Category: Analytical Methods

Rakesh Dhawan, BTech, MSEE, MBA

In brushless permanent magnet (PM) motors, the air gap is crucial in determining motor efficiency, torque production, and overall electromagnetic performance. Ideally, a smaller air gap is preferred because it results in higher magnetic flux density, stronger coupling between rotor and stator, improved efficiency, and reduced weight. However, in practical applications, manufacturing limitations and mechanical constraints prevent us from reducing the air gap beyond a specific limit.

Similarly, the slot width directly impacts the fringing effect, which occurs when magnetic flux bulges outward at the slot openings before crossing the air gap. The wider the slot opening, the more significant the fringing impact, which increases the effective air gap length and introduces additional reluctance to the magnetic circuit. This leads to flux leakage, reduced torque production, and increased harmonics in the back EMF waveform.

Challenges in Minimizing the Air Gap

1. Manufacturing Tolerances
   • Achieving a minimal air gap requires high-precision machining and tighter assembly tolerances.
   • Even minor misalignments in the rotor-stator assembly can cause uneven air gaps, leading to localized flux variations and torque ripple.
2. Mechanical Stability and Thermal Expansion
   • A minimal air gap makes the motor more susceptible to rotor eccentricity and thermal expansion, leading to unwanted physical contact between the rotor and stator.
   • To prevent this, designers must allow for a manufacturing tolerance in the air gap, balancing performance with mechanical reliability.
3. Vibration and Noise Considerations
   • Reducing the air gap increases magnetic attraction forces between the rotor and stator teeth, leading to higher cogging torque, vibration, and acoustic noise.
   • This requires additional design optimizations like skewing the stator slots or using non-uniform tooth designs.

Importance of Minimizing Slot Width

While reducing the air gap is limited by mechanical factors, minimizing slot width is a practical approach to reducing the fringing effect. Narrower slot openings result in:

1. Lower Effective Air Gap
   • A narrower slot opening means that flux lines do not need to bend as much, reducing the bulging and fringing effect and maintaining a more uniform field distribution across the air gap.
2. Reduced Harmonics and Torque Ripple
   • Wide slot openings cause flux pulsations and introduce harmonics, contributing to torque ripple and motor noise.
   • A smaller slot width results in smoother flux transitions, improving torque stability.
3. Higher Flux Linkage and Efficiency
   • When slot openings are narrow, more flux is directed into the rotor rather than leaking outward.
   • This improves flux linkage and ensures that more magnetic energy is converted into proper torque.

Practical Trade-offs in Slot Design

• If the slot width is too narrow, winding the stator becomes challenging, which can lead to difficulties in inserting and cooling the copper coils unless one uses a segmented design.
• If the slot width is too wide, flux leakage increases, requiring Carter’s coefficient correction to adjust for increased air gap reluctance.

Conclusion

While a smaller air gap is ideal for optimal motor performance, manufacturing constraints, and mechanical considerations set practical limits. To compensate for this, reducing slot width effectively minimizes fringing effects and ensures efficient flux utilization. Proper design trade-offs, including optimized slot geometry, precision machining, and skewing techniques, are essential to achieving a high-performance brushless PM motor.

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.

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

k_c=g \g_eff​

Where:

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

The value of k_c​ 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.

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.

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.

Let us consider an example as follows:

g = 1 mm
W_s = 3 mm
W_t = 19.5 mm
T_s = 22.5 mm

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.

Rakesh Dhawan, BTech, MSEE, MBA

Abstract

Due to their efficiency and reliability, Brushless Permanent Magnet (PM) Motors are widely used in high-performance applications. This paper explores analytical methods for modeling and designing such motors. It discusses key aspects like air gap modeling, slot modeling, core loss analysis, and permanent magnet circuit modeling. These analytical techniques help optimize motor performance and improve design efficiency. The study also addresses critical design implications such as air gap corrections, slot width considerations, and the impact of permanent magnets on motor operation.

1. Introduction

Brushless PM motors offer power density, efficiency, and control precision advantages. However, their design complexity requires robust analytical techniques. This paper provides a detailed study of fundamental analytical methods for designing and optimizing these motors. The key focus areas include air gap modeling, slot modeling, and the magnetic circuit representation of PM materials.

1.1 Importance of Analytical Methods in PM Motor Design

Analytical modeling provides insight into the motor’s electromagnetic behavior, reducing reliance on extensive finite element simulations. This enables quick design iteration, leading to more efficient and cost-effective development.

