Parametric Modeling of Mid-Drive PMSMs for Electric Vehicles: Scaling Laws, Thermal Networks, and Design Application

1. Introduction

Mid-drive permanent magnet synchronous motors (PMSMs) have become the dominant choice for light electric vehicles (LEVs) — from e-bikes to electric scooters and cargo bikes.

By placing the motor at the crank and driving the wheel through the existing bicycle drivetrain, the motor always operates near its efficiency sweet spot regardless of terrain or rider speed. This contrasts with hub motors, which are locked to wheel speed and can be pushed into low-efficiency regions on hills or at very low speeds.

Yet for system-level designers who need to model, simulate, or compare multiple motor options, a fundamental problem exists: you rarely have access to the internal geometry of a commercial mid-drive unit.

Manufacturers do not publish winding cross-sections, stator lamination geometries, or magnet dimensions. Without this, traditional finite element analysis (FEA) is impossible.

This paper solves that problem with a geometry-independent parametric model for mid-drive PMSMs in the 240–1000 W class.

Every key parameter — resistance, inductance, flux linkage, inertia — is expressed as a function of one number: the motor’s rated power P_n.

The scaling laws are derived from log-log regression on a dataset of commercial motors spanning the full power range.

The result is a single unified model that can represent any motor in its class by changing one input. A full worked numerical example using the TO7Motor DM02 at 500W is provided in Section 8 to demonstrate the framework in practice.

2. Electromagnetic Formulation

The motor is modeled in the synchronous dq reference frame — a standard mathematical transformation that converts the three rotating AC phase voltages into two steady DC-like quantities (d-axis and q-axis).

This makes the equations far simpler to work with and enables straightforward torque control implementation in MATLAB/Simulink.

The four key parameters — phase resistance R, d-axis inductance L_d, q-axis inductance L_q, and permanent magnet flux linkage λ_m — are all treated as functions of rated power P_n via the scaling laws in Section 6.

2.1 Stator Voltage Equations

2.2 Control Utilization Factor and Torque

In a real drive system the motor is bounded by three hard operating limits: maximum inverter voltage V_max, maximum phase current I_max, and a thermal derating function f_derate(T_w) that reduces torque as winding temperature approaches its limit.

These are combined into a single scalar control utilization factor γ_c — a standard implementation found in all field-oriented control (FOC) motor drives:

3. Mechanical Dynamics and Transmission

Mid-drive motors include an internal reduction stage — typically a planetary gearset — before the output sprocket.

Rotor speed and output shaft speed are therefore different, and the wheel torque is amplified by the gear ratio. Two equations govern this behavior.

Table I — Typical Mechanical Parameter Values for Commercial Mid-Drive Units

4. Comprehensive Loss Modeling

Losses are not just wasted energy — they are the heat sources that drive the thermal model.

Getting the loss decomposition right is what allows the electrical and thermal domains to remain energy-consistent. Total losses are split into four components:

4.1 Copper Losses

Copper loss is the heat generated by current flowing through the winding resistance.

Resistance rises with temperature, creating a positive feedback loop: more current → more heat → higher resistance → even more heat. The model captures this with temperature-dependent resistance:

4.2 Core Losses — Modified Steinmetz Model

Core losses in the stator iron are modeled using the modified Steinmetz equation, separating hysteresis, classical eddy-current, and anomalous excess loss contributions:

Table II — Typical Steinmetz Coefficients for Common Motor Lamination Steel Grades

4.3 Mechanical and Stray Losses

5. Three-Node Thermal Network

A single-node thermal model cannot capture the temperature gradient between the hottest point (winding copper) and the cooler outer housing.

In a PMSM, the winding hotspot temperature T_w is the critical reliability limit — exceeding it causes insulation degradation and magnet demagnetization. Three thermal nodes are used: winding (T_w), stator core (T_s), and housing (T_h).

