Version information

%matplotlib notebook
from PySide2.QtWidgets import *
from datetime import date
print("Running date:","%B %d, %Y"))
import pyleecan
print("Pyleecan version:" + pyleecan.__version__)
import SciDataTool
print("SciDataTool version:" + SciDataTool.__version__)
Running date: April 27, 2023
Pyleecan version:1.5.0
SciDataTool version:2.5.0

How to compute currents, voltage and torque using the Electrical Module

This tutorial explains how to use the Electrical Module to compute currents, voltage and torque, using a simple electrical equivalent circuit. The idea is to provide insight on how to implement other methods.

The reference used to validate this tutorial is: Z. Yang, M. Krishnamurthy and I. P. Brown, “Electromagnetic and vibrational characteristic of IPM over full torque-speed range,” 2013 International Electric Machines & Drives Conference, Chicago, IL, 2013, pp. 295-302, doi: 10.1109/IEMDC.2013.6556267.

Machine and Simulation definition

This tutorial use the machine IPMSM_A (Prius 2004) defined in the “How to define a machine” tutorial. The configuration is similar to the one used in the “How to set the Operating Point” tutorial.

%matplotlib notebook
# Load the machine
from os.path import join
from pyleecan.Functions.load import load
from pyleecan.definitions import DATA_DIR

Toyota_Prius = load(join(DATA_DIR, "Machine", "Toyota_Prius.json"))
fig, ax = Toyota_Prius.plot()

Electrical Equivalent Circuit (EEC)

The electrical module is defined with the object EEC_PMSM that correspond to the electrical equivalent circuit for synchronous machine from “Advanced Electrical Drives, analysis, modeling, control”, Rik de doncker, Duco W.J. Pulle, Andre Veltman, Springer edition, is then used for the computation of Ud/Uq or Id/Iq (see schematics hereafter).

To compute the parameters of the EEC, you must first set a Magnetic model to compute the flux linkage. In this example we use a MagFEMM object to compute the flux linkage and the magnetic inductances using FEMM simulations (with symmetries and number of time steps). For the flux linkage computation, the currents are set to 0 [A].

Once the parameter of the EEC are known, the voltage can be computed. The electrical torque is then computed according to the formula: \(T_{em}=\frac{P-RI^2}{\Omega}\), where \(P\) is the magnetic power \(P=\frac{q_s}{2}\Re(VI^*)\)

             <---                               --->
 -----R-----wsLqIq----              -----R-----wsLdId----
|                     |            |                     |
|                     |            |                    BEMF
|                     |            |                     |
 ---------Id----------              ---------Iq----------

         --->                               --->
          Ud                                 Uq
from pyleecan.Classes.Simu1 import Simu1
from pyleecan.Classes.Electrical import Electrical
from pyleecan.Classes.EEC_PMSM import EEC_PMSM
from pyleecan.Classes.MagFEMM import MagFEMM

# Initialization of the Simulation
simu = Simu1(name="tuto_elec", machine=Toyota_Prius)
# Definition of the Electrical Equivalent Circuit
eec = EEC_PMSM(  # No parameter enforced => compute all
    fluxlink=MagFEMM(is_periodicity_a=True, T_mag=60),  # Magnetic model to compute fluxlinkage
# The Electrical module is defined with the EEC
simu.elec = Electrical(eec=eec)

# Run only Electrical module
simu.mag = None
simu.force = None
simu.struct = None

Defining starting point with InputElec or InputCurrent

The starting point of the simulation is defined with InputVoltage or InputCurrent. These objects will create an OutElec object and initialize it with the provided values for Id/Iq, and/or Ud/Uq. InputCurrent enables to define in several ways the starting point (see “How to set the Operating Point”).

