Effect of Saturation on the Slot Leakage Inductance of the Stator Winding of a Shock Generator

V. S. Baklin

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

During a sudden short circuit of a shock generator, the transverse slot leakage fluxes substantially saturate the stator teeth, which leads to a reduction in the slot leakage inductance of the stator winding. To estimate the effect of saturation on the slot leakage inductance, we shall use the method for calculating the nonlinear magnetic field in the stator slot of a shock generator, with all the assumptions and boundary conditions adopted in [7]. In order to reduce machine time, the present work uses a mesh with 198 internal nodes.

In [7] it is assumed that the line \(EC\) at the middle of the air gap is a line of zero potential. This makes it possible to express the energy of the magnetic field in the stator slot of a shock generator in the form

\[ W = \frac{1}{8} LI\delta \sum_s (p_i + p_i - 1)\cdot(q_k + q_k - 1) = \]

\[ = \frac{1}{8}\delta l \sum_s A_{i,k}(p_i + p_i - 1)\cdot(q_k + q_k - 1), \tag{1} \]

where \(L\) is the inductance of the conductor in the slot; \(I\) is the current in the conductor of the slot; \(\delta\) is the current density; \(l\) is the active length of the machine; \(A_{i,k}\) is the value of the vector potential at the mesh nodes; \(p_i, q_k\) are the dimensions of the mesh cells; \(s\) is the cross section of the conductor in the slot. From expression (1) we find the inductance of the conductor in the stator slot:

\[ L = \frac{ l \sum_s A_{i,k}(p_i + p_i - 1)\cdot(q_k + q_k - 1) }{ I \cdot \sum_s (p_i + p_i - 1)\cdot(q_k + q_k - 1) } \tag{2} \]

The total permeance coefficient of slot leakage is

\[ \lambda_{\text{tot}} = \frac{ \sum_s A_{i,k}(p_i + p_i - 1)\cdot(q_k + q_k - 1) }{ \mu_0 I \sum_s (p_i + p_i - 1)\cdot(q_k + q_k - 1) }, \tag{3} \]

where \(\mu_0\) is the magnetic constant.

According to the boundary conditions adopted in [7], \(\lambda_{\text{tot}}\) includes the slot-leakage permeance coefficient \(\lambda_{\text{s}}\) and the permeance coefficient over the tooth tips \(\lambda_{\text{k}}\). It is of interest to separate its components from \(\lambda_{\text{tot}}\).

Usually, in deriving formulas for calculating the permeance coefficients \(\lambda_{\mathrm{p}}\) and \(\lambda_{\mathrm{k}}\), the magnetic fields in the region of the slot and of the air gap are considered independently of one another, assuming that the line bounding the slot along the stator bore is a line of equal potential. Taking this assumption into account, analytical methods for calculating the permeance coefficient \(\lambda_{\mathrm{k}}\) can be found in [2], and the permeance coefficient \(\lambda_{\mathrm{p}}\) in [5, 6].

Calculation of \(\lambda_{\text{total}}\) by the finite-difference method shows that the line bounding the slot along the stator bore is not a line of equal potential. A bulging of the lines of force of the slot leakage flux toward the air gap is observed.

In deriving the calculation formula for the permeance coefficient \(\lambda_{\mathrm{k}}\), in [3] the leakage field in the air gap is considered jointly with the leakage field in the slot. From the resulting total permeance of the slot and tooth crowns, the slot permeance \(\lambda_{\mathrm{p}}\), determined by the leakage flux entering the slot wall, is subtracted. According to such a division of the fields, the value of the permeance coefficient over the tooth crowns can be found from the expression

\[ \lambda_{\mathrm{k}}=\frac{A_a}{\mu_0 I}, \tag{4} \]

where \(A_a\) is the value of the vector potential at point \(a\) in Fig. 1. We define the permeance coefficient of the slot leakage as

\[ \lambda_{\mathrm{p}}=\lambda_{\text{total}}-\lambda_{\mathrm{k}}. \tag{5} \]

Fig. 1. Leakage flux over the tooth crowns \(\Phi_{\mathrm{k}}\) and slot leakage flux \(\Phi_{\mathrm{p}}\)

Sometimes the influence of magnetic lines of force passing in the air gap while bypassing several slots is neglected, and it is assumed that the slots are not linked with one another by the leakage field over the crowns [2]. The permeance coefficient corresponding to this representation of the leakage field in the air gap can be determined from the expression

\[ \lambda_{\mathrm{k}1}=\frac{A_a-A_b}{\mu_0 I}. \tag{6} \]

The results of calculating the permeance coefficients \(\lambda_{\mathrm{p}}\), \(\lambda_{\mathrm{k}}\), and \(\lambda_{\mathrm{k}1}\) as functions of the current \(I\) in the slot conductor are presented in Figs. 2 and 3. In this case the slot dimensions remained unchanged and equal to: wid-

per slot \(b_{\text{p}}=5\ \text{cm}\), slot height \(h_{\text{p}}=10\ \text{cm}\), wedge height together with insulation \(h_{\text{k}}=2\ \text{cm}\), air-gap value \(\delta/2=1.5\ \text{cm}\). The tooth width was varied and took the following values: for curves 1, \(2.5\ \text{cm}\); for curves 2, \(3.5\ \text{cm}\); for curves 3, \(5\ \text{cm}\); for curves 4, \(7.5\ \text{cm}\); and for curves 5, \(10\ \text{cm}\). Figure 2 (straight line 6) presents the result of calculating the slot-leakage permeance coefficient from expression [5, 6].

\[ \lambda_{\text{p}}=\frac{h_{\text{p}}-h_{\text{k}}}{3b_{\text{p}}}+\frac{h_{\text{k}}}{b_{\text{p}}}, \tag{7} \]

If tooth saturation is absent, then the difference in the values of the coefficient \(\lambda_{\text{p}}\), calculated from expression (5) or (7), does not exceed 2%.

