ARMATURE REACTION OF A SHOCK GENERATOR

A. V. LOOS, G. A. SIPAILOV, K. A. KHOR’KOV

Presented by the scientific seminar of the departments of Electrical Machines and General Electrical Engineering

A shock-power generator is a single-phase generator of the turbogenerator type operating in the regime of a sudden short circuit or of connection to a load whose resistance is commensurate with the generator’s subtransient resistance. In the rotor slots of a shock generator, along with the field winding, there is a powerful damper winding uniformly distributed over the entire circumference of the rotor.

The study of the armature reaction of a single-phase machine has its own peculiarities and difficulties in comparison with a three-phase machine; the mathematical treatment of the armature reaction of shock generators is especially complex.

A short circuit of the stator winding is made at the instant when the generator e.m.f. passes through zero; in this case the stator-winding current contains the maximum possible aperiodic component. The shock current of the generator creates a magnetomotive force that is stationary relative to the \(d\)-axis of the stator winding and varies with time. The armature-reaction flux induces electromotive forces and currents in the rotor windings that counteract the penetration of the armature-reaction flux into the circuits of the rotor windings; moreover, at the initial instant the values of these currents strictly correspond to the condition of conservation of the constancy of the flux linkages of the circuits. The armature-reaction flux is thereby forced into leakage paths and does not penetrate into the body of the rotor.

As a first approximation, the rotor windings may be regarded as superconducting circuits, and the armature-reaction flux may be neglected in calculations of the generator e.m.f. In this case the damper winding, located closer to the air gap of the machine and having the smaller leakage resistance, compensates the greater part of the armature-reaction flux, relieving the field winding of additional currents. However, such a treatment is permissible only for circuits whose active resistance is considerably smaller than the inductive leakage resistance.

In real cases, the presence of active resistances causes the decay of free currents and the gradual penetration of the reaction flux—

Fig. 1. Arrangement of the stator and rotor windings at the moment of short circuit.

the armature reaction into the rotor circuits. The armature reaction will demagnetize the machine, thereby reducing the e.m.f. of the generator stator winding.

The article considers questions concerning the influence of armature reaction on the e.m.f. of the generator during the first period of a sudden short circuit, as a function of the parameters of the shock generator. The study of armature reaction is carried out using the equations of the electromagnetic processes of the shock generator. The winding diagram of the generator is shown in Fig. 1. The principal feature of the investigation method is the use of an additional measuring winding, electromagnetically linked with the resultant flux of the machine. The measuring winding is located on the stator and has the same mutual-inductance coefficients with the rotor windings as the stator winding. The measur—

changes in the emf of the auxiliary winding reflect the change in the emf of the stator winding under the action of armature reaction.

The electromagnetic processes of the shock generator are described by the system of equations:

\[ \begin{aligned} i_c r_c+\frac{d\psi_c}{dt}&=0;\\ i_{\mathrm{B}} r_{\mathrm{B}}+\frac{d\psi_{\mathrm{B}}}{dt}&=U_{\mathrm{B}};\\ i_{\mathrm{D}d} r_{\mathrm{D}d}+\frac{d\psi_{\mathrm{D}d}}{dt}&=0;\\ i_{\mathrm{D}q} r_{\mathrm{D}q}+\frac{d\psi_{\mathrm{D}q}}{dt}&=0. \end{aligned} \tag{1} \]

The values of the flux linkages are:

\[ \begin{aligned} \psi_c&=i_c L_c+i_{\mathrm{B}}M\cos\omega t+i_{\mathrm{D}d}M\cos\omega t+i_{\mathrm{D}q}M\sin\omega t;\\ \psi_{\mathrm{B}}&=i_cM\cos\omega t+i_{\mathrm{B}}L_{\mathrm{B}}+i_{\mathrm{D}d}M;\\ \psi_{\mathrm{D}d}&=i_cM\cos\omega t+i_{\mathrm{B}}M+L_{\mathrm{D}d}i_{\mathrm{D}d};\\ \psi_{\mathrm{D}q}&=i_cM\sin\omega t+i_{\mathrm{D}q}L_{\mathrm{D}q}. \end{aligned} \tag{2} \]