Note that Id/Iq are required to accurately compute the magnetic inductances, so that if only Ud/Uq is provided, a current Id=1A and Iq=1A will be used for the computation of Ld and Lq. A more satisfactory method would be to compute a map of Ld and Lq over Id/Iq with the LUTdq object.

from pyleecan.Classes.InputCurrent import InputCurrent
from pyleecan.Classes.OPdq import OPdq
from numpy import sqrt, exp, pi

# Definition of a sinusoidal current
simu.input = InputCurrent()

I0_rms = 250/sqrt(2) # Maximum current [Arms]
Phi0 = 140*pi/180  # Maximum Torque Per Amp

simu.input.OP = OPdq(N0 = 2000)
simu.input.OP.set_I0_Phi0(I0_rms, Phi0)

simu.input.Nt_tot = 50 # Number of time steps
simu.input.Na_tot = 2048 # Spatial discretization

(-135.4188051049254, 113.62986941801093)

Running the simulation and postprocessings

out =
[16:14:49] Starting running simulation tuto_elec (machine=Toyota_Prius)
[16:14:49] Starting Electric module
[16:14:49] Starting running simulation tuto_elec_FluxLinkFEMM (machine=Toyota_Prius)
[16:14:49] Starting Magnetic module
[16:14:50] Solving time steps: 0%
[16:14:58] Solving time steps: 25%
[16:15:05] Solving time steps: 50%
[16:15:10] Solving time steps: 75%
[16:15:16] Solving time step: 100%
[16:15:16] End of simulation tuto_elec_FluxLinkFEMM
[16:15:16] Starting running simulation tuto_elec_FluxLinkFEMM_FluxLinkFEMM (machine=Toyota_Prius)
[16:15:16] Starting Magnetic module
[16:15:17] Solving time steps: 0%
[16:15:25] Solving time steps: 25%
[16:15:33] Solving time steps: 50%
[16:15:40] Solving time steps: 75%
[16:15:47] Solving time step: 100%
[16:15:47] End of simulation tuto_elec_FluxLinkFEMM_FluxLinkFEMM
[16:15:47] End of simulation tuto_elec
# Print voltage and torque
print("Ud: "+str(out.elec.OP.Ud_ref))
print("Uq: "+str(out.elec.OP.Uq_ref))
print("Tem: "+str(out.elec.Tem_av))

# Electrical Equivalent Circuit parameters are stored in:
print("\nLd: "+str(out.elec.eec.Ld))
print("Lq: "+str(out.elec.eec.Lq))
print("R1: "+str(out.elec.eec.R1))
Ud: -201.40443033554803
Uq: -12.313080032430975
Tem: 353.7141872050232

Ld: 0.0011908433790078954
Lq: 0.0020622123849967674
R1: 0.03778895717396608
# Plot the currents
out.elec.get_Is().plot_2D_Data("time", "phase")
# Plot the voltages
out.elec.get_Us().plot_2D_Data("time", "phase")

Iterating on several Operating Points

Reference torque and current angle vector are:

from numpy import linspace, array, pi
Tem_av_ref = array([79, 125, 160, 192, 237, 281, 319, 343, 353, 332, 266, 164, 22]) # Yang et al, 2013
Phi0_ref = linspace(60 * pi / 180, 180 * pi / 180, Tem_av_ref.size)
N_simu = Tem_av_ref.size

The multisimulation is similar to the one described in “How to set the Operating Point”, except that we are now computing the electrical torque instead of the magnetic torque.

from pyleecan.Classes.VarLoadCurrent import VarLoadCurrent
from pyleecan.Classes.DataKeeper import DataKeeper
from numpy import zeros, ones, linspace, array, sqrt, arange

simu2 = simu.copy()
# Enforce the EEC parameters from previous run
simu2.elec.eec = out.elec.eec

# Define the Variable speed simulation
varload = VarLoadCurrent()
simu2.var_simu = varload

# Creating the Operating point matrix
OP_matrix = zeros((N_simu,4))

# Set N0 = 2000 [rpm] for all simulation
OP_matrix[:,0] = 2000 * ones((N_simu))

# Set I0 = 250 / sqrt(2) [A] (RMS) for all simulation
OP_matrix[:,1] = I0_rms * ones((N_simu))

# Set Phi0 from 60° to 180°
OP_matrix[:,2] = Phi0_ref

# Set reference torque from Yang et al, 2013
OP_matrix[:,3] = Tem_av_ref
varload.set_OP_array(OP_matrix, "N0", "I0", "Phi0", "Tem")

