MOTOR-22 — Electromagnetic Design Principles

FACT — Pick up a traction motor’s stator stack and at first glance you see nothing but slot count, magnet count, and wire gauge — yet these apparently simple geometric choices — 48 slots or 36, magnets skewed by $2\text{ mm}$ or not, an air gap of $0.7\text{ mm}$ or $1\text{ mm}$ — determine torque smoothness, efficiency, acoustic noise, and magnet longevity from the ground up. This lesson quantifies the geometry that produces torque, in engineering terms.

In this lesson we cover:

1. Pole/Slot Count Combinations ($N_s / 2p$) and Torque Ripple
2. Winding Factor: $k_w = k_p \cdot k_d$
3. Stator Tooth Skewing: Continuous vs. Step-Skew
4. Cogging Torque Minimization Techniques
5. Armature Reaction and Air-Gap Flux Distribution
6. Air-Gap Sizing and Lamination Thickness Tradeoffs
7. Worked Example: 48-Slot/8-Pole Traction Motor

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1. Pole/Slot Count Combinations and Torque Ripple

In a three-phase synchronous machine, the ratio between stator slot count $N_s$ and pole-pair count $p$ (pole count $2p$) sets the machine’s fundamental electromagnetic identity. Slots per pole per phase:

$q = \frac{N_s}{2p \cdot m}, \quad m = 3 \text{ (phases)}$

An integer $q$ yields an integral-slot winding; a fractional $q$ yields a fractional-slot winding.

CRITICAL — Fractional-Slot Concentrated Windings (FSCW) have become dominant in EV traction motors because: (1) end-turn length is very short → lower copper loss and shorter axial length; (2) high winding factors enable compact designs. However, the MMF waveform develops unwanted sub-harmonics, which drive extra eddy-current loss in the rotor magnets and back-iron, and additional torque ripple.

The most common traction-motor combination is $48\text{ slots} / 8\text{ poles}$ ($q = 48/(8\cdot3) = 2$, integral-slot, distributed winding). The Tesla Model 3/Y rear motor and many modern IPMSM traction designs use slot/pole ratios in this family.

ASSUMPTION — The exact slot/pole count of specific OEM motors is rarely confirmed in public technical documentation; the “48/8” figure cited here represents a widely referenced design family in the industry, not a verified specification of a specific manufacturer’s model.

Torque ripple originates from the harmonic components (particularly orders $6k \pm 1$, $k=1,2,3...$) produced by the product of the air-gap MMF distribution and the magnet flux waveform, generating oscillating torque components on the rotor:

$T_{ripple}(\theta) = \sum_{n} T_n \cos(n \cdot N_{LCM} \cdot \theta_e + \phi_n)$

Here $N_{LCM} = \text{lcm}(N_s, 2p)$ — the least common multiple of slot count and pole count. The larger $N_{LCM}$ is, the higher the cogging/ripple frequency, and the smaller its amplitude (mechanical inertia filters high-frequency content more effectively).

Combination ($N_s$/$2p$)	$q$	$N_{LCM}$	Torque Ripple	Typical Use
36/6	2	36	Moderate	Industrial PMSM
48/8	2	48	Low-Moderate	EV traction (common)
12/10	0.4	60	Low (FSCW)	Compact hub motor
9/8	0.375	72	Very Low	High-pole-count FSCW

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2. Winding Factor: $k_w = k_p \cdot k_d$

The effective fundamental voltage/MMF a winding produces is always less than an idealized full-pitch, single-slot winding — because real coils are both short-pitched and distributed across multiple slots. This reduction is captured by the winding factor $k_w$:

$k_w = k_p \cdot k_d$

A. Pitch Factor $k_p$

When coil span is chosen shorter than the pole pitch ($y < \tau_p$, “chorded winding”), harmonics are suppressed, but the fundamental is also somewhat reduced:

$k_p = \sin\left(\frac{y}{\tau_p} \cdot \frac{\pi}{2}\right)$

For example, a winding with $y/\tau_p = 5/6$ gives $k_p = \sin(75^\circ) = 0.966$; this particular span ratio nearly fully cancels the 5th and 7th harmonics.

