9.3 MMF, Flux Density, Permeability, Hysteresis & Eddy Currents
Key Takeaways
- Magnetomotive force MMF = N × I (ampere-turns); magnetic field strength H is MMF per length of magnetic path (A/m)
- Flux density B = Φ/A (tesla); in magnetic materials B = μH where permeability μ = μrμ0
- The hysteresis loop shows B versus H through a magnetising cycle; retentivity (remanence) and coercive force are loop intercepts
- Reluctance opposes flux (magnetic analogue of resistance); saturation means further H gives little rise in B
- Eddy currents are induced loops in conducting cores under changing flux; laminations and ferrite cores reduce eddy-current losses
9.3 MMF, Flux Density, Permeability, Hysteresis & Eddy Currents
Quick Answer: MMF = NI (ampere-turns). H = NI/ℓ (A/m). B = Φ/A (T) and B = μH with μ = μrμ0. Reluctance resists flux; saturation flattens B–H. The hysteresis loop gives remanence and coercive force. Changing flux induces eddy currents in solid cores — use laminations or ferrites to cut losses.
Qualitative magnets and hand rules are not enough for Module 3. Topic 3.10 expects the magnetic-circuit quantities, the B–H (hysteresis) story, and eddy-current losses that reappear in transformers, inductors, and AC machines.
Magnetic Quantities and Units (Memorise This Table)
| Quantity | Symbol | Defining idea | SI unit |
|---|---|---|---|
| Magnetomotive force | MMF or ℱ | Cause of flux in a magnetic circuit; NI | ampere-turn (A) |
| Magnetic field strength | H | MMF per metre of path; H = NI/ℓ | ampere per metre (A/m) |
| Magnetic flux | Φ (phi) | “Amount” of magnetism through a surface | weber (Wb) |
| Flux density | B | Flux per unit area; B = Φ/A | tesla (T) = Wb/m² |
| Permeability | μ | B = μH in linear media | henry per metre (H/m) |
| Relative permeability | μr | μ = μrμ0 | dimensionless |
| Free-space permeability | μ0 | 4π × 10⁻⁷ H/m | H/m |
| Reluctance | ℛ | Magnetic opposition; ℛ = ℓ/(μA) | A/Wb (or H⁻¹) |
| Reluctance “Ohm’s law” | — | Φ = MMF / ℛ | — |
Analogy to the electric circuit (exam favourite):
| Electric | Magnetic |
|---|---|
| EMF (V) | MMF (ampere-turns) |
| Current I | Flux Φ |
| Resistance R | Reluctance ℛ |
| Conductivity / low R path | High μ / low ℛ path |
Magnetomotive Force and Field Strength
MMF = N × I
- N = number of turns
- I = current in amperes
- Product = ampere-turns
Worked example 1. A coil of 500 turns carries 0.4 A.
MMF = 500 × 0.4 = 200 ampere-turns.
If the mean magnetic path length in the core is ℓ = 0.25 m and the path is uniform:
H = MMF / ℓ = 200 / 0.25 = 800 A/m.
Field strength H depends on geometry and ampere-turns; it is not the same as flux density B. Materials respond to H by producing B according to their permeability and saturation behaviour.
Flux, Flux Density, and Permeability
Φ = B × A for uniform B perpendicular to area A, or B = Φ/A.
Worked example 2. Flux Φ = 0.006 Wb through a core cross-section A = 0.002 m².
B = 0.006 / 0.002 = 3.0 T (a high but illustrative training number — many machines run at lower peak B).
In a material (linear region):
B = μH = μrμ0 H
| Material type | Typical μr (order) | Comment |
|---|---|---|
| Free space / air | 1 | Reference |
| Soft iron / silicon steel | Hundreds to thousands | Cores, good flux concentrators |
| Ferrites | Tens to thousands (grade-dependent) | High-frequency cores |
| Permanent-magnet materials | Effective behaviour is nonlinear / different design focus | Hard magnets |
Worked example 3. In air, μr ≈ 1, μ0 = 4π × 10⁻⁷. For H = 800 A/m:
B ≈ μ0H ≈ 4π × 10⁻⁷ × 800 ≈ 1.0 × 10⁻³ T (about 1 mT).
Same H in soft iron with μr = 2000 would give B ≈ 2000 times larger in the linear model — until saturation intervenes. That is why cores matter.
Reluctance and Magnetic Circuits
ℛ = ℓ / (μA)
| Change | Effect on reluctance |
|---|---|
| Longer path ℓ | ℛ increases |
| Larger area A | ℛ decreases |
| Higher μ | ℛ decreases |
| Air gap inserted | ℛ rises sharply (μ ≈ μ0 in the gap) |
Φ = MMF / ℛ — larger ampere-turns or lower reluctance → more flux.
Air-gap insight. Relay and contactor magnetic circuits include an air gap when open. Large gap → high reluctance → less flux → weaker pull until the armature closes and reluctance falls. That matches the tactile “pull-in” behaviour of many devices.
