Magnetic Materials and Magnetic Circuits

Name	Formula	Related Formulae	Variables/Constants
B, H and magnetization	$\vec{B}={\mu }_0\left(\vec{H}+\vec{M}\right)$	$\vec{M}={\chi }_m\vec{H},\vec{B}=\mu \vec{H},\mu ={\mu }_0{\mu }_r={\mu }_0\left(1+{\chi }_m\right)$	$\vec{B}:magneticfluxdensity(T)$ $\vec{H}:magneticfieldstrength(A/m)$ $\vec{M}:magnetization(A/m)$ ${\chi }_m:magneticsusceptibility$ $\mu :permeability$ ${\mu }_r:relativepermeability$
Ampere’s law for magnetic circuits	$\oint \vec{H}\cdot d\vec{l}=NI$	$\mathcal{F}=NI={\sum }_k{H}_k{l}_k$	$\vec{H}:magneticfieldstrength$ $N:numberofturns$ $I:current$ $\mathcal{F}:magnetomotiveforce(ampere-turns)$ $H_k,l_k:fieldandpathlengthinsectionk$
Reluctance	$\mathcal{R}=\frac{l}{\mu A}$	$\mathcal{F}=\Phi \mathcal{R},{\mathcal{R}}_{total}={\mathcal{R}}_{core}+{\mathcal{R}}_{gap}$	$\mathcal{R}:reluctance(A/Wb)$ $l:magneticpathlength$ $\mu :permeability$ $A:cross-sectionalarea$ $\Phi :magneticflux$
Inductance from reluctance	$L=\frac{N^2}{\mathcal{R}}$	$\lambda =N\Phi =LI$	$L:inductance$ $N:numberofturns$ $\mathcal{R}:totalreluctance$ $\lambda :fluxlinkage$ $\Phi :flux$ $I:current$
Hysteresis loss	$P_h=f{V}_{core}\oint HdB$	$P_h\approx k_{h}f{B}_{max}^n(\text{Steinmetz, }n\approx 1.6\text{to}2)$	$P_h:hysteresispowerloss$ $f:frequency$ $V_{core}:corevolume,\oint HdB:B-Hlooparea$ $k_h:materialconstant$ $B_{max}:peakfluxdensity$ $n:Steinmetzexponent$
Eddy-current loss	$P_e=k_e{f}^2{B}_{max}^2{t}^2$	$P_{core}=P_h+P_e$	$P_e:eddy-currentpowerloss$ $\begin{array}{r}{k}_e:constantdependingonresistivityand\\ volume\end{array}$ $f:frequency$ $B_{max}:peakfluxdensity$ $t:laminationthickness$ $P_{core}:totalcoreloss$
Generalized Steinmetz equation	$P_v=k{f}^{\alpha }{B}_{pk}^{\beta }$	$\text{valid for sinusoidal excitation; }\alpha \approx 1\text{to}2,\beta \approx 2\text{to}3$	$P_v:corelossperunitvolume$ $k,\alpha ,\beta :empiricalmaterialconstants$ $f:frequency$ $B_{pk}:peakfluxdensity$
Magnetic energy density	$w_m=\frac{B^2}{2\mu }=\frac{1}{2}BH$	$W=\frac{1}{2}L{I}^2$	$w_m:energyperunitvolume$ $B:fluxdensity$ $H:fieldstrength$ $\mu :permeability$ $W:storedenergy$

Important Questions

Ques: What is the difference between B and H?

H is the field set up by free currents, and B is the total flux density, including the contribution of the material’s magnetization: $\vec{B}={\mu }_0(\vec{H}+\vec{M})$. Ampere’s law in magnetic circuits is written with H, because $\oint \vec{H}\cdot d\vec{l}$ equals the ampere-turns $NI$, while flux and Faraday’s law use B. In a linear material $\vec{M}={\chi }_m\vec{H}$, so $B=\mu H$ with $\mu ={\mu }_0(1+{\chi }_m)$. B is measured in tesla and H in amperes per metre; in vacuum 1 T corresponds to about 796 kA/m (computed). In iron, the relative permeability can reach thousands, so a small H produces a large B until the material saturates. In everyday use both are loosely called the magnetic field, so the units show which is meant. Reference: D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, 2013); S. J. Chapman, Electric Machinery Fundamentals, 5th ed. (McGraw-Hill, 2012).