B. Distribution Factor $k_d$

When coils are spread across $q$ adjacent slots rather than concentrated in one, an electrical angle $\gamma$ separates the voltage phasors of each slot, and their phasor sum falls short of the simple arithmetic sum:

$k_d = \frac{\sin(q\gamma/2)}{q \sin(\gamma/2)}, \quad \gamma = \frac{2\pi p}{N_s}$

EXAMPLE — For $N_s=48$, $2p=8$ ($p=4$), $q=2$: $\gamma = 2\pi \cdot 4/48 = 30^\circ$ electrical. Then: $k_d = \frac{\sin(2 \cdot 15^\circ)}{2\sin(15^\circ)} = \frac{\sin 30^\circ}{2\sin 15^\circ} = \frac{0.5}{2 \times 0.2588} = 0.966$ Taking $k_p \approx 0.966$ (5/6-pitch): $k_w = 0.966 \times 0.966 \approx 0.933$.

$k_w$ appears as a direct multiplier in the Back-EMF and torque equations:

$E_{phase} = \sqrt{2}\, \pi f N_{ph} k_w \Phi_{pm}$

CRITICAL — Every 1% lost in $k_w$ is a direct 1% torque loss for the same magnet volume and current — this is why designers target $k_w > 0.93$, while balancing it against harmonic suppression (5th/7th harmonics); span ratio and $q$ selection sit at the center of this tradeoff.

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3. Stator Tooth Skewing: Continuous vs. Step-Skew

The strongest mechanical remedy for cogging torque and torque ripple is skewing the stator slots or rotor magnets relative to the rotor axis.

Skew Types (axial cross-section, rotor magnet arrangement):

Continuous Skew:                            Step-Skew (3 steps):
   ______                                     ____
  /     /|                                   |    |
 /     / |  <- helical, continuous           |____|  <- segment 1 (0°)
/_____/  |     angle change                     ____
|     |  |                                      |    |
|  θ  | /                                       |____|  <- segment 2 (+θ/2)
|_____|/                                            ____
                                                     |    |
                                                     |____|  <- segment 3 (+θ)

Skew angle is typically set to one stator slot pitch ($\theta_{slot} = 360^\circ/N_s$); this choice theoretically drives the dominant cogging harmonic toward zero. Skew factor:

$k_{sk} = \frac{\sin(n \theta_{sk}/2)}{n \theta_{sk}/2}$

where $\theta_{sk}$ is the skew angle in electrical radians and $n$ is the harmonic order.

Method	Cogging/Ripple Reduction	Back-EMF/Torque Loss	Manufacturing Complexity
No skew (baseline)	—	0%	Low
Continuous skew (1 slot pitch)	Highest (>90%)	~2-4%	High (helical winding/tooling)
Step-skew (2-3 steps, rotor magnets)	Moderate-High (60-80%)	~1-2%	Moderate (segment alignment)
No skew + magnet pole-arc optimization	Moderate	~0.5%	Low

WARNING — Skewing is not “free”: it multiplies the fundamental itself by $k_{sk}$, trading away torque and Back-EMF magnitude. High-performance traction motors therefore typically favor partial skew rather than a full slot pitch, or step-skewed magnet segmentation on the rotor — which is also considerably cheaper to manufacture than continuous helical skew, since the lamination stack is built from flat slices, each rotated a small step.

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4. Cogging Torque Minimization Techniques

Cogging torque is the position-dependent torque component produced even at zero current, arising from the varying reluctance seen by the magnets as they pass the stator teeth — the source of the “detent” feel a driver can sense even coasting with no throttle. Its frequency:

$f_{cogging} = N_{LCM} \cdot n_{rpm}/60$

The primary minimization techniques, roughly in order of effectiveness:

1. Slot/pole combination selection — maximizing $N_{LCM}$ (Section 1).
2. Skewing — Section 3.
3. Magnet pole-arc optimization — the ratio of magnet angular width to pole pitch ($\alpha_p$) is chosen to cancel dominant cogging harmonics; the optimum typically falls around $\alpha_p \approx 0.7\text{–}0.85$ (a magnet spanning the full pole pitch usually produces the worst cogging).
4. Stator tooth-tip notching — small auxiliary notches cut into tooth tips virtually increase the effective slot count, a low-cost cogging suppression technique.

FACT — These techniques are largely non-exclusive; real traction motor design typically combines (slot/pole selection + mild magnet pole-arc optimization + step-skew), tuned iteratively via finite-element analysis (FEA). Analytical formulas give a first estimate; final verification always requires FEA.

