COMPARISON OF MAXIMUM-CAPACITY TRANSFORMERS WITH COIL AND MULTILAYER HV WINDINGS

I. D. KUTYAVIN, G. V. DEL, L. I. DEL

The article presents the results of an exploratory comparison of maximum-capacity transformers with identical helical LV windings and with concentric multilayer and helical coil HV windings.

Coil windings (helical, spiral, disc) have a very high copper fill factor for their area, and therefore this class of windings proves economical for voltages of \(6\)—\(220\) kV.

Fig. 1.

At higher rated voltages of the HV winding, the use of a coil winding for it proves disadvantageous because of the large insulation distances between the LV and HV windings, its low electrodynamic strength, etc. In this case a multilayer concentric HV winding proves considerably more advantageous than a coil winding. Since networks of 110 kV and above operate with a solidly grounded neutral, one end of the inner layer will be grounded; therefore the insulation distance between the LV and HV windings may be minimal, corresponding to the voltage at the other end of the layer. With \(m\) layers in the winding, the voltage between adjacent layers connected according to the circuit of Fig. 1a is

\[ U_{\mathrm{sl}}=\frac{1.1U_{\mathrm{n}}}{\sqrt{3}\,m}. \tag{1} \]

When the layers are connected according to the circuit of Fig. 1b,

\[ U_{\mathrm{sl}}=\frac{2.2U_{\mathrm{n}}}{\sqrt{3}\,m}. \tag{2} \]

The insulation of the outer layer of the HV winding from the LV winding increases stepwise owing to the interlayer insulation, and large insulating gaps ...

ducts are needed only up to the yokes and between the HV windings of different phases. The axial height of the layer decreases from the inner layer to the outer one, taking into account the increase in the voltage of the layers (Fig. 1c).

To determine the limiting power of the transformer limb, a method was used that is based on maximizing the formula for the limb power and that assumes maximum utilization of the active materials and of the permissible railway clearances. This method, in simplified form \((x_1 = x_3; y_1 = y_2; b_1 = b_2)\), for transformers with coil LV and HV windings is described in [1]. Here it is assumed that \(x_1 \ne x_2\), \(b_1 \ne b_2\), and \(y_1 = y_2 = 1.45\).

Then the expressions for the limiting diameter of the limb \(d\) and the limiting height of the winding \(h\) will have the form (all notation coincides with [1]):

\[ d = A - 2(b_1 + b_2); \qquad h = B + 2(b_1 + b_2). \tag{3} \]

The remaining winding dimensions were found by the joint solution of equations obtained from the short-circuit voltage formula, the equality of the magnetomotive forces (MMF), and relation (17) of [1].

The equation from the short-circuit voltage formula, after substituting \(d\) from (3), is:

\[ \kappa_c B U_p [A - 2(b_1 + b_2)]^2 \cdot 10^4 - 2.4 \kappa_R (A - 2b_2 + 2\delta_{01} + \delta_{12})(b_1 + b_2 + 3\delta_{12}) \varphi_2 = 0, \tag{4} \]

where \(\varphi_2\), on the basis of (10) and (11) of [1], is

\[ \varphi_2 = \sqrt{ \frac{\alpha_2 P_2 b_2 x_2 y} {k_{r2}(x_2 + i_2)(y + \delta_2)^2} }. \tag{5} \]

The equality of the MMF for windings of equal height, with \(y_1 = y_2 = y = 1.45\), has the form

\[ \frac{\alpha_1 P_1 b_1 x_1 y} {k_{r1}(x_1 + i_1)(y + \delta_1)^2} = \frac{\alpha_2 P_2 b_2 x_2 y} {k_{r2}(x_2 + i_2)(y + \delta_2)^2} = \varphi_2^2 . \tag{6} \]

From relation (17) of [1] we find \(b_1\):

\[ b_1 = \frac{3(x_1 + i_1)(y + \delta_1)} {C x_1^2 y \sqrt{\dfrac{2x_1}{i_1} + 3}} . \tag{7} \]

Each \(b\) corresponds to a certain optimal \(x\), determined by relation (17) of [1], or by (7).

From equation (4) we find the dependence \(b_1(b_2)\). To do this, we prescribe a series of values of \(b_2\) within the design limits, find for them, using the method of [1], the corresponding optimal \(x_2\), \(\kappa_{r2}\), \(\varphi_2\), etc., and plot the curve \(b_1(b_2)\).

From (7) we find a series of dependences \(b_1(x_1)\), which we successively substitute into (6) until this equality is satisfied for each value of \(b_2\) and \(\varphi_2\), and thus we find a new dependence \(b_1(b_2)\). The intersection of the curves \(b_1(b_2)\) makes it possible to find the optimal \(b_1\) and \(b_2\) and the corresponding \(x_1\) and \(x_2\) for each prescribed value of \(u_p\). After this the values \(d\) and \(h\) are determined, and from them the limiting power of the limb is determined from the expression

\[ S_c = K_c d^2 h \varphi_1, \tag{8} \]

\[ K_c = 1.11 f K_c \pi B \cdot 10^{-7}. \tag{9} \]

The limiting powers of the limb for this combination of windings of 500- and 750-kV transformers are shown in the table.

