Normalized-Characteristic-Current-Based Methodology for Quantifying Active Winding Reconfiguration Benefits in Permanent Magnet Synchronous Machines

1. Introduction

Primary traction electric drives for electrified mobility must balance cost, sustainability, and high-speed performance under a fixed DC-link voltage constraint. Several rare-earth-free rotor solutions exist, including induction machines, synchronous reluctance machines (SynRM), permanent magnet (PM) assisted SynRM (PMaSynRM), and externally excited machines. Each introduces varying trade-offs with torque density, efficiency, volume, or thermal margin for reduced critical-material content [1,2,3]. Among these, ferrite-magnet permanent magnet synchronous machines (PMSMs) require a larger machine volume and higher inverter current rating to match rare-earth performance [4,5,6,7,8].

Active winding reconfiguration (AWR) has regained attention as a means to address this trade-off. Prior works cover reconfigurable hardware topologies, low-level reconfiguration control, and fault-tolerant operation, but design studies remain largely single-point demonstrations[9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Reported gains are substantial and span a wide-range, including 20 percent to 80 percent increases in torque or power density and up to 60 percent reductions in total losses [23]. Yet, there is a lack of clear design parameters or threshold ranges to predict this benefit without a full design study [23,24]. Identifying this benefit currently requires a full AWR design study. Conventional multi-objective optimization methods require numerous detail design evaluations [25]. While data-driven approaches reduce this evaluation count [26,27,28], such design studies benefit from analytical methods to further reduce computational effort.

A sensitivity study [29] found that weak rotor magnetic loading benefits more from AWR based on a single rotor design. However, the trend starts to reverse at very low magnetic loading and does not generalize further. Generalized design-space explorations have instead used normalized parameter-plane (NPP) models based on the saliency–flux-linkage NPP model [30,31,32]. Works in [12,33,34,35,36,37] identify regions of AWR improvement and discuss design boundaries along with validation against models and measurements considering saturation effects. However, the boundaries identified shift based on the input parameters, requiring iterative solving or lookup tables for each design [37]. In contrast, normalized characteristic current ($\widehat{I_{\mathrm{ch}}}$) based NPP model [38] expresses a design-independent, closed-form boundary at $\widehat{I_{\mathrm{ch}}}=1$. This threshold value represents theoretical infinite CPSR capability. Since AWR benefit is fundamentally governed by field-weakening capability, this type of NPP model is well suited for extension to AWR topologies, though [38] did not originally address AWR.

One study [39] extended the $\widehat{I_{\mathrm{ch}}}$ based NPP model to AWR by introducing a third parameter, normalized field-weakening current, for series-parallel reconfiguration topology. However, the study omitted the effect of turns ratio on PM flux linkage, as otherwise expected [12,35]. The same single threshold from [38], $\widehat{I_{\mathrm{ch}}}=1$, is identified qualitatively, without any figure of merit (FoM) to quantify torque or power gain over the CPSR. Validation is limited to one design at $\widehat{I_{\mathrm{ch}}}=4$, a machine with inherently narrow field-weakening. The recommendation is that machines with narrow field-weakening (large $\widehat{I_{\mathrm{ch}}}$ and low saliency) primarily benefit from AWR and contradicts the findings in [29]. Broader case-study evidence [23] also indicate substantial AWR benefit for machines with wide field-weakening and CPSR capability ($\widehat{I_{\mathrm{ch}}}\approx 1$) [38,40,41,42].

The gap therefore remains threefold: an AWR-extended $\widehat{I_{\mathrm{ch}}}$ based NPP model that accounts for the turns-ratio effect on PM flux linkage, expression of reconfiguration benefit through quantified FoMs, and validation across the wide-field-weakening region near $\widehat{I_{\mathrm{ch}}}\approx 1$.

The goals of this work are:

1.

Extend the two-parameter NPP model from [38] with turns-ratio effect to obtain a geometry-independent analytical model that predicts the torque-speed-power behavior of AWR topologies over the design space.

2.

Derive a reduced set of FoMs from this model that quantify the benefit of AWR reconfiguration.

3.

Identify explicit parameter thresholds that separate AWR-favorable from AWR-unfavorable design regions.

