What is the formula for torque ripple ratio?

The torque ripple ratio is defined as (T_max - T_min) / T_avg × 100%, where T_max is the maximum torque, T_min is the minimum torque, and T_avg is the average torque. For typical IPMSMs, it is generally around 5–15%.

How is torque ripple calculated using FEM?

It is primarily calculated using the Maxwell stress tensor method and area integration over the air gap. A transient analysis is performed by rotating the rotor in minute angular steps to obtain the torque waveform over one electrical cycle, which is then analyzed using FFT.

What design techniques are used to reduce torque ripple?

Common techniques include skewing the rotor or stator, optimizing the slot opening width, asymmetrical shaping of the magnets, adopting fractional-slot windings, and injecting harmonic currents from the control side.

Torque Ripple Analysis — Torque Pulsation Mechanisms and Reduction Methods for Electric Motors

Q: What is the difference between torque ripple and cogging torque?

Cogging torque is a magnetic pulsation that occurs even without energization, caused by the interaction between the permanent magnets and the stator slot openings. Torque ripple is the overall torque fluctuation that includes cogging torque plus additional components from current harmonics during energization and PWM switching from the inverter.

Category: Electromagnetic Field Analysis > Motor Design | Updated 2026-04-11

Torque ripple waveform analysis showing harmonic components in permanent magnet synchronous motor — Visualization of torque ripple waveform and harmonic components in an IPMSM. It can be seen that the 6th and 12th order components are dominant.

Theory and Physics

What is Torque Ripple?

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Professor, I often hear the term "torque ripple," but roughly speaking, what's the problem with it?

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Simply put, it's the phenomenon where the torque produced by a motor "fluctuates" depending on the rotation angle. An ideal motor should produce constant torque, but in reality, due to the influence of magnet arrangement and current patterns in the coils, the torque periodically increases and decreases.

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Why is that a problem? It seems like it would be fine as long as the average torque is produced...

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In the case of EVs, torque ripple is a major cause of unpleasant vibration and noise inside the vehicle. For electric power steering, it's transmitted to the driver as a "grating" sensation in the steering wheel. For industrial robots, it directly leads to degraded positioning accuracy. That's why predicting the magnitude of torque ripple numerically and reducing it at the design stage is one of the most important tasks in motor development.

The magnitude of torque ripple is quantified by the Torque Ripple Ratio:

$$ \text{Torque Ripple} [\%] = \frac{T_{\max} - T_{\min}}{T_{\text{avg}}} \times 100 $$

Here $T_{\max}$ and $T_{\min}$ are the maximum and minimum values of the torque waveform, respectively, and $T_{\text{avg}}$ is the average torque. For typical interior permanent magnet synchronous motors (IPMSM), it's often 5-15%, with high-quality designs aiming for below 3%.

Difference Between Cogging Torque and Torque Ripple

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What's the difference between torque ripple and cogging torque? Even when I ask my seniors, they say "well, they're kind of similar"...

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They're completely different! Cogging torque is a magnetic pulsation that occurs even without current, caused solely by the interaction between the permanent magnets and the slot openings. That "cogging" sensation you feel when you manually turn a motor shaft, that's cogging torque.

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On the other hand, torque ripple is the comprehensive torque fluctuation that occurs during current flow. In addition to cogging torque, it includes distortion of magnetomotive force due to current harmonics, ripple current caused by inverter PWM switching, and even pulsating components of reluctance torque. In other words, cogging torque is just one component of torque ripple.

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I see! So cogging torque is a problem of the "geometric compatibility" between magnets and slots, while torque ripple is like a "mixed martial arts" match that adds electrical factors to that.

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Good analogy. To summarize:

Item	Cogging Torque	Torque Ripple (Broad Sense)
Occurrence Condition	Occurs even without current	Occurs during current flow
Main Cause	Magnet ⇔ Slot interaction	Cogging + Current Harmonics + PWM + Reluctance Pulsation
Period	Function of LCM(Pole Number, Slot Number)	Electrical 6th, 12th orders are dominant
Reduction Methods	Slot shape, magnet shape	Above + Current control, winding design

Harmonic Decomposition of Torque Waveform

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Even if we say torque "fluctuates," can we break down the content of that wave further?

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Of course. By expanding the torque waveform into a Fourier series, we can see which harmonic orders are contained and by how much. For three-phase motors, due to symmetry, integer multiples of the electrical 6th order (6th, 12th, 18th...) become dominant.

The harmonic decomposition of the torque waveform is expressed as follows:

$$ T(\theta_e) = T_{\text{avg}} + \sum_{n=1}^{\infty} T_{6n} \sin(6n\theta_e + \phi_{6n}) $$

Here $\theta_e$ is the electrical angle, $T_{6n}$ is the torque amplitude of the $6n$th harmonic, and $\phi_{6n}$ is the phase angle. Identifying the source of each harmonic component is the starting point for implementing effective reduction measures.

