On the Limiting Power of Compensating Chokes (Reactors) with an Air Gap

I. D. Kutyavin, N. P. Kostritskaya

(Tomsk Polytechnic Institute)

This article presents the results of a study of compensating chokes with an air gap equal to the height of the core window (Fig. 1). The core consists only of an armored yoke assembled from C-shaped sections made with radial laminations of transformer steel. This core design was chosen in order to minimize additional losses in the pressing devices of the core and in the tank.

The article investigates chokes with two winding variants: multilayer and coil windings.

Fig. 1.

A multilayer winding with solid grounding of the outer terminal (end). The inner end is brought out through the central opening in the yoke and is the working terminal. The height of the winding layers decreases toward the center; therefore the winding insulation increases in stages from the outer layer to the inner one.

Regulation of the reactor is not considered.

The basic relationships between the dimensions of the reactor (Fig. 1).

Radial width of the winding cross section:

\[ b = mx + (m - 1)\delta, \tag{1} \]

where \(m\) is the number of layers in the winding;

\(x\) is the radial width of the copper of a layer;

\(\delta\) is the radial distance between the copper of adjacent winding layers, consisting of the width of the axial cooling duct and the conductor insulation on two sides—\(i\).

Length of the mean turn of the winding:

\[ l_{\mathrm{m}} = \pi(d+b)=\pi(D-b). \tag{2} \]

\(D;\ d\) are the outer and inner diameters of the winding.

Cross-sectional area of the winding material:

\[ q_{\mathrm{m}}=\frac{mxhy}{y+i}; \tag{3} \]

where \(h\) is the height of the mean winding layer;

\(y\) is the axial dimension of an elementary conductor.

Width of the core window:

\[ a = D + 2\delta_{01} = d + 2(b+\delta_{01}). \tag{4} \]

Height of the core window:

\[ H = h + 2l_{\mathrm{u}}; \tag{5} \]

where

\[ l_{\mathrm{u}}=\frac{l_{0}+l}{2}. \tag{6} \]

Width of the steel of split yokes:

\[ \tau = D\sin\frac{180}{n}. \tag{7} \]

Cross-sectional area of the yoke steel:

\[ q_{\mathrm{c}}=k_{\mathrm{c}}nr\tau = k_{\mathrm{c}}nrD\sin\frac{180}{n}, \tag{8} \]

where \(n\) is the number of branches of the split yoke. For Fig. 1, \(n=8\). BBC uses \(n=6 \div 12\).

Calculated length of the core steel, referred to \(q_{\mathrm{c}}\):

\[ l_{\mathrm{c}}=H+2r+2\delta_{01}+0.5D\cos\frac{180}{n}. \tag{9} \]

Equation of the heat balance of the mean winding layer per one running centimeter of turn:

\[ k_{\mathrm{p}}2he=\rho k_{\mathrm{r}}\Delta^{2}\frac{hxy}{y+i}. \tag{10} \]

or

\[ \frac{2k_{\mathrm{p}}\varepsilon}{\rho}=a=k_{\mathrm{r}}\Delta^2\frac{xy}{y+i}. \tag{11} \]

The current density in the winding in A/cm\(^2\):

\[ \Delta=\sqrt{\frac{a(y+i)}{k_{\mathrm{r}}xy}}, \tag{12} \]

\(\varepsilon\) — density of the heat flux from the winding surface;
\(k_{\mathrm{p}}\) — coefficient taking into account the parts of the winding-layer surfaces covered by insulating details;
\(\rho\) — resistivity of the winding material at the design temperature;
\(k_{\mathrm{r}}\) — coefficient of increase in the active resistance of the winding due to the skin effect;

\[ k_{\mathrm{r}}=1+\frac{c^2m^2x^4y^2}{9(y+i)^2}, \tag{13} \]

where

\[ c=\pi f\sqrt{\frac{\mu_0}{\rho}}, \tag{14} \]

\(f\) — number of current periods (Hz) and \(\mu_0=4\pi\,10^{-9}\).

