List of Calculation Formulae

List of Calculation Formulae#

This page lists formulae used in EMSolution electromagnetic-field analysis. Each formula is referenced from User Guide.

[EQ1]

B_INTEG calculation formula#

Spatial magnetic flux density due to magnetization and current in B_INTEG is calculated using the following formulae.

For three dimensions:

(1)#\[\pmb{B} (r_p) = \frac{\mu_0}{4\pi} \int_{\Omega} \frac{J (R) \times r}{r^3} dv - \frac{1}{4\pi} \int_{\Omega} \left( \frac{M}{r^3} - \frac{3(\pmb{M} \cdot \pmb{r}) \pmb{r}}{r^5} \right) dv\]

For two dimensions:

(2)#\[\pmb{B} (r_p) = \frac{\mu_0}{2\pi} \int_{\Omega} \frac{J (R) \times r}{r^2} dv - \frac{1}{2\pi} \int_{\Omega} \left( \frac{M}{r^2} - \frac{2(\pmb{M} \cdot \pmb{r}) \pmb{r}}{r^4} \right) dv\]
[EQ2]

Heat-generation calculation formula#

Heat generation (Joule heating) in conductors can be calculated with the following formula.

(3)#\[W_{e} = \int_{cond} \frac{\pmb{J}_e^2}{\sigma} dv = \int_{cond} \pmb{E} \cdot \pmb{J}_e dv\]

Taking the time average yields time-averaged heat generation.

[EQ3]

Linking-flux calculation formula#

Linking flux can be calculated from the line integral of magnetic vector potential.

(4)#\[\psi = \int_{C} \pmb{A} dl\]
[EQ4]

Nodal-force calculation formula#

(5)#\[f = - \int_{} \pmb{T}_{ik} \frac{\partial \omega_n}{\partial x_k} d \Omega\]
(6)#\[\pmb{T}_{ik} = \pmb{H}_i \pmb{B}_k - \delta_{ik} \int_{0}^{H} \pmb{B} d\pmb{H}\]
[EQ5]

Magnetic-energy calculation formula#

Magnetic energy is calculated using the following formulae.

(7)#\[ \begin{align}\begin{aligned}\text{co_energy} = \int_{v} \left( \int_{} \pmb{B}d \pmb{H} \right) dv\\\text{energy} = \int_{v} \left( \int_{} \pmb{H}d \pmb{B} \right) dv\\\frac{BH}{2} = \int_{v} \left( \frac{\pmb{B} \pmb{H}}{2} \right) dv\end{aligned}\end{align} \]
[EQ6]

Iron-loss calculation formula#

Two methods are available for calculating iron loss in laminated cores. Input parameters configured for iron-loss calculation are used by the method based on maximum magnetic flux density.

Method based on maximum magnetic flux density (Steinmetz equation)

When an alternating magnetic field with a single frequency \(f\) and amplitude \(B_{max}\) (maximum magnetic flux density) is applied, iron loss is calculated using the Steinmetz equation below.

Enter \(f\) and the exponents \(\alpha, \beta, \gamma\) of \(f\) and \(B_{max}\) for in-plane eddy-current loss \(w_{e\perp}\) and hysteresis loss \(w_h\). A simplified setting uses \(\alpha = \beta = \gamma = 2.0\).

(8)#\[W_i=W_{e||}+(w_{e⊥}+w_h)ΔV_i = W_{e||} + (K_e f^{\alpha} B_{max}^{\beta} + K_h f B_{max}^{\gamma} )ΔV_i\]

In-plane eddy-current loss \(W_{e||}\) can be calculated directly from FEA results using the following formula.

(9)#\[ \begin{align}\begin{aligned}W_{e||} = \frac{1}{T} \int_{0}^{T} \int_{iron} \frac{|J_e|^2}{\Sigma_{||}} dvdt\\\Sigma_{||} = \alpha \sigma, \Sigma_{\perp} = 0\end{aligned}\end{align} \]

Method based on magnetic flux density waveforms

The method based on magnetic flux density waveforms was proposed by Professor Yamasaki of Chiba Institute of Technology. It can approximately consider waveform distortion and minor loops. It is calculated using the following formulae, where the z direction is the lamination direction.

(10)#\[ \begin{align}\begin{aligned}W_{e\perp}=\frac{K_e D}{2π^2} \int_{iron} \frac{1}{N} \sum_{k=1}^N \left\{ \left( \frac{B_x^{k+1} - B_x^k}{Δt} \right)^2 + \left( \frac{B_y^{k+1} - B_y^k}{Δt} \right)^2 \right\} dv\\W_h=\frac{K_h D}{T} \sum_{i=1}^{N_E} \frac{ΔV^i}{2} \left\{ \sum_{j=1}^{N_{px}^i} (B_{mx}^{ij})^2 + \sum_{j=1}^{N_{py}^i} (B_{my}^{ij})^2 + \sum_{j=1}^{N_{pz}^i} (B_{mz}^{ij})^2 \right\}\end{aligned}\end{align} \]

For AC steady-state analysis (AC(2)=2), core loss per unit time and unit volume (\(W/m^3\)) is calculated using the imaginary part of complex relative permeability \(\mu^{''}\) as follows.

(11)#\[W = \frac{\omega}{2\pi} \int_{} \pmb{H}d\pmb{B} = \frac{1}{2} \omega \mu^{''} \pmb{H}_0^2\]

Here, \(\omega\) is angular frequency and \(\pmb{H}_n\) is the amplitude of the AC magnetic field (\(\pmb{H} = \pmb{H}_0 e^{j\omega t}\)).

[EQ7]

Nonlinear surface impedance#

General nonlinear surface impedance is defined by the following formula.

(12)#\[Z_s = \sqrt{\frac{j\omega \mu}{\sigma}} \left( 1 + j \frac{\sigma}{\omega \mu} \right)\]

EMSolution uses the following formulae according to IMP_TYPE.

  • 1: nonlinear, sinusoidal magnetic field

(13)#\[ \begin{align}\begin{aligned}Z_m = \frac{16}{3\pi}\frac{1}{\sigma \delta_{nl}} \left( 1 + 0.5 j \right)\\\delta_{nl} = \sqrt{\frac{2 H_s}{\sigma \omega f B(H_s)}}\end{aligned}\end{align} \]

  • 2: nonlinear, sinusoidal electric field

(14)#\[Z_e = \frac{27 \pi^3}{2 \sqrt{5}}\frac{1}{\sigma \delta_{nl}} \left( 1 + \frac{4}{3 \pi} j \right)\]

  • 3: mixture of 0 and 1 (\(Z_{nl} = Z_m\))

  • 4: mixture of 0 and 2 (\(Z_{nl} = Z_e\))

(15)#\[ \begin{align}\begin{aligned}Z = f(H_s)Z_l + (1-f(H_s))Z_{nl}\\f(H_s) = \frac{1}{1 + \left( k \frac{H_s}{H_{k}} \right)}\\Z_l = \frac{1 + j}{\sigma \delta_{l}}\\\delta_l = \sqrt{\frac{2}{\sigma \omega \mu}}\end{aligned}\end{align} \]