Maximum Power of Core-Type Smoothing Reactors
Abstract Generated abstract
This article derives an expression for the maximum rating of core-type smoothing reactors with a distributed magnetic gap and coil winding. Using equivalent reactor power, flux-linkage components, an idealized plane-parallel magnetic field, and a winding heat-balance relation, it reduces the rating formula to a function of principal geometric and conductor parameters. The analysis shows that, while some dimensions require additional constraints such as weight and overall size, the reactor power has a maximum with respect to the axial conductor dimension, yielding an explicit optimum value for that dimension.
Full Text
Maximum Power Rating of Core-Type Smoothing Reactors
L. I. Piletskii
(Presented by the scientific seminar of the departments of electric power stations and electrical systems and networks)
This article considers reactors of core-type construction, whose magnetic core contains a core magnetic gap distributed over its height. The general form of such a reactor is shown in Fig. 1. The winding of the reactor under study is of the coil type. Similar reactor designs are widely used both in the USSR \[1, 2, 3\] and abroad \[4, 5\].
Fig. 1
In studying reactors it is more convenient to operate not with complex parameters, but with certain equivalent quantities (effective values of current and voltage) that fully correspond to the electromagnetic loading of the reactor $LI$. In order to apply to reactors the regularities known from transformer theory, it is necessary to introduce the conventional concept of the reactor’s rated power. The concept of the rated, or overall, power of a reactor is convenient because it can be used to represent the physical dimensions (weight, overall dimensions) and, correspondingly, the calculated costs of manufacturing and operating the reactor. Based on the specified
of the reactor inductance and the dc component of the current, the typical power of the reactor \(S\) can be represented in the following form:
\[ S=\omega L I_0 I_{\mathrm{p}}, \tag{1} \]
where
\[ \begin{aligned} \omega&\text{ — the industrial-frequency current frequency;}\\ I_0&\text{ — the dc component of the rectified current;}\\ I_{\mathrm{p}}&\text{ — the rms value of the rectified current;}\\ L&\text{ — the reactor inductance due to the industrial-frequency current.} \end{aligned} \]
For smoothing reactors, according to [1], one may take
\[ I_{\mathrm{p}}=I_0. \tag{2} \]
Then, expressing the inductance through flux linkage and current, the power of the reactor, taking (2) into account, takes the following form:
\[ S=\omega\psi I_0, \tag{3} \]
where \(\psi\) is the total flux linkage
\[ \psi=\psi_{\mathrm{c}}+\psi_{\mathrm{b}}+\psi_{\mathrm{w}}, \tag{4} \]
\(\psi_{\mathrm{c}}\) — flux linkage due to the flux passing only through the core of the magnetic circuit;
\(\psi_{\mathrm{b}}\) — flux linkage due to the flux passing between the core and the inner diameter of the winding;
\(\psi_{\mathrm{w}}\) — flux linkage due to the flux flowing exclusively across the winding width \(b\).
Fig. 2
In considering the magnetic-field pattern it was assumed:
- The magnetic induction across the winding width in section \(a-a\) (Fig. 1) is distributed according to a linear law (Fig. 2).
- The magnetic field is idealized, i.e., it is regarded as plane-parallel.
- The magnetic permeability of the magnetic circuit is assumed to be infinitely large.
Then, proceeding from the above assumptions and the diagram of the distribution of induction across the winding width (Fig. 2), the separate components of the total flux linkage can be determined as
\[ \psi_{\mathrm{c}}=B_{\mathrm{z}}Wq_{\mathrm{z}}, \tag{5} \]
where \(q_{\mathrm{z}}\) is the calculated cross section of the gap, taking account of the bulging of the lines of force,
\[ \psi_{\mathrm{b}}=\pi B_{\mathrm{sh}} W \delta_0(d+\delta_0). \tag{6} \]
The third component, \(\psi_{\mathrm{w}}\), is the integral sum of the linkages of the flux flowing across the winding width with its turns.
The law of variation of induction across the winding width can be expressed in the following form:
\[ B_x=\frac{B_{\mathrm{sh}}}{b}\left(\frac{d+2\delta_0+2b}{2}-x\right). \tag{7} \]
The magnetic flux of an elementary tube \(d\Phi_x\), produced by an infinitely thin layer of the winding located at a distance \(x\) from the winding axis, is equal to:
\[ d\Phi_x = B_x \cdot 2\pi x\,dx . \tag{8} \]
This flux \(d\Phi_x\) links with all turns located toward the outer diameter of the winding from the elementary tube, that is, with the number of turns
\[ \frac{W}{b}\left(\frac{d+2\delta_0+2b}{2}-x\right). \]
Then the total flux linkage over the width of the winding is equal to
\[ \psi_b = \frac{2\pi B_{\mathrm{sh}}W}{b^2} \int_{\frac{d}{2}+\delta_0}^{\frac{d}{2}+\delta_0+b} \left(\frac{d+2\delta_0+2b}{2}-x\right)^2 x\,dx = \pi B_{\mathrm{sh}}W\frac{b}{3}(d+2\delta_0+0.5b). \tag{9} \]
In expressions (5), (6), (7), and (9), \(B_z\) is the calculated induction in the gap, taking into account the bulging of the magnetic lines,
\[ B_z=\frac{\mu_0 I W q_c}{n z k_c q_z}, \tag{10} \]
where
\[ \begin{aligned} \mu_0 &\text{ is the relative magnetic permeability in the core gap;}\\ z &\text{ is the length of a unit gap;}\\ n &\text{ is the number of unit gaps;}\\ q_c &\text{ is the active cross section of the steel of the limb;}\\ k_c &\text{ is the coefficient of filling with steel of the area of a circle of diameter } d, \end{aligned} \]
\[ q_c=\frac{\pi d^2}{4}k_c; \tag{11} \]
\(d\) is the diameter of the magnetic-core limb;
\(W\) is the number of turns of the winding.
