Motor NVH Analysis (Electromagnetic Excitation Force)
Theory and Physics
Why Motor Noise is a Problem in EVs
EVs are supposed to be quiet, so why is motor noise a problem?
Because there is no engine noise, high-frequency motor sounds (hum, whine) become more noticeable. Especially when the 48th order component of the electromagnetic excitation force falls in the 2-4 kHz range, it becomes unpleasant. The early Tesla Model 3 also had this issue.
What is the 48th order component? Why does such a specific order become a problem?
"Order" refers to the frequency multiplier of the electromagnetic force relative to the motor's rotation. For example, in a 48-slot motor, the 48th order becomes the most dominant excitation force component. If the motor is rotating at 5,000 rpm, the frequency of the 48th order component is $48 \times \frac{5000}{60} = 4{,}000\ \text{Hz}$. That's right in the 2-4 kHz band where the human ear is most sensitive.
I see, so at highway cruising speeds it hits that unpleasant frequency range. Wasn't this noticeable in engine vehicles?
Engine noise is broadband noise, so it "masked" pure tones at specific frequencies. In EVs, that masking effect is gone, so the motor's tonal noise (pure tone noise) feels like it's piercing the driver's ears. The early Nissan Leaf also had its "whine" sound during acceleration become a topic of discussion.
Fundamentals of Radial Electromagnetic Force
How does such vibration force get generated inside the motor?
The magnetic flux density $B_r(\theta, t)$ in the air gap exerts radial pressure on the stator tooth surfaces. This is the radial component of the Maxwell stress tensor, expressed by the following formula:
Here $B_r$ is the radial magnetic flux density in the air gap, and $\mu_0 = 4\pi \times 10^{-7}$ H/m is the permeability of free space. Because it is proportional to the square of the magnetic flux density, the product of fundamental wave $f_1$ components contained in the flux density generates an excitation force at $2f_1$, and the product of different harmonics $f_m$ and $f_n$ generates components at $f_m + f_n$ and $|f_m - f_n|$.
So squaring it increases the frequency components. Is it like distortion on a guitar?
Good analogy. It's exactly the same principle where harmonics are generated by nonlinear processing. Practically, it's important to expand the air gap flux density as a product of spatial harmonics and temporal harmonics. The flux density can generally be written as:
Here $\nu$ is the order of the spatial harmonic, and $\omega_\nu$ is the corresponding angular frequency. $\nu = p$ (number of pole pairs) is the fundamental wave, and $\nu = p \pm kQ_s$ ($Q_s$: number of slots, $k$: positive integer) are the slot-induced harmonics.
Force Order and Modes
Could you explain in more detail how those harmonics lead to stator vibration?
When magnetic flux density components of different spatial orders $\nu_1$ and $\nu_2$ are multiplied, the force mode order (circumferential mode order) $r$ of the radial force is determined:
This $r$ corresponds to the stator's vibration mode (circumferential deformation pattern). $r = 0$ is the breathing mode (stator uniformly expands/contracts), $r = 2$ is the elliptical mode, $r = 3$ is the triangular mode, and so on.
The natural frequency of the stator's $r$-th order mode can be estimated using the thin-walled cylindrical shell approximation formula:
Here $D_s$ is the bending stiffness, $\rho$ is the density, $h$ is the yoke thickness, $R$ is the representative radius of the stator inner diameter. The $r = 0$ and $r = 2$ modes have the highest acoustic radiation efficiency, so significant noise occurs when the frequency of the electromagnetic excitation force matches these modes under operating conditions.
So lower mode orders are more dangerous. Is the $r = 0$ breathing mode that significant?
$r = 0$ involves the entire stator uniformly expanding and contracting, so its acoustic radiation efficiency is nearly 100%. In practice, the $r = 0$ component is often relatively small, but the $r = 2$ elliptical mode has both high force level and high radiation efficiency, making it the most problematic pattern in practice. For example, in an 8-pole 48-slot IPMSM, the combination $\nu_1 = 4$, $\nu_2 = 4$ produces $r = 0$ and $r = 8$, and the combination $\nu_1 = 4$, $\nu_2 = 44$ produces $r = 48$.
Campbell Diagram
I often hear about Campbell diagrams. What is that diagram for?
