Electrodynamic Forces in the End Windings of a Pulse Generator

V. I. Popov, G. A. Sipailov

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

In sudden short circuits in the windings of electrical machines, significant electrodynamic forces arise. Still greater electrodynamic forces (EDF) act on the stator windings of pulse and interrupting-power generators (PG). Thus, in the domestic PG of type TI-75-2, at a current amplitude of \(170\ \mathrm{kA}\), the EDF acting on the end part of a bar lying at the edge of a phase zone is \(1000\ \mathrm{N/cm}\), and the total force on the entire end part is \(7.7\ \mathrm{t}\) [2]. In order to withstand such enormous dynamic loads, the end part of the PG stator winding must be securely fastened. The development of the most advanced fastening can be carried out only when there is a sufficiently clear and complete picture of the distribution of the mechanical loads acting on the end part.

On the basis of [1], for the components of the forces acting on elementary volumes of the conductor of the end part, in a cylindrical coordinate system we have:

for curvilinear sections

\[ dF_z = \frac{I}{ab} B_{r\mathrm{II}}\,dv,\qquad \mathrm{N}; \tag{1} \]

\[ dF_{r\mathrm{k}} = \frac{I}{ab} B_{z\mathrm{II}}\,dv,\qquad \mathrm{N}. \tag{2} \]

for rectilinear sections

\[ dF_{r\mathrm{p}} = \frac{I}{ab} B_{\varphi\mathrm{II}}\,dv',\qquad \mathrm{N}; \tag{3} \]

\[ dF_{\varphi} = \frac{I}{ab} B_{r\mathrm{II}}\,dv',\qquad \mathrm{N}, \tag{4} \]

where

\[ \begin{aligned} F_z &\text{ — axial force in the curvilinear part of the winding;}\\ F_{r\mathrm{k}} &\text{ — radial force in the curvilinear part of the winding;}\\ F_{r\mathrm{p}} &\text{ — radial force acting on the straight sections of the bars;}\\ F_{\varphi} &\text{ — tangential force acting on the straight sections of the bars;}\\ v &\text{ — integration volume in the curvilinear part of the winding;} \end{aligned} \]

\(v'\) — the same, for the rectilinear part;
\(I\) — the total current of the bar;
\(ab\) — the cross section of the bar;
\(B_{r\mathrm{II}}, B_{z\mathrm{II}}, B_{\varphi\mathrm{II}}\) — the induction components for the corresponding points at which the electrodynamic forces are determined (region II) [1].

To determine the total electrodynamic forces acting on the bar, it is necessary to sum the elementary forces over the volume of the bar. After integrating expressions (1), (2), (3), (4) and transformations, we obtain formulas for computing the total forces acting on an arbitrary volume of the end portion.

\[ \begin{aligned} F_z &= \frac{32 I^2 R_k^2 \mu_0 q}{\pi^2 a(C_2-C_1)p} \sum_{n=1}^{\infty} \Biggl[ \left(A_{1vn}+\frac{R_k+R_1}{4R_k}A_{1n}\right) \frac{1}{n} \cos\frac{\pi n(C_2+C_1)}{2C} \\ &\qquad {}\times \sin\frac{\pi n(C_2-C_1)}{2C} \sin\frac{\pi(1-0.5\gamma)}{2} - \left(A_{1wn}+\frac{R_k+R_1}{4R_k}A_{1n}\right) \\ &\qquad {}\times \frac{p^2(C_2-C_1)Ck_p}{\pi R_k^2} \frac{1}{n^2} \cdot \sin^2\frac{\pi n(C_2+C_1)}{4C} \Biggr] \cdot \sin\frac{\pi n(z_2+z_1)}{2C} \\ &\qquad {}\times \sin\frac{\pi n(z_2-z_1)}{2C} \cdot \sin\frac{p(\varphi_2+\varphi_1)}{2} \sin\frac{p(\varphi_2-\varphi_1)}{2}, \end{aligned} \tag{5} \]