The power of the choke in kVA:

\[ S=4{,}44f\,10^{-11}B_{\mathrm{v}}q_{\mathrm{v}}\Delta q_{\mathrm{m}}. \tag{15} \]

The reduced area of the air gap:

\[ q_{\mathrm{v}}=\frac{\pi}{4}(D-b)^2. \tag{16} \]

Induction in the air gap:

\[ B_{\mathrm{v}}=\frac{0{,}4\pi IW\sqrt{2}}{H} =\frac{0{,}4\pi\sqrt{2}\,\Delta q}{H}. \tag{17} \]

Expression for the choke power after substituting the variables:

\[ S=K\frac{m^2h^2xy(D-b)^2}{k_{\mathrm{r}}H(y+i)}, \tag{18} \]

\[ K=4{,}44\cdot10^{-12}\sqrt{2}\,\pi^2 a. \tag{19} \]

The choke power (18) is a function of five variables: \(D\), \(h\), \(m\), \(x\), and \(y\), and has neither a maximum nor a minimum with respect to the variables \(D\), \(h\), and \(y\). It increases as these variables increase. However, for certain values of \(m\) and \(x\), expression (18) has a maximum. To determine the critical values of \(m\) and \(x\), we use the condition:

\[ \frac{\partial s}{\partial m}=0,\quad \text{and}\quad \frac{\partial s}{\partial x}=0. \tag{20} \]

As a result, we obtain two equations:

\[ 2k_{\mathrm{r}}\left[D-2m(x+\delta)+\delta\right] -mk'_{\mathrm{r}m}\left[D-m(x+\delta)+\delta\right]=0, \tag{21} \]

\[ k_{\mathrm{r}}\left[D-3mx-(m-1)\delta\right] -xk'_{\mathrm{r}x}\left[D-m(x+\delta)+\delta\right]=0, \tag{22} \]

where \(k'_{\mathrm{r}m}\) and \(k'_{\mathrm{r}x}\) are the derivatives of \(k_{\mathrm{r}}\) with respect to \(m\) and \(x\).

The simultaneous solution of these equations gives the following relations:

\[ m_0=\frac{3(D+\delta)}{5x+7\delta}; \tag{23} \]

\[ \frac{\sqrt{3cy\delta(D+\delta)}}{y+i} =U=\frac{(5v+7)}{v^2}\sqrt{\frac{1-v}{1+v}}, \tag{24} \]

where

\[ v=x:\delta<1. \tag{25} \]

To determine \(x\), one may use the curve in Fig. 2, constructed for \(U(v)\) from the right-hand side of (24). In this case \(U\) is determined from the left-hand side of (24).

The power \(S\) depends only weakly on \(y\); therefore the axial dimension of the conductor may be adopted within very wide limits.

When determining the limiting power of a reactor under transport-restriction conditions, the dimensions \(D\) and \(H\) will be limited by the maximum railway clearances.

When designing a reactor for a specified power, the dimension \(D\) can be found from (18), while the height \(H\) is determined by some condition, for example, the condition of the minimum total weight of active materials (steel—\(Q_c\) and copper—\(Q_m\)), or the condition of the minimum cost of the reactor, or the condition of minimum calculated expenditures. These questions are not considered in the present article.

Fig. 2.

The power of the reactor has a very strong dependence on the dimension \(\delta\), decreasing as it increases. This is evident from item 11 of Table 1, the data of which were obtained for \(y=3\); \(i=0.2\); \(D=200\); \(h=200\); \(l_0=10\); \(l=30\); \(\delta_{01}=10\); \(\delta=1.2;\ 2;\ 2.5\) (all dimensions in centimeters); \(k_c=0.8\); \(k_p=0.6\);

\[ B=16.5\,\text{kGs};\quad \varepsilon=0.16\,\frac{\text{W}}{\text{cm}^2};\quad n=8;\quad c=0.92\ \text{and}\ \alpha=9\cdot10^4 \]
for copper, and \(c=0.575\);

\[ \alpha=5.6\cdot10^4 \]
for aluminum.

