ON THE LIMITING POWER OF SMOOTHING REACTORS WITH A LINEAR CHARACTERISTIC

L. I. Piletskii, I. D. Kutyavin

The article investigates the influence of the principal geometrical dimensions of smoothing reactors on their maximum power. The reactor considered has a linear characteristic (the air gap is equal to the height of the core window).

The design of the reactor magnetic system resembles a hollow multifaceted prism assembled from U-shaped packets made of radially laminated transformer steel (Fig. 1). The winding is located inside. Such a design makes it possible to minimize the additional losses in the tank and clamping device caused by the pulsating flux.

The relationship between the electrical parameters and the geometrical dimensions of the reactor.

The energy stored in the reactor is:

\[ W_{\text{р}} = L \cdot \frac{I^2}{2}, \tag{1} \]

where \(I\) is the constant component of the current flowing through the reactor, A;
\(L\) is the inductance of the reactor, H;

\[ L=\frac{B_c \cdot q_c w}{I}; \tag{2} \]

\(B_c\) is the induction in the steel, G;
\(w\) is the number of turns of the reactor winding;
\(q_c\) is the cross-sectional area of the steel of the outer yoke.

Fig. 1

We idealize the case under consideration by assuming that the magnetic flux passes only through the steel. Therefore

\[ \Phi_{\text{steel}}=\Phi_{\text{gap}} \]

or

\[ q_c=\frac{B_{\text{gap}}\cdot q_{\text{gap}}}{B_c}, \tag{3} \]

where \(B_{\text{gap}}\) and \(q_{\text{gap}}\) are the induction and the cross section in the air gap. Then the standard power of the reactor in kva is

\[ S=\omega L\frac{I^2}{2}=\frac{\omega}{2}\cdot B_c\cdot q_c\cdot \Delta\cdot q_m\cdot 10^{-11}; \tag{4} \]

where \(\omega\) is the frequency of the supply network;
\(q_m\) is the cross-sectional area of the winding material.

The steel cross section \(q_c\) can be expressed through the geometrical dimensions

\[ q_c=\kappa_c\cdot n\cdot r\cdot \tau =\kappa_c\cdot n\cdot r\cdot D\cdot \operatorname{tg}\frac{180^\circ}{n}, \tag{5} \]

since

\[ \tau=D\cdot \operatorname{tg}\frac{180^\circ}{n}. \]

In expression (5)

\(n\) is the number of packets of the split core within the range \(=6\div 12\);
\(\kappa_c\) is the steel filling factor of the yoke;
\(r\) is the radial dimension of the core packet, cm;
\(D\) is the outside diameter of the winding, cm.

The induction in the air gap, for its length equal to \(H\), is

\[ B_{\text{gap}}=\frac{0.4\pi\cdot I\cdot w}{H} =\frac{0.4\pi\cdot \Delta\cdot q_m}{H}. \tag{6} \]

The cross-sectional area of the winding material is expressed through the geometrical dimensions in the following form:

\[ q_m=\frac{\kappa_0\cdot b\cdot x\cdot y\cdot h}{(x+i)\cdot (y+\delta)}, \tag{7} \]

here \(\kappa_0\) is a coefficient accounting for the presence of axial cooling ducts in the coil every \(8\div 10\) cm (see Fig. 2),

\[ \kappa_0=\frac{b_1}{b_1+v}, \tag{8} \]

Fig. 2

\(b\) is the radial width of one side of the winding, cm;
\(x\) is the radial dimension of the copper of an elementary conductor, cm;
\(i\) is the thickness of the insulation on the two sides of the conductor, cm;
\(y\) is the axial dimension of the conductor copper, cm;
\(\delta\) is the axial distance between the (aluminum) copper of adjacent coils, including the width of the radial cooling duct, cm;
\(h\) is the axial height of the winding.

The area of the air gap, reduced to the mean turn of the winding, taking into account the axial cooling ducts, is

\[ q_{\text{gap}}=\frac{\pi}{4}\cdot(D-b)^2 . \tag{9} \]

Neglecting the increase in the active resistance of the winding due to the alternating component of the current flowing through the reactor, we write the heat-balance equation of the coil per one running centimeter of the mean turn:

\[ 2\varepsilon\left[\kappa_{\mathrm{v}}\cdot\kappa_0\cdot b+\kappa_{\mathrm{y}}\cdot y\cdot \frac{b}{b_1+v}\right] = \rho\cdot\Delta^2\cdot \frac{\kappa_0\cdot b\cdot x\cdot y}{(x+i)} \tag{10} \]

or

\[ 2\varepsilon\left(\kappa_{\mathrm{v}}+\kappa_{\mathrm{y}}\cdot y\frac{1}{b_1}\right) = \rho\cdot\Delta^2\frac{xy}{(x+i)}, \]

where \(\varepsilon\) is the heat-flux density from the winding surface, W/cm\(^2\);
\(\kappa_{\mathrm{v}}\) and \(\kappa_{\mathrm{y}}\) are coefficients allowing for the parts of the surfaces \(b\) and \(y\) covered by insulating components;
\(\rho\) is the resistivity of the winding material at the design temperature, \(\Omega\cdot\text{cm}\).

