time-varying components for enhancing wireless transfer of

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.

Powered by TCPDF (www.tcpdf.org)

This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.

Jayathurathnage, Prasad; Liu, Fu; Mirmoosa, Mohammad S.; Wang, Xuchen; Fleury, Romain;Tretyakov, Sergei A.Time-Varying Components for Enhancing Wireless Transfer of Power and Information

Published in:Physical Review Applied

DOI:10.1103/PhysRevApplied.16.014017

Published: 07/07/2021

Document VersionPublisher's PDF, also known as Version of record

Please cite the original version:Jayathurathnage, P., Liu, F., Mirmoosa, M. S., Wang, X., Fleury, R., & Tretyakov, S. A. (2021). Time-VaryingComponents for Enhancing Wireless Transfer of Power and Information. Physical Review Applied, 16(1),[014017]. https://doi.org/10.1103/PhysRevApplied.16.014017

https://doi.org/10.1103/PhysRevApplied.16.014017


P H Y SI C A L R E VI E W A P P LI E D 1 6, 0 1 4 0 1 7 ( 2 0 2 1)

Ti m e- V a r yi n g C o m p o n e nts f o r E n h a n ci n g Wi r el ess T r a nsf e r of P o w e r a n dI nf o r m ati o n

Pr as a d J a y at h ur at h n a g e ,1 F u Li u ,1, 2, * M o h a m m a d S. Mir m o os a ,1, 3 X u c h e n Wa n g ,1

R o m ai n Fl e ur y, 3 a n d S er g ei A. Tr et y a k o v 1

1D e p art m e nt of El e ctr o ni cs a n d N a n o e n gi n e eri n g, A alt o U ni v ersit y, P. O. B o x 1 5 5 0 0, FI- 0 0 0 7 6 A alt o, Fi nl a n d2K e y L a b or at or y f or P h ysi c al El e ctr o ni cs a n d D e vi c es of t h e Mi nistr y of E d u c ati o n a n d S h a a n xi K e y L a b ofI nf or m ati o n P h ot o ni c T e c h ni q u e, S c h o ol of El e ctr o ni c S ci e n c e a n d E n gi n e eri n g, F a c ult y of El e ctr o ni c a n d

I nf or m ati o n E n gi n e eri n g, Xi’ a n Ji a ot o n g U ni v ersit y, Xi’ a n 7 1 0 0 4 9, C hi n a3L a b or at or y of W a v e E n gi n e eri n g, S wiss F e d er al I nstit ut e of T e c h n ol o g y i n L a us a n n e ( E P F L), C H- 1 0 1 5

L a us a n n e, S witz erl a n d

( R e c ei v e d 2 1 J a n u ar y 2 0 2 1; r e vis e d 7 M a y 2 0 2 1; a c c e pt e d 1 0 J u n e 2 0 2 1; p u blis h e d 7 J ul y 2 0 2 1)

T e m p or al m o d ul ati o n of c o m p o n e nts of el e ctr o m a g n eti c s yst e ms pr o vi d es a n e x c e pti o n al o p p or-

t u nit y t o e n gi n e er t h e r es p o ns e of t h os e s yst e ms i n a d esir e d f as hi o n, b ot h i n t h e ti m e a n d

fr e q u e n c y d o m ai ns. F or e n gi n e eri n g ti m e- m o d ul at e d s yst e ms, o n e n e e ds t o t h or o u g hl y st u d y t h e b asi c

c o n c e pts a n d u n d erst a n d t h e s ali e nt c h ar a ct eristi cs of t e m p or al m o d ul ati o n. I n t his p a p er, w e c ar e-

f ull y st u d y p h ysi c al m o d els of b asi c b ul k cir c uit el e m e nts — c a p a cit ors, i n d u ct ors, a n d r esist ors — as

fr e q u e n c y dis p ersi v e a n d ti m e- v ar yi n g c o m p o n e nts a n d st u d y t h eir e ff e cts i n t h e c as e of p eri o di-

c al ti m e m o d ul ati o ns. We d e v el o p a s oli d t h e or y f or u n d erst a n di n g t h es e el e m e nts, a n d a p pl y it t o

t w o i m p ort a nt a p pli c ati o ns: wir el ess p o w er tr a nsf er a n d a nt e n n as. F or t h e first a p pli c ati o n, w e s h o w

t h at, b y p eri o di c all y m o d ul ati n g t h e m ut u al i n d u ct a n c e b et w e e n t h e tr a ns mitt er a n d r e c ei v er, t h e

f u n d a m e nt al li mits of cl assi c al wir el ess p o w er tr a nsf er s yst e ms c a n b e o v er c o m e. R e g ar di n g t h e s e c-

o n d a p pli c ati o n, w e c o nsi d er a ti m e- v ar yi n g s o ur c e f or el e ctri c all y s m all di p ol e a nt e n n as a n d s h o w

h o w ti m e m o d ul ati o n c a n e n h a n c e t h e a nt e n n a p erf or m a n c e. T h e d e v el o p e d t h e or y of el e ctr o m a g-

n eti c s yst e ms e n gi n e er e d b y t e m p or al m o d ul ati o n is a p pli c a bl e fr o m r a di o fr e q u e n ci es t o o pti c al

w a v el e n gt hs.

D OI: 1 0. 1 1 0 3/ P h ys R e v A p pli e d. 1 6. 0 1 4 0 1 7

I. I N T R O D U C TI O N

Alt h o u g h t h e c o n c e pt of “t e m p or al m o d ul ati o n ” [ 1 ] ofr a di o a n d o pti c al c o m p o n e nts w as pr o p o u n d e d d e c a d esa g o (s e e, e. g., R efs. [ 2 – 1 2 ]), a gr e at d e al of r e n e w e datt e nti o n h as r e c e ntl y b e e n gi v e n t o t his c o n c e pt, a n d,t o d a y, it is o n e of t h e m ai n r es e ar c h t o pi cs i n t h e el e c-tr o m a g n eti cs c o m m u nit y. T his is d u e t o t h e f a ct t h atti m e m o d ul ati o n of ( e ff e cti v e) p ar a m et ers of el e ctri c cir-c uits, tr a ns missi o n li n es, m e di a [ 1 3 – 1 7 ], m et a- at o ms [1 8 ],m et as urf a c es, m et a m at eri als [ 1 9 – 2 1 ], et c. c a n r es ult i no v er c o mi n g f u n d a m e nt al li mit ati o ns s u c h as r e ci pr o cit y[2 2 – 2 9 ], e n er g y a c c u m ul ati o n [3 0 ], b a n d wi dt h [3 1 ], a n di m p e d a n c e m at c hi n g [3 2 ]. M or e o v er, a d diti o n al f u n c-ti o n aliti es b e c o m e a v ail a bl e, s u c h as fr e q u e n c y c o n v er-si o n a n d g e n er ati o n of hi g h er- or d er fr e q u e n c y h ar m o ni csof w a v es [ 3 3 ], w a v efr o nt e n gi n e eri n g [3 3 – 3 5 ], o n e- w a yb e a m s plitti n g [ 3 6 ], p ar a m etri c a m pli fi c ati o n of w a v es

* f u.li u @ xjt u. e d u. c n

[3 7 ,3 8 ], a n d c o ntr olli n g i nst a nt a n e o us r a di ati o n fr o m s m allp arti cl es [ 1 8 ,3 9 ].

