.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "examples/plot_3kw_bsyrm_gradnet_maps.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_examples_plot_3kw_bsyrm_gradnet_maps.py: 2.6-kW bearingless SyRM, GradNet magnetic maps ============================================== This example visualizes the GradNet magnetic maps [#Li2026]_ of a 2.6-kW six-phase bearingless synchronous reluctance machine (SyRM), trained on finite-element data. As in *motulator*, the machine model uses the energy-based current map, with or without the spatial harmonics, and the control system uses the co-energy-based flux map without the rotor angle, which approximates the average over the angle [#Das2026]_. In the figures, the markers show the finite-element results, averaged over the rotor angle unless otherwise noted, and the lines show the networks. The markers are not used in the training. .. GENERATED FROM PYTHON SOURCE LINES 17-27 .. code-block:: Python from pathlib import Path from typing import cast import matplotlib.pyplot as plt import numpy as np from mpl_toolkits.mplot3d import Axes3D import motulator_bearingless.gradnet as gn from motulator_bearingless import model, utils .. GENERATED FROM PYTHON SOURCE LINES 28-29 Load the dataset and the networks. .. GENERATED FROM PYTHON SOURCE LINES 29-56 .. code-block:: Python p = Path(__file__).resolve().parent if "__file__" in globals() else Path.cwd() R_s = 0.215 # Stator resistance from [#Das2026]_ path = "trained_models/bsyrm_fem_curr_map_harm_softmax_d128.npz" harm_map = gn.CurrentMapWithHarmonics(gn.load_gradnet(p / path)) path = "trained_models/bsyrm_fem_curr_map_softmax_d64.npz" current_map = gn.CurrentMap(gn.load_gradnet(p / path)) path = "trained_models/bsyrm_fem_flux_map_softmax_d128.npz" est_par = model.SaturatedBearinglessSynchronousMachinePars( R_s, psi_s_fcn=gn.FluxMap(gn.load_gradnet(p / path)) ) data = dict(np.load(p / "datasets" / "bsyrm_fem.npz")) utils.set_style() def averaged(sel: np.ndarray, *keys: str) -> dict[str, np.ndarray]: """Average the given signals over the rotor angle, the points being grouped.""" keys_in = ("i_s1", "i_s2", "q_M") arr = np.stack([np.round(data[k][sel], 9) for k in keys_in], axis=-1) uniq, inv = np.unique(arr, axis=0, return_inverse=True) out = {k: uniq[:, m] for m, k in enumerate(keys_in)} for key in keys: vals = data[key][sel] out[key] = np.array([np.mean(vals[inv == n]) for n in range(len(uniq))]) return out .. GENERATED FROM PYTHON SOURCE LINES 57-60 Current map of the machine model: the four-pole flux linkage versus the four-pole current at a centric rotor without the two-pole current. The lines are the currents given by the network at the finite-element flux linkages. .. GENERATED FROM PYTHON SOURCE LINES 60-83 .. code-block:: Python fem = averaged(data["set"] == 2, "psi_s1", "psi_s2") _, i_2, _ = current_map(fem["psi_s1"], fem["psi_s2"], fem["q_M"]) i_d, i_q = np.real(fem["i_s2"]), np.imag(fem["i_s2"]) psi_d, psi_q = np.real(fem["psi_s2"]), np.imag(fem["psi_s2"]) fig, ax = plt.subplots() for level, ls in ((0, "-"), (24, "--")): m = np.isclose(i_q, level) & (i_d <= 24) ax.plot(i_d[m], psi_d[m], ".", c="r") label = rf"$\psi_\mathrm{{s2d}}(i_\mathrm{{s2d}}, {level}\,\mathrm{{A}})$" ax.plot(np.real(i_2)[m], psi_d[m], ls, c="r", label=label) for level, ls in ((0, "-"), (24, "--")): m = np.isclose(i_d, level) & (i_q <= 24) ax.plot(i_q[m], psi_q[m], ".", c="b") label = rf"$\psi_\mathrm{{s2q}}({level}\,\mathrm{{A}}, i_\mathrm{{s2q}})$" ax.plot(np.imag(i_2)[m], psi_q[m], ls, c="b", label=label) ax.set_xlim(0, 24) ax.set_ylim(0, None) ax.set_xlabel(r"$i_\mathrm{s2d}$, $i_\mathrm{s2q}$ (A)") ax.set_ylabel(r"$\psi_\mathrm{s2d}$, $\psi_\mathrm{s2q}$ (Vs)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_001.