On One Method for Reducing Electrodynamic Forces in Transformer Windings
Abstract Generated abstract
The paper examines a winding arrangement intended to reduce electrodynamic forces in high-power two-winding transformers during external three-phase short circuits. It develops design relations for a transformer in which a two-layer low-voltage winding embraces the high-voltage winding, then combines thermal, magnetomotive-force, short-circuit voltage, and cost equations to determine principal dimensions and current densities. A numerical study for 110 kV substation transformers rated 60 to 240 MVA indicates that the proposed multilayer arrangement substantially lowers mechanical stresses in the low-voltage winding compared with single-layer designs, with calculated outer-layer stresses from 480 to 960 kg/cm². The results also suggest that the mechanically compressed high-voltage winding can be made of aluminum while the more highly stressed low-voltage winding is made of copper, potentially reducing copper consumption in large transformers.
Full Text
ON ONE METHOD FOR REDUCING ELECTRODYNAMIC FORCES IN TRANSFORMER WINDINGS
Doctor of Technical Sciences I. D. KUTYAVIN, postgraduate student G. V. DEL,
postgraduate student V. P. KRASNOV.
The design of high-power power transformers encounters difficulties owing to the appearance in the windings of large electrodynamic forces during a three-phase short circuit beyond the transformer. In this connection, it becomes necessary to seek methods for reducing electrodynamic forces in transformer windings. One such method, partially considered in the present article, is the use of multilayer continuous coil or helical windings.
Figure 1 shows the arrangement of the windings of one phase in the window of the core of a two-winding transformer without a device for voltage regulation. The low-voltage winding (LV) consists of two layers embracing the high-voltage winding (HV). The transposition of the LV winding is made at the transition from one layer to the other; therefore both phase windings constitute a homogeneous structure over the entire height of the limb, with the exception of the end coils.
Using the method given in [1] and [2] and Fig. 1, we find the relations for the principal design dimensions of the transformer.
The width of the transformer-core window is
\[ a = 2(x_1 + 2x_2 + l_r) \tag{1} \]
where
\[ l_r = 0.5\,\delta_{11} + 2\delta_{12} + \delta_{02} \]
The window height is
\[ H = h + 2l_u \tag{2} \]
The length of the average turn of the phase windings is
\[ l_{\mathrm{M}} = \pi(d + x_1 + 2x_2 + l_r) \tag{3} \]
where
\[ l_z = 2(\delta_{12} + \delta_{02}). \]
We shall use the equation of the heat balance of a coil [2] to determine the current densities of the phase windings:
\[ \left. \begin{aligned} \Delta_1&=\sqrt{\frac{2\sigma_1 k_{y1}(k_{m1}x_1+y_1)}{\rho_1 k_1 x_1 y_1}} =a_1\sqrt{\frac{z_1}{x_1y_1}}\\ \Delta_2&=\sqrt{\frac{2\sigma_2 k_{y2}(k_{m2}x_2+y_2)}{\rho_2 k_2 x_2 y_2}} =a_2\sqrt{\frac{z_2}{x_2y_2}} \end{aligned} \right\} \tag{4} \]
where
\[ a_1=\sqrt{\frac{2\sigma_1 k_{y1}}{\rho_1 k_1}};\qquad a_2=\sqrt{\frac{2\sigma_2 k_{y2}}{\rho_2 k_2}}. \]
Fig. 1.
The cross-sectional areas of the phase windings are
\[ q_{m1}=\frac{k_1 x_1 y_1}{y_1+\delta_1}\,h;\qquad q_{m2}=\frac{k_2 x_2 y_2}{y_2\delta_2}\,h. \tag{5} \]
The condition for equality of the magnetomotive forces of the phase windings:
\[ q_{m1}\Delta_1=q_{m2}\Delta_2. \tag{6} \]
Let \(\Delta_2=b\Delta_1\).