2. Materials and Methods

2.1. Active Winding Reconfiguration Topologies and Notation

Most conventional traction machines use a three-phase winding driven by a two-level voltage source inverter (VSI) with one end of the phases joined at a common neutral point, i.e., in a star (${\rm Y}$) configuration. Actively reconfiguring the windings from this state changes the voltage utilization and field-weakening capability. The following topologies are considered in this work, as summarized in Table 1.

• Star (${\rm Y}$): The anchor ($anc$) configuration; the three phases are joined at a common neutral point, driven by a single inverter.
• Delta ($\Delta$): The three phases form a closed triangular loop, requiring a minimum of five switches [23,43]. Relative to ${\rm Y}$, this increases effective phase voltage by $\sqrt{3}$ and reduces effective phase current by $1/\sqrt{3}$.
• Parallel ($\Pi$): Each phase winding is reconnected from series to parallel, giving an effective turns ratio $k_{\mathrm{t}}=1/2$ relative to ${\rm Y}$; current per parallel branch is halved while the induced voltage is unchanged. This typically requires nine switches [23].

Figure 1. Machine winding topologies considered in this work connected to a two-level voltage source inverter.

Figure 1. Machine winding topologies considered in this work connected to a two-level voltage source inverter.

2.2. Normalized Parameter-Plane Model

Following standard assumptions apply to the simplified d-q model representation of IPMSM machines [30,32,35,38]:

1.

Lossless machine: stator (phase) resistance and all loss terms (iron loss, friction, leakage etc.) are set to zero.

2.

Fundamental-wave, sinusoidal operation: higher harmonics are neglected, and sinusoidal quantities are assumed.

3.

No cross-coupling: cross-coupling is neglected. ${\Psi }_{\mathrm{d}}$ and ${\Psi }_{\mathrm{q}}$ are solely dependent on $I_{\mathrm{d}}$ and $I_{\mathrm{q}}$, respectively.

4.

Linear magnetic properties: saturation effects and temperature-dependent changes of PM flux are neglected.

5.

Steady-state operation: time-derivative terms in the voltage equations are zero.

The normalized voltage and torque equations, derived from the established absolute d-q model is described in [38], with normalization base quantities reproduced in Appendix A, as used in this work are:

$$\begin{array}{cc}\hfill \widehat{U_{\mathrm{d}}}& =-\widehat{{\Psi }_{\mathrm{q}}}\widehat{{\omega }_{\mathrm{e}}}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \widehat{U_{\mathrm{q}}}& =+\widehat{{\Psi }_{\mathrm{d}}}\widehat{{\omega }_{\mathrm{e}}}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \widehat{{\Psi }_{\mathrm{d}}}& =\widehat{L_{\mathrm{d}}}\widehat{I_{\mathrm{d}}}+\widehat{{\Psi }_{\mathrm{PM}}}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \widehat{{\Psi }_{\mathrm{q}}}& =\widehat{L_{\mathrm{q}}}\widehat{I_{\mathrm{q}}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \widehat{T_{\mathrm{e}}}& =\widehat{{\Psi }_{\mathrm{PM}}}\widehat{I_{\mathrm{q}}}+\left(\widehat{L_{\mathrm{d}}}-\widehat{L_{\mathrm{q}}}\right)\widehat{I_{\mathrm{d}}}\widehat{I_{\mathrm{q}}}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \widehat{P}& =\widehat{T_{\mathrm{e}}}\widehat{{\omega }_{\mathrm{e}}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \widehat{I_{\lim }}& =\sqrt{{\widehat{I_{\mathrm{d}}}}^2+{\widehat{I_{\mathrm{q}}}}^2}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \widehat{U_{\lim }}& =\sqrt{{\widehat{U_{\mathrm{d}}}}^2+{\widehat{U_{\mathrm{q}}}}^2}\hfill \end{array}$$

(8)

Voltage limit equations show an ellipse whose d- and q-axes diameters are inversely proportional to the product of machine speed ${\omega }_{\mathrm{e}}$ and the respective inductances, $L_{\mathrm{d}}$ and $L_{\mathrm{q}}$. The center of these ellipses is given by the characteristic current, $I_{\mathrm{ch}}$, and the ratio of the q- and d-axis inductances defines the saliency, $\xi$, of the machine.

$$\begin{array}{cc}\hfill \xi & ={\displaystyle \frac{\widehat{L_{\mathrm{q}}}}{\widehat{L_{\mathrm{d}}}}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \widehat{I_{\mathrm{ch}}}& =-{\displaystyle \frac{\widehat{{\Psi }_{\mathrm{PM}}}}{\widehat{L_{\mathrm{d}}}}}\hfill \end{array}$$

(10)

The normalization of any given reference drive and reference machine is made with the drive current and voltage limits set to one. The base speed of the reference machine, defined as the speed at which the voltage and current limits of the reference drive under MTPA control are reached, is normalized to one. With this convention, the reference drive and machine transitions from current-limited operation to FW at this normalized speed value of one. This convention is consistent with standard parameter-plane formulations [30,31,32,35,42] and coincides with the method of normalization in [38].

