Toward a Techno-Economic Determination of the Optimal Dimensions of Single-Phase Two-Winding Transformers
Abstract Generated abstract
This article extends a previously developed techno-economic optimization method for three-phase two-winding transformers to single-phase 110/10 kV substation transformers rated from 10.5 to 50 MVA without voltage regulation. It derives dimensional, thermal, electromagnetic, material-weight, loss, and cost relations for shell-type and core-type magnetic circuits, then reduces the cost minimization problem under the simplifying assumption of equal current densities in the two windings. Numerical calculations using specified design and economic parameters determine near-optimal winding dimensions and principal transformer characteristics for both core arrangements. The results indicate that moderate deviations from the calculated optimum have little effect on total calculated costs, and that core-type designs in the examined cases have lower calculated costs despite greater conductor weight.
Full Text
Toward a Techno-Economic Determination of the Optimal Sizes of Single-Phase Two-Winding Transformers
Doctor of Technical Sciences I. D. KUTYAVIN, postgraduate student G. V. DEL,
postgraduate student V. P. KRASNOV.
The methodology for the techno-economic determination of the optimal sizes of three-phase two-winding transformers, indicated in [1 and 2], can also be applied to single-phase transformers.
This article presents the results of calculations obtained by applying the above-mentioned methodology to substation single-phase two-winding transformers rated at 110/10 kV, with capacities of 10.5; 20; 30; 40, and 50 MVA, without a voltage-regulating device.
Figure 1 shows the arrangement of the windings, the dimensions, and the insulation clearances for a transformer with a shell-type core (the designations in parentheses refer to a transformer with a core-type core) [3].
The high-voltage winding may be made as a continuous coil winding, and the low-voltage winding as a helical winding [3].
The basic relations connecting the dimensions of the transformer are presented in the following form (all dimensions are in cm).
Height of the core window:
\[ H = h + 2l_{\text{и}} \tag{1} \]
where \(h\) is the height of the winding;
\(l_{\text{и}}\) is the insulation distance from the winding to the yoke.
Width of the core window for a transformer with a shell-type core:
\[ a = x_1 + x_2 + \delta_{02} + \delta_{12} + \delta_{10}, \tag{2} \]
for a transformer with a core-type core:
\[ a = 2(x_1 + x_2 + \delta_{02} + \delta_{12} + 0.5\delta_{11}), \tag{2a} \]
where \(x_1\) and \(x_2\) are the widths of the coils of the phase winding;
\(\delta_{02}\) is the insulation distance from the limb to the secondary winding;
\(\delta_{12}\) is the insulation distance between the primary and secondary windings;
\(\delta_{10}\) is the insulation distance from the primary winding to the limb;
\(\delta_{11}\) is the insulation distance between the primary windings.
Average turn length of the primary winding
\[ l_{m1}=\pi\bigl(d+x_1+2x_2+2\delta_{02}+2\delta_{12}\bigr). \tag{3} \]
Average turn length of the secondary winding
\[ l_{m2}=\pi\bigl(d+x_2+2\delta_{02}\bigr), \tag{4} \]
where \(d\) is the diameter of the limb.
Fig. 1.
The length of core steel, referred to the cross-sectional area of the limb,
for transformers with a shell-type core
\[ l_c=(h+2l_u)(1+k_{\text{я}})+k_{\text{я}}\bigl[2(x_1+x_2+\delta_{02}+\delta_{12}+\delta_{10})+1.8d\bigr] \tag{5} \]
and for transformers with a core-type magnetic circuit
\[ l_c=2h+4l_u+k_{\text{я}}\bigl[4(x_1+x_2+\delta_{02}+\delta_{12}+0.5\delta_{11})+3.6d\bigr], \tag{5a} \]
where \(k_{\text{я}}\) is the coefficient of increase of the cross-sectional area by the yokes.
