It is importa,nt to note tl1a.t the cxpoucntia.l t,rrrn must be taken out- side of the iutcgral in order t,o rcwrit.e the contribu- tious from the vector pot~ciitial portiou of ...
,
Recent
Im p r o v e m e n ts
J a n G a r r e tt IHM A S /d 0 0 Division R o c h c s”1 P I-7 M N , 5 5 9 0 1
in P E E C A lbert
M o d e ling
Rueldi
IB M R e s e a r c h Division Y o r k t o w n I-lcights, N Y 1 0 5 9 8
A ccuracy Clayton
Paul
University o f K e n tucky Lexington, K Y 4 0 5 0 6
s u g g e s te d to o v e r c o m e th e s e instabilities. In p a r ticular, [7 ] u s e s a c o n j u g a ,te g r a d i e n t m e th o d , [8 ] u s e s a filte r i n g te c h n i q u e , a n d [9 ] u s e s th e m a trix pencil te c h n i q u e to eliminate late tim e instabilities. W h ile all o f th e s e te c h n i q u e s h e l p eliminate s o m e o f stability p r o b l e m s fo r specific cases, s o m e sources o f instabilitie s r e m a i n . In th e w o r k p r e s e n te d in this p a p e r w e u s e a diffe r e n t a p p r o a c h . In a r e c e n t p a p e r o n stability [lo ] it w a s o b s e r v e d th a t th e discretization o f th e c o n tin u o u s tim e electric fie l d integral e q u a tio n ( E F IE ) l e a d s to a n u n s ta b l e discrete system since th e m o d e l h a s p o l e s in th e right half p l a n e . W e k n o w from circuit th e o r y th a t th e P E E C m o d e ls zoithozlt delays a r e p a s sive a n d th a t th e y a r e always sta b l e . H e n c e , it is e v i d e n t th a t th e instabilities a r e c a u s e d b y th e delays b e t w e e n th e p a r tial inducta,nces a n d th e c o e fficie n ts o f p o te n tial. F r o m circuit th e o r y , g i v e n a passive circuit, th e i n p u t i m p e d a n c e m u s t b e positive real. In this work, w e m o n ito r th e r e a l p a r t o f th e i n p u t i m p e d a .n c e R e [Z(.s)] 2 0 fo r frequencies s = jw 2 0 to e n s u r e th a t th e te r m i n a l i m p e d a n c e is positive r e a l 1 In tro d u c tio n fo r a n a n te n n a . T h e r e a l p o r tio n o f th e i n p u t i m p e d a n c e fo r a c e n T h e P E N ! m e t h o d is lmscd o n a n I’lcctric Field In te r fe d p a tch a n te n n a is s h o w n in Fig. 1 fo r two tcg r a .l E q u a tjio n ( E l:IlC) full w a .vc fo r u ~ u h tio n w h i c h O n e discretization is fo r 2 O F at is p a .rticula,rly uscflll fo r I~ m l d i n g clcct,rollla.gnctics discretizations. 2 0 G H z , th e s e c o n d discretization is fo r 2 5 9 a t 2 0 prolhiis. ‘I’h C iI.ppliCihiiitJ o f l.llC I’klfX m o d e l to 1 , it is evib o th tim e s o d frequc~ncy tlo m a i 1 1 p r o b l e m s is very G IIz. F r o m th e results s h o w n in Fig. d e n t th a t th e i n p u t i m p e d a n c e a b o v e 2 0 G H z is b o th uscfrii. S i m p l e I’E IX m o tlcls w h i c h d o n o t involve d e l a ,ys c a n b e a .pplictd to b 0 t.h d o n m ins u s i n g a c o n - n e g a tive a n d h a s various f& e r e s o n a n c e s . S imilar v e n ti0 u a .l circuit. solver like S p ice [l] o r A S T A P [a ]. b e h a ,vio r h a s also b e e n o b s e r v e d u s i n g o th e r fo r m u l a In this p a p e r , w e a .re restricting ourselves to th e fre- tio n s . O n e solution w o u l d b e to fu r th e r discretize th e q u e n c y d o n m in. H o w e v e r , w e arc also very interested g e o m e try into m o r e cells, h o w e v e r , this is very u n in th e sta .bility issue fo r t.hct tim e d o m a in w h i c h to d e s i r a b l e since th e n u m b e r o f u n k n o w n s is i n c r e a s e d d a .te h a .s n o t b e e n solv(\d fo r g t\n f\ral p r o b l e m s . Insta.