To determine \(B_{\mathrm{sh}}\), the induction in the channel between the core and the inner diameter of the winding, it is assumed that the magnetic flux passing through this channel, as well as the flux passing across the width of the winding, closes not through the magnetic core, but through air.
Then, on the basis of the total-current law,
\[ B_{\mathrm{sh}}=\frac{\mu_0 I W k_p}{2h}, \tag{12} \]
where \(k_p\) is Rogowski’s coefficient [6].
It should be noted that a small fraction of the flux passing through the axial channel and across the width of the winding will also close through the yokes of the magnetic core. This applies to the portion of the winding located in one plane with the magnetic core. In further investigations this small fraction of the flux, because of its small magnitude, may be neglected.
Taking into account (5), (6), (7), (9), (10), and (12), and expressing the ampere-turns \(I_0W\) through the current density \(\Delta\) and the copper cross section of the winding \(q_M\), the reactor power may be represented in the form:
\[ S=\pi\omega\mu_0\Delta^2 q_M^2 \left\{ \frac{d^2}{4nz} + \frac{k_p}{6} \left[ 3\delta_0(d+\delta_0)+b(d+2\delta_0+0.5b) \right] \right\}. \tag{13} \]
Copper cross-section with allowance for the variable fill factor of the conductor metal
\[ q_{\mathrm{m}}=\frac{\varepsilon h b x y}{(x+i)(y+\delta)}, \tag{14} \]
where \(x\) and \(y\) are, respectively, the radial and axial dimensions of the elementary conductor;
\(i\) is the insulation thickness on two sides of the conductor;
\(\delta\) is the axial distance between the conductor material of adjacent coils, including the width of the cooling radial duct. With allowance for (14), the reactor rating (13) is a function of seven variables \(d\), \(b\), \(h\), \(nz\), \(x\), \(y\), and \(\Delta\).
If the relationship between current density and the specific heat removal from the winding surface is used, then the number of variables entering into (13) can be reduced.
The heat-balance equation of the coil per one linear centimeter of the mean turn, without allowance for heat transfer from the surface of the magnetic core, is
\[ 2\varepsilon(k_b b+k_y y)=\rho \Delta^2 \frac{bxy}{x+i}, \tag{15} \]
where \(\varepsilon\) is the heat-flux density from the winding surface, W/cm\(^2\);
\(k_b,\ k_y\) are coefficients taking into account the parts of the surfaces \(b\) and \(y\) covered by insulating parts;
\(\rho\) is the resistivity of the winding material at the design temperature, ohm·cm.
Denoting
\[ \frac{2\varepsilon k_y}{\rho}=a;\quad k_{\mathrm{p}}=\frac{k_b}{k_y};\quad k_{\mathrm{p}}b+y=p, \tag{16} \]
the current density can be determined from (15):
\[ \Delta=\sqrt{\frac{ap(x+i)}{bxy}}. \tag{17} \]
Using (17), the reactor rating in kilovolt-amperes can be expressed through the principal geometric dimensions as
\[ S=\frac{Kbh^2xyP}{(x+i)(y+\delta)^2} \left\{ \frac{d^2}{4nz} +\frac{k_{\mathrm{p}}}{h}\left[3\delta_0(d+\delta_0)+bd+2\delta_0+0.5b\right] \right\}, \tag{18} \]
where \(K=4\pi\mu_0\omega a\,10^{-3}\).
Expression (18) is a function of six variables: \(d\), \(b\), \(h\), \(x\), \(y\), and \(nz\). As \(d\), \(b\), and \(h\) increase, the rating increases without bound; with respect to the variable \(nz\), an inverse proportionality is observed. For \(x \gg i\), the reactor rating is practically independent of this variable. With respect to the variable \(y\), the reactor rating has a maximum. Using the condition
\[ \frac{\partial S}{\partial y}=0, \tag{19} \]
one can obtain the optimum value \(y_0\), at which the maximum rating is attained. The solution of (19) gives
\[ y_0=\frac{k_{\mathrm{p}}b\delta}{k_{\mathrm{p}}b-2\delta}. \tag{20} \]
The nature of the change in power as a function of the axial dimension of the conductor \(y\) is shown in Fig. 3, where the reactor power at \(y = y_0\) is taken as 100%.
Fig. 3
To determine the other variables \(d\), \(b\), and \(h\), it is necessary to use additional constraints: a) a constraint on the weight of the reactor; b) a constraint on the overall dimensions of the reactor.
Conclusion
A maximum is observed in the power of a core-type smoothing reactor with a coil winding at a definite axial dimension of the conductor.
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