MC-A1 · Magnetic circuit

Magnetic Circuits and the Magnetic Ohm’s Law

The way the flux made by a coil travels once around an iron core looks just like current flowing around a circuit. Pin down what takes the place of current in a magnetic circuit, and learn to read magnetic circuits the way you read electric ones.

What plays the role of current in a magnetic circuit?

The coil wound on the left leg pushes flux into the iron core, and that flux runs once around the core and back to the coil. This flow takes the same place as current in an electric circuit, and the first question is which of the three tabs (field strength, flux density, flux) fills that place exactly. Pick a tab and the part of the picture that quantity refers to turns gold.

Air gap g0.0 mm

The tabs show where H/B/Φ act on the core; widening the gap in the right leg with the slider raises the reluctance and lowers the flux Φ.

Flux Φ = F / R (falls as the gap widens)

Φ ≈ 100%  ·  g = 0.0 mm

The flux Φ is the total passing through the whole cross-section and going once around the circuit, so it stands in exactly the place of current. Φ = NI / R is the magnetic Ohm's law.

Electric and magnetic circuits are look-alikes

Just as voltage pushes current, the magnetomotive force F = NI pushes flux, and just as resistance holds back current, reluctance holds back flux. Put Φ = F/R in the place of Ohm's law I = V/R and the magnetic circuit is solved. Magnetomotive force is counted in ampere-turns because how many times the coil is wound and how much current it carries together set the push. The two are not alike everywhere, though. Air around a wire almost blocks current but does not block flux, so some flux leaves the core and leaks through the air as leakage flux; and the permeability of iron drops as the flux grows, which is saturation, so a magnetic circuit is not as straight a proportion as an electric one.

ObserveF = N I

Magnetomotive force corresponds to voltage: it is what pushes flux.

What sets the reluctance

Reluctance has exactly the same form as electrical resistance R = l/(σA): R = l/(μA). It grows with the length of the path and shrinks with a larger permeability μ or a wider cross-section A. Iron's μ is thousands of times that of air, so the same magnetomotive force drives far more flux through it, which is why the flux gathers along the core and forms a circuit. That ratio of thousands is tiny next to the gap in electrical conductivity between a metal and an insulator, so flux is never confined to its path as neatly as current is.

ChooseR = l / ?

Reluctance is proportional to length and inversely proportional to permeability and cross-sectional area.

Fill inΦ = F / ?

The magnetic Ohm's law: flux = magnetomotive force / reluctance.

A tiny air gap dominates the reluctance

Cut a gap of just 1 mm into the core and, because air's μ is so small, the reluctance of that short gap matches or exceeds that of the whole long iron path. In the picture, too, the flux arrows thin out and fade as the gap widens, because reluctances add in series and the largest term sets the flow. That is why motors and generators keep their air gaps as narrow as possible, while inductors and flyback transformers, on the contrary, include a gap on purpose so their cores do not saturate. At the edges of a gap the flux bulges outward (fringing), so the effective gap area is taken as a little larger than the core's.

On your ownΦ = NI / ?

Expand the reluctance and the flux collapses to μANI/l.

Back to the first screen

The quantity that filled the place of current exactly was the flux Φ. H, the pressure pushing the flow along the path, sits on the voltage side, and B, the density of the flow, sits on the current-density side, so their places did not match. Only the total going once around the circuit plays the role of current, and that total is Φ = NI / R, magnetomotive force divided by reluctance. Widening the gap reduced the flux because it made the denominator of this formula larger.

The magnetic Ohm’s law: flux Φ = F / R. The magnetomotive force F = NI drives the flux and the reluctance R = l / (μA) opposes it, so Φ = μANI / l. With voltage→MMF, current→flux, resistance→reluctance, a magnetic circuit solves like an electric one.

Once you hold this handle

The magnetic Ohm's law is the foundation of this whole subject. In the next lesson, a coil's inductance hangs directly on reluctance as L = N²/R (MC-A2), and the flux paths of transformers, motors, and generators are all analyzed as series and parallel magnetic circuits like these. The sense that the air gap dominates the reluctance leads straight into the gap design of rotating machines.