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 84-87 Current map with the spatial harmonics: the torque versus the rotor angle at the rated peak current, evaluated at the finite-element flux linkages. The finite-element results are not averaged here. .. GENERATED FROM PYTHON SOURCE LINES 87-107 .. code-block:: Python sel = data["set"] == 5 theta, i_1, q_M = data["theta_M"][sel], data["i_s1"][sel], data["q_M"][sel] psi_1, psi_2 = data["psi_s1"][sel], data["psi_s2"][sel] _, _, f_h, tau_h = harm_map(psi_1, psi_2, q_M, np.exp(1j * theta)) _, _, f_a = current_map(psi_1, psi_2, q_M) deg = np.rad2deg(theta) m = np.isclose(i_1, 0) & np.isclose(q_M, 0) fig, ax = plt.subplots() ax.plot(deg[m], tau_h[m], c="m", lw=1.5, label="Current map with harmonics") ax.plot(deg[m], data["tau_M"][sel][m], ".", c="b", label="Finite-element model") ax.set_xlim(0, 60) ax.set_xticks(np.arange(0, 61, 10)) ax.set_ylim(17.5, 22.5) ax.set_xlabel(r"$\vartheta_\mathrm{M}$ (deg)") ax.set_ylabel(r"$\tau_\mathrm{M}$ (Nm)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_002.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 108-111 The magnitude of the radial force at `i_s1 = 2 A`. The force ripple appears when the two-pole current produces the force. The current map without the harmonics gives the average. .. GENERATED FROM PYTHON SOURCE LINES 111-125 .. code-block:: Python m = np.isclose(i_1, 2) fig, ax = plt.subplots() ax.plot(deg[m], np.abs(f_h[m]), c="m", lw=1.5, label="Current map with harmonics") ax.plot(deg[m], np.abs(f_a[m]), "--", c="r", lw=1.5, label="Current map") ax.plot(deg[m], np.abs(data["f_M"][sel][m]), ".", c="b", label="Finite-element model") ax.set_xlim(0, 60) ax.set_xticks(np.arange(0, 61, 10)) ax.set_ylim(740, 900) ax.set_xlabel(r"$\vartheta_\mathrm{M}$ (deg)") ax.set_ylabel(r"$|f_\mathrm{M}|$ (N)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_003.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 126-128 Flux map of the control system: the two-pole flux linkage versus the two-pole current at a centric rotor, for different four-pole currents. .. GENERATED FROM PYTHON SOURCE LINES 128-152 .. code-block:: Python ops = (5 + 0j, 15 + 0j, 12 + 12j, 20 + 20j) colors = ("b", "r", "m", "c") labels = [rf"$i_\mathrm{{s2}}$ = {i.real:.0f} + j{i.imag:.0f} A" for i in ops] fem = averaged(data["set"] == 3, "psi_s1", "f_M") i_1d = np.linspace(0, 8.5, 35) i_1 = i_1d + 0j zeros = np.zeros_like(i_1) psi_1 = [est_par.psi_s(i_1, i_2 + zeros, zeros)[0] for i_2 in ops] f = [est_par.f_M(i_1, i_2 + zeros, zeros) for i_2 in ops] fem_masks = [ np.isclose(fem["i_s2"], i_2) & np.isclose(np.imag(fem["i_s1"]), 0) for i_2 in ops ] fig, ax = plt.subplots() for m, psi, c, label in zip(fem_masks, psi_1, colors, labels, strict=True): ax.plot(np.real(fem["i_s1"][m]), np.real(fem["psi_s1"][m]), ".", c=c) ax.plot(i_1d, np.real(psi), c=c, label=label) ax.set_xlim(0, 8.5) ax.set_xlabel(r"$i_\mathrm{s1d}$ (A)") ax.set_ylabel(r"$\psi_\mathrm{s1d}$ (Vs)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_004.