SUBTRANSIENT INDUCTIVE REACTANCE OF A SHOCK GENERATOR WITH A NONMAGNETIC ACTIVE LAYER

G. A. Sipailov, V. F. Kulakov

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

One of the main tasks in the design of shock generators is to ensure as small as possible a subtransient inductive reactance, which determines the magnitude of the shock current. The subtransient inductance consists of the leakage inductance of the stator winding and an additional inductance caused by the rotor contours. To reduce the latter, the rotor is provided with a powerful damping winding of copper, uniformly distributed in slots near the surface. The leakage inductance of the stator winding constitutes the main part of the subtransient reactance and includes the leakage inductances of the slot and end portions of the winding, as well as differential leakage, determined by the harmonic composition of the field. Slot and end leakage are usually reduced by decreasing the ratio of the slot height to its width and by shielding the end portions of the winding with caps made of a material with good electrical conductivity.

A further reduction of the subtransient reactance can be achieved by moving the stator winding into the air gap. Owing to the absence of teeth on the stator, it becomes possible to increase the induction in the air gap and thereby substantially increase the shock power. A similar winding design is also being designed for high-power turbogenerators [1].

Therefore, the problem of determining the subtransient reactance in machines with such a design of the active zone is of considerable importance and interest. It is convenient to solve it with the aid of the vector-potential function.

In Fig. 1, a, the arrangement of the winding phase over the pole pitch $\tau$ is shown schematically. In the presence of transposition, the current density $\Delta$, distributed uniformly over the winding height $b$, may be considered uniform; then the vector potential $A_{\mathrm{II}}$ in this region is found from Poisson’s equation

$$ \frac{\partial^2 A_{\mathrm{II}}}{\partial x^2}+\frac{\partial^2 A_{\mathrm{II}}}{\partial y^2}=-\mu_0\Delta, \tag{1} $$

where the current density

$$ \Delta=\frac{4}{\pi}\Delta_0 \sum \frac{1}{n}\cos ma \sin mx\, e^{j\omega t} \tag{2} $$

is represented in the form of a series.

Fig. 1

For the regions without current I and III

\[ \frac{\partial^2 A}{\partial x^2}+\frac{\partial^2 A}{\partial y^2}=0. \tag{3} \]

The surface of the rotor, which in the general case is a composition of the steel of the forging and the copper of the damper system, may be taken as homogeneous with a certain average electrical characteristic, called the penetration depth

\[ d=\sqrt{\frac{2}{\omega \gamma \mu}}, \tag{4} \]

where \(\gamma\) and \(\mu\) are the electrical conductivity and magnetic permeability of the material, and \(\omega\) is the angular frequency.

Calculations show that in the transverse magnetic field of armature reaction, saturated steel has a penetration depth very close to the penetration depth of copper; in particular, at \(f=50\ \text{Hz}\), \(d\approx 1\ \text{cm}\).

It can also be shown that, in real designs, the time constant of the equivalent damper circuit

\[ T_{\mathrm{e}}=\frac{\tau}{2\pi}\,\mu\gamma d \tag{5} \]

is several times greater than the period of the emf, and therefore the calculation may be carried out as for a quasi-stationary process [2]. Then, writing for the rotor region IV the Poisson equation for the eddy-current field

\[ \frac{\partial^2 A_{\mathrm{IV}}}{\partial x^2}+\frac{\partial^2 A_{\mathrm{IV}}}{\partial y^2}=jk^2 A_{\mathrm{IV}}, \tag{6} \]

we find the expressions for the vector potentials in regions I—IV in the form

\[ A_{\mathrm{I}}=\mu_r\mu_0\sum B e^{my}\sin mx, \tag{7} \]

where \(y\) lies within the limits from \(0\) to \(-\infty\);

\[ A_{\mathrm{II}}=\mu_0\sum (De^{my}+Ee^{-my}+I)\sin mx, \tag{8} \]

where \(b>y>0\);

\[ A_{\mathrm{III}}=\mu_0 \sum \left(F e^{my}+Qe^{-my}\right)\sin mx, \tag{9} \]

where \(\delta>y>b\);

\[ A_{\mathrm{IV}}=\mu_0 \sum L e^{-\sqrt{m^2+jk^2}\,y}\sin mx \tag{10} \]

for \(\infty>y>\delta\).

