Dirich1: Time- and Coordinate-Dependent Boundary Condition

Conductive cylinder in rotating magnetic field.

Problem Type:
A plane problem of transient magnetics.

Geometry:

Given:
Relative magnetic permeability of air μ = 1;
Relative magnetic permeability of conductor μ = 1;
Conductivity of conductor σ = 6.3·10⁷ S/m;
Magnitude of external field B₀ = 1 T;
Number of poles 2p = 6;
Frequency f = 50 Hz.

Solution:
To specify rotating magnetic field on the outer boundary of the region, B_n = B₀ sin (ωt - pφ), we apply the Dirichlet boundary condition, using the formula: A = cos (18000*t - 3*atan2 (y/x)) / 60

The coefficient A₀ = 1/60 arises from consideration

B_n = (1/r)(∂A/∂φ) = A₀p·sin(ωt - pφ) / r

and

A₀ = B₀·r/p

Due to periodicity of the problem, only half of the model is presented, and odd periodic boundary condition A₁ = -A₂ is applied on the cut. In fact, it would be enough to simulate just 60° sector of the model. In time domain, problem is simulated with automatic adaptive time step, up to 0.05 seconds (approx. 2.5 periods).

Results:
t = 0.0002 sec:

t = 0.048 sec:

t = 0.05 sec:

See the Dirich1.pbm problem in the Examples folder.