2. Analytical Methods for Brushless PM Motors

2.1 Magnetic Circuit Concepts

The design of brushless PM motors relies on understanding the fundamental properties of magnetic circuits. The governing principles include:

• Magnetic flux paths
• Permeance variations
• Magnetic reluctance calculations

2.2 Air Gap Modeling

Permeance of an Air Gap measures the ease with which a magnetic flux can pass through an air gap in a magnetic circuit. It is the reciprocal of reluctance and is given by:

P=μ₀A/g​ … Equation 1

where:

• P = Permeance (measured in Henries, H)
• μ₀ = Permeability of free space (4π×10⁻⁷ H/m)
• A = Cross-sectional area of the air gap (m²)
• g = Length of the air gap (m)

Since air has a much lower permeability than ferromagnetic materials, the air gap introduces a significant reluctance in the magnetic circuit, which affects the overall magnetic performance of devices like transformers, inductors, and electric motors.

The air gap is critical in defining the motor’s magnetic field distribution. The air gap permeance is determined using the same equation as above:

P_g​=μ₀​A​/g … Equation 2

Where Pg​ is the air gap permeance, μ₀​ is the permeability of free space, A is the cross-sectional area, and g is the air gap length.

Here are the permeance values for different air gap lengths, assuming a cross-sectional area of 100 mm² (0.0001 m²):

• 1 mm air gap → 1.26×10⁻⁷H
• 2 mm air gap → 6.28×10⁻⁸ H
• 3 mm air gap → 4.19×10⁻⁸ H

As expected, the permeance decreases as the air gap increases, reducing the ease of magnetic flux flow.

What happens in a transformer with an air gap?

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In Fig. 1, our primary focus is on the leakage flux around the air gap. If we define a leakage radius of x at the end of the air gap, the permeance for a differential segment dx can be calculated. The effective length of the flux path becomes g + πx, while the additional effective area to consider is L·dx. P_f represents the additional contribution to the total permeance due to the leakage flux.

Let us consider an example:

L = 12 mm
g = 2mm
A = 50 mm²
Pg = 3.14*10^-05 H

If we account for x in the leakage flux, we can use the above equation to determine Pf, representing the additional permeance the leakage flux contributed. The graph at the top (burgundy) shows the percentage contribution, and the graph at the bottom (sky blue) shows the actual value of the contribution. 50% contribution value is achieved at x= 15mm

Effects of Higher Permeance Due to Leakage Flux on Transformer Performance

When higher permeance is caused by leakage flux, it introduces several adverse effects on transformer performance:

1. Reduced Magnetic Coupling
   • Leakage flux does not effectively link both primary and secondary windings, leading to weaker electromagnetic coupling.
   • This results in reduced energy transfer efficiency and increased losses.
2. Increased Leakage Inductance
   • Higher leakage permeance leads to increased leakage inductance, which opposes rapid changes in current.
   • This can cause voltage spikes and transients, affecting the performance of power electronics circuits.
3. Poor Voltage Regulation
   • With increased leakage inductance, the output voltage drops significantly under load conditions.
   • This makes the transformerless effective in applications requiring stable voltage output.
4. Higher Core Losses & Heating
   • More leakage flux results in additional eddy current and hysteresis losses, leading to excessive heating.
   • Increased temperature can reduce the lifespan of the transformer and degrade insulation materials.
5. Reduced Power Transfer Efficiency
   • Since leakage flux does not contribute to power transfer, some magnetic energy is wasted.
   • This decreases the overall efficiency of the transformer.
6. Potential Electromagnetic Interference (EMI)
   • Stray magnetic fields from leakage flux can interfere with nearby electronic components.
   • This is especially problematic in high-frequency transformers used in power electronics.

Key takeaways:

1. As we move further from the air gap, the contribution of differential permeances decreases.
2. The exact values chosen are not that critical.
3. As x increases beyond 10g, the total air gap permeance changes little.
4. Higher permeance due to leakage flux negatively impacts transformer performance by increasing losses, reducing voltage regulation, and decreasing efficiency.
5. To mitigate these effects, we need to control and optimize the leakage flux using better winding arrangements, magnetic shielding, or air gaps.

DISCUSSION: Discuss the implications of large air gaps concerning the motor design with thin stacks. How do you balance stack Height vs. air Gap? See the answer here.