Flux approximation script#

Note

This is a Placeholder and part of a different Pull-Request. It is here to clarify how the docs pages would interact with each other.

Script to approximate flux-linkages from a 3D-Lookup-Table. To enable this dynamic implementation, the flux-linkages are approximated using analytic-Prototype functions. This is based on the approach and findings from [1]. The flux-linkages can be approximated using the following equations.

\[\hat{\psi}_{d}(i_{d},i_{q}) = \hat{\psi}_{self}^{d}(i_{d}) - \hat{\psi}_{cross}^{d}(i_{d},i_{q})\]
\[\hat{\psi}_{q}(i_{d},i_{q}) = \hat{\psi}_{self}^{q}(i_{q})-\hat{\psi}_{cross}^{q}(i_{d},i_{q})\]

For the self-axis saturation prototype function a hyperbolic tangent function and a linear function to mimic the saturation effect in a single axis can be employed [1].

\[\hat\psi_{self}^{d} = \hat\psi_{d}(i_{d},i_{q}=0) = a_{d1} \cdot \tanh(a_{d2} \cdot (i_{d}-a_{d3}))\]
\[\hat\psi_{self}^{q} = \hat\psi_{q}(i_{d}=0,i_{q}) = a_{q1} \cdot \tanh(a_{q2} \cdot i_{q})+ i_{q} \cdot a_{q3}\]

The cross-coupling saturation terms can be found if the the flux-linkages are evaluated at their maximum cross-coupling currents \(I_{d1}\) and \(I_{d1}\).

\[\hat{\psi}_{cross}^{d,s1}(i_{d},i_{q}=I_{q1}) = \hat{\psi}_{self}^{d}(i_{d})-\psi_{d,s1}(i_{d},I_{q1})\]
\[\hat{\psi}_{cross}^{q,s1}(i_{d}=I_{d1},i_{q}) = \hat{\psi}_{self}^{q}(i_{q})-\psi_{q,s1}(I_{d1},i_{q})\]

The flux-linkages in these points can be approximated with the following equations [1]:

\[\hat\psi_{d,s1}(i_{d}) = \hat\psi_{d}(i_{d},i_{q}=I_{q1}) = a_{d4} \cdot \tanh(a_{d5} \cdot (i_{d}-a_{d6}))\]
\[\hat\psi_{q,s1}(i_{q}) = \hat\psi_{q}(i_{d}=I_{d1},i_{q}) = a_{q4} \cdot \tanh(a_{q5} \cdot i_{q})+ i_{q} \cdot a_{q6}\]

With that the cross-coupling terms are resulting.

\[\hat\psi_{cross}^{d,s1}(i_{d},i_{q}=I_{q1}) = a_{d1} \cdot \tanh(a_{d2} \cdot (i_{d}-a_{d3})) - a_{d4} \cdot \tanh(a_{d5} \cdot (i_{d}-a_{d6})\]
\[\hat\psi_{cross}^{q,s1}(i_{d}=I_{d1},i_{q}) = a_{q1} \cdot \tanh(a_{q2} \cdot i_{q})+ i_{q} \cdot a_{q3} - a_{q4} \cdot \tanh(a{q5} \cdot i_{q})- i_{q} \cdot a_{q6}\]

These have to be integrated [1] , which yields.

\[\int \hat{\psi}_{cross}^{d,s1}(i_{d}) \, di_{d} = \frac{a_{d1}}{a_{d2}} \cdot \log(\cosh(a_{d2}(i_{d}-a_{d3}))) - \frac{a_{d4}}{a_{d5}} \cdot \log(\cosh(a_{d5}(i_{d}-a_{d6})))\]
\[\int \hat{\psi}_{cross}^{q,s1}(i_{q}) \, di_{q} = \frac{1}{2}(a_{q3}-a_{q6}) \cdot i_{q}^2 + \frac{a_{q1}}{a_{q2}} \cdot \log(\cosh(a_{q2}i_{q})) - \frac{a_{q4}}{a_{q5}} \cdot \log(\cosh(a_{q5}i_{q}))\]