Fig. 2. Dependence of the slot-leakage permeance coefficient \(\lambda_{\text{p}}\) on the current in the slot \(I\)

There are various formulas for calculating the unsaturated value of the permeance coefficient at the tooth tips. Thus, for example, in [1]

\[ \lambda_{\text{k}}=\frac{5\delta}{5b_{\text{p}}+4\delta}; \tag{8} \]

in [2]

\[ \lambda_{\text{k}}=\frac{10\delta}{7\delta+18b_{\text{p}}}; \tag{9} \]

in [3]

\[ \lambda_{\text{k}}= \frac{1}{\pi}\ln \frac{\sqrt{1-\left(\frac{2\delta}{b_{\text{p}}}\right)^2}}{2} -\frac{2\delta}{\pi b_{\text{p}}}\arctg\frac{2\delta}{b_{\text{p}}} +\frac{2\delta}{b_{\text{p}}} +\frac{\delta}{b_{\text{p}}}. \tag{10} \]

Table 1 presents the results of calculating the permeance coefficient at the tooth tips by formulas (8), (9), (10) and by a numerical method. Here \(b_{\text{p}}=b_z=5\ \text{cm}\). Analysis shows that, as the air gap increases, the difference between \(\lambda_{\text{k}}\) and \(\lambda_{\text{k}}\) [2] increases, while the difference between \(\lambda_{\text{k}}\) and \(\lambda_{\text{k}}\) [1] decreases somewhat. The difference in the values of \(\lambda_{\text{k}}\) and \(\lambda_{\text{k}}\) [2] can be explained by the fact that expression (9) does not take into account the leakage flux in the air gap linking the stator slots, which,

Table 1

as shown below, constitutes a significant part of the total leakage flux in the air gap. The leakage flux in the air gap linking the stator phases can be estimated from the difference between the coefficients $\lambda_k$ and $\lambda_{k1}$. Figure 3 shows that, as the tooth width decreases, the difference in the values of $\lambda_k$ and $\lambda_{k1}$ increases. For example, at $b_z = 2.5$ cm (curves 1) it reaches 35% and increases even more if tooth saturation is taken into account.

Fig. 3. Dependence of the permeance coefficient along the tooth crowns $\lambda_k$ and of the permeance coefficient $\lambda_{k1}$ on the phase current $I$.

Table 2 gives some results of a study of the influence of the air-gap magnitude and saturation on the permeance coefficients $\lambda_{\mathrm{total}}$, $\lambda_k$, and $\lambda_{k1}$. In this case the phase dimensions remained the same, $b_z = 3.5$ cm.

Table 2

If, for \(\delta/2=1.5\) cm, the difference between \(\lambda_k\) and \(\lambda_{k1}\) is 25%, then for \(\delta/2=3\) cm it is already more than 50%. Consequently, the leakage flux in the air gap linking the stator slots may constitute a significant part of the total leakage flux in the air gap, which causes an error in calculating the permeance coefficient along the tooth crowns if expression (9) is used.

As the calculations have shown, for the unsaturated value of the permeance coefficient along the tooth crowns \(\lambda_k\), expression (9) may be used when \(b_z>b_p\), and expression (8) when \(b_z<b_p\).

The influence of saturation in calculating the permeance coefficient along the tooth crowns should be taken into account only for relatively wide teeth \((b_z>b_p)\). In this case the unsaturated value \(\lambda_k\) obtained from expression (9) should be reduced by \((10\text{--}15\%)\).

In engineering practice, approximate methods are used to calculate the permeance coefficient of slot leakage with allowance for saturation; these methods may conventionally be divided into three groups:

1. A method that takes into account only the presence of the transverse slot leakage flux [4].
2. A method that, in addition to the transverse slot leakage flux, takes into account the presence of the main field [5].
3. A method that takes into account only the longitudinal slot-leakage flux closing through the teeth at the boundaries of the phase zones [6].

The last method assumes a dependence of the saturation coefficient on the number of slots per pole and phase, \(q\). The assumptions in [7] are adopted in such a way that it is not possible to compare the calculation results for \(\lambda_p\) obtained in the present work with those obtained by the procedure of [6]. The closest agreement with the \(\lambda_p\) values obtained in the present work is given by the procedure of [4], Table 3. The procedure of [5] gives a somewhat underestimated value, since, in addition to the transverse slot leakage flux, it takes into account the presence of the main field.

Table 3

When calculating \(\lambda_p\) by the procedure of [4], the induction in the region of the tooth head, \(B_k\), was determined in the present work from the expression

\[ B_k=\frac{\mu_0\cdot I+b_z\cdot B_{\mathrm{cr}}}{t}, \tag{11} \]

where \(t\) is the tooth pitch; \(B_{\mathrm{cr}}\) is the critical induction. For electrical steel E41, \(B_{\mathrm{cr}}=1.914\) T. To determine the relative magnetic permeability \(\mu_r\) in the region of the tooth head, for \(B_k\leq 2.05\) T the actual magnetization characteristic of electrical steel E41 was used; and for \(B_k>2.05\) T, \(\mu_r\) was found from the expression

\[ \mu_r=\frac{B_k}{B_k-B_{\mathrm{cr}}}. \tag{12} \]

Thus, the use of the finite-difference method for calculating the slot leakage inductance made it possible not only to take into account the saturation of the steel sections and the actual configuration of the boundary conditions ...

investigated region, but also to verify the validity of those assumptions that underlie simplified analytical methods.

In conclusion, the author expresses his gratitude to Assoc. Prof. K. A. Khorkov for valuable advice concerning the content of the article.

REFERENCES

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