Calculation of self and mutual inductance for PMSM machines?

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zmladen
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Calculation of self and mutual inductance for PMSM machines?

Post by zmladen »

Hello,

is there a way to calculate self and mutual inductances and flux-linkages in a 3-phase PMSM machines using elmers transient 2D electromagnetic solver? In my previous post I was able, with the help of the forum, to calculate induced voltage using circuits modul (see: Induced voltage in PMSM without Paraview). This raised again some interest in elmer.

According to Ansys help the calculation of inductances in Maxwell is based on the "Frozen Permeability" method. Hear the detailed description from Ansys help:
Inductance Computation for 2D and 3D Transient Solutions
Because both 2D and 3D transient solutions are nonlinear, for inductance computation, Maxwell provides two options: apparent inductance and incremental inductance. After the FEA normal solution is completed at each time step, the permeability associated with apparent inductance, or the differential permeability associated with incremental inductance, of each element, is frozen – which is equivalent to freezing the coefficient matrix (the left side of the equation to be solved). In order to obtain winding self-inductance and mutual inductance, the solver sets an excitation current of 1A in the first winding, while setting all other winding currents to zero (permanent magnet effects are excluded). This excitation assignment corresponds to one source vector on the right-hand side of the equation to be solved. As a result, the calculated flux linkage provides the self-inductance for the first winding with 1A excitation current, and the calculated flux linkages represent mutual inductance for all other windings with zero current. Next, the solver excites the second winding with a current of 1A, while setting all other winding currents to zero (permanent magnet effects are excluded). This excitation assignment corresponds to another source vector on the right-hand side of the equation to be solved. The solver continues this process until all windings have been assigned 1A current in turn.
Would something like this be possible to implement in elmerFEM? I would really like continue using elmer for some calculations. As far as know this should be possible in FEMM.

Regards,
Mladen
zmladen
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by zmladen »

Hello,

here are some more inputs.

In order to understand how the calculation of inductances in Maxwell works I have run two simulations for the motor shown in Fig 1. The first simulation represents the so-called no-load simulation, where I calculated the self and mutual inductances of phase A (LAA, LAB and LAC) using the method implemented by Ansys.
Fig1.png
Fig1.png (25.97 KiB) Viewed 1696 times
In the second simulation I supplied phase A with 1A constant current and other two phases (B and C) with 0A. The simulation runs for half of electrical period of the machine. Instead of inductance the flux-linkage of the corresponding phases was plotted. In both simulations the magnet excitation was turned off.
Fig2.png
Fig2.png (124.25 KiB) Viewed 1696 times
From the results it can be seen that self and mutual inductances of the phase A (from the no-load simulation) are actually the flux-linkages of the machine when phase A is supplied with 1A current, others with 0A (see Fig. 2). Inductances of the other phases can be calculated in the same way as flux-linkage by supplying the corresponding phases with 1A current.

I think that, up to this part the same simulation can be done with elmer using circuits solver.

I am not sure is how the effect of magnets can be included in elmer. The Maxwell applies the so-called frozen permeability method, where in the first step the permeability matrix is calculated under magnet excitation. Then in the next step it keeps this matrix frozen and applies the same procedure explained above.

I am not sure how this can be done in elmer. Any ideas on how to do this?

Regards,
Mladen
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by raback »

Hi Mladen

We could perhaps extend some library features for this.

We started once with constrained modes that are used for Craig-Bampton reduction in mechanics. You set load in one node, solved the equation, and register the load to nodes where the object is attached.

Then we extended the feature for capacitance matrices. Permutate over N object, register the induced surface charged, and thereby solve the NxN capacitance matrix with N linear solutions.

There is some preliminary code to extend this to Helmholtz type of wave equations. Permute velocity load over N port and register the pressures at each port to create NxN impedance matrix.

I guess this would be very similar. Permute over N currents for linear system Ax=b keeping the A=A(x) fixed (frozen) and only playing with the r.h.s. and register some integrals \Phi. What integral exactly? Would it be line integral over vector potential or surface integral over magnetic flux density? And for then define some matrix L_ij=\Phi_i/I_j? I think this would be extension of the above except the circuit simulation creates some challenges.

-Peter
zmladen
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by zmladen »

Hello Peter,

it would be great if this feature could be added to the library soon. I think with this implemented, Elmer could be effecively used to generate machine parameters required for the circuit simulators, such as Matlab or Modelica. Connected with this topic, I have shared the the working case for "induced voltage withoud paraview" with the forum. The results are the same as obtained in Maxwell. The circuits helped calculating the BEMF directly during the simulation.

Concerning your question I think that the needed flux linkage could be calculated as line integral over vector potential. Pavel also did this in his example "FEM Modeling of PMSM Using Elmer". Is shown in my previous example in case when fome of the phases (e.g. phase A) is supplied with 1A current the flux linkages are LAA = Fluxlinkage_A, LAB = Fluxlinkage_B and LAC=Fluxlinkage_C.

