An EV Motor Makes Torque Two Different Ways at Once
The first term is current interacting with magnet flux. The second is free torque from geometry alone — available whether or not there is a magnet present, and worth 30 to 50 per cent of the total.
Magnets and Motors · Part 3 — The Machine · Chapter 6 · 21 min read
Part 3 — The Machine
Where torque actually comes from
The first term is the one everybody knows. The second is free torque from geometry alone, and in a modern traction motor it supplies a third to a half of the output.
An EV motor makes torque two different ways at once. The first is current interacting with magnet flux. The second is available whether or not there is a magnet present — and it is the reason traction motors bury their magnets rather than gluing them to the surface.
30–50 %
Of traction torque from reluctance
~30–35°
Current angle at the MTPA peak
D²L
How torque scales
K_t = K_e
Not a coincidence
6.1 — The torque equation has two terms
T = (3/2) · p · [ ψ_m·i_q + (L_d − L_q)·i_d·i_q ]
p = pole pairs · ψ_m = magnet flux linkage (Wb) · i_d, i_q = direct- and quadrature-axis currents · L_d, L_q = inductance along the magnet axis and 90° electrical from it.
- •First term — magnet torque. Proportional to magnet strength and to i_q. This is the one everybody knows.
- •Second term — reluctance torque. Proportional to the difference in inductance between the two axes, and it needs a negative i_d to be positive.
6.2 — Reluctance torque, without the algebra
In plain English
Magnetic flux takes the easiest path, the way current takes the least resistance. If a rotor is magnetically asymmetric — easy in one direction, hard in another — then it experiences a torque pulling it into alignment with the stator field, exactly the way a compass needle made of plain iron aligns in a field despite having no magnetisation of its own.
Building that asymmetry is the entire design of an interior-permanent-magnet rotor. Magnets buried in slots create flux barriers, because a magnet has roughly the permeability of air. Flux flows easily around them and poorly through them, and the difference between those two paths is L_q − L_d.
Important
A surface-magnet rotor cannot do this. With magnets glued to the outside, both axes see roughly the same air path, so L_d ≈ L_q and the reluctance term is zero. That is the main reason traction motors bury their magnets and machine tools often do not.
6.3 — Why MTPA exists
Why this matters
At constant current magnitude, sweeping the current angle trades magnet torque against reluctance torque, and the peak of the sum sits near 30 to 35 degrees rather than at zero. Maximum Torque Per Amp control exists to hold the drive on that peak at every operating point — worth tens of per cent over feeding pure q-axis current, from the same copper and the same magnets, in software.
6.4 — Torque is a surface phenomenon
Torque is produced in the airgap, as a shear stress acting on the rotor surface. That gives a scaling law worth carrying around.
T = σ · (π D L) · (D/2) = σ · π · D² · L / 2
σ = airgap shear stress (kPa) · D = rotor diameter · L = stack length. Typical σ: 20–35 kPa air-cooled, 40–70 kPa liquid-cooled, up to 100 kPa peak.
Torque scales with D²L — diameter squared, length once. Doubling diameter quadruples torque; doubling length only doubles it. Pancake-shaped motors make torque, and that is why axial-flux machines exist.
Technical framing
Power, however, is torque × speed. So a small motor spun fast delivers the same power as a large one spun slowly — which is why traction motors run at 12,000 to 20,000 rpm behind an 8:1 to 12:1 reduction gear instead of turning the wheels directly. The gearbox is there to let the motor be small.
6.5 — The constant that is two constants
A rotating magnet induces voltage in the stator windings — back-EMF, proportional to speed. It opposes the applied voltage, and at some speed it equals everything the inverter can supply. That is the base speed, and it would be the top speed if nothing were done about it.
K_t [N·m/A] = K_e [V·s/rad]
Mechanical power out equals electrical power in: T·ω = E·I, so T/I = E/ω.
This is not a coincidence — it is conservation of energy. A strong magnet therefore buys torque per amp and costs you top speed, in exactly the same ratio. You cannot have one without the other.
