Expressions for the Current Density of Ferromagnetic Devices
Abstract Generated abstract
This paper derives expressions for selecting the current density of ferromagnetic devices under thermal and geometric constraints. Starting from classical formulas for coil and core overheating, active material volumes, and cooling surfaces, it obtains relations for current density for devices up to 10 kVA at 50 to 100 Hz and for higher power devices operating at frequencies above 100 Hz, with a corresponding expression for magnetic induction. The resulting formulas relate current density to permissible overheating, heat transfer, material resistivity, filling factors, cooling surfaces, geometry, power, and frequency, and are intended to support calculation, analysis, and synthesis of ferromagnetic device designs.
Full Text
CURRENT DENSITY EXPRESSIONS FOR FERROMAGNETIC DEVICES
V. P. OBRUSNIK, G. A. KHAIDUROVA
(Presented by the scientific-technical seminar of the NII AEM)
In calculations of ferromagnetic devices (FD), the current density is usually chosen to be maximal, since this always leads to a reduction in the specific economic index (weight, dimensions, and cost per unit power). The most significant limitation on its limiting values is the permissible overheating of the coils, which is closely related to many physical and geometrical parameters of the FD. Below are expressions that make it possible to calculate the current density optimally. They are obtained on the basis of classical formulas:
a) for the overheating temperature of FD coils [1, 2]
\[ \Delta \tau_k = j^2 \cdot \frac{\rho_k \cdot K_{zk} \cdot V_k}{\sigma_k \cdot \Pi_{ok}} ; \tag{1} \]
b) for the overheating temperature of cores [1, 2]
\[ \Delta \tau_c = B^2 \left( \frac{f_1}{50} \right)^{1.5} \cdot \frac{g_c \cdot K_{zc} \cdot \rho_c \cdot V_c}{\sigma_c \cdot \Pi_{oc}} . \tag{2} \]
c) for the volumes of coils and cores [3, 4]
\[ V_k = \frac{D_0^3}{\sqrt[4]{B^3 \cdot j^3}} \cdot l_k \sqrt{ \frac{x \cdot z}{y \cdot \kappa_0} } . \tag{3} \]
\[ V_c = \frac{D_0^3}{\sqrt[4]{B^3 \cdot j^3}} \cdot \frac{l_c}{ \sqrt{ \frac{x \cdot z}{y \cdot \kappa_0} } } , \tag{4} \]
where
\[ D_0 = \sqrt[4]{ \frac{P_{r\phi}}{ 4 \cdot K_f \cdot n_k \cdot K_{zc} \cdot K_{zk} \cdot f_1 } } ; \tag{5} \]
d) for the cooling surfaces of coils and cores [3, 4]
\[ \Pi_{ok} = \frac{D_0^2}{\sqrt{B \cdot j}} \cdot \frac{ z + \frac{x}{\kappa_0} }{ \sqrt[4]{ \frac{x \cdot z \cdot y}{\kappa_0} } } \cdot 2 \cdot l_k \cdot \varphi_{pk}, \tag{6} \]
\[ \Pi_{\mathrm{oc}}=\frac{D_0^2}{\sqrt{B\cdot j}}\cdot \sqrt[4]{\frac{1-y}{\dfrac{x\cdot z\cdot y}{\kappa_0}}}\cdot 2\cdot l_c\cdot \varphi_{\mathrm{pc}}. \tag{7} \]
In expressions (1)—(7) and below the following notation is adopted:
\(B, j\)—induction and current density;
\(\rho_k, \sigma_k, V_k, \Pi_{ok}, k_{zk}\)—specific resistance, heat-transfer coefficient, volume, cooling surface, and fill factor of the active materials of the FU coils;
\(\rho_c, g_c, \sigma_c, V_c, \Pi_{oc}, k_{zc}\)—specific losses in the steel (at a frequency of 50 Hz), specific weight, heat-transfer coefficient, volume, cooling surface, and fill factor of the FU magnetic core;
\(l_c, l_k\)—relative length of the cores and coils [4] (hereinafter abbreviated);
\(f_1, k_f\)—frequency and waveform coefficient of the line voltage;
\(P_{\mathrm{gf}}\)—overall power of one phase of the FU;
\(\varphi_{pk}, \varphi_{pc}\)—coefficients of the relative cooling surface of the coils and cores, dimensionless quantities [4];
\(n_k\)—number of coils of the alternating-current winding connected to the network;
\(\kappa_0\)—coefficient showing how many times the window of the FU magnetic core is wider than the thickness of one coil of the primary winding; for example, for a BT \(\kappa=2.5\div 3.3\); for \(\kappa_0=5\div 5.5\), etc.