2.2.1. Significance of Normalized Characteristic Current

Previous research established normalized parameter-plane models for describing the FW capability of IPMSM. In this work, the formulation based on normalized characteristic current $\widehat{I_{\mathrm{ch}}}$ and saliency ratio $\xi$ is used. The characteristic current is a fundamental machine parameter and forms a key boundary recognized for its key role in determining field weakening performance [38,40,41,44]. In its normalized form (i.e., $\widehat{I_{\mathrm{ch}}}$), values above one indicate that the voltage limit ellipse intersect is outside the current limit circle, so the machine becomes speed-limited, whereas values below one allow continued field weakening toward higher speed with diminishing torque. Saliency $\xi$ modifies the shape of the voltage-limit ellipse and therefore the effectiveness of current utilization in FW; higher saliency reduces the d-axis current needed to remain within the voltage limit and improves torque-per-ampere output.

The characteristic current represents the ratio of the magnet flux to the d-axis inductance. When normalized against maximum stator current limit, it points to the machine’s relative magnetic loading and current loading, both fundamental design parameters in preliminary, analytical machine design.

The characteristic current also equals the balanced three-phase steady-state short-circuit current of the machine at high-speeds under idealized assumptions. $\widehat{I_{\mathrm{ch}}}$ placed near the maximum current limit therefore implies that a stator short circuit alone can approach the magnet’s demagnetizing field. Reduced- and rare-earth-free magnet designs, with lower demagnetization resistance, thus often target $\widehat{I_{\mathrm{ch}}}$ below this limit, most notably in ferrite-assisted synchronous reluctance machines. $\widehat{I_{\mathrm{ch}}}$ thus also captures the trade-off between demagnetization resistance and FW capability.

Together, $\widehat{I_{\mathrm{ch}}}$ and $\xi$ provide a physically grounded way to locate traction-machine designs in the normalized parameter plane. In this plane, most wide-CPSR machines, including those in automotive traction use, typically occupy the region with characteristic current around the rated maximum current limit (i.e., $0.8<\widehat{I_{\mathrm{ch}}}<1.2$) and a saliency, $\xi >1$ [38,40,44].

2.3. Extension of the $\widehat{I_{\mathrm{ch}}}$–$\xi$ NPP Model to Active Winding Reconfiguration

The equivalent effect of winding reconfiguration on the voltage, current and torque equations is represented by a factor, $k_{\mathrm{w},\mathrm{rec}}$. The anchor state (anc) is the ${\rm Y}$ configuration, and the reconfigured states (rec) are the $\Delta$ and $\Pi$ configurations.

The peak phase MMF ($\mathcal{F}$) of the anchor configuration is defined as:

$$\begin{array}{cc}\hfill {\widehat{\mathcal{F}}}_{\mathrm{anc}}& =N_{\mathrm{t}}·\widehat{I_{\lim }}\hfill \end{array}$$

(11)

where $N_{\mathrm{t}}$ is the number of series turns per phase and $\widehat{I_{\lim }}$ is the normalized inverter current limit.

The peak phase MMF ($\widehat{\mathcal{F}}$) of the reconfigured state is then defined as:

$$\begin{array}{c}\hfill {\widehat{\mathcal{F}}}_{\mathrm{rec}}={\displaystyle \frac{{\widehat{\mathcal{F}}}_{\mathrm{anc}}}{k_{\mathrm{w},\mathrm{rec}}}}\end{array}$$

(12)

The turns-ratio also affects the flux linkage and the inductances. The underlying reluctance and the PM flux remain unchanged , but the effect of the turns ratio appears in the flux linkage and in the inductance: the flux linkage scales proportionally with the turns ratio, while the inductance scales quadratically [12,35,43].

