Solve the magnetic flux flowing through the iron core of an electromagnet or inductor as a magnetic circuit built from reluctances. Change the coil turns, current, core dimensions and air-gap length to see the magnetomotive force, reluctance, flux, flux density and inductance update in real time, and watch how a thin air gap dominates the circuit.
Parameters
Coil turns N
turns
Current I
A
Core magnetic path length ℓ
mm
Length of the path the flux follows around the core
Core cross-sectional area A
mm²
Core relative permeability μ_r
About 2000-6000 for electrical steel (linear model)
Air-gap length ℓ_g
mm
Air gap in the path. 0 = all-iron circuit
Results
—
MMF (A·turns)
—
Total reluctance ℳ (×10⁶ A/Wb)
—
Magnetic flux Φ (µWb)
—
Flux density B (T)
—
Air-gap reluctance share (%)
—
Inductance L (mH)
—
Magnetic circuit and flux flow
The magnetomotive force N·I is produced by the coil winding, and the flux circulates around the iron-core loop and crosses the air gap. The number of flux lines scales with the flux Φ.
The flux Φ equals the magnetomotive force ℳℱ (= coil turns N × current I) divided by the reluctance ℳ. ℳ depends on the path length ℓ, area A and permeability (μ₀μr). This is the "magnetic Ohm's law" (Hopkinson's law).
Series reluctances add. The air gap (μr = 1) has far more reluctance than the iron and tends to dominate even when thin. Flux density B = Φ/A, coil inductance L = N²/ℳtotal.
What is a Magnetic Circuit and Reluctance?
🙋
An electromagnet is just a coil that becomes a magnet when you run current through it, right? But "magnetic circuit" suddenly sounds hard...
🎓
Relax — the idea is very simple. A magnetic circuit is just the "magnetic version of an electric circuit". In an electric circuit, voltage drives current through resistance. In a magnetic circuit, the magnetomotive force (coil turns times current) drives magnetic flux through reluctance. Just swap voltage with MMF, current with flux, and resistance with reluctance. The equation Phi = MMF/R has exactly the same form as Ohm's law I = V/R.
🙋
Wait, it really is the same form! So is "reluctance" set by the material and shape, like electrical resistance?
🎓
Exactly. Electrical resistance was R = rho l/A. Reluctance is R = l/(mu0 mu_r A) — also proportional to length and inversely proportional to area. The difference is that the permeability mu sits in the denominator. A material with high permeability — iron, which conducts flux easily — has low reluctance. The relative permeability of iron is around 2000-5000, so an iron core is like a "thick conductor for magnetic flux".
🙋
I see. When I move the "air-gap length" slider from 1 mm, it says the gap is already 87% of the total reluctance. But the iron core is 300 mm long — isn't that strange?
🎓
Great catch — that is the single most important thing to see in this tool. Air has a relative permeability of 1, the iron core has 2000. Put them into the same formula R = l/(mu0 mu_r A), and the reluctance per unit length of air is 2000 times that of iron. So 1 mm of air is equivalent to "2000 mm of iron". A 300 mm iron core loses easily. This is the key point of magnetic circuits: a thin air gap dominates the whole circuit.
🙋
If it gets in the way that much, shouldn't we just remove the air gap?
🎓
Here is the interesting part — we often add an air gap on purpose. Take an inductor. With iron only, as current rises it quickly magnetically saturates and the inductance collapses. Add an air gap and most of the reluctance is set by the "linear air", so the characteristics become stable and harder to saturate. Conversely, in transformers and motors where you want to push as much flux as possible, the gap is kept as close to zero as you can. The air-gap length is a parameter the designer chooses deliberately.
🙋
Adding a gap reduces the flux, doesn't it? How does it still make things harder to saturate?
🎓
Because the flux density B drops. Adding a gap raises the total reluctance, so for the same MMF the flux Phi falls. Since B = Phi/A, the flux density falls too. Iron saturates when B reaches about 1.6-2 T. If the gap keeps B around 1 T, you have headroom so a moderate current rise will not saturate it. This tool also warns once B exceeds about 1.6 T. An air gap is a trick that "sacrifices flux to buy stability and margin".
Frequently Asked Questions
A magnetic circuit is the magnetic analogue of an electric circuit. Just as voltage drives current through resistance, the magnetomotive force (MMF = coil turns x current) drives magnetic flux through reluctance. The relation Phi = MMF / R has exactly the same form as Ohm's law I = V / R and is called Hopkinson's law. Once you remember the mapping voltage<->MMF, current<->flux, resistance<->reluctance, designing electromagnets and transformers feels like solving a DC circuit.
The reluctance of a magnetic-path segment is R = l / (mu0 x mu_r x A), where l is the path length, A the cross-sectional area, mu0 the permeability of free space (4 pi x 10^-7) and mu_r the relative permeability. This has the same structure as electrical resistance R = rho l / A: proportional to length and inversely proportional to area. Reluctances of series-connected segments add, just like resistors, so for an iron core and an air gap in series R_total = R_core + R_gap.
Air has a relative permeability of 1, while an iron core has mu_r of order 1000-5000. Since reluctance is R = l / (mu0 mu_r A), air has thousands of times the reluctance of iron for the same area. As a result, a 1 mm air gap can contribute more reluctance than a 300 mm iron path. At the default settings the gap accounts for about 87% of the total. Air gaps are introduced deliberately to linearise an inductor, store energy or prevent saturation, and minimised in transformers and motors where strong flux is wanted.