R e g ar di n g li n e ar a n d t e m p or all y m o d ul at e d el e ctri c cir-c uits, t h e r e a cti v e el e m e nts ( c a p a cit ors a n d i n d u ct ors) a n dt h e dissi p ati v e el e m e nt (r esist ors) ar e f or c e d t o c h a n g e i nti m e i n a d esir e d w a y. F or e x a m pl e, t his is t h e k e yst o n eof cl assi c al p ar a m etri c a m pli fi ers i n w hi c h, b y p eri o di-c all y m o d ul ati n g a r e a cti v e el e m e nt, it b e c o m es p ossi bl et o a m plif y si g n als p assi n g t hr o u g h t his el e m e nt. I n f a ct, t h er e as o n is t h at s u c h t e m p or al m o d ul ati o n e m ul at es a n e g a-ti v e r esist a n c e, w hi c h all o ws o n e t o p arti all y c o m p e ns at et h e i nt er n al r esist a n c e of t h e s o ur c e as w ell as ot h er l oss es.T h er ef or e, t h er e is t h e or eti c all y n o f u n d a m e nt al li mit o nt h e a m plit u d e of t h e c urr e nt dr a w n fr o m a v olt a g e s o ur c e( of c o urs e, t h er e ar e pr a cti c al li mit ati o ns d u e t o p ot e nti ali nst a biliti es, a v ail a bl e p o w er, a n d s o f ort h). A p art fr o mt h es e p ar a m etri c s yst e ms t h at e m ul at e n e g ati v e r esist a n c e,it a p p e ars t h at t h er e ar e ot h er p ot e nti all y us ef ul f e at ur esa n d e ff e cts i n ti m e- m o d ul at e d el e ctri c cir c uits t h at h a v e n oty et b e e n c o nsi d er e d. T o r e ali z e t h os e c h ar a ct eristi cs a n de ff e cts, o n e n e e ds t o est a blis h a g e n er al t h e or y a n d pr a cti c al

2 3 3 1- 7 0 1 9/ 2 1/ 1 6( 1)/ 0 1 4 0 1 7( 1 5) 0 1 4 0 1 7- 1 © 2 0 2 1 A m eri c a n P h ysi c al S o ci et y

https://orcid.org/0000-0002-6732-8348

https://orcid.org/0000-0002-7393-2963

https://orcid.org/0000-0002-4989-7651

https://orcid.org/0000-0002-5997-1130

https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevApplied.16.014017&domain=pdf&date_stamp=2021-07-07

http://dx.doi.org/10.1103/PhysRevApplied.16.014017

P R A S A D J A Y A T H U R A T H N A G E et al. P H Y S. R E V. A P P LI E D 1 6, 0 1 4 0 1 7 ( 2 0 2 1)

d esi g n t o ols t h at c a n b e us e d f or t h or o u g hl y d es cri b-i n g v ari o us ti m e- v ar yi n g el e m e nts, i n cl u di n g dissi p ati v ea n d r e a cti v e o n es. I n t his p a p er w e f o c us o n t his i m p or-t a nt iss u e. We i ntr o d u c e a g e n er al a n al yti c al a p pr o a c h f ora n al ysis of ti m e- m o d ul at e d str u ct ur es a n d dis c uss s e v er ali ntri g ui n g pr o p erti es t h at aris e d u e t o t e m p or al v ari ati o nsof cir c uit el e m e nts. We b eli e v e t h at t his st u d y will p a v e t h er o a d t o w ar d fi n di n g alt er n at e p ossi biliti es or e n gi n e eri n gel e ctr o m a g n eti c s yst e ms t h at c a n e ff e cti v el y us e m o d u-l at e d el e ctri c cir c uits f or i m pr o vi n g t h eir e ffi ci e n c y a n dp erf or m a n c e.

I n t his p a p er, w e c o nt e m pl at e w h at a ti m e- v ar yi n g r e a c-ti v e el e m e nt m e a ns i n g e n er al, fr o m t h e f u n d a m e nt alp oi nt of vi e w. We writ e a li n e ar, c a us al, a n d n o nst ati o n-ar y r el ati o n b et w e e n t h e el e ctri c c h ar g e a n d v olt a g e, a n d,c o ns e q u e ntl y, i ntr o d u c e t h e c a p a cit a n c e k er n el. A n a n al-o g o us r el ati o n r el at es m a g n eti c fl u x a n d el e ctri c c urr e nti n t er ms of t h e i n d u ct a n c e k er n el. B as e d o n t h e n oti o n oft h es e k er n els, w e d e fi n e t h e c o n c e pts of “t e m p or al c a p a c-it a n c e ” a n d “t e m p or al i n d u ct a n c e ” as w ell as e q ui v al e ntr e a ct a n c es. T h es e n oti o ns all o w us t o dis c uss t h e f u n d a-m e nt al ass u m pti o ns b e hi n d t h e cir c uit m o d els of ti m e-m o d ul at e d c o m p o n e nts. We pr es e nt a g e n er al a n al yti c alm et h o d f or t h e a n al ysis of li n e ar ti m e- m o d ul at e d cir c uitswit h ar bitr ar y p eri o di c m o d ul ati o n of el e m e nts i n cl u di n gt h e r esisti v e el e m e nt. T his m et h o d is b as e d o n t h e us e ofe ff e cti v e m atri x cir c uit p ar a m et ers ( m atri x i m p e d a n c es ora d mitt a n c es) t h at r el at e fr e q u e n c y h ar m o ni cs of v olt a g esa n d c urr e nts.

We e m pl o y t his a p pr o a c h t o st u d y t w o di ff er e nt p ossi bil-iti es o ff er e d b y ti m e m o d ul ati o n. As t h e first p ossi bilit y, w ec o nsi d er n e ar- fi el d wir el ess p o w er tr a nsf er ( W P T) s yst e msi n w hi c h t h e t e m p or al m o d ul ati o n of m ut u al i n d u ct a n c e oft h e t w o c oils all o ws e n h a n c e m e nt of d eli v er e d p o w er wit h-o ut c o m pr o misi n g p o w er e ffi ci e n c y, w hi c h is n ot p ossi bl ei n c o n v e nti o n al s yst e ms. As t h e s e c o n d p ossi bilit y, w e fi n dt h at a ti m e- m o d ul at e d r esist a n c e dr a m ati c all y a ff e cts t h et ot al e q ui v al e nt r e a ct a n c e of a R L C r es o n a nt cir c uit wit h-o ut a d di n g e xt er n al r e a cti v e el e m e nts t o t h e cir c uit. T hish as a si g ni fi c a nt i m p a ct, f or e x a m pl e, i n s m all a nt e n n asf or t u ni n g a n d e n h a n ci n g t h e a nt e n n a b a n d wi dt h. N ot e t h ats u c h r esist a n c e c a n b e pr o vi d e d e xt er n all y ( c o n n e ct e d t ot h e i n p ut t er mi n als of t h e s m all a nt e n n a as t h e i nt er n alr esist a n c e of t h e si g n al s o ur c e) or b y m o d ul ati n g t h e s h a p eof t h e a nt e n n a.