\[ \begin{aligned} F_{rk} &= \frac{2I^2\mu_0 q(z_2-z_1)}{\pi ab^2 p^2 C} \Biggl[ A_{0p} + \frac{16bR_kC^2p}{\pi^2(z_2-z_1)(C_2-C_1)} \sum_{n=1}^{\infty} (A_{2vn}+A_{2n}) \\ &\qquad {}\times \frac{1}{n^2} \cos\frac{\pi n(z_2+z_1)}{2C} \cdot \sin\frac{\pi n(z_2-z_1)}{2C} \cos\frac{\pi n(C_2+C_1)}{2C} \\ &\qquad {}\times \sin\frac{\pi n(C_2-C_1)}{2C} \Biggr] \sin\frac{\pi(1-0.5\gamma)}{2} \cdot \sin\frac{p(\varphi_2+\varphi_1)}{2} \sin\frac{p(\varphi_2-\varphi_1)}{2}, \end{aligned} \tag{6} \]

\[ \begin{aligned} F_{r\mathrm{II}} &= \frac{16I^2\mu_0 q k_p C}{\pi^2 ab} \sum_{n=1}^{\infty} \left( A_{2wn}+A_{2n} - b\frac{R_k+R_1}{4R_k}A_{1n} \right) \\ &\qquad {}\times \frac{1}{n^2} \sin^2\frac{\pi n(C_2+C_1)}{4C} \sin\frac{\pi n(z_2'+z_1')}{2C} \cdot \sin\frac{\pi n(z_2'-z_1')}{2C} \\ &\qquad {}\times \cos\frac{p(\varphi_2'+\varphi_1')}{2} \sin\frac{p(\varphi_2'-\varphi_1')}{2}, \end{aligned} \tag{7} \]

\[ \begin{aligned} F_\varphi &= \frac{32I^2R_k^2\mu_0 q}{\pi^2a(C_2-C_1)p} \sum_{n=1}^{\infty} \Biggl[ \left( A_{1vn}+\frac{R_k+R_1}{4R_k}A_{1n} \right) \frac{1}{n} \cos\frac{\pi n(C_2+C_1)}{2C} \\ &\qquad {}\times \sin\frac{\pi n(C_2-C_1)}{2C} \sin\frac{\pi(1-0.5\gamma)}{2} - \left( A_{1wn}+\frac{R_k+R_1}{4R_k}A_{1n} \right) \\ &\qquad {}\times \frac{p^2(C_2-C_1)Ck_p}{\pi R_k^2} \frac{1}{n^2} \sin^2\frac{\pi n(C_2+C_1)}{4C} \Biggr] \sin\frac{\pi n\cdot(z_2'+z_1')}{2C} \\ &\qquad {}\times \sin\frac{\pi n(z_2'-z_1')}{2C} \sin\frac{p(\varphi_2'+\varphi_1')}{2} \sin\frac{p(\varphi_2'-\varphi_1')}{2}, \end{aligned} \tag{8} \]

where

\[ \Lambda_{1v_n} = \frac{f_v K_p(mR_k)+\varepsilon_v I_p(mR_k)} {\beta_v f_v-\varepsilon_v \lambda_v} \left[\beta_v I_p(mR_k)+\lambda_v K_p(mR_k)\right], \]

\[ \Lambda_{1w_n} = \frac{f_w K_p(mR_k)+\varepsilon_w I_p(mR_k)} {\beta_w f_w-\varepsilon_w \lambda_w} \left[\beta_w I_p(mR_k)+\lambda_w K_p(mR_k)\right], \]

\[ \Lambda_{2v_n} = \frac{f_v K_p(mR_k)+\varepsilon_v I_p(mR_k)} {\beta_v f_\Lambda-\varepsilon_v \lambda_v} \times \]

\[ \times \left[\beta_v\{R_2 I_p(mR_2)-R_1 I_p(mR_1)\} +\lambda_v\{R_2 K_p(mR_2)-R_1 K_p(mR_1)\}\right], \]

