Techno-economic studies of single-phase three-winding transformers
Abstract Generated abstract
This paper develops a techno-economic optimization procedure for single-phase three-winding power transformers with high-voltage and medium-voltage disk windings and a low-voltage helical winding. A cost function combining manufacturing and operating costs is expressed through transformer dimensions, active-material weights, current densities, thermal constraints, power relations, and short-circuit voltage requirements, then minimized numerically because the general analytical problem leads to high-order equations. Calculations for 60, 200, and 400 MVA per-phase transformers compare core-type and split-core designs, conductor-size constraints, and economic coefficients. The results indicate that core-type designs have lower calculated costs but larger transport-limiting dimensions, standard conductor restrictions increase height and leakage voltage, and the studied price coefficients have little effect on optimal winding dimensions or transformer weight.
Full Text
TECHNO-ECONOMIC STUDIES OF SINGLE-PHASE THREE-WINDING TRANSFORMERS
I. D. KUTYAVIN, V. P. KRASNOV, G. V. DEL
(Presented by the scientific seminar of the Department of Electrical Stations and Networks)
This paper presents the results of minimizing the expression for calculated costs [1], used for three-winding transformers. Figure 1 shows the arrangement of the windings in the core window, where the high-voltage (HV) and medium-voltage (MV) windings are continuous disk windings, and the low-voltage (LV) winding is a helical winding [2].
Fig. 1.
The equation of the costs for manufacture and operation of a three-winding transformer has the form:
\[ Z=(A_1+DB^2)Q_c+\left[\beta A_2+\varepsilon(\Delta_1K_{31})^2\right]Q_{m1} +\left[\beta A_2+\varepsilon(\Delta_2K_{32})^2\right]Q_{m2} +\left[\beta A_3+\varepsilon(\Delta_3K_3)^2\right]Q_{m3}. \tag{1} \]
Here \(A_1;\ A_2;\ D;\ \varepsilon;\ \beta\) are constants [1];
\[
\Delta_1;\ \Delta_2;\ \Delta_3
\]
are the current densities in the windings (HV; MV; LV), \((\text{A}/\text{cm}^2)\);
\(Q_c\) is the weight of the core steel, kg;
\(Q_{m1};\ Q_{m2};\ Q_{m3}\) are the weights of the conducting material of the windings (HV; MV; LV), kg;
\(K_{31};\ K_{32};\ K_{33}\) are the transformer load coefficients.
To obtain equation (1) as a function of the parameters and dimensions of the transformer, one may use the following basic transformer equations.
The heat-balance equation for coils (windings) per 1 cm of the mean turn length:
\[ \left. \begin{aligned} 2\sigma_1(x_1K_{x1}+y_1K_{y1})&=\rho_1\Delta_1^2x_1y_1K_1;\\ 2\sigma_2(x_2K_{x2}+y_2K_{y2})&=\rho_2\Delta_2^2x_2y_2K_2;\\ 2\sigma_3(x_3K_{x3}+y_3K_{y3})&=\rho_3\Delta_3^2x_3y_3K_3. \end{aligned} \right\} \tag{2} \]
The power of the transformer windings (per limb)
\[ \left. \begin{aligned} S_1&=K\Delta_1q_{m1}q_c;\\ S_2&=K\Delta_2q_{m2}q_c;\\ S_3&=K\Delta_3q_{m3}q_c. \end{aligned} \right\} \tag{3} \]
The reactive components of the short-circuit voltage
