MOTOR-25 — Efficiency and Torque-Speed Maps

FACT — A “97.5% peak efficiency” figure on an EV traction motor’s datasheet doesn’t directly determine the vehicle’s real-world range. Range is determined by how much time the motor spends at which torque-speed points during actual driving (stop-and-go city traffic, highway cruising, hill climbing) and the time-weighted average of the efficiency at those points. This lesson teaches how to read that map and the loss physics underneath it.

The previous lesson (MOTOR-24) covered mechanical limits. This lesson covers the motor’s performance map and the loss mechanisms that shape it:

1. Efficiency Contours in the T-N Plane: The “Sweet Spot” Island
2. Loss Breakdown I: Stator Copper Loss
3. Loss Breakdown II: Iron Loss (Hysteresis and Eddy-Current)
4. Loss Breakdown III: Rotor Magnet Eddy-Current Losses
5. Loss Breakdown IV: Bearing Friction and Windage Losses
6. Illustrative Loss-Breakdown Table
7. Cycle-Weighted Efficiency: WLTP and EPA

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1. Efficiency Contours in the T-N Plane: The “Sweet Spot” Island

A traction motor’s torque-speed (T-N) map defines the reachable operating envelope produced by the current/voltage limits from MOTOR-19 (MTPA, field weakening, MTPV): a constant peak-torque region at low speed, and a constant peak-power (CPSR) region above base speed.

The efficiency map overlays that same T-N plane with iso-efficiency contours — each contour line represents a set of points where the motor operates at the same efficiency (analogous to elevation contour lines on a topographic map).

FACT — In modern traction PMSMs, peak efficiency typically exceeds $97\%$, and this peak doesn’t sit at the exact center of the map — it appears as a small “sweet spot” island in the mid-torque, mid-speed region. In the map below you can see the contracting contours surrounding this island and how efficiency falls off toward the edges of the map.

Efficiency Map in the Torque-Speed Plane (conceptual):
  Torque
  (Nm)
   ^
   |‾‾‾‾‾‾‾‾╲                        Constant-Torque Region (Region I)
   |  90%     ╲___
   |    94%      ╲___  97%+ "sweet spot" island
   |      96%  ┌──╲──┐___
   |       97%+│ ●  │    ╲___        Constant-Power Region (Region II,
   |      96%  └────┘        ╲___     field weakening)
   |    94%                      ╲___
   |  90%                            ╲___
   +---------------------------------------> Speed (rpm)
   0        n_base                 n_max
   (High torque at low speed,  (Field weakening at high speed,
    high I → copper loss        high f_e → iron loss
    dominant)                    dominant)

INTERPRETATION — The island’s location isn’t accidental: motor designers deliberately align this “sweet spot” with the operating point the vehicle is expected to spend the most time at (typical highway/city cruising speed and torque demand) — this is the design-stage counterpart of the drive-cycle weighting we cover in Section 7.

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2. Loss Breakdown I: Stator Copper Loss

Total motor loss decomposes into four main components: copper loss, iron loss, magnet eddy-current loss, and mechanical (bearing + windage) loss. Each comes from a distinct physical mechanism and scales differently — which is why the dominant loss type shifts across different regions of the map.

Stator copper loss (Joule/ohmic loss):

$P_{Cu} = 3 \, I_{rms}^2 \, R_s$

where $I_{rms}$ is the RMS phase current and $R_s$ is the phase resistance (rises with temperature, see MOTOR-23).

CRITICAL — Copper loss scales with the square of current, and current is largely proportional to demanded torque ($T_e \propto I_q$, approximately, per the MTPA relation from MOTOR-19). Copper loss therefore dominates in the high-torque / low-speed region (the upper-left corner of the T-N map — launch, hill climbing, rapid acceleration). Even near zero speed, a large torque demand keeps copper loss high.

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3. Loss Breakdown II: Iron Loss (Hysteresis and Eddy-Current)

The time-varying magnetic flux in the stator and rotor iron core produces loss through two distinct mechanisms: hysteresis loss (the energy spent realigning the material’s magnetic domains each cycle) and eddy-current loss (the ohmic heating of circulating currents that the changing flux induces within the laminated steel).