With that the entire range of the flux-linkage is found. Note that the terms \(\int \hat{\psi}_{cross}^{q,s1}(I_{q1}) di_{q}\) and \(\int \hat{\psi}_{cross}^{d,s1}(I_{d1}) di_{d}\) are constant values and will be used in the fitting parameters.

\[\hat{\psi}_{d}(i_{d},i_{q}) = \hat{\psi}_{d,self}(i_{d}) - \underbrace{\frac{1}{\int \hat{\psi}_{cross}^{q,s1}(i_{q}) \, di_{q}} \left( \hat{\psi}_{cross}^{d,s1}(i_{d},i_{q}=I_{q1}) \right) \left( \int \hat{\psi}_{cross}^{q,s1}(i_{q}) \, di_{q} \right)}_{=\hat{\psi}_{cross}^{d}(i_{d},i_{q})}\]
\[\hat{\psi}_{q}(i_{d},i_{q}) = \hat{\psi}_{q,self}(i_{q}) - \underbrace{\frac{1}{\int \hat{\psi}_{cross}^{d,s1}(i_{d}) \, di_{d}} \left( \hat{\psi}_{cross}^{q,s1}(i_{d}=I_{d1},i_{q}) \right) \left( \int \hat{\psi}_{cross}^{d,s1}(i_{d}) \, di_{d} \right)}_{=\hat{\psi}_{cross}^{q}(i_{d},i_{q})}\]

To find the fitting-Parameter the following nonlinear-square Problems have to be minimized. For that the MATLAB nonlinear-regression function lsqnonlin with the Levenberg-Marquart algorithm is used.

\[\min_{a_{d1},a_{d2},a_{d3}} \sum_{j=1}^{m} \left[ \psi_{d} \left(i_{d,j}, 0\right) - \hat{\psi}_{d,self}\left(i_{d,j},a_{d1},a_{d2},a_{d3}\right) \right]^2\]
\[\min_{a_{q1},a_{q2},a_{q3}} \sum_{k=1}^{n} \left[ \psi_{q} \left( 0, i_{q,k}\right) - \hat{\psi}_{q,self}\left(i_{q,k},a_{q1},a_{q2},a_{q3}\right) \right]^2\]
\[\min_{a_{d4},a_{d5},a_{d6}} \sum_{j=1}^{m} \left[ \psi_{d} \left(i_{d,j}, I_{q1}\right) - \hat{\psi}_{d,s1}\left(i_{d,j},a_{d4},a_{d5},a_{d6}\right) \right]^2\]
\[\min_{a_{q4},a_{q5},a_{q6}} \sum_{k=1}^{n} \left[ \psi_{d} \left(I_{d1}, i_{q,k}\right) - \hat{\psi}_{q,s1}\left(i_{d},a_{q4},a_{q5},a_{q6}\right) \right]^2\]

Example usage#

In this example usage, flux-linkages of a example motor are getting approximated.

  • There needs to be a Excel data sheet in the same directory as the PMSM IP-Core at ultrazohm_sw\ip_cores\uz_pmsm_model.

  • The naming in the script has to be adjusted.

Listing 98 Example to get data out of a Excel data sheet.#
1...
2FluxMapData = readtable('FluxMapData_Prototyp_1000rpm_');
3...

  • Afterwards the area where the Array is in the excel sheet has to be specified.

Listing 99 Example to specify array location and size.#
1...
2% Currents
3id = FluxMapData{1,1:20};
4iq = FluxMapData{22:41,1};
5%Psi_d
6psi_d = FluxMapData{43:62,1:20}*(1e-3);
7%Psi_q
8psi_q = FluxMapData{108:127,1:20}*(1e-3);
9...

  • To run the approximation script, first the uz_pmsm_model_init_parameter.m file has to be ran.

  • If the the script ran successfully the fitting parameters are in the MATLAB workspace an can be used in the IP-Core for nonlinear behavior or for different use in the sw-framework.

Sources#