Curl-Curl-Eigenvalue Equation - Institut für Allgemeine ...

Solution Space of Eigenvalue Problem
Im{ λ}
N-fold eigenvalue 0 (static solutions)
2 2
2 Re
ω 1
ω
2
… ω
{ λ}
2 N
smallest (non-zero) eigenvalue = fundamental oscillation
Symmetry and positive semi-definiteness of A'
⇒ orthogonality of the modes:
e' ⋅ e' = 0 ∀i, j withλ ≠λ
i j i j
Orthogonality condition after back-transformation:
ei⋅ Mεej = 0 and ei⋅ d
j
= 0 ( i ≠ j)
4 x stored electric energy (non-zero only for i = j)
Ursula van Rienen, Universität Rostock, Institut für Allgemeine Elektrotechnik, AG Computational Electrodynamics
Consequently, we should search for some transformation which reduces the dimension of the
system matrix from 3N P to 2N P . Building in the discrete Gauss law for electricity with
vanishing right hand side (no charges) would yield such a transformation but destroy the
band structure of the matrix affording higher computational effort for it´s storage.
Instead of this it is more elegant to control the static modes by a modified formulation of the
wave equation and/or by specific algebraic solution methods which will be treated later.
Another property of the solution of the Curl-Curl matrix A´. For two eigenvectors, each, of
this matrix orthogonality holds if the corresponding eigenvalues are not equal to each other.
Hence, orthogonality holds for all pairs of non-degenerated modes – also for each pair build
by one dynamic and one static mode. For degenerated dynamic modes (with λ i =λ j ≠0)
orthogonal linear combinations can always be found.
After back-transformation from A` to A cc we obtain the orthogonality relation for dynamic
eigenmodes given above.
The scalar product of the vectors of electric grid voltage and electric grid flux formally
corresponds to four times of the stored electric energy which is non-vanishing only if both
vectors belong to the same field.