Harmonic Order	Main Source	Typical Magnitude
6th	Interaction between 5th/7th harmonics of back-EMF and fundamental current	Largest (50-70% of total ripple)
12th	11th/13th harmonics of back-EMF, influence of magnetic saturation	Medium (20-30% of total ripple)
18th and above	Slot harmonics, PWM ripple current	Small (usually negligible)

Torque Calculation via Maxwell Stress Tensor

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What kind of formula is used to calculate torque in FEM analysis?

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The most widely used is the Maxwell Stress Tensor method. It integrates electromagnetic force over a closed surface $S$ in the air gap, from the tangential component $B_t$ and normal component $B_n$ of the magnetic flux density. In 2D analysis, the torque is:

$$ T = \frac{L_{\text{stk}}}{\mu_0} \oint_S r \cdot B_n(\theta) \cdot B_t(\theta) \, dS $$

Here $L_{\text{stk}}$ is the stack length (motor axial length), $r$ is the radius of the integration path, and $\mu_0$ is the permeability of free space.

In the general 3D form, the components of the Maxwell stress tensor $\overleftrightarrow{T}$ are:

$$ T_{ij} = \frac{1}{\mu_0}\left(B_i B_j - \frac{1}{2}\delta_{ij}|\mathbf{B}|^2\right) $$

Torque is obtained by performing a surface integral of this tensor over a closed surface in the air gap.

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Are there other methods for calculating torque?

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Yes. Another important method is based on the principle of virtual work. Torque is derived from the energy change when the rotor is rotated by a small angle $\Delta\theta$:

$$ T = -\frac{\partial W_{\text{co}}}{\partial \theta}\bigg|_{\Psi=\text{const}} $$

Here $W_{\text{co}}$ is the co-energy and $\Psi$ is the flux linkage. This method has the advantage of being less mesh-dependent than the Maxwell Stress Tensor method.

Classification of Ripple Sources

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So, how many types of "culprits" are there for torque ripple? I want to organize the overall picture.

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There are four main sources. They occur independently and superimpose, so in analysis it's important to determine which is dominant.

Cogging Torque: Caused by changes in the magnetic energy of permanent magnets at slot openings. Appears as harmonics of order $\text{LCM}(P, Q)$ ($P$: pole number, $Q$: slot number).
Magnetomotive Force (MMF) Harmonics: Because the spatial distribution of windings is not an ideal sine wave, spatial harmonics of 5th, 7th, 11th, 13th... orders are generated, causing torque pulsation through interaction with the fundamental flux.
Distortion due to Magnetic Saturation: Magnetic saturation of the iron core makes the B-H curve nonlinear, distorting the air gap flux density distribution. The heavier the load, the greater the influence.
Inverter-Induced Ripple: Current ripple due to PWM switching generates high-frequency torque pulsation. Appears in frequency bands around twice the carrier frequency $f_c$.

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So it's not a case of "just dealing with cogging torque is OK." The ripple during current flow is the real deal...

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Exactly. In actual EV motor development, cogging torque often accounts for only about 20-30% of the total torque ripple. The rest is occupied by MMF harmonics and saturation effects. That's why in FEM analysis, it's necessary to always evaluate torque ripple using transient analysis under current flow conditions.

Governing Equation: Magnetic Vector Potential Method

In 2D electromagnetic field analysis of motors, the diffusion equation for the magnetic vector potential $A_z$ is solved:

$$ \nabla \times \left(\frac{1}{\mu}\nabla \times \mathbf{A}\right) = \mathbf{J}_s - \sigma\frac{\partial \mathbf{A}}{\partial t} + \nabla \times \mathbf{M} $$

Here $\mathbf{J}_s$ is the source current density, $\sigma$ is the conductivity (eddy current term), and $\mathbf{M}$ is the magnetization vector of the permanent magnet. By discretizing this equation with FEM and solving it with time stepping, the torque at each rotation angle is calculated.

Dimensional Analysis and Typical Values

Physical Quantity	SI Unit	Typical Value in IPMSM
Air Gap Flux Density $B_g$	T (Tesla)	0.7〜1.0 T
Remanent Flux Density $B_r$ (NdFeB)	T	1.1〜1.4 T
Iron Core Saturation Flux Density	T	1.6〜2.0 T (Silicon Steel)
Air Gap Length $g$	mm	0.5〜1.5 mm
Torque Ripple Ratio	%	3〜15% (Design Target <5%)
Cogging Torque	% of rated torque	1〜5%

Numerical Methods and Implementation

Torque Calculation Methods Using FEM

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What specific steps are involved in actually calculating torque ripple with FEM?

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The basic flow is three steps. First, Step 1: Create a 2D cross-sectional model of the motor and mesh the area around the air gap sufficiently finely. Step 2: Rotate the rotor in small angle increments, solving the magnetic field equation with nonlinear iteration at each step. Step 3: Integrate the Maxwell stress tensor over a closed surface in the air gap...