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Analytical Methods for Brushless Permanent Magnet Motors – Slot Modeling

Rakesh Dhawan, BTech, MSEE, MBA

In brushless permanent magnet (PM) motor design, accurate modeling of the air gap is critical for predicting magnetic field distribution and optimizing motor performance. One of the key challenges in slot modeling is the effect of slot openings on the effective air gap. Due to the presence of stator slots, the magnetic field distribution in the air gap is not uniform, and flux tends to bulge outward before entering the rotor. This phenomenon, known as the fringing or slotting effect, increases the reluctance of the air gap and alters the motor’s magnetic characteristics. Carter’s coefficient corrects this non-uniform flux distribution and determines the effective air gap length, directly influencing torque production, efficiency, and inductance calculations.

Looking at the image above, you can tell that the flux has to travel a longer path in the slot openings. Intuitively, the effective air gap is increased, and we can also expect distortion of the flux distribution at the slot opening boundaries. This is inside the motor.
Looking at the image above, you can tell that the flux has to travel a longer path in the slot openings. Intuitively, the effective air gap is increased, and we can also expect distortion of the flux distribution at the slot opening boundaries. This is inside the motor.
Looking at the motor from a side view also indicated flux leakage and distortion due to the slot opening and magnet structure.
Looking at the motor from a side view also indicated flux leakage and distortion due to the slot opening and magnet structure.

Carter’s coefficient (kc​) is a correction factor that accounts for the influence of slot openings on the air gap reluctance inside the motor. It does not take into account the flux leakage, as shown above in the side view of the motor. Additional correction factors may need to be considered because of the 3-D geometry.

Carter’s coefficient (kc​) is given by:

kc=g \geff​

Where:

  • g = actual physical air gap,
  • geff​ = effective air gap larger than g due to slotting effects.

The value of kc​ depends on the ratio of slot opening width to slot pitch and the relative permeability of the iron core. A larger slot opening increases kck_ckc​, effectively increasing the air gap reluctance and reducing flux linkage. This impacts motor performance by decreasing torque and increasing harmonics in the back electromotive force (EMF). We use Carter’s coefficient to adjust the motor’s magnetic circuit model, ensuring more accurate predictions of inductance, air gap flux density, and overall efficiency. By correctly accounting for slot effects, we can optimize slot geometry, improve flux distribution, and achieve better control over motor performance.

Slot Modeling in an Exterior Rotor Permanent Magnet Motor.
Slot Modeling in an Exterior Rotor Permanent Magnet Motor.
Front View of the Electromagnetic Architecture defining Slot Width, Tooth Width, Slot Pitch, etc. for Slot Modeling.
Front View of the Electromagnetic Architecture defining Slot Width, Tooth Width, Slot Pitch, etc. for Slot Modeling.

Conformal Mapping Technique for Slot Modeling by Carter

The conformal mapping technique is a mathematical transformation method used to model the influence of stator slots in electrical machines. It plays a crucial role in calculating the effective air gap length in slot modeling for brushless permanent magnet (PM) motors. Carter’s Method, which applies conformal mapping, corrects for the impact of slot openings on the motor’s magnetic circuit by transforming the complex slot geometry into a simplified equivalent form, making analytical calculations more feasible.

Concept of Conformal Mapping in Slot Modeling

Conformal mapping preserves angles and transforms complex geometries into more manageable shapes while maintaining the relative proportions of the electromagnetic field. In slot modeling, the technique helps in:

  • Converting the actual slotted air gap into an equivalent smooth air gap.
  • Correcting for flux bulging and fringing at the slot openings to estimate magnetic reluctance.
  • Improving the accuracy of air gap permeance calculations.

The conformal mapping transformation modifies the shape of the slot into an equivalent smooth surface, allowing the air gap field distribution to be analyzed without complex numerical methods.

Application in Brushless PM Motors

  • Accurate Magnetic Field Distribution: By modeling the actual effect of slots, designers can predict the precise flux distribution in the air gap, reducing errors in torque and inductance calculations.
  • Reduction of Slot Harmonics: The fringing effect caused by slot openings introduces harmonics in the air gap flux. Conformal mapping helps in optimizing slot shapes to minimize these harmonics.
  • Performance Optimization: This technique allows us to design optimal slot geometries that reduce unwanted losses and improve motor efficiency.

In summary, Carter’s conformal mapping technique allows us to correct slot effects in brushless PM motors, ensuring accurate modeling of air gap behavior and improved performance predictions.

Simplified Geometry of a Complex Electromagnetic Architecture to Analyze Air Gap Flux Distribution.
Simplified Geometry of a Complex Electromagnetic Architecture to Analyze Air Gap Flux Distribution.

Flux Crossing the Gap Over the Slot: Increased Path Length and Its Implications

In brushless permanent magnet (PM) motors, the presence of stator slots introduces a discontinuity in the air gap, affecting the way magnetic flux travels across the gap. Unlike flux crossing the air gap directly over solid iron surfaces, the flux that crosses over a slot opening follows a longer, more resistive path before reaching the highly permeable material on the other side. This is due to the slotting effect, which causes magnetic flux to bulge outward (fringing effect), increasing the effective air gap length and thereby impacting motor performance.

In conclusion, the effective air gap is always more significant than the actual air gap as Kc > 1.

This fundamental equation allows us to determine the contribution to magnetic permeance due to slot openings.
This fundamental equation allows us to determine the contribution to magnetic permeance due to slot openings.
Carter presented three analytical expressions.
Carter presented three analytical expressions.
The above plot has the x-axis as the ratio of air gap and slot pitch and plots Kc1, Kc2, and Kc3. For smaller air gaps, the fringing flux is significant, reducing the effective air gap by up to 9%.
The above plot has the x-axis as the ratio of air gap and slot pitch and plots Kc1, Kc2, and Kc3. For smaller air gaps, the fringing flux is significant, reducing the effective air gap by up to 9%.

Let us consider an example as follows:

g = 1 mm
Ws = 3 mm
Wt = 19.5 mm
Ts = 22.5 mm

The Air Gap is varied while keeping the slot width and pitch constant. Smaller air gaps contribute more to magnetic permeance.
The Air Gap is varied while keeping the slot width and pitch constant. Smaller air gaps contribute more to magnetic permeance.
Permeance contribution is related to the variation of the air gap.
Permeance contribution is related to the variation of the air gap.
The air gap is kept constant as we vary the slot width. As slot width increases, the magnetic permeance contribution increases.
The air gap is kept constant as we vary the slot width. As slot width increases, the magnetic permeance contribution increases.
Permeance contribution as slot width is varied and magnetic permeance is kept constant.
Permeance contribution as slot width is varied and magnetic permeance is kept constant.

Conclusion: Ideally, we want the smallest possible air gap. However, manufacturing considerations prevent us from doing so. Having the smallest possible slot width helps us minimize the slot width’s fringing effect. See a more detailed conclusion here.