Ques: How does a magnetic circuit mirror an electric circuit?

Flux plays the role of current, magnetomotive force $NI$ the role of voltage and reluctance $l/(\mu A)$ the role of resistance, so $\mathcal{F}=\Phi \mathcal{R}$ is the magnetic version of Ohm’s law. Reluctances in series add, like resistors, and flux divides between parallel paths. Combining with $\lambda =N\Phi$ gives $L=N^2/\mathcal{R}$. For an iron path 0.1 m long with ${\mu }_r=2000$ and a 1 cm^2 cross-section, the reluctance is about $4.0\times 10^5$ A/Wb, and 100 turns give about 25 mH (computed). The analogy is only approximate: iron saturates, so its reluctance rises at high flux, and some flux leaks around the core, which has no counterpart in a wire. Reference: S. J. Chapman, Electric Machinery Fundamentals, 5th ed. (McGraw-Hill, 2012); A. E. Fitzgerald, C. Kingsley Jr. and S. D. Umans, Electric Machinery, 7th ed. (McGraw-Hill, 2014).

Ques: Why do inductor cores have an air gap?

To make the inductance predictable and to store energy without saturating the core. The gap’s reluctance $g/({\mu }_{0}A)$ is large because air has ${\mu }_r=1$: a 1 mm gap in a 1 cm^2 core adds about $8.0\times 10^6$ A/Wb, twenty times the $4.0\times 10^5$ A/Wb of a 0.1 m iron path with ${\mu }_r=2000$ (computed). The inductance falls, here from about 25 to 1.2 mH, but it now depends mainly on the gap rather than on the iron’s variable permeability. The gap also raises the current the core can carry before saturating, so most of the stored energy $\frac{1}{2}L{I}^2$ ends up in the gap. Fringing flux around the gap makes the real reluctance slightly lower than the simple formula. Reference: R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 3rd ed. (Springer, 2020); S. J. Chapman, Electric Machinery Fundamentals, 5th ed. (McGraw-Hill, 2012).

Ques: Where do core losses come from, and how are they reduced?

From hysteresis and eddy currents. Hysteresis loss is the energy needed to cycle the material around its B-H loop, equal to the loop area per cycle, so it rises in proportion to frequency; soft magnetic materials with narrow loops keep it low. Eddy-current loss comes from currents that the changing flux induces in a conducting core; it rises with the square of frequency, flux density and lamination thickness, so halving the lamination thickness cuts it to a quarter (computed). Transformers therefore use thin insulated silicon-steel laminations at mains frequency and high-resistivity ferrites at switching frequencies. The empirical Steinmetz equation $k{f}^{\alpha }{B}_{pk}^{\beta }$ fits the total loss for sinusoidal excitation. Reference: S. J. Chapman, Electric Machinery Fundamentals, 5th ed. (McGraw-Hill, 2012); R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 3rd ed. (Springer, 2020); A. E. Fitzgerald, C. Kingsley Jr. and S. D. Umans, Electric Machinery, 7th ed. (McGraw-Hill, 2014).

References

1. D. J. Griffiths, Introduction to Electrodynamics, 4th ed. Boston: Pearson, 2013.
2. S. J. Chapman, Electric Machinery Fundamentals, 5th ed. New York: McGraw-Hill, 2012.
3. A. E. Fitzgerald, C. Kingsley Jr. and S. D. Umans, Electric Machinery, 7th ed. New York: McGraw-Hill, 2014.
4. R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 3rd ed. Cham: Springer, 2020.