- drastically w h i c h directly i m p a c ts th e solution tim e bilitics fo r th e tim e d o m a .in intc> g r a d e q u a .tio n fo r m u - a n d p o te n tially th e size o f th e p r o b l e m w h i c h c a n b e la.tions fo r E M p r o b l c ~ tn s h a ,vc b e e n o b s e r v e d b y n m n y solved. a u th o r s w h e r e sotnc o f t.hctcarlicr w o r k is g i v e n in [3 ], In this p a ,p e r , w e p r e s e n t n e w r e fin e m e n ts to th e [4 ], [5 ], a n d [ri]. hllatiy tliffm ~ lit, lkdir~iqucs h a v e l m n p a r titio n i n g te c h n i q u e a s it w a s d e s c r i b e d in [ll]. T h e
A b s tract
T h e P i3rtia.l I% w c n l ~ l+ luiva.lcnt (‘ircuil (I’E E C ) te c h n i q u e is a circuit, b a .sctl f0 r u tula.t i o n w h i c h is n u m e r i ca.lly cquivalc>nt, to a full w a v e m r th o d o f m o m e n ts solution with G a .lerkirr m a .t.dliIlg. In 1 ,hisp q w r , n u m e r ical i m p r o v e m e n ts a r e m a .tlc to th e P E E C fo r m u l a .tio n th r o u g h a p a ,rtitio n i n g s c h e m e th a t increases th e solution accura.cy without illcrc~asillg th e Ilu m b e r o f u n k n o w n s . This p a r titio n i n g S C ~ C IIICis u s e ful in b o th th e tim e a n d f r e q u e n c y d o n A n . In a d d i tio n , in th e tim e d o n i a ,in , intcgra.1 r q u a tio u te c h n i q u e s h a v e sta .bility p r o b l e m s d u e cithcr to 1 ,h c n u m e r i c a l te c h n i q u e u s e d , o r to p r o b l e m s c r e a ~ t(\d b y f.h c discrete r e p r c s e n ta tio n o f th e n u tm h i solutiotl o f th e p r o b l e m . This p a ,rtit,io n i n g n i q u ~ ca.n b e u s e d to eliminate th e insta.bilit its d u e to th e discrctc rcprcscnt a .lio n o f th e n u n 1 e r i c a .l solution. This w o r k is o f p a ,rticular interest w h e n u s i n g niodcl rctluct.ion te c h riiqucs a .u d also fo r tim e d o m a in a.lli~.lysis whctrr 1 ,lio 1 a .t.ct,im c instability is well k n o w n .
t.c d ~
347
o - 7 8 0 3 - 4 1 40-6/97/$10.00
X
Figure
1: Resonances
Figure
for 2 discretizations.
partitioning technique was shower in [ 1 l] to both eliminate the false resouances, a.ud more importantly result in a, posit.ivo real input impedance over an very large frcqucury ra.ngc. In this work, t.he effects of the partitiouing scheme on the part.ial inducta,nce term is shown. In a.ddition, a revised pa.rt,itioning scheme is introduced which is computatioually more efficient. Results are shown for a simple center fed pa.tc.11antenna. In section 2, we derive the PElX model for the partial inductance Icrm from the vect.or potential portion of the electric firltl iutegra.1 cqua.tiou. Then irk section 3, the pa,rtitioning scheme is presented a.long with silnplifica.t,iotls. Da.t.a. is also presented which shows the effects of the partitioning scheme on these coefficients. Finally, in section 4, results are presented comparing the partitioned PEEC with both a tra,ditional PEEC formulation and MOM results.