Saturation
As H increases, B rises steeply at first in ferromagnetic materials, then flattens: almost all domains are aligned. Further increases in current (H) produce little extra B. That region is magnetic saturation.
| Consequence | Practical meaning |
|---|---|
| Diminishing returns | More coil current buys little extra flux |
| Distortion | AC flux waveforms can distort when driven into saturation |
| Heating / surge currents | Transformer magnetising current spikes if saturated |
| Design limit | Machines and transformers are rated with peak B below deep saturation |
Hysteresis Loop, Retentivity, and Coercive Force
Plot B against H while cycling the magnetising current through positive and negative peaks. Ferromagnetic materials trace a closed hysteresis loop, not a single-valued straight line.
| Loop feature | Meaning |
|---|---|
| Rising/falling paths differ | B lags H — energy is dissipated per cycle (hysteresis loss) |
| Remanence / retentivity (Br) | Flux density remaining when H returns to 0 after magnetisation |
| Coercive force (Hc) | Reverse H needed to drive B back to 0 |
| Loop area | Proportional to hysteresis energy loss per cycle per unit volume |
| Wide loop / high Hc | Hard magnetic material — permanent magnets |
| Narrow loop / low Hc | Soft magnetic material — cores, low hysteresis loss |
Definitions in exam wording:
- Retentivity (remanence): ability to retain magnetism when the magnetising force is removed — high in permanent magnets.
- Coercive force (coercivity): magnetising force needed to remove residual magnetism — high in hard magnets (hard to demagnetise), low in soft cores (easy to demagnetise).
Worked concept. Soft silicon steel for a transformer: narrow loop → low hysteresis loss at 400 Hz or 50/60 Hz as applicable. Hard magnet alloy: wide loop → stays magnetised for a speaker or tach generator magnet.
Eddy Currents
A changing magnetic flux through a conducting core induces EMFs (Faraday — expanded in the inductance chapter). Those EMFs drive circulating eddy currents within the bulk metal. Eddy currents:
- Produce I²R heating (eddy-current loss)
- Create opposing fields (Lenz) that can be useful in damping instruments but are unwanted in transformer cores
Reducing eddy currents
| Method | How it helps |
|---|---|
| Laminated cores | Thin sheets insulated from each other interrupt large eddy paths; eddy loops confined to thin laminations |
| Higher resistivity core materials | Ferrites — poor electrical conductors, low eddy loss at high frequency |
| Thinner laminations | Smaller eddy loop area / path → less loss (especially as frequency rises) |
Aircraft angle. Many aircraft AC systems use 400 Hz. Eddy and hysteresis losses scale unfavourably with frequency if cores are poorly designed — hence laminated silicon steel or appropriate magnetic materials in transformers, instrument transformers, and machine stators/rotors.
Worked qualitative stem. “Why laminate a transformer core?” → reduce eddy currents / eddy-current loss by breaking up conducting paths perpendicular to the flux, while still providing a high-μ path for flux along the lamination plane.
Combined Loss Picture
| Loss type | Cause | Reduced by |
|---|---|---|
| Hysteresis loss | Domain friction / loop area each AC cycle | Soft magnetic materials (narrow loop) |
| Eddy-current loss | Induced loops in conducting core | Laminations, ferrites, higher resistivity |
| Copper loss (preview) | I²R in windings | Wire size, lower current — not a core magnetic loss |
Total core loss ≈ hysteresis + eddy. Module 3 wants you to name and mitigate the magnetic parts here.
Formula Drill Summary
| Need | Use |
|---|---|
| Ampere-turns | MMF = NI |
| Field strength | H = NI/ℓ |
| Flux density from flux | B = Φ/A |
| Material link | B = μH = μrμ0H |
| Reluctance | ℛ = ℓ/(μA) |
| Flux from MMF | Φ = MMF/ℛ |
Exam Scenario Set
Scenario A — Same NI, add air gap. Reluctance up → Φ down → weaker electromagnet pull.
Scenario B — Soft vs hard on the loop. Soft: small Br and Hc after cycling used as a core. Hard: large Br — permanent magnet behaviour.
Scenario C — Solid vs laminated core at 400 Hz. Solid steel overheats from eddy currents; laminated core runs cooler for the same AC flux.
Scenario D — Saturation. Doubling current near saturation does not double B; magnetising current becomes disproportionate.
Lock the units table, the electric–magnetic analogy, loop vocabulary (retentivity / coercive force), and eddy-current laminations. Together with §9.1 materials and §9.2 electromagnets and hand rules, that is the complete Module 3 magnetism package before inductance and machines build on it.
A coil of 250 turns carries 2.0 A. What is the magnetomotive force?
In SI units, magnetic flux density B is measured in which unit?
On a hysteresis loop, coercive force is best described as:
Why are transformer and inductor cores often laminated?