On the limiting power of transformers with multilayer windings

ON THE LIMITING POWER OF TRANSFORMERS WITH MULTILAYER WINDINGS

I. D. Kutyavin, L. I. Del.

(Presented by the scientific seminar of the Department of Electric Power Stations and the Department of Electrical Systems and Networks)

In addition to [1], this article considers the determination of the limiting power of transformers with a generalized coil (continuous coil, disk, and helical) LV winding (low voltage) and a multilayer HV winding (high voltage).

For the investigation we shall use the expression for the power of a transformer per one limb (MVA)

\[ S_c = K \cdot \Delta \cdot B q_m q_c, \tag{1} \]

where \(\Delta\) is the current density in the winding, \(a/cm^2\);
\(B\) is the induction in the limb steel, gs;
\(q_c\) is the cross-sectional area of the limb steel, \(cm^2\);
\(q_m\) is the total cross-sectional area of the winding material, \(cm^2\);

\[ K = 4.44 f \cdot 10^{-11}. \tag{2} \]

We adopt the following constraints and simplifications:

1. The specific heat transfer from the winding surface is limited, \(\varepsilon\; wt/cm^2\).
2. The short-circuit voltage \(U_k\) is limited in accordance with GOST;
3. The transport dimensions of the transformer are limited;
4. Additional losses from circulating currents in parallel conductors are not taken into account, assuming ideal transposition.

For layer windings, the heat-balance equation per unit length of the middle layer according to [2] is:

\[ 2 h \varepsilon k_{n2} = \rho k \Gamma 2 \Delta_2^2 \frac{x_2 y_2}{(y_2 + i_2)} \cdot h, \tag{3} \]

where \(k_{n2}\) is a coefficient taking into account the covering of part of the layer surface by insulating parts;
\(\rho\) is the specific resistance of the conductor, \(ohm \cdot cm\);
\(h\) is the axial height of the winding, \(cm\);
\(x_2\) is the radial width of the copper of the winding layer, \(cm\);
\(y_2\) is the axial dimension of the conductor copper, \(cm\);
\(i_2\) is the thickness of the conductor insulation on two sides, \(cm\).

The coefficient of increase in the active resistance of the winding material due to the skin effect for layer windings on the basis of [3] will be expressed as follows:

\[ k_{r2}=1+\frac{c^2x_2^4y_2^2m_2^2}{9(y_2+i_2)^2}, \tag{4} \]

\[ c=\pi f\frac{\mu_0}{\rho}, \tag{5} \]

where \(f\) is the current frequency, Hz;
\(\mu_0\) is the magnetic permeability of air.

The current density in the winding is determined from (3):

\[ \Delta_2=\sqrt{\frac{a_2(y_2+i_2)}{x_2y_2k_{r2}}}, \tag{6} \]

where

\[ a_2=\frac{2\varepsilon k_{n2}}{\rho}. \tag{7} \]

The cross-sectional area of the copper of the winding is

\[ q_{m2}=\frac{x_2y_2m_2}{(y_2+i_2)}\,h, \tag{8} \]

where \(m_2\) is the number of layers of the HV winding.

The cross-sectional area of the steel of the core is

\[ q_c=k_c\frac{\pi}{4}d^2, \tag{9} \]

where \(d\) is the diameter of the circle circumscribed about the stepped core, cm;
\(k_c\) is the coefficient of filling, with steel, of the area of the circle of diameter \(d\).

The expression for the power (1), taking into account (6), (8), (9), takes the form

\[ S_c=KB\frac{k_c\pi}{4}d^2h\cdot \sqrt{\frac{a_2m_2^2x_2y_2}{k_{r2}(y_2+i_2)}}. \tag{10} \]

As the variables \(d, h\) increase, the transformer power increases without bound. When determining the limiting power from transport dimensions, the quantities \(d\) and \(h\) are determined by these dimensions [1]. With increasing \(m_2\), the power tends to a certain limit

\[ \left. \sqrt{\frac{m_2^2x_2y_2}{k_{r2}(y_2+i_2)}} \right|_{m_2\to\infty} = \sqrt{\frac{9(y_2+i_2)}{c^2x_2^3y_2}}. \tag{11} \]

For given values of \(\Delta_2\) and \(x_2\), the quantity \(y_2\) is determined by the rated current of the secondary winding of the core

\[ I_{nc2}=x_2y_2\Delta_2. \tag{12} \]

As a result, the power expression (10) has a maximum only with respect to the variable \(x_2\).

