Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

4.3 Magnetic Circuits

An iron core with a few windings on it is a circuit: the flux plays the part of the current, the ampere-turns play the part of the battery, and every branch has a reluctance, so transformers, motors, recording heads and lifting magnets can be analysed with the bookkeeping of a resistor network.

16 solved examples and exercises3 interactive tools

In this section you will learn

how Ampère's law $\oint_{C} H \cdot d l = N I$ and the flux law $\oint_{S} B \cdot d S = 0$ , written branch by branch along an iron core, become a circuit with a loop rule and a node rule,
what the reluctance $Z = l / (μ S)$ and the magnetomotive force $F_{m} = N I$ are, and how far the analogy with Ohm's law and Kirchhoff's laws can be trusted,
how reluctances combine in series and in parallel, and why a millimetre of air can matter more than half a metre of iron,
how to solve a circuit whose iron is nonlinear, by reading $H$ from the $B$ – $H$ curve of the material instead of using a constant $μ$ ,
what saturation, hysteresis, fringing and leakage do to the simple picture, and
how transformers, motors, recording heads, reluctance motors and lifting magnets are magnetic circuits.

In Section 4.1 and Section 4.2 we learnt to compute the magnetic field of a current in free space and in a magnetic material, and we saw that in iron with $μ_{r} \sim 1 0^{3}$ – $1 0^{5}$ the flux of $B$ is guided along the iron much as a current is guided along a copper wire. This section exploits that observation. An iron core with a few windings on it is treated as a circuit: the flux $Φ$ plays the part of the current, the ampere-turns $N I$ of the winding play the part of the battery, and each piece of the core has a reluctance $Z = l / (μ S)$ that plays the part of a resistance. Nothing new is assumed. Every statement of the circuit model is one of the two fundamental equations of magnetostatics applied to one branch or one node of the core. What we gain is speed: a transformer core, a motor, a lifting magnet or the write head of a hard disk can be analysed in a few lines, with the same bookkeeping as a resistor network.

We shall also meet the limits of the model. Iron saturates, so $μ$ is not a constant and the circuit becomes nonlinear, and the problems at the end of the section show how one works with the $B$ – $H$ curve of the material instead. Flux leaks out of the iron and spreads out at every air gap, and we shall estimate how much.

4.3 Magnetic Circuits

Contents of Section 4.3

Problems · 3

Home exercises · 3

Further exercises · 10