Let

\[ \frac{2\varepsilon\cdot\kappa_{\mathrm{y}}}{\rho}=\alpha, \qquad \frac{\kappa_{\mathrm{v}}}{\kappa_{\mathrm{y}}}=\kappa_{\mathrm{p}}, \]

then the current density \(\Delta\) from (10) is

\[ \Delta= \sqrt{\frac{\alpha\cdot(x+i)}{x\cdot y} \left(\kappa_{\mathrm{p}}+\frac{y}{b_1}\right)} . \tag{11} \]

Substituting (6), (7), and (11) into (3), and solving (3) jointly with (5) with respect to \(r\), we determine

\[ r= \frac{0.1\cdot\pi^2\cdot\Delta\cdot q_{\mathrm{m}}\cdot(D-b)^2} {H\cdot B_{\mathrm{c}}\cdot\kappa_{\mathrm{c}}\cdot n\cdot D\cdot \tg\frac{180^\circ}{n}} . \tag{12} \]

And expression (5) will take the form

\[ q_{\mathrm{c}}= \frac{0.1\cdot\pi^2\cdot\Delta\cdot q_{\mathrm{m}}\cdot(D-b)^2} {H\cdot B_{\mathrm{c}}}. \tag{13} \]

After substituting (7), (11), and (13) into (4), the reactor power is expressed through its geometric dimensions:

\[ S= K\, \frac{\kappa_0^2\cdot b^2\cdot h^2\cdot x\cdot y\cdot(D-b)^2} {H(x+i)(y-\delta)^2} \left(\kappa_{\mathrm{p}}+\frac{y}{b_1}\right). \tag{14} \]

Here

\[ K=\pi^3\cdot f\cdot\alpha\cdot10^{-12}. \]

The reactor power (14) is a function of five variables \(D,h,b,x,y\). With increasing \(D\), \(h\), and \(x\), the power increases without having a maximum. The variables \(b\) and \(y\) affect the growth of power differently. With increasing variables \(b\) and \(y\), the power increases, reaches a maximum value, and then decreases.

To determine the values of \(b\) and \(y\) at which the power reaches a maximum, we use the condition

\[ \frac{\partial S}{\partial b}=0 \qquad \text{and} \qquad \frac{\partial S}{\partial y}=0, \]

\[ \frac{\partial S}{\partial b}=2b(D-b)^2-2b^2(D-b)=0, \tag{15} \]

\[ \frac{\partial S}{\partial y}= \frac{\left(\kappa_{\mathrm{p}}+2\frac{y}{b_1}\right)(y+\delta)-2y\left(\kappa_{\mathrm{p}}+\frac{y}{b_1}\right)} {(y+\delta)^3}=0. \tag{16} \]

Solving (15) and (16), we determine the optimal \(b_0\) and \(y_0\):

\[ b_0=\frac{D}{2}, \tag{17} \]

\[ y_0=\frac{\kappa_{\mathrm{p}} b_1 \delta}{\kappa_{\mathrm{p}} b_1-2\delta}. \tag{18} \]

Analyzing expression (14), one may note that the power depends only weakly on the radial dimension of the conductor and increases as it is increased.

Table 1 shows the influence of \(b_1\) on the reactor power for the following values of the remaining variables and constants entering into [14]:
\(y=y_0;\ b=100\ \mathrm{cm};\ D=200\ \mathrm{cm};\ h=40\ \mathrm{cm};\ H=h+2l=60\ \mathrm{cm};\ \delta=1\ \mathrm{cm};\)
\(\vartheta=1\ \mathrm{cm};\ n=8;\ K=2.1\cdot 10^{-4};\ \varepsilon=0.16\ \mathrm{W/cm^2};\ \kappa_{\mathrm{v}}=0.7;\ \kappa_y=0.9\)
and \(\rho=2.1\cdot 10^{-6}\ \Omega\cdot\mathrm{cm}\).

Table 1

When designing a reactor for a specified power, the dimension \(D\) can be determined from (14), while for determining \(h\) some additional condition is necessary, for example the condition of minimum weight of active materials or minimum calculated costs. Since the winding width from (17),

\[ b_0=\frac{D}{2}, \]

is not technological, it is necessary to take

\[ b<\frac{D}{2}. \]

Figure 3 shows the influence of the reactor winding width on its power. The reactor power at \(b=D/2\) is taken as 100%. From the graph it is seen that the reactor power decreases only slightly for

\[ b=(0.4\div0.45)D. \]

Fig. 3