In the design of single-phase induction motor. The flux density in the stator core is given by

Question

Examoogle · Accepted Answer

Option A: 𝑩𝒄𝒔 = ∅𝒎 / ( 𝟐 × 𝒅𝒄𝒔 × 𝑳𝒊 )

Explanation & Concepts:
In a single-phase induction motor, the magnetic flux $\Phi_m$ splits into two paths through the stator core. The flux density $B_{cs}$ is calculated as the flux divided by twice the cross-sectional area of the core path, where the area is the product of the core depth $d_{cs}$ and the net length $L_i$.

Working Principle: The flux path in a stator core is typically closed. Since the magnetic circuit is divided into two parallel paths, the total flux $\Phi_m$ is shared equally between these paths. Thus, the flux in each path is $\frac{\Phi_m}{2}$. The flux density is defined as the flux per unit area, resulting in the expression $B_{cs} = \frac{\Phi_m}{2 	imes d_{cs} 	imes L_i}$.

Key Formulas: $B_{cs} = \frac{\Phi_m}{2 \cdot d_{cs} \cdot L_i}$ — Flux density in the stator core; $L_i = L \cdot K_i$ — Net core length (where $K_i$ is the stacking factor)

Key Points: Flux density calculation considers the splitting of the total flux into two core branches.; $d_{cs}$ represents the depth of the stator core behind the slots.; $L_i$ is the effective length of the iron core after accounting for the stacking factor ($L_i = L 	imes K_i$).; Proper calculation of $B_{cs}$ is essential to avoid saturation in the stator magnetic circuit.

Advantages: Ensures optimal material utilization for the stator core.; Prevents overheating due to excessive hysteresis and eddy current losses.

Disadvantages: Assumes uniform flux distribution, which may vary slightly at the tooth tips.; Neglects leakage flux which technically exists but is excluded in the primary core flux density design.

Applications: Design of fractional horsepower motors.; Magnetic circuit optimization in single-phase induction machines.

Additional Info: Option B is incorrect as it uses gross length L instead of net length $L_i$.. Stacking factor $K_i$ is typically in the range of 0.9 to 0.95 for electrical steel laminations.

Answer: A is correct — The formula accounts for the bifurcation of the magnetic flux into two parallel paths in the stator yoke.

Exam Tip: Always remember to use the 'net' length $L_i$ (stacking factor) rather than the 'gross' length $L$ whenever calculating flux density in electrical machines.

Examoogle · Answer

Option B: 𝑩𝒄𝒔 = ∅𝒎 / ( 𝟐 × 𝒅𝒄𝒔 × 𝑳 )

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Option C: 𝑩𝒄𝒔 = ∅𝒎 × 𝟐 × 𝒅𝒄𝒔 × 𝑳𝒊

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Option D: 𝑩𝒄𝒔 = ∅𝒎 × 𝟐 × 𝒅𝒄𝒔 × 𝑳

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