The cross-sectional area of the material of the phase windings is
\[ q_{m1}=k_1x_1y_1\,\frac{h}{y_1+\delta_1};\qquad q_{m2}=k_2x_2y_2\,\frac{d}{y_2+\delta_2}, \tag{6} \]
where \(k_1\) and \(k_2\) are the coefficients of filling, by conductor material, of the cross-sectional areas of the coils of the primary and secondary windings;
\(y_1\) and \(y_2\) are the heights of the winding coils;
\(\delta_1\) and \(\delta_2\) are the heights of the radial cooling ducts between the coils of the primary and secondary windings.
The cross-sectional area of the steel of the core is
\[ q_c=k_c\frac{\pi d^2}{4}, \tag{7} \]
where \(k_c\) is the coefficient of filling by steel of the area of a circle with diameter \(d\).
The heat-balance equation of the winding coil (per running cm of the mean turn)
\[ \left. \begin{aligned} 2\sigma_1(k_{x1}x_1+k_{y1}y_1)&=\rho \Delta_1^2 k_1x_1y_1,\\ 2\sigma_2(k_{x2}x_2+k_{y2}y_2)&=\rho \Delta_2^2 k_2x_2y_2 \end{aligned} \right\} \tag{8} \]
where \(\sigma_1\) and \(\sigma_2\) are the calculated heat-flux density from the surface of the coil of the primary and secondary windings, W/cm;
\(\rho\) is the specific resistance of the winding material, ohm cm;
\(k_{x1}, k_{x2}, k_{y1}, k_{y2}\) are coefficients allowing for the coverage of the coil surfaces \(x_1, y_1, x_2, y_2\) by insulating parts.
On the basis of equations (8) one may write an expression for the current density of the primary \(\Delta_1\) and secondary \(\Delta_2\) windings
\[ \left. \begin{aligned} \Delta_1&=\sqrt{\frac{2\sigma_1(k_{x1}x_1+k_{y1}y_1)}{\rho k_1x_1y_1}} =\alpha_1\sqrt{\frac{z_1}{x_1y_1}},\\ \Delta_2&=\sqrt{\frac{2\sigma_2(k_{x2}x_2+k_{y2}y_2)}{\rho k_2x_2y_2}} =\alpha_2\sqrt{\frac{z_2}{x_2y_2}} \end{aligned} \right\} \tag{9} \]
where
\[ \left. \begin{gathered} z_1=k_{n1}x_1+y_1;\qquad z_2=k_{n2}x_2+y_2;\qquad k_{n1}=\frac{k_{x1}}{k_{y1}},\\ \alpha_1=\sqrt{\frac{2\sigma_1k_{y1}}{\rho k_1}};\qquad \alpha_2=\sqrt{\frac{2\sigma_2k_{y2}}{\rho k_2}};\qquad k_{n2}=\frac{k_{x2}}{k_{y2}} \end{gathered} \right\} \tag{10} \]
The expression for the transformer power in kVA is
\[ S=K\Delta_1q_{m1}q_cn_c=K\Delta_2q_{m2}q_cn_c . \tag{11} \]
Substituting into (11) the values \(\Delta_1, q_{m1}\), and \(q_c\) from (9), (6), and (7), we find the expression for the winding height
\[ h=\frac{N_2(y_2+\delta_2)}{d^2\sqrt{z_2x_2y_2}} =\frac{N_1(y_1+\delta_1)}{d^2\sqrt{z_1x_1y_1}} . \tag{12} \]
For transformers with a shell-type core,
\[ N_2=\frac{4S}{Kk_c\,k_2\pi a_2};\qquad N_1=\frac{4S}{Kk_c\,k_1\pi a_1}, \tag{12a} \]
and for transformers with a core-type core,
\[ N_2=\frac{2S}{Kk_c\,k_2\pi a_2};\qquad N_1=\frac{2S}{Kk_c\,k_1\pi a_1}, \tag{13a} \]
here
\[ K=4.44\,fB\cdot10^{-11} \]
where \(f\) is the frequency, Hz;
\(B\) is the induction in the core, G;
\(n_c\) is the number of transformer limbs carrying a winding.