It's a diagram with rotational speed on the horizontal axis and frequency on the vertical axis, plotting each order component of the electromagnetic excitation force as straight lines. The frequency of the $n$-th order component is $f = n \cdot N / 60$, so it becomes a straight line with slope $n/60$. If you overlay the stator's natural frequencies as horizontal lines, the intersections become resonance points.
So more intersections mean more danger?
Yes. However, not all intersections are problematic. What's important are three conditions: (1) Is the excitation force level of that order large? (2) Is the acoustic radiation efficiency of the resonant mode high? (3) Is it within the common operating speed range? For automotive applications, ideally there should be no intersections below 2 kHz in the 2,000–10,000 rpm range, but that's almost impossible in practice, so the design aims to minimize the force level at the intersections as much as possible.
Structure-Acoustic Coupling
Assuming the stator vibrates, how does that become noise inside the vehicle?
The vibration velocity $v_n$ of the stator surface pushes the air away and radiates sound waves. The radiated sound power is expressed as:
Here $\sigma_{\text{rad}}$ is the radiation efficiency (depends on mode order and frequency), $\rho_0 c_0$ is the characteristic impedance of air (approx. 415 Pa·s/m), $S$ is the stator outer surface area, and $\langle v_n^2 \rangle$ is the area-averaged mean square of the normal direction vibration velocity.
The key point is the radiation efficiency $\sigma_{\text{rad}}$. For low-order modes ($r = 0, 2$) it's close to 1, but for high-order modes, the radiation efficiency drops dramatically at low frequencies. This means that even at the same vibration level, the actual sound loudness can be completely different depending on the mode order. This is the reason for cases where "large force levels don't produce sound" and "small forces are noisy".
There's also a path where stator vibration is transmitted to the housing, and sound comes from there, right?
Exactly. The actual transmission path is primarily the structure-borne path: stator → housing → mounts → vehicle body → interior air. There is also an air-borne path directly from the stator to the air, but usually the structure-borne path is dominant. Therefore, for NVH design, "system-level" evaluation including mount rubber hardness and housing stiffness design is essential.
The Tesla Model 3 "Hum" and the Inside Story of the OTA Fix
The high-frequency "hum" sound that became famous in the early Tesla Model 3 (2017~) is said to have been caused by the 48th order excitation force component of the IPMSM resonating with the stator's elliptical mode ($r=2$) in the mid-speed range. Interestingly, Tesla used an OTA (Over-The-Air) software update to change the inverter's PWM carrier frequency, reducing the perceived discomfort. This approach of changing the "perceived sound" via software without changing hardware overturned conventional NVH design wisdom. However, the root cause—the electromagnetic excitation force itself—remained unchanged, so tonal noise still persists during acceleration in specific speed ranges.
Full Form of Maxwell Stress Tensor and Tangential Force
- Radial component $\sigma_r = \frac{B_r^2 - B_\theta^2}{2\mu_0}$: The main excitation force for NVH. Dominated by the square of the radial magnetic flux density $B_r$, but the tangential magnetic flux density $B_\theta$ also contributes via subtraction. Usually $B_r \gg B_\theta$, so it is approximated as $B_r^2 / (2\mu_0)$.
- Tangential component $\sigma_\theta = \frac{B_r B_\theta}{\mu_0}$: The component directly linked to torque generation. Its contribution to NVH appears as rotational vibration (torsional vibration) through torque ripple. It can be transmitted via gears to the drive shaft and enter the cabin as gear noise.
Assumptions and Applicability Limits
- 2D Cross-Sectional Analysis Assumption: Ignores end effects (axial magnetic field at coil ends). 3D analysis is required when axial forces become problematic (skewed motors, axial flux types).
- Breakdown of Linear Material Assumption: When magnetic saturation of the iron core makes the $B$-$H$ curve nonlinear, harmonic components increase significantly. Nonlinear analysis is essential for NVH analysis under high load (high torque) conditions.
- Rigid Rotor Assumption: Rotor deformation is usually ignored in NVH analysis, but for high-speed rotors, centrifugal deformation changes the air gap length, affecting the magnetic flux density distribution.
- Steady-State Assumption: During rapid acceleration/deceleration transients, the effects of current control delay and PWM switching add up, creating more complex excitation patterns than steady-state analysis.
Dimensional Analysis: Guideline for Excitation Force Level
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