Eddy-current loss dependence on flux density If B_max is the maximum flux density in a magnetic core, how does the eddy-current power loss vary with B_max (keeping frequency and lamination thickness constant)?

Introduction / Context:Eddy-current losses arise from circulating currents induced in conductive core materials by time-varying magnetic flux. Recognizing the correct dependence on flux density is essential for core design, lamination choices, and loss estimation.

Given Data / Assumptions:

• Laminated core with fixed lamination thickness t and electrical conductivity.
• Sinusoidal excitation at fixed frequency f.
• Maximum flux density B_max is the variable of interest.

Concept / Approach:Classical eddy-current loss per unit volume follows P_e ∝ f^2 * t^2 * B_max^2 for sinusoidal excitation. The square dependence on B_max comes from Faraday’s law (induced emf proportional to dB/dt) and Joule heating proportional to current squared.

Step-by-Step Solution:Induced emf ∝ dΦ/dt ∝ A * dB/dt ∝ A * ω * B_max.Eddy current density J ∝ induced emf / resistance ∝ ω * B_max (for fixed geometry).Power density P ∝ J^2 * ρ_eff^−1 → P_e ∝ (ω^2) * B_max^2.At fixed f and t, P_e ∝ B_max^2.

Verification / Alternative check:Standard core-loss separation uses Steinmetz-like forms: total loss = hysteresis (∝ f * B_max^n with n ≈ 1.6–2.0) plus eddy-current (∝ f^2 * B_max^2). Holding f constant isolates the B_max^2 term for eddy currents.

Why Other Options Are Wrong:Linear or cubic dependencies do not follow from the classical derivation. The 1.6 exponent pertains to hysteresis loss, not eddy-current loss. Independence from B_max contradicts Faraday induction.

Common Pitfalls:

• Mixing hysteresis and eddy-current exponents.
• Ignoring the role of lamination thickness and frequency in the full expression.

Final Answer:Proportional to B_max^2