6.6 — Field weakening — driving the magnet backwards on purpose
Above base speed, the drive injects negative i_d: current whose field opposes the rotor magnets. This cuts the net flux linkage, which cuts the back-EMF, which lets the motor keep accelerating on the same DC bus voltage. Torque falls as flux falls, so power stays roughly constant.
6.6.1 — Two things follow directly
- •The constant-power speed range is set by how much flux you can cancel, which is a function of L_d — a design parameter, not an afterthought.
- •That negative i_d is exactly the demagnetising field the magnets have to survive, at the exact moment the motor is hottest. Chapter 9 is about what happens when that goes wrong.
Quick check: test yourself
1.Why do traction motors bury their magnets while many industrial servos do not?
Show answer
2.A motor doubles in diameter and halves in length. What happens to torque?
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3.A customer wants more torque per amp and the same top speed. What is wrong with that request?
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Chapter summary
- ✓Torque has two terms: magnet torque from flux and current, and reluctance torque from the inductance difference between the two axes.
- ✓Reluctance torque needs magnetic asymmetry, which is why traction rotors bury their magnets — a surface-magnet rotor gets none of it.
- ✓The torque peak sits near 30 to 35 degrees of current angle, and MTPA control exists to hold the drive there — tens of per cent more torque from the same hardware, in software.
- ✓Torque is an airgap shear stress and scales as D²L, so traction motors are small and fast behind a reduction gear rather than large and slow.
- ✓K_t equals K_e by conservation of energy, so magnet strength trades torque per amp against top speed one-for-one — and field weakening buys speed back by applying a demagnetising field on purpose.
Frequently asked questions
What is reluctance torque?+
Torque produced by magnetic asymmetry rather than by magnets. Flux takes the easiest path, so a rotor that is magnetically easy in one direction and hard in another experiences a torque pulling it into alignment with the stator field — the way a plain iron compass needle aligns despite having no magnetisation of its own. In the torque equation it is the (L_d − L_q)·i_d·i_q term, it needs a negative i_d to be positive, and in a modern traction motor it supplies 30 to 50 per cent of total output.
Why do traction motors bury their magnets?+
Because buried magnets create flux barriers — a magnet has roughly the permeability of air, so flux flows easily around it and poorly through it, and the difference between those two paths is L_q − L_d. That difference is what produces reluctance torque. With magnets glued to the rotor surface, both axes see roughly the same air path, L_d ≈ L_q, and the reluctance term is zero. The V-shaped arrangement also holds the magnets against 20,000 rpm centrifugal load.
Why does torque scale as D²L?+
Because torque is produced in the airgap as a shear stress acting on the rotor surface: T = σ·(πDL)·(D/2). Diameter appears twice — once in the surface area and once in the lever arm — while length appears once. Doubling diameter quadruples torque; doubling length only doubles it. That is why pancake-shaped machines make torque, and why axial-flux motors exist. Typical airgap shear stress runs 20 to 35 kPa air-cooled and 40 to 70 kPa liquid-cooled.
Why is the torque constant numerically equal to the back-EMF constant?+
Conservation of energy, not coincidence. Mechanical power out equals electrical power in, so T·ω = E·I, which rearranges to T/I = E/ω. K_t in N·m/A and K_e in V·s/rad are therefore the same number in SI units. The consequence is that a stronger magnet buys torque per amp and costs top speed in exactly the same ratio — you cannot have one without the other.
What is field weakening and why is it risky?+
Above base speed the drive injects negative d-axis current whose field opposes the rotor magnets, cutting net flux linkage and therefore back-EMF, which lets the motor keep accelerating on the same DC bus voltage. Torque falls as flux falls, so power stays roughly constant. The risk is that this negative i_d is exactly the demagnetising field the magnets have to survive, applied deliberately and continuously at the top of the speed range where losses and temperatures are highest.
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Magnets and Motors is an original educational series on permanent magnets and electric machines. Datasheet ranges, temperature coefficients and efficiency figures are representative standard-condition literature values, not measurements of a specific product, and sources differ on several. Always verify against the specific magnet and lamination datasheets in use before making design, procurement or certification decisions.