Then
\[ q_{m1}=bq_{m2}. \tag{7} \]
Substituting into (7) the values of \(q_{m1}\) and \(q_{m2}\) from (5), we find:
\[ y_1=\frac{2kb\delta_1x_2y_2}{x_1(y_2+\delta_2)-2kbx_2y_2} \tag{8} \]
where
\[ k=\frac{k_2}{k_1}. \]
Let us also find the values of \(x_1\), substituting into (7) the expressions (4) for the current densities:
\[ x_1 = \frac{ a \pm \sqrt{a^2 - 2 k b \delta_1 k_{\mathrm{n}1} x_2 y_2 (y_2+\delta_2)} }{ k_{\mathrm{n}1}(y_2+\delta_2) }; \tag{9} \]
where:
\[ a = k b \left( k_{\mathrm{n}1} x_2 y_2 + \delta_1 z_2 \frac{k_{y2}\sigma_2\rho_2}{k b^2 k_{y1}\sigma_1\rho_1} \right). \]
Using the expression for the rated power of a transformer [1], we find the height of the phase windings:
\[ h = \frac{ N (y_2+\delta_2) }{ d^2 \sqrt{x_2 y_2 z_2} }; \tag{10} \]
where:
\[ N = \frac{2S}{K k_c k_2 a_2 \pi}. \]
From the expression for the reactive component of the transformer short-circuit voltage, one can find a formula for determining the core diameter [1]:
\[ d = \frac{ U + \sqrt{U^2 + 4 P U V (y_2+\delta_2)} }{ 2P(y_2+\delta_2) } \tag{11} \]
where:
\[ U = \sqrt{x_2 y_2 z_2}\,(0.5x_1+x_2+3\delta_{12}); \]
\[ V = x_1 + 2x_2 + l_z; \]
\[ P = \frac{3.33 B k_c U_p}{16 k_r k_2 a_2}; \]
The expression for the calculated costs for a transformer with windings made of different materials [1]:
\[ З = (A_1 + D B^2)Q_c + (\beta_1 A_2 + E_1 \Delta_1)Q_{\mathrm{m}1} + (\beta_2 A_2 + E_2 \Delta_2)Q_{\mathrm{m}2}; \tag{12} \]
where:
\[ E_1 = (C_1 p_{\mathrm{n}} k_{\mathrm{m}}^2 + C''_2 \tau k_{\mathrm{n}}^2) \left(\frac{S_{\mathrm{m}}}{S}\right)^2 \alpha_{\mathrm{m}1}; \]
\[ E_2 = (C_1 p_{\mathrm{n}} k_{\mathrm{m}}^2 + C''_2 \tau k_{\mathrm{n}}^2) \left(\frac{S_{\mathrm{m}}}{S}\right)^2 \alpha_{\mathrm{m}2}. \]
After substituting the variables and carrying out certain transformations, the calculated costs (12) can be brought to the following form:
\[ З = \frac{k_c \pi \gamma}{4} (A_1 + D B^2)\cdot 10^{-3}\, \varphi(x_2,y_2) \tag{13} \]
where:
\[ \varphi(x_2,y_2) = 3N(y_2+\delta_2) \left[ \frac{1}{\sqrt{x_2 y_2 z_2}} + \frac{ 8k_2P\left(\dfrac{n}{z_2}+\dfrac{m}{x_2y_2}\right) }{ k_c\gamma_c(0.5x_1+x_2+3\delta_{12}) } \right] + d^2\{k_a[8(x_1+2x_2+l_r)+5.6d]+6l_u\}; \tag{14} \]
\[ n = \frac{ A_2(\beta_{\mathrm{m}1}\gamma_{\mathrm{m}} + b\beta_{\mathrm{m}2}\gamma_{\mathrm{m}}) }{ A_1 + D B^2 }; \tag{15} \]
\[ m = \frac{ (b\gamma_{\mathrm{m}1}E_{\mathrm{m}1}+\gamma_{\mathrm{m}2}E_{\mathrm{m}2})a_2^2 }{ b(A_1 + D B^2) }. \tag{16} \]
A numerical study of function (14) was carried out for 110 kV substation two-winding transformers with ratings from 60 to 240 MVA, with cores made of cold-rolled steel, with an aluminum primary HV winding and a copper secondary LV winding. The initial data were taken to be the same as in [2]. The study showed that the minimum of (14) occurs at \(y_2 \simeq 1.5\) cm and at \(x_2\) depending on the transformer rating. The dependence of the critical values of \(x_2\) on transformer rating is shown in Fig. 2.
Fig. 2.