2
Derivation
of the PEEC
model
The purpose of this scctiotl is to derive a.n a.ppropria.te integral cqua.tiou for the PEEC interpretation, and then from this defiuc the pa.rtial inductance terms. This section will provide the founda.tion for the PEEC advancements in the uext section. 2.1
Integral
The electric space is
Equation
ra.diating
0
=
jwp
e)dv/ J G( F,e)J( VI
1-x
Q s VI
G(v, ~/)q(rr)dv~
P-2)
where J is the current density, q is the charge density, and G is the free space Greens function as given above.
2.2
Partial
Element
Equivalent
Circuit
Models In the PEEC formulation, both J and q are unknowns and are related by the continuity equation V . J + jwq = 0. The continuity equation is enforced by writing Kirchoff’s current law (KCL) at each node. The two unknowns, J and q, are each expanded into a series of pulse basis functions with an unknown amplitude. Pulse functions are also selected for the testing functions. The inner product is defined as the following < f,s >= J,fgdv=
in free
Example
where j is the current density in the conductor, Q is the charge density, and G is the free space Green’s function defined as G(p, ?) = s where R is given by R =I r - ? 1, r is the observation point, and F’ is the source point. For a perfect electrical conductor (PEC), f goes to zero, and the total electric field. within the conductor is 0. With this, Eq. 2.1 becomes
Formulation
field due to a currcut
2: Conductor
where v is the volume, area of a cell. 2.2.1
Partial
+
(2.3)
and a is the cross sectional
Inductance
The vector potential term in the integral equation 2.2 will be shown to correspond with partial Eq.
348
A
2 shows a. perfectly conducting inductances. Fig. strip which has been subdivided into 2 volume cells a and /3. Iiccpiug in mind the current density J is co~wt,a.i~tover each cell md 1; = .Jznccl[, where uccll is the cross scct.ioual area. of a cell, am1 by a’p plying Eq. 2.3 with respect to cell CY, the vector potential term in Eq. 2.2 hvx~~llcs
G(r, iv)dP,dV,l flaacvi V,l un j+ I t-z ssup/ vm G( 1‘, j’l)dv,dvpl apa,
1 +
LP+
.iWll 1 -SJ
From [12], the partial 111ut.ua.1iuductaucc cells cr a.nd ij is defiuctl a.s Lp(yp = .k -
4x ~q
1
1 -dv,dvup
ssvo! v8 R
Figure 3: Partitioning
between
(2.5)
Using this definition, filling in for the Green’s Function, and approsima.t.ing the in1.egra.l of the exponentia.l term by using the ccut.er to ccuter distance for R, equation 2.4 ca.u be rewritten as jw l~p~yuI, + jti Lp~Y,~I~jc-jpria~
(2.6)
where the first term is the part.ia.l self iiiducta,ncc of the cell a aud tllc second term represents the inductive coupling t,o cell Q from a. current in cell ,0 in Fig. 2. The dela,y between cells a and /3 are given by the phase term F -jpRd where R,.,,j is the center to center dista.nce between cells o a.nd 13, Similarly, by test.iug wit,h cell /?, the following relationships result (2.7) where the terms a.rc given as before. It is importa,nt to note tl1a.t the cxpoucntia.l t,rrrn must be taken outside of the iutcgral in order t,o rcwrit.e the contributious from the vector pot~ciitial portiou of the integral equation in terms of t.hc tlchnitjiou for pa,rtia.l inductmre in Eq. 2.5. In addition, this ospoucntial term is not integra.i.cd as it correspond t,o the time delay between cells in the time doma.iu I’E EC formulation. From both Eq. 2.6 am1 Eq. 2.7, it is cvideut that only the n1utua.l coupliugs between cells have a contribution due to the expotic1it~ia.i term. Therefore, by appr0xinia.t iiig the cxpoucutial term, there is no contribution it1 the calculation of the pa.rtial inductance for each self term. 349
of Cell
(2.4)
3
PEEC
Model
Advancements
