SOME RESULTS OF A STUDY OF LOW-POWER HIGH-VOLTAGE TRANSFORMERS

I. D. Kutyavin, D. I. Anderman

In practice there often arises a need to manufacture transformers of small power—on the order of several kVA—for voltages up to 100 kV. Such transformers are used as laboratory transformers and in a number of other cases. When they are manufactured in single quantities, the most economical insulation design is the oil-barrier type. The basic requirement in design is minimization of the weight or cost of the active materials.

The available literature considers either transformers with power above 40,000 kVA for the 110 kV voltage class [1], or transformers with power up to 1 kVA for voltages not exceeding 36 kV [2].

Fig. 1. Sketch of a longitudinal section of transformer windings.

We have adopted the winding design shown in Fig. 1. A cascade design of the insulation of the secondary (high-voltage) winding makes it possible to reduce the average thickness of the interlayer insulation \(\delta_{\text{sl. av}}\) and to increase the copper fill factor both for the area of the secondary winding \(k_{m2}\) and for the entire window as a whole \(k_m\). When the entire layer is divided into \(n\) cascades, the thickness of the interlayer insulation of the first cascade must be calculated for the voltage \(\frac{2U_{\text{sl}}}{n}\), where \(U_{\text{sl}}\) is the layer voltage; of the second cascade, \(2 \cdot \frac{2U_{\text{sl}}}{n}\); of the \(n\)-th cascade, \(2U_{\text{sl}}\). The average thickness of the interlayer insulation is calculated for the voltage

\[ 2U_{\text{sl. av}} = 2U_{\text{sl}} \cdot \frac{1}{2}\left(\frac{1}{n}+1\right). \tag{1} \]

When \(n\) varies from 1 to \(\infty\), \(2U_{\text{sl. av}}\) varies in the range from \(2U_{\text{sl}}\) to \(U_{\text{sl}}\). The most rational division is to divide the layer into 2–4 cascades. Then, using the recommended [1, p. 153] thickness of interlayer insulation, the following insulation-design scheme may be recommended.

Table 1

Approximating \(\delta_{\text{sl. av}}(2U_{\text{sl}})\), we obtain the straight line

\[ \delta_{\text{sl. av}} = 0.15 + 10^{-4}\cdot 2U_{\text{sl}} \;(\text{mm}). \]

Let us express the weights of the active materials through the dimensions of the transformer (Fig. 1). The weight of the magnetic-core steel is

\[ G_c=\gamma_c \cdot q_c \cdot \{h+2l_i+k_{\text{ya}}\cdot [h+2l_i+D+2(r+a)]\}, \tag{2} \]

where

\[ q_c \text{ is the cross-sectional area of the core} \]

\[ q_c = k_c \cdot \frac{\pi D^2}{4}; \]

\(D\) is the diameter of the circle circumscribed about the stepped core;

\(k_c = k_{\text{kr}}\cdot k_{\text{zs}}\) is the steel fill factor of the area of this circle, equal to the product of the interlamination-insulation coefficient \(k_{\text{zs}}\) and the shape factor of the core section \(k_{\text{kr}}\) [1];

\(h\) is the height of the windings,

\(l_i\) is the insulation gap,

\(k_{\text{ya}}\) is the yoke reinforcement coefficient,

\(a\) is the window width, \(a=\delta_{01}+b_1+\delta_{12}+b_2+\delta_{02}\);

\(b_1\) and \(b_2\) are the radial widths of the windings,

\[ b_1=n_1(x_1+i_1), \]

\[ \theta_2 = n_2(p + t_2 + \delta_{\text{sl. av}})\cdot k_{\text{v}2}, \]

\(n_1, n_2\) are the numbers of layers, respectively, of the primary and secondary windings,

\(k_{\text{v}2}\) is the bulging coefficient of the secondary winding [2],

\(r\) is the yoke thickness, which, for a yoke width \(\tau\), we can determine from the equality

\[ k_{\text{zs}}\cdot \tau \cdot r = 0.5\cdot k_{\text{ya}}\cdot \frac{\pi D^2}{4}\cdot k_{\text{zs}}\cdot k_{\text{kr}}, \]

whence

\[ r = 0.125\cdot k_{\text{ya}}\cdot k_{\text{kr}}\cdot \frac{\pi D^2}{\tau}. \]

