Magnetic Circuits and Reluctance
Ohm's law for magnetism — and the three places the analogy breaks, which are what actually size a machine.
Skip to the animationMMF drives flux against reluctance exactly as EMF drives current against resistance, which lets a magnetic circuit be solved with Ohm's law — and the three places that analogy breaks are what actually determine a machine's size.
The analogy, term for term
| Electric | Magnetic | Note |
|---|---|---|
| EMF, V | MMF = N·I | Ampere-turns — turns and current are interchangeable |
| Current, I | Flux, Φ | Webers |
| Resistance, R = ρl/A | Reluctance, S = l/μA | Same form, different constant |
| V = I·R | MMF = Φ·S | Ohm's law for magnetism |
Reluctances add in series and combine in parallel exactly as resistances do, so a magnetic circuit is solved with the methods of a first circuits course.
Why cores are iron, and why the air gap dominates
Iron's relative permeability is a few thousand, so its reluctance is thousands of times lower than air's. Flux therefore stays in the core instead of spreading — which is the only reason a magnetic *circuit* can be spoken of at all.
The corollary is startling: an air gap 1 mm long in a 300 mm iron path can need more MMF than all the iron together, because reluctance goes as 1/μ. Every rotating machine's design is really an argument about its air gap, which is made as small as the bearings allow.
Where the analogy breaks
- 1Saturation. Iron's permeability is not constant. Below the knee of the B-H curve the analogy holds; past it, doubling the MMF barely raises the flux. No resistor changes value with the current through it.
- 2Leakage. There is no magnetic insulator, so some flux always escapes through air — which is why leakage reactance appears in every transformer and machine model.
- 3Losses. A steady flux dissipates nothing, unlike a steady current. Core losses come from the flux *changing*, which is why they scale with frequency.
Saturation is the one that sets machine size: it puts a ceiling on flux, so torque cannot be increased indefinitely by driving more field current.
The numbers you will be asked for
- Ohm's law for magnetism
Φ = MMF / S = N·I / S
Same algebra as an electric circuit.
- Reluctance
S = l / (μ₀·μᵣ·A)
Longer, thinner or less permeable means more.
- Flux density
B = Φ / A
Saturation is a limit on B, not on Φ directly.
- Field strength
H = N·I / l
The horizontal axis of the B-H curve.
Advantages and disadvantages
Advantages
- Reuses circuit methods you already know.
- Series and parallel reluctances combine like resistances.
- Makes the dominance of the air gap immediately visible.
- Good accuracy below the knee of the B-H curve.
Disadvantages
- Saturation makes reluctance load-dependent, which has no electrical analogue.
- Leakage flux always exists, since there is no magnetic insulator.
- The loss mechanisms are entirely different from I²R.
- μᵣ varies with operating point, so a single number is an approximation.
Watch it work
Check yourself
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