Differentiating (10) with respect to \(x_2\), after transformations we obtain the equation

\[ k_{r2}-x_2k'_{r2}(x)=0, \tag{13} \]

where \(k'_{r(x)}\) is the derivative of \(k_r\) with respect to \(x_2\).

Solving equation (13), we obtain

\[ x_{20}=\sqrt{\frac{\sqrt{3}\,(y_2+i_2)}{cm_2y_2}}. \tag{14} \]

To determine \(y_2\), we use the expression for the current of the core of the secondary winding (12).

\[ y_2=\frac{I_{nc2}}{x_2\Delta_2}. \tag{15} \]

Taking into account (6) and (14), formula (15) takes the form

\[ y_2=I_{nc2}\sqrt[4]{\frac{cm_2r^2}{\sqrt{3}\,a^2}}-i_2, \tag{16} \]

where

\[ I_{\mathrm{sc}2}=\frac{\sqrt{3}\,S_c}{U}. \tag{17} \]

In deriving formula (16) it is assumed that

\[ y_2(y_2+i_2)^3 \cong (y_2+i_2)^4 . \tag{18} \]

The error obtained with such a substitution, for \(i_2=0.1\div 0.3\) and \(y_2 \gg 1\), does not exceed \((1\div 7)\) percent.

Substituting (14) into (10), we obtain

\[ S_c=KBq_{ch}\cdot m_2^{3/4}\cdot \sqrt[4]{\frac{\sqrt{3}\,a_2}{ck_{r2}^{\,2}}}. \tag{19} \]

The coefficient \(k_{r2}\) for layered windings, upon substitution of the optimal \(x_2\) from (14), will be equal to

\[ k_{r2}=1+\frac{1}{3}. \tag{20} \]

In the case of splitting the conductor of thickness \(x_2\) into \(k\) parallel branches, it is shown in [3] that the optimal coefficient is expressed as

\[ k_{r2k}=1+\frac{1}{3k^2}. \tag{21} \]

In Table 1, items 1 through 9 determine the geometric dimensions and parameters of transformers of limiting powers with layered windings on the HV side for the following specified constant quantities:

Table 1.

\[ \begin{gathered} h=260\ \mathrm{cm};\quad d=120\ \mathrm{cm};\quad \rho=2.14\cdot10^{-6}\ \mathrm{ohm\cdot cm};\quad k_c=0.8;\\ k_R=0.95;\quad B=17\,500\ \mathrm{gs};\quad f=50\ \mathrm{Hz};\quad k_{p2}=0.75;\quad i_2=0.3\ \mathrm{cm};\\ C=0.92;\quad \varepsilon=0.3\ \mathrm{W/cm^2};\quad \delta_{02}=10\ \mathrm{cm}. \end{gathered} \]

The parallel conductors are made with enamel insulation of thickness \(0.01\ \mathrm{cm}\), and the turn insulation is made of electrical cardboard.

As noted above, the rod power \(S_c\) increases with the number of winding layers \(m_2\). Along with this, the short-circuit voltage also rises sharply, reaching an inadmissible value, as was shown in [1]. To reduce the magnitude of the short-circuit voltage, the high-voltage winding is made with two concentrics, as in Fig. 1. Then the short-circuit voltage will be determined by the formula

\[ U_p=\frac{k_R\Delta_2 q_{m2} l_m \delta^1}{3.33Bq_c\cdot h}, \tag{22} \]

Fig. 1

where

\[ \delta^{1}=0.5\,b_{1}+0.5\,b_{2}+3\delta_{12}; \tag{23} \]

\[ l_{m}=\pi d_{12}; \tag{24} \]

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

\(\delta'\) is the reduced width of the leakage channel, cm;