The combined empirical scaling of these two losses is approximated by a Steinmetz-style expression:

$P_{Fe} \approx k_h \, f \, B^{n} + k_e \, f^2 \, B^2 \quad \Rightarrow \quad P_{Fe} \propto f^{1.6} \, B^2 \text{ (approximate combined scaling)}$

where $f$ is electrical frequency, $B$ is peak flux density, and $k_h$, $k_e$ are material-specific hysteresis/eddy-current coefficients. ($n \approx 1.6$-$2$ is the typical hysteresis exponent; the $f^{1.6}$ approximation in the combined expression represents the practically observed average behavior of summing the hysteresis and eddy-current terms, which individually scale as $f^1$ and $f^2$.)

CRITICAL — Iron loss scales strongly with frequency (hence motor speed, $f_e = \frac{p \cdot n}{120}$) and is independent of current/torque — iron loss continues even at zero current, as long as the magnets are spinning past the stator core (hence it is sometimes called “spin loss”). Iron loss therefore dominates in the low-torque / high-speed region (the lower-right corner of the T-N map — steady highway cruising, light load).

Together with Section 2, this explains the loss asymmetry across the T-N map: the map carries a diagonal “sweet spot” band because copper loss dominates the upper-left and iron loss dominates the lower-right; the middle region, where both are relatively low, yields the highest efficiency.

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4. Loss Breakdown III: Rotor Magnet Eddy-Current Losses

Rotor magnets (especially electrically conductive materials like sintered NdFeB) can also generate eddy-current loss internally. This loss stems from two harmonic disturbances:

1. Stator MMF spatial harmonics — a finite number of slots and a discrete winding layout produce a non-ideal, non-sinusoidal magnetomotive force (MMF) waveform; these spatial harmonics appear as a time-varying field in the rotor’s reference frame and induce eddy currents in the magnets.
2. PWM switching harmonics — the inverter’s pulse-width modulation injects high-frequency voltage/current harmonics well above the fundamental (typical carrier frequency $8$-$20\text{ kHz}$); these high-frequency components can produce disproportionately large eddy-current loss in the magnets, because loss scales with the square of frequency ($P_{mag} \propto f^2$).

FACT — Although magnet eddy-current loss is a small percentage of total loss (see the table in Section 6), its practical significance is disproportionately large: this loss turns directly into heat inside the magnet, and magnets risk permanent demagnetization at elevated temperature (see MOTOR-23). This is why magnets are typically divided into segments (magnet segmentation) to interrupt eddy-current paths — electrically isolated small pieces instead of one large magnet block.

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5. Loss Breakdown IV: Bearing Friction and Windage Losses

The final loss category is purely mechanical, with two sub-components:

• Bearing friction loss: arises from viscous friction in the bearings, scaling roughly linearly with speed ($P_{bearing} \propto n$).
• Windage (aerodynamic) loss: the aerodynamic/hydrodynamic drag the rotor encounters spinning through air (or coolant, in oil-cooled motors); scales with the cube of speed ($P_{windage} \propto n^3$).

CRITICAL — Because of the $n^3$ scaling, windage loss is negligible at low-to-moderate speeds but suddenly becomes significant at the extreme-speed class ($18,000$-$22,000\text{ rpm}$) covered in MOTOR-24. This explains why aerodynamically smoothing the rotor surface at high speed (an incidental benefit of the carbon-fiber sleeve) becomes a design goal.

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6. Illustrative Loss-Breakdown Table

The table below shows the loss distribution at a typical highway-cruise operating point (mid-high speed, mid-low torque — near the start of the CPSR region). These figures apply to one representative operating point only — the proportions shift entirely across different regions of the T-N map (as explained in Sections 2 and 3).

Loss Category	Physical Mechanism	Scaling	Representative Share (~)
Stator copper loss	$I^2 R$ Joule heating	$\propto I^2$	$40\%$
Iron loss (hysteresis + eddy)	Time-varying flux, domain shift + eddy currents	$\propto f^{1.6} B^2$	$35\%$
Rotor magnet eddy-current	MMF/PWM harmonics	$\propto f^2$ (harmonic component)	$10\%$
Bearing + windage (mechanical)	Viscous friction + aerodynamic drag	$\propto n$ and $\propto n^3$	$15\%$

EXAMPLE — Evaluate the same motor at low speed under full-torque launch (hill climbing) instead, and the table flips entirely: copper loss share rises to $65\%$-$75\%$ while iron loss share drops below $10\%$, and mechanical loss share nearly vanishes (low speed → low $n^3$). This is concrete evidence of why a full map — not a single efficiency number — is necessary.