# Add DataKeeper to track I0/Phi0/Tem
I0_dk = DataKeeper(
    name="Stator current rms amplitude",
    keeper="lambda output: output.elec.OP.get_I0_Phi0()['I0']",
Phi0_dk = DataKeeper(
    name="Stator current phase",
    keeper="lambda output: output.elec.OP.get_I0_Phi0()['Phi0']",
Tem_dk = DataKeeper(
    name="Computed torque",
    keeper="lambda output: output.elec.Tem_av",
simu2.var_simu.datakeeper_list = [I0_dk, Phi0_dk, Tem_dk]
[[ 2.00000000e+03  8.83883476e+01  1.53093109e+02  7.90000000e+01]
 [ 2.00000000e+03  6.04611907e+01  1.66115756e+02  1.25000000e+02]
 [ 2.00000000e+03  3.06969510e+01  1.74091060e+02  1.60000000e+02]
 [ 2.00000000e+03  1.08244507e-14  1.76776695e+02  1.92000000e+02]
 [ 2.00000000e+03 -3.06969510e+01  1.74091060e+02  2.37000000e+02]
 [ 2.00000000e+03 -6.04611907e+01  1.66115756e+02  2.81000000e+02]
 [ 2.00000000e+03 -8.83883476e+01  1.53093109e+02  3.19000000e+02]
 [ 2.00000000e+03 -1.13629869e+02  1.35418805e+02  3.43000000e+02]
 [ 2.00000000e+03 -1.35418805e+02  1.13629869e+02  3.53000000e+02]
 [ 2.00000000e+03 -1.53093109e+02  8.83883476e+01  3.32000000e+02]
 [ 2.00000000e+03 -1.66115756e+02  6.04611907e+01  2.66000000e+02]
 [ 2.00000000e+03 -1.74091060e+02  3.06969510e+01  1.64000000e+02]
 [ 2.00000000e+03 -1.76776695e+02  2.16489014e-14  2.20000000e+01]]
# Run multisimulation
Xout =
[16:16:09] Starting running simulation tuto_elec (machine=Toyota_Prius)
[16:16:09] Computing reference simulation for Variable Load
[16:16:09] Starting Electric module
[16:16:09] Variable Load Reference Results: I0=176.8 [Arms], Phi0=1.047 [rad], Tem_av=-284.9 [rad], N0=2000 [rpm], Id=88.39 [Arms], Iq=153.1 [Arms], Tem_av_ref=79 [N.m]
[===                                               ]   7%
[16:16:09] Running simulation 1/13 with N0=2000 [rpm], Id=88.39 [Arms], Iq=153.1 [Arms]
[16:16:09] Simulation matches reference one: Skipping computation
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=1.047 [rad], Tem_av=-284.9 [rad], N0=2000 [rpm], Id=88.39 [Arms], Iq=153.1 [Arms], Tem_av_ref=79 [N.m]
[=======                                           ]  14%
[16:16:09] Running simulation 2/13 with N0=2000 [rpm], Id=60.46 [Arms], Iq=166.1 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=1.222 [rad], Tem_av=-209.5 [rad], N0=2000 [rpm], Id=60.46 [Arms], Iq=166.1 [Arms], Tem_av_ref=125 [N.m]
[==========                                        ]  21%
[16:16:09] Running simulation 3/13 with N0=2000 [rpm], Id=30.7 [Arms], Iq=174.1 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=1.396 [rad], Tem_av=-127.7 [rad], N0=2000 [rpm], Id=30.7 [Arms], Iq=174.1 [Arms], Tem_av_ref=160 [N.m]
[==============                                    ]  28%
[16:16:09] Running simulation 4/13 with N0=2000 [rpm], Id=1.082e-14 [Arms], Iq=176.8 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=1.571 [rad], Tem_av=-42.05 [rad], N0=2000 [rpm], Id=1.082e-14 [Arms], Iq=176.8 [Arms], Tem_av_ref=192 [N.m]
[=================                                 ]  35%
[16:16:09] Running simulation 5/13 with N0=2000 [rpm], Id=-30.7 [Arms], Iq=174.1 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=1.745 [rad], Tem_av=44.9 [rad], N0=2000 [rpm], Id=-30.7 [Arms], Iq=174.1 [Arms], Tem_av_ref=237 [N.m]