The reactive component of the short-circuit voltage [3] (in relative units) is reduced to the following form:
\[ u_p = \frac{ 7{,}92 k_r \, \Delta_2 q_m \, 2\pi (d+2x_2+2\delta_{02}+\delta_{12})(x_1+x_2+3\delta_{12}) }{ 13{,}32 B q_c h }, \tag{15} \]
where \(k_r\) is the Rogowski coefficient.
After substituting the variables into (15), we find an expression for the diameter of the core limb:
\[ d = \frac{ U\sqrt{U^2+4PUV(y_2+\delta_2)} }{ 2P(y_2+\delta_2) }, \tag{16} \]
where
\[ \left. \begin{aligned} U &= (x_1+x_2+3\delta_{12})\sqrt{z_2y_2x_2},\\ V &= 2x_2+2\delta_{02}+\delta_{12},\\ P &= \frac{13{,}32 B k_c u_p}{7{,}92 k_r k_2 a_2}. \end{aligned} \right\} \tag{17} \]
Weight of the core steel, kg,
\[ Q_c = \gamma_c q_c l_c 10^{-3} \tag{18} \]
for transformers with a shell-type core,
\[ Q_c = \frac{k_c \pi \gamma_c 10^{-3}}{4} \left\{ \frac{N_2(y_2+\delta_2)(1+k_q)}{\sqrt{z_2x_2y_2}} + 2l_u(1+k_q)d^2 + k_q d^2 \left[ 2(x_1+x_2+\delta_{02}+\delta_{12}+\delta_{10})+1{,}8d \right] \right\} \tag{19} \]
for transformers with a core-type core,
\[ Q_c = \frac{K_c \pi \gamma_c 10^{-3}}{4} \left\{ \frac{2N_2(y_2+\delta_2)}{\sqrt{z_2x_2y_2}} + d^2 \left[ 4k_q(x_1+x_2+\delta_{02}+\delta_{12}+0{,}5\delta_{11}) + 3{,}6k_q d + 4l_u \right] \right\}. \tag{20} \]
Weight of the conductor material of the transformer windings, kg,
\[ Q_m = \gamma_m 10^{-3}(q_{m1}l_{m1}+q_{m2}l_{m2}). \tag{21} \]
Substituting the values of the variables into (21), we find:
\[ Q_m = \frac{ 3{,}33\gamma_m u_p B k_c \pi 10^{-3} }{ 7{,}92k_r (d+2x_2+\delta_{02}+\delta_{12})(x_1+x_2+3\delta_{12}) } \cdot \left[ \frac{ N_1(y_1+\delta_1)(d+x_1+2x_2+2\delta_{02}+2\delta_{12}) }{ a_1z_1 } + \frac{ N_2(y_2+\delta_2)(d+x_2+2\delta_{02}) }{ a_2z_2 } \right] \tag{22} \]
Expressions (21) and (22) contain four variables \((x_1, y_1, x_2, y_2)\). One of the variables can be eliminated by using the equality of the magnetomotive forces of the windings, which has the following form:
\[ \frac{k_1x_1y_1\Delta_1}{y_1+\delta_1} = \frac{k_2x_2y_2\Delta_2}{y_2+\delta_2}. \tag{23} \]
As a result, three variables remain.
The expression for the calculated costs, presented in the form [1],
\[ Z=(A_1+DB^2)Q_c+(\beta A_2+E\Delta_1^2)Q_{\mathrm{M}1} +(\beta A_2+E\Delta_2^2)Q_{\mathrm{M}2}, \tag{24} \]
where \(A_1, A_2, D, E, \beta\) are constants [1], is a function of three independent variables.
Investigation of equation (24) for the minimum of the calculated costs is rather difficult. In the present article a particular case is considered, when \(\Delta_1=\Delta_2=\Delta\).
