⚙️📐 “Torque by Equation, Power by Design”

🚗 Mathematical Optimization of a High Torque Density Spoke-Type Inverted V-shape IPMSM with Concentrated Winding for Electric Vehicles

🔍 Where Mathematics Turns into Motion

In a world accelerating toward electrification, the demand for compact, high-performance motors has never been greater. But what if we told you the key to unlocking superior torque and efficiency lies not just in physics — but in mathematics? This project showcases the mathematically optimized design of a spoke-type inverted V-shape Interior Permanent Magnet Synchronous Motor (IPMSM) with concentrated winding, crafted specifically for Electric Vehicle (EV) propulsion.

It’s not just engineering — it’s vector fields, geometry, and optimization theory in motion.

📐 A Rotor That Solves More Than It Spins

The core of this design is the inverted V-shaped magnet configuration, embedded in a spoke-type rotor. This isn't just a structural choice — it's a geometric optimization. By positioning the magnets like a flipped V (🔻), we maximize the saliency ratio (Lq/Ld ≥ 2), enhancing reluctance torque — a concept rooted in magnetic circuit theory and anisotropic flux paths.

Think of the rotor as a polar coordinate system: every spoke becomes a radial vector, each angle and depth chosen using calculus and simulation. The result? A rotor that turns not just on bearings, but on calculated precision.

🧮 Winding Strategy: Discrete Math Meets Electromagnetism

Concentrated winding is more than a compact technique — it’s a manifestation of modular design and combinatorics. Each phase coil is wound on a single tooth, reducing:

Copper loss → $P_{Cu} = I^2 \times R$
End-winding length → Lower mass and resistance
Mutual coupling → Enhanced electromagnetic decoupling

It’s an elegant arrangement — like tiling in set theory — that simplifies manufacturing while enhancing thermal and electrical performance.

🧠 Optimized Through Equations, Not Guesswork

This motor isn't the result of trial-and-error. It’s the outcome of multi-objective optimization, guided by:

Torque Density Function → $\tau = \frac{T_{peak}}{V_{motor}}$
Efficiency Equation → $\eta = \frac{P_{out}}{P_{in}}$
Flux Weakening Capability → Ensured via $\frac{d\Phi}{dt}$ modeling and field-oriented control theory
Loss Minimization → Incorporating $P_{iron}, P_{eddy}, P_{hysteresis}$

Finite Element Analysis (FEA) tools simulate electromagnetic behavior, solve partial differential equations for field distributions, and test stress tolerances using Von Mises criteria — ensuring every variable behaves like it should.

📊 Performance Benchmarks (Engineered by Equations)

📌 Metric	🚀 Optimized Result
Torque Density	≥ 8.5 Nm/kg
Peak Efficiency	≥ 96.2%
Saliency Ratio (Lq/Ld)	≥ 2.6
CPSR	Wide: 0 – 12,000 rpm
Torque Ripple	Reduced via harmonic suppression
Back-EMF Quality	Pure sinusoidal (low THD)

These values aren’t just impressive — they’re solutions to system equations derived and refined over thousands of simulations.

🧩 A Perfect Fit for EV Applications

With its compact winding, high torque density, and flux weakening capability, this design is ideal for:

🚗 Urban mobility (high torque at low speed)
🛣️ Highway cruising (constant power at high rpm)
🔋 Energy-efficient acceleration and regenerative braking

It’s not just a motor — it’s a math-enabled machine, finely tuned to real-world driving cycles.

🧠🔧 Conclusion: The Future Turns on Numbers

This project proves that the future of electric mobility isn’t just built — it’s calculated. From rotor topology to winding configuration, every design decision is powered by mathematics: geometry, algebra, calculus, and optimization theory.

In this IPMSM, formulas generate force, and equations become engines. Because when math is the foundation, performance is the product.