Characteristic mode analysis
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 the 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
and being wavenumber. The integro-differential operator is the one to be diagonalized via characteristic modes.
The governing equation of the CM decomposition is
with and being real and imaginary parts of impedance operator , respectively, defined as
The outcome of (1) is a set of characteristic modes , , accompanied by associated characteristic numbers . 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.
Discretization of the body of the scatterer into subdomains as and using a set of linearly independent piece-wise continuous functions , , allows current density to be represented as
and by applying the Galerkin method, the impedance operator (2)
The eigenvalue problem (1) is then recast into its matrix form
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 and associated characteristic numbers . The properties of the CM decomposition are investigated below.
The properties of CM decomposition are demonstrated in its matrix form.
First, recall that the bilinear forms
where superscript denotes the Hermitian transpose and where 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 is theoretically positive definite and is indefinite. The Rayleigh quotient
then spans the range of and indicates whether the characteristic mode is capacitive (), inductive (), or in resonance (). In reality, the Rayleigh quotient is limited by the numerical dynamics of the machine precision used and the number of correctly found modes is limited.
- The characteristic numbers evolve with frequency, i.e., , 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 . 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, . 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
This last relation presents the ability of characteristic modes to diagonalize the impedance operator (2) and demonstrates far field orthogonality, i.e.,
The modal currents can be used to evaluate antenna parameters in their modal form, for example:
- modal far-field ( — polarization, — direction), ,
- modal directivity ,
- modal radiation efficiency ,
- modal quality factor ,
- modal impedance .
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 antennas,
- compact antenna design (RFID, Wifi),
- 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,
- utilization of Combined Field Integral Equation,
- periodic structures,
- formulation for arrays.
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 ti is often better to use other formulations such as energy modes, radiation modes, stored energy modes or radiation efficiency modes.
- Garbacz, R. J., "Modal expansions for resonance scattering phenomena," Proc. IEEE, vol.53, pp. 856–864, August 1965, doi: 10.1109/PROC.1965.4064.
- Garbacz, R. J., "A Generalized Expansion for Radiated and Scattered Fields," Ph.D. thesis, Department of Electrical Engineering, The Ohio State Univ., 1968.
- Harrington, R. F.; Mautz, J. R., "Theory of characteristic modes for conducting bodies," IEEE Transactions on Antennas and Propagation, vol.19, no.5, pp. 622–628, September 1971, doi: 10.1109/TAP.1971.1139999.
- Harrington, R. F.; Mautz, J. R., "Computation of characteristic modes for conducting bodies," IEEE Transactions on Antennas and Propagation, vol.19, no.5, pp. 629–639, September 1971, doi: 10.1109/TAP.1971.1139990.
- Chang, Y.; Harrington, R. F., "A surface formulation for characteristic modes of material bodies," IEEE Transactions on Antennas and Propagation, vol.25, no. 6, pp. 789–795, November 1977, doi: 10.1109/TAP.1977.1141685.
- Harrington, R. F.; Mautz, J. R., "Characteristic modes for aperture problems," IEEE Transactions on Microwave Theory and Techniques, vol.33, no.6, pp. 500–505, June 1985, doi: 10.1109/TMTT.1985.1133105.
- Harrington, R. F.; Mautz, J.R.; Chang, Y., "Characteristic modes for dielectric and magnetic bodies," IEEE Transactions on Antennas and Propagation, vol.20, no.2, pp. 194–198, March 1972, doi:10.1109/TAP.1972.1140154.
- El-Hajj, A.; Kabalan, K.; Harrington, R. F., "Characteristic mode analysis of electromagnetic coupling through multiple slots in a conducting plane," IEE Proceedings H - Microwaves, Antennas and Propagation, vol.140, no.6, pp. 421–425, December 1993, doi:10.1049/ip-h-2.1993.0069.
- Montgomery, C. G.; Dicke, R.H.; Purcell, E. M., Principles of Microwave Circuits, Section 9.24, New York, United States: McGraw-Hill, 1948.
- Cabedo-Fabres, M.; Antonino-Daviu, E.; Valero-Nogueira, A.; Bataller, M. F., "The theory of characteristic modes revisited: A contribution to the design of antennas for modern applications," IEEE Antennas and Propagation Magazine, vol.49, no.5, pp. 52–68, October 2007, doi:10.1109/MAP.2007.4395295.
- Capek, M.; Losenicky, V.; Jelinek, L.; Gustafsson, M., "Validating the characteristic modes solvers," IEEE Transactions on Antennas and Propagation, vol.PP, no.99, pp. 1–12, 2017, doi:10.1109/TAP.2017.2708094.