The emf of the measuring winding is equal to:

\[ e=-\frac{d\psi}{dt}; \]

where the flux linkage \(\psi\), associated with the measuring winding, unlike \(\psi_c\), is determined by the expression:

\[ \psi=i_cM+Mi_{\mathrm{B}}\cos\omega t+Mi_{\mathrm{D}d}\cos\omega t+i_{\mathrm{D}q}M\sin\omega t. \tag{3} \]

Using the first equations of systems (1), (2) and equation (3), one can express the emf of the auxiliary winding through \(i_c\):

\[ e=i_c r_{\mathrm{B}}+L_{cs}\frac{di_c}{dt}; \tag{4} \]

where \(L_{cs}=L_c-M\) is the leakage inductance of the stator winding.

The system of equations (1) has no solution in general form when active resistances are taken into account. However, methods exist for an approximate solution of this system. We apply the method of successive approximations, representing the unknown currents in the form of an expansion in a series in the small parameter [1]. For this it is necessary to transform system (1), dividing each of the equations respectively by \(L_c\), \(L_{\mathrm{B}}\), \(L_{\mathrm{D}d}\), \(L_{\mathrm{D}q}\) and multiplying by \(T\), where \(T\) is the period of the emf.

\[ \begin{aligned} \frac{T\,dI_c}{dt}+\alpha i_c&=0;\\ \frac{T\,dI_{\mathrm{B}}}{dt}+\gamma_{\mathrm{B}}\alpha i_{\mathrm{B}}&=\frac{T}{L_{\mathrm{B}}}U_{\mathrm{B}};\\ \frac{T\,dI_{\mathrm{D}d}}{dt}+\gamma_{\mathrm{D}d}\alpha i_{\mathrm{D}d}&=0;\\ \frac{T\,dI_{\mathrm{D}q}}{dt}+\gamma_{\mathrm{D}q}\alpha i_{\mathrm{D}q}&=0. \end{aligned} \tag{5} \]

Here

\[ \begin{aligned} I_c &= i_c+i_{\mathrm{B}}\,\frac{M}{L_c}\cos\omega t +i_{\mathrm{Dd}}\,\frac{M}{L_c}\cos\omega t +i_{\mathrm{Dq}}\,\frac{M}{L_c}\sin\omega t;\\ I_{\mathrm{B}} &= i_c\,\frac{M}{L_{\mathrm{B}}}\cos\omega t +i_{\mathrm{B}}+i_{\mathrm{Dd}}\,\frac{M}{L_{\mathrm{B}}};\\ I_{\mathrm{Dd}} &= i_c\,\frac{M}{L_{\mathrm{Dd}}}\cos\omega t +i_{\mathrm{B}}\,\frac{M}{L_{\mathrm{Dd}}}+i_{\mathrm{Dd}};\\ I_{\mathrm{Dq}} &= i_c\,\frac{M}{L_{\mathrm{Dq}}}\sin\omega t+i_{\mathrm{Dq}}; \end{aligned} \tag{6} \]

\[ \alpha=\frac{T r_c}{L_c} \]
is a small parameter, since
\[ \frac{L_c}{r_c}=T_c\gg T; \]

\[ \gamma_{\mathrm{B}}=\frac{T_c}{T_{\mathrm{B}}},\quad \gamma_{\mathrm{Dd}}=\frac{T_c}{T_{\mathrm{Dd}}},\quad \gamma_{\mathrm{Dq}}=\frac{T_c}{T_{\mathrm{Dq}}},\quad T_{\mathrm{B}}=\frac{L_{\mathrm{B}}}{r_{\mathrm{B}}},\quad T_{\mathrm{Dd}}=\frac{L_{\mathrm{Dd}}}{r_{\mathrm{Dd}}}, \]

\[ T_{\mathrm{Dq}}=\frac{L_{\mathrm{Dq}}}{r_{\mathrm{Dq}}}. \]