T h e p a p er is or g a ni z e d as f oll o ws. I n S e c. II, w e gi v ea g e n er al e x pl a n ati o n of t h e n at ur e a n d m o d els of n o n-st ati o n ar y r e a cti v e el e m e nts, a n d, i n S e cs. III a n d I V, w edis c uss t h e g e n er al a p pr o a c h t o m o d el a n d u n d erst a n dt h e p eri o di c all y m o d ul at e d cir c uit c o m p o n e nts. I n S e c. V ,t h e e n h a n c e m e nt of n e ar- fi el d W P T is st u di e d, a n d, i nS e c. VI , w e s h o w h o w a ti m e- v ar yi n g r esist or h el ps t oi m pr o v e i m p e d a n c e m at c hi n g. Fi n all y, S e c. VII c o n cl u d est h e p a p er.

II. B A SI C C O N C E P T S: TI M E- V A R YI N GR E A C TI V E E L E M E N T S

I n t his s e cti o n, w e dis c uss t h e ass u m pti o ns m a d e i nm o d eli n g b ul k ( el e ctri c all y s m all) c o m p o n e nts w h os ep ar a m et ers v ar y i n ti m e d u e t o m o d ul ati o n b y e xt er n alf or c es or fi el ds. T his dis c ussi o n is i m p ort a nt t o d e fi n et h e a p pli c a bilit y r e gi o n of t h e t h e or eti c al m o d els t h at w ed e v el o p a n d us e i n t his p a p er. F urt h er m or e, w e i ntr o d u c ee ff e cti v e p ar a m et ers of ti m e- m o d ul at e d el e m e nts (t e m p or alc a p a cit a n c e, i n d u ct a n c e, a n d r esist a n c e) t h at will b e us e di n st u d yi n g e n h a n c e d wir el ess p o w er tr a nsf er d e vi c es a n da nt e n n a s yst e ms.

A. Ti m e- v a r yi n g c a p a cit a n c e

L et us c o nsi d er a si m pl e c a p a cit or f or m e d b y t w o m et alpl at es ( m a d e of p erf e ct el e ctri c c o n d u ct ors) c o n n e ct e d t o ati m e- v ar yi n g v olt a g e s o ur c e, w hi c h e x erts a ti m e- v ar yi n gel e ctri c fi el d E (t) b et w e e n t h e m. T h e s p a c e b et w e e n t h epl at es is fill e d b y a di el e ctri c m at eri al. At t his p oi nt, w ec o nsi d er a n ar bitr ar y t e m p or al f u n cti o n f or t h e v olt a g es o ur c e.

T h e el e ctri c pr o p erti es of c a p a cit ors c a n b e m o d ul at e di n ti m e usi n g s e v er al m e a ns. O n e a p pr o a c h is t o c h a n g e t h eg e o m etri c al p ar a m et ers (s u c h as t h e dist a n c e b et w e e n t h epl at es) b y a p pl yi n g s o m e m e c h a ni c al f or c e. T his is p ossi-bl e e v e n if t h er e is n o m at eri al filli n g t h e c a p a cit or. A n ot h era p pr o a c h is t o c h a n g e t h e pr o p erti es of t h e m e di u m t h atfills t h e c a p a cit or v ol u m e b y a p pl yi n g a str o n g e n o u g he xt er n al v olt a g e, or ill u mi n ati n g b y li g ht, or h e ati n g a n dc o oli n g t h e c a p a cit or ( n ot e t h at t h e m e di u m r es p o ns e t ot h e si g n al v olt a g e s o ur c e is still li n e ar). T his is t h e c as e ofa v ar a ct or, f or e x a m pl e.

T h e first ass u m pti o n t h at w e m a k e is t h at t h e c a p a ci-t or is a b ul k el e m e nt. T h at is, its si z e is s m all c o m p ar e dt o t h e w a v el e n gt h at all r el e v a nt fr e q u e n ci es. T his is t h eus u al ass u m pti o n i n cir c uit t h e or y, j ustif yi n g t h e us e oft h e n oti o ns of v olt a g e a n d m a g n eti c fl u x, a n d all o wi n g t h ewriti n g of r el ati o ns b et w e e n v olt a g es a n d c urr e nts. I n t h ec as e of ti m e m o d ul ati o ns or t h e pr es e n c e of n o nli n e ar el e-m e nts, hi g h er- or d er fr e q u e n c y h ar m o ni cs ar e g e n er at e d,a n d t h e si z e of t h e c o m p o n e nt is ass u m e d t o b e s m allat all fr e q u e n ci es w h er e t h e os cill ati o ns ar e si g ni fi c a nt.B e c a us e of t h e e n er g y c o ns er v ati o n a n d t h e fi nit e i n p ute n er g y fr o m t h e m o d ul ati n g m e c h a nis m, t h e a m plit u d e ofr es p o ns e at hi g h- or d er h ar m o ni cs m ust d e cr e as e, a n d e v e n-t u all y it t e n ds t o z er o at hi g h fr e q u e n ci es, e v e n wit h ad e e p m o d ul ati o n, as s h o w n i n R ef. [ 2 9 ]. F or t his r e as o n,t h e b ul k- c o m p o n e nt m o d el a n d t h e n oti o ns of c a p a ci-t a n c e, i n d u ct a n c e, a n d r esist a n c e c a n als o b e us e d f orti m e- m o d ul at e d el e m e nts of el e ctri c all y s m all si z es at allr el e v a nt fr e q u e n ci es.

L et us first dis c uss t h e m o d ul ati o n b y m e c h a ni c al m e a ns,s a y, b y c h a n gi n g t h e dist a n c e b et w e e n t h e pl at es i n t h e

0 1 4 0 1 7- 2

TI M E- V A R YI N G C O M P O N E N T S... P H Y S. R E V. A P P LI E D 1 6, 0 1 4 0 1 7 ( 2 0 2 1)

a bs e n c e of m at eri al filli n g. T h e r el ati o n b et w e e n t h e s ur-f a c e c h ar g e d e nsit y a n d t h e n or m al c o m p o n e nt of el e ctri cfl u x d e nsit y ( w hi c h i n t his c as e is e q u al t o 0 m ulti pli e d b yt h e n or m al c o m p o n e nt of t h e el e ctri c fi el d at t h e pl at es) isi nst a nt a n e o us, as it f oll o ws fr o m i nt e gr ati o n of ∇ · D = ρo v er a b o x of a n e gli gi bl e t hi c k n ess. B ut, si n c e w e h a v eass u m e d t h at t h e el e ctri c al si z e of t h e c a p a cit or is n e gli-gi bl e, w e c a n n e gl e ct t h e ti m e d el a y b et w e e n t h e c h a n g eof t h e el e ctri c fi el d ri g ht at t h e pl at es a n d t h e c orr es p o n d-i n g c h a n g e of t h e v olt a g e b et w e e n t h e pl at es. T h us, t h er el ati o n b et w e e n t h e c h ar g e a n d v olt a g e c a n b e ass u m e dt o b e i nst a nt a n e o us, a n d i n t his c as e it is p ossi bl e t o us et h e c o n v e nti o n al d e fi niti o n of c a p a cit a n c e, writi n g Q (t) =C (t)V (t), w h er e c a p a cit a n c e C (t) d e p e n ds o n ti m e d u e t oe xt er n al m o d ul ati o n.