\[ \left. \begin{aligned} u_{p1-3}&=\frac{7.92K_r\Delta_1q_{m1}l_{13}a_{p13}}{4.44Bhq_c} =\frac{7.92K_r\Delta_3q_{m3}l_{13}a_{p13}}{4.44Bhq_c};\\[4pt] u_{p2-3}&=\frac{7.92K_r\Delta_2q_{m2}l_{23}a_{p23}}{4.44Bhq_c} =\frac{7.92K_r\Delta_3q_{m3}l_{23}a_{p23}}{4.44Bhq_c};\\[4pt] u_{p1-2}&=\frac{7.92K_r\Delta_1q_{m1}l_{12}a_{p12}}{4.44Bhq_c} =\frac{7.92K_r\Delta_2q_{m2}l_{12}a_{p12}}{4.44Bhq_c}. \end{aligned} \right\} \tag{4} \]
Since the number of starts of the LV helical winding is an integer, the equation of the magnetomotive force of a turn may be written as
\[ I_{n3}W_{k3}=x_3y_3\Delta_3K_3. \tag{5} \]
In these equations:
\(\sigma_1;\ \sigma_2;\ \sigma_3\) are the calculated heat-flux densities from the surface of the coil \((\text{W}/\text{cm}^2)\);
\(K_{x1};\ K_{x2};\ K_{x3};\ K_{y1};\ K_{y2};\ K_{y3}\) are coefficients taking into account the covering of the coil surface by insulating materials;
\(\rho_1;\ \rho_2;\ \rho_3\) are the specific resistivities of the winding material, \(\Omega\cdot\text{cm}\);
\(K_1;\ K_2;\ K_3\) are coefficients of filling the cross-sectional area of the coils (windings) with conducting material [3].
\[ \begin{gathered} K_1=\frac{1}{1+\lambda_1\Delta_1 y_1};\qquad \lambda_1=\frac{m_1 i_1}{I_{\mathrm{n}1}};\\ K_2=\frac{1}{1+\lambda_2\Delta_2 y_2};\qquad \lambda_2=\frac{m_2 i_2}{I_{\mathrm{n}2}};\\ K_3=\frac{1}{1+\lambda_3\Delta_3 y_3};\qquad \lambda_3=\frac{m_3 i_3}{I_{\mathrm{n}3}}. \end{gathered} \tag{6} \]
\(I_{\mathrm{n}1};\ I_{\mathrm{n}2};\ I_{\mathrm{n}3}\) — nominal current in the winding (per one limb), and
\(m_1;\ m_2;\ m_3\) — number of parallel conductors in a branch of the winding;
\(i_1;\ i_2;\ i_3\) — thickness of the conductor insulation on two sides (cm);
\(K = 4.44 f B\,10^{-7}\);
\(f\) — frequency, (Hz);
\(B\) — induction in the limb, (T);
\(q_{\mathrm{m}1};\ q_{\mathrm{m}2};\ q_{\mathrm{m}3}\) — cross-sectional area of the winding material.
\[ \begin{gathered} q_{\mathrm{m}1}=K_1 x_1 y_1\frac{h}{y_1+\delta_1} =\frac{x_1 y_1 h}{(y_1+\delta_1)(1+\lambda_1\Delta_1 y_1)};\\ q_{\mathrm{m}2}=K_2 x_2 y_2\frac{h}{y_2+\delta_2} =\frac{x_2 y_2 h}{(y_2+\delta_2)(1+\lambda_2\Delta_2 y_2)};\\ q_{\mathrm{m}3}=K_3 x_3 y_3\frac{h}{y_3+\delta_3} =\frac{x_3 y_3 h}{(y_3+\delta_3)(1+\lambda_3\Delta_3 y_3)}. \end{gathered} \tag{7} \]
\(\delta_1;\ \delta_2;\ \delta_3\) — axial distance between the coils of the windings, taking into account conductor insulation, (cm);
\(q_{\mathrm{c}}\) — cross-sectional area of the steel of the limb, (cm\(^2\));
\[ q_{\mathrm{c}}=K_{\mathrm{c}}\frac{\pi d^2}{4}; \tag{8} \]
\(K_{\mathrm{c}}\) — coefficient of filling with steel the area of the circle (of diameter \(d\)) circumscribed about the limb;
\(K_{\mathrm{r}}\) — Rogowski coefficient;
\(l_{13};\ l_{12};\ l_{23}\) — mean diameter of the leakage channel between windings, (cm);
\(a_{\mathrm{p}13};\ a_{\mathrm{p}12};\ a_{\mathrm{p}23}\) — reduced width of the leakage channel between the corresponding windings;
\(h\) — winding height, (cm).