\[ \Lambda_{2w_n} = \frac{f_w K_p(mR_k)+\varepsilon_w I_p(mR_k)} {\beta_w f_w-\varepsilon_w \lambda_w} \left[ \beta_w\{R_2 I_p(mR_2)-R_1 I_p(mR_1)-b I_p(mR_k)\} +\right. \]

\[ \left. +\lambda_w\{R_2 K_p(mR_2)-R_1 K_p(mR_1)-b K_p(mR_k)\} \right], \]

\[ \Lambda_{1n} = I_p\left(m\frac{R_k+R_1}{2}\right)K_p(mR_k) - I_p(mR_k)K_p\left(m\frac{R_k+R_1}{2}\right), \]

\[ \Lambda_{2n} = R_2\left[I_p(mR_k)K_p(mR_2)-I_p(mR_2)K_p(mR_k)\right], \]

\[ \Lambda_{0p} = \frac{R_2^{\,2-p}-R_1^{\,2-p}}{2-p} \left[ \frac{r_2^{2p}(R_2^{1+p}-R_1^{1+p})}{r_2^{2p}-r_1^{2p}} + \right. \]

\[ \left. +\frac{p+1}{p-1} \frac{(r_1r_2)^{2p}(R_2^{1-p}-R_1^{1-p})}{r_2^{2p}-r_1^{2p}} -R_2^{1+p} \right] + \]

\[ + \frac{R_2^{\,2+p}-R_1^{\,2+p}}{2+p} \left[ \frac{r_1^{2p}(R_2^{1-p}-R_1^{1-p})}{r_2^{2p}-r_1^{2p}} - \frac{p-1}{p+1} \frac{R_2^{1+p}-R_1^{1+p}}{r_2^{2p}-r_1^{2p}} +R_2^{1-p} \right], \]

\(q\) — number of bars per pole and phase,

\(z_1, z_2, \varphi_1, \varphi_2\) — limits of integration with respect to the variables \(z\) and \(\varphi\) in the curvilinear part of the winding,

\(z_1', z_2', \varphi_1', \varphi_2'\) — the same, in the rectilinear part.

The problem of determining the leakage field in the end zone has been solved in general form, i.e., for any machine polarity [1]; therefore, when calculating the coefficient \(\Lambda_{0p}\) for \(p=1\) and \(p=2\), indeterminacies of the form \(0/0\) appear. By using l’Hôpital’s rule, one can eliminate this indeterminacy. For example, for \(p=1\) we have

\[ \Lambda_{01} = b\left[ \frac{r_2^2(R_2^2-R_1^2)}{r_2^2-r_1^2} + \frac{2(r_1r_2)^2}{r_2^2-r_1^2}\ln\frac{R_1}{R_2} -R_2^2 \right] + \frac{R_2^3-R_1^3}{3}. \]

Because of the nonuniformity of the leakage field in the end zone, the bars experience unequal forces. Therefore, in order to identify the most highly stressed locations in the end portion, it is necessary to determine the forces acting on each bar individually.

For this purpose, we introduce the following notation [Fig. 1]:

\(n_c\) — bar number, counted from the active steel of the stator,

\(b_p\) — width of the stator slot,

\(h\) — thickness of the spacing strip between the bars in the curvilinear part,

\(\Delta_{\text{ins}}\) — thickness of the one-sided insulation of the bar.

From Fig. 1, taking into account the introduced notation, it can be shown that for

\[ \gamma = \frac{2}{3}. \]

Fig. 1. For determining the geometric dimensions of the end-winding part of the pulse generator

\[ \frac{z_2+z_1}{2}=z_2'=C_1-h-0.5b_{\mathrm{p}}+n_c(h+b_{\mathrm{p}}), \]

\[ z_2-z_1=a, \]

\[ z_1'=0. \]

\[ \frac{\varphi_2'+\varphi_1'}{2}=\frac{\pi}{3q}(q+1-2n_c), \]

\[ \frac{\varphi_2'-\varphi_1'}{2}=\frac{a}{2R_k}, \]

\[ \frac{\varphi_2-\varphi_1}{2} = \frac{\pi}{3p}\left[\frac{1}{2}+\frac{1}{q}(2n_c-1)\right], \]

\[ \frac{\varphi_2+\varphi_1}{2}=\frac{\pi}{2p}. \]