Taking into account that in equations (5) \(\alpha\) is a small parameter, the solution for the currents can be found in the form of an expansion in a series [1]:

\[ \begin{aligned} i_c &= i_{0c}+\alpha i_{1c}+\alpha^2 i_{2c}+\ldots;\\ i_{\mathrm{B}} &= i_{0\mathrm{B}}+\alpha i_{1\mathrm{B}}+\alpha^2 i_{2\mathrm{B}}+\ldots;\\ i_{\mathrm{Dd}} &= i_{0\mathrm{Dd}}+\alpha i_{1\mathrm{Dd}}+\alpha^2 i_{2\mathrm{Dd}}+\ldots;\\ i_{\mathrm{Dq}} &= i_{0\mathrm{Dq}}+\alpha i_{1\mathrm{Dq}}+\alpha^2 i_{2\mathrm{Dq}}+\ldots; \end{aligned} \tag{7} \]

When solving system (1), it is sufficient to restrict ourselves to the first two terms of expressions (7), since \(\alpha^2\ll\alpha\), and each subsequent term gives only a small correction to the principal solution. Substituting expressions (7) into (6), we obtain a system of equations for \(I_c\), \(I_{\mathrm{B}}\), \(I_{\mathrm{Dd}}\), and \(I_{\mathrm{Dq}}\); moreover, taking into account their identical form, we shall write only the equation for \(I_c\):

\[ \begin{aligned} I_c={}&\left(i_{0c}+i_{0\mathrm{B}}\frac{M}{L_c}\cos\omega t +i_{0\mathrm{Dd}}\frac{M}{L_c}\cos\omega t +i_{0\mathrm{Dq}}\frac{M}{L_c}\sin\omega t\right)+\\ &+\alpha\left(i_{1c}+i_{1\mathrm{B}}\frac{M}{L_c}\cos\omega t +i_{1\mathrm{Dd}}\frac{M}{L_c}\cos\omega t +i_{1\mathrm{Dq}}\frac{M}{L_c}\sin\omega t\right)+\ldots\\ ={}&I_{0c}+\alpha I_{1c}+\ldots \end{aligned} \tag{8} \]

Substituting the equations of system (8) into (5), we obtain differential equations for each circuit:

\[ T\frac{dI_{0c}}{dt}+\alpha T\frac{dI_{1c}}{dt}+\alpha i_{0c}+\ldots=0. \tag{9} \]

In this case, for the field winding the right-hand side will be equal to

\[ \frac{T}{L_{\mathrm{B}}}\,U_{\mathrm{B}}. \]

To solve the system consisting of equations of the form (9), we apply the method of successive approximations, neglecting in the zeroth approximation all terms containing \(a\); then

\[ \frac{T\,d I_{0C}}{dt}=0,\qquad \frac{T\,d I_{0B}}{dt}=\frac{T U_B}{L_B}, \]

\[ \frac{T\,d I_{0Dd}}{dt}=0,\qquad \frac{T\,d I_{0Dq}}{dt}=0. \tag{10} \]

Whence

\[ i_{0C}=C_{0C},\qquad I_{0B}=C_{0B}+\frac{U_B}{L_B}\,t, \]

\[ I_{0Dd}=C_{0Dd},\qquad I_{0Dq}=C_{0Dq}. \tag{11} \]

Here \(C_{0C}, C_{0B}, C_{0Dd}, C_{0Dq}\) are constants of integration determined from the initial conditions. For \(t=0\), \(i_C=0,\ i_B=i_{B0},\ i_{Dd}=0,\ i_{Dq}=0\), hence

\[ C_{0C}=i_{B0}\frac{M}{L_C};\qquad C_{0B}=i_{B0};\qquad C_{0Dd}=i_{B0}\frac{M}{L_{Dd}};\qquad C_{0Dq}=0. \]

Solving system (11), we find the currents \(i_{0C}, i_{0B}, i_{0Dd}, i_{0Dq}\).