L et us n o w c o nsi d er t h e c as e w h e n t h e s p a c e b et w e e nt h e pl at es is fill e d wit h a n el e ctri c all y p ol ari z a bl e m e di u mw h os e pr o p erti es ar e c h a n gi n g i n ti m e d u e t o s o m e e xt er n alf or c e. I n t his c as e, w e n e e d t o a c c o u nt f or ti m e (fr e-q u e n c y) dis p ersi o n of t h e filli n g m at eri al. Ass u mi n g t h att h e r es p o ns e is li n e ar ( a n d, of c o urs e, c a us al), t h e p ol ar-i z ati o n d e nsit y P (t) b et w e e n t h e pl at es is c o u pl e d wit h t h eel e ctri c fi el d as

P (t) = 0

∞

0

χ ( γ , t)E (t − γ ) d γ ( 1)

wit h t h e r e al- v al u e d f u n cti o n χ ( γ , t) b ei n g t h e s us c e pti-bilit y k er n el. Writi n g t h e d e fi niti o n of v e ct or D , D (t) =

0 E (t) + P (t), w e r e a dil y c o n cl u d e t h at t h e el e ctri c fl u xd e nsit y D (t) is

D (t) = 0

∞

0k ( γ , t)E (t − γ ) d γ , ( 2)

w h er e

k ( γ , t) = δ ( γ ) + χ ( γ , t) ( 3)

is t h e r el ati v e p er mitti vit y k er n el i n t h e ti m e d o m ai n. T h en or m al c o m p o n e nt of t h e el e ctri c fl u x d e nsit y e q u als t h es urf a c e c h ar g e d e nsit y o n t h e pl at es. B y i nt e gr ati n g t h e s ur-f a c e c h ar g e d e nsit y o v er t h e s urf a c e of t h e pl at e, w e fi n dt h e t ot al fr e e c h ar g e o n t h e pl at es. T h er ef or e, b y ass u mi n gh o m o g e n e o us c h ar g e distri b uti o n o n t h e pl at es, w e h a v et h e i nst a nt a n e o us c h ar g e Q (t) o n t h e pl at e as

Q (t) = A 0

∞

0k ( γ , t)[a n · E (t − γ ) ]d γ , ( 4)

w h er e A a n d a n ar e t h e ar e a a n d t h e u nit v e ct or n or m al t ot h e s urf a c e, r es p e cti v el y.

O n t h e ot h er h a n d, t h e i nt e gr ati o n of E (t) al o n g ar o ut e b et w e e n t h e t w o pl at es gi v es t h e v olt a g e di ff er e n c eV (t) b et w e e n t h e t w o pl at es. R e c alli n g t h e ass u m pti o n

of n e gli gi bl e el e ctri c si z e of t h e c a p a cit or at all r el e-v a nt fr e q u e n ci es, w e us e t h e q u asist ati c a n d h o m o g e n e o usa p pr o xi m ati o n, w h er e t h e v olt a g e is si m pl y t h e m ulti-pli c ati o n of t h e n or m al el e ctri c fi el d a n d t h e dist a n c e db et w e e n t h e t w o pl at es. T h us, t h e a b o v e e q u ati o n c a n b er e writt e n as

Q (t) =∞

0

C k ( γ , t)V (t − γ ) d γ , ( 5)

w h er e

C k ( γ , t) = 0 k ( γ , t)A

d( 6)

is t h e c a p a cit a n c e k er n el i n t h e ti m e d o m ai n. T his is ag e n er al r el ati o n t h at c o n n e cts t h e c h ar g e o n t h e pl at esa n d t h e v olt a g e di ff er e n c e b et w e e n t h e m. H er e, C k ( γ , t)d e p e n ds o n t h e o bs er v ati o n ti m e t, at w hi c h w e m e as ur et h e el e ctri c c h ar g e, a n d t h e d el a y ti m e γ b et w e e n t h ec h ar g e m e as ur e m e nt a n d t h e a p pli e d ti m e- v ar yi n g v olt a g edi ff er e n c e.

N o w, s u p p os e t h at t h e s o ur c e v olt a g e is ti m e h ar m o ni c;i n ot h er w or ds,

V s (t) = R e[ V 0 ej ω t], ( 7)

w h er e V 0 i s t h e c o m pl e x a m plit u d e, ω is t h e a n g ul ar fr e-q u e n c y of t h e si g n al, a n d R e[ ·] d e n ot es t h e r e al p art oft h e e x pr essi o n i nsi d e t h e br a c k et. B as e d o n t h e a b o v e t w oe q u ati o ns, w e fi n d t h at t h e c h ar g e is si m pl y

Q (t) = R e[ C ( ω , t)V 0 ej ω t], ( 8)

w h er e

C ( ω , t) =∞

0

C k ( γ , t)e − j ω γ d γ . ( 9)

T h e ri g ht- h a n d si d e of E q. ( 9) is t h e F o uri er tr a nsf or m oft h e c a p a cit a n c e k er n el C k ( γ , t) wit h r es p e ct t o t h e ti m ed el a y v ari a bl e γ . T h er ef or e, t h e f u n cti o n C ( ω , t) o n t h el eft- h a n d si d e e x pli citl y d e p e n ds o n t h e a n g ul ar fr e q u e n c yd u e t o t h e i n h er e nt dis p ersi o n of t h e m at eri al filli n g, i na d diti o n t o t h e d e p e n d e n c y o n ti m e d u e t o t h e n o nst ati o n-arit y. If t h er e is n o ti m e m o d ul ati o n of t h e m at eri al filli n gt h e s p a c e b et w e e n t h e t w o pl at es, C ( ω , t) is o nl y a f u n cti o nof t h e a n g ul ar fr e q u e n c y a n d ∂ C ( ω , t) / ∂ t = 0.

We c all C ( ω , t) t e m p or al c a p a cit a n c e. Si mil arl y, t e m p o-r al p er mitti vit y of t h e filli n g m e di u m is d e fi n e d as

( ω , t) =∞

0k ( γ , t)e − j ω γ d γ . ( 1 0)

I n vi e w of E q. ( 6), w e c a n e asil y c o n cl u d e t h at, f or p ar all el-pl at e c a p a cit ors, t h e t e m p or al c a p a cit a n c e a n d t h e t e m p or al

0 1 4 0 1 7- 3


p er mitti vit y ar e r el at e d i n t h e us u al w a y as

C ( ω , t) = 0 ( ω , t)A

d. ( 1 1)

T h e el e ctri c c urr e nt t hr o u g h t h e c a p a cit or is t h e ti m ed eri v ati v e of t h e c h ar g e, i. e., I (t) = d Q (t) /dt . H e n c e, b ye m pl o yi n g E q. ( 8), w e d e d u c e t h at

I (t) = R e j ω C ( ω , t) +∂ C ( ω , t)

∂ tV 0 e

j ω t . ( 1 2)

I n t h e st ati o n ar y c as e w h e n t h e c a p a cit or pr o p erti es ar eti m e i n v ari a nt, t h e ti m e d eri v ati v e of t h e t e m p or al c a p a ci-t a n c e is z er o, a n d w e g et t h e us u al e x pr essi o n: I0 = j ω C V 0

i n w hi c h I0 i s t h e c o m pl e x a m plit u d e of t h e el e ctri c c urr e nt.L et us i ntr o d u c e t h e d e fi niti o n

C e q ( ω , t) = C ( ω , t) +1

j ω

∂ C ( ω , t)