\(x, y, z\)—geometric ratios of the FU dimensions.
\[
x=\frac{c}{a},\quad z=\frac{h}{a},\quad y=\frac{b}{a}
\]
for a rectangular cross section;
\[
x=\frac{c}{d_0},\quad z=\frac{h}{d_0},\quad y=1
\]
for stepped cross sections,
where \(a, b, d_0\) are the width, thickness, or mean diameter of the main core of the magnetic core,
\(h, c\)—the height and width of the window of the magnetic core.
The simultaneous solution of (1—7) makes it possible to obtain, for FU up to 10 kVA, operating at a frequency of \(50\div 100\) Hz, when the induction may be chosen as the maximum on the knee of the magnetization curve:
\[ j= \sqrt[7]{ \left(\frac{\sigma_k\cdot \Delta\tau_k}{\rho_k\cdot k_{zk}}\right)^4 \cdot \frac{B}{P_{\mathrm{gf}}} \cdot 4k_f\cdot f_1\cdot k_{zc}\cdot k_{zk}\cdot n_k(2\varphi_{pk})^4 \cdot \frac{z\cdot y\cdot \kappa_0^3}{x^3} }, \tag{8} \]
for FU of all ratings above 10 kVA, operating at increased frequency (from 100 Hz and above):
\[ j= \sqrt[12]{ \left(\frac{\sigma_k\cdot \Delta\tau_k}{\rho_k\cdot k_{zk}}\right)^7 \frac{\sigma_c\cdot \Delta\tau_c\cdot k_{zc}}{\rho_c\cdot g_c} \left(\frac{50}{f_1}\right)^{1.5} \left(\frac{4k_f\cdot k_{zk}\cdot n_k\cdot f_1}{P_{\mathrm{gf}}}\right)^2 \cdot \frac{z^2\cdot y\sqrt{y\cdot \kappa_0^5}}{x^5} } \cdot \sqrt{y^{\pm 1}}(2\varphi_{pk})^7\,2\varphi_{pc}. \tag{9} \]
Here the induction must likewise be calculated:
\[ B= \sqrt[12]{ \frac{\sigma_k\cdot \Delta\tau_k\cdot k_{zk}}{\rho_k\cdot P_{\mathrm{gf}}^2} (4k_f\cdot k_{zc}\cdot n_k f_1)^2 \left[ \frac{\sigma_c\cdot \Delta\tau_c}{\rho_c\cdot k_{zc}\cdot g_c} \cdot \left(\frac{50}{f_1}\right)^{1.5} \right]^7 } \cdot 2\varphi_{pk}(2\varphi_{pc})^7 \cdot \frac{x\cdot z^2}{\sqrt{y^3\cdot \kappa_0^2}} \sqrt{y^{\pm 1}}. \tag{10} \]
Expressions (8, 9) give complete information on the relation of the current density of a ferromagnetic device to all its parameters, and make it possible to carry out sufficiently accurate calculations and to analyze and synthesize these calculations.
In conclusion, let us note the principal properties of the current density of a ferromagnetic device that follow from (8, 9).
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Most significantly, the density depends on the permissible overheating temperature, the heat-transfer coefficient, the specific resistance, and the coil fill factor.
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As the thickness of the coils increases, it decreases approximately in proportion to \(1/\sqrt{x}\).
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The cooling surface has an effective influence on the current density; therefore, one should always strive for the participation in cooling of all sections of the coils, introduce additional cooling channels, etc.
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The current density decreases slowly, but still decreases, with an increase in the power of the ferromagnetic device, and increases slightly with an increase in the frequency of the supply network voltage.
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