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5. Armature Reaction and Air-Gap Flux Distribution

When the motor is loaded, current flowing through the stator windings generates its own MMF, which superimposes on the flux distribution produced by the rotor magnet — this is armature reaction.

At no-load, the air-gap flux density is a symmetric waveform; under load:

$B_{gap}(\theta) = B_{pm}(\theta) + B_{armature}(\theta)$

The $B_{armature}$ component shifts the flux wave forward or backward in the direction of rotor rotation (depending on load angle and the $I_d/I_q$ ratio) and skews the waveform asymmetrically — flux concentrates at one pole tip while thinning at the other.

CRITICAL — In this concentrated region, local flux density can exceed the knee of the iron saturation curve; this drives both extra iron loss and local magnet demagnetization risk — because the demagnetizing component of armature reaction (large negative $I_d$ during field weakening, see MOTOR-19) can coincide with the hottest, most vulnerable region of the magnet. This is why magnet demagnetization analysis always combines both temperature (MOTOR-23) and worst-case armature reaction (maximum negative $I_d$).

The magnitude of armature reaction is set by the ratio of stator MMF to magnet MMF; this ratio is bounded in design by a short-circuit-ratio-like safety margin — high-current-density, low-air-gap designs are more sensitive to armature reaction.

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6. Air-Gap Sizing and Lamination Thickness Tradeoffs

A. Mechanical Air Gap ($g$)

Typical traction motors use a mechanical air gap $g = 0.7\text{–}1.2\text{ mm}$. This range sits at the intersection of several conflicting constraints:

Gap Reduced ($g \downarrow$)	Gap Increased ($g \uparrow$)
Magnetizing current (reactive component) falls → efficiency ↑	Magnetizing current rises → efficiency ↓
Torque density rises (higher $B_{gap}$ for same MMF)	Torque density falls
Armature reaction and slot harmonics felt more strongly → cogging/ripple ↑	Harmonics “smoothed” more → ripple ↓
Rotor eccentricity/mechanical tolerance margin shrinks → contact risk ↑	Mechanical safety margin ↑
Margin against thermal expansion mismatch (stator/rotor) shrinks	Margin increases

WARNING — A narrow air gap like $0.7\text{ mm}$, combined with rotor centrifugal growth at high speed (>15,000 rpm), bearing tolerances, and differential thermal expansion, carries a rotor-stator rub risk. The air gap cannot be shrunk without verifying worst-case tolerance stack-up.

B. Lamination Thickness and Core Loss

Stator/rotor cores are built from thin, electrically insulated non-oriented (NO) silicon-steel laminations to limit eddy currents. Typical thickness ranges $0.25\text{–}0.35\text{ mm}$ (traction motors also use “high-frequency” grades down to 0.20mm given their high electrical frequency).

Eddy-current loss scales with the square of lamination thickness:

$P_{eddy} \propto t^2 \cdot f^2 \cdot B_{max}^2$

Hysteresis loss, by contrast, is thickness-independent and scales roughly linearly with frequency:

$P_{hyst} \propto f \cdot B_{max}^{1.6-2.0}$

Total core loss is conventionally specified as $W_{1.5/50}$: loss in $\text{W/kg}$ at $1.5\text{ T}$ peak flux density and $50\text{ Hz}$. For example, a grade like M270-35A at $0.35\text{ mm}$ thickness may sit around $W_{1.5/50} \approx 2.7\text{ W/kg}$; a thinner variant like M250-27AP (0.27mm) yields lower $W_{1.5/50}$ but costs more and has a slightly lower stacking factor (fill factor).

INTERPRETATION — Because traction motors run at high electrical fundamental frequencies ($500\text{–}1500\text{ Hz}$, from high pole count × high rpm), thinner laminations ($0.20\text{–}0.30\text{ mm}$) are preferred over the $0.35\text{–}0.50\text{ mm}$ typical of $50\text{/}60\text{ Hz}$ grid motors — eddy loss, growing with the square of thickness, dominates at these frequencies. Thinner steel is more expensive and has a lower fill factor (more insulation layers per stack), so the choice is a cost/efficiency tradeoff, not a pure performance one.

