24.4. Exercises

24.4.1. permanent magnet

Replace the coil by a cylindrical bar magnet. The VF is

\[ \int \mu^{-1} \operatorname{curl} A \cdot \operatorname{curl} v \, dx = \int M \cdot \operatorname{curl} v \, dx, \]

where the magnetization \(M\) is a vectorfield pointing in the direction of the bar magnet.

24.4.2. coil with iron core

put a highly permeable iron core (\(\mu_r = 10^4\)) inside a coil. Plot the magnetic field

24.4.3. model a simple transformer

It consists of a primary and a secondary coil, with cylindrical iron cores (limbs) through them. To form a closed core, connect the cores with iron bars at top and bottom. A real transformer is shown here: https://jschoeberl.github.io/talk-pdesoft/transformer/transformer.html

What is the generated voltage measured at the secondary coil ?

Use a time-harmonic current source \(j(x,t) = e^{i \omega t} j(x)\), which leads to a time-harmonic vector potential \(A(x,t) = e^{i \omega t} A(x)\).

The voltage is the electric field integrated along the wire: \(\int_C E \cdot d\tau\). Assume the secondary coil consists of \(N\) closed turns.

\[ \int_{\partial S} E \cdot \tau = \frac{d}{dt} \int_S B \cdot n = \frac{d}{dt} \int_S \operatorname{curl} A \cdot n = i \omega \int_{\partial S} A \cdot \tau \]

24.4.4. Eddy current problem

Assume we have a coil with AC current \(j(x,t) = e^{i \omega t} j(x)\), and we have a conduction (an aluminium block) next to the coil.

The Eddy current approximation in frequency domain is

\[ i \omega \sigma A + \operatorname{curl} \frac{1}{\mu} \operatorname{curl} A = j_{coil} \]

The term \(j_{Eddy} := i \omega \sigma A\) are the induced Eddy currents in the conductor. An example is here: https://ngsolve.github.io/TEAM-problems/TEAM-7/team7.html