

Taking into account each of the indicated factors—the change in generator rotational speed and saturation—requires going beyond the framework of the linear theory of differential equations. Modern mathematics makes it possible to consider transient processes with these factors taken into account. However, the complexity of the original system of equations, the need to account for various nonlinearities, the large research program, etc., make this problem rather laborious. Therefore, it is most expedient to carry out all investigations on an analog computing machine.

In the theory of synchronous machines, the method of linear transformations of equations is widely used; it consists in replacing the original variables in the equations by new variables linearly related to the original ones. After the introduction of the new variables, the transformed equations are then solved, and the new variables become the unknowns. With a successful choice of coef-

coefficients of linear transformations can substantially facilitate the solution and investigation of differential equations. However, the use of linear transformations does not make it possible to eliminate variable coefficients in the differential equations of a single-phase synchronous machine [2, 3].

A single-phase synchronous machine having a powerful damper system, when operating in steady state, may be regarded as a three-phase machine with a negative-sequence inductive reactance \(x_2\) connected in front of it. Such a representation of a single-phase synchronous machine will also be valid for the study of transient processes, since at slips of the asynchronous field close to two, the values \(x_d(j2) \approx x_d''\) and \(x_q(j2) \approx x_q''\) vary very little [1, 2, 3, 4]. This representation of a single-phase synchronous machine makes it possible to use the known form of the equations of a synchronous machine in the \(d\) and \(q\) axes rigidly connected with the rotor, which to a considerable extent simplifies the form of the original system of equations:

\[ \begin{gathered} - p\psi_d - (1+S)\psi_q - (r+r_{\mathrm n})i_d = 0,\\ (1+S)\psi_d - p\psi_q - (r+r_{\mathrm n})i_q = 0,\\ p\psi_{\mathrm B} + r_{\mathrm B} i_{\mathrm B} = U_{\mathrm B},\\ p\psi_{Dd} + r_{Dd} i_{Dd} = 0,\\ p\psi_{Dq} + r_{Dq} i_{Dq} = 0,\\ T_I pS + \psi_d i_q - \psi_q i_d = m_T . \end{gathered} \tag{1} \]

The flux linkages here are equal to:

\[ \begin{gathered} \psi_d = (x_a + 0.5x_{s1} + 0.5x_2 + x_{\mathrm n}) i_d + x_a i_{\mathrm B} + x_a i_{Dd},\\ \psi_q = (x_a + 0.5x_{s1} + 0.5x_2 + x_{\mathrm n}) i_q + x_a i_{Dq},\\ \psi_{\mathrm B} = x_a i_d + x_{\mathrm B} i_{\mathrm B} + x_a i_{Dd},\\ \psi_{Dd} = x_a i_d + x_{Dd} i_{Dd} + x_a i_{\mathrm B},\\ \psi_{Dq} = x_a i_q + x_{Dq} i_{Dq}, \end{gathered} \tag{2} \]

where \(x_a\) is the inductive reactance of the armature reaction;

\(x_{\mathrm B}, x_{Dd}, x_{Dq}\) are the inductive reactances of the field winding and the damper circuits;

\(T_I\) is the inertial constant of the rotor;

\(S\) is the slip of the machine rotor relative to the synchronous axis rotating with constant angular velocity \(\omega_c\);

\(i_d, i_q\) are the direct and quadrature components of the stator current;

\(i_{\mathrm B}, i_{Dd}, i_{Dq}\) are the currents in the field winding and in the damper circuits along the direct and quadrature axes;

\(r_{\mathrm n}, r, r_{\mathrm B}, r_{Dd}, r_{Dq}\) are the active resistances of the load, stator, field winding, and damper circuits along the direct and quadrature axes;

\(m_T\) is the torque of the prime mover;

\(x_{s1}\) is the leakage inductive reactance of the stator winding of the single-phase impact generator;

\(x_2\) is the negative-sequence inductive reactance of the single-phase impact generator.