The weight of copper in the primary winding is

\[ G_{\text{m}1} = \gamma_{\text{m}}\cdot \pi D_{\text{sr}1}\cdot h\cdot \theta_1\cdot k_{\text{m}1}, \tag{3} \]

and in the secondary winding

\[ G_{\text{m}2} = \gamma_{\text{m}}\cdot \pi D_{\text{sr}2}\cdot h\cdot \theta_2\cdot k_{\text{m}2}, \tag{4} \]

where

\(D_{\text{sr}1}, D_{\text{sr}2}\) are the average turn lengths of the primary and secondary windings:

\[ D_{\text{sr}1} = D + 2\delta_{01} + \theta_1, \]

\[ D_{\text{sr}2} = D + 2\delta_{01} + 2\theta_1 + 2\delta_{12} + \theta_2; \]

\(k_{\text{m}1}\) and \(k_{\text{m}2}\) are the coefficients of filling the area of the primary and secondary windings with copper:

\[ k_{\text{m}1} = \frac{I_1}{\Delta_1 (y_1+i_1)(x_1+i_1)k_{\text{y}1}}, \]

\[ k_{\text{m}2} = \frac{I_2}{\Delta_2\cdot (d+i_2)(d+i_2+\delta_{\text{sl. av}})k_{\text{y}2}\cdot k_{\text{v}2}}, \]

\(I_1\) and \(I_2\) are the primary and secondary currents;

\(\Delta_1\) and \(\Delta_2\) are the current densities in the windings;

\(k_{\text{y}1}\) and \(k_{\text{y}2}\) are winding laying coefficients [1, 2].

The cost of the active materials is

\[ C_{\text{a}} = \alpha G_{\text{c}} + \beta G_{\text{m}}. \tag{5} \]

The values of \(\alpha\) and \(\beta\) are given in [1].

From the expressions obtained for the weights of the active materials and their cost, it is seen that the transformer will be designed in the best way at the optimal values of the quantities

\[ D,\ h,\ x_1,\ y_1,\ d,\ n_1,\ n_2,\ \Delta_1\ \text{and}\ \Delta_2, \]

corresponding to the minimum weight or cost of the active materials. We shall call the listed quantities variables. The functional relationship between the variables and the technical characteristics will be specified through “physical constraints”:

1. Transformer power

\[ S = K D^2 h \cdot \frac{\Delta_1\cdot x_1\cdot y_1\cdot n_1}{(y_1+i_1)\cdot k_{\text{y}1}}, \tag{6} \]

where

\[ K = 1.11\pi\cdot B\cdot f\cdot k_{\text{c}}\cdot 10^{-4}. \]

1. Reactive component of the short-circuit voltage

\[ u_{\text{p}} = \frac{7.92\cdot f\cdot S\cdot \pi\cdot D_{12}\cdot \delta_{\text{p}}\cdot k_R\cdot 10^{-8}} {h u_{\text{v}}^2}, \tag{7} \]

where

\[ D_{12}=D+2b_{01}+2\delta_i+\delta_{12}, \]

\[ \delta_p=\frac{\delta_{12}D_{12}+D_{\mathrm{cp}1}\cdot \frac{\varepsilon_1}{3}+D_{\mathrm{cp}2}\cdot \frac{\varepsilon_2}{3}}{D_{12}}, \]

\[ U_{\mathrm{B}}=KD^2, \]

\(k_R\) is the Rogowski coefficient.

1. The permissible heat-flux density from the outer surface of the LV winding.

\[ \sigma_1=\frac{\rho_m\cdot \Delta_1\cdot I_1\cdot n_1}{(y_1+i_1)\cdot k_{y1}\cdot k_{p1}}, \tag{8} \]

where

\(k_{p1}\) is the coefficient of coverage of the winding surface by slats and other insulating parts.