Bearing in mind that the coefficient of filling the coil cross section of the secondary winding with conductor material \(k_2\) is practically independent of the transformer power, it is convenient to take \(x_2\) and \(y_2\) as the independent variables. From (23) we have
\[ x_1=\frac{k_2x_2y_2(y_1+\delta_1)} {(y_2+\delta_2)k_1y_1}. \tag{25} \]
Solving (8) and (25) jointly, we find the expression for \(y_1\):
\[ y_1= \frac{(kx_2y_2-\sigma z_2\delta_1)\pm \sqrt{(kx_2y_2-\sigma z_2\delta_1)^2 -4kk_y\delta_1x_2y_2(y_2+\delta_2)}} {2\{\sigma z_2-k_y(y_2+\delta_2)\}}, \tag{26} \]
where
\[ k=k_{x1}\frac{k_2}{k_1}; \qquad \sigma=\frac{\varepsilon_2}{\varepsilon_1}. \]
The expression for the calculated costs (24) under the adopted conditions has the form:
\[ Z=(A_1+DB^2)Q_c+(\beta A_2+E\Delta^2)Q_{\mathrm{M}} = \frac{k_c\pi\gamma_c\cdot 10^{-3}}{4}\,(A_1+DB^2)\,\varphi_1(x_2,y_2) \tag{28} \]
For transformers with a shell-type core,
\[ \varphi_1(x_2,y_2)= N_2(y_2+\delta_2) \left[ \frac{2}{\sqrt{z_2x_2y_2}}+ \right. \]
\[ \left. +\frac{ M(2d+x_1+3x_2+4\delta_{02}+2\delta_{12}) \left(n+m\frac{z_2}{\lambda y_2}\right) }{ z_2(d+2x_2+2\delta_{02}+\delta_{12})(x_1+x_2+3\delta_{12}) } \right] +d^2\left\{ k_{\mathrm{я}}\,[2(x_1+x_2+\delta_{02}+ \delta_{12}+\delta_{10})+1.8d]+2l_u(1+k_{\mathrm{я}}) \right\} \tag{29} \]
and for transformers with a core-type core,
\[ \varphi_1(x_2,y_2)= N_2(y_2+\delta_2) \left[ \frac{1+k_{\mathrm{я}}}{\sqrt{z_2x_2y_2}}+ \right. \]
\[ \left. +\frac{ M(2d+x_1+3x_2+4\delta_{02}+2\delta_{12}) \left(n+m\frac{z_2}{x_2y_2}\right) }{ z_2(d+2x_2+2\delta_{02}+\delta_{12})(x_1+x_2+3\delta_{12}) } \right] + \]
\[ +d^2[4k_{\mathrm{я}}(x_1+x_2+\delta_{02}+\delta_{12}+0.5\delta_{11}) +3.6k_{\mathrm{я}}d+4l_u], \tag{29a} \]
where
\[ M=\frac{P\gamma_{\mathrm{M}}}{k_c}\,\frac{k_2}{j_c}; \qquad n=\frac{\beta A_2}{A_1+DB^2}; \qquad m=\frac{E\alpha^2}{A_1+DB^2}. \tag{30} \]
Since a general analytical investigation of the minimum of functions (29) and (29a) is complicated, a numerical determination of the opt-
optimal values of \(x_2\) and \(y_2\) for transformers with a core made of cold-rolled steel.
