ENERGETICS OF THE TRANSIENT PROCESS WHEN THE FIELD WINDING OF AN IMPULSE GENERATOR IS SWITCHED ON TO A DC VOLTAGE

A. I. VERKHOTUROV, G. S. SIGAILOV

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

In impulse generators, especially under an intermittent excitation regime, the rotor windings operate in transient modes; in this case it is important that the rise of voltage from zero to a specified value be accompanied by the release of minimal losses in the rotor windings.

In the present article the problem is posed of determining the conditions under which the rise of voltage from zero to a specified value occurs with the release of minimal losses in the rotor windings.

To solve the problem posed, let us consider the energetics of the transient process occurring when the field winding of an impulse generator is switched on to the DC voltage \(U_2\); here, for simplification, we assume that the magnetic permeability of the steel is constant and that the coupling coefficient between the rotor circuits is equal to unity.

The transient process occurring in the rotor windings is described by the following system of differential equations:

\[ \left. \begin{aligned} i_2 r_2 + L_2 \frac{d i_2}{dt} + \sqrt{L_2 L_3}\,\frac{d i_3}{dt} &= U_2,\\ i_3 r_3 + L_3 \frac{d i_3}{dt} + \sqrt{L_2 L_3}\,\frac{d i_2}{dt} &= 0 \end{aligned} \right\}, \tag{1} \]

where

\(r_2, L_2\) are the parameters of the field winding, taking the exciter into account,

\(r_3, L_3\) are the parameters of the damper winding.

The expressions for the currents have the form:

\[ i_2 = \frac{U_2}{r_2}\left(1 - B e^{-\frac{t}{T}}\right), \tag{2} \]

\[ i_3 = -\,\frac{U_2}{r_2}(1 - B)e^{-\frac{t}{T}}, \tag{3} \]

where

\[ T = T_2 + T_3 \]

is the total time constant of the rotor circuits,

\[ B = \frac{T_2}{T} = \frac{1}{1 + \frac{T_3}{T_2}} \]

is a quantity depending on the ratio of the time constants of the damping winding and the excitation winding.

The flux linkage with the stator winding is proportional to the total current of the rotor windings and to the coefficient of mutual inductance between the rotor and the stator \(M_{12}\):

\[ \Psi = \frac{U_2}{r_2} M_{12}\left(1 - e^{-\frac{t}{T}}\right). \tag{4} \]

The power supplied to the excitation winding is

\[ P = u_2 i_2 = \frac{u_2^2}{r_2}\left(1 - B e^{-\frac{t}{T}}\right). \tag{5} \]

Part of this power is converted into electromagnetic power and goes to create the excitation field:

\[ P_M = u_m i = \frac{u_2^2}{r_2} B e^{-\frac{t}{T}}\left(1 - e^{-\frac{t}{T}}\right), \tag{6} \]

where

\[ u_m = \frac{d\Psi}{dt} = \frac{u_2 M_{12}}{r_2 T} e^{-\frac{t}{T}} \simeq u_2 B e^{-\frac{t}{T}}. \tag{7} \]

Investigation of equation (6) shows that \(P_M\) reaches a maximum at \(t = 0.693T\). The remaining part \((P - P_M)\) is released as heat in the excitation and damping windings:

\[ P_2 = i_2^2 r_2 = \frac{u_2^2}{r_2}\left(1 - B e^{-\frac{t}{T}}\right)^2 \tag{8} \]

\[ P_3 = i_3^2 r_3 = \frac{u_2^2}{r_2} B(1 - B)e^{-\frac{2t}{T}}. \tag{9} \]

The instantaneous value of the efficiency of converting the energy of the electric field into the energy of the magnetic field is

\[ \eta_t = \frac{P_M}{P} = \frac{B e^{-\frac{t}{T}}\left(1 - e^{-\frac{t}{T}}\right)} {1 - B e^{-\frac{t}{T}}}. \tag{10} \]

Let us examine the expression for \(\eta_M\) under the condition of a maximum:

\[ \frac{d\eta_M}{dt} = B e^{-\frac{2t}{T}} - 2e^{-\frac{t}{T}} + 1 = 0. \tag{11} \]

Solving equation (11), we obtain the time at which \(\eta_M\) reaches a maximum.

\[ t_{\mathrm{M}}=T\ln \frac{B}{1\pm \sqrt{1-B}} . \tag{12} \]

In the last expression the quantity \(B\) may vary from unity (a machine without a damper winding) to zero (the time constant of the damper winding is equal to infinity, \(r_3=0\)).

Since in our case time cannot be negative, we obtain one value for \(t_{\mathrm{M}}\) with the positive value of the root:

\[ t_{\mathrm{M}}=T\ln \frac{B}{1-\sqrt{1-B}} . \tag{13} \]

The results of calculations by (13) show that when \(\dfrac{T_3}{T_2}\) changes from zero to unity, the quantity \(t_{\mathrm{M}}\) changes from 0 to \(0.536T\), respectively.