The same for the surface of the HV winding:

\[ \sigma_2=\frac{\rho_m\cdot \Delta_2\cdot n_2\cdot I_2}{k_{p2}\cdot k_{y2}\cdot (d+i_2)}. \tag{9} \]

1. Equality of the magnetomotive forces of the primary and secondary windings

\[ \frac{I_1\cdot n_1}{(y_1+i_1)k_{y1}}= \frac{I_2\cdot n_2}{(d+i_2)\cdot k_{y2}}. \tag{10} \]

1. The currents in the windings are related to the wire dimensions by the relations:

\[ I_1=\Delta_1\cdot x_1\cdot y_1, \tag{11} \]

\[ I_2=\Delta_2\cdot \frac{\pi d^2}{4}. \tag{12} \]

Using the seven equations given above (6)—(12), seven variables can be eliminated; one of them, namely \(n_1\), can be taken as the initial quantity. Then optimization of the functions of weight, cost of active materials, and total losses reduces to their minimization with respect to one independent variable, chosen to be the current density \(\Delta_1\). In addition, it is desirable to investigate the dependence of the functions \(G\), \(C_a\), and \(\sum P\) on the reactive component of the short-circuit voltage. In this connection, minimization of the functions was carried out simultaneously with respect to both variables \(\Delta_1\) and \(U_p\).

The power range studied was from 1.6 to 160 kVA, with \(\Delta_1\) varying within the limits from 1.6 to 2.6 A/mm² and \(u_p\) from 4 to 40%.

The effect of variation of \(u_p\) and \(\Delta_1\) on the weight of a 16.2 kVA transformer, the cost of the active materials, and the total losses is shown in Fig. 2. The minimum weight is observed at \(u_p=8\%\), \(\Delta_1=2\) A/mm²; the minimum cost of active materials at \(u_p=4\%\) and \(\Delta_1=2.2\) A/mm²; and the minimum total losses at \(u_p=6\%\) and \(\Delta_1=1.6\) A/mm². As the power is increased, the minimum of the weight, the cost of active materials, and the losses shifts toward smaller \(u_p\), while the optimum current density decreases (Fig. 3).

In the variant close to optimum in weight and material cost, the following dimensions were obtained for a 16.2 kVA transformer: \(D=8.408\) cm; \(h=70.9\) cm; \(x_1=3.378\) mm; \(y_1=9.08\) mm; \(d=0.265\) mm; \(b_1=0.7756\) cm; \(b_2=3.8\) cm, \(\Delta_2=3.2\) A/mm² at \(u_p=6\%\), \(\Delta_1=2.4\) A/mm². The weight of steel is 75.01 kg, the weight of copper 30.59 kg, copper losses 611 W, and steel losses 108 W. We note that at \(u_p=10\%\) it is possible to design a transformer of approximately the same weight and with the same losses as in

Fig. 2

Fig. 3. Values of \(U_p\) and \(\Delta_1\) corresponding to the minimum of weight, cost of active materials, and total losses as the power changes.

in the variant cited, but the cost of active materials is 20% higher, owing to the increase in the weight of copper.

In the calculation we adopted the following initial data: \(B = 1.5\ \text{T}\); \(k_{3c}=0.93\); \(k_{\mathrm{kr}}=0.908\); \(n_1=2\); \(\sigma_1=\sigma_2=0.1\ \text{W}/\text{cm}^2\); \(k_{p1}=0.75\); \(k_{p2}=1\); \(k_{v1}=1.05\); \(k_{v2}=1.05\); \(k_{v2}=1.12\); \(l_1=4\ \text{cm}\); \(\sigma_{01}=0.8\ \text{cm}\); \(\sigma_{12}=2\ \text{cm}\) when \(v_2>3\ \text{cm}\), and \(\sigma_{12}=5-v_2\) when \(v_2<3\ \text{cm}\); \(\sigma_{02}=6\ \text{cm}\); \(k_R=0.94\); \(k_{\mathrm{ya}}=1.1\); \(i_1=0.05\ \text{cm}\); \(i_2=0.009\ \text{cm}\).

On the basis of the investigations carried out, the following conclusions may be drawn:

1) the optimum value of the short-circuit voltage for low-power transformers with a voltage on the order of 100 kV depends on the power and increases as the latter decreases;

2) the optimum current density decreases somewhat as the power increases.

References

1. P. M. Tikhomirov. Calculation of Transformers. Energiya Publishing House, Moscow, 1968.

2. I. I. Belopolsky, L. G. Pikalova. Calculation of Low-Power Transformers and Chokes. Gosenergoizdat, 1963.