The following initial data were adopted in the calculation:
\[ \begin{gathered} k_c=0.82;\quad k_{x1}=k_{x2}=0.7;\quad k_{y1}=k_{y2}=0.9;\quad k_{\mathrm{я}}=1.05;\quad k_2=0.82;\quad k_m=0.7;\\ k_{\mathrm{н}}=0.8;\quad k_{\mathrm{г}}=0.95;\quad k_{\mathrm{т}}=0.7\ \text{rubles/kg};\quad \delta_1=0.75\ \text{cm};\quad \delta_2=0.6\ \text{cm};\\ \delta_{02}=1.5\ \text{cm};\quad \delta_{12}=5\ \text{cm};\quad \delta_{11}=6\ \text{cm};\quad \delta_{10}=9\ \text{cm};\quad l_0=9\ \text{cm};\quad u_p=0.105;\\ B=16500\ \text{G};\quad \alpha_c=0.8\cdot10^{-11};\quad \gamma_c=7.65\ \text{kg/cm}^3;\quad P_{\mathrm{н}}=0.125;\quad P_a=0.06;\\ C_1=124\ \text{rubles/kW};\quad C'_2=0.007\ \text{rubles/kWh};\quad C''_2=0.009\ \text{rubles/kWh};\\ \tau=4000\ \text{h};\quad t=8700\ \text{h};\quad \left(\frac{k_{\mathrm{н}}S_m}{S}\right)^2=0.4;\quad Z_p=1.5\ \text{rubles/kvar};\\ a_1=0.046\ \text{kVA/kg};\quad \rho_m=2.14\cdot10^{-6}\ \Omega\cdot\text{cm};\quad \alpha_m=2.4\cdot10^{-7};\quad \gamma_m=8.9\ \text{kg/cm}^3;\\ \beta=3.2;\quad \sigma_1=\sigma_2=0.15\ \text{W/cm};\quad k_1=0.64\ \text{for transformers rated }10.5\text{ and }20\ \text{MVA}\\ \text{and }k_1=0.52\ \text{for transformers rated }30,\ 40,\text{ and }50\ \text{MVA}. \end{gathered} \]
Fig. 2. Curve 1 for a 10.5 MVA transformer;
2 — 20; 3 — 30; 4 — 40; 5 — 50 MVA.
The investigation of equation (28) was carried out for \(x_2 = 2 \div 8\) with a step of 1 and for \(y_2 = 0.5 \div 6\) with a step of 0.5.
Figure 2 shows the dependence \(Z = f_1(x_2 y_2)\) for \(y_2 = 1.5\) cm for transformers with a shell-type core, and Fig. 3—for transformers with a core-type core.
Fig. 3. Curve 1 for a 10.5 MVA transformer;
2—20; 3—30; 4—40; 5—50 MVA.
Figure 4 shows the dependence of the optimal value on power, found from the curves in Figures 2 and 3. Curve 1 in Fig. 4 corresponds to transformers with a shell-type core, curve 2—to transformers with a core-type core.
Figure 5 gives the dependence of costs on \(y_2\) for \(x_2\) corresponding to the optimal value. Curves 1, 2, 3, 4, and 5 correspond to transformers with a shell-type core; curves 6, 7, 8, 9, and 10—to transformers with a core-type core, respectively, for ratings of 10.5; 20; 30; 40, and 50 MVA.
From the curves in Figs. 2, 3, and 5 it is evident that the calculated values \(x_2\) and \(y_2\) may be taken as differing considerably from the optimum values, which will not lead to a substantial increase in costs.
Table 1 gives the principal dimensions of transformers with a shell-type core, calculated for various values of the variables \(x_2\) and \(y_2\); Table 2 gives the same for transformers with a core-type core.
Fig. 4. Curve 1 for a transformer with a shell-type core;
2 — with a core-type core.
In item 8 of Tables 1 and 2 the height is given as \(h' = 1.05h\). The increase in the winding height is made to accommodate transposition of the low-voltage winding and to strengthen the insulation and reduce the current density in the end coils of the high-voltage winding.
Item 17 indicates the mechanical stress in the winding material caused by the radial forces arising during a short circuit beyond the transformer.
The expression for determining the mechanical forces [3] may be brought to the following form:
\[ \sigma_{\mathrm{p}} = C \frac{\Delta_1^{2} q_{\mathrm{m}1} \pi \bigl(d + 2x_2 + [[\text{unclear: term in numerator}]]\bigr)}{h}, \tag{31} \]
where \(C \simeq 5.7 \cdot 10^{-6}\) is a constant.
Fig. 5. Curves 1, 2, 3, 4, and 5 for transformers with a shell-type core; 6, 7, 8, 9, and 10—with a core-type core.