Fig. 2 also shows the dependence of the stress in the material of the outer layer of the LV winding \((\sigma_p)\), caused by radial forces during a short circuit beyond the transformer, on its rating \((S)\). To calculate \(\sigma_p\), expression (27) from [2] was used, reduced to the form
\[ \sigma_p = 2.27\, \frac{(k_{\mathrm{п}2}x_2+y_2)(d+2x_1+2x_2+18)} {y_2+b_2} \tag{17} \]
The ratio of the stress in the material of the outer layer to the stress of the inner LV winding is
\[ \sigma_{pн}:\sigma_{pв}=(d+2x_1+2x_2+18):(d+2x_2+8). \]
It is seen from Fig. 2 that, for the adopted mutual arrangement of the LV and HV windings, the stresses in the material of the outer layer of the LV winding for a 60 MVA transformer amount to \(480\ \text{kg}/\text{cm}^2\), and for a transfor-
of a 240 MVA transformer—960 kg/cm², that is, considerably less than for transformers with single-layer windings [2].
For the winding arrangement under consideration, only the LV winding experiences significant mechanical stress and must be made of copper. The HV winding works in compression and may be made of aluminum. This can reduce copper expenditures in the production of large power transformers.
Table 1.
| Transformer rating, MVA | 60 | 90 | 120 | 180 | 240 |
|---|---|---|---|---|---|
| 1. \(y_2\) adopted, cm | 1 | 1 | 1 | 1 | 1 |
| 2. \(x_2\) from Fig. 2, cm | 3 | 3.2 | 3.5 | 4 | 4.5 |
| 3. \(\Delta_2\) from (14), A/cm² | 418 | 416 | 412 | 405 | 400 |
| 4. \(\Delta_1\) at \(\sigma=2.5\), A/cm² | 167 | 166 | 165 | 162 | 160 |
| 5. \(y_1\) from (8), cm | 3.5 | 3.7 | 3.8 | 3.7 | 3.7 |
| 6. \(x_1\) from (9), cm | 12.8 | 13.5 | 14.7 | 17 | 19 |
| 7. \(d\) from (11), cm | 65.2 | 71 | 78 | 91 | 104 |
| 8. \(h\) from (10), cm | 179 | 214 | 216 | 216 | 214 |
| 9. \(H=h+2l_{\text{ц}}\), cm | 197 | 232 | 234 | 234 | 232 |
| 10. \(a\) from (1), cm | 66.7 | 69.5 | 73 | 79 | 86 |
| 11. \(Q_c=\gamma_c q_c l_c\), tons | 26 | 34.4 | 43.7 | 62.8 | 88 |
| 12. \(Q_{\text{м}}=3\gamma_a q_2 l_{\text{м}}\), tons | 3.9 | 5.3 | 6.4 | 8.2 | 10.5 |
| 13. \(Q_a=3\gamma_a q_1 l_{\text{м}}\), tons | 2.95 | 4.0 | 4.85 | 6.27 | 7.9 |
| 14. \(\sigma_p\) from (17), kg/cm² | 480 | 540 | 620 | 780 | 960 |
Let us note here that, in transformers with single-layer windings, the inner winding (lying on the core limb) experiences compression and, evidently, can be calculated for bending as a multi-supported arched beam. The role of the supports will be played by the strips. Consequently, this winding will always have a margin of strength and therefore may be made of aluminum.
In conclusion, the table gives the principal dimensions of large substation transformers with a two-layer copper LV winding and a single-layer aluminum HV winding. The initial data are taken from [2].
The calculated value \(y_2\) is taken equal to 1 cm, with the calculated costs exceeding the minimum by no more than 1%. The weight of the LV winding made of copper is denoted by \(Q_{\text{м}}\) and is given in item 12, while the weight of the HV winding made of aluminum, \(Q_a\), is given in item 13. The stress in the material of the outer layer of the LV winding is given in item 14.
As is evident from this table, three quarters of the windings of a transformer of any rating can be made of aluminum, if it were possible to make one of the windings two-layer.
REFERENCES
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Kutyavin I. D., On determining the optimum dimensions of three-phase two-winding transformers. In this same collection.
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Del G. V., Krasnov V. P., Kutyavin I. D., On the techno-economic determination of the optimum dimensions of large-capacity substation three-phase two-winding transformers. In this same collection.