To determine the limiting power of the limb with a helical LV winding and a multilayer HV winding, one may use the materials of [2, 3].

The equality of the MMF, taking into account the trapezoidal nature of the cross-sectional area of the multilayer HV winding (Fig. 1), is

\[ \varphi_1 = \frac{h_{\mathrm{cp}}}{h}\varphi_2 = \frac{C_2 + 2(b_1 + b_2)} {C_1 + 2(b_1 + b_2)} \varphi_2; \tag{10} \]

\[ C_1 = H_{\mathrm{v}} - l_{i1} - A; \qquad C_2 = H_{\mathrm{v}} - (l_{i1} + l_{i2}) - A; \tag{11} \]

\[ A = D_{\mathrm{v}} - 2\delta_{12} - 2\delta_{10}; \qquad B = H_{\mathrm{v}} - 2l_{\mathrm{i}} - A. \tag{12} \]

The specific short-circuit force of the multilayer winding from (6) and (8) [2]:

\[ \varphi_2=\frac{q_{\mathrm{m}2}\Delta_2}{h_{\mathrm{sr}}} = m_2 \sqrt{\frac{\alpha_2 x_2 y_2}{k_{r2}(y_2+i_2)}} ; \tag{13} \]

\[ b_2=m_2 x_2 M \rho_2 (m_2-1) \tag{14} \]

The coefficient of increase of the active resistance of the copper of a multilayer winding from (23) [3]:

\[ k_{r2}=1+\frac{1}{3p^2}. \tag{15} \]

The optimum value of the copper thickness of a layer [3]:

\[ x_2=\sqrt{\frac{1.73(y_2+i_2)}{C m_2 y_2}}. \tag{16} \]

Specifying several values of \(m_2\) within the range (4--12), we find for them \(x_2\) from (16) and \(b_2\) from (14). Using relation (27) [3], we find the corresponding values \(l_2\) and \(p_2\). Then we determine \(k_{r2}\) from (15) and \(\varphi_2\) from (13), after which (4), (6), and (7) can be solved jointly to find the optimum \(b_1\), \(b_2\), and \(x_1\).

The limiting powers per limb of 500 and 750 kV for the second combination of winding types are given in the last three columns of the table.

Table 1

To compile the table, the following initial data were adopted: general data and for the LV winding:
\(D_{\mathrm{v}}=260\); \(H_{\mathrm{v}}=450\); \(B=1.7\ \mathrm{T}\); \(k_c=0.83\); \(y_1=1.45\); \(\delta_1=0.8\); \(i_1=0.1\); \(\delta_{01}=5\); \(C=0.95\); \(\alpha_1=25\cdot10^4\); \(\varepsilon_1=0.3\); \(k_{\mathrm{p}}=0.78\); \(k_v=0.9\).

For the 500-kV coil winding:
\(i_2=0.3\); \(y_2=1.45\); \(\delta_2=1.2\); \(\delta_{12}=14\); \(l_{\mathrm{i}}=25\); \(\varepsilon_2=0.3\); \(k_{\mathrm{p}}=0.78\); \(\alpha_2=25\cdot10^4\); for the multilayer winding -- \(i_2=0.3\); \(y_2=2.0\); \(\delta_2=3.0\); \(\delta_{12}=5\); \(l_{\mathrm{i}2}=25\); \(l_{\mathrm{i}1}=12\); \(\varepsilon_2=0.3\); \(k_{\mathrm{p}}=0.57\); \(\alpha_2=16\cdot10^4\);

For the 750-kV coil winding:
\(i_2=0.4\); \(y_2=1.45\); \(\delta_2=1.2\); \(\delta_{12}=17\), \(l_{\mathrm{i}}=3.8\); \(\varepsilon_2=0.3\); \(\alpha_2=25\cdot10^4\).

For the multilayer winding:
\(i_2=0.4\); \(y_2=2\); \(\delta_2=3.0\); \(\delta_{12}=6\); \(l_{\mathrm{i}1}=12\); \(l_{\mathrm{i}2}=38\), \(\varepsilon_2=0.3\); \(k_{\mathrm{p}}=0.57\); \(\alpha_2=16\cdot10^4\) (dimensions in cm). The number of layers does not

rounded, and the dimensions of the conductors were not deliberately brought into conformity with the standard.

A comparison of the limb capacities for 500- and 750-kV transformers with coil and multilayer HV windings, as given in the table, speaks in favor of multilayer windings, despite the fact that we adopted the interlayer distance with a large margin \((\delta_2 = 3\ \text{cm})\).

REFERENCES

1. I. D. Kutyavin, L. I. Del’. On the limiting capacity of a transformer. Izvestiya TPI, vol. 172, 1967.

2. I. D. Kutyavin, L. I. Del’. On the limiting capacity of a transformer with multilayer windings. Izvestiya TPI, vol. 191, 1969.

3. I. D. Kutyavin. Design of transformer windings with minimum copper losses. Elektrotekhnika, No. 7, 1969.