Hysteresis loss in ferromagnetic cores – frequency dependence Hysteresis power loss in a magnetic core operated at fixed flux density waveform amplitude is:

Introduction / Context:
Core losses in transformers and inductors comprise hysteresis loss and eddy-current loss. Distinguishing their frequency dependence is essential for material selection and for estimating heat dissipation over operating ranges.

Given Data / Assumptions:

• Magnetic core cycled sinusoidally at peak flux density B_max within the material's normal operating range.
• Waveform and B_max are held constant while frequency f varies.
• Hysteresis is characterized by a loop area proportional to B_max^n (n ≈ 1.6–2 for many steels).

Concept / Approach:

The energy dissipated per cycle due to hysteresis equals the loop area in the B–H plane and is approximately k_h * B_max^n per unit volume (Steinmetz form). Power loss equals energy per cycle times the number of cycles per second, so P_h ∝ f * B_max^n. At fixed B_max, this reduces to P_h ∝ f, i.e., proportional to frequency.

Step-by-Step Solution:

Verification / Alternative check:

Measurements on silicon steel and ferrites show linear frequency scaling of hysteresis loss over practical ranges when B_max is held constant, while eddy-current loss scales as f^2.

Why Other Options Are Wrong:

(b) ignores cycling rate; (c) corresponds to eddy-current loss, not hysteresis; (d) states the general Steinmetz form but the question fixes B_max, reducing it to (a).

Common Pitfalls:

Mixing hysteresis and eddy losses or comparing at fixed applied voltage rather than fixed B_max (which changes the scaling).

Final Answer:

proportional to frequency