ON THE DETERMINATION OF OPTIMAL DIMENSIONS OF THREE-PHASE TWO-WINDING TRANSFORMERS

Doctor of Technical Sciences I. D. Kutyavin.

The article presents the results of the author’s attempt to find a more rigorous method for determining the most advantageous dimensions of the indicated transformers, proceeding from the following specified parameters:

1. Transformer power \(S\), kva.
2. Magnetic induction in the core \(B\), gs.
3. Short-circuit voltage \(U_k\).
4. Permissible density of heat flow from the winding surface \(\sigma\), W/cm\(^2\).

For the techno-economic determination of the optimal dimensions of transformers in accordance with the official method proposed by P. G. Grudinskii [3], the following costs are taken into account:

1. The calculated price of the transformer with depreciation deductions.
2. The calculated cost of the additional system capacity required to cover losses in the transformer during the period of maximum load in the system.
3. The calculated cost of electric-energy losses in the transformer.
4. The calculated costs for compensation of the reactive power of no-load operation. Costs for compensation of the reactive power lost in leakage reactance are not taken into account, because they do not depend on the dimensions of the transformer.

The inclusion of the calculated price of the transformer instead of the calculated cost according to the consolidated indicators of TEPa is explained by the fact that the cost of delivery and installation of the transformer enters into the total costs as a constant quantity and does not affect the position of the minimum of these costs.

The calculated price of a transformer may, with great approximation, be taken as proportional to the total weight of the active materials, reduced to steel,

\[ Z_t = k_t (Q_c + \beta Q_m) / k_t Q_n , \tag{1} \]

where
\(k_t\) — the calculated specific price of the transformer in rubles/kg;
\(Q_c\) and \(Q_m\) — weights of steel and copper in kg;
\(\beta\) — the ratio of the specific calculated cost of copper to the cost of steel in the product.

Approximate values of \(k_t\) for some transformers with cores of hot-rolled steel and without devices for voltage regulation (at \(\beta = 3.65\)) are given in Table 1.

Table 1.

| \multicolumn{2}{c}{10 kV} | \multicolumn{2}{c}{35 kV} | \multicolumn{2}{c}{110 kV} |
|---:|---:|---:|---:|---:|---:|
| $S$, kVA | $k_{\mathrm{t}}$ | $S$, MVA | $k_{\mathrm{t}}$ | $S$, MVA | $k_{\mathrm{t}}$ |
| 560 | 0.52 | | | 10 | 0.65 |
| 750 | 0.53 | 7.5 | 0.50 | 15 | 0.58 |
| 1000 | 0.51 | 10 | 0.47 | 20 | 0.53 |
| 1350 | 0.52 | 15 | 0.42 | 31.5 | 0.49 |
| 1800 | 0.54 | 20 | 0.41 | 40.5 | 0.46 |
| 3200 | 0.50 | 31.5 | 0.39 | 60 | 0.43 |

Costs (1), with depreciation deductions, referred to one year of the calculation period $T_{\mathrm{p}} = 8$ years:

\[ Z_{\mathrm{ta}} = k_{\mathrm{t}} Q_{\mathrm{p}} (P_{\mathrm{n}} + P_{\mathrm{a}}) \tag{2} \]

where $P_{\mathrm{n}} = \dfrac{1}{T_{\mathrm{p}}}$ is the standard efficiency coefficient;

$P_{\mathrm{a}}$ — depreciation deductions in fractions of unity.

The calculated costs for creating additional capacity in the system [4], referred to one year of the calculation period, are:

\[ Z_{\mathrm{s}} = p_{\mathrm{n}} C_{1} \left[ P_{\mathrm{c}} + P_{\mathrm{m}} \left( \frac{k_{\mathrm{m}} S_{\mathrm{m}}}{S} \right)^{2} \right], \tag{3} \]

where $P_{\mathrm{c}}$ and $P_{\mathrm{m}}$ are losses in steel and copper, kW;

$S_{\mathrm{m}}$ — calculated maximum transformer load, in kVA, at the end of the planning period;

$k_{\mathrm{m}}$ — coefficient of coincidence with the system maximum.