In the above equations, all quantities of the rotor circuits are referred to the stator. The equations are written in the system of relative units, whose base quantities are chosen so that the form of writing in relative units coincides with the form of writing in

Fig. 1. Block diagram of the solution on an analog computer.

of the named quantities, and the equations had physical clarity [2].

In the system of equations (1) and (2), all generator parameters are assumed to be independent of saturation. To take saturation into account it is necessary to account for the dependence of the parameters on the currents in the windings. During short circuits of synchronous machines, local saturation arises, caused mainly by fluxes around the slots. In this case it may be assumed that only the leakage reactances of the stator change [5, 6, 7]. Such an assumption will be especially valid for impact generators, in which the equivalent inductive reactance in the supertransient regime, taking into account the shielding influence of the circuits of the damper system and the field winding, is determined mainly by the stator leakage reactance \(x_{\nu p}=1.15x_{s1}\)

[8]. Therefore we shall restrict ourselves to taking account of the influence of saturation only along the stator leakage paths. For this purpose, the dependence of the leakage reactance of the stator circuit on the current must be known: \((x_{s1}+x_2)0.5+x_n=f(i)\), or the leakage flux linkage on the current, \(\psi_{s1}=f(i)\).

For convenience in allowing for saturation along the leakage paths, it is advisable to divide the flux linkages of the stator winding \(\psi_d\) and \(\psi_q\) into the flux linkages along the path of the main flux and along the leakage paths:

\[ \psi_d=\psi_{бd}+\psi_{sd}=x_a(i_d+i_{\text{в}}+i_{Dq})+i_d f(i), \]

\[ \psi_q=\psi_{бq}+\psi_{sq}=x_a(i_q+i_{Dq})+i_q f(i). \tag{3} \]

From expression (3) one can find the currents \(i_d\) and \(i_q\):

\[ i_d=\frac{1}{f(i)}(\psi_d-\psi_{бd}), \]

\[ i_q=\frac{1}{f(i)}(\psi_q-\psi_{бq}). \tag{4} \]

The total stator current is determined from the known \(i_d\) and \(i_q\):

\[ i=\sqrt{i_d^2+i_q^2} \tag{5} \]

Thus, the complete system of differential equations of the impact generator, compiled with allowance for saturation and variation of the rotational speed, is determined by expressions (1), where the values of the flux linkages \(\psi_d\) and \(\psi_q\) are found from expressions (3), (4), and (5).

Fig. 2. Oscillogram of the solution with allowance for saturation and variation of rotational speed.

In the part concerning the modeling of the equations of a synchronous machine of the form (1) and (2), insurmountable difficulties usually do not arise. More difficult is taking account of the various nonlinearities and setting up the resulting block diagram, since the total amount of equipment required for the solution is quite large [5]. Reduced to a form convenient for drawing up the block diagram of the solution, the initial system of equations takes the form:

\[ \begin{gathered} p\psi_d=-(1+S)\psi_q-(r+r_n)i_d,\\ p\psi_q=(1+S)\psi_d-(r+r_n)i_q,\\ p\psi_v=U_v-i_v r_v,\\ p\psi_{Dd}=-r_{Dd}i_{Dd},\\ p\psi_{Dq}=-r_{Dq}i_{Dq},\\ pS=-\frac{1}{T_J}\psi_d i_q+\frac{1}{T_J}\psi_q i_d+\frac{m_T}{T_J}. \end{gathered} \tag{6} \]

Assuming that from equations (6) one can find the flux linkages \(\psi_d,\ \psi_q,\ \psi_v,\ \psi_{Dd}\) and \(\psi_{Dq}\), the values of the currents \(i,\ i_d,\ i_q,\ i_v,\ i_{Dd},\ i_{Dq}\), taking into account expressions (2), (3), (4), (5), will be equal to:

\[ \begin{gathered} i_d=\frac{1}{f(i)}(\psi_d-\psi_{bd}),\\[4pt] i_q=-\frac{1}{f(i)}(\psi_q-\psi_{bq}),\\[4pt] i=\sqrt{i_d^2+i_q^2},\\[4pt] i_v=\frac{1}{x_v}\psi_v-\frac{x_a}{x_v}i_d-\frac{x_a}{x_v}i_{Dd},\\[4pt] i_{Dd}=\frac{1}{x_{Dd}}\psi_{Dd}-\frac{x_a}{x_{Dd}}i_d-\frac{x_a}{x_{Dd}}i_v,\\[4pt] i_{Dq}=\frac{1}{x_{Dq}}\psi_{Dq}-\frac{x_a}{x_{Dq}}i_q. \end{gathered} \tag{7} \]