Substituting the relations obtained into (5), (6), (7), (8), we obtain final expressions for calculating the total components of the electrodynamic force on any bar of the end-winding part of a pulse generator having \(2/3\)-filled stator slots.

\[ F_z= \frac{16\sqrt{3}\,I^2R_k^2\mu_0q}{\pi^2a(C_2-C_1)p} \sum_{n=1}^{\infty} \left[ \left( A_{\nu n}+\frac{R_k+R_1}{4R_k}A_{1n} \right) \frac{1}{n} \right]\times \]

\[ \begin{aligned} &\times \cos \frac{\pi n(C_2+C_1)}{2C}\, \sin \frac{\pi n(C_2-C_1)}{2C} -\left(\Lambda_{1w n}+\frac{R_k+R_1}{4R_k}\Lambda_{1n}\right)\times\\ &\times \frac{2p^2(C_2-C_1)Ck_p}{\sqrt{3}\,\pi R_k^2}\, \frac{1}{n^2}\, \sin^2\frac{\pi n(C_2+C_1)}{4C} \bigg]\sin \frac{\pi[0.5q+(2n_c-1)]}{3q}\times\\ &\times \sin \frac{\pi n[C_1-h-0.5b_{\mathrm{п}}+n_c(h+b_{\mathrm{п}})]}{C}\, \sin \frac{\pi n a}{2C}, \end{aligned} \tag{9} \]

\[ \begin{aligned} F_{rk}={}& \frac{\sqrt{3}\,I^2\mu_0 q}{\pi b^2p^2C} \left[ \Lambda_{0p} +\frac{16bR_kpC^2}{\pi^2a(C_2-C_1)} \sum_{n=1}^{\infty} (\Lambda_{2vn}+\Lambda_{2n})\frac{1}{n^2}\times \right.\\ &\left. \times \cos \frac{\pi n[C_1-h-0.5b_{\mathrm{п}}+n_c(h+b_{\mathrm{п}})]}{C}\, \sin \frac{\pi n a}{2C}\times \right.\\ &\left. \times \cos \frac{\pi n(C_2+C_1)}{2C}\, \sin \frac{\pi n(C_2-C_1)}{2C} \right] \sin \frac{\pi[0.5q+(2n_c-1)]}{3q}, \end{aligned} \tag{10} \]

\[ \begin{aligned} F_{\mathrm{п}}={}& \frac{32I^2\mu_0 q\,Ck_p}{\pi^2ab} \sum_{n=1}^{\infty} \left( \Lambda_{2wn}+\Lambda_{2n} -\frac{R_k+R_1}{4R_k}\Lambda_{1n} \right) \frac{1}{n^2}\times\\ &\times \sin^2\frac{\pi n(C_2+C_1)}{4C}\cdot \sin^2\frac{\pi n[C_1-h-0.5b_{\mathrm{п}}+n_c(h+b_{\mathrm{п}})]}{2C}\times\\ &\times \cos \frac{p\pi(q+1-2n_c)}{3q}\cdot \sin \frac{pa}{2R_k}, \end{aligned} \tag{11} \]

\[ \begin{aligned} F_\varphi={}& \frac{16\sqrt{3}\,I^2R_k^2\mu_0 q}{\pi^2a(C_2-C_1)p} \sum_{n=1}^{\infty} \bigg[ \left( \Lambda_{1vn}+\frac{R_k+R_1}{4R_k}\Lambda_{1n} \right)\frac{1}{n}\times\\ &\times \cos \frac{\pi n(C_2+C_1)}{2C}\, \sin \frac{\pi n(C_2-C_1)}{2C} -\left( \Lambda_{1wn}+\frac{R_k+R_1}{4R_k}\Lambda_{1n} \right)\times\\ &\times \frac{2p^2(C_2-C_1)k_p}{\sqrt{3}\,\pi R_k^2}\, \frac{1}{n^2}\, \sin^2\frac{\pi n(C_2+C_1)}{4C} \bigg]\times\\ &\times \sin^2 \frac{\pi n[C_1-h-0.5b_{\mathrm{п}}+n_c(h+b_{\mathrm{п}})]}{2C}\, \sin \frac{\pi p(q+1-2n_c)}{3q}\times\\ &\times \sin \frac{pa}{2R_k}. \end{aligned} \tag{12} \]