When the core flux density B exceeds roughly 1.6-2.0 T, the iron enters magnetic saturation. In saturation the relative permeability mu_r drops sharply, so increasing the MMF barely increases the flux. This tool uses a linear model with a constant mu_r, so it warns once B exceeds about 1.6 T because the calculation then diverges from real hardware. Real electromagnets and transformers are designed to keep the flux density well below saturation (around 1.0-1.5 T for electrical steel). To avoid saturation, increase the cross-sectional area or add an air gap to lower B.
Real-World Applications
Electromagnets, solenoids and relays: Lifting electromagnets, industrial solenoid valves and electromechanical relays are designed by finding the core flux from the MMF N·I and the reluctance. The pull force on a movable armature scales with the square of the flux density, so as the gap closes the reluctance drops, the flux rises and the pull force strengthens sharply — the "pull-in" characteristic. Sweep the air-gap length in this tool to follow that transition.
Inductors and choke coils: The inductor in a switching power supply deliberately includes an air gap so that most of the reluctance is set by the air. This stabilises L = N²/ℳ against current and resists saturation even under a DC bias. The gap is also where energy is stored — most of the stored energy sits in the magnetic field of the gap. The gap length is the key parameter for balancing inductance value against saturation current.
Transformer and motor core design: In transformers and rotating machines the reluctance is kept low so flux flows efficiently. A transformer's laminated core has essentially zero gap, and motors keep the gap to the minimum needed for rotation (a few hundred micrometres). The cross-sectional area is sized so the flux density stays below saturation, and the operating flux density is chosen by trading off against core loss (hysteresis and eddy-current loss).
Equivalent magnetic-circuit modelling: Before running a finite-element (FEM) electromagnetic analysis, a magnetic-circuit model gives a quick estimate of the flux and inductance. Replacing a complex shape with a network of series and parallel reluctances lets you get a fast first read with the feel of solving a DC circuit. If the FEM result differs from this estimate by an order of magnitude, it is a sanity check pointing to overlooked leakage flux, fringing, or a boundary-condition mistake.
Common Misconceptions and Pitfalls
The biggest pitfall is assuming the relative permeability mu_r is a fixed constant. This tool uses a linear model with a constant mu_r, which is only valid at low flux density. In a real core, as the flux density approaches saturation (about 1.8-2.0 T for electrical steel, about 0.4 T for ferrite) mu_r drops sharply and the flux barely grows even as the MMF rises. The linear-model claim that "doubling the current doubles the flux" is a gross overestimate in the saturation region. Once the flux density B exceeds about 1.6 T, treat this tool's values as a rough reference only and verify against a real B-H curve.
Next, assuming leakage flux and fringing can be ignored. This tool uses an ideal model in which all flux stays inside the iron-core loop and the cross-sectional area is unchanged across the gap. In reality, part of the flux runs outside the core as leakage flux, and at the gap the flux lines bulge outward (fringing), widening the effective area. The longer the gap, the less fringing can be ignored: the real reluctance is smaller than calculated and the real flux is larger. When the gap length becomes comparable to the cross-section dimension, a fringing correction is essential.
Finally, the complacent thought that "a magnetic circuit is exactly like Ohm's law, so it is easy". The form is the same, but there are decisive differences from electric circuits. First, a magnetic circuit has no analogue of Joule heating from the flux; instead core loss (hysteresis and eddy-current loss) appears under AC operation. Second, reluctance is a non-linear element that depends on flux density — it is not a constant like electrical resistance. Third, flux is never perfectly confined and always leaks. The magnetic-circuit model is powerful for "DC, low-flux-density estimates", but remember it cannot be used as-is when AC loss or saturation matter.
How to Use
Enter the number of coil turns (typically 100–5000 for inductors) and the DC current flowing through the winding in amperes.
Set the iron core length and cross-sectional area; optionally define an air gap to model reluctance in series (e.g., 0.5 mm for a transformer core joint).
The simulator calculates magnetomotive force (MMF = N·I), total reluctance ℜ from both core and air-gap permeances, magnetic flux Φ, flux density B, and self-inductance L using the reluctance model.
Worked Example
Consider an iron-core solenoid with N=500 turns, I=2 A, core length=0.15 m, core cross-section=80 mm², and a 1 mm air gap. MMF=1000 A·turns. Iron reluctance (μᵣ≈2000 for mild steel) dominates; air-gap reluctance ≈15% of total. Total reluctance ℜ≈8.5×10⁵ A/Wb yields flux Φ≈1.18 mWb, flux density B≈14.75 mT, and inductance L≈295 mH. Reducing the air gap to 0.1 mm increases L to ≈850 mH, demonstrating air-gap sensitivity.
Practical Notes
Air gaps introduce disproportionate reluctance; a 1 mm air gap in a 150 mm steel core can consume 10–20% of total MMF despite occupying <1% of path length.
Saturation effects are ignored here; real iron cores (B_sat≈2.0 T for steel) limit flux linearly only at low MMF levels.
Laminated cores reduce eddy-current losses but require stacking-factor correction (typically 0.90–0.98) to effective area.
Inductance L=N²/ℜ scales with turn count squared; doubling turns quadruples inductance if reluctance remains constant.