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# SSN JMAG Spatial Harmonic PMSM Block

# Library

ARTEMiS/SSN Machines

# Blocks

# Description

Implement a 3-phase permanent magnet synchronous machine with inductance and flux provided by JMAG-RT in the form of 3-D tables.

The model can also support tables from other companies such as Infolytica and ANSYS with minimum modifications.

# Mask

# Parameters

**RT data file**: JMAG-RT data file (entered with ' ' and .rtt extension)

**RT data file type**: Type of .rtt file (set to JMAG-RT Spatial Harmonic)

**Mechanical offset angle (deg):** the mechanical angle offset of the rotor in degrees.

**Re-Sample SH tables:** When this option is checked, the SH tables are resampled with the user-selected number of levels for each dimension. This option is provided to mimic similar re-sampling on FPGAsim SH motor models.

**Re-sampling levels** (currents amplitude, mechanical angle, current beta angle):

**Sample Time (s):** sample time of the model in seconds.

# Input and Output signals

**Simulink connection points:**

**SHdata**: Simulink bus containing the following signals:

Rs: stator resistance in Ohms. This is a 3x3 matrix.

Te: electric torque in N.m.

dphi0/dt: this is the apparent back-EMF of the motor at the point of operation in Volts. mechanical speed of the machine in rad/s. This speed is usually computed from a mechanical model that includes the machine and other devices torques and inertias.

Iabc: ABC terminal currents in Amperes.

**Physical Modeling connection points:**

**a,b,c**: stator phase connection points.

**N**: the star neutral point (4-wire model only). This point should always be connected to a high impedance to the ground (ex: 1e8 Ohms)

# Example

An example model is provided in the Rotating Machines demo section of ARTEMiS.

# Discretized PMSM-SH Model Theory

The so-called Spatial Harmonic PMSM (SH-PMSM) model is provided by JMAG as tables of inductance and total flux. We derive here the discrete model for the SH-PMSM model.

### JMAG table format

The JMAG-RT tables for the SH-PMSM come in the form of 3-D tables that provides the total flux Ψ, the differential inductance Ldiff and the motor torque as a function of 3 variables: the current amplitude Iamp, the current angle β and the rotor angle, using the orthonormal Park transform to obtain dq-domain values.

### Differential inductance interpretation

The inductance table is provided as a 3x3 matrix. It does not assume a specific winding connection but floating-wye is the most common. This inductance Ldiff is the linearized inductance at the point of operation. This linearization as the effect of creating a flux offset Ψo which must be carefully considered.

Flux-current relationship in a Spatial-Harmonic model with saturation

### Differential equation of the SH-PMSM model

with Ψ (vector of 3) as the total flux, Ψo (vector of 3) as the flux offset at the point of operation, Ldiff is differential inductance matrix (3x3), that is the local slope of the total flux at the point of operation, Iabc (or *I* in the figure) are the coil currents (vector of 3).

In equations 1-2, one must consider that the null-current magnet flux Ψp (if any, depending on the rotor angle) is included in Ψo. Also note that in a non-saturated PMSM, Ψo = Ψp and Ldiff is constant.

### Discretization of the model and use into a nodal admittance solver

A central difference method is applied on equation 2 to obtain the discrete nodal model:

With *(n-1),* *(n) *and *(n+1) *representing to time instant separated by the time step length* h*. Flux and inductance values are obtained from the table with a one time-step delay so their time indices could be *(n)* and *(n-1)* instead of *(n+1) *and *(n)* above. In the equation, *V _{abc}* is the vector of coil voltages, which is different than the node voltages.

The ‘discrete admittance’ of the equation is therefore:

In order to obtain the coil voltages in the floating Y configuration of the PMSM, we have to find the neutral point voltage, and therefore, a system with 4 terminal needs to be created, which a very high impedance to ground on the neutral point.

Star-connected PMSM-SH with neutral point as SSN node.

With Ihist_{neutral} = -ΣIhist_{abc} since all phases are connected together on the neutral side.

## 3-wire model

It is possible to get rid of the neutral node with some gymnastics, and thus hopefully increase the calculation speed by the reduction of the global admittance matrix. The 3-wire model reduction is obtained first by adding the neutral admittance into the extended Y matrix:

Then I_{neutral}=0, we can solve for V_{neutral} and the system can be reduced to a 3x3 system, the discrete Norton model being equal to:

The neutral voltage, still required to find the coil voltages, is equal to

# References

[1] C. Dufour, S. Cense, T. Yamada, R. Imamura, J. Belanger, “FPGA Permanent Magnet Synchronous Motor Floating-Point Models with Variable-DQ and Spatial Harmonic Finite-Element Analysis Solvers”, 15th Int. Power Electronics and Motion Control Conference, EPE-PEMC 2012, Novi Sad, Serbia, Sept. 4-6, 2012

[2] https://www.jmag-international.com/modellibrary (for JMAG-RT models)

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