Here, for simplicity, the following notation has been adopted:

\[ m=\frac{\pi}{\tau}n. \]

\[ I=\frac{4A_0}{\pi n m^2}\cos ma = \frac{4}{3m^2}\frac{iqk_w}{(\tau-2a)b}, \tag{11} \]

\[ k=\sqrt{\omega\gamma\mu}, \]

\[ k_w=\frac{3\cos ma}{\pi n}, \]

where \(\tau\) is the pole pitch,
\(n\) is any integer,
\(i\) is the armature current,
\(q\) is the number of turns per pair of poles,
\(p\) is the number of pairs of poles.

Equating the tangential components of the field intensity \(\dfrac{1}{\mu}\dfrac{\partial A}{\partial y}\) and the normal components of the induction \(-\dfrac{\partial A}{\partial x}\) at the boundaries of the regions, it is not difficult to obtain a system of equations, which, after equating the coefficients of \(\sin mx\) and \(\cos mx\), we write in matrix form:

where

\[ \lambda= \frac{\sqrt{m^2+jk^2}}{m}\, e^{-\sqrt{m^2+jk^2}\,\delta}, \]

\[ \eta=e^{-\sqrt{m^2+jk^2}\,\delta}. \]

Since \(\mu_r\) in the stator iron is very large in comparison with unity, the coefficients in the expressions for the vector potentials will be

\[ B=0; \qquad D=E=\frac{I}{2}\dot R, \]

\[ F=\frac{I}{2}\left(R+e^{-mb}\right), \qquad Q=\frac{I}{2}\left(\dot R+e^{mb}\right), \tag{11} \]

\[ L=\frac{I}{\eta-\lambda}\,e^{m\delta}\left(R+e^{-mb}\right), \]

where

\[ \dot R= \frac{(e^{mb}-1)\left[\dfrac{\eta-\lambda}{\eta+\lambda}+e^{m(2\delta-b)}\right]} {e^{2m\delta}-\dfrac{\eta-\lambda}{\eta+\lambda}} -1. \]

Taking into account that the penetration depth \(d=\dfrac{\sqrt{2}}{k}\) is much less than the pole pitch \(\tau\) \(\left[\dfrac{k}{m}\gg 1\right]\), the complex coefficient \(\dot R\) is transformed into a real one:

\[ R=-\frac{e^{m(2\delta-b)}+e^{mb}}{e^{2m\delta}+1}. \tag{12} \]

Then the expressions for the vector potentials are:

\[ A_I=0, \tag{13} \]

\[ A_{II}=\mu_0\sum I\left(R\,\operatorname{ch} my+1\right)\sin mx, \tag{14} \]

\[ A_{III}=\mu_0\sum I\left[R\operatorname{ch} my+\operatorname{ch}m(y-b)\right]\sin mx, \tag{15} \]

\[ A_{IV}=-\mu_0\sum I\frac{md}{\sqrt{2}}\,e^{m\delta}\left(R+e^{-mb}\right) e^{-j\frac{\pi}{4}}e^{-(1+j)\frac{y-\delta}{d}}\sin mx. \tag{16} \]

By differentiating the expressions, the components of the induction are found:

\[ B_x=\frac{\partial A}{\partial y}, \qquad B_y=-\frac{\partial A}{\partial x}; \tag{17} \]

Integrating the square of the induction \(B^2=B_x^2+B_y^2\) over the volume, we determine the field energy \(W\), and find the inductance \(L\) as the ratio of the energy to the square of the current,

\[ L=\frac{2W}{i^2} = \frac{2l\displaystyle\int_0^{\tau}\int_0^{y} B^2\,dx\,dy}{\mu_0 i^2}. \tag{18} \]

After integration we obtain

\[ L_n''= \frac{8}{9}\,\mu_0 l q^2 \frac{\tau p}{(\tau-2a)^2 b^2} \sum \frac{k_w^2}{m^3} \left(N_{II}+N_{III}+N_{IV}\right), \tag{19} \]

where

\[ N_{II}=R^2\operatorname{sh}2mb+4\operatorname{sh}mb\cdot R+2mb, \]

\[ N_{III}=-2R^2\operatorname{sh}m(\delta-b)\operatorname{ch}m(\delta+b)+4R\operatorname{sh}m(\delta-b)\operatorname{ch}m\delta+\operatorname{sh}2m(\delta-b), \]

\[ \dot N_{IV}=\frac{md}{\sqrt{2}}\,e^{2mb}(R+e^{-mb})^2e^{-j\frac{\pi}{4}}. \]

The short-circuit inductance consists of two real parts \(N_{II}\) and \(N_{III}\) and one complex part \(\dot N_{IV}\). The imaginary component of this term \(\operatorname{Im}\dot N_{IV}\) indicates an increase in active resistance, the so-called introduced active resistance, caused by energy dissipation in the damper system.