The calculated cost of the installed kilowatt of additional capacity $C_{1}$, taking into account the auxiliary consumption of power plants, the required capacity reserve in the system, and the costs of constructing the station and its fuel base, is:

\[ C_{1} = k_{\mathrm{sn}} (k_{\mathrm{r}} C_{\mathrm{c}} + q C_{\mathrm{t}}), \tag{4} \]

where $k_{\mathrm{sn}}$ is a coefficient taking into account consumption for auxiliary needs;

$k_{\mathrm{r}}$ — reserve coefficient;

$C_{\mathrm{c}}$ — average calculated cost of an installed kilowatt, in rubles, of newly commissioned power-plant capacities in the systems of the Soviet Union;

$C_{\mathrm{t}}$ — average calculated costs, in rubles, for creating the fuel base, referred to one ton of annual production of standard fuel;

$q$ — annual consumption of standard fuel per one kilowatt of additional capacity, in tons.

Table 2 gives the values of $C_{1}$ for $k_{\mathrm{sn}} = 1.1$; $k_{\mathrm{r}} = 1.1$; $C_{\mathrm{t}} = 20$ rubles/t and for a specific consumption of standard fuel per one kilowatt-hour of 400 g.

Table 2.

| $C_{\mathrm{c}}$, rubles/kW | \multicolumn{5}{c}{Duration of use $T_{\mathrm{m}}$, hours} |
|---:|---:|---:|---:|---:|---:|
| | 2000 | 3000 | 4000 | 5000 | 7000 |
| 60 | 90.2 | 99.0 | 107.8 | 116.6 | 134.2 |
| 80 | 114.4 | 123.2 | 132.0 | 140.8 | 158.4 |
| 100 | 138.6 | 147.4 | 156.2 | 165.0 | 182.6 |
| 120 | 162.6 | 171.4 | 180.2 | 189.0 | 206.6 |

As the calculated value of \(C_{\mathrm{c}}\), one should take not the design value, but the actual value.

\[ Z_{\mathrm{p}}=C'_2 P_{\mathrm{c}} t + C''_2 P_{\mathrm{m}}\tau \left(\frac{k_{\mathrm{n}}S_{\mathrm{m}}}{S}\right)^2 \tag{5} \]

The calculated annual cost of losses in the transformer [4]:

where \(C'_2\) and \(C''_2\) are the calculated costs of one kilowatt-hour of losses in steel and copper, in rubles/kWh;

\(\tau\) is the annual loss time, hours;

\(t\) is the operating time of the transformer per year, hours;

\(k_{\mathrm{n}}\) is the load growth coefficient [3].

The calculated expenditures for compensating the reactive power of magnetization of the core steel are

\[ Z_{\mathrm{n}}=Z_{\mathrm{r}}\alpha_{\mathrm{r}}Q_{\mathrm{c}}, \]

where \(\alpha_{\mathrm{r}}\) is the specific loss of reactive power in steel, kvar/kg,

\(Z_{\mathrm{r}}\simeq 1.5\) rubles/kvar is the specific cost of compensation [3].

If the induction in the yoke and in the limb is assumed to be the same, then the losses in the core steel, in kW, are:

\[ P_{\mathrm{c}}=\alpha_{\mathrm{c}}B^2 Q_{\mathrm{c}}. \tag{6} \]

The losses in copper (kW), for different current densities in the phase windings, are:

\[ P_{\mathrm{m}}=\alpha_{\mathrm{m}}\left(\Delta_1^2 Q_{\mathrm{m}1}+\Delta_2^2 Q_{\mathrm{m}2}\right), \tag{7} \]

where \(\alpha_{\mathrm{c}}\) and \(\alpha_{\mathrm{m}}\) are the loss coefficients in steel and copper;

\(\Delta_1\) and \(\Delta_2\) are the current densities of the primary and secondary windings, A/cm\(^2\).