Historically PEEC has used the definition of partial inductance as it was defined in [12], which is based on analytically integrating the k portion of the Green’s function, and approximating the exponential term by setting R equal to the center to center distance between two cells. This approximation, while less accura.te than a full integration, has provided very good answers in the active frequency range. Throughout this work, the active frequency range corresponds to a wavelength where all the cells are of a size such that they are 2 20 cells / A. It is the purpose of this section to introduce the partitioning technique, and some computational refinements for calculating the partial inductance terms in order to increase the accuracy of the solution for frequencies in the extended frequency range without introducing more unknowns. The extended frequency ra,nge, fe, is defined as frequencies greater than fmam, where fill,, is the highest frequency in the active frequency range. 3.1
Derivation
of Advancement
By subdividing ea.ch cell in Fig. 2 into a finite number of partitions as shown in Fig. 3, we can introduce a new way of computing the partial inductance terms for each cell. Each of these cells can be divided into a finite number of partitions given by A = $ where c is the speed of light in free space, fe is the upper end of the extended frequency range, and n = 1,2,3, . . . . This partitioning scheme is based on that fact that the cell is being subdivided into smaller partitions which a,re on the order of a wavelength at the highest frequency of interest, fe. The pa.rtitioning technique results in computing
Figure
4: Real
(sLp)
comparison
as n varies
the partial inductance for ea.& cell based on a summation over all the partitions in each cell. Taking an inductance cell a.s an csa.mplc, Eq. 2.5 becomes L],,f
= 2.l
..k&
5
E
5
5
Figure varies.
5:
Imaginary
comparison
as n
mutual coupling terms, the k term may be approximated using the center to center distance between the partitions resulting in Mi
e-.ipR.k,kk,q,qq
Lpt.
4 .I k=l ,kk=l q=l qq=l
where cells %and j a,re partitioned into AJi by Ni sections, a,nd A,f*i by Nj sections, respectively. The phase term is not under the integral, but is a.pproximated by defining R a?.sthe dist.a.nce bct.wccn the centroid of pa.rtition k, X:X:and of pa,rtition (/, (IQ. It is iruporta.nt to uutlcrst,a.ntl t hc effects of the partitioning schcute on t.hc parl.ia.1 self inductance term, since this is the term tl1a.t did not have a contribution due to the espourut.ia.l approximation. The real portion of sI,p 1ia.s been plol,ted in Fig. 4 for a, self pa.rtia.l inductanc~c term. As is evident, with no partitions, this port,ion of the pa.rt.ia.1 induct.a.nce term is 0. In a.dtlitiot~, one can n0t.c that through pa,rtitioning, t.hc self pa.rt.ial iutlucta,ncc term begins to converge for n = 3. The ima.ginary portion of sLp is shown in Fig. 5. This also shows convergence a,t approximately n = 3 for t,he partitioning. It should adso be noticed the pa.rtial inductance term does not va.ry.greatly below 20 GHz, but only in the extended region. From this da.ta., it is evident t,ha.t the partitioning does significantly improve the solution in the extended frequency ra.ugc. It should be noted tha,t the computation of the partial inclucta.nce has changed, although the number of unknowns renla*ins consta,nt. Since the partitioning scheme is effectively integrating the & term also, the pa.rtitioned partial inductance term in Eq. 3.8 ma,y be modified. For all the
(sLp)
23
Ni
Mj
Nj
’
=
where R is the center to center distance between partition L, Kk: and q, qq. The h term is integrated for each self partition calculation, where i = j, k = q, and kL = qq, using Eq. 3.8. With this approximation, there is only one integra,tion per cell, just as with the traditional PEEC formulation.