- Harrington, R. F., Field Computation by Moment Methods, Wiley -- IEEE Press, 1993.
- Schab, K. R.; Bernhard, J. T., "A group theory rule for predicting eigenvalue crossings in characteristic mode analyses," IEEE Antennas and Wireless Propagation Letters, vol.16, pp.944–947, 2017, doi: 10.1109/LAWP.2016.2615041.
- Capek, M.; Hazdra, P.; Hamouz, P.; Eichler, J., "A method for tracking characteristic numbers and vectors," Progress In Electromagnetics Research B, vol.33, pp. 115–134, 2011, doi: 10.2528/PIERB11060209.
- Raines, B. D.; Rojas, R. G., "Wideband characteristic mode tracking," IEEE Transactions on Antennas and Propagation, vol.60, no.7, pp. 3537–3541, July 2012, doi: 10.1109/TAP.2012.2196914.
- Ludick, D. J.; Jakobus, U.; Vogel, M., "A tracking algorithm for the eigenvectors calculated with characteristic mode analysis," in Proceedings of the 8th European Conference on Antennas and Propagation (EuCAP 2015), pp. 569–572, 2014, doi: 10.1109/EuCAP.2014.6901820.
- Miers, Z.; Lau, B. K., "Wide band characteristic mode tracking utilizing far-field patterns," IEEE Antennas and Wireless Propagation Letters, vol.14, pp.1658–1661, 2015, doi:10.1109/LAWP.2015.2417351.
- Safin, E.; Manteuffel, D., "Advanced eigenvalue tracking of characteristic modes," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp.~2628–2636, July 2016, doi: 10.1109/TAP.2016.2556698.
- Capek, M.; Eichler, J.; Hazdra, P., "Evaluating radiation efficiency from characteristic currents," IET Microwaves, Antennas & Propagation, vol.9, no.1, pp. 10–15, January 2015, doi: 10.1049/iet-map.2013.0473.
- Capek, M.; Hazdra, P.; Eichler, J., "A method for the evaluation of radiation Q based on modal approach," IEEE Transactions on Antennas and Propagation, vol.60, no.10, pp. 4556–4567, October 2012, doi: 10.1109/TAP.2012.2207329.
- Wu, Q.; Su, D., "A broadband model of the characteristic currents for rectangular plates," IEEE Transactions on Electromagnetic Compatibility, vol.55, no.4, pp. 725–732, August 2013, doi: 10.1109/TEMC.2012.2221718.
- Vogel, M.; Gampala, G.; Ludick, D.; Jakobus, U.; Reddy, C., "Characteristic mode analysis: Putting physics back into simulation," IEEE Antennas and Propagation Magazine, vol.57, no.2, pp. 307–317, April 2015, doi: 10.1109/MAP.2015.2414670.
- Yang, B.; Adams, J. J., "Computing and visualizing the input parameters of arbitrary planar antennas via eigenfunctions," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp.2707–2718, July 2016, doi: 10.1109/TAP.2016.2554604.
- Li, H.; Miers, Z.; Lau, B. K., "Design of orthogonal mimo handset antennas based on characteristic mode manipulation at frequency bands below 1GHz," IEEE Transactions on Antennas and Propagation, vol.62, no.5, pp.2756–5766, May 2014, doi: 10.1109/TAP.2014.2308530.
- Deng, C.; Feng, Z.; Hum, S. V., "MIMO mobile handset antenna merging characteristic modes for increased bandwidth," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp.2660–2667, July 2016, doi: 10.1109/TAP.2016.2537358.
- Yang, B.; Adams, J. J., "Systematic shape optimization of symmetric mimo antennas using characteristic modes," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp. 2668–2678, July 2016, doi: 10.1109/TAP.2015.2473703.
- Eichler, J.; Hazdra, P.; Capek, M.; Korinek, T.; Hamouz, P., "Design of a dual-band orthogonally polarized L-probe-fed fractal patch antenna using modal methods," IEEE Antennas and Wireless Propagation Letters, vol.10, no.10, pp. 1389–1392, 2011, doi: 10.1109/LAWP.2011.2178811.
- Rezaiesarlak, R.; Manteghi, M., "Design of chipless rfid tags based on characteristic mode theory (CMT)," IEEE Transactions on Antennas and Propagation, vol.63, no.2, pp. 711–718, February 2015, doi: 10.1109/TAP.2014.2382640.