To characterize the transient process as a whole, we introduce the concept of the energy coefficient of the conversion process

\[ \eta_{\mathrm{e}}= \frac{\displaystyle \int_0^t P_{\mathrm{M}}\cdot dt} {\displaystyle \int_0^t P\cdot dt} = \frac{0.5\cdot B\left(1-e^{-\frac{t}{T}}\right)^2} {t\cdot T^{-1}+B\left(e^{-\frac{t}{T}}-1\right)} . \tag{14} \]

The results of calculations by equation (14) are presented in Fig. 1,a. The curves of the dependence of \(\eta_{\mathrm{e}}\) on time have an extremal character. The value of the maximum efficiency can be found by setting the derivative of \(\eta_{\mathrm{e}}\) with respect to time equal to zero:

\[ \frac{d\eta_{\mathrm{e}}}{dt} = \ln\left(\frac{2t}{T}+Be^{-\frac{2t}{T}}+B+1\right) -\frac{t}{T} =0 . \tag{15} \]

The transcendental equation (15) can be solved approximately, with any desired degree of accuracy, with respect to time for known values of \(B\). The results of solving the equation are presented in Fig. 1. In Fig. 2a and b, for comparison, the dependences of the efficiency, power, voltage, and losses released in the rotor windings on time are presented for a machine without a damper winding, \(\dfrac{T_3}{T_2}=0\), and with a damper winding, \(\dfrac{T_3}{T_2}=1\).

Analyzing the curves of Fig. 2,a, the following conclusions can be drawn regarding the unforced course of the transient process when the field winding is connected to the voltage \(u_2\) in the absence of a damper winding, \(\dfrac{T_3}{T_2}=0\):

a) the power developed by the power source at the beginning of the transient process is significantly less than its rated power;

b) the efficiency of the process of converting electrical energy into magnetic-field energy is high only at the beginning of the increase of the magnetic flux;

c) to increase the efficiency of the process of converting the exciter’s electrical energy into magnetic-field energy, it is necessary to force the transient processes; moreover, the faster the transient process proceeds, the higher the conversion efficiency.

In the presence of a damper winding, the energetics of the course of the transient process is somewhat different (Fig. 2,b). Since the coupling coefficient between the circuits is equal to unity, the power of the source at the begin-

Fig. 1. a) Dependence of \(\eta_{\mathrm{e}}\) on time for various ratios \(\dfrac{T_{\mathrm{e}}}{T_2}\)

b) For determining the time at which the maximum values of \(\eta_{\mathrm{e}}\) are reached

At the beginning of the process it increases rapidly; however, it is spent on covering losses in the damper winding. The efficiency of the conversion of electrical energy into magnetic-field energy at the beginning of the process (at \(t=0\)) is equal to zero, then reaches a maximum, after which it decreases again. It should be noted that the more powerful the damper winding, the lower, for one and the same time, is the efficiency of conversion of electrical energy into magnetic-field energy (Fig. 2b).

Thus, in the presence of a damper winding and with an unforced course of the transient process into magnetic-field energy, pre-

Fig. 2. Characteristics of the course of the transient process in machines without a damping winding (a) and with a damping winding (b)

only 9 percent is produced; for

\[ \frac{T_3}{T_2}=1 \]

and up to 17 percent for

\[ \frac{T_3}{T_2}=0.2 \]

of the exciter’s electrical energy. If the transient process is forced, the efficiency can be increased from 15 percent at

\[ \frac{T_3}{T_2}=1 \]

to 39 percent at

\[ \frac{T_3}{T_2}=0.2. \]

Studies have shown that one of the most effective means of accelerating transient processes for generators with a damping winding is acceleration by increasing the excitation flux. With this method of accelerating transient processes, the system of differential equations for switching the excitation winding to a constant voltage will be analogous to (1), if \(ku_2\) is substituted for \(u_2\). The expressions for the currents take the form:

\[ i_2=\frac{k u_2}{r_2}\left(1-Be^{-\frac{t}{T}}\right), \tag{16} \]

\[ i_3=-\frac{k u_2}{r_2}(1-B)e^{-\frac{t}{T}}, \tag{17} \]

where \(k\) is the multiple of the voltage applied to the excitation winding in comparison with the rated voltage.

Multiplying the total current of the rotor windings by the coefficient of mutual inductance between the rotor and the stator \(M_{12}\), we obtain an expression for the total flux linkage

\[ \psi=\frac{k u_2}{r_2}M_{12}\left(1-e^{-\frac{t}{T}}\right). \tag{18} \]

The time required for the flux to rise to a specified value can be determined from the condition that the total current of the windings is equal to the value \(\frac{u_2}{r_2}\), whence

\[ t=T\ln\frac{k}{k-1}, \tag{19} \]

Expression (19) for \(K=1\) becomes infinite, i.e., without forcing of excitation the flux reaches the steady-state value in \(t=\infty\). In practical calculations it is assumed that the flux rises to the steady-state value in the time \(t=3T\); then expression (19) can be rewritten in the following form:

\[ t'=T\ln\frac{k}{k-0.95}. \tag{20} \]

