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Ferromagnetic cores and saturation

Key idea: A ferromagnetic core increases flux linkage and inductance. Near saturation, further current gives little extra magnetisation, so a constant-inductance model no longer applies.

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  • GCE A-Level H3 Physics 2027
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An air-cored coil and a coil with a ferromagnetic core can respond differently to the same current change. Recall that a larger induced e.m.f. for the same current gradient means a larger self-inductance. A core can add a great deal of inductance, but that enhancement cannot keep growing in proportion to the current.

Check your understanding 1 Predict first

An unsaturated iron core is slid into a coil. The current through the coil is then changed at the same rate as before. How does the induced e.m.f. compare with the air-cored coil?

Induced e.m.f. with the core
Show answer

Larger. The core’s domains align with the coil’s field and add to it, so the flux linkage changes more for the same current change: the inductance is larger. This boost shrinks as the core approaches saturation.

A ferromagnetic core enhances inductance

Ferromagnetic materials such as iron contain magnetic domains: regions with aligned magnetic moments. A coil’s field can change their relative alignment and sizes. The core then adds to the magnetic field and linked flux produced by the current.

For the same rate of current change, the enhanced flux-linkage response generally produces a larger induced e.m.f. and hence a larger effective inductance than the corresponding air-cored coil. Compare equal current gradients, rather than merely equal total current changes.

Near saturation, much of the available magnetisation is already aligned. A further current increase then produces less additional magnetic response from the core. Saturation does not mean that the total magnetic field becomes zero or that further current changes produce no field change at all.

Guided practice 1

Comparing a core’s incremental response

About 5 min

Problem

A coil’s current rises uniformly by 0.40 A in 0.10 s in each of two small intervals. Far from saturation its flux linkage increases by 0.80 Wb turns; near saturation it increases by 0.20 Wb turns. Estimate the incremental inductance and induced-e.m.f. magnitude in each interval. Does the smaller second response imply that the existing field has vanished?

Try this before viewing the solution

Unit: H
Unit: H
Unit: V
Unit: V
Has the existing field vanished near saturation?

Hints

Hint 1

track incremental response

Compare flux-linkage change per unit current change, then per unit time.

Show solution step by step

  1. Compare incremental inductances

    Method

    The estimates are 2.0 H far from saturation and 0.50 H near saturation.

    Reason

    Incremental inductance is the small flux-linkage change divided by the current change.

    Working

    Answer

    L₁ = 0.80/0.40 = 2.0 H; L₂ = 0.20/0.40 = 0.50 H.

  2. Find e.m.f. and interpret saturation

    Method

    The induced-e.m.f. magnitudes are 8.0 V and 2.0 V. The field has not vanished.

    Reason

    Faraday’s law uses the rate of flux-linkage change; saturation reduces the additional response rather than removing the existing magnetisation.

    Working

    Answer

    |E₁| = 0.80/0.10 = 8.0 V and |E₂| = 0.20/0.10 = 2.0 V. Equal current gradients produce different responses, so one constant inductance does not fit both intervals.

Scroll across the figure to read all labels.

Schematic domain model, not measured data. Near saturation, little extra alignment remains available. The core still has magnetisation; its additional response is smaller.

Incremental inductance means the coil’s response to a small additional current change at its present operating condition. A smaller additional flux-linkage change for the same current change means a smaller incremental inductance. This is different from saying that the existing field has vanished.

Apply the limitation to a circuit model

For a constant-L RL circuit, the time constant is τ = L/R. Compare local behaviour at the same current, resistance and applied voltage; otherwise a changed current gradient could have several causes.

Try it yourself 2

Why a core can change RL transient behaviour (qualitative)

Minimal support

Problem

For a constant-L resistor–inductor circuit, the current-rise time constant is τ = L/R. A real circuit instead uses a saturating ferromagnetic core. Compare at the same instantaneous current, resistance and applied voltage. Infer how a falling incremental inductance affects the local current-rise rate. Does one constant time constant still describe the whole rise?

Try this before viewing the solution

At the same current, R and applied voltage, a smaller incremental L makes the current rise
Does one constant τ = L/R describe the whole rise?

Hints

Hint 1

hold the driving effect fixed

Write the circuit equation and keep current, resistance and applied voltage fixed.
Show solution step by step

  1. Apply saturation

    Method

    Effective incremental L can decrease as current rises.

    Reason

    The core adds progressively less flux response.

    Working

    Answer

    I↑ ⇒ L_eff↓

  2. Hold the circuit conditions fixed

    Method

    The driving voltage left after the resistive drop is unchanged in this comparison.

    Reason

    The current, resistance and applied voltage are the same; only the incremental inductance differs.

    Working

    Answer

    I, R, V_applied held fixed

  3. Infer transient shape

    Method

    For the same instantaneous current and applied voltage, current rises faster than a model retaining the larger unsaturated inductance predicts.

    Reason

    The circuit becomes less inductive as saturation develops.

    Working

    Answer

    non-linear current growth

Common mistakes

  • Saying a saturated core has no magnetic field. It is strongly magnetised; only its additional response to a current increase is small.
  • Comparing inductances using equal total current changes instead of equal current gradients.
  • Using one constant time constant τ = L/R for a coil whose core saturates during the current rise.
  • Assuming a ferromagnetic core adds a fixed amount of inductance at every current.

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