Then the total calculated expenditures for the purchase and operation of the transformer can be expressed through the weights of the active materials:

\[ Z=Z_{\mathrm{ta}}+Z_{\mathrm{c}}+Z_{\mathrm{p}} =(A_1+DB^2)Q_{\mathrm{c}}+(\beta A_2+E\Delta_1^2)Q_{\mathrm{m}1} +(\beta A_2+E\Delta_2^2)Q_{\mathrm{m}2}, \tag{8} \]

where:

\[ \left. \begin{aligned} A_2&=k_{\mathrm{t}}(p_{\mathrm{n}}+p_{\mathrm{a}});\qquad A_1=Z_{\mathrm{r}}\alpha_{\mathrm{r}}+A_2,\\ D&=(C_1p_{\mathrm{n}}+C'_2t)\alpha_{\mathrm{c}},\\ E&=(C_1p_{\mathrm{n}}k_{\mathrm{m}}^2+C''_2\tau k_{\mathrm{n}}^2)\left(\frac{S_{\mathrm{m}}}{S}\right)^2\alpha_{\mathrm{m}}. \end{aligned} \right\} \tag{9} \]

If now the weights of the active materials \(Q_{\mathrm{c}}\) and \(Q_{\mathrm{m}}\) are expressed through the principal dimensions of the transformer and substituted into (8), then one can determine the optimal value of these dimensions, as well as the optimal current densities \(\Delta_1\) and \(\Delta_2\).

Within the scope of this article, let us consider the problem of determining the optimal dimensions of low-power transformers with cylindrical windings cooled by axial ducts. Suppose that the primary winding (the high-voltage winding) has \(n_1\) layers, separated by axial ducts of width \(\delta_1\). The width of each layer of this winding is \(x_1\). Then the total width of the primary winding (all dimensions in centimeters) is:

\[ e_1=n_1x_1+(n_1-1)\delta \tag{10} \]

and of the secondary winding

\[ e_2=n_2x_2+(n_2-1)\delta \tag{11} \]

provided that \(\delta_1=\delta_2\).

The width of the core window is

\[ a=2(e_1+e_2+l_{\mathrm{r}}\delta); \tag{12} \]

where \(l_{\mathrm{r}}\) is one half of the sum of the radial insulation distances in the window.

The height of the core window

\[ H=h+2l_{\mathrm{и}}, \tag{13} \]

where \(h\) is the winding height and \(l_{\mathrm{и}}\) is the insulation distance from the windings to the yoke.

The length of the core steel, reduced to the cross-sectional area of the limb:

\[ l_c=3H+k_{\mathrm{я}}(4a+5{,}6d) =3h+k_{\mathrm{я}}\bigl[8(b_1+b_2)+5{,}6d\bigr]+l, \tag{14} \]

where \(k_{\mathrm{я}}\) is the coefficient of increase of the yoke cross section,
\(d\) is the limb diameter, and

\[ l=6l_{\mathrm{н}}+8k_{\mathrm{я}}l_{\mathrm{г}}. \tag{15} \]

The length of the mean turn of the phase windings

\[ l_{\mathrm{м}}=\pi(d+b_1+b_2+l_{\mathrm{г}}). \tag{16} \]

The cross-sectional areas of the material of the phase windings

\[ q_{\mathrm{м}1}=k_1 n_1 x_1 h;\qquad q_{\mathrm{м}2}=k_2 n_2 x_2 h. \tag{17} \]

where \(k_1\) and \(k_2\) are the coefficients of filling the area of the winding layer with conducting material.

The cross-sectional area of the limb steel

\[ q_c=k_c\frac{\pi d^2}{4}, \tag{18} \]

where \(k_c\) is the coefficient of filling with steel the area of a circle of diameter \(d\).