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7. Worked Example: 48-Slot/8-Pole Traction Motor

Using a simplified Esson’s-coefficient-style torque density estimate:

$T_e = 2 \pi \cdot k_w \cdot B_{gap} \cdot A_s \cdot D^2 \cdot L$

where $D$ is stator bore diameter, $L$ is stack axial length, and $A_s$ is electric loading (linear current density, A/m).

EXAMPLE — For $D = 200\text{ mm}$, $L = 150\text{ mm}$, $B_{gap} = 0.85\text{ T}$, $A_s = 45{,}000\text{ A/m}$, $k_w = 0.933$ (from Section 2), simplifying the proportionality constant gives an approximate torque: $T_e \approx k \cdot k_w \cdot B_{gap} \cdot A_s \cdot D^2 \cdot L \approx 250\text{–}300\text{ Nm}$ (The exact constant $k$ depends on machine geometry and winding type; this example only illustrates the relative sensitivity to $k_w$, $B_{gap}$, and $A_s$.) If $k_w$ drops from $0.933 \to 0.90$ due to skewing (~3.5% loss), torque drops by roughly the same proportion for the same current and geometry — numerical proof that skewing is not “free.”

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8. Frequently Asked Questions (FAQ)

Why doesn’t every traction motor use 48/8 slot-pole?

FACT — Slot/pole selection balances target electrical frequency, end-turn length constraints, manufacturing method (hairpin vs. round wire), target $k_w$, and torque ripple goals. FSCW designs (e.g., 12/10, 9/8) offer a more compact end-turn but carry higher MMF sub-harmonic content, which can increase rotor magnet eddy loss.

Is skewing always necessary?

INTERPRETATION — No. If the slot/pole combination naturally yields a high $N_{LCM}$ (e.g., FSCW designs) and magnet pole-arc is already optimized, the extra torque/efficiency loss from skewing may be unjustified. The skewing decision is always an FEA-verified tradeoff analysis.

Is thinner lamination always better?

FACT — For efficiency, yes (lower eddy loss), but not for cost or fill factor. In low-speed/low-frequency applications, thicker steel (0.35mm) is adequate and cheaper; in high-electrical-frequency traction motors, thinner steel (0.20-0.27mm) is justified.

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9. Summary

• Pole/slot combination ($N_s/2p$) and the resulting $N_{LCM}$ set torque ripple frequency and amplitude; FSCW designs offer compact end-turns but carry MMF sub-harmonics.
• Winding factor $k_w = k_p \cdot k_d$ (pitch factor × distribution factor) is a direct multiplier in Back-EMF and torque equations — a tradeoff between harmonic suppression and torque magnitude.
• Skewing (continuous or step) strongly suppresses cogging/torque ripple but also reduces the fundamental component via $k_{sk}$ — it is not free.
• Cogging torque is minimized through a combination of slot/pole selection, skewing, magnet pole-arc optimization, and tooth-tip notching.
• Armature reaction distorts air-gap flux asymmetrically under load, raising local saturation and demagnetization risk.
• Air-gap sizing ($0.7$–$1.2\text{ mm}$) and lamination thickness ($0.25$–$0.35\text{ mm}$) selections are multidimensional tradeoffs among efficiency, torque density, mechanical tolerance, and core loss; $W_{1.5/50}$ is the common language for comparing steel grades.

10. Sources and Verification Note

• Lipo — Introduction to AC Machine Design (winding factor, torque equations).
• Pyrhönen, Jokinen & Hrabovcová — Design of Rotating Electrical Machines (slot/pole combinations, skewing).
• Hendershot & Miller — Design of Brushless Permanent-Magnet Machines (cogging torque minimization, FSCW design).
• IEC 60034-1 (rotating electrical machine rating and performance terminology).

ASSUMPTION — The numerical examples in this lesson (48/8 slot-pole, $B_{gap}=0.85\text{ T}$, $A_s=45{,}000\text{ A/m}$, etc.) represent industry-typical reference ranges, not a verified specification of any specific manufacturer’s model; real designs require FEA verification.

Next Lesson

• MOTOR-23 — Thermal Design and Magnet Temperature: We turn to the heat sources (copper/iron/magnet losses) of the machine we just sized electromagnetically, model the heat path with a Lumped Parameter Thermal Network (LPTN), examine winding-head hotspots, magnet knee-point temperature, and compare cooling method choices (liquid jacket vs. oil spray).