Table 1

As can be seen from the table, some of the optimal dimensions and parameters of the choke variants considered turned out to be unusual. A large value was obtained for the additional losses:

\[ p_q \% = (k_t - 1)\,100 = 24 \div 29, \tag{26} \]

apparently inherent in a multilayer winding. The current density in the winding (item 7) also proved to be too high, which caused increased copper losses.

The weights of the active materials in tons (items 16 and 17) were determined from the expressions:

\[ Q_c = \gamma_c q_c l_c 10^{-6} \quad \text{and} \quad Q_m = \gamma_m q_m l_m 10^{-6}. \tag{27} \]

The cross-sectional area of the yoke steel and the radial dimension of the yoke \(r\) were determined from the condition of constancy of the magnetic flux at a specified induction in the steel \(B_c\):

\[ q_c = \frac{q_v B_v}{B_c}, \tag{28} \]

\[ r = \frac{q_c}{k_c n D \sin \frac{180}{n}}. \tag{29} \]

After substituting into (28) and (29) the expressions for the quantities \(q_c\), \(q_v\), and \(B_v\), we find

\[ r = \frac{0.1\pi^2\sqrt{2}\, m h (D-b)^2} {k_c B_c n H D \sin \frac{180}{n}} \sqrt{\frac{a}{\dfrac{x y}{k_r (y+i)}}}; \tag{30} \]

Let us now consider chokes with continuous coil, disk, or helical windings.

Basic relations of dimensions and parameters. The radial width of one side of the winding is

\[ b = m(x+i), \tag{31} \]

where \(m\) is the number of elementary conductors in the coil;
\(x\) is the radial dimension of the copper of an elementary conductor.

The total cross-sectional area of copper on one side of the winding is

\[ q_m = mxy \frac{h}{y+\delta}, \tag{32} \]

where \(\delta\) is the axial distance between the copper of adjacent coils, including the width of the radial cooling duct.

The heat-balance equation of the coil per one running centimeter of its length is:

\[ 2\varepsilon (k_b b + k_y y) = \rho k_t \Delta^2 mxy, \tag{33} \]

where \(k_b\), \(k_y\) are coefficients taking into account the portions of the surfaces \(b\) and \(y\) covered by insulating parts.

Let us denote

\[ \frac{2\varepsilon k_y}{\rho} = \alpha; \quad k_b : k_y = k_p \quad \text{and} \quad k_p b + y = P. \tag{34} \]

Then the current density in the winding from (33) is

\[ \Delta = \sqrt{\frac{\alpha P(x+i)}{k_t Bxy}}. \tag{35} \]

The coefficient of increase of the active resistance is reduced to the following form:

\[ k_{\Gamma}=1+\frac{c^{2}b^{2}x^{4}y^{2}}{9(x+i)^{2}(y+\delta)^{2}} . \tag{36} \]

Then the expression for the power for a reactor with a coil winding is:

\[ S=K\frac{bxyh^{2}P(D-b)^{2}}{k_{\Gamma}H(x+i)(y+\delta)^{2}} . \tag{37} \]

In this case the power of the reactor is a function of the variables \(D, h, b, x\), and \(y\). With increasing \(D\) and \(h\) the power of the reactor increases continuously, while when the variables \(b, x\), and \(y\) are varied, expression (37) has a maximum. To determine the critical (optimal) values of these variables one may use the condition:

\[ \frac{\partial S}{\partial b}=0;\quad \frac{\partial S}{\partial x}=0:\quad \frac{\partial S}{\partial y}=0, \tag{38} \]

This gives three equations:

\[ k_{\Gamma}\left[(k_{\mathrm{p}}b+P)(D-b)-2bP\right]-bP(D-b)k'_{\Gamma b}=0, \tag{39} \]

\[ k_{\Gamma}i-x(x+i)k'_{\Gamma x}=0, \tag{40} \]

\[ k_{\Gamma}\left[\delta(y+P)-k_{\mathrm{p}}by\right]-y(y+\delta)Pk'_{\Gamma y}=0; \tag{41} \]

where

\[ k'_{\Gamma b},\ k'_{\Gamma x}\ \text{and}\ k'_{\Gamma y} \]

are the derivatives of \(k\) with respect to \(b, x\), and \(y\).