The heat-balance equation for the winding layer per one centimeter of mean turn

\[ 2hk_{\mathrm{п}1}\sigma_1=\rho\Delta_1^{\,2}k_1x_1h, \tag{19} \]

where \(k_{\mathrm{п}1}\) is a coefficient allowing for the covering of part of the layer surfaces by insulating parts;
\(\sigma_1\) is the permissible heat-flux density from the surface of the winding, W/cm\(^2\);
\(\rho\) is the specific resistance of the conductor, ohm·cm.

The current densities of the windings, on the basis of (19), are

\[ \Delta_1=\sqrt{\frac{2k_{\mathrm{п}1}\sigma_1}{\rho k_1x_1}} =\frac{a_1}{\sqrt{x_1}}; \qquad \Delta_2=\sqrt{\frac{2k_{\mathrm{п}2}\sigma_2}{\rho k_2x_2}} =\frac{a_2}{\sqrt{x_2}}, \tag{20} \]

where

\[ a_1=\sqrt{\frac{2k_{\mathrm{п}1}\sigma_1}{\rho k_1}} \qquad \text{and} \qquad a_2=\sqrt{\frac{2k_{\mathrm{п}2}\sigma_2}{\rho k_2}}. \tag{21} \]

The condition of equality of the magnetomotive forces of the windings:

\[ k_1 n_1 x_1\Delta_1 h=k_2 n_2 x_2\Delta_2 h. \tag{22} \]

Substituting here the current densities from (20), we find:

\[ x_1=bx_2 \tag{23} \]

where

\[ b=\left(\frac{k_2n_2a_2}{k_1n_1a_1}\right)^2. \tag{24} \]

The expression for the transformer rating in kVA may be represented in the following form:

\[ S=K\Delta_2 q_{\mathrm{м}2}q_c, \tag{25} \]

where

\[ K=13{,}32 fB10^{-11}. \tag{26} \]

Substituting the variables here, we find the winding height

\[ h=\frac{N}{d^2\sqrt{x_2}}, \tag{27} \]

where

\[ N=\frac{4S}{Kk_c\,k_2 n_2 a_2\pi}. \tag{28} \]

The reactive component of the short-circuit voltage in relative units [1] can be reduced to the following form:

\[ u_p=\frac{2k_r\Delta_2 q_{m2}l_m\left(b_1+b_2+3\delta_{12}\right)} {3.33\,Bq_c\,h}, \tag{29} \]

where \(k_r\) is Rogowski’s coefficient;

\(\delta_{12}\) is the insulation distance between the windings.

After substituting the variables into (29), we find the core diameter:

\[ d=\frac{U+\sqrt{U^2+4PUV}}{2P}, \tag{30} \]

where

\[ U=\sqrt{x_2}\left[x_2(bn_1+n_2)+(n_1+n_2-2)\delta+3\delta_{12}\right]; \tag{31} \]

\[ V=x_2(bn_1+n_2)+(n_1+n_2-2)\delta+l_r \tag{32} \]

\[ P=\frac{3.33k_c\,Bu_p}{8k_r\,k_2 n_2 a_2}. \tag{33} \]

Now all the dimensions of the transformer can be expressed through the independent variable \(x_2\).

The weight of the core steel in kg:

\[ Q_c=\gamma_c q_c l_c\,10^{-3} =\frac{k_c\pi\gamma_c\,10^{-3}}{4} \left\{ \frac{3N}{\sqrt{x_2}}+ d^2\left[k_a(8V+5.6d)+6l_u\right] \right\}. \tag{34} \]

The weight of the core steel in kg:

\[ Q_m=3\gamma_m l_m(q_{m1}+q_{m2})10^{-3} =3\gamma_m\pi N(bk_1n_1+k_2n_2)\, \frac{P}{a_1+a_2x_2}. \tag{35} \]

