On the Selection of the Air Gap Size of a Surge Generator

G. A. Sipailov, K. A. Khorkov

(Submitted by the scientific seminar of the departments of electrical machines and general electrical engineering)

In turbogenerators of normal design, the minimum size of the air gap is determined by two principal factors: the condition of stability during parallel operation in the power system, and the permissible level of additional pulsation losses on the rotor surface due to the stator slotting. Fulfillment of these conditions leads to the fact that, in high-power turbogenerators, the air gap reaches 50 mm and more. To overcome the magnetic reluctance of such a gap, a large magnetomotive force of excitation is required, in connection with which, from the thermal point of view, the field winding is the most heavily loaded part of the turbogenerator.

In generators of breaking and surge power, which are special machines operating isolated from the power system, other conditions, determined by the purpose and nature of operation of the generator, take the place of the stability condition. It is known that in generators of breaking and surge power, in order to obtain the maximum amplitude of the surge current, and in single-phase design also to suppress the reverse-synchronous field in the rotor slots, together with the field winding a powerful damper winding is installed, occupying in some cases up to 40% of the slot area. In this connection the problem of unloading the field winding becomes especially acute, which can be achieved most effectively by reducing the size of the air gap.

In existing generators of breaking and surge power the air-gap size is approximately two times smaller than in turbogenerators (table). In the first row of Table 1, for turbogenerators the power is given in MW, and for generators of breaking and surge power—the generator type. However, the air gap in magnitude still considerably exceeds the value permissible under the conditions of the mechanical strength of the rotor. The reason limiting any further decrease in the air-gap size is the permissible level of pulsations in the e.m.f. curve of the stator winding. For example, for surge-power generators intended to operate as storage devices and energy sources for impulse feeding of the excitation windings of charged-particle accelerators, the requirement is imposed that in the current curve producing the controlling magnetic field of the accelerator there be no dips caused by higher harmonics. This is especially important

for the initial segment of the rising part of the current and, correspondingly, of the magnetic field, up to a value determined by the injection level. Therefore, when selecting the air-gap size of surge generators, the question of the influence of higher harmonics, including sawtooth harmonics, on the shape of the current curve should be considered in detail.

Table 1

Taking into account that in surge generators operating on an inductive load, the active resistance of both the generator and the load is considerably smaller than the reactance, one may, allowing for a certain error in order to simplify the mathematical calculations, take the damping coefficient of the current to be equal to unity. Then the expression for the current can be written in the following form:

$$ i=\sum_{\nu=1}^{\infty}\left[\frac{E_{\nu}}{\nu\omega L}\cos\psi_0-\frac{E_{\nu}}{\nu\omega L}\cos(\nu\omega t+\psi_0)\right], \tag{1} $$

where $L=L_g+L_n$ is the total inductance of the generator and the load,
$\nu$ is the order of the harmonic,
$\omega$ is the angular frequency,
$\psi_0$ is the initial switching phase.

The operating mode of the surge generator will be optimal if its connection to the load occurs at the instant when the generator e.m.f. $E_1$ passes through zero, that is, at $t=0$, $E_1=0$ and $\psi_0=0$. Then

$$ i=\sum_{\nu=1}^{\infty}\frac{E_{\nu}}{\nu\omega L}(1-\cos\nu\omega t). \tag{1a} $$

Using equation (1a), one can find the condition under which the current curve, representing the sum of the fundamental and the $\nu$-th harmonics, will have no dips. For this it is necessary that the derivative at any point of the rising part of the current curve be greater than zero,

$$ \frac{d}{dt}\sum_{\nu=1}^{\infty}\frac{E_{\nu}}{\nu\omega L}(1-\cos\nu\omega t)>0. \tag{2} $$

Taking into account that the inductance $L$ has only positive values, it can be excluded from inequality (2). Then, after differentiating this inequality and separating out the fundamental harmonic from the total sum, we obtain:

$$ E_1\sin\omega t+\sum_{\nu=3}^{\infty}E_{\nu}\sin\nu\omega t>0. \tag{3} $$

Since \(E_1 \sin \omega t\) for \(0 < \omega t < \pi\) is always a positive quantity, condition (3) will be satisfied if

\[ E_1 \sin \omega t > \left| \sum_{\nu=3}^{\infty} E_\nu \sin \nu \omega t \right|. \tag{4} \]

Of the entire sum of the higher harmonics of the e.m.f., the largest weight (up to 90–95% of \(\sum_{\nu=3}^{\infty} E_\nu \sin \nu \omega t\)) is carried by the tooth harmonics; therefore it is advisable to consider the interaction of the fundamental current harmonic specifically with them. Condition (4) for the fundamental harmonic and for some one \(\nu\)-th harmonic having a predominant value will have the form:

\[ E_1 \sin \omega t > |E_\nu \sin \nu \omega t|, \]

whence

\[ \frac{E_\nu}{E_1} < \left| \frac{\sin \omega t}{\sin \nu \omega t} \right|. \tag{5} \]

