DETERMINATION OF THE LIMITING POWER OF TRANSFORMERS TAKING INTO ACCOUNT SHORT-CIRCUIT VOLTAGE

I. D. Kutyavin, L. I. Del

In [1], studies are presented of transformers of limiting power with a restriction on heating. In this case the reactive component of the short-circuit voltage \(u_p\) was found to be higher than that recommended by the GOST standard.

Therefore, in the proposed article, studies are presented of transformers of limiting power with a specified value of \(u_p\).

Using expressions (7) and (11) from [1], we write

\[ \Delta=\sqrt{\frac{\alpha P(x+l)}{k_r bxy}}, \tag{1} \]

\[ \varphi=\sqrt{\frac{Pxy}{k_r(x+i)(y+\delta)^2}}. \tag{2} \]

All designations correspond to [1]. To simplify the solution of the problem, the following assumptions were initially adopted: \(i_1=i_2=i\); \(\varepsilon_1=\varepsilon_2=\varepsilon\); \(\delta_1=\delta_2=\delta\); \(h_1=h_2=h\), and the equalities following from them, \(x_1=x_2=x\); \(y_1=y_2=y\); \(b_1=b_2=b\).

The independent variables adopted are the radial width of an elementary conductor \(x\), the axial height of the copper of an elementary conductor \(y\), and the radial width of the winding \(b\).

Let us use the expression for the reactive component of the short-circuit voltage of a transformer (in relative units):

\[ u_p=\frac{8k_R\pi d_{12}\delta h\varphi}{3{,}33Bk_c\pi d^2h} =\frac{8k_R d_{12}\delta'\varphi}{3{,}33Bk_c d^2}, \tag{3} \]

where

\[ \delta'=2b+3\delta_{12}, \tag{4} \]

\(d_{12}\) is the mean diameter of the leakage channel, cm, equal to

\[ d_{12}=d+2b+2\delta_{01}+\delta_{12}. \tag{5} \]

To shorten the notation, let us denote

\[ l=2\delta_{01}+\delta_{12}. \tag{6} \]

\[ p=\frac{3{,}33Bk_cu_p}{8k_R}. \tag{7} \]

From expression (3) we find the diameter of the core:

\[ d=\frac{\delta'\varphi}{2p}\left[1+\sqrt{1+\frac{4p}{\delta'\varphi}(2b+l)}\right]. \tag{8} \]

Denote

\[ 1+\sqrt{1+\frac{4p}{\delta'\varphi}(2b+l)}=z. \tag{9} \]

After replacing the variables by independent ones, the limiting power of core 1 [1] can be determined from the expression

\[ S_c=K_s h\,\frac{\delta'^2\varphi^3}{4p^2}\,z^2, \tag{10} \]

when the maximum permissible values of the quantities entering into (10) are substituted, for example on the basis of railway clearances,

\[ K_s=4{,}44f\frac{\pi}{4}\kappa_c. \tag{11} \]

The power expression (10) has a maximum at certain values of \(x\) and \(y\), which can be found from the condition

\[ \frac{\partial S}{\partial y}=0,\qquad \frac{\partial S}{\partial x}=0. \tag{12} \]

As a result of solving (12), a number of relations are obtained for determining the critical values of \(x\) and \(y\):

\[ 1+\sqrt{1+\frac{4p}{\delta'\varphi}(2b+l)}=0, \tag{13} \]

\[ \varphi^2=0, \tag{14} \]

\[ 3\left[1+\sqrt{1+\frac{4p}{\delta'\varphi}(2b+l)}\right] -\frac{4p(2b+l)}{\delta'\varphi\sqrt{1+\frac{4p}{\delta'\varphi}(2b+l)}}=0, \tag{15} \]

\[ \varphi(x)=0, \tag{16} \]

\[ \varphi(y)=0. \tag{17} \]

Considering (13), (14), and (15) successively, we arrive at the conclusion that the roots of these equations have no physical meaning.

Expressions (16) and (17) give the following equations:

\[ \kappa_r i-x(x+i)\kappa_r(x)=0, \tag{18} \]

\[ \kappa_r\left[\delta(\kappa_n b+2y)-\kappa_n\delta y\right] -Py(y+\delta)\kappa_r(y)=0. \tag{19} \]

Equations (18), (19) are identical with (13) and (14) [1]. Hence it may be concluded that the optimal values \(x_0\) and \(y_0\), corresponding to the maximum power under the constraint on \(u_p\) and on heating, coincide with the optimal values \(x_0\) and \(y_0\) under the constraint only on heating.

In items 6 and 9 of Tables 1 and 2, the calculation of the core diameter and limiting power per core is given for the following constant quantities.

For Table 1:

\[ h=260\ \text{cm};\quad B=17500\ \text{G};\quad u_p=0{,}2;\quad i=0{,}01\ \text{cm}; \]

\[ \delta=0{,}8\ \text{cm};\quad \delta_{01}=5\ \text{cm};\quad \delta_{12}=18\ \text{cm};\quad \varepsilon=0{,}3\ \text{W/cm}^2; \]

\[ \rho=2{,}14\cdot10^{-6}\ \Omega\cdot\text{cm};\quad \kappa_n=0{,}78;\quad \kappa_c=0{,}8; \]

\[ \kappa_R=0{,}95;\quad f=50\ \text{Hz};\quad C=0{,}92. \]

winding material—copper.

Table 1

For Table 2:

\[ h = 260\ \text{cm};\quad B = 17500\ \text{Hz};\quad u_p = 0{,}15;\quad i = 0{,}01\ \text{cm}; \]

\[ \delta = 1{,}0\ \text{cm};\quad \delta_{01} = 5\ \text{cm};\quad \delta_{12} = 18\ \text{cm};\quad \varepsilon = 0{,}3\ \text{W}/\text{cm}^2; \]

\[ \rho = 3{,}5 \cdot 10^{-6}\ \Omega\!\cdot\!\text{cm};\quad C = 0{,}562;\quad \kappa_c = 0{,}8;\quad \kappa_R = 0{,}95; \]

\[ f = 50\ \text{Hz};\quad \kappa_n = 0{,}78. \]

the winding material is aluminum.

Figure 1 shows the dependence \(d = f(u_p)\) for various values of \(b\), from which it follows that, as \(b\) increases, the core diameter becomes impractically large.

Fig. 1. \(1 — b = 7\ \text{cm},\ 2 — b = 8\ \text{cm},\ 3 — b = 9\ \text{cm},\ 4 — b = 10\ \text{cm},\ 5 — b = 12\ \text{cm}\)

Therefore, a satisfactory dimension is a coil width \(b\) on the order of \(8\ \text{cm}\).

Table 2

LITERATURE

1. I. D. Kutyavin, L. I. Del. On the limiting power of a transformer. Proceedings of the Tomsk Polytechnic Institute, vol. 172.
2. S. A. Farbman, A. Yu. Bun. Repair and modernization of transformers.