The weight of the active materials, reduced to steel, in kg:

\[ Q_{\mathrm{p}}=Q_c+\beta Q_m =\frac{k_c\pi\gamma_c}{4}\,f(x_2), \quad \text{where} \tag{36} \]

\[ f(x_2)=10^{-3} \left\{ 3N\left(\frac{1}{\sqrt{x_2}}+\frac{M}{a_1+a_2x_2}\right) +d^2\left[k_a(8V+5.6d)+6l_u\right] \right\} \tag{37} \]

\[ \left. \begin{aligned} M&=\frac{4P\beta\gamma_m(bk_1n_1+k_2n_2)}{k_c\gamma_c},\\ a_1&=(n_1+n_2)\delta+3\delta_{12};\qquad a_2=bn_1+n_2 \end{aligned} \right\} \tag{38} \]

\(\gamma_c\) and \(\gamma_m\) are the specific weights of steel and of the conductor material.

Investigation of the expression for the reduced weight (36) is of considerable interest for comparing transformer dimensions, obtain-

estimated from the condition of minimum consumption, referred to steel, of active materials, with the dimensions found from the condition of minimum total costs. In addition, for \(\beta=1\), expression (36) gives the total weight of natural active materials and makes it possible to find the dimensions of the lightest transformer, which may be of interest for mobile installations.

However, determining the minimum of function (37) in the general form, after substituting the values \(d(x_2)\) and \(V(x_2)\), is a very cumbersome problem, which can be solved most simply with the aid of mathematical machines. Therefore we shall confine ourselves here to a numerical investigation of (37) for a transformer of \(560\ \text{kVA}\), \(10/0.4\ \text{kV}\), with a core of hot-rolled steel and with copper and aluminum used for the windings. For this purpose we take the following initial data (the dimensions are indicated in the text):

\[ \begin{gathered} u_p=0.055;\quad l_u=3.5;\quad l_k=4;\quad \delta_{12}=1.7;\quad n_1=n_2=2;\quad k_1=0.6;\quad k_2=0.8;\\ k_c=0.84;\quad k_{n1}=k_{n2}=0.7;\quad \kappa_r=0.95;\quad \kappa_y=1;\quad \kappa_m=0.70;\quad k_n=1;\\ k_t=0.52;\quad \delta_1=0.12;\quad \delta_2=0.14;\quad \delta=0.8;\quad \gamma_c=7.5;\quad B=14500;\quad p_i=\\ =0.125;\quad p_a=0.06;\quad t=8700;\quad \tau=3000;\quad \gamma_c=1.35\cdot10^{-11};\quad C_1=124;\quad C'_2=\\ 0.008;\quad C''_2=0.012;\quad \left(\frac{k_n S_m}{S}\right)^2=0.8\ [4]. \end{gathered} \]

It was calculated that

\[ A_1=0.1457;\quad A_2=0.0962;\quad DB^2=0.242;\quad K=9.65\cdot10^{-5};\quad \varepsilon=1.55. \]

Adopted and calculated for copper:

\[ \gamma_m=8.9;\quad \rho=2.14\cdot10^{-6};\quad \alpha_m=2.4\cdot10^{-7};\quad \beta=3.65;\quad \alpha_1=362;\quad \alpha_2=340;\quad P=0.54;\quad M=38.5;\quad E=83.6\cdot10^{-7};\quad N= \]
\[ =1.62\cdot10^4;\quad a_1=6.7;\quad a_2=5.1;\quad a_3=0.512;\quad a_4=0.22. \]

Adopted and calculated for aluminum:

\[ \gamma_m=2.7;\quad \rho=3.6\cdot10^{-6};\quad \alpha_m=1.33\cdot10^{-6};\quad \beta=4.5;\quad \alpha_1=280;\quad \alpha_2=261;\quad P=0.704;\quad M=18.8; \]
\[ E=4.64\cdot10^{-5};\quad N=2.10\cdot10^4;\quad a_1=6.7;\quad a_2=5.1;\quad a_3=1.38;\quad a_4=0.22. \]

Table 3 gives the values of \(d(x_2)\) and \(f(x_2)\), calculated respectively from (30) and (37) for a transformer with copper and aluminum windings. The critical value of \(x_2\) for copper was found to be \(0.45\ \text{cm}\), and for aluminum, \(0.55\ \text{cm}\). To clarify the character of the curve, Table 3 gives the percentage expression of \(f(x_2)\). As can be seen from Table 3, the curve \(f(x_2)\) has a sufficiently flat minimum; therefore considerable deviations of \(x_2\) from its critical value give a consumption of referred active materials \(Q_p\) close to the minimum.

