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Permanent Magnet Synchronous Machine 12-phase
ehs solver
This block implements a 12 Phase permanent magnet synchronous machine.
The PMSM block implements a twelve-phase Permanent Magnet Synchronous Motors (12-Ph PMSM) model with resolvers and encoders.
Equations & Characteristics
General PMSM Solver Equation
The equation of the PMSM model can be expressed as follows:
where Labcdefuvwxyz is the time-varying inductance matrix (global inductance for DQ0 and VDQ models), Iabcdefuvwxyz is the stator current inside the winding, R is the stator resistance, Vabcdefuvwxyz is the voltage across the stator windings and ψabcdefuvwxyz defines the magnet flux linked into the stator windings.
The solver computes the outputs in phase domain, using phase domain states. The integration type used is Backward Euler.
Standard DQ0 Motor Characteristics
Under normal conditions, the ideal sinusoidal stator voltages of the PMSM, back-EMFs, and inductances all have sinusoidal shapes. One can transform the equation using the Park transformation with a referential locked on the rotor position θ using the following two equations:
and
The Park transform (also called ‘DQ’ transform) reduces sinusoidal varying quantities of inductances, flux, current, and voltage to constant values in the D-Q frame thus greatly facilitating the analysis and control of the device under study.
It is important to note that there are many different types of Park transforms and this often leads to confusion when interpreting the motor states inside the D-Q frame. The one used here presents the advantage of being amplitude-invariant which means that the peak amplitude of the signals in the D-Q frame by performing a transformation will be numerically equal to the one computed in the phase domain.
The corresponding equations in the DQ0 domain, for representation, are:
All zero sequence inductances have a value of 1e15.
With this transform (and only this transform) the PMSM torque can be expressed by the following equation, where pp is the number of pole pairs.
where
same for Phase b,c,d,e,f,u,v,w,x,y,z.
and ua is the unit vector template of the BackEMF profile and is the flux linkage constant of the Winding Set.
The synchronous component of Torque includes impact due to Zero Sequence Current. i=1,2,34 , are ideally the same but provided separately.
Figure 1 explains the principle of the Park transform. Considering fixed ABC referential with all quantities ( Vbemf, motor current I) rotating at the electric frequency ω, if we observe these quantities in a D-Q frame turning at the same speed we can see that the motor quantities will be constant.
This is easy to see for the Back-EMF voltage Vbemf that directly follows the Q-axis (because the magnet flux is on the D-axis by definition). In Figure 1, I leads the Q-axis by an angle called β (beta). The modulus of the vector I is called Iamp. In the figure below, θ is the rotor angle, aligned with the D-axis.
The above figure is for D1Q1 reference frame, the D2Q2 reference frame has a phase shift of , which is the phase shift between the winding sets. Every corresponding reference frame is phase shifted by
Parameters & Measurements
The PMSM's parameters and measurements are separated in 4 different tabs, Electrical, Mechanical, Resolver and Encoder.
Electrical Parameters & Measurements
Symbol | Name | Description | Unit | Type |
|---|---|---|---|---|
Mconf | Motor configuration type | Type of motor : Constant DQ |
| Input |
Rs1 | Stator resistances - Windings 1 | Resistances of the stator windings specified for every phase, A, B and C. | Ω | Input |
Rs2 | Stator resistances - Windings 2 | Resistances of the stator windings specified for every phase, D, E and F. | Ω | Input |
Rs3 | Stator resistances - Windings 3 | Resistances of the stator windings specified for every phase, U, V and W. | Ω | Input |
Rs4 | Stator resistances - Windings 4 | Resistances of the stator windings specified for every phase, X, Y and Z. | Ω | Input |
Ldq1 | Stator inductances (DQ) - Winding 1 | d and q axis inductances of Winding 1 | H | Input |
Ldq2 | Stator inductances (DQ) - Winding 2 | d and q axis inductances of Winding 2 | H | Input |
Ldq3 | Stator inductances (DQ) - Winding 3 | d and q axis inductances of Winding 3 | H | Input |
Ldq4 | Stator inductances (DQ) - Winding 4 | d and q axis inductances of Winding 4 | H | Input |