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 153-155 The radial force versus the two-pole current. The force depends nearly linearly on the two-pole current, while its slope depends strongly on the four-pole current. .. GENERATED FROM PYTHON SOURCE LINES 155-166 .. code-block:: Python fig, ax = plt.subplots() for m, f_i, c, label in zip(fem_masks, f, colors, labels, strict=True): ax.plot(np.real(fem["i_s1"][m]), np.real(fem["f_M"][m]), ".", c=c) ax.plot(i_1d, np.real(f_i), c=c, label=label) ax.set_xlim(0, 8.5) ax.set_xlabel(r"$i_\mathrm{s1d}$ (A)") ax.set_ylabel(r"$f_\mathrm{M,i}$ (N)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_005.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_005.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 167-169 Flux map: the unbalanced magnetic pull versus the displacement, without the two-pole current. The force is in the direction of the displacement. .. GENERATED FROM PYTHON SOURCE LINES 169-190 .. code-block:: Python fem = averaged(data["set"] == 4, "psi_s1", "f_M") q_i = np.linspace(0, 400e-6, 41) q = q_i + 0j zeros = np.zeros_like(q) f = [est_par.f_M(zeros, i_2 + zeros, q) for i_2 in ops] psi_1 = [est_par.psi_s(zeros, i_2 + zeros, q)[0] for i_2 in ops] fem_masks = [ np.isclose(fem["i_s2"], i_2) & np.isclose(np.imag(fem["q_M"]), 0) for i_2 in ops ] fig, ax = plt.subplots() for m, f_i, c, label in zip(fem_masks, f, colors, labels, strict=True): ax.plot(1e6 * np.real(fem["q_M"][m]), np.real(fem["f_M"][m]), ".", c=c) ax.plot(1e6 * q_i, np.real(f_i), c=c, label=label) ax.set_xlim(0, 400) ax.set_xlabel(r"$q_\mathrm{M,i}$ (µm)") ax.set_ylabel(r"$f_\mathrm{M,i}$ (N)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_006.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_006.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 191-192 The two-pole flux linkage induced by the displacement. .. GENERATED FROM PYTHON SOURCE LINES 192-203 .. code-block:: Python fig, ax = plt.subplots() for m, psi, c, label in zip(fem_masks, psi_1, colors, labels, strict=True): ax.plot(1e6 * np.real(fem["q_M"][m]), 1e3 * np.real(fem["psi_s1"][m]), ".", c=c) ax.plot(1e6 * q_i, 1e3 * np.real(psi), c=c, label=label) ax.set_xlim(0, 400) ax.set_xlabel(r"$q_\mathrm{M,i}$ (µm)") ax.set_ylabel(r"$\psi_\mathrm{s1d}$ (mVs)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_007.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_007.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 204-211 The unbalanced magnetic pull at the rated peak current, at constant currents according to the flux map and at constant flux linkages according to the current map. The flux linkages are those of the finite-element model at a centric rotor. The pull is destabilizing also at constant flux linkages, i.e., the energy is concave in the displacement. Therefore, the current map represents the shifted energy `W + k_q*abs(q_M)**2/2`, which is convex where `k_q` exceeds the stiffness of the pull at constant currents. .. GENERATED FROM PYTHON SOURCE LINES 211-231 .. code-block:: Python i_2 = 12 + 12j fem_0 = averaged((data["set"] == 2) & np.isclose(data["i_s2"], i_2), "psi_s2") _, _, f_psi = current_map(zeros, fem_0["psi_s2"][0] + zeros, q) f_i = est_par.f_M(zeros, i_2 + zeros, q) k_q = current_map.fcn.k_q * current_map.f_base / current_map.q_base m = fem_masks[ops.index(i_2)] fig, ax = plt.subplots() ax.plot(1e6 * np.real(fem["q_M"][m]), np.real(fem["f_M"][m]), ".", c="b") ax.plot(1e6 * q_i, np.real(f_i), c="b", label="Flux map (constant currents)") ax.plot(1e6 * q_i, np.real(f_psi), c="r", label="Current map (constant flux linkages)") ax.plot(1e6 * q_i, k_q * q_i, "--", c="m", label=r"$k_q q_\mathrm{M,i}$") ax.set_xlim(0, 400) ax.set_ylim(0, None) ax.set_xlabel(r"$q_\mathrm{M,i}$ (µm)") ax.set_ylabel(r"$f_\mathrm{M,i}$ (N)") ax.legend() plt.show() .