Characteristic modes (CM) form a set of functions which, under specific boundary conditions, diagonalizes operator relating field and induced sources. Under certain conditions, the set of the CM is unique and complete (at least theoretically) and thereby capable of describing the behavior of a studied object in full.
This article deals with characteristic mode decomposition in electromagnetics, a domain in which the CM theory has originally been proposed.
CM decomposition was originally introduced as set of modes diagonalizing a scattering matrix. The theory has, subsequently, been generalized by Harrington and Mautz for antennas. Harrington, Mautz and their students also successively developed several other extensions of the theory. Even though some precursors were published back in the late 1940s, the full potential of CM has remained unrecognized for an additional 40 years. The capabilities of CM were revisited in 2007 and, since then, interest in CM has dramatically increased. The subsequent boom of CM theory is reflected by the number of prominent publications and applications.
For simplicity, only the original form of the CM – formulated for perfectly electrically conducting (PEC) bodies in free space — will be treated in this article. The electromagnetic quantities will solely be represented as Fourier's images in frequency domain. Lorenz's gauge is used.
The scattering of an electromagnetic wave on a PEC body is represented via a boundary condition on the PEC body, namely
\boldsymbol{\hat{n}} x \boldsymbol{E}i=-\boldsymbol{\hat{n}} x \boldsymbol{E}s,
with
\boldsymbol{\hat{n}}
\boldsymbol{E}i
\boldsymbol{E}s
\boldsymbol{E}s=-j\omega\boldsymbol{A}-\nabla\varphi,
with
j
\omega
\boldsymbol{A}
\boldsymbol{A}\left(\boldsymbol{r}\right)=\mu0\int\limits\Omega\boldsymbol{J}\left(\boldsymbol{r}'\right)G\left(\boldsymbol{r},\boldsymbol{r}'\right)dS,
\mu0
\varphi
\varphi\left(\boldsymbol{r}\right)=-
1 | |
j\omega\epsilon0 |
\int\limits\Omega\nabla ⋅ \boldsymbol{J}\left(\boldsymbol{r}'\right)G\left(\boldsymbol{r},\boldsymbol{r}'\right)dS,
\epsilon0
G\left(\boldsymbol{r},\boldsymbol{r}'\right)
G\left(\boldsymbol{r},\boldsymbol{r}'\right)=
e-jk\left|\boldsymbol{r-\boldsymbol{r | |
'\right|}}{4\pi |
\left|\boldsymbol{r}-\boldsymbol{r}'\right|}
and
k
\boldsymbol{\hat{n}} x \boldsymbol{E}s\left(\boldsymbol{J}\right)
The governing equation of the CM decomposition is
l{X}\left(\boldsymbol{J}n\right)=λnl{R}\left(\boldsymbol{J}n\right) (1)
with
l{R}
l{X}
l{Z}(⋅)=l{R}(⋅)+jl{X}(⋅).
l{Z}
l{Z}\left(\boldsymbol{J}\right)=\boldsymbol{\hat{n}} x \boldsymbol{\hat{n}} x \boldsymbol{E}s\left(\boldsymbol{J}\right). (2)
The outcome of (1) is a set of characteristic modes
\left\{\boldsymbol{J}n\right\}
n\in\left\{1,2,...\right\}
\left\{λn\right\}
Discretization
l{D}
\Omega
M
\OmegaM=l{D}\left(\Omega\right)
\left\{\boldsymbol{\psi}n\right\}
n\in\left\{1,...,N\right\}
\boldsymbol{J}
\boldsymbol{J}\left(\boldsymbol{r}\right) ≈
N | |
\sum\limits | |
n=1 |
In\boldsymbol{\psi}n\left(\boldsymbol{r}\right)
and by applying the Galerkin method, the impedance operator (2)
Z=R+jX=\left[Zuv\right]=\left[\int\limits\Omega
\ast | |
\boldsymbol{\psi} | |
u |
⋅ l{Z}\left(\boldsymbol{\psi}v\right)dS\right].
The eigenvalue problem (1) is then recast into its matrix form
XIn=λnRIn,
which can easily be solved using, e.g., the generalized Schur decomposition or the implicitly restarted Arnoldi method yielding a finite set of expansion coefficients
\left\{In\right\}
\left\{λn\right\}
The properties of CM decomposition are demonstrated in its matrix form.
First, recall that the bilinear forms
Pr ≈
1 | |
2 |
IHRI\geq0
and
2\omega\left(Wm-We\right) ≈
1 | |
2 |
IHXI,
where superscript
H
I
R
X
λn ≈
| |||||||
|
then spans the range of
-infty\leqλn\leqinfty
λn<0
λn>0
λn=0
λn=λn\left(\omega\right)
λn\left(\omega\right)
In\inRN x
1 | |
2 |
H | |
I | |
m |
ZIn ≈ \left(1+jλn\right)\deltamn.
This last relation presents the ability of characteristic modes to diagonalize the impedance operator (2) and demonstrates far field orthogonality, i.e.,
1 | \sqrt{ | |
2 |
\varepsilon0 | |
\mu0 |
The modal currents can be used to evaluate antenna parameters in their modal form, for example:
\boldsymbol{F}n\left(\boldsymbol{\hat{e}},\boldsymbol{\hat{r}}\right)
\boldsymbol{\hat{e}}
\boldsymbol{\hat{r}}
\boldsymbol{D}n\left(\boldsymbol{\hat{e}},\boldsymbol{\hat{r}}\right)
ηn
Qn
Zn
These quantities can be used for analysis, feeding synthesis, radiator's shape optimization, or antenna characterization.
The number of potential applications is enormous and still growing:
The prospective topics include
CM decomposition has recently been implemented in major electromagnetic simulators, namely in FEKO, CST-MWS, and WIPL-D. Other packages are about to support it soon, for example HFSS and CEM One. In addition, there is a plethora of in-house and academic packages which are capable of evaluating CM and many associated parameters.
CM are useful to understand radiator's operation better. They have been used with great success for many practical purposes. However, it is important to stress that they are not perfect and it is often better to use other formulations such as energy modes, radiation modes, stored energy modes or radiation efficiency modes.