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7. Cycle-Weighted Efficiency: WLTP and EPA

A single peak-efficiency figure is insufficient to estimate a vehicle’s real-world range — you need to know how much time/energy the motor spends at which T-N points over a standardized drive cycle (WLTP in Europe, EPA in the US).

The method works as follows:

1. The standard cycle’s speed-time profile (e.g., WLTP Class 3) is converted, via a vehicle dynamics model (mass, aerodynamic drag, rolling resistance, road grade), into required torque and speed at every instant.
2. This produces a “trajectory” the motor traces through the T-N plane over the course of the cycle — this trajectory is overlaid on the efficiency map.
3. The energy consumed in each segment of the trajectory is weighted by the local efficiency value at that segment and summed:

$\eta_{cycle} = \frac{\sum_i E_{out,i}}{\sum_i E_{out,i} / \eta_i} = \frac{\sum_i E_{out,i}}{\sum_i E_{in,i}}$

where $i$ indexes each time step of the cycle, $E_{out,i}$ is the mechanical output energy at that step, and $\eta_i$ is the local efficiency (read from the map) at that step.

CRITICAL — The result is a single number called cycle-weighted efficiency, and it is far more meaningful for real-world range than peak efficiency — because a motor hitting $97.5\%$ in one small corner of its T-N map contributes almost nothing to range if the vehicle rarely operates in that corner during daily use. Even a motor with a lower peak efficiency can post a higher cycle-weighted efficiency than a rival motor, if its “sweet spot” overlaps the region the cycle spends the most time in.

ASSUMPTION — Exactly which T-N map regions the WLTP Class 3 cycle’s urban/rural/highway sub-segments correspond to depends on vehicle mass, gear ratio, and tire radius; the trajectory concept presented in this lesson is a general methodology — a specific vehicle’s actual WLTP trajectory requires a separate simulation.

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8. Frequently Asked Questions (FAQ)

What is the efficiency map for?

FACT — It shows which operating point is efficient; it feeds directly into motor design, control algorithm tuning (MTPA/field-weakening transitions), and range estimation/simulation.

Where is the most efficient region, and why the middle?

FACT — Usually the mid-torque, mid-to-high-speed region, because neither copper loss (dominant at high torque) nor iron loss (dominant at high speed) dominates alone there — both are balanced at relatively low levels.

Why can peak efficiency and cycle-weighted efficiency differ?

FACT — Peak efficiency describes a single point on the map; cycle-weighted efficiency is the weighted average across ALL points the vehicle actually visits during real use. If the “sweet spot” location doesn’t overlap the driving profile, the two figures can diverge significantly.

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9. Summary

• The T-N efficiency map shows the motor’s “sweet spot” ($\ge 97\%$ in modern PMSMs) via iso-efficiency contours; this island typically sits in the mid-torque/mid-speed region.
• Copper loss scales with $I^2R$ and dominates at high torque/low speed; iron loss scales roughly as $f^{1.6}B^2$ (Steinmetz-style) and dominates at high speed/low torque.
• Rotor magnet eddy-current losses come from stator MMF and PWM harmonics; though a small loss share, they demand magnet segmentation due to demagnetization risk.
• Bearing friction scales with speed, windage loss with the cube of speed; the latter becomes significant only in the extreme-speed class covered in MOTOR-24.
• WLTP/EPA drive-cycle points are overlaid on the efficiency map to compute cycle-weighted efficiency — far more decisive for real-world range than peak efficiency alone.

10. Sources and Verification Note

• Lipo — Introduction to AC Machine Design (efficiency maps, loss mechanisms).
• Husain — Electric and Hybrid Vehicles: Design Fundamentals (drive cycle and efficiency relationship).
• Krishnan — Permanent Magnet Synchronous and Brushless DC Motor Drives (iron loss and magnet eddy-current loss modeling, Steinmetz scaling).

ASSUMPTION — The loss-breakdown table in Section 6 is an order-of-magnitude representative example for a typical modern PMSM traction motor; actual percentages vary significantly with motor geometry, magnet material, switching frequency, and cooling method. Verify against test/simulation data for a specific motor design before publication.

Next Lesson

• MOTOR-26 — Traction Motor: Operating Regions: How vehicle-level torque/power demand (launch, acceleration, steady-speed cruising, regenerative braking) maps onto the motor’s T-N map, and how vehicle performance targets translate into motor selection.