Magnetism & Magnetic Circuit Formulas

Important Nomenclature and Notation

\[ \begin{aligned} \phi & = \text{magnetic flux in Wb (webers)} \\ B & = \text{magnetic flux density in T (tesla)} \\ H & = \text{magnetic field intensity in AT/m} \\ 1~\mathrm{T} & = 1~\mathrm{Wb/m^2} = 10,000~\text{gauss} \\ A & = \text{cross-sectional area in}~\mathrm{m^2} \\ \mathcal{F} & = \text{magnetomotive force (MMF) in AT (amp-turns)} \\ \mathcal{R} & = \text{reluctance of the material in AT/Wb} \\ N & = \text{number of loops or turns in the coil} \\ I & = \text{current in the coil in A (amperes)} \\ l & = \text{average length of the material in m (meters)} \\ L & = \text{inductance of the coil in H (henry) } \\ \mu & = \text{permeability of the material in H/m} \\ \mu_0 & = \text{permeability of free space} = 4\pi \times 10^{-7}~\mathrm{H/m} \\ \mu_r & = \text{relative permeability (constant) of the material} \\ \end{aligned} \]

Analogy Terms - Electric Circuit Vs Magnetic Circuit

\[ \begin{aligned} \text{EMF (Electromotive force)} ~\mathcal{E} & ~\rightarrow ~\text{MMF} ~\mathcal{F} \\ \text{Current}~i & ~\rightarrow ~\text{Flux} ~\Phi\\ \text{Resistance}~R & ~\rightarrow ~\text{Reluctance} ~\mathcal{R}\\ \text{Conductance} ~G& ~\rightarrow ~\text{Permeance} ~P\\ \text{Current density}~J & ~\rightarrow ~\text{Flux density}~B \\ \end{aligned} \]
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Magnetic Circuit Excited by a Coil

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Fundamental Formulas

\[ \begin{aligned} \mathcal{F} & = NI \\ H & = \dfrac{F}{l} = \dfrac{NI}{l}~\mathrm{AT/m} \\ \phi & = \dfrac{NI}{\mathcal{R}} \\ B & = \dfrac{\phi}{A} \\ B & = \mu H \\ \mu & = \mu_0\mu_r\\ \mathcal{R} & = \dfrac{l}{\mu_0 \mu_r A} \\ e & = N \cdot \dfrac{d\phi}{dt} = L \cdot \dfrac{di}{dt} \\ L & = N \cdot \dfrac{d\phi}{di}\\ L & = \dfrac{\mu_0\mu_rN^2A}{l} \\ \end{aligned} \]

B-H Curve of a Magnetic Material

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Magnetic Circuit With and Without Air-gap

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\[ NI = H_cl_c+H_gl_g \quad (\text{with air gap}) \]
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Reluctances in Series

\[ \begin{aligned} & \Re_{\mathrm{T}}=\Re_1+\Re_2+\Re_3+\cdots+\Re_{\mathrm{N}} \\ & \Phi_{\mathrm{T}}=\Phi_1=\Phi_2=\Phi_3=\cdots=\Phi_{\mathrm{N}} \end{aligned} \]
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Reluctances in Parallel

\[ \begin{aligned} & \Re_{\mathrm{T}}=\frac{1}{\frac{1}{\Re_1}+\frac{1}{\Re_2}+\frac{1}{\Re_3}+\cdots+\frac{1}{\Re_{\mathrm{N}}}} \\ & \Phi_{\mathrm{T}}=\Phi_1+\Phi_2+\Phi_3+\cdots+\Phi_{\mathrm{N}} \end{aligned} \]
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Energy Stored in a Magnetic Circuit

\[ \begin{aligned} W_m & =\frac{1}{2} L i^2 \\ & =\frac{1}{2}\left[\frac{\mu N^2 A}{L}\right]\left[\frac{H \cdot l}{N}\right]^2 \\ & =\frac{1}{2}\left[\frac{\mu N^2 A}{l}\right]\left[\frac{H^2 l^2}{N^2}\right] \\ & =\frac{1}{2}\left[\frac{B^2lA}{\mu}\right]\\ & = \dfrac{1}{2}\mathcal{R}\Phi^2 \end{aligned} \]