The weights of the active material of the transformer are
\[ Q_{\mathrm{c}}=\gamma_{\mathrm{c}} q_{\mathrm{c}} l_{\mathrm{c}}\,10^{-3}; \tag{9} \]
\[ \begin{gathered} Q_{\mathrm{m}1}=n_{\mathrm{c}}\gamma_{\mathrm{m}1}q_{\mathrm{m}1}l_{\mathrm{m}1}\cdot10^{-3};\\ Q_{\mathrm{m}2}=n_{\mathrm{c}}\gamma_{\mathrm{m}2}q_{\mathrm{m}2}l_{\mathrm{m}2}\cdot10^{-3};\\ Q_{\mathrm{m}3}=n_{\mathrm{c}}\gamma_{\mathrm{m}3}q_{\mathrm{m}3}l_{\mathrm{m}3}\cdot10^{-3}, \end{gathered} \tag{10} \]
where \(\gamma_{\mathrm{c}}\) — specific weight of the core steel, g/cm\(^3\);
\(\gamma_{\mathrm{m}1};\ \gamma_{\mathrm{m}2};\ \gamma_{\mathrm{m}3}\) — specific weight of the conductor material of the windings, (g/cm\(^3\));
\(l_{\mathrm{c}}\) — length of the core steel, referred to the cross-sectional area of the limb, (cm).
\[ l_{\mathrm{c}}=(2+n_{\mathrm{c}})(h+2l_{\mathrm{u}})+K_{\mathrm{a}}\bigl[2n_{\mathrm{c}}(x_1+x_2+x_3+\delta_{03}+\delta_{23}+\delta_{12} +n_{10}\delta_{10}+n_{11}\delta_{11})+n_{\mathrm{d}}d\bigr]; \tag{11} \]
\(l_{m1};\ l_{m2};\ l_{m3}\)—average length of the windings, (cm).
\[ \begin{aligned} l_{m1} &= \pi \left(d+x_1+2x_2+2x_3+2\delta_{03}+2\delta_{23}+2\delta_{12}\right);\\ l_{m2} &= \pi \left(d+x_2+2x_3+2\delta_{03}+2\delta_{23}\right);\\ l_{m3} &= \pi \left(d+x_3+2\delta_{03}\right); \end{aligned} \tag{12} \]
Here \(n_c;\ n_\sigma;\ n_{10};\ n_{11};\ n_d\) are coefficients taking into account the type of core; for a core-type core (Fig. 1, a) \(n_c=0;\ n_\sigma=1;\ n_{10}=0;\ n_{11}=0.5;\ n_d=3.6\); for a core with a split yoke (Fig. 1, b) \(n_c=1.05;\ n_\sigma=2;\ n_{10}=0.5;\ n_{11}=0.25;\ n_d=2.8\); \(l_u\) is the average insulation distance from the winding to the yoke, (cm); \(K_y\) is the coefficient of increase of the yoke cross section; \(\delta_{03};\ \delta_{23};\ \delta_{12};\ \delta_{10};\ \delta_{11}\) are shown in Fig. 1.
Minimization of equation (1) in general form represents a rather complex problem, since equations are obtained with powers of the unknowns equal to 16 and higher. Therefore, it is more convenient to carry out the investigation by a numerical method.
