Understanding Post-Processing Formulas

This section is about finding and understanding the formulas used by the Magnetic solver in post processing. They can be found in the physical master library, that contains the physical formulations. This file is found in your Magnetics Installation in folder ’templates’ and is named ’lib_Master.pro’.

Magnetostatic

1. Electrode Voltage:
   \(ResName\_globalvoltage =~ ’Voltage’, UnitName\_globalvoltage =~ ’V’\)

   • The applied voltage on the electrode surface is set to the specified value and is calculated if it arises.

   Electrode Current:
   \(ResName\_globalcurrent = 'Current', UnitName\_globalcurrent = 'A',\)

   • The applied current on the electrode surface is set to the specified value and is calculated if it arises.

2. Electrode Power:
   \(Name: Power; Value: \{ Term\{ \{ CoefPower*\{I\}*\{U\} \}\}; In DomainC\)

   • P = U \(\cdot\) I \(\cdot\) CoefPower,
     with CoeffPower = 0.5 in frequency solutions and CoeffPower = 1 in non frequency solutions

3. Circuit Voltage:
   \(ResName\_voltagecircuits =~ 'VoltageCircuits', UnitName\_voltagecircuits =~ 'V',\)

   • The applied voltage on the circuit is set to the specified value and is calculated if it arises.

4. Circuit Current:
   \(ResName\_currentcircuits =~ 'CurrentCircuits', UnitName\_currentcircuits =~ 'A',\)

   • The applied current on the circuit is set to the specified value and is calculated if it arises.

5. Circuit Power:
   \(ResName\_powercircuits = 'PowerCircuits', UnitName\_powercircuits =~ 'W',\)

   • The applied power on the circuit is set to the specified value and is calculated if it arises.

Magnetodynamic Frequency

1. EddyCurrentLosses [P]:
   \(\int~\{~[~ \text{CoefPower}*sigma[\{T\}]*SquNorm[\{ D_t[\{a\}] + dv\}]*\text{GeoCoeff}]~\}; In \#\{DomainC,-DomainV\}\)

   • \(sigma[\{T\}]\), electrical conductivity of the material \([\sigma]\)

   • \(Dt[\{a\}]\), time derivative of the magnetic vectorpotential \([a]\)

   • \(\{d v\}\), gradient of electric scalarpotential

   • \(SquNorm[ Dt[\{a\}]+\{d v\} ]\) , corresponds to the squared magnitude of the electric field strength [\(E^2\)]

   • so \(P~=~\sigma \cdot |E^2| \cdot CoeffPower\) is the basic formula. This could also be expressed as \(P = \frac{1}{R}\cdot(U^2) \cdot CoeffPower\) what equals \(P = U\cdot I\cdot CoeffPower\).

2. Core Losses:
   \(kh: \int~\{~[ Kh~[\{T\}]*UpdateFreq\texttt{\^{}}KhExpA [\{T\}]* Norm[\{d a\}]\texttt{\^{}}KhExpB[\{T)\}]*\{GeoCoeff\}];\)
   \(~~~~~~ In Region[\{DomainNonMoving\}]\)

   • kh: \(Kh[\{T\})\), is the hysteresis coefficient, which can be temperature dependend

   • \(UpdateFreq\), represents the frequency

   • \(KhExpA[\{T\}]\), is the exponent of kh

   • \(Norm[\{d~a\}]\), is the magnetic fluxdensity b, because b = rot a

   \(ke: \int\{ ~[ Ke[\{T\}] * (\$UpdateFreq * Norm[\{d a\}])\texttt{\^{}}2\)

   • \(Ke[\{T\}\), describes the eddy current loss coefficient in the core

   \(kx: \int\{ ~[ Kx[\{T\}] * (\$UpdateFreq * Norm[\{d a\}])\texttt{\^{}}1.5\)

   • \(Kx(T)\) is the excess loss coefficient

   From the Steinmetz equation: Kh, Ke, and Kx represent hysteresis losses, eddy current losses in the core, and excess losses (relevant at high frequencies in the MHz range). Combined they form the core losses.

3. Total Losses:
   The total losses consist of the sum of eddy current losses and core losses.

4. Ohm Resistance:
   In 2D:
   \(Name Resistance;~ Value~ \{ Term \{ [ -\{U\} *Conj[\{I\} ] /SquNorm[\{I\} ] ];~ In DomainC; Jacobian Vol; \}\)

   • \(U\), is the voltage

   • \(Conj[\{I\}]\), is the conjugated current. This is necessary to involve potential complex numbers in the calculation

   • \(SquNorm[\{I\} ]\), is the squared magnitude of the current

   • For complexe calculations the conjugated current enables complex solutions. For non-complexe numbers, cancelling the current leads to the basic formula:  \(R = \frac{U}{I}\).

   In 3D:
   \(\text{Resistance}; \, \text{Value} \{ \, \text{Term} \{ \, \left[ \{-U\} / \{I\} \right] \, \}; \, in~ Surf\_elec;~Jacobian~Vol \}\)

   • In 3D, because of the 3D model properties the resistance is calculated by the basic formula: \(R = \frac{U}{I}\)

5. Inductivity:
   \(Name Inductivity; Value \{ Term \{ [ ((-Im[\{U\} ]/Re[\{I\} ])/(2*Pi*UpdateFreq)) ];\)
   \(In DomainC; Jacobian Vol; \}\)

   • \(-Im[\{U\}]\), describes the imaginary part of the voltage

   • \(Re[\{I\}]\), describes the real part of the current

   • \(2*Pi*\$UpdateFreq\), corresponds to the angular frequency \(\omega =2\cdot \pi \cdot f\)

   • in mathematical notation: \(L~=~\frac{-Im(U)}{Re(I)\cdot \omega}\)

   • basic formula: \(L~=~\frac{U}{j \cdot \omega \cdot I} = \frac{-j \cdot U}{\omega \cdot I}\)

   • Known is : \(U~=~j\omega L \cdot I\), this shows that "U" is imaginary and "I" is real. If "I" was imaginary, \(j^2 =~-1\), leads to: \(U ~=~ \omega L\)

   • ’j’ describes a phase shift of \(90^\circ\) (U is \(90^\circ\) phase shifted to I) and is left out of the formula

   • So: \(L~=~\frac{-Im(U)}{I\cdot \omega}\)

   • Because the real component (Re(I)) determines the inductance, it is the decisive factor. The reactive component (Im(I)) does not play a role there (no net power).

   • So: \(L~=~\frac{-Im(U)}{Re(I)\cdot \omega}\)

6. Phase Shift:
   \(Name PhaseShift\_\_Unit\_deg Term \{ [ ((Atan[Im[\{Ub\}]/Re[\{Ub\}]])-(Atan[Im[Ib\}]/Re[\{Ib\}]]))*(180/Pi) ];In DomainB; Jacobian Vol; \}\)

   • \(Atan[Im[\{Ub\}]/Re[\{Ub\}]])\) describes the phase shift of the voltage "U"

   • \(Atan[Im[Ib\}]/Re[\{Ib\}]])\) describes the phase shift of the current "I"

   • \(*(180/Pi)\), to convert from radians to degrees

   • the basic formula is \(\varphi = atan(\frac{Im(U)}{Re(U)})-atan(\frac{Im(I)}{Re(I)}) = \varphi_{U} - \varphi_{I}\)