The structural solution diagram compiled from equations (6) and (7) is presented in Fig. 1. The flux linkages \(\psi_d,\ \psi_q,\ \psi_v,\ \psi_{Dd},\ \psi_{Dq}\) are obtained at the output of operational amplifiers 1, 2, 3, 4, 5 by integrating the right-hand side of equations (6). The currents of the rotor circuits \(i_v,\ i_{Dd},\ i_{Dq}\) are determined at the output of the corresponding summing amplifiers. The equation of motion of the rotor is modeled by means of two multiplication blocks, MBU, and integrating amplifier 12, at whose output the slip \(S\) is obtained.

The flux linkages \(\psi_{sd}\) and \(\psi_{sq}\) are obtained at the output of the corresponding summing amplifiers according to the expressions:

\[ \begin{gathered} \psi_{sd}=\psi_d-\psi_{bd},\\ \psi_{sq}=\psi_q-\psi_{bq}. \end{gathered} \tag{8} \]

From the obtained flux linkages \(\psi_{sd}\) and \(\psi_{sq}\), the square of the total flux linkage is found: \(\psi_{s1}^2=\psi_{sd}^2+\psi_{sq}^2\). On the nonlinearity block, the dependence

\[ \frac{1}{(x_{s1}+x_2)\,0.5+x_n}=F(\psi_{s1}^2) \]

is set. The obtained value

\[ \frac{1}{(x_{s1}+x_2)\,0.5+x_n} \]

is multiplied in two multiplication blocks by \(\psi_{sd}\) and \(\psi_{sq}\). At the output of the multiplication blocks, the currents \(i_d\) and \(i_q\) are obtained, which are then used in the solution circuit according to the original equations.

The developed structural diagram makes it possible to solve the original system of equations also without taking saturation into account. To do this, it is necessary to disconnect the nonlinearity block from the solution circuit, and to feed the flux linkages \(\psi_{sd}\) and \(\psi_{sq}\), obtained at the output of the summing amplifiers, through a transfer coefficient proportional to

\[ \frac{1}{(x_{s1}+x_2)\,0.5+x_n} \]

(which will correspond to the values \(i_d\) and \(i_q\)) to blocks 1, 2, 13, 14, 15, 16, 17.

To solve the original system without taking into account the change in speed, it is necessary to disconnect from the solution circuit the blocks that model the equation of motion.

The choice of scales for representing the variables and of the transmission coefficients of the operational amplifiers is well known and, in the present case, presents no particular difficulties.

Using the developed structural diagram of the operation of an impact generator on an inductive load, investigations were carried out of the influence of saturation and of the change in rotational speed on the stator current. Figure 2 gives an oscillogram of the solution, taking into account saturation and the change in speed, for a short circuit at the terminals of a generator having the following parameter values: \(x_a = 1.0\); \(x_{s1} = 0.05\); \(x_{Dd} = x_{Dq} = 1.03\); \(x_B = 1.1\); \(r_c = 0.005\); \(r_B = 0.02\); \(r_{Dd} = r_{Dq} = 0.01\); \(T_I = 6\) sec.

As a result of the solution, projections of the stator current on the \(d\)- and \(q\)-axes, \(i_d\) and \(i_q\), were obtained, as well as the change in rotor slip \(s\). Using the formulas of linear transformations, it is not difficult, from the known \(i_d\) and \(i_q\), to find the actual value of the stator current.

REFERENCES

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