∂ t

= 0 ( ω , t) +1

j ω

∂ ( ω , t)

∂ t

A

d. ( 1 3)

I n t his w a y, w e c a n als o us e t h e c o n v e nti o n al e x pr es-si o n f or t h e el e ctri c c urr e nt f or ti m e- m o d ul at e d c a p a cit ors,writi n g

I (t) = R e[ j ω C e q ( ω , t)V 0 ej ω t]. ( 1 4)

I n ot h er w or ds, w e c a n d e fi n e t e m p or al a d mitt a n c e as

Y ( ω , t) = j ω C e q ( ω , t). ( 1 5)

I nt er esti n gl y, t h e e q ui v al e nt t e m p or al c a p a cit a n c e C e q ( ω , t)is c o m pl e x v al u e d r at h er t h a n a r e al v al u e. If w e ass u m et h at t h e t e m p or al c a p a cit a n c e C ( ω , t) is r e al wit hi n a c ert ai nfr e q u e n c y r a n g e, t h e r e al a n d i m a gi n ar y p arts of C e q ( ω , t)ar e as f oll o ws:

R e[ C e q ( ω , t)] = C ( ω , t),

I m[C e q ( ω , t)] = −1

ω

∂ C ( ω , t)

∂ t.

( 1 6)

T h er ef or e, t h e i m a gi n ar y p art of t h e e q ui v al e nt t e m p o-r al c a p a cit a n c e is pr o p orti o n al t o t h e ti m e d eri v ati v e oft h e t e m p or al c a p a cit a n c e. R e c alli n g E q. ( 1 5), w e fi n d t h att h e r e al p art of t h e t e m p or al a d mitt a n c e is R e[Y ( ω , t)] =∂ C ( ω , t) / ∂ t. W h at d o es t his m e a n ? It s h o ws t h at t h e ti m ed eri v ati v e of t h e t e m p or al c a p a cit a n c e C ( ω , t) c a n b e i nt er-pr et e d as a n e ff e cti v e p ositi v e or n e g ati v e t e m p or al r esis-t a n c e. If t h e ti m e d eri v ati v e is p ositi v e, w e h a v e a p ositi v er esist a n c e, a n d t h er e is a n e ff e cti v e ti m e- v ar yi n g n e g ati v er esist a n c e i n t h e c as e of a n e g ati v e ti m e d eri v ati v e.

B as e d o n t h e a b o v e r es ults, w e c a n us e E qs. ( 1 2) a n d( 1 4) t o fi n d c urr e nt r el at e d t o all fr e q u e n c y c o m p o n e nts of

t h e v olt a g e. I m p ort a ntl y, b e c a us e t h e s yst e m is li n e ar, w ec a n us e t his t e c h ni q u e t o fi n d r es p o ns es t o a n y p eri o di c all yv ar yi n g v olt a g e. T o d o t h at, w e e x p a n d t h e v olt a g e f u n cti o ni nt o t h e F o uri er s eri es a n d us e t h e t e m p or al i m p e d a n c e asa f u n cti o n of t h e fr e q u e n c y.

B. Ti m e- v a r yi n g i n d u ct a n c e

Si mil arl y t o t h e c as e of a c a p a cit or, f or a ti m e- m o d ul at e di n d u ct or, t h e first ass u m pti o n is t h at t h e si z e of t h e i n d u c-t or is el e ctri c all y s m all c o m p ar e d t o t h e w a v el e n gt h at allr el e v a nt fr e q u e n ci es. T h at is, t h e i n d u ct or c a n b e vi e w e d asa b ul k c o m p o n e nt. Als o i n t his c as e, ti m e m o d ul ati o n c a nb e r e ali z e d eit h er b y c h a n gi n g t h e c oil si z es or b y c h a n g-i n g t h e m a g n eti c pr o p erti es of t h e c oil c or e. I n t h e firstc as e, t h e r el ati o n b et w e e n t h e m a g n eti c fl u x a n d t h e c ur-r e nt c a n b e ass u m e d t o b e i nst a nt a n e o us, a n d w e c a n writ eϕ ( t) = L (t)I (t). If t h e c or e is fill e d wit h a dis p ersi v e m a g-n eti c m at eri al, t h e r el ati o n t a k es t h e g e n er al li n e ar c a us alf or m

ϕ ( t) =∞

0

L k ( γ , t)I (t − γ ) d γ , ( 1 7)

w h er e L k ( γ , t) is t h e i n d u ct a n c e k er n el i n t h e ti m e d o m ai n.Usi n g t h e F o uri er tr a nsf or m, w e d e fi n e t e m p or al i n d u c-t a n c e as

L ( ω , t) =∞

0

L k ( γ , t)e − j ω γ d γ . ( 1 8)

Ass u mi n g a ti m e- h ar m o ni c c urr e nt s o ur c e Is (t) = R e[I0 e

j ω t], w e r e a dil y c o n cl u d e t h at

ϕ ( t) = R e[ L ( ω , t)I0 ej ω t]. ( 1 9)

Si n c e t h e v olt a g e di ff er e n c e o v er t h e el e m e nt V (t) is t h eti m e d eri v ati v e of t h e m a g n eti c fl u x, aft er si m pl e al g e br ai cm a ni p ul ati o ns, w e o bt ai n

V (t) = R e[ j ω L e q ( ω , t)I0 ej ω t], ( 2 0)

w h er e t h e e q ui v al e nt t e m p or al i n d u ct a n c e L e q ( ω , t) r e a ds

L e q ( ω , t) = L ( ω , t) +1

j ω

∂ L ( ω , t)

∂ t. ( 2 1)

As c a n b e s e e n, L e q ( ω , t) is c o m pl e x v al u e d, a n d, b as e do n t h e a b o v e t w o e q u ati o ns, w e d e fi n e t h e t e m p or ali m p e d a n c e Z( ω , t) as

Z ( ω , t) = j ω L e q ( ω , t)

=∂ L ( ω , t)

∂ t+ j ω L ( ω , t). ( 2 2)

It is cl e ar t h at p ositi v e ( n e g ati v e) t e m p or al r esist a n c eis a c hi e v a bl e w h e n t h e ti m e d eri v ati v e of t h e t e m p or ali n d u ct a n c e is p ositi v e ( n e g ati v e).

0 1 4 0 1 7- 4


III. A G E N E R A L T O O L F O R A N A L Y ZI N GTI M E- M O D U L A T E D E L E M E N T S

As w e h a v e s h o w n a b o v e, t h e r es p o ns e of ti m e-m o d ul at e d cir c uit c o m p o n e nts is d es cri b e d b y t e m p or alc a p a cit a n c e, i n d u ct a n c e, a n d r esist a n c e t h at d e p e n d b ot ho n ti m e a n d t h e fr e q u e n c y. We h a v e s e e n t h at t h e fr e-q u e n c y d e p e n d e n c e a p p e ars d u e t o fr e q u e n c y dis p ersi o nof m at eri als fr o m w hi c h t h e c o m p o n e nts ar e m a d e ( us u-all y di el e ctri c filli n gs of c a p a cit ors a n d m a g n eti c filli n gsof i n d u ct ors). If t h e p er mitti vit y a n d p er m e a bilit y of t h em at eri al filli n g h as n e gli gi bl e fr e q u e n c y dis p ersi o n i n t h er el e v a nt fr e q u e n c y r a n g e, w e c a n c o nsi d er t e m p or al c a p a c-it a n c e, t e m p or al a d mitt a n c e, a n d t e m p or al r esist a n c e asf u n cti o ns of ti m e o nl y.