Table 1
| No. | Transformer power, MVA | 10.5 I | 20 I | 20 II | 20 III | 30 I | 40 I | 50 I |
|---|---|---|---|---|---|---|---|---|
| 1 | \(y_2\) adopted, cm | 1,5 | 1,5 | 1,4 | 1,9 | 1,5 | 1,5 | 1,5 |
| 2 | \(x_2\) adopted, cm | 5,2 | 6,0 | 5,8 | 6,1 | 6,4 | 6,6 | 7,0 |
| 3 | \(x_1\) from (25), cm | 6,61 | 7,61 | 7,37 | 7,76 | 9,37 | 9,75 | 10,3 |
| 4 | \(y_1\) from (26), cm | 1,93 | 1,93 | 1,8 | 2,44 | 2,42 | 2,42 | 2,42 |
| 5 | \(\Delta_1=\Delta_2\) from (9), A/cm² | 330 | 325 | 335 | 296 | 322 | 320 | 318 |
| 6 | \(d\) from (16), cm | 56,7 | 67,1 | 63,6 | 67,1 | 75,3 | 78,2 | 84,7 |
| 7 | \(h\) from (12), cm | 138,5 | 165,5 | 189,5 | 167,5 | 187 | 224,5 | 227 |
| 8 | \(h^1=1{,}05h\), cm | 145,5 | 174 | 199 | 176 | 196 | 235 | 238 |
| 9 | \(H\) from (1), cm | 163,5 | 192 | 217 | 194 | 214 | 253 | 258 |
| 10 | \(a\) from (2), cm | 27,3 | 29,1 | 28,7 | 29,4 | 30,4 | 31,8 | 32,8 |
| 11 | \(l_{\mathrm{m}1}\) from (3), cm | 272 | 313 | 308 | 315 | 347 | 358 | 383 |
| 12 | \(l_{\mathrm{m}2}\) from (4), cm | 204 | 239 | 232 | 240 | 266 | 276 | 298 |
| 13 | \(l_c\) from (5), cm | 485 | 565 | 588 | 569 | 628 | 716 | 731 |
| 14 | \(q_{\mathrm{m}1}=q_{\mathrm{m}2}\) from (6), cm² | 842 | 1162 | 1196 | 1274 | 1400 | 1736 | 1862 |
| 15 | \(q_c\) from (7), cm² | 2070 | 2910 | 2730 | 2910 | 3660 | 3940 | 4620 |
| 16 | \(Q_{\mathrm{m}}\) from (21), t | 1,78 | 2,85 | 2,87 | 3,14 | 3,84 | 4,89 | 5,64 |
| 17 | \(Q_c\) from (18), t | 7,69 | 12,5 | 12,5 | 12,6 | 17,55 | 21,6 | 25,8 |
| 18 | \(\sigma_p\) from (31), kg/cm² | 445 | 579 | 568 | 523 | 668 | 705 | 792 |
| 19 | \(p_c\) (losses in steel), kW | 16,7 | 27,3 | 27,2 | 27,4 | 38,1 | 47,0 | 56,1 |
| 20 | \(p_{\mathrm{m}}\) (losses in copper), kW | 46,5 | 72,2 | 77,5 | 66,1 | 95,0 | 118,0 | 137,0 |
| 21 | \(Z\) from (28), rub. \(10^3\) | 4,45 | 7,17 | 7,26 | 7,2 | 9,84 | 12,2 | 14,44 |
Table 2
| No. | Transformer power, MVA | 10.5 I | 20 I | 20 II | 20 III | 30 I | 40 I | 50 I |
|---|---|---|---|---|---|---|---|---|
| 1 | \(y_2\) adopted, cm | 1,5 | 1,5 | 1,4 | 1,9 | 1,5 | 1,5 | 1,5 |
| 2 | \(x_2\) adopted, cm | 4,4 | 5,1 | 5,0 | 5,3 | 5,3 | 5,7 | 6,1 |
| 3 | \(x_1\) from (25), cm | 5,6 | 6,5 | 6,34 | 6,73 | 7,81 | 8,4 | 9,0 |
| 4 | \(y_1\) from (26), cm | 1,93 | 1,92 | 1,8 | 2,45 | 2,43 | 2,43 | 2,43 |
| 5 | \(\Delta_1=\Delta_2\) from (9), | 338 | 331 | 351 | 304 | 329 | 326 | 324 |
| 6 | \(d\) from (16), cm | 47 | 55,4 | 54,4 | 56,9 | 59,9 | 65,2 | 71 |
| 7 | \(h\) from (12), cm | 116 | 140,5 | 147 | 131,5 | 174 | 184 | 183 |