$$\begin{array}{cc}\hfill k_{\mathrm{t}}& ={\displaystyle \frac{N_{\mathrm{t},\mathrm{rec}}}{N_{\mathrm{t},\mathrm{anc}}}}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\Psi }_{\mathrm{PM}},\mathrm{rec}& =k_{\mathrm{t}}·{\Psi }_{\mathrm{PM}},\mathrm{anc}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill L_{\mathrm{d}},\mathrm{rec}& =k_{\mathrm{t}}^2·L_{\mathrm{d}},\mathrm{anc}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill L_{\mathrm{q}},\mathrm{rec}& =k_{\mathrm{t}}^2·L_{\mathrm{q}},\mathrm{anc}\hfill \end{array}$$

(16)

For any configuration, the onset of field weakening is the point at which the induced voltage reaches the voltage limit. The reconfiguration factor can be used to relate the voltage limit of the reconfigured state to that of the anchor state as follows:

$$\begin{array}{cc}\hfill {\widehat{U}}_{\mathrm{ind}},\max & =\widehat{U_{\lim }}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\widehat{U}}_{\lim ,\Delta }& =k_{\mathrm{w},\mathrm{rec},\Delta }·{\widehat{U}}_{\lim ,{\rm Y}}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\widehat{U}}_{\mathrm{ind},\max ,\Pi }& =k_{\mathrm{w},\mathrm{rec},\Pi }·{\widehat{U}}_{\mathrm{ind},\max ,{\rm Y}}\hfill \end{array}$$

(19)

For the $\Delta$ configuration, $k_{\mathrm{w},\mathrm{rec}}$ is the ratio of the current and voltage limits between line and phase quantities. For the $\Pi$ configuration, $k_{\mathrm{w},\mathrm{rec}}$ is the inverse of the turns-ratio between the reconfigured and anchor windings. The flux linkage, inductances, the voltage- and current-limits collectively make up the torque-speed equations in eq:normdq. Thus, the $k_{\mathrm{w},\mathrm{rec}}$ factor is incorporated into those equations using these relationships, as summarized in Table 2.

Analysis is then carried out pointwise in the $\left(\widehat{I_{\mathrm{ch}}},\xi \right)$ plane. Each point representing the parameter pair defines an anchor machine in the ${\rm Y}$ configuration. For each anchor machine, the AWR topologies defined in Section 2.1 are evaluated as alternative reconfigured operating states of the underlying anchor machine by applying topology-dependent changes in current limit, voltage limit, or effective turns per phase, as summarized in the Table 2. This interpretation reflects the physical premise of AWR: the winding connection is reconfigured during operation whereas the machine geometry and material system remain unchanged.

The normalization established for the ${\rm Y}$ configuration remains applicable across all reconfigured states. Thus, keeping a normalized circle diagram from the ${\rm Y}$ configuration as the anchoring reference, MTPA and MTPV loci are generated for each topology, along with the resulting torque–speed envelopes, following the methodology in [38]. The corner-speed of the anchoring ${\rm Y}$ configuration always appears at the normalized value of unity, whereas the corner-speed of the reconfigured topologies appears at different normalized values.

The resulting comparison at each point in the parameter plane represents the effect of reconfiguration on one underlying machine design. This avoids renormalization per topology and ensures that all the corresponding torque–speed envelopes of the underlying machine design are directly comparable.

The comparison of torque–speed envelopes at selected $\widehat{I_{\mathrm{ch}}}$–$\xi$ value pairs (Figure 3) provides only a preliminary view of the effect of reconfiguration. In the next section, scalar figures of merit are defined to systematically identify beneficial design regions across the entire parameter plane and to facilitate machine-level benchmarking.

2.4. Figures of Merit for Evaluating Reconfiguration Benefit

Figures of merit (FoMs) are defined to systematically identify thresholds and design-rules across the parameter plane.

A summary of the different FoMs are given in Table 3 and Table 4, while Figure 2 illustrates the reference points used in their definitions. The FoMs are primarily grouped in to four types.

1.

Maximum performance

2.

Crossover point

3.

CPSR and field-weakening range

4.