5.1 Nodal ODEs

5.2 Steady-State Hotspot Expression

Table III — Typical Thermal Resistance Values for Mid-Drive PMSMs (Natural Convection)

6. Parametric Scaling Laws

All four key parameters use a single power-law form referenced to a 250W baseline motor:

6.1 Calibration Dataset

The exponents are derived from log-log regression on an expanded dataset of commercial motors. The dataset has been deliberately expanded beyond the original 17-motor set to improve power-range coverage and reduce clustering at 250W — a known weakness of earlier versions of this framework. TO7Motor mid-drive and hub motor data has been incorporated as it provides well-documented specifications across a broad power range.

Table IV — Expanded Commercial Motor Dataset (240–1000W)

6.2 Fitted Scaling Laws

7. Sensitivity Analysis

To quantify model robustness, each scaling exponent b_X is perturbed by ±20% while keeping the reference value X₂₅₀ fixed, and the resulting parameter error is evaluated at P_n = 1000 W — the point of maximum extrapolation from the 250W baseline.

Table V — Exponent Sensitivity at P_n = 1000 W

8. Worked Numerical Example

Every equation in this framework is now evaluated step by step using the TO7Motor DM02 at its 500W rated operating point. This example demonstrates how to go from a single known value — rated power — to a complete set of motor parameters and a steady-state thermal prediction.

Step 1 — Scaling Laws (Eq. 16–19)

Substituting P_n = 500W into the four scaling laws:

Step 2 — Phase Current Estimate

At rated load, the RMS phase current is estimated from the power balance:

Step 3 — Copper Loss (Eq. 8 + 9)

First calculate resistance at an assumed initial winding temperature of 60°C (a reasonable mid-load estimate), then compute copper loss:

Step 4 — Core and Mechanical Losses

Step 5 — Steady-State Winding Temperature (Eq. 14)

Using mid-range thermal resistance values from Table III (R_ws = 0.5, R_sh = 0.2, R_ha = 1.2 K/W) and T_amb = 25°C:

9. Validation and Calibration Range

9.1 Validation Scope

The scaling laws are validated at the parameter level — the regression reproduces the commercial motor dataset with R² ≥ 0.953 across all four parameters. The worked example in Section 8 provides an additional check: the predicted phase resistance for the DM02 (43.0 mΩ) matches the measured datasheet value (41 mΩ) to within 5%, which lies comfortably within the stated ±15–20% model tolerance. This is one data point, not a statistical validation — but it confirms the scaling laws produce physically reasonable results for a real motor in the calibrated range.

Full validation against operating-point efficiency maps and thermal transient data would require time-domain measurements at multiple torque-speed points and is reserved for future work. Until then, the model should be used for early-stage sizing and comparative analysis, not for final design sign-off.

9.2 Calibration Range

10. Conclusion and Future Work

This paper has presented a complete, geometry-independent parametric model for mid-drive PMSMs in the 240–1000W class.

By expressing all key motor parameters as power-law functions of rated power — calibrated from an expanded commercial dataset with good power-range coverage — the framework enables rapid system-level modeling without requiring internal geometry or FEA.

The model is fully coupled across electromagnetic, mechanical, loss, and thermal domains, provides all necessary coefficient ranges for immediate use, and is demonstrated through a complete numerical worked example using the TO7Motor DM02 at 500W.

10.1 Future Work

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  Operating-Point Validation: Collect or source torque-speed-efficiency map data for multiple commercial mid-drives to validate the coupled model at actual operating points beyond the current parameter-level regression check.
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  Thermal Transient Validation: Compare three-node ODE predictions against thermocouple or infrared winding temperature measurements during realistic LEV duty cycles — hill climbs, stop-start urban, sustained assist.
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  MATLAB/Simulink Implementation Guide: Publish a block-by-block Simulink implementation of the full coupled model, enabling practitioners to adopt the framework directly without re-deriving equations from scratch.
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  Extended Power Range: Expand the calibration dataset beyond 1000W — up to 1500–2000W — to cover emerging high-power LEV platforms such as speed pedelecs and light electric motorcycles, where the current framework is explicitly out-of-calibration.
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  Saturation and Saliency Modeling: Extend the electromagnetic sub-model to account for magnetic saturation at high current densities and saliency variation with operating point — both of which become significant above 750W in compact mid-drive architectures.

References