- Bohannon, N. L.; Bernhard, J. T., "Design guidelines using characteristic mode theory for improving the bandwidth of PIFAs," IEEE Transactions on Antennas and Propagation, vol.63, no.2, pp. 459–465, February 2015, doi: 10.1109/TAP.2014.2374213.
- Chen, Y.; Wang, C.-F., "Electrically small UAV antenna design using characteristic modes," IEEE Transactions on Antennas and Propagation, vol.62, no.2, pp.535–545, February 2014, doi: 10.1109/TAP.2013.2289999.
- Austin, B. A.; Murray, K. P., "The application of characteristic-mode techniques to vehicle-mounted NVIS antennas," IEEE Antennas and Propagation Magazine, vol.40, no.1, pp. 7–21, February 1998, doi: 10.1109/74.667319.
- Gustafsson, M.; Tayli, D.; Ehrenborg, C.; Cismasu, M.; Norbedo, S., "Antenna current optimization using MATLAB and CVX," FERMAT, vol.15, pp. 1–29, May–June 2016.
- Adams, J. J.; Bernhard, J. T., "Broadband equivalent circuit models for antenna impedances and fields using characteristic modes," IEEE Transactions on Antennas and Propagation, vol.61, no.8, pp. 3985–3994, August 2013, doi: 10.1109/TAP.2013.2261852.
- Safin, E.; Manteuffel, D., "Manipulation of characteristic wave modes by impedance loading," IEEE Transactions on Antennas and Propagation, vol.63, no.4, pp. 1756–1764, April 2015, doi: 10.1109/TAP.2015.2401586.
- Hassan, A. M.; Vargas-Lara, F.; Douglas, J. F.; Garboczi, E. J., "Electromagnetic resonances of individual single-walled carbon nanotubes with realistic shapes: A characteristic modes approach," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp. 2743–2757, July 2016, doi: 10.1109/TAP.2016.2526046.
- Rabah, M. H.; Seetharamdoo, D.; Berbineau, M., "Analysis of miniature metamaterial and magnetodielectric arbitrary-shaped patch antennas using characteristic modes: Evaluation of the Q factor," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp. 2719–2731, July 2016, doi: 10.1109/TAP.2016.2571723.
- Rabah, M. H.; Seetharamdoo, D.; Berbineau, M.; De Lustrac, A., "New metrics for artificial magnetism from metal-dielectric metamaterial based on the theory of characteristic modes," IEEE Antennas and Wireless Propagation Letters, vol.15, pp.460–463, 2016, doi: 10.1109/LAWP.2015.2452269.
- Dai, Q. I.; Wu, J. W.; Gan, H.; Liu, Q. S.; Chew, W. C.; Sha, W. E. I., "Large-scale characteristic mode analysis with fast multipole algorithms," IEEE Transactions on Antennas and Propagation, vol.64, no.7, pp.2608–2616, July 2016, doi: 10.1109/TAP.2016.2526083.
- Guo, L.; Chen, Y.; Yang, S., "Characteristic mode formulation for dielectric coated conducting bodies," IEEE Transactions on Antennas and Propagation, vol.65, no.3, pp. 1248–1258, March 2017, doi: 10.1109/TAP.2016.2647687.
- Dai, Q. I.; Liu, Q. S.; Gan, H. U. I.; Chew, W. C., "Combined field integral equation-based theory of characteristic mode," IEEE Transactions on Antennas and Propagation, vol.63, no.9, pp. 3973–3981, September 2016, doi: 10.1109/TAP.2015.2452938.
- Tzanidis, I.; Sertel, K.; Volakis, J. L., "Characteristic excitation taper for ultrawideband tightly coupled antenna arrays," IEEE Transactions on Antennas and Propagation, vol.60, no.4, pp. 1777–1784, April 2012, doi: 10.1109/TAP.2012.2186269.
- Altair, [Online: FEKO], 2017.
- Dassault Systemes, CST Computer Simulation Technology, [Online: CST-MWS], 2017.
- WIPL-D d.o.o., [Online: WIPL-D], 2017.
- ANSYS, [Online: HFSS], 2017.
- ESI Group, [Online: CEM One], 2017.
- Schab, K. R.; Bernhard, J. T., "Radiation and energy storage current modes on conducting structures," IEEE Transactions on Antennas and Propagation, vol.63, no.12, pp. 5601–5611, December 2015, doi: 10.1109/TAP.2015.2490664.
- Jelinek, L.; Capek, M., "Optimal currents on arbitrarily shaped surfaces," IEEE Transactions on Antennas and Propagation, vol.65, no.1, pp. 329–341, January 2017, doi: 10.1109/TAP.2016.2624735.