Part-load performance

2.5. Baseline Traction-Machine Set and Finite-Element Models

The results from the analytical extended NPP model, and the FoMs derived from it, are evaluated using numerical models of for representative automotive traction machines. Using detailed cross-sectional geometry, FEA models can provide sufficient fidelity in capturing nonlinear, saturation-dependent electromagnetic behavior across the full operational range. The evaluation is to quantify the correlation of important trends between the analytical model and the numerical models, and to identify any limitations of the analytical model. This section describes in brief the approach to finite-element-analysis (FEA) of the four representative automotive traction machines and two benchmark virtual designs of automotive traction machines (Table 5).

2.5.1. FEM Modeling

Electromagnetic modeling and FEA of the baseline machines is performed with the commercial software Ansys Motor-CAD [45] using its 2D transient finite-element-method solver. The geometry of machines M1–M3 is reconstructed after direct measurement from the machines’ active parts. Whereas, the geometry and model parameters of M4–M6 are derived from manufacturer design data. Background information on the FEA modeling approach and the assumptions made in the modeling of the baseline machines is provided in Appendix C.

2.6. Measurement of Machine M3

Machine M3, the device under test (DUT) is an open-end winding machine that can be reconfigured between ${\rm Y}$ and $\Delta$ configurations. It was measured on a dynamometer test bench to validate the FEA model and to provide experimental data for comparison with the analytical model. The machine was operated using field-oriented control, while the dynamometer imposed the speed setpoint. Manual reconfiguration between ${\rm Y}$ and $\Delta$ was performed using the terminal box shown in Figure 4. Shaft torque, shaft speed, three-phase line currents and voltages were recorded using the measurement equipment summarized in Appendix B.

3. Results

Three thresholds of $\widehat{I_{\mathrm{ch}}}$ are identified in the resulting FoMs across the parameter plane and marked in the subsequent figures.

1.

Threshold $t_1$: $\widehat{I_{\mathrm{ch}}}=1/k_{\mathrm{w},\mathrm{rec}}$

2.

Threshold $t_2$: $\widehat{I_{\mathrm{ch}}}=1$

3.

Threshold $t_3$: $\widehat{I_{\mathrm{ch}}}={\displaystyle \frac{k_{\mathrm{w},\mathrm{rec}}}{\sqrt{k_{\mathrm{w},\mathrm{rec}}^2-1}}}$

3.1. Maximum Performance FoMs

The inverse torque ratio in Figure 5 transitions between two limits. For $\xi =1$, torque depends only on $\widehat{I_{\mathrm{q}}}$. Consequently, the inverse torque ratio equals $k_{\mathrm{w},\mathrm{rec}}$. For $\widehat{I_{\mathrm{ch}}}=0$, the optimum occurs at $\widehat{I_{\mathrm{d}}}=\widehat{I_{\mathrm{q}}}$. As a result, torque becomes proportional to the square of the current magnitude and the inverse torque ratio becomes $k_{\mathrm{w},\mathrm{rec}}^2$.

Peak power and its FoM,${\kappa }_{P,max}$, in Figure 5 shows three distinct regions of behavior with respect to $\widehat{I_{\mathrm{ch}}}$ and $k_{\mathrm{w},\mathrm{rec}}$:

• For $\widehat{I_{\mathrm{ch}}}=0$, reconfiguration only shifts the speed at which maximum power occurs, resulting in ${\kappa }_{P,max}=1$.
• For $\widehat{I_{\mathrm{ch}}}\le 1/k_{\mathrm{w},\mathrm{rec}}$, the reconfigured state can raise maximum power, and ${\kappa }_{P,max}\simeq k_{\mathrm{w},\mathrm{rec}}$ for $\xi =1$. Machines with $\xi >1$ show improvement of more than 40% in peak power.
• For $\widehat{I_{\mathrm{ch}}}>1/k_{\mathrm{w},\mathrm{rec}}$, the benefit vanishes towards ${\kappa }_{P,max}\le 1$ at a threshold of $\widehat{I_{\mathrm{ch}}}\simeq 1$, and stays below unity for higher $\widehat{I_{\mathrm{ch}}}$.

3.2. Crossover Point FoMs

The crossover FoMs identify where the reconfigured envelope begins to dominate the anchor envelope and the voltage stress associated with such a mode transition. Values of ${\kappa }_{u,x,1}$ above unity indicate that the open-circuit induced voltage exceeds the available DC-link voltage. Under a short-circuit condition, this can lead to a risk of uncontrolled generating operation if no active control is applied. In an open-circuit condition, the resulting terminal voltage must remain below the blocking-voltage rating of the inverter switching devices, as observed in [35].