From Fig. 1, which presents the current curve obtained as a result of adding the fundamental and \(\nu\)-th current harmonics, it follows that the most dangerous point, with respect to the occurrence of breakdown on the rising part of the current, is point \((b)\), corresponding to the time instant \(t = T_\nu\), where \(T_\nu\) is the period of the \(\nu\)-th harmonic. The condition for absence of breakdown at point \((b)\) will be sufficient if the tangents to any point of the curve segment between points \((a)\) and \((b)\) have

Fig. 1. On the selection of the permissible amplitude of the \(\nu\)-th harmonic.

In connection with the fact that, in the general case, it is impossible to give permissible values for the amplitudes of all higher harmonics, since they depend not only on the ratio between the numbers of stator and rotor teeth, but also on the slot openings, the magnitude of the air gap, the number of slots filled by the winding, etc., it will be rational to introduce a distortion coefficient for the emf curve, which would also determine the permissible magnitude of the pulsations. In determining the distortion coefficient \(k_i\), it is advisable to exclude the higher harmonics with a relatively large period \(\nu = 3,\ 5,\ 7,\ 9\), as well as the harmonics 11, 13, and 15, whose amplitude values are close to zero. Thus

\[ k_i = \frac{ \sqrt{E_{17}^{2} + E_{19}^{2} + \cdots E_{z'_1 \pm 1}^{2} + \cdots E_{z'_2 \pm 1}^{2} + \cdots} }{E_1}. \tag{7} \]

The values of the permissible distortion coefficient of the emf curve for the ratio between the numbers of stator and rotor teeth

\[ z'_2 = z'_1 \pm 4, \]

which may be regarded as the most favorable for surge-power generators, are presented in Table 3.

Table 3

Since the presence of higher harmonics in the stator emf curve depends mainly on the magnitude of the air gap and on the degree of opening of the stator slots \(b_{\mathrm{oc}}\) and rotor slots \(b_{\mathrm{op}}\), it is most convenient to relate the choice of the gap magnitude to the relative value of the opening of the stator and rotor slots

\[ b_{\mathrm{rel}} = \frac{b_{\mathrm{oc}} + b_{\mathrm{op}}}{2\delta}. \tag{8} \]

Experimental investigations carried out on the installation described earlier [1] make it possible to establish the relationship between the distortion coefficient of the emf curve of the stator winding and the relative value of the opening of the stator and rotor slots, which is necessary for determining \(\delta\). The generator under investigation, with a single-phase single-layer winding on the stator, at \(z'_1 = 24\) and \(b_{\mathrm{oc}} = 3.5\) mm, had three replaceable rotors with \(z'_2 = 20,\ 26\), and 28 and, respectively, \(b_{\mathrm{op}} = 6.5;\ 6\), and 6.5 mm. The change in the relative opening of the machine slots was achieved, for each individual rotor, by increasing the air gap of the machine by turning down the rotor. In order to maintain the value of the induction in the gap at one level, simultaneously with the increase in the gap the excitation current was increased accordingly. The results of the experimental investigations are presented in Table 4 in the form of the percentage content of higher harmonics in the emf curve of the stator winding; in this table the values of the higher harmonics of the emf from 9 through 15 inclusive are omitted, since their amplitudes are close to zero. From the data of Table 4, dependences of the distortion coefficient on the relative slot opening were constructed (Fig. 2). Using,

In view of the fact that in the general case it is impossible to give permissible values for the amplitudes of all higher harmonics, since they depend not only on the ratio between the numbers of stator and rotor teeth, but also on the slot openings, the magnitude of the air gap, the number of slots filled by the winding, etc., it is reasonable to introduce a distortion coefficient for the e.m.f. curve, which would determine the permissible magnitude of the pulsations. In determining the distortion coefficient \(k_i\), it is advisable to exclude the higher harmonics with relatively large periods \(\nu = 3, 5, 7, 9\), as well as harmonics 11, 13, and 15, whose amplitude values are close to zero. Thus,

\[ k_i=\frac{\sqrt{E_{17}^{2}+E_{19}^{2}+\cdots E_{z'_1\pm1}^{2}+\cdots E_{z'_2\pm1}^{2}+\cdots}}{E_1}. \tag{7} \]

The values of the permissible distortion coefficient of the e.m.f. curve for the ratio between the numbers of stator and rotor teeth

\[ z'_2=z'_1\pm4, \]

which may be regarded as the most favorable for surge generators, are presented in Table 3.