4
Results
In this section, we give experimental evidence of the benefits of the model improvements. Each example is discretized for 20 cells / X for the active frequency range, where the active frequency range is given by fmas = 20 GHz. The extended frequency range is given by f, = 1000 GHz. First, we would like to compare our results with a carefully implemented MOM formulation [13]. Our first comparison is for a patch antenna where we compare a traditional PEEC formulation with a carefully implemented MOM formulation as shown in Fig. 6. The patch antenna. is center fed and has a length of 9 mm and a width of 4.5 mm. It is clear that both approaches result in similar results for the active frequency range. For this same dipole example, and for the same discretization, a comparison is made between results from a traditional PEEC formulation and the new pa,rtitioned PEEC formulation for n = 1, referred to
350
0.B . ...:
0.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.
. ..
700
Boa
.
. ,
.
0.4 0.3 0.2 Y CT
0.1 0 -0.1 -0.2 -0.3 0
Figure 6: MOM
and
PEEC
for the active
range
200
100
300
600
400 Frequenoy
800 (GHr)
Figure 8: PEEC and Lp+ Z over the active range
Comparison
for Real
1 0.8 0.6
Figure
7: PEEC
and
Lp+
for the active
. .
j. . . . . . . .:.
.I..
. . .v . . .I..
. ..‘.‘..‘. . .
y . . . . .
. . .. . . . .
. . . I-““i”
PEEC --
,:
.:...,.
. . . . . . . . . . ;..
range
7 here as .Ly+. These results a.ro shown in Fig. over the a.ctive frequency ra.ngc. 11; is evident that the results wc wry similar. More intcrc~sting, howev~, a.rc tllc> results over the exknded frcquttnc’y ra.ngc. ‘I‘lwc results a,rc shown in Fig. 8 for 1,tic real porl,ioti of t hc inpub inipecla.nce, and in Fig. 9 for the imaginary portion of the input imp&nce for the same patch antcnua, a,nd the sa.me discrctiza.t,ioii. Frotn tlictsc rCsult.s, it is clear tha.t the partitioning schcm~ has climinatcd t,hc spurious resonances in the exi,cntlcd frequency ra.nge. In addition, results for the I-‘E EC method show tha,t t.he real input inipetlancc~ is ucgativc at approximately 150 GHz, w1ierca.s pa.rtitionetl results show that the real input impcda.nce positive is up to 800 C:IIz! This is a remsrkrtble improvcmc~nt in cxt.cnding the useful frequency ra,ngc, a.ntl in improving t.he sta,bility for the time doma.in. 3.8 for These ca.lcula.t.ions were made using Eq. only the self partition terms, a.ntl using Eq. 3.9 for all the mutual partition terms. It is important, to evaluatc the impact of only infCcgral.titlg the + term for the self pa.rt,it iolj t.crms. ‘J ’lw results in Fig: 10 show a. comparison of int rtgrat.ing 111~ + krms for all pa,rtitions, only for a.ll the self cc\11partitions, a.nd only for the self partit,iolr t crm. As is evitlcnt from the 351
. . . .
Figure 9: PEEC and Lpf inary Z over the active
Comparison range
for Imag-
data, integrating either all the partitions in a problem, or all the partitions within one celI results in a very similar solution. In addition, by only integrating the & term for the self pa.rtition cells, it is clear that a very good solution is obtained, while only having one integration per cell. From the results in Fig. 4 and Fig. 5, it was shown that the coefficients began to converge for n = 3. It is of interest to show the impact of varying the partitioning from 7~ = 2 to n = 4. These results for the real portion of the input impedance are shown 11. From these results, it is evident that in Fig.
.
. .>.
.
.:. .
. . . . . . . . . ;..
.
