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?
125 ampere-turns
500 ampere-turns
250 ampere-turns
2 ampere-turns
In SI units, magnetic flux density B is measured in which unit?
Weber (Wb)
Ampere per metre (A/m)
Henry per metre (H/m)
Tesla (T)
On a hysteresis loop, coercive force is best described as:
The flux density remaining when magnetising force H is reduced to zero
The reverse magnetising force H required to reduce flux density B to zero
The reluctance of an air gap in the magnetic circuit
The permeability of free space μ0
Why are transformer and inductor cores often laminated?
To increase eddy currents so the core heats and dries out moisture
To raise reluctance by forcing all flux through air gaps between sheets
To interrupt large circulating eddy-current paths and thereby reduce eddy-current losses
To convert the core into a permanent magnet with high coercive force
Sections you finish are checked off in the contents.