N e xt, w e d e v el o p a g e n er al t o ol f or a n al y zi n g el e ctri c als yst e ms c o nsisti n g of p eri o di c all y ti m e- m o d ul at e d dis p er-si v e el e m e nts. T h e m o d ul ati o n f u n cti o n c a n b e ar bitr ar y,li mit e d o nl y b y t h e ass u m pti o n of el e ctri c all y s m all c o m-p o n e nts. A si mil ar m atri x f or m ul ati o n wit h o ut c o nsi d eri n gt h e dis p ersi o n e ff e ct h as b e e n pr es e nt e d i n e arl y st u di es[1 2 ]. H er e, w e s yst e m ati c all y d e v el o p t h e g e n er al t h e-or y as t h e f o u n d ati o n of t h e pr o p os e d a p pli c ati o ns i n t hism a n us cri pt. We first a n al y z e t h e e ff e ct of ti m e- m o d ul at e dl u m p e d el e m e nts, i. e., i n d u ct or, c a p a cit or, a n d r esist or,wit h t h e v olt a g e- c urr e nt r el ati o n i n t h e ti m e d o m ai n, a n dt h e n i n v esti g at e t h eir e ff e cts i n cir c uits.

A. Ti m e- m o d ul at e d i n d u ct o rs

L et us c o nsi d er a ti m e- v ar yi n g i n d u ct or, m o d el e d b y itst e m p or al i n d u ct a n c e ( 1 8). If t h e m o d ul ati o n is p eri o di c alwit h p eri o d T , t h e i n d u ct a n c e f u n cti o n s atis fi es L ( ω , t) =L ( ω , t + n T ) wit h n ∈ Z . T h er ef or e, t h e t e m p or all y v ar yi n gi n d u ct a n c e c a n b e e x p a n d e d i n c o m pl e x F o uri er s eri es as

L ( ω , t) =

+ ∞

p = − ∞

lp ( ω )e j pω M t, ( 2 3)

w h er e ω M = 2 π / T is t h e f u n d a m e nt al a n g ul ar fr e q u e n c yof m o d ul ati o n, a n d t h e c o m pl e x c o e ffi ci e nts lp ( ω ) =

(1 / T )T

0 L ( ω , t)e − j pω M tdt ar e t h e m o d ul ati o n c o e ffi ci e nts( a m plit u d es a n d p h as es) at a n g ul ar fr e q u e n ci es p ω M . Ifl oss es i n t h e i n d u ct or m at eri al c a n b e n e gl e ct e d, L ( ω , t)is a r e al- v al u e d f u n cti o n, a n d t h e m o d ul ati o n c o e ffi ci e ntss atisf y t h e c o n diti o n lp ( ω ) = l∗− p ( ω ). I n t h e g e n er al c as e,L ( ω , t) is a c o m pl e x- v al u e d c a us al r es p o ns e f u n cti o n,w hi c h s atis fi es t h e Kr a m ers- Kr o ni g r el ati o n at e v er y v al u eof t h e ti m e ar g u m e nt t [1 6 ].

L et us ass u m e t h at a p eri o di c all y ti m e- m o d ul at e d i n d u c-t or is c o n n e ct e d t o a n el e ctri c al cir c uit t h at is e x cit e d b ye xt er n al s o ur c es. A c c or di n g t o t h e Fl o q u et t h e or e m, if t h ee xt er n al e x cit ati o n h as a ti m e- h ar m o ni c c o m p o n e nt e j ω s t

at a n g ul ar fr e q u e n c y ω s , t h e c orr es p o n di n g v olt a g e V (t)a n d c urr e nt I (t) a cr oss t his ti m e- m o d ul at e d i n d u ct or will

e x hi bit a n i n fi nit e n u m b er of Fl o q u et h ar m o ni cs, a n d t h e yc a n b e e x pr ess e d b y [ 4 1 ,4 2 ]

V (t) =

+ ∞

n = − ∞

v n ej ω n t ( 2 4)

a n d

I (t) =

+ ∞

n = − ∞

in ej ω n t, ( 2 5)

w h er e v n a n d in ar e c o m pl e x- v al u e d c o e ffi ci e nts, a n d

ω n = ω s + n ω M . ( 2 6)

N ot e t h at i n E qs. ( 2 4) a n d ( 2 5) w e o mit t h e o p er at orR e[ ·] i n fr o nt of t h e s u m m ati o n si g n f or m at h e m ati c alc o n v e ni e n c e i n t h e f oll o wi n g d eri v ati o ns.

If t h e e x cit ati o n c o nt ai ns s e v er al ti m e- h ar m o ni c c o m p o-n e nts at di ff er e nt a n g ul ar fr e q u e n ci es, t h e v olt a g e a cr oss( a n d c urr e nt t hr o u g h) t h e ti m e- m o d ul at e d i n d u ct or will b ea s u p er p ositi o n of Fl o q u et h ar m o ni cs of E qs. ( 2 4) a n d ( 2 5)f or m ulti pl e fr e q u e n c y s ets i n E q. ( 2 6), as f oll o ws fr o m t h eli n e arit y of t h e cir c uit.

F or e a c h fr e q u e n c y h ar m o ni c of t h e c urr e nt, t h e c orr e-s p o n di n g v olt a g e is gi v e n b y E q. ( 2 0). L et us c o nsi d er t h en t h c urr e nt h ar m o ni c, In (t) = in e

j ω n t. F or t his h ar m o ni c,t h e c orr es p o n di n g v olt a g e a cr oss t h e ti m e- v ar yi n g i n d u c-t or c a n b e o bt ai n e d b y s u bstit uti n g In (t) a n d t h e F o uri ere x p a nsi o n of t h e i n d u ct a n c e ( 2 3) i nt o E q. ( 2 0), w hi c h gi v es

V n (t) =p

j ( ωn + p ω M )lp ( ωn )ej pω M t in e

j ω n t. ( 2 7)

B e c a us e of t h e li n e arit y, w e c a n fi n d t h e t ot al v olt a g e [s e eE q. ( 2 4)] b y s u m mi n g t h e c o ntri b uti o ns i n d u c e d b y all t h efr e q u e n c y h ar m o ni cs of t h e c urr e nt,

n

v n ej ω n t =

n p

j ω n + p lp ( ωn )in ej ω n + p t, ( 2 8)

w h er e t h e s u m m ati o n o v er n a n d p is fr o m − ∞ t o + ∞ .B y r e pl a ci n g t h e i n d e x n b y n − p o n t h e ri g ht- h a n d si d eof E q. ( 2 8), w e h a v e

n

v n ej ω n t =

n p

j ω n lp ( ωn − p )in − p e j ω n t. ( 2 9)

N o w w e c a n s e e t h at E q. ( 2 9) s h ar es t h e s a m e b asis o n b ot hsi d es, a n d w e c a n e q u at e t h e c orr es p o n di n g c o e ffi ci e nts:

v n =p

j ω n lp ( ωn − p )in − p ( 3 0)

f or p fr o m − ∞ t o + ∞ . Fr o m t his e q u ati o n w e cl e arl y s e et h at, w h e n t h e i n d u ct or is m o d ul at e d i n ti m e, t h e v olt a g e

0 1 4 0 1 7- 5


c o m p o n e nt at fr e q u e n c y ω n i s n ot o nl y r el at e d t o t h e c ur-r e nt at t h e s a m e fr e q u e n c y ω n , b ut als o t o t h e c urr e nts atot h er fr e q u e n ci es ω n − p , wit h t h e c o e ffi ci e nts j ω n lp ( ωn − p )pl a yi n g t h e r ol e of m ut u al i m p e d a n c es.