| 8 | \(h^1=1{,}05h\), cm | 122 | 147,5 | 154 | 138 | 183 | 193 | 192 |
| 9 | \(H\) from (1), cm | 140 | 165 | 172 | 156 | 201 | 211 | 210 |
| 10 | \(a\) from (2a), cm | 39 | 42,2 | 41,7 | 43 | 45,5 | 47,2 | 49,2 |
| 11 | \(l_{\mathrm{m}1}\) from (3), cm | 234 | 265 | 263 | 274 | 287 | 308 | 325 |
| 12 | \(l_{\mathrm{m}2}\) from (4), cm | 171 | 196 | 196 | 204 | 214 | 232 | 246 |
| 13 | \(l_c\) from (5a), cm | 527 | 615 | 619 | 604 | 705 | 749 | 772 |
| 14 | \(q_{\mathrm{m}1}=q_{\mathrm{m}2}\) from (6), cm² | 598 | 838 | 842 | 868 | 1080 | 1230 | 1302 |
| 15 | \(q_c\) from (7), cm² | 1425 | 1980 | 1900 | 2080 | 2310 | 2440 | 3250 |
| 16 | \(Q_{\mathrm{m}}\) from (21), t | 2,15 | 3,44 | 3,42 | 3,68 | 4,81 | 5,91 | 6,62 |
| 17 | \(Q_c\) from (18), t | 5,75 | 9,31 | 9,02 | 9,6 | 12,45 | 15,7 | 19,15 |
| 18 | \(\sigma_p\) from (30), kg/cm² | 336 | 430 | 432 | 411 | 472 | 534 | 612 |
| 19 | \(p_c\) (losses in steel), kW | 12,5 | 21 | 19,6 | 20,8 | 27,2 | 34,1 | 41,7 |
| 20 | \(p_{\mathrm{m}}\) (losses in copper), kW | 59 | 90,5 | 95,5 | 81,7 | 125 | 150,9 | 167 |
| 21 | \(Z\) from (28), rub. \(10^3\) | 4,15 | 6,59 | 6,58 | 6,64 | 8,99 | 11,14 | 13,0 |
Table 3.
| No. | Transformer rating, MVA | 10.5 | 10.5 | 20 | 20 | 30 | 30 | 40 | 40 | 50 | 50 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Core type | I | II | I | II | I | II | I | II | I | II | |
| 1 | $Q_{\mathrm{m}}$, % | 100 | 121 | 100 | 121 | 100 | 125 | 100 | 121 | 100 | 118 |
| 2 | $Q_{\mathrm{s}}$, % | 100 | 75 | 100 | 74.5 | 100 | 71 | 100 | 72.8 | 100 | 74.2 |
| 3 | $Q_{\mathrm{p}}$, % | 100 | 94 | 100 | 93.7 | 100 | 93.3 | 100 | 93 | 100 | 92 |
| 4 | $Z$, % | 100 | 93.3 | 100 | 92 | 100 | 91.3 | 100 | 90.6 | 100 | 90 |
Table 3 gives data for comparing transformers with shell-type (column I) and core-type (column II) magnetic cores.
In conclusion, it may be noted that, despite its complexity, the method [1, 2] can be applied to techno-economic studies of single-phase power transformers when digital computers are used.
REFERENCES
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Kutyavin I. D., On the determination of the optimal dimensions of three-phase two-winding transformers (see above).
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Kutyavin I. D., Del G. V., Krasnov V. P., Toward a techno-economic determination of the optimal dimensions of substation two-winding three-phase transformers of high power (see above).
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Tikhomirov P. M., Calculation of Transformers. Gosenergoizdat, 1963.