The series entering expressions (9), (10), (11), (12) converge rapidly because of the presence of terms with \(\frac{1}{n^2}\); therefore, when calculating the sums of the series it is sufficient to take 2–3 terms. The following figures may be given for confirmation. In determining the forces in the machine taken as an example, five terms were taken in the series. If the sum of five terms is taken as \(100\%\), then, for example, for the first bar the forces in the axial direction amounted, respectively, for \(n=1,2,3,4,5\) to \(+10.88\%\), \(90.2\%\), \(+0.67\%\), \(-1.48\%\), \(-0.25\%\).

Using (9), (10), (11), (12), the total and specific forces acting on the bars of the end winding of a hydro-generator were calculated; the generator has the following data:

\(D_c = 1.125\ \text{m};\ D_p = 1.075\ \text{m};\ \beta = 0.3;\ 2p = 2;\ q = 16;\ B_\delta = 1\ \text{T};\ B_{zc} = 2\ \text{T};\)

\(C = 0.55\ \text{m};\ C_1 = 0.13\ \text{m};\ C_2 = 0.42\ \text{m};\ R_1 = 0.575\ \text{m};\ R_2 = 0.62\ \text{m};\ r_1 = 0.525\ \text{m};\)

\(r_2 = 1.38\ \text{m};\ a = 0.0187\ \text{m};\ h = 0.\)

The calculated curves of the total and specific loads on the bars of the end-winding portion of the hydro-generator are shown in Figs. 2 and 3. As was to be expected, the greatest

Fig. 2. Distribution of total forces over the bars of the end-winding portion of the hydro-generator:
1, 2 — axial and radial forces in the curvilinear part of the winding; 3, 4 — tangential and radial forces on the rectilinear sections of the bars

specific loads occur on the bars lying closer to the active steel of the stator. The decrease in tangential load with increasing bar number is explained by the decrease in the radial component of the leakage-field induction. The axial loads in the curvilinear part of the winding are predominant. Thus, at a current of \(200\ \text{kA}\), for this machine the axial specific load on the first bar is \(5300\ \text{N/cm}\). The total force acting on the entire end-winding portion of the machine in the axial direction reaches \(11\ \text{t}\).

By this method it is possible to calculate in a single step both the total and the specific forces acting on any bar of the end winding; therefore there is no need to divide the curvilinear part of the winding into rectilinear segments and calculate the forces for each segment separately.

Fig. 3. Distribution of specific mechanical loads over the bars of the end winding: 1, 2 — axial and radial loads in the curvilinear part of the winding; 3, 4 — radial and tangential loads on the straight sections of the bars

of the segment through its interaction with all the other conductors. In determining the forces from expressions (9), (10), (11), and (12), the influence of the surrounding ferromagnetic surfaces, the dimensions of the cross sections of the current leads and the geometry of the end-winding space, as well as the influence of all currents in the bars of the end winding, are taken into account quite fully.

The expressions obtained make it possible to determine the forces in the end winding of an unfolded machine with any number of poles.

REFERENCES

1. V. I. Popov, G. A. Sipaylov. Calculation of leakage fields in the end zones of single-layer windings (present collection).
2. M. F. Yudov. Vibration and Deformation of the Stator Windings of Synchronous Machines. “Energiya,” 1968.
3. E. Ya. Kazovskii, A. A. Chistyakov. Designs for fastening the stator windings of turbogenerators of foreign manufacture. Informstandartelektro, Moscow, 1968.