Table 3

Since the presence of higher harmonics in the stator e.m.f. curve depends mainly on the magnitude of the air gap and on the degree of opening of the stator slots \(b_{\mathrm{os}}\) and rotor slots \(b_{\mathrm{or}}\), it is most convenient to relate the choice of the gap size to the relative magnitude of the opening of the stator and rotor slots

\[ b_{\mathrm{rel}}=\frac{b_{\mathrm{os}}+b_{\mathrm{or}}}{2\delta}. \tag{8} \]

Experimental investigations carried out on the installation described earlier [1] make it possible to establish a relationship between the distortion coefficient of the e.m.f. curve of the stator winding and the relative magnitude of the opening of the stator and rotor slots, which is necessary for determining \(\delta\). The generator under study, with a single-phase single-layer winding on the stator at \(z'_1=24\) and \(b_{\mathrm{os}}=3.5\ \mathrm{mm}\), had three interchangeable rotors with \(z'_2=20,\ 26,\) and \(28\), and, respectively, \(b_{\mathrm{or}}=6.5;\ 6;\) and \(6.5\ \mathrm{mm}\). The change in the relative opening of the machine slots was achieved, for each individual rotor, by increasing the air gap of the machine by turning the rotor. To maintain the value of the induction in the gap at the same level, simultaneously with the increase in the gap the excitation current was increased correspondingly. The results of the experimental investigations are presented in Table 4 in the form of the percentage content of higher harmonics in the e.m.f. curve of the stator winding; in the table the values of the higher harmonics of the e.m.f. from 9 to 15 inclusive are omitted, since their amplitudes are close to zero. From the data of Table 4, dependences of the distortion coefficient on the relative slot opening were constructed (Fig. 2). Using,

Continuation of Table 4

with these dependencies one can determine the value \(b_{\mathrm{rel}}\) at which the pulsations of the emf for each particular case will lie within the permissible limits.

[Figure: graph with vertical axis \(k_{\mathrm{u}}, \%\) and horizontal axis \(b_{\mathrm{rel}}\); two straight lines are marked \(z'_1+4\) and \(z'_1-4\).]

Figure 2. Dependence \(k_{\mathrm{u}} = f(b_{\mathrm{rel}})\) for \(z'_2 = z'_1 \pm 4\).

By substituting into expression (8) the values

\[ b_{\mathrm{os}}=\kappa_{\mathrm{p}1}\frac{\pi D}{z'_1} \quad \text{and} \quad b_{\mathrm{or}}=\kappa_{\mathrm{p}2}\frac{\pi(D-2\delta)}{z'_2}, \]

one can obtain a calculation formula for determining the size of the air gap

\[ \delta \geq \frac{b_{\mathrm{os}}+b_{\mathrm{or}}}{2b_{\mathrm{rel}}} = \frac{\pi D}{z'_1}\, \frac{\kappa_{\mathrm{p}1}z'_2+\kappa_{\mathrm{p}2}z'_1} {2\left(\pi\kappa_{\mathrm{p}2}+b_{\mathrm{rel}}z'_2\right)}, \tag{9} \]

where \(\kappa_{\mathrm{p}1}\) and \(\kappa_{\mathrm{p}2}\) are the ratio of the slot width of the stator and rotor to the corresponding tooth pitch.

Without introducing appreciable errors into formula (9), one may neglect the quantity \(\pi\kappa_{\mathrm{p}2}\leq 1.57\), which appears in the parentheses of the denominator, in comparison with the quantity \(b_{\mathrm{rel}}z'_2\); then

\[ \delta \geq \frac{\pi D}{2b_{\mathrm{rel}}}\, \frac{\kappa_{\mathrm{p}1}z'_2+\kappa_{\mathrm{p}2}z'_1} {z'_1 z'_2}. \tag{10} \]

Using the curves (Fig. 2) and formula (10), the size of the air gap of the generator under consideration can be expressed through its diameter. For example, for the permissible value \(k_{\mathrm{u}}=4\%\) (Fig. 2), with open slots on the stator and rotor \((\kappa_{\mathrm{p}1}=0.5\) and \(\kappa_{\mathrm{p}2}=0.5)\), we obtain

\[ \delta \approx 0.01D . \]

The established relationship between the permissible degree of distortion of the e.m.f. curve of the stator winding and the relative magnitude of the opening of the stator and rotor slots makes it possible to determine the optimum gap size when designing surge-power generators.

REFERENCES

1. G. A. Sipailov, K. A. Khor’kov. Questions of the theory and design of electrical machines. Interuniversity collection of papers, issue 3, Novosibirsk, 1963.