[=====================                             ]  42%
[16:16:09] Running simulation 6/13 with N0=2000 [rpm], Id=-60.46 [Arms], Iq=166.1 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=1.92 [rad], Tem_av=130.5 [rad], N0=2000 [rpm], Id=-60.46 [Arms], Iq=166.1 [Arms], Tem_av_ref=281 [N.m]
[=========================                         ]  50%
[16:16:09] Running simulation 7/13 with N0=2000 [rpm], Id=-88.39 [Arms], Iq=153.1 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=2.094 [rad], Tem_av=212.1 [rad], N0=2000 [rpm], Id=-88.39 [Arms], Iq=153.1 [Arms], Tem_av_ref=319 [N.m]
[============================                      ]  57%
[16:16:09] Running simulation 8/13 with N0=2000 [rpm], Id=-113.6 [Arms], Iq=135.4 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=2.269 [rad], Tem_av=287.3 [rad], N0=2000 [rpm], Id=-113.6 [Arms], Iq=135.4 [Arms], Tem_av_ref=343 [N.m]
[================================                  ]  64%
[16:16:09] Running simulation 9/13 with N0=2000 [rpm], Id=-135.4 [Arms], Iq=113.6 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=2.443 [rad], Tem_av=353.7 [rad], N0=2000 [rpm], Id=-135.4 [Arms], Iq=113.6 [Arms], Tem_av_ref=353 [N.m]
[===================================               ]  71%
[16:16:09] Running simulation 10/13 with N0=2000 [rpm], Id=-153.1 [Arms], Iq=88.39 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=2.618 [rad], Tem_av=409.4 [rad], N0=2000 [rpm], Id=-153.1 [Arms], Iq=88.39 [Arms], Tem_av_ref=332 [N.m]
[=======================================           ]  78%
[16:16:09] Running simulation 11/13 with N0=2000 [rpm], Id=-166.1 [Arms], Iq=60.46 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=2.793 [rad], Tem_av=452.7 [rad], N0=2000 [rpm], Id=-166.1 [Arms], Iq=60.46 [Arms], Tem_av_ref=266 [N.m]
[==========================================        ]  85%
[16:16:09] Running simulation 12/13 with N0=2000 [rpm], Id=-174.1 [Arms], Iq=30.7 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=2.967 [rad], Tem_av=482.2 [rad], N0=2000 [rpm], Id=-174.1 [Arms], Iq=30.7 [Arms], Tem_av_ref=164 [N.m]
[==============================================    ]  92%
[16:16:09] Running simulation 13/13 with N0=2000 [rpm], Id=-176.8 [Arms], Iq=2.165e-14 [Arms]
[16:16:09] Starting Electric module
[16:16:09] Variable Load Results: I0=176.8 [Arms], Phi0=3.142 [rad], Tem_av=497 [rad], N0=2000 [rpm], Id=-176.8 [Arms], Iq=2.165e-14 [Arms], Tem_av_ref=22 [N.m]
[==================================================] 100%
[16:16:09] End of simulation tuto_elec

Once the simulation is done, the torque as a function of Phi0 can be plotted with:

# Plot torque as a function of Phi0
from SciDataTool.Functions.Plot.plot_2D import plot_2D
from numpy import array

    array([x*180/pi for x in Xout.xoutput_dict["Phi0"].result]),
    [Xout.xoutput_dict["Tem_av"].result, Tem_av_ref],
    legend_list=["Pyleecan", "Yang et al, 2013"],
    xlabel="Current angle [°]",
    ylabel="Electrical torque [N.m]",
    title="Electrical torque vs current angle",
[ ]: