Computatonal Electromagnetcs n Antenna Analyss and Desgn Introducton It s rare for real-lfe EM problems to fall neatly nto a class that can be solved by the analytcal methods presented n the precedng lectures. Classcal approaches may fal f: the materal s not lnear and cannot be lnearzed wthout serously affectng the result the soluton regon s complex (.e. the varous boundares do not concde wth any well descrbed coordnate system). the boundary condtons are tme-dependent the medum s nhomogeneous or ansotropc Whenever a problem wth such complexty arses numercal solutons must be employed. Fortunately there are a large number of very good commercal programs avalable for solvng antenna problems.
Computatonal Electromagnetcs computatonal electromagnetcs rgorous methods Hgh frequency IE DE VM TD FD TD FD feld based current based MoM FEM FDTD TLM GO/GTD PO/PTD
Comparson of Methods Doman Generalty Accuracy Memory N= number of elements MoM Frequency Homogeneous or dscretely homogeneous regons FDTD Tme (all frequences n one run) Very general. nhomogeneous dspersve ansotropc FEM Frequency Very general. nhomogeneous dspersve ansotropc Hgh Frequency Methods Frequency Only good for structures much larger than the wavelength Antenna Types Very accurate @(N ) All but harder for large reflector antennas Moderately accurate @(N) All but harder for large reflector antennas Very accurate @(N log N) All but harder for large reflector antennas Only accurate for large structures @(N) Only good for large antennas (mostly used for reflectors)
Goal of all of these methods Approxmate these B D E = M H = t t D = ρ B = ρ Wth ths m J [ A ] [ x] = [ b]
Fnte Dfference Tme Doman FDTD
Reason for nterest n FDTD In the tme doman Maxwell s equatons gve rse to PDEs nvolvng tme and spatal dervatves. Some good reasons for dealng wth PDE s are: Complex-value materals easly accommodated. Computer resources are adequate. PDE solutons are robust. Tme doman PDE methods usually have no matrces. Geometres to be solved can be more vared.
Some bg advantages Broadband response wth a sngle exctaton. 3D models easly. Memory requrement scales lnearly wth problem sze Frequency dependent materals accommodated. Most parameters can be generated e.g. Scattered felds antenna patters RCS S-parameters etc..
How does t wor? Based on the two Maxwell curl equatons n dervatve form. These are lnearzed by central fnte dfferencng. We only consder nearest neghbor nteractons because all the felds are advanced temporally n dscrete tme steps over spatal cells. e we sample n space & tme embeddng of an antenna n a FDTD space lattce (note that the whole volume s meshed!)
Dscretze Obects n Space usng Cartesan Grd D Dscretzaton 3D Dscretzaton z x D Dscretzaton Z z = 0 Ex () zt z = Z
Defne Locatons of Feld Components: FDTD Cell called Yee Cell Fnte-Dfference Space s dvded nto small cells One Cell: (dx)(dy)(dz) E and H components are dstrbuted n space around the Yee cell (note: feld components are not collocated) FDTD: Yee K. S.: Numercal soluton of ntal boundary value problems nvolvng Maxwell's equatons n sotropc meda. IEEE Transactons on Antennas Propagaton Vol. AP-4 pp. 30-307 966.
3D formulaton y x z x y x z y y x z z E H E H t z y H E E H t x z E E H H t y x ρ µ ρ µ ρ µ = = = y x z x y x z y y x z z H E H E t y z E H H E t z x H H E E t x y σ ε σ ε σ ε = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = z E E x E E H H y E E z E E H H n x n x n z n z r n x n x n z n z n y n y r n x n x 0 0 µ µ µ µ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = 0 0 E E z H H y H H t E E x E E y E E t H H n x n x n y n y n z n z r n x n x n y n y n x n x r n z n z σ ε ε µ µ Convert equatons le these To ones le these
QucWave (http://www.qwed.eu/) Commercal FDTD pacage wth CAD nterface Uses conformal FDTD mesh Many specal features for antenna problems Wrtten and supported by QWED Poland Runs under 3/x64 bt Wndows platforms and Lnux
QucWave (http://www.qwed.eu/) Examples:
QucWave (http://www.qwed.eu/) Examples:
XFdtd (http://www.remcom.com /) Commercal FDTD pacage wth CAD nterface Probably the most popular FDTD pacage for antenna problems Many specal features for antenna problems ncludng full human body mesh Wrtten and supported by Remcom USA Runs under 3/x64 bt Wndows Mac OS X and Lnux
XFdtd (http://www.remcom.edu /)
Lumercal (http://www.lumercal.com /) Commercal FDTD pacage wth CAD nterface Popular FDTD pacage for the optcs fols (.e. ntegrated optcs lght scatterng plasmoncs) Large lbrary of materals at optcal wavelengths Has very nce scrptng language for defnng large complcated problems. Has nce bult n optmzaton Runs under 3/x64 bt Wndows Mac OS X and Lnux
Lumercal (http://www.lumercal.com /)
Fnte Element Method
Fnte Element Method
Varatonal Approach In solvng problems arsng n physcs and engneerng t s often possble to replace the problem of ntegratng a dfferental equaton by the equvalent problem of seeng a functon that gves a mnmum value of some ntegral. Problems of ths type are called varatonal problems. The methods that allow us to reduce the problem of ntegratng a dfferental equaton to the equvalent varatonal problem are usually called varatonal methods.