Increasing $\xi$ improves the anchor-state field-weakening capability and therefore shifts the crossover to higher speed, while the dependence on $\widehat{I_{\mathrm{ch}}}$ produces a ridge near $\widehat{I_{\mathrm{ch}}}\approx 1$. Below this region, the reconfigured state extends useful high-speed operation; above it, the field-weakening benefit weakens and the crossover speed decreases. The voltage-stress factor follows the same ridge qualitatively, but is moderated by the reduction in $\widehat{{\Psi }_{\mathrm{PM}}}$ with increasing $\xi$ in the normalized model.

3.3. Constant-Power over Speed Range FoMs

The impact of the characteristic current $\widehat{I_{\mathrm{ch}}}$ on the FW capability, and hence on CPSR and the torque–speed envelope FoM, as seen in Figure 6, is established with three thresholds defined by $\widehat{I_{\mathrm{ch}}}$ and $k_{\mathrm{w},\mathrm{rec}}$:

• For $\widehat{I_{\mathrm{ch}}}\le 1/k_{\mathrm{w},\mathrm{rec}}$, both ${\kappa }_{\mathrm{CPSR},\mathrm{env}}$ and ${\kappa }_{T,\mathrm{env}}$ increase with $\widehat{I_{\mathrm{ch}}}$. However, the gain decreases with increasing $\xi$, since $\xi$ improves the anchor state’s own CPSR. Machines with moderate $\xi$ show improvements of up to four times in CPSR. Machines with low $\xi$ and low $\widehat{I_{\mathrm{ch}}}$ show the largest improvement in ${\kappa }_{T,\mathrm{env}}$ of more than 20% over the anchor state.
• For $1/k_{\mathrm{w},\mathrm{rec}}<\widehat{I_{\mathrm{ch}}}\le k_{\mathrm{w},\mathrm{rec}}/\sqrt{k_{\mathrm{w},\mathrm{rec}}^2-1}$, the reconfigured state’s finite maximum speed limits its advantage, and both ratios remain close to unity.
• For $\widehat{I_{\mathrm{ch}}}>k_{\mathrm{w},\mathrm{rec}}/\sqrt{k_{\mathrm{w},\mathrm{rec}}^2-1}$, both states are speed-limited with a narrow CPSR; the ratios rise slightly above unity, but the absolute improvement stays limited.

The composite envelope FoMs show that the largest AWR benefit occurs when reconfiguration moves the machine from a voltage-limited anchor state into a more favorable high-speed operating regime. This confirms that AWR should be evaluated by the composite envelope rather than by peak values alone.

3.4. Part-Load Performance FoMs

The FoMs described so far characterize performance along the rated torque–speed envelope only. To address part-load performance, two additional FoMs, ${\overline{\kappa }}_{PF}$ and ${\overline{\kappa }}_{Cu}$, are defined based on indicative power factor (PF) and copper losses, respectively. Though stator ohmic losses were neglected in the derivation of the normalized voltage and torque equations, a representation of these losses is useful for design analysis and is post-processed from the analytical model results [38].

Both FoMs are area-weighted means of the pointwise difference between the reconfigured and anchor states, normalized over the common feasible operating region ${\Omega }_{\mathrm{common}}$; positive values indicate that reconfiguration is favorable. ${\overline{\kappa }}_{PF}$ captures the reduction in apparent power demand on the inverter for a given torque and speed, while ${\overline{\kappa }}_{Cu}$ captures the improvement in part-load efficiency through reduced stator ohmic (DC) loss.

Figure 6 shows ${\overline{\kappa }}_{PF}$ in the base-speed region improves with $\widehat{I_{\mathrm{ch}}}$ but degrades with $\xi$, whereas in deep FW the trend reverses. Since ${\Omega }_{\mathrm{common}}$ is dominated by the base-speed region, and predominantly by that of the reconfigured state, ${\overline{\kappa }}_{PF}$ follows the base-speed trend: it decreases with $\xi$ and increases with $\widehat{I_{\mathrm{ch}}}$.