Next, for the first approximation we take into account the terms containing \(a\).

\[ \frac{T\,d I_{1C}}{dt}+i_{0C}=0,\qquad \frac{T\,d I_{1B}}{dt}+\gamma_B i_{0B}=0, \]

\[ \frac{T d I_{1Dd}}{dt}+\gamma_{Dd} i_{0Dd}=0,\qquad \frac{T d I_{1Dq}}{dt}+\gamma_{Dq} i_{0Dq}=0. \tag{12} \]

Whence

\[ I_{1C}=-\frac{1}{T}\int_{0}^{t} i_{0C}\,dt, \]

\[ I_{1B}=-\frac{1}{T}\gamma_B\int_{0}^{t} i_{0B}\,dt, \]

\[ I_{1Dd}=-\frac{1}{T}\gamma_{Dd}\int_{0}^{t} i_{0Dd}\,dt, \tag{13} \]

\[ I_{1Dq}=-\frac{1}{T}\gamma_{Dq}\int_{0}^{t} i_{0Dq}\,dt. \]

The solution for the currents in the windings in the zeroth approximation has the form:

\[ i_{0C}=\frac{1}{A-B\cos^2\omega t}\left[ i_{\mathrm{BO}}C\frac{M}{L_C}(1-\cos\omega t) -i_{\mathrm{BO}}\frac{M}{L_C}\frac{t}{T_B}D\cos\omega t \right], \]

\[ i_{0B}=\frac{1}{A-B\cos^2\omega t}\left[ i_{\mathrm{BO}}C -i_{\mathrm{BO}}\frac{M^2C}{L_C L_{Dq}}\sin^2\omega t -i_{\mathrm{BO}}\frac{M^2D}{L_C L_{Dd}}\cos^2\omega t -\right. \]

\[ \left. -i_{\mathrm{BO}}\frac{M^2D}{L_C L_B}\cos\omega t -i_{\mathrm{BO}}\frac{t}{T_B}\frac{M^2}{L_C L_{Dq}} \right], \]

\[ i_{0Dd}=\frac{1}{A-B\cos^2\omega t}\left[ -i_{\mathrm{BO}}\frac{M^2F}{L_C L_{Dd}}\cos\omega t +i_{\mathrm{BO}}\frac{M^2D}{L_C L_{Dd}}\cos^2\omega t +\right. \]

\[ \left. +i_{\mathrm{BO}}\frac{t}{T_B}\frac{M^3}{L_C L_B L_{Dq}}\sin\omega t -i_{\mathrm{BO}}\frac{t}{T_B}\frac{M}{L_{Dd}} \right], \]

\[ i_{0Dq}=\frac{1}{A-B\cos^2\omega t}\left[ i_{\mathrm{BO}}\frac{M^2}{L_C L_{Dd}}\sin\omega t\cos\omega t \left(C+\frac{t}{T_B}\right) -\right. \]

\[ \left. -i_{\mathrm{BO}}\frac{M^2C}{L_C L_{Dd}}\sin\omega t \right]. \tag{14} \]

where

\[ A=\left(1-\frac{M^2}{L_C L_{Dd}}\right) \left(1-\frac{M^2}{L_B L_{Dd}}\right); \]

\[ B=\frac{M^2}{L_C L_B}\left(1-\frac{M^2}{L_{Dd}}\right)^2,\qquad C=1-\frac{M^2}{L_C L_B}, \]

\[ D=1-\frac{M}{L_{Dd}};\qquad F=1-\frac{M}{L_B}. \tag{15} \]

Solving equations (13), we obtain:

\[ i_{1C}=\frac{1}{A-B\cos^2\omega t}\left[ C\left(\frac{M}{L_C}\frac{T_C}{T_{Dq}}\frac{1}{T}\sin\omega t \int_0^t i_{0Dq}\,dt -\frac{1}{T}\int_0^t i_{0C}\,dt\right)+ \right. \]

\[ \left. +\frac{M}{L_C}\cos\omega t\left( D\frac{T_C}{T_B}\frac{1}{T}\int_0^t i_{0B}\,dt +F\frac{T_C}{T_{Dq}}\frac{1}{T}\int_0^t i_{0Dd}\,dt \right)\right]; \tag{16} \]

similarly for \(i_{1B}\), \(i_{1Dd}\), and \(i_{1Dq}\).