Varatonal Approach Name of equatons PDE Varatonal prncple Homogeneous wave equaton wth sources Homogeneous wave equaton wthout sources Dffuson equaton Posson s equaton Homogenous Laplace s equaton Φ Φ = Φ Φ Φ = Φ t Φ = Φ = g 0 = g 0 0 [ I( Φ) = Φ Φ gφ]dv v I( Φ) = v [ Φ Φ ]dv Φ I( Φ) = Φ Φ dvdt t t v [ I( Φ) = Φ gφ]dv v I( Φ) = v [ Φ ]dv
Fnte Element Method The fnte element method (FEM) has ts orgn n the feld of structural analyss. However snce then the method has been employed n nearly all areas of computatonal physcs and engneerng. The FEM method whle more dffcult to program than ether the fnte dfference (FD) or method of moments (MOM) s a more powerful and versatle numercal technque for handlng problems nvolvng complex geometres and nhomogeneous meda.
Basc concept Although the behavour may be complex when vewed over a large regon a smple approxmaton may suffce over a small subregon. The regon s dvded up nto fnte elements. (usually trangles or squares but can be more complcated) Regardless of the shape the feld s approxmated by a dfferent expresson over each element mantanng contnuty at adonng elements.
Soluton Strategy: Varatonal Approach The equatons to be solved are usually stated not n terms of feld the varables but n terms of an ntegral-type functonal such as energy. The functonal s chosen such that the feld soluton maes the functonal statonary The total functonal s the sum of the ntegral over each element
Fnte Element Method The fnte element method (FEM) nvolves bascally four steps: () Dscretze the soluton regon nto a fnte number of subregons or elements () Derve the governng equatons for each element based on ether a varatonal approach or Galern s method (3) Assemble all the elements together n the soluton space. (4) Solve the resultng system of equatons
HFSS (http://www.ansoft.com/) Commercal FEM pacage wth CAD nterface Uses adaptve meshng Probably the most popular commercal pacage for antenna applcatons. Wrtten and supported by Ansoft USA Runs under 3/x64 bt Wndows platforms Redhat Lnux Solars (Sun worstatons). Has ntegrated hybrd fnte element / boundary ntegral methods (MoM) Knd of expensve! Optonal optmzaton pacage (optmetrcs)
HFSS (http://www.ansoft.com/)
CST Mcrowave Studo (http://www.cst.com/) Commercal FEM MoM and TLM pacage wth CAD nterface Mature and easy to use nterface Popular program for mcrowave crcut applcatons but also very useful for antennas. Wrtten and supported by CST Internatonal Runs under 3/x64 bt Wndows platforms and Redhat Lnux Knd of expensve! Has a specfc antenna desgn opton called Magus
CST Mcrowave Studo (http://www.cst.com/)
Comsol Multphyscs(http://www.comsol.com/) Commercal FEM pacage wth CAD nterface Is best nown for ts ablty to solve multphyscs problems Becomng a very popular program. Lns ncely wth Matlab Easy to learn nterface Runs under 3/x64 bt Wndows platforms Moderately prces
Comsol Multphyscs(http://www.comsol.com/)
References:. Clemson ste lsts all the free EM modelng tools http://www.clemson.edu/ces/cvel/modelng/emag/free-codes.html. Clemson ste lsts all the commercal EM modelng tools http://www.clemson.edu/ces/cvel/modelng/emag/csoft.html