\(\delta_{01}\) is the insulation distance from the core to the winding, cm;

\(\delta_{01}\) is the insulation distance from the core to the winding, cm;

\(b_{2}\) is the width of the high-voltage winding, cm;

\[ b_{2}=m_{2}x_{2}+\delta_{2}(m_{2}-1), \tag{25} \]

\(\delta_{2}\) is the axial duct of the high-voltage winding, determined by the interlayer voltage and the breakdown voltage per 1 cm (breakdown voltage for oil is 25 kV/cm), cm;

\(b_{1}\) is the width of the low-voltage winding, cm.

To determine the dimensions of the “generalized coil” low-voltage winding we use Fig. 2, which presents the dependences \(b_{1}=f(S_{c})\); \(x_{1}=f(S_{c})\); \(y_{1}=f(S_{c})\) and \(\Delta_{1}=f(S_{c})\), constructed from the data of [1].

Fig. 2

From \(S_c\), taken from Table 1, using Fig. 2, we determine the corresponding dimensions of the LV winding, given in items 10–17 of Table 1. From the data for the HV and LV windings, the voltage \(u_p\), shown in item 18, is determined.

When determining the limiting power of transformers, the quantities \(d\) and \(h\) are found from the permissible railway clearances.

Taking the foregoing into account, it is possible to determine the dimensions of the transformer for maximum utilization of the active materials, for a specified power and short-circuit voltage, in the case of using a coil-type LV winding and a multilayer HV winding.

According to [1], \(x_1;\ y_1;\ \Delta_1\) are determined and the ratio \(\dfrac{S_c}{d^2h}\) is found

\[ \frac{S_c}{d^2h} = \frac{K \cdot B \cdot k_c \cdot \pi \cdot b_1 \cdot x_1 y_1 \Delta_1} {4(x_1+i_1)(y_1+\delta_1)} . \tag{26} \]

Next, the dependences are constructed

\[ x_1=f\!\left(\frac{S_c}{d^2h}\right);\quad y_1=f\!\left(\frac{S_c}{d^2h}\right);\quad \Delta_1=f\!\left(\frac{S_c}{d^2h}\right);\quad b_1=f\!\left(\frac{S_c}{d^2h}\right). \]

Using the specified core power, the short-circuit voltage, and the adopted number of layers of the layer winding, the parameters of the high-voltage winding \((y_2;\ x_2;\ b_2;\ \Delta_2)\) are calculated successively, and the ratio is determined:

\[ \frac{S_c}{d^2h} = \frac{KB \cdot k_c \pi}{4} \cdot \frac{x_2 y_2 \cdot \Delta_2 m_2}{(y_2+i_2)} . \tag{27} \]

From the obtained ratio \(\left(\dfrac{S_c}{d^2h}\right)\), \(b_1\) is determined from the graph (Fig. 3).

Solving jointly the power equation (10) and the voltage equation—

Fig. 3

Table 2

of short-circuit voltage (the HV winding is made with one concentric winding)

\[ u_p=\frac{2k_R\Delta_2 q_{m2}\cdot l_m\cdot \delta'}{3{,}33Bq_c h}, \tag{28} \]

we successively find the winding height and the core diameter.

The calculation of the transformer for a specified power and short-circuit voltage is presented in Table 2 for the following adopted constants: $\varepsilon=0{,}26\ \mathrm{W/cm^2}$; $U_p=0{,}13$; $U=750\ \mathrm{kV}$; $B=17\,500\ \mathrm{Gs}$; $\delta_{01}=5\ \mathrm{cm}$. The remaining constants are the same as for Table 1.

REFERENCES

1. I. D. Kutyavin, L. I. Del. “On the limiting power of a transformer.” Izv. TPI, vol. 172.

2. I. D. Kutyavin. “On the determination of the optimal dimensions of three-phase two-winding transformers.” Izv. TPI, vol. 130, 1964.

Dietrich W. Auslegung von Transformatorenwicklungen mit kleinstom Wirkwiderstand, “Elektrotechnische Leitschrift”, No. 6, 1965.