Reconfiguration reduces phase current in the $\Delta$ state and the resistance $R_{\mathrm{eff}}$ in the $\Pi$ state. As copper loss scales with $I^2{R}_{\mathrm{eff}}$, the advantage is largest near the current limit and shrinks at part load. ${\overline{\kappa }}_{Cu}$ is further constrained by the definition of ${\Omega }_{\mathrm{common}}$ itself: for $\widehat{I_{\mathrm{ch}}}\le 1/k_{\mathrm{w},\mathrm{rec}}$, a large fraction of the reconfigured envelope lies outside ${\Omega }_{\mathrm{common}}$. Therefore, the advantage is only partially captured, whereas at higher $\widehat{I_{\mathrm{ch}}}$ the overlap grows and the copper-loss reduction is more fully reflected in ${\overline{\kappa }}_{Cu}$.

3.5. Verification on Baseline Machines

The baseline machines M1–M6 (Table 5) are used to compare nonlinear FEA results against the extended NPP model at matched normalized design points. Measurement results of machine M3 are additionally included. The original winding connection is treated as the ${\rm Y}$ anchor, and the $\Delta$ state is obtained by reconfiguring the FEA models and the test machine M3 while holding geometry, materials, temperatures, system voltage and current limits as fixed boundary conditions. The torque-speed curves from the FEA results and measurement results are normalized in the same normalization framework as the analytical model as from [38], restated in Appendix A. Corner speed of the anchor configuration is taken as the speed normalization reference (${\omega }_{\mathrm{m},0}$). Torque is normalized to the corner point speed with unity power factor electrical power ($T_0$). Whereas the FEA-derived normalized characteristic current ($\widehat{I_{\mathrm{ch}}}$) and corner-point $\xi$ locate each machine on the normalized parameter plane to generate the corresponding analytical envelopes and FoMs.

The envelopes in Figure 7 show the largest discrepancies in the base-speed region, with good agreement in field weakening, consistent with prior literature [32,38,42]. The analytical model underpredicts peak torque for the reconfigured state, particularly at low $\widehat{I_{\mathrm{ch}}}$ and high $\xi$ (Figure 9, machine M6), because the constant-inductance NPP model cannot capture current-dependent saturation resolved by FEA, an effect also reported in other works utilizing the analytical NPP model [38,42,46]. The measurement results from machine M3 show good agreement with the FEA results in field weakening region for both configurations. For the anchor configuration, the maximum torque is in between the analytical and FEA results, while for the reconfigured configuration, the maximum torque matches the FEA results.

Higher $\xi$ amplifies this nonlinearity (Figure 9), therefore, the analytical model consistently overestimates ${\kappa }_{T,c}^{-1}$ at low $\widehat{I_{\mathrm{ch}}}$ and high $\xi$, resulting in the predicted crossover speed being lower than in FEA.

Despite this, Figure 8 shows strong correlation for system-level FoMs (composite CPSR, envelope expansion, peak-power ratio, crossover voltage-stress factor, relative copper-loss improvement). Agreement is weakest for the corner-point torque ratio and inverter utilization FoMs, reflecting their sensitivity to nonlinear saturation and making ${\kappa }_{T,c}$ the least reliable metric under the analytical model’s constant-inductance assumption. Data from the FoMs for machine M3 are provided in Table 6 and for all the machines in Appendix D.

4. Discussion

The relevant transition regions in the normalized design space occur at three characteristic levels:

1.

$\widehat{I_{\mathrm{ch}}}\le {\displaystyle \frac{1}{k_{\mathrm{w},\mathrm{rec}}}}$ marks the region of maximum CPSR and torque–speed envelope improvement.

2.

$\widehat{I_{\mathrm{ch}}}\approx 1$ is where maximum-power improvement and crossover voltage stress become critical.

3.

$\widehat{I_{\mathrm{ch}}}\approx {\displaystyle \frac{k_{\mathrm{w},\mathrm{rec}}}{\sqrt{k_{\mathrm{w},\mathrm{rec}}^2-1}}}$ is where torque–speed envelope improvement is least.