In the investigations it is assumed that the coils are cooled mainly from the surfaces \(x\). As a result of solving the system of equations (2, 3, 4, 5, 6, 7, and 8) with respect to \(y_2\) and \(y_3\), the following calculation formulas were obtained (from equation 4 the first two were taken, since they include all the necessary dimensions).
\[ \Delta_2=\frac{\lambda_2+\sqrt{\lambda_2^2+\dfrac{4a_2}{y_2}}}{2a_2}, \tag{13} \]
\[ x_2=\frac{S_2 I_{n3} W_{k3}(y_2+\delta_2)(1+\lambda_2\Delta_2 y_2)} {S_3 (y_3+\delta_3)\Delta_2 y_2}; \tag{14} \]
\[ x_3=\frac{W_{k3}\left[m_3 i_3+\sqrt{(m_3 i_3)^2+\dfrac{4a_3 I_{n3}^2}{y_3}}\right]}{2}; \tag{15} \]
\[ \Delta_3=\frac{x_3}{a_3 I_{n3} W_{k3}}; \tag{16} \]
\[ d=\frac{U_1+\sqrt{U_1^2+4P_1U_1v_1}}{2p_1}; \tag{17} \]
\[ x_1=\frac{P_2}{U_2}-v_2; \tag{18} \]
\[ \Delta_1=\frac{(\delta_1\lambda_1+M)-\sqrt{(\delta_1\lambda_1+M)^2-4\delta_1(M\lambda_1+a_1)}}{2\delta_1a_1}; \tag{19} \]
\[ y_1=\frac{1}{\Delta_1(a_1\Delta_1-\lambda_1)}; \tag{20} \]
\[ h=\frac{S_3(y_3+\delta_3)} {1.743\cdot 10^{-5} B d^2 I_{n3} W_{k3}}; \tag{21} \]
Here
\[ a_1=\frac{\rho_1}{2\sigma_1 K_{x1}};\qquad a_2=\frac{\rho_2}{2\sigma_2 K_{x2}}; \]
\[ a_3=\frac{\rho_3}{2\sigma_3 K_{x3}};\qquad p_1=\frac{U_{p2-3}K_c B (y_3+\delta_3)} {2.38\cdot 10^{-4}K_r I_{n3} W_{k3}}; \]
\[ U_1=x_2+x_3+3\delta_{23};\quad v_1=2x_3+2\delta_{03}+\delta_{23}; \]
\[ P_2=\frac{P_1 d^2 U_{\mathrm{p}1-3}}{U_{\mathrm{p}2-3}};\quad M=\frac{S_3 x_1 (y_3+\delta_3)}{S_1 I_{\mathrm{n}3} W_{\mathrm{k}3}}; \]
\[ U_2=v_1+d+x_2+\delta_{23};\quad v_2=3x_2+x_3+3(\delta_{12}+\delta_{23}). \]
Of great importance for high-power transformers is the overall height dimension. To determine the overall dimension, the total height of the transformer core was calculated (\(H\) in Fig. 1)
\[ H=h+2l_{\mathrm{i}}+K_d d. \tag{22} \]
Here \(K_d\) is a coefficient allowing for the height of the core yokes.
Investigations of the cost function (1) were carried out on an electronic digital computer. In compiling the program it was taken into account that, in expressions (18) and (20), negative values of \(x_1\) and \(y_1\) may occur, which has no physical meaning.