The inductance of the armature winding located in the gap changes from \(L_n\) at no load to \(L_n''\) under short circuit. The inductance of the end portions \(L_l\) enters practically unchanged into the total inductances \(L=L_n+L_l\), synchronous, and \(L''=L_n''+L_l\), short-circuit. Taking into account the variety of machine geometries and, for convenience of analysis and practical use of the results, it is advisable to find the relative change in inductance

\[ \frac{L_n''}{L_n}. \]

The expression for the vector potentials \(A_{II}\), \(A_{III}\) for the case of a winding located between two ferromagnetic surfaces, for which \(\mu_r\gg 1\), can be obtained in the form

\[ A_{II}=\psi_0\sum I\left[1-\frac{\operatorname{sh}m(\delta-b)}{\operatorname{sh}m\delta}\operatorname{ch}my\right]\sin mx, \tag{20} \]

\[ A_{III}=\psi_0\sum I\frac{\operatorname{sh}mb}{\operatorname{sh}m\delta}\operatorname{ch}m(y-\delta)\sin mx. \tag{21} \]

Then the inductance according to (18) is equal to

\[ L_n=\frac{8}{9}\mu_0lq^2\frac{\tau p}{(\tau-2a)^2b} \sum \frac{k_w^2}{m^3} \left[ \frac{\operatorname{sh}^2m(\delta-b)}{\operatorname{sh}^2m\delta}\operatorname{sh}2mb - 4\frac{\operatorname{sh}m(\delta-b)}{\operatorname{sh}m\delta}\operatorname{sh}mb +2mb +\frac{\operatorname{sh}^2mb}{\operatorname{sh}^2m\delta}\operatorname{sh}2m(\delta-b) \right]. \tag{22} \]

The equipotential lines constructed from equations (13)—(16), (20)—(21) show the character of the field at “no load” and under sudden short circuit (Figs. 1, b, 1, c); graphs of the distribution of the induction components \(B_x\), \(B_y\) in the gap are also given there.

Figure 2 presents the relative change in inductance

\[ \frac{L_n''}{L_n} \]

as a function of

\[ \frac{\delta}{\tau} \]

at a frequency of 50 Hz and a pole pitch \(\tau=1.8\ \text{m}\).

An approximately quadratic dependence was obtained, which is distorted when the magnitude of the air gap becomes comparable with the parameter \(d\) and in the region of large values of

\[ \frac{\delta}{\tau}. \]

Experimental verification of the parameters was carried out on a special model of a generator with a smooth armature and a set of rotors at different air gaps. The general view of the machine stator is shown in Fig. 3. The winding is single-layer, concentric, with a \(120^\circ\) phase zone. The end portions of the winding are bent at right angles and can be shielded by copper caps. The geometric dimensions of the machine and the modeling frequency were chosen according to approximate-similarity criteria.

In Fig. 2, along with the curves characterizing the relative change in inductance and obtained by calculation on a digital computer, points are plotted that were taken on the experimental setup for the corresponding values

Fig. 2. Dependence of the relative value of the subtransient inductance on the parameters [[unclear: τδ]]

Fig. 3. General view of the stator of the model with a smooth armature

of the parameters. The discrepancy between the calculated and experimental data is insignificant, which confirms the validity of the method considered.

The advantages of machines with a nonmagnetic active layer, consisting in the possibility of increasing the induction in the air gap and in a very small inductance, can be effectively used in the design of high-power shock generators.

REFERENCES

1. Electrical Review, 1968, 8, p. 363, “Air Gap Windings Could Reduce Costs.”
2. K. Simonyi. Theoretical Electrical Engineering. Mir, 1964.
3. V. A. Venikov. Theory of Similarity and Modeling. Vysshaya Shkola, 1966.