Characteristic mode analysis
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Characteristic modes (CM) form a set of functions which, under specific boundary conditions, diagonalizes operator relating
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and induced
sources Source may refer to: Research * Historical document * Historical source * Source (intelligence) or sub source, typically a confidential provider of non open-source intelligence * Source (journalism), a person, publication, publishing institute o ...
. 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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, a domain in which the CM theory has originally been proposed.


Background

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.


Definition

For simplicity, only the original form of the CM — formulated for perfectly electrically conducting (PEC) bodies in
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
— will be treated in this article. The electromagnetic quantities will solely be represented as Fourier's images in
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
. Lorenz's gauge is used. The scattering of an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
on a PEC body is represented via a boundary condition on the PEC body, namely : \boldsymbol \times \boldsymbol^\mathrm = -\boldsymbol \times \boldsymbol^\mathrm, with \boldsymbol representing unitary normal to the PEC surface, \boldsymbol^\mathrm representing incident electric field intensity, and \boldsymbol^\mathrm representing scattered electric field intensity defined as :\boldsymbol^\mathrm = -\mathrm\omega\boldsymbol - \nabla\varphi, with \mathrm being
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, \omega being
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
, \boldsymbol being
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
: \boldsymbol \left(\boldsymbol\right) = \mu_0 \int\limits_\Omega \boldsymbol \left(\boldsymbol'\right) G \left(\boldsymbol, \boldsymbol'\right) \, \mathrmS, \mu_0 being
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
, \varphi being
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
: \varphi \left(\boldsymbol\right) = - \frac \int\limits_\Omega \nabla\cdot\boldsymbol \left(\boldsymbol'\right) G \left(\boldsymbol, \boldsymbol'\right) \, \mathrmS, \epsilon_0 being vacuum permittivity, G \left(\boldsymbol,\boldsymbol'\right) being scalar
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
: G \left(\boldsymbol,\boldsymbol'\right) = \frac and k being
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
. The integro-differential operator \boldsymbol \times \boldsymbol^\mathrm \left(\boldsymbol \right) is the one to be diagonalized via characteristic modes. The governing equation of the CM decomposition is : \mathcal \left(\boldsymbol_n\right) = \lambda_n \mathcal \left(\boldsymbol_n\right) \qquad\mathrm with \mathcal and \mathcal being real and imaginary parts of impedance operator, respectively: \mathcal(\cdot) = \mathcal(\cdot) + \mathrm\mathcal(\cdot)\,. The operator, \mathcal is defined by : \mathcal \left(\boldsymbol\right) = \boldsymbol \times \boldsymbol \times \boldsymbol^\mathrm \left(\boldsymbol\right). \qquad\mathrm The outcome of (1) is a set of characteristic modes \left\, n\in \left\, accompanied by associated characteristic numbers \left\. Clearly, (1) is a generalized eigenvalue problem, which, however, cannot be analytically solved (except for a few canonical bodies). Therefore, the numerical solution described in the following paragraph is commonly employed.


Matrix formulation

Discretization \mathcal of the body of the scatterer \Omega into M subdomains as \Omega^M = \mathcal\left(\Omega\right) and using a set of linearly independent piece-wise continuous functions \left\, n\in\left\, allows current density \boldsymbol to be represented as : \boldsymbol \left(\boldsymbol\right) \approx \sum\limits_^N I_n \boldsymbol_n \left(\boldsymbol\right) and by applying the
Galerkin method In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
, the impedance operator (2) : \mathbf = \mathbf + \mathrm \mathbf = \left _\right= \left ,\int\limits_\Omega \boldsymbol_u^\ast \cdot \mathcal \left(\boldsymbol_v\right) \, \mathrmS\right The eigenvalue problem (1) is then recast into its matrix form : \mathbf \mathbf_n = \lambda_n \mathbf\mathbf_n, 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\ and associated characteristic numbers \left\. The properties of the CM decomposition are investigated below.


Properties

The properties of CM decomposition are demonstrated in its matrix form. First, recall that the bilinear forms : P_\mathrm \approx \frac \mathbf^\mathrm \mathbf \mathbf \geq 0 and : 2\omega\left(W_\mathrm - W_\mathrm\right) \approx \frac \mathbf^\mathrm \mathbf \mathbf, where superscript ^\mathrm denotes the
Hermitian transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex co ...
and where \mathbf represents an arbitrary surface current distribution, correspond to the radiated power and the reactive net power, respectively. The following properties can then be easily distilled: * The weighting matrix \mathbf is theoretically positive definite and \mathbf is indefinite. The
Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
: \lambda_n \approx \frac then spans the range of -\infty \leq \lambda_n \leq \infty and indicates whether the characteristic mode is capacitive (\lambda_n < 0), inductive (\lambda_n > 0), or in resonance (\lambda_n = 0). In reality, the Rayleigh quotient is limited by the numerical dynamics of the
machine precision Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subj ...
used and the number of correctly found modes is limited. * The characteristic numbers evolve with frequency, i.e., \lambda_n = \lambda_n \left(\omega\right), they can cross each other, or they can be the same (in case of degeneracies ). For this reason, the tracking of modes is often applied in order to get smooth curves \lambda_n \left(\omega\right). Unfortunately, this process is partly heuristic and the tracking algorithms are still far from perfection. * The characteristic modes can be chosen as real-valued functions, \mathbf_n \in \mathbb^. In other words, characteristic modes form a set of equiphase currents. * The CM decomposition is invariant with respect to the amplitude of the characteristic modes. This fact is used to normalize the current so that they radiate unitary radiated power : \frac \mathbf_m^\mathrm \mathbf \mathbf_n \approx \left(1 + \mathrm \lambda_n\right) \delta_. This last relation presents the ability of characteristic modes to diagonalize the impedance operator (2) and demonstrates far field orthogonality, i.e., : \frac \sqrt \int\limits_0^ \int\limits_0^\pi \boldsymbol_m^\ast \cdot \boldsymbol_n \sin \vartheta \, \mathrm \vartheta \, \mathrm \varphi = \delta_.


Modal quantities

The modal currents can be used to evaluate antenna parameters in their modal form, for example: * modal far-field \boldsymbol_n \left(\boldsymbol, \boldsymbol\right) (\boldsymbolpolarization, \boldsymbol — direction), * modal directivity \boldsymbol_n \left(\boldsymbol, \boldsymbol\right), * modal radiation efficiency \eta_n, * modal quality factor Q_n, * modal impedance Z_n. These quantities can be utilized for analysis, feeding synthesis, radiator's shape optimization, or antenna characterization.


Applications and further development

The number of potential applications is enormous and still growing: * antenna analysis and synthesis, * design of
MIMO In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wi ...
antennas, * compact antenna design ( RFID,
Wi-Fi Wi-Fi () is a family of wireless network protocols, based on the IEEE 802.11 family of standards, which are commonly used for local area networking of devices and Internet access, allowing nearby digital devices to exchange data by radio wav ...
), * UAV antennas, * selective excitation of chassis and platforms, * model order reduction, * bandwidth enhancement, * nanotubes and metamaterials, * validation of computational electromagnetics codes. The prospective topics include * electrically large structures calculated using MLFMA, * dielectrics, * utilization of Combined Field Integral Equation, * periodic structures, * formulation for arrays.


Software

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.


Alternative bases

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.


References

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