Saliency $\xi$ modifies the magnitude of the predicted benefit and shifts the crossover behavior, but does not change the dominant role of $\widehat{I_{\mathrm{ch}}}$ in the normalized design space. The key implications are summarized in Table 7. Based on the FoMs several design recommendations can be made for automotive traction machines with reconfigurable windings:

• Low ($\widehat{I_{\mathrm{ch}}}$) designs benefit most from reconfiguration in terms of CPSR and torque–speed envelope expansion. The maximum benefit occurs up to a threshold value of $\widehat{I_{\mathrm{ch}}}<{\displaystyle \frac{1}{k_{\mathrm{w},\mathrm{rec}}}}$. This observation motivates a design approach of reducing magnetic loading relative to current-loading, which in turn, can move the machine into the low-$\widehat{I_{\mathrm{ch}}}$ region where reconfiguration yields stronger high-speed benefits. Following designs benefit most:

  SynRM

  designs without PM

  PMaSynRM

  designs with weaker magnets and higher current loading, and

  IPMSM

  designs with significantly reduced rare-earth magnet content.

  SPMSM

  designs with weaker magnets and higher current loading.

  In all four cases, conventional designs required a trade-off with high speed performance and low-end torque. Reconfiguration significantly enhances the machine’s high-speed performance and extends its operational range while saving on magnet material costs. The torque ratio from FEA models suggest availability of around $k_{\mathrm{w},\mathrm{rec}}$ times the torque in the anchor state as compared to the reconfigured state. This means the benefit in high-speed FoMs is achieved without compromising the low-speed torque requirements for traction applications.
• Designs with $\widehat{I_{\mathrm{ch}}}\approx 1$, which make up most conventional rare-earth magnet based traction IPMSM designs, show limited gains in CPSR and torque–speed envelope from reconfiguration. However, they benefit in part-load efficiency and inverter loading, particularly during high-speed field-weakening operation. Thus, the motivation here is that strong-magnet machines suffer higher losses in field weakening. Reconfiguration mitigates these losses by optimizing the operating point. An alternative design approach is to reduce magnetic strength and material cost to utilize the potential of reconfiguration as mentioned in the previous point.
• For machines with high $\widehat{I_{\mathrm{ch}}}$, i.e., with high magnetic strength and low current loading with limited field-weakening capability, reconfiguration may not improve the rated maximum power. It does, however, improve the torque–speed envelope and part-load efficiency, thereby providing a viable route to extend speed range despite limited field-weakening capability.
• Choice between reconfiguration pairs: The absolute improvement in high speed performance and torque–speed envelope FoMs is higher for ${\rm Y}$–$\Pi$ reconfiguration pair than the ${\rm Y}$–$\Delta$ pair. This follows from the larger effective reconfiguration factor $k_{\mathrm{w},\mathrm{rec}}$ of the ${\rm Y}$–$\Pi$ pair. However, following aspects favor ${\rm Y}$–$\Delta$ pair:

  1.

  The crossover speed is lower for the ${\rm Y}$–$\Delta$ pair, hence, the benefits are already available at lower speeds.

  2.

  The crossover voltage-stress is lower for the ${\rm Y}$–$\Delta$ pair, hence, the risk of mode-shift is lower.

  3.

  The threshold value for CPSR and torque–speed envelope improvement is inversely proportional to $k_{\mathrm{w},\mathrm{rec}}$. More machine designs are covered by the ${\rm Y}$–$\Delta$ pair than the ${\rm Y}$–$\Pi$ pair.

5. Conclusions

This work extended the characteristic-current-based normalized parameter-plane model to active winding reconfiguration by introducing topology-dependent reconfiguration factors for the ${\rm Y}$–$\Delta$ and ${\rm Y}$–$\Pi$ configurations. The resulting model keeps the ${\rm Y}$ machine as the normalization anchor and predicts the torque-speed-power behavior of AWR topologies over the $\widehat{I_{\mathrm{ch}}}$–$\xi$ design space.

The normalized characteristic current is the central design parameter because it captures the relative strength of PM flux linkage, or magnetic loading, with respect to current loading. Together with $\xi$ and $k_{\mathrm{w},\mathrm{rec}}$, it gives physically meaningful thresholds that separate favorable and unfavorable regions for AWR. The figures of merit defined in this work quantify reconfiguration benefit across peak performance, crossover behavior, constant-power speed range, and part-load performance.

Comparison with non-linear FEA results for six representative traction-machine designs, and with measurement results from one machine (M3), confirmed that the analytical model captures the main system-level trends. The extended NPP model is therefore suitable for early-stage design screening and for identifying machine concepts that can benefit from reconfiguration. Final performance prediction still requires nonlinear FEA due to saturation-dependent inductance effects influencing the torque ratio and power-factor metrics.