For the investigations, transformers of 60, 200, and 400 MVA (per phase) were taken, with voltages: HV-242, MV-121, LV-20 kV and with the following data:
\[ \rho_1=\rho_2=\rho_3=2{,}14\cdot 10^{-6};\quad \sigma_1=\sigma_2=\sigma_3=0{,}2;\quad i_1=0{,}2;\quad i_2=0{,}2;\quad i_3=0{,}055; \]
\[ K_c=0{,}8;\quad B=1{,}65;\quad K_r=0{,}95;\quad \delta_1=1{,}6;\quad \delta_2=1{,}4;\quad \delta_3=1{,}2;\quad \delta_{03}=2{,}5; \]
\[ \delta_{23}=5;\quad \delta_{12}=10;\quad \delta_{10}^{*}=14;\quad \delta_{11}=20;\quad \gamma_{\mathrm{m}1}=\gamma_{\mathrm{m}2}=\gamma_{\mathrm{m}3}=8{,}9; \]
\[ \gamma_c=7{,}65;\quad K_T=0{,}7;\quad \beta=3;\quad P_{\mathrm{n}}=0{,}125;\quad P_a=0{,}06; \]
\[ Z_{\mathrm{p}}=8{,}10;\quad a_{\mathrm{p}}=0{,}0193;\quad P_{\mathrm{ak}}=0{,}06;\quad C_1=150;\quad K_{\mathrm{ya}}=1{,}05; \]
\[ C_2'=C_2''=0{,}25\cdot 10^{-2};\quad t=0{,}7\cdot 10^4;\quad \alpha_c=0{,}6\cdot 10^{-3}; \]
\[ K_{\mathrm{m}}=1;\quad \tau=0{,}6\cdot 10^4;\quad K_{\mathrm{n}}=1;\quad \alpha_{\mathrm{m}}=0{,}24\cdot 10^{-6}; \]
\[ K_{31}=0{,}5;\quad K_{32}=0{,}5;\quad K_{33}=1;\quad U_{\mathrm{p}2-3}=0{,}07;\quad U_{\mathrm{p}1-3}=0{,}24;\quad l_{\mathrm{i}}=18. \]
The remaining data are given in Table 1.
Table 1
| \(S\) (MVA) | \(K_{x1(2)}\) | \(K_{x3}\) | \(S_1\) | \(S_2\) | \(S_3\) | \(I_{\mathrm{n}1}\) | \(I_{\mathrm{n}2}\) | \(I_{\mathrm{n}3}\) | \(m_1\) | \(m_2\) | \(m_3\) | \(W_{\mathrm{k}3}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 30 | 0,75 | 0,7 | 30 | 30 | 30 | 215 | 430 | 1500 | 3 | 5 | 16 | 1 |
| 100 | 0,7 | 0,7 | 0,7 | 100 | 100 | 715 | 1430 | 5000 | 4 | 8 | 16 | 0,5 |
| 200 | 0,7 | 0,7 | 0,7 | 200 | 200 | 1430 | 2860 | 10000 | 4 | 12 | 32 | 0,25 |
Table 2 gives the optimal dimensions, parameters, weights of active materials, and costs for the indicated transformers with different types of core, obtained by minimizing expression (1) with respect to the independent variables \(y_2\) and \(y_3\).
It is seen from Table 2 that the height of the HV winding coil (\(y_1\)) is greater than the maximum standard value (2.2 cm), and the current density is comparatively low. This indicates insufficient utilization of the copper of the HV winding, which may constitute a reserve for reducing the weight of the transformers under consideration.
To compare the calculated costs for a transformer with arbitrarily obtained dimensions and with dimensions constrained by standard wires, Table 3 gives the optimal dimensions and parameters of the transformer for \(y_1=2{,}2\) cm. The change in dimensions is also shown here
and costs at different values of \(y_2\) and \(y_3\) for a 200 MVA transformer (per phase) with a core made according to Fig. 1, b.