Si n c e all t h e v olt a g e h ar m o ni cs s atisf y E q. ( 3 0), w e c a nwrit e t his li n e ar e q u ati o n i n m atri x f or m as

v = ¯̄Z L · i, ( 3 1)

w h er e v e ct ors v = { v − ∞ , . . . , v − 1 , v 0 , v 1 , . . . , v ∞ } a n di = { i− ∞ , . . . , i− 1 , i0 , i1 , . . . , i∞ } r es e m bl e t h e c o m pl e xs p e ctr a of t h e v olt a g e a n d c urr e nt at fr e q u e n ci es

{ω − ∞ , . . . , ω − 1 , ω s , ω 1 , . . . , ω ∞ }. T h e m atri x ¯̄Z L i s t h er e-f or e t h e i m p e d a n c e m atri x t h at r el at es t h e v olt a g e a n dc urr e nt i n t h e fr e q u e n c y d o m ai n. T his i m p e d a n c e m atri xc o nt ai ns o ff- di a g o n al t er ms d u e t o t h e m o d e c o u pli n gi ntr o d u c e d b y ti m e m o d ul ati o n.

Us u all y, t h e m o d ul ati o n d e pt h is r el ati v el y s m all a n dt h e m o d e c o u pli n g m ai nl y t a k es pl a c e at fr e q u e n ci es cl os et o ω s . I n t his c as e, w e c a n si m plif y t h e s yst e m a n d c o n-si d er o nl y t h e h ar m o ni cs wit h n fr o m − N t o + N wit h t h etr u n c ati o n i n d e x N b ei n g a fi nit e i nt e g er. I n f a ct, it is n e c es-s ar y t o li mit t h e or d er of h ar m o ni cs i n or d er t o e ns ur e t h atw e st a y wit hi n t h e ass u m pti o n of el e ctri c all y s m all, b ul kc o m p o n e nts. Si n c e t h e c o m p o n e nt si z e h as b e e n ass u m e dt o b e m u c h s m all er t h a n t h e w a v el e n gt h at all r el e v a ntfr e q u e n ci es, t h e tr u n c ati o n i n d e x m ust s atisf y

Nc

Df s, ( 3 2)

w h er e c is t h e s p e e d of li g ht, D is t h e si z e of t h e c o m-p o n e nt, a n d fs = ω s / 2 π is t h e f u n d a m e nt al fr e q u e n c y. I npr a cti c e, t h e l o w er b o u n d o n t h e tr u n c ati o n n u m b er Nt h at is d e fi n e d b y t h e str e n gt hs of cr oss- c o u pli n g b et w e e nhi g h er- or d er h ar m o ni cs is c o nsi d er a bl y s m all er t h a n t h eb o u n d gi v e n b y E q. ( 3 2). T his pr a cti c al b o u n d d e p e n dsm ai nl y o n t h e m o d ul ati o n d e pt h, as w e dis c uss i n S e c. I V.

T h e v olt a g e a n d c urr e nt v e ct ors t h us b e c o m e fi nit e-di m e nsi o n 2 N + 1 v e ct ors. T h e i m p e d a n c e m atri x i n E q.( 3 1) is a 2N + 1 b y 2 N + 1 m atri x i nst e a d of a n i n fi nit el yl ar g e o n e, a n d it r e a ds

¯̄Z L = j ¯̄W

⎛

⎜⎜⎝

l0 ( ω− N ) l− 1 ( ω1 − N ) · · · l− 2 N ( ω N )l1 ( ω− N ) l0 ( ω1 − N ) · · · l1 − 2 N ( ω N )

......

......

l2 N ( ω− N ) l2 N − 1 ( ω 1 − N ) · · · l0 ( ωN )

⎞

⎟⎟⎠ ,

( 3 3)

w h er e

¯̄W = di a g {ω − N , . . . , ω − 1 , ω s , ω 1 , . . . , ω N } ( 3 4)

is a di a g o n al m atri x wit h t h e el e m e nts b ei n g t h e m o d e fr e-q u e n ci es d e fi n e d i n E q. ( 2 6). We n ot e t h at t his i m p e d a n c e

m atri x r e d u c es t o t h e c o n v e nti o n al s c al ar m o d el w h e nt h e i n d u ct or is n ot m o d ul at e d i n ti m e a n d is dis p ersi o n-l ess. Wit h o ut m o d ul ati o n a n d fr e q u e n c y dis p ersi o n, t h ei n d u ct a n c e is a c o nst a nt v al u e L 0 a n d, t h er ef or e, t h e c o e ffi-ci e nts i n E q. ( 2 3) si m plif y t o t h e Kr o n e c k er d elt a f u n cti o n:lp = δ p L 0 . As a r es ult, t h e a b o v e i m p e d a n c e m atri x is adi a g o n al o n e, a n d t h e v olt a g e at fr e q u e n c y ω s i s r el at e do nl y wit h t h e c urr e nt at t h e s a m e fr e q u e n c y b y i m p e d a n c ej ω s L 0 .

B. Ti m e- m o d ul at e d c a p a cit o rs

Si mil arl y, f or a p eri o di c all y ti m e- m o d ul at e d dis p ersi v ec a p a cit or, w e c a n d o t h e s a m e t o c h ar a ct eri z e it i n t h efr e q u e n c y d o m ai n. H o w e v er, i nst e a d of t h e i m p e d a n c em atri x, it is m u c h m or e c o n v e ni e nt t o us e t h e a d mitt a n c em atri x, c o nsi d eri n g t h e c urr e nt as a f u n cti o n of v olt a g e (t h ec urr e nt is t h e ti m e d eri v ati v e of t h e m ulti pli c ati o n of c a p a c-it a n c e a n d v olt a g e). If w e ass u m e t h at t h e c a p a cit a n c e ism o d ul at e d wit h t h e s a m e p eri o d T t h e n it c a n b e e x pr ess e di n F o uri er s eri es as

C ( ω , t) =

+ ∞

p = − ∞

c p ( ω )e j pω M t, ( 3 5)

w h er e c p ( ω ) = (1 / T )T

0 C ( ω , t)e − j pω M tdt ar e t h e c o m pl e xm o d ul ati o n c o e ffi ci e nts t h at d e p e n d o n t h e fr e q u e n c y.T h e n, u n d er a n e xt er n al e x cit ati o n wit h a ti m e- h ar m o ni cc o m p o n e nt e j ω s t at t h e a n g ul ar fr e q u e n c y ω s , t h e v olt a g es( c urr e nts) a cr oss (t hr o u g h) t h e c a p a cit or c a n b e e x pr ess e di n t h e s a m e w a y as i n E qs. ( 2 4) a n d ( 2 5). Usi n g t h e r el ati o ni n E q. ( 1 4) a n d f oll o wi n g t h e s a m e d eri v ati o n pr o c e d ur eas i n S e c. III A, w e g et a si mil ar m atri x v olt a g e- c urr e ntr el ati o n i n t h e fr e q u e n c y d o m ai n, i. e.,

i = ¯̄Y C · v , ( 3 6)

w h er e ¯̄Y C i s t h e a d mitt a n c e m atri x f or t h e ti m e- m o d ul at e dc a p a cit or. C o nsi d eri n g a fi nit e n u m b er of h ar m o ni cs wit h− N ≤ n ≤ + N , it c a n b e writt e n as

¯̄Y C = j ¯̄W

⎛

⎜⎜⎝

c 0 ( ω− N ) c − 1 ( ω1 − N ) · · · c − 2 N ( ω N )c 1 ( ω− N ) c 0 ( ω1 − N ) · · · c 1 − 2 N ( ω N )

......