Regards,
Mladen
FabianH
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by FabianH »

Hi Mladen,

I think the calculation of the machine parameter can be quiet hard. There are simply too many possibilities in modelling, especially for the control engineers. What your goal?

The frozen permeability method is to seperate the field excitation by stator and magnets. The sum of both fields is equivalent to your results. This is often used for saturated machines. One example is here https://www.femm.info/wiki/FrozenBenchmark..

The apparent inductance is simplifies the flow in the operation point divide by the current -> Flux Linkage Frame of Reference.
The differential inductance is the slope of the tangent in the operation point. Your state variables are the phase currents -> Current Frame of
Reference.
Ansys uses now the frozen permabilty method to have the right sum of all your separated fields and to calculate the correct voltage of this operation point.

I personally typically calculate the parameters for a dq model. This is usually the best place for control engineers to start. To account for saturation I record the magnetic flux without id. This results in a characteristic depending on iq (due to the saturation by the q-current, the flux of the magnet is reduced). This is taken into account with this dependency. Ld and Lq are then simply calculated by connecting flux and current (depending on saturation) for all operation points.
zmladen
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by zmladen »

Hello FabianH,

My initial goal is to calculate the machine parameters for a linear machine model in the dq frame. For that I need ke (speed constant), R-phase resistance and Ld /Lq inductances. The first two I could calculate using Elmer and geometry data. I am currently investigating how I can use Elmer to calculate Ld and Lq. My first intention was to calculate the self and mutual inductance matrix in abc frame and apply Park-Clark transformation to obtain Ld and Lq. This can be done quite easily with Maxwell. In fact I have already shared the data.

When this is done, the next step for me would be to calculate the so-called current-to-flux tables and develop the circuit model that could take into account cogging torque, non-linearities,... This can be also done relatively easily with Maxwell (i could provide the data as reference for Elmer). The procedure is to apply id/iq current to motor phases and sweep them, together with the position within a needed range. The resulting table could look like:

id; iq; angle; fluxA, fluxB, fluxC, torque

See:
https://de.mathworks.com/help/sps/ug/im ... xwell.html

Since the derivation of flux in circuit simulators can cause numerical problems, it is common to use incremental inductances instead. This is based on the total derivative of the flux linkage found in the dq motor model. Thus, I would need to calculate the Ldd, Ldq, Lqd and Lqq in dependance of id, iq and position.

I know that frozen permeability method is used to separate field excitations by coming from current and magnets. What I don't know is how Elmer could be used to do this. I think that Peter gave a good suggestion on how to proceed. It would be great if he could support that as I do not have enough experience in elmer to even try that.

I don't know if I really understand your last paragraph. You say that it would be enough to sweep only iq and set id to 0 to calculate the flux including the saturation. In my opinion this is a special case of what I described above where I would need to sweep id, iq and position. Your approach would only apply to machines that do not have reluctance and would not resolve cogging torque additionally. But it does make sense for some machines. Could be also used to calculate the so-called efficiency maps. I think though, that calculation of inductance is any easier that way. Can you please provide some more details on that.

Mladen
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by raback »

Hi Mladen,

Could you define really stupid reference test for flux linkage, say a cross section with a few wires in a circular domain?

-Peter
FabianH
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by FabianH »

zmladen wrote: 22 Jan 2023, 13:09 I don't know if I really understand your last paragraph. You say that it would be enough to sweep only iq and set id to 0 to calculate the flux including the saturation. In my opinion this is a special case of what I described above where I would need to sweep id, iq and position. Your approach would only apply to machines that do not have reluctance and would not resolve cogging torque additionally. But it does make sense for some machines. Could be also used to calculate the so-called efficiency maps. I think though, that calculation of inductance is any easier that way. Can you please provide some more details on that.
Hi,

I was just saying that I would consider saturation in the permanent magnet flux as well. Simplified I do the following to calculate Ld and Lq:

1. Sweep iq from 0 to max with id = 0 to get PsiPM. -> PsiPM(iq)
2. Create FluxMap with all needed id, iq pairs -> Psi(id, iq)
3. Calculate Ld and Lq with PsiD = Ld*id + PsiPM(iq) and PsiQ = Lq*iq

For the simple purpose of a e.g. simulink model, you take the mean values and get the mean torque for every current (lorentz and reluctance torque). For the case of cogging torque, u also need the position and have to look at your model. Thats depents in some cases. That works for the common machine types.
jiana02
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Re: Calculation of self and mutual inductance for PMSM machines?

Post by jiana02 »

Yes, it's possible to calculate self and mutual inductances as well as flux-linkages in 3-phase PMSM machines using Elmer's transient 2D electromagnetic solver. You can implement a methodology similar to Ansys Maxwell's "Frozen Permeability" method. Set up the solver, freeze permeability after each solution, apply excitation currents sequentially to each winding, and solve for flux linkage. This approach allows you to use Elmer for your calculations effectively.
https://www.elmerfem.org/forum/viewtopic.php?p=27746#p27746gcp
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