Table 2
| Core according to Fig. 1, 60 MVA | Core according to Fig. 1, 200 MVA | Core according to Fig. 1, 400 MVA | Core according to Fig. 2, 60 MVA | Core according to Fig. 2, 200 MVA | Core according to Fig. 2, 400 MVA | |
|---|---|---|---|---|---|---|
| \(y_1,\ \mathrm{cm}\) | 2,51 | 7,26 | 10,34 | 2,53 | 7,14 | 9,96 |
| \(y_2,\ \mathrm{cm}\) | 0,6 | 0,6 | 0,6 | 0,6 | 0,6 | 0,6 |
| \(y_3,\ \mathrm{cm}\) | 0,6 | 1,0 | 0,7 | 0,6 | 0,95 | 0,7 |
| \(x_1,\ \mathrm{cm}\) | 12,28 | 17,11 | 21,2 | 12,32 | 17,63 | 21,1 |
| \(x_2,\ \mathrm{cm}\) | 8,08 | 9,48 | 10,36 | 8,0 | 9,7 | 10,56 |
| \(x_3,\ \mathrm{cm}\) | 5,81 | 7,13 | 8,48 | 5,81 | 7,3 | 8,48 |
| \(\Delta_1,\ \mathrm{A/cm^2}\) | 502 | 238 | 189,3 | 502 | 227 | 182 |
| \(\Delta_2,\ \mathrm{A/cm^2}\) | 661 | 546 | 537 | 673 | 546 | 525 |
| \(\Delta_3,\ \mathrm{A/cm^2}\) | 507 | 373 | 444 | 507 | 383 | 444 |
| \(d,\ \mathrm{cm}\) | 75,73 | 107,7 | 131,4 | 75,57 | 111,3 | 132 |
| \(h,\ \mathrm{cm}\) | 272,8 | 329,6 | 382,8 | 274 | 301,9 | 379 |
| \(Q_c,\ \mathrm{T}\) | 28,03 | 70,67 | 122,4 | 38,04 | 97,09 | 161,4 |
| \(Q_m,\ \mathrm{T}\) | 8,111 | 29,34 | 51,8 | 8,098 | 28,87 | 53,1 |
| \(З,\ \mathrm{rub.}\cdot 10^3\) | 14,53 | 35,67 | 64,33 | 16,8 | 41,59 | 73,16 |
| \(U_{p\,1-2},\ \%\) | 16,1 | 15,7 | 15,6 | 16,08 | 15,7 | 15,53 |
| \(H,\ \mathrm{cm}\) | 460,3 | 581 | 681,6 | 385,6 | 431,2 | 547 |
Comparing the data of Tables 2 and 3, it may be noted that the introduction of restrictions on the size \(y_1\) leads to an increase in costs and in the overall dimension \((H)\).
Table 3
| \(y_1,\ \mathrm{cm}\) | 2,2 | 2,2 | 2,2 | 2,2 |
| \(y_2,\ \mathrm{cm}\) | 0,1 | 0,6 | 0,65 | 1,1 |
| \(y_3,\ \mathrm{cm}\) | 1,2 | 0,95 | 0,7 | 0,9 |
| \(x_1,\ \mathrm{cm}\) | 12,79 | 17,63 | 21,06 | 18,29 |
| \(x_2,\ \mathrm{cm}\) | 14,56 | 9,7 | 10,88 | 9,68 |
| \(x_3,\ \mathrm{cm}\) | 6,53 | 7,31 | 8,48 | 7,51 |
| \(\Delta_1,\ \mathrm{A/cm^2}\) | 255,7 | 227,1 | 221,6 | 225,7 |
| \(\Delta_2,\ \mathrm{A/cm^2}\) | 1219 | 545,9 | 528 | 425,8 |
| \(\Delta_3,\ \mathrm{A/cm^2}\) | 342 | 382,8 | 444 | 393 |
| \(d,\ \mathrm{cm}\) | 111,1 | 111,3 | 133,1 | 114,3 |
| \(h,\ \mathrm{cm}\) | 338 | 301,9 | 186,6 | 279,4 |
| \(Q_c,\ \mathrm{T}\) | 103,0 | 97,08 | 116,9 | 99,01 |
| \(Q_m,\ \mathrm{T}\) | 26,82 | 33,64 | 27,86 | 34,22 |
| \(З,\ \mathrm{rub.}\cdot 10^3\) | 42,69 | 44,09 | 45,74 | 44,53 |
| \(U_{p\,1-2},\ \%\) | 21,0 | 28,8 | 28,8 | 28,8 |
| \(H,\ \mathrm{cm}\) | 485,1 | 439,2 | 355,7 | 429,7 |
Table 4 presents studies of a transformer rated at 60 MVA (per phase) with a core made according to Fig. 1a, in terms of the calculated specific price of the transformer \((K_{\mathrm{t}})\) and the ratio of the calculated specific cost of copper to the cost of steel in the product—\(\beta\).