Table 3.

Expression for the total calculated costs upon substituting into (8) the values of \(Q_c\), \(Q_m\), and \(\Delta\):

\[ \mathcal{Z}=\frac{k_c \pi \gamma_c}{4}\left(A+DB^2\right)\varphi(x_2); \tag{39} \]

where

\[ \varphi(x_2)=10^{-3}\left\{3N\left[\frac{1}{\sqrt{x_2}}+ \frac{M(a_3+a_4x_2)}{x_2(a_1+a_2x_2)}\right] +d^2\left[k_q(8V+5.6d)+6l_{\mathrm{c}}\right]\right\}, \tag{40} \]

\[ a_3=\frac{k_1 n_1 a_1^2}{\mu k_2 n_2}\left(\frac{E}{A_1+DB^2}\right); \qquad a_4=-\frac{A_2}{A_1+DB^2}. \tag{41} \]

The values of \(\varphi(x_2)\), calculated from (40) for the adopted initial data, are given in Table 3 in absolute values and in percent. The critical value of \(x_2\) for a transformer with copper windings is \(0.7\ \mathrm{cm}\), and with aluminum windings \(0.9\ \mathrm{cm}\), and it can be changed by \(\pm 0.1\ \mathrm{cm}\) without a noticeable increase in the calculated costs.

In Fig. 1 the dependence \(d(x_2)\) is plotted for a transformer with copper windings at \(k_1=0.5;\ 0.6\), and \(0.7\), and in Fig. 2 for a transformer with aluminum windings and for the same values of \(k_1\). It should be noted that the coefficient \(k_1\) depends rather strongly on the transformer power, whereas \(k_2\) remains practically constant.

Fig. 1. Curve 1 for \(k_1=0.5\); 2—0.6; 3—0.7.

The character of the curves in Figures 1 and 2 shows that expression (30) can be replaced by the equation of a straight line of the following form:

\[ d=a_5+a_6x_2 \tag{42} \]

The constants \(a_5\) and \(a_6\) for copper and aluminum, calculated from the curves (Figs. 1 and 2) for various \(k_1\), are given in Table 4. The use of the replacement of expression (30) by the equation of the straight line (42) greatly simplifies the solution of the problem.

Figure labels: \(a_5\); aluminum; copper.

Fig. 2. Curve 1 for \(k_1=0.5\); 2—0.6; 3—0.7.

Table 4.

Then function (37) can be put in the form:

\[ f(x_2)=10^{-3}\left[ 3N\left(\frac{1}{\sqrt{x_2}}+\frac{M}{a_1+a_2x_2}\right) +(a_5+a_6x_2)^2(a_7+a_8x_2) \right], \tag{43} \]

where:

\[ \begin{aligned} a_7&=8k_{\mathrm{ya}}\,[\delta(n_1+n_2-2)+0.7a_5+l_{\mathrm{g}}]+6l_{\mathrm{i}},\\ a_8&=8k_{\mathrm{a}}\,(\delta n_1+n_2+0.7a_6). \end{aligned} \tag{44} \]

Function (40) is reduced to the form:

\[ \varphi(x_2)=10^{-3}\left\{ 3N\left[ \frac{1}{\sqrt{x_2}}+ \frac{M(a_3+a_4x_2)}{x_2(a_1+a_2x_2)} \right] +(a_5+a_6x_2)^2(a_7+a_8x_2) \right\}. \tag{45} \]

To determine the critical values of \(x_2\) corresponding to the minima of the simplified functions (43) and (45), we set the first derivative of these functions equal to zero and hence find an expression for the constant \(N\), which depends on the transformer rating.