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_008.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_008.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 232-236 Characteristics of the flux map over the four-pole current at a centric rotor without the two-pole current: the torque, the incremental inductance of the two-pole plane, the slope of the radial force with respect to the two-pole current, and the stiffness of the unbalanced magnetic pull at constant currents. The rated peak current is 17 A. .. GENERATED FROM PYTHON SOURCE LINES 236-269 .. code-block:: Python d = np.linspace(0, 25, 21) i_d, i_q = np.meshgrid(d, d, indexing="ij") i_2 = np.ravel(i_d + 1j * i_q) zeros = np.zeros_like(i_2) _, psi_2 = est_par.psi_s(zeros, i_2, zeros) L_s = est_par.incr_ind_mat(zeros, i_2, zeros) eps_i, eps_q = 1e-3, 1e-6 maps = ( (6 * np.imag(i_2 * np.conj(psi_2)), r"$\tau_\mathrm{M}$ (Nm)"), (1e3 * 0.5 * (L_s[:, 0, 0] + L_s[:, 1, 1]), r"$L_1$ (mH)"), ( np.abs(est_par.f_M(zeros + eps_i, i_2, zeros)) / eps_i, r"$|\partial f_\mathrm{M}/\partial i_\mathrm{s1}|$ (N/A)", ), ( 1e-6 * np.real(est_par.f_M(zeros, i_2, zeros + eps_q)) / eps_q, r"$\partial f_\mathrm{M,i}/\partial q_\mathrm{M,i}$ (N/µm)", ), ) for z, zlabel in maps: fig, ax = plt.subplots(subplot_kw={"projection": "3d"}) ax = cast(Axes3D, ax) z_grid = np.reshape(z, i_d.shape) ax.plot_surface(i_d, i_q, z_grid, cmap="viridis", alpha=0.5) ax.plot_wireframe(i_d, i_q, z_grid, color="black", linewidth=0.5, alpha=0.5) ax.set_xlabel(r"$i_\mathrm{s2d}$ (A)") ax.set_ylabel(r"$i_\mathrm{s2q}$ (A)") ax.zaxis.set_rotate_label(False) ax.set_zlabel(zlabel, rotation=90) ax.view_init(elev=15, azim=-135) plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_009.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_009.png :class: sphx-glr-multi-img * .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_010.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_010.png :class: sphx-glr-multi-img * .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_011.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_011.png :class: sphx-glr-multi-img * .. image-sg:: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_012.png :alt: plot 3kw bsyrm gradnet maps :srcset: /examples/images/sphx_glr_plot_3kw_bsyrm_gradnet_maps_012.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 270-278 .. rubric:: References .. [#Li2026] Li, Foissner, Martin, Piippo, Hinkkanen, "Gradient networks for universal magnetic modeling of synchronous machines," 2026, https://arxiv.org/abs/2602.14947 .. [#Das2026] Das, Martin, Kontodinas, Hinkkanen, "Direct force control of six-phase bearingless synchronous machines," Proc. ICEM, 2026, https://doi.org/10.1109/ICEM68283.2026.11757980 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.820 seconds) .. _sphx_glr_download_examples_plot_3kw_bsyrm_gradnet_maps.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_3kw_bsyrm_gradnet_maps.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_3kw_bsyrm_gradnet_maps.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_3kw_bsyrm_gradnet_maps.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_