dΦ/dθ | Back EMF profile | Profile of the back EMF, either Sinusoidal |
| Input |
λm1 | Permanent magnet flux linkage - Winding 1 | Amplitude of the rotor permanent magnet flux for Winding 1 | Wb | Input |
λm2 | Permanent magnet flux linkage - Winding 2 | Amplitude of the rotor permanent magnet flux for Winding 2 | Wb | Input |
λm3 | Permanent magnet flux linkage - Winding 3 | Amplitude of the rotor permanent magnet flux for Winding 3 | Wb | Input |
λm4 | Permanent magnet flux linkage - Winding 4 | Amplitude of the rotor permanent magnet flux for Winding 4 | Wb | Input |
ε | Phase shift between Winding | Phase shift between any 2 sets of windings | ° | Input |
pp | Number of pole pairs | Number of pole pairs |
| Input |
dq0 | DQ Transform | Amplitude Invariant or Power Invariant (see Permanent Magnet Synchronous Machine 12-phase | Limitations ) |
| Input |
θdq0 | DQ Transform Angle | A aligned to D / A aligned to Q (see Permanent Magnet Synchronous Machine 12-phase | Limitations ) |
| Input |
is1 | Stator currents - Winding 1 | Currents measured at phases A, B and C of the stator | A | Measurement |
is2 | Stator currents - Winding 2 | Currents measured at phases D, E and F of the stator | A | Measurement |
is3 | Stator currents - Winding 3 | Currents measured at phases U, V and W of the stator | A | Measurement |
is4 | Stator currents - Winding 4 | Currents measured at phases X, Y and Z of the stator | A | Measurement |
isdq1 | Stator currents (DQ) - Winding 1 | Currents measured of axis d1, q1 | A | Measurement |
isdq2 | Stator currents (DQ) - Winding 2 | Currents measured of axis d2, q2 | A | Measurement |
isdq3 | Stator currents (DQ) - Winding 3 | Currents measured of axis d3, q3 | A | Measurement |
isdq4 | Stator currents (DQ) - Winding 4 | Currents measured of axis d4, q4 | A | Measurement |
Vs1 | Stator voltages - Winding 1 | Voltages measured at phases A, B and C of the stator | V | Measurement |
Vs2 | Stator voltages - Winding 2 | Voltages measured at phases D, E and F of the stator | V | Measurement |
Vs3 | Stator voltages - Winding 3 | Voltages measured at phases U, V and W of the stator | V | Measurement |
Vs4 | Stator voltages - Winding 4 | Voltages measured at phases X, Y and Z of the stator | V | Measurement |
Bemf1 | Back EMF voltages - Winding 1 | Phase to neutral voltage generated from the permanent magnet flux linkage for Winding Set 1 | V | Measurement |
Bemf2 | Back EMF voltages - Winding 2 | Phase to neutral voltage generated from the permanent magnet flux linkage for Winding Set 2 | V | Measurement |
Bemf3 | Back EMF voltages - Winding 3 | Phase to neutral voltage generated from the permanent magnet flux linkage for Winding Set 3 | V | Measurement |
Bemf4 | Back EMF voltages - Winding 4 | Phase to neutral voltage generated from the permanent magnet flux linkage for Winding Set 4 | V | Measurement |
θe | Electrical rotor position | Position of the rotor from 0 to 360 degrees | ° | Measurement |
Rsn | Snubber resistance | Resistances of the snubber on phase A, B, C, D, E, F, U, V, W, X, Y and Z | Ω | Input |
Csn | Snubber capacitance | Capacitance of the snubber on phase A, B, C, D, E, F, U, V, W, X, Y and Z | F | Input |
Mechanical Parameters & Measurements
Symbol | Name | Description | Unit | Type |
|---|---|---|---|---|
J | Rotor inertia | Moment of inertia of the rotor (only available when Torque Mode is selected) | kg*m2 | Input |
Fv | Viscous friction coefficient | Viscous friction (only available when Torque Mode is selected) | N*m*s/rad | Input |
Fs | Static friction torque | Static friction | N*m | Input |
ctrl | Mechanical control mode | Control mode of the mechanical model. Has two possible values: speed or torque. In speed mode, the mechanical model is bypassed and the speed command is sent directly. In torque mode, the torque command is used to measure the speed using the mechanical parameters of the machine. |
| Input |
T | Torque command | Torque command sent to the mechanical model | N*m | Input |
ωr0 | Initial Rotor Speed Command | Setpoint for Initial Rotor speed | rpm | Input |
ωrc | Rotor speed command | Speed command sent to the mechanical model | rpm | Input |
ωr | Rotor speed | Speed of the rotor | rpm | Measurement |
θ0 | Initial rotor angle | Rotor position at time t = 0 | ° | Input |
θ | Rotor angle | Rotor position from 0 to 360 degrees | ° | Measurement |
Te | Electromagnetic torque | Torque measured at the rotor | N*m | Measurement |