To obtain the dependence of the energy efficiency factor as a function of the forcing coefficient, it is necessary in expression (14) to substitute, in place of \(t\), its value from (19); then, after transformation, we obtain

\[ \eta_{\mathrm{e}}= \frac{0.5}{k^2\ln\frac{k}{k-1}-Bk}. \tag{21} \]

Expression (21) is meaningful for all \(k\), except \(k=1\). To calculate \(\eta_{\mathrm{e}}\) for \(k=1\), it is necessary in expression (14), instead of \(t\), to substitute its value from (20). The results of calculations by (21) are presented in Fig. 3a. From the curves in Fig. 3a it is seen that, as the degree of forcing increases, \(\eta_{\mathrm{e}}\) first increases, reaches a maximum, and, with a further increase in \(k\), decreases. For one and the same \(k\), the energy efficiency factor is the higher, the smaller the ratio \(\frac{T_3}{T_2}\). The value of the forcing coefficient \(k\) at which \(\eta_{\mathrm{e}}\) reaches a maximum can be determined if

\[ \frac{d\eta_{\mathrm{e}}}{dk} \]

is set equal to zero:

\[ 2k\ln\frac{k}{k-1}-\frac{k}{k-1}-B=0. \tag{22} \]

The results of an approximate solution of the transcendental equation pred-

Fig. 3. a) Dependence of \(\eta_{\mathrm{э}}\) on the forcing coefficient

b) For determining the optimal values of the forcing coefficients from the condition of minimum losses in the rotor windings.

plotted in Fig. 3b for various ratios \(\dfrac{T_3}{T_2}\). The forcing coefficient at which \(\eta_\mathrm{e}\) reaches its maximum value will be denoted by \(\kappa_\mathrm{e}\).

Thus, in machines with a damping winding, the electrical energy of the exciter can be converted into the energy of the magnetic field with the least electrical losses in the rotor windings only at a definite forcing coefficient.

It can be shown that, when the magnetic flux rises from zero to a specified value, the losses in the rotor windings at the optimal forcing coefficient will indeed be smaller than at any other forcing coefficient. To this end, let us express the energy of losses in the rotor windings as a function of the forcing coefficient:

\[ Q_{\phi\kappa} = \int_0^t i_2^2 r_2\,dt + \int_0^t i_3^2 r_3\,dt = -\frac{u_2^2 T}{r_2} \left( \kappa^2 \ln \frac{\kappa-1}{\kappa} + B\kappa + 0.5B \right). \tag{23} \]

If in equation (23) we substitute \(t'\) from (20) in place of \(t\), then the expression for the energy of losses in the rotor windings as the flux rises from zero to the specified value takes the form

\[ Q'_{\phi\kappa} = -\frac{u_2^2 T}{r_2} \left( \kappa^2 \ln \frac{\kappa-0.95}{\kappa} + 0.95B\kappa + 0.45B \right). \tag{24} \]

Let us denote by \(\kappa_{\phi m}\) the ratio of the energy of losses in the rotor windings under forcing to the energy of losses in them at \(\kappa=1\); then

\[ \kappa_{\phi m} = \frac{ \kappa^2 \ln \dfrac{\kappa}{\kappa-0.95} + 0.95B\kappa + 0.45B }{ 1.4B-3 }. \tag{25} \]

The results of calculations according to (25) are presented in Fig. 4 for various ratios \(\dfrac{T_3}{T_2}\). As is seen from Fig. 4, the curves have an extremal character. To find the forcing coefficient at which the losses will be minimal, it is necessary to take the first derivative of \(Q_{\phi\kappa}\) with respect to \(\kappa\) and set it equal to zero:

\[ 2\kappa \ln \frac{\kappa}{\kappa-1} - \frac{\kappa}{\kappa-1} - B = 0. \tag{26} \]

This equation coincides exactly with equation (22), whence it follows that, at the optimal forcing coefficient, the losses in the rotor windings will be minimal.

Thus, when the voltage at the generator terminals rises from zero to the specified value, in machines with a damping winding it is possible to choose such a forcing coefficient that the conversion of the electrical energy of the exciter into the energy of the magnetic field of excitation of the machine will occur with the least electrical losses dissipated in the rotor windings. In this case the energy efficiency of the conversion of the exciter’s electrical energy into magnetic-field energy at the optimal forcing coefficient will be approximately twice as high as during the transient process at \(\kappa=1\).

If it is assumed that, when the voltage at the generator terminals rises from zero to the specified value at the optimal forcing coefficient, the same amount of losses is released in the rotor windings as at \(\kappa=1\), then when operating with the optimal coefficient

Fig. 4. Dependence of \(K_{\phi m}\) on the forcing coefficient

With forcing it is possible to excite the machine to a higher voltage than at \(k = 1\), and thereby significantly increase the surge power of the generator, which depends on the square of the voltage.