By eliminating \(k_{\Gamma}\) from (39) and (40), and from (40) and (41), we find two new equations:

\[ P(D-b)i-(x+2i)\left[(k_{\mathrm{p}}b+P)(D-b)-2bP\right]=0; \tag{42} \]

\[ (x+2i)\left[\delta(y+P)-k_{\mathrm{p}}by\right]-i\delta P=0. \tag{43} \]

This makes it possible to replace the complex system of equations (39)—(41) by the simpler system (40), (42), and (43). However, this system also proves very difficult to solve simultaneously. Therefore we shall restrict ourselves to the general simultaneous solution of the two equations (40) and (43) with respect to \(x\) and \(y\), and the investigation of expression (37) for a maximum with respect to the variable \(b\) will be carried out numerically.

From equation (40), after substituting the values of \(k_{\Gamma}\) and \(k'_{\Gamma x}\), we find

\[ 3(x+i)(y+\delta)=cbx^{2}y\sqrt{\frac{2x+3i}{i}} . \tag{44} \]

The value of \(y\) from (43):

\[ y=\frac{-k_{\mathrm{p}}b\delta(x+i)}{k_{\mathrm{p}}b(x+2i)-\delta(2x+3i)} . \tag{45} \]

Substituting \(y\) from (45) into (44), we find

\[ \sqrt{\frac{3(k_{\mathrm{p}}b-\delta)}{k_{\mathrm{p}}b^{2}ci}}=U=\frac{v}{\sqrt[4]{2v+3}}, \tag{46} \]

\[ v=x:i. \]

Figure 3 shows the curve constructed from the right-hand side of expression (46). Having determined \(U\) (for known \(b\)) from the left-hand side of (46), from the curve in Fig. 3 we find \(v\) and, consequently, \(x\).

The results of investigating expression (37) for a maximum with respect to the variable \(b\) are given in Table 2. The investigation was carried out for \(k_{\mathrm{p}}=0.78;\ k_y=0.9;\ \delta=1;\ 1.2;\ i=0.01;\ 0.1;\ a=13.5\cdot10^{4};\ i_{\mathrm{n}}=\delta_{01}=10\). The remaining initial data are equal to those adopted earlier.

Table 2

As is seen from Table 2, the maximum choke power \(S(b)\) is reached at \(b \simeq 90\). However, for \(b > 70 \div 80\) the choke power increases only slightly; therefore the calculated value is \(b \simeq (0.35 \div 0.4)D\).

Fig. 3.

The investigations showed the effectiveness of reducing the insulation thickness of elementary conductors in order to increase the limiting power of the choke. For insulating parallel conductors from one another it is necessary to restrict oneself to enamel; turn insulation should preferably be made as a strip of electrical pressboard or another material.

From a comparison of Tables 1 and 2 it is evident that, under the same conditions, a multilayer winding gives a lower reactor power than a coil winding.

To increase the maximum limiting power of the reactor in comparison with the data of Table 2, the overall dimensions of the removable part, \(H_{\mathrm{v}}\) and \(D_{\mathrm{v}}\) (items 7 and 17), may be increased up to the limits permitted by railway clearances; the specific heat transfer from the surface of the windings may also be increased from the adopted value \(\varepsilon = 0.16\ \frac{\mathrm{W}}{\mathrm{cm}^{2}}\) to 0.24 by applying forced oil circulation.