Using known methods of expansion into a series of functions of the form

\[ \frac{f(\omega t)} {(x''_d+x''_q)-(x''_d-x''_q)\cos 2\omega t}, \qquad \text{where } f(\omega t)=\cos\omega t;\ \sin\omega t;\ 1, \]

equations (14) and (16) can be brought to a form convenient for integration and, for \(x''_d=x''_q\), substantially simplified. For example, the first equation of system (14) can be reduced to the form:

\[ i_{oc}=\frac{E_0}{\sqrt{x''_d x''_q}}\left(1+2b\cos 2\omega t+2b^2\cos 4\omega t+\ldots\right)- \]

\[ -\frac{2E_0}{x''_d+\sqrt{x''_d x''_q}}\left(\cos \omega t+b\cos 3\omega t+b^2\cos 5\omega t+\ldots\right), \]

\[ \text{where } b=\frac{\sqrt{x''_q}-\sqrt{x''_d}}{\sqrt{x''_q}+\sqrt{x''_d}}; \]

\[ x''_d=(L_c-M)+\frac{1}{\dfrac{1}{M}+\dfrac{1}{L_c-M}+\dfrac{1}{L_{Dd}-M}}. \]

For \(x''_d=x''_q\):

\[ b=0 \text{ and } i_{oc}=\frac{E_0}{x''_d}(1-\cos \omega t). \]

Total stator current with allowance for active resistances

\[ i_c=i_{oc}+a i_{1c}. \tag{17} \]

The generator emf, taking into account the influence of armature reaction on the magnitude of the flux in the machine air gap, is determined from expression (4) by substituting the value of the stator current with allowance for active resistances (17).

Using the method set forth above, a calculation was made of the emf of the stator winding with allowance for armature reaction and for the active resistances of the windings. The calculation was performed for the following winding parameters: \(L_c=1.05\); \(L_B=1.1\); \(L_{Dd}=L_{Dq}=1.03\); \(M=1\). The results of the calculation are presented in the form of curves in Fig. 2 for various values of the active resistances of the stator and rotor windings.

As follows from the curves of Fig. 2, the reduction of the emf due to armature reaction may be neglected only for the first half-wave of the emf, provided that \(r_D\), in relative units, has a value of the order of 0.005. In calculating the emf of the second and subsequent half-periods, neglecting armature reaction leads to substantial errors.

Studies of the influence of the active resistance of the damper winding on the magnitude of the emf were also carried out on the IPT-5 analog computer according to the method set forth in [2] for the winding parameters given above, with \(r_D\) varied within the limits \(0.001 \div 0.05\). The results of the studies are presented in Table 1. In addition, for two values of the damper-winding resistance, \(r_D=0.001\) and \(r_D=0.02\), Fig. 3 presents oscillograms of the emf and currents.

Fig. 2. Calculated curves of the stator surge current and the e.m.f. of the measuring winding:

1. \(r_C = 0.001;\quad r_B = 0.001;\quad r_{Dd}=r_{Dq}=0.001.\)

2. \(r_C = 0.005;\quad r_B = 0.001;\quad r_{Dd}=r_{Dq}=0.01.\)

Fig. 3. Oscillograms of the stator shock current and the emf of the measuring winding.

a. \(r_C = 0.001;\quad r_{\mathrm{B}} = 0.001;\quad r_{Dd} = r_{Dq} = 0.001.\)

b. \(r_C = 0.001;\quad r_{\mathrm{B}} = 0.001;\quad r_{Dd} = r_{Dq} = 0.02.\)

Table 1

Using Table 1, for a known value of $r_D$ one can estimate the influence of armature reaction on the emf amplitude of a shock generator without carrying out complex mathematical calculations requiring large expenditures of time.

REFERENCES

1. N. N. Bogolyubov, Asymptotic Methods in the Theory of Nonlinear Oscillations, Fizmatgiz, 1952.

2. A. V. Loos, G. A. Sipailov. Mathematical modeling of the electromagnetic processes of a shock generator. Izvestiya TPI, vol. 152, 1965.