The expression for the constant \(N\) for the minimum of function (43):

\[ N_{\text{п}}= \frac{ 2x_2\sqrt{x_2}\,(a_1+a_2x_2)^2\,(a_5+a_6x_2)\,(2a_6a_7+a_5a_8+3a_6a_8x_2) }{ 3\left[(a_1+a_2x_2)^2+2Ma_2x_2\sqrt{x_2}\right] } \tag{46} \]

The same for the minimum of function (45):

\[ N_{\text{э}}= \frac{ 2x_2^2(a_1+a_2x_2)^2(a_5+a_6x_2)(2a_6a_7+a_5a_8+3a_6a_8x_2) }{ 3\left\{\sqrt{x_2}(a_1+a_2x_2)^2+2M\left[a_3(a_1+a_2x_2)+a_2x_2(a_3+a_4x_2)\right]\right\} } \tag{47} \]

The curves (Fig. 3), constructed respectively for copper and aluminum from the data calculated from (47), can be used to determine the critical value of \(x_2\). Having calculated \(N\) from (28), we find \(x_2\) (for the adopted winding material and coefficient \(\kappa_1\)) corresponding to the minimum of \(\varphi(x_2)\).

Fig. 3. Curve 1 for copper and 2 for aluminum.

The indicated curves are constructed for two-layer windings with two-sided cooling and are suitable for transformers with ratings from 240 to 560 kVA.

The construction of similar curves from expressions (46) and (47) for other initial conditions presents no special difficulty.

The critical values of \(x_2\), determined from the curves of Fig. 3 for the problem under consideration, are given in Table 5. The remaining dimensions

for a 560 kVA transformer have been calculated by the formulas given above.

As can be seen from Table 5, the transformer with copper windings that satisfies the condition of minimum calculated costs has an excess expenditure of active materials of 4%, and the transformer with aluminum windings—of 12%. This excess expenditure relates to steel, since the expenditure of conductor material, in comparison with the condition of minimum \(Q_{\mathrm{p}}\), decreases. The minimum calculated costs for the transformer with aluminum windings are only 7% greater than with copper windings. This indicates the advisability of using aluminum in low-power transformers.

According to data in [2], the weight of the core steel of currently manufactured 560 kVA transformers with copper windings is 810 kg, and the weight of copper is 260 kg. At \(\beta = 3.65\), the reduced weight is \(Q_{\mathrm{p}} = 1760\) kg.

Table 5.

The losses in steel are 2.5 kW and in copper 9.4 kW. The total calculated costs for the factory-produced transformer are determined from the expression:

\[ Z = Z_{\mathrm{ta}} + (C_1 p_{\mathrm{n}} + C'_2 t)p_{\mathrm{c}} + \left(C_1 p_{\mathrm{k}} k_{\mathrm{m}}^2 + C''_2 t k_{\mathrm{n}}^2\right) \left(\frac{S_{\mathrm{m}}}{S}\right)^2 p_{\mathrm{m}} + Z_{\mathrm{n}} \tag{48} \]

and amount to 760 rubles. An economical transformer with a copper winding has total costs of 667 rubles and a saving of reduced active materials of 50 kg (see Table 5).

In conclusion it is necessary to note that the proposed method is applicable to transformers with cylindrical windings, which, according to [1], are made up to a power of 560 kVA.

Despite a certain cumbersomeness, this method has high accuracy and may prove useful. It should also be taken into account that determining the dimensions of a series of transformers in the overall complex

designing this series take a very small share in terms of time and labor expenditure. The use of computing machines can simplify this task.

References

1. Postnikov I. M., Design of Electrical Machines. Kiev, 1960.
2. Petrov G. N., Electrical Machines, Part I. Gosenergoizdat, 1956.
3. Tikhomirov P. M., Calculation of Transformers. Gosenergoizdat, 1962.
4. Kutyavin I. D., On the economical choice of transformer ratings. Energetik, No. 1, 1962.