Table 4
| \(K_{\mathrm{t}}=1\) | \(K_{\mathrm{t}}=1.3\) | \(K_{\mathrm{t}}=1.5\) | \(\beta=2\) | \(\beta=4\) | \(\beta=6\) | |
|---|---|---|---|---|---|---|
| \(y_1,\ \text{cm}\) | 2,51 | 2,51 | 2,51 | 0,806 | 2,51 | 2,51 |
| \(y_2,\ \text{cm}\) | 0,6 | 0,6 | 0,6 | 2,9 | 0,6 | 0,6 |
| \(y_3,\ \text{cm}\) | 0,6 | 0,6 | 0,6 | 0,8 | 0,6 | 0,6 |
| \(x_1,\ \text{cm}\) | 12,27 | 12,27 | 12,23 | 8,45 | 12,27 | 12,27 |
| \(x_2,\ \text{cm}\) | 8,08 | 8,08 | 8,08 | 10,1 | 8,08 | 8,08 |
| \(x_3,\ \text{cm}\) | 5,81 | 5,81 | 5,81 | 5,10 | 5,81 | 5,81 |
| \(\Delta_1,\ \text{A}/\text{cm}^2\) | 502 | 502 | 502 | 656 | 502 | 502 |
| \(\Delta_2,\ \text{A}/\text{cm}^2\) | 661 | 661 | 661 | 427 | 661 | 661 |
| \(\Delta_3,\ \text{A}/\text{cm}^2\) | 507 | 507 | 507 | 445 | 507 | 507 |
| \(d,\ \text{cm}\) | 75,73 | 75,73 | 75,73 | 71,15 | 75,73 | 75,73 |
| \(h,\ \text{cm}\) | 272,8 | 272,8 | 272,8 | 343,4 | 272,8 | 272,8 |
| \(Q_{\mathrm{s}},\ \text{t}\) | 28,03 | 28,03 | 28,03 | 27,63 | 28,03 | 28,03 |
| \(Q_{\mathrm{m}},\ \text{t}\) | 8,11 | 8,11 | 8,11 | 9,40 | 8,11 | 8,11 |
| \(Z,\ \text{rub}\cdot10^3\) | 15,56 | 18,46 | 20,4 | 11,54 | 13,70 | 15,80 |
| \(U_{p\,1-2}^{*},\ \%\) | 16,06 | 16,06 | 16,06 | 15,55 | 16,06 | 16,06 |
| \(H,\ \text{cm}\) | 384,5 | 384,5 | 384,5 | 450,5 | 384,5 | 384,5 |
As can be seen from Table 4, the coefficient \(K_{\mathrm{t}}\) and \(\beta\) have practically no effect on the elementary dimensions of the transformer windings or on the overall dimensions. Significant changes in dimensions for \(\beta<3\) will not have a substantial effect on the transformers under study, since for high-power transformers \(\beta\) is usually greater than 3.
On the basis of the studies carried out, the following conclusions may be drawn:
-
Despite lower costs, transformers with a core-type core (Fig. 1a) have larger overall dimensions, which makes their transportation by rail more difficult. Therefore, it is apparently more expedient to adopt transformers having a split core (Fig. 1b), despite their greater weight and costs.
-
Introducing restrictions on the dimensions of existing winding conductors makes it possible to manufacture the winding more simply from the standpoint of technology, but this leads to an increase in the overall height of the transformer and in the leakage voltage.
-
Since the quantities \(K_{\mathrm{t}}\) and \(\beta\) have practically no effect on the elementary dimensions of the windings and the weight of the transformer, there is no need, for studies, to strive for an exact determination of their values.
REFERENCES
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I. D. Kutyavin. On the determination of the optimal dimensions of three-phase two-winding transformers. Izv. TPI, vol. 130, 1964.
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P. M. Tikhomirov. Calculation of Transformers. Gosenergoizdat, 1962.
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G. V. Del, V. P. Krasnov. Techno-economic study of the optimal dimensions of power transformers. Izv. TPI, vol. 132, 1964.