......

c 2 N ( ω− N ) c 2 N − 1 ( ω 1 − N ) · · · c 0 ( ωN )

⎞

⎟⎟⎠ ,

( 3 7)

w hi c h is si mil ar t o E q. ( 3 3). We als o n ot e t h at, w h e n t h ec a p a cit or h as a c o nst a nt c a p a cit a n c e C 0 at all fr e q u e n-ci es ( n o ti m e m o d ul ati o n a n d fr e q u e n c y dis p ersi o n), t h ea d mitt a n c e m atri x r e d u c es t o a di a g o n al o n e wit h el e m e ntsj ω s C 0 .

0 1 4 0 1 7- 6


C. Ti m e- m o d ul at e d r esist o r

Fi n all y, f or a p eri o di c all y ti m e- m o d ul at e d r esist or, t h etr e at m e nt is e asi er as t h e ti m e- d o m ai n V -I r el ati o n is m u c hsi m pl er, i. e., V (t) = R ( ω , t)I (t). F oll o wi n g t h e s a m e pr o-c e d ur e a n d ass u mi n g t h at t h e p eri o di c all y ti m e- m o d ul at e dr esist a n c e is e x p a n d e d as

R ( ω , t) =

+ ∞

p = − ∞

r p ( ω )e j pω M t, ( 3 8)

t h e i m p e d a n c e m atri x of a ti m e- m o d ul at e d r esist or iso bt ai n e d as

¯̄Z R =

⎛

⎜⎜⎝

r 0 ( ω− N ) r − 1 ( ω1 − N ) · · · r − 2 N ( ω N )r 1 ( ω− N ) r 0 ( ω1 − N ) · · · r 1 − 2 N ( ω N )

......

......

r 2 N ( ω− N ) r 2 N − 1 ( ω 1 − N ) · · · r 0 ( ωN )

⎞

⎟⎟⎠ ,

( 3 9)

s atisf yi n g v = ¯̄Z R · i. We n ot e t h at t h e m o d ul ati o n c o e ffi-ci e nts r p ( ω ) ar e c o m pl e x n u m b ers, m e a ni n g t h at t h e ti m e-m o d ul at e d r esist a n c e c a n pr o d u c e a n e ff e cti v e r e a ct a n c e.W h e n t h e r esist a n c e is ti m e i n v ari a nt a n d n o n dis p ersi v e,t h e i m p e d a n c e m atri x r e d u c es t o a di a g o n al o n e wit h t h es a m e di a g o n al v al u e r 0 = R 0 .

N o w, w e c a n c h ar a ct eri z e ti m e- m o d ul at e d el e m e ntswit h i m p e d a n c e or a d mitt a n c e m atri c es i n t h e fr e q u e n c yd o m ai n. T his is a p o w erf ul t o ol t o a n al y z e a n y el e ctri c alcir c uit t h at c o nt ai ns p eri o di c all y m o d ul at e d el e m e nts. I nt h e f oll o wi n g, w e c o nsi d er a f e w i m p ort a nt e x a m pl es t ov erif y t h e d e v el o p e d t o ol o n pr a cti c all y r el e v a nt e x a m pl esof us a g e of ti m e- m o d ul at e d c o m p o n e nts.

I V. TI M E- M O D U L A T E D I N D U C T O R I N A R L CCI R C UI T

L et us first c o nsi d er a R L C cir c uit dri v e n b y a v olt a g es o ur c e at fr e q u e n c y ω s ,

V s (t) = R e[ V 0 ej ω s t], ( 4 0)

wit h L b ei n g ti m e m o d ul at e d. T h e cir c uit u n d er st u d y iss h o w n i n Fi g. 1 o n t h e l eft. L et us ass u m e t h at t h e i n d u c-t or is dis p ersi v e a n d m o d ul at e d i n a h ar m o ni c w a y ar o u n dits n o mi n al v al u es L 0 ( ω ) at fr e q u e n c y ω M wit h t h e s a m em o d ul ati o n d e pt h m M :

L ( ω , t) = L 0 ( ω )[ 1 + m M c os ( ωM t + φ M )]. ( 4 1)

T h e v ari a bl es m M , φ M , a n d ω M i n di c at e t h e m o d ul ati o nd e pt h, p h as e, a n d fr e q u e n c y, r es p e cti v el y (t his c o n v e nti o nis us e d t hr o u g h o ut t h e p a p er). Si mil arl y t o E q. ( 2 3), w e

FI G. 1. A R L C cir c uit wit h a ti m e- m o d ul at e d i n d u ct or a n d itse q ui v al e nt cir c uit r e pr es e nt ati o n.

writ e t his f u n cti o n i n t h e e x p o n e nti al f or m

L ( ω , t) = l0 ( ω ) + l− 1 ( ω )e − j ω M t + l1 ( ω )e j ω M t, ( 4 2)

w h er e l0 = L 0 ( ω ) a n d l± 1 ( ω ) = 12m M L 0 ( ω )e ± j φ M ar e t h e

c o m pl e x m o d ul ati o n c o e ffi ci e nts at t h e m o d ul ati o n fr e-q u e n ci es ± ω M , r es p e cti v el y.

A. T h e i m p e d a n c e m at ri c es a n d t h e m ast e r e q u ati o n

F or t his si m pl e ti m e- m o d ul at e d i n d u ct or, w e fi n d fr o mE q. ( 3 3) t h at its i m p e d a n c e m atri x is a 2N + 1 b y 2 N + 1tri di a g o n al m atri x,

¯̄Z L = j ¯̄W

⎛

⎜⎜⎜⎜⎜⎝

l0 ( ω− N ) l− 1 ( ω1 − N ) 0 · · · 0l1 ( ω− N ) l0 ( ω1 − N ) l− 1 ( ω2 − N ) · · · 0

0 l1 ( ω1 − N ) l0 ( ω2 − N )... 0

......

......

...0 0 0 · · · l0 ( ωN )

⎞

⎟⎟⎟⎟⎟⎠

.

( 4 3)

We c a n s e e t h at fr e q u e n c y c o m p o n e nt c o u pli n g t a k es pl a c eo nl y at t h e s a m e a n d n e ar est fr e q u e n ci es.

I n g e n er al, all t h e fr e q u e n ci es ar e di ff er e nt, i. e., ω n a n dω n + 1 h a v e di ff er e nt ( a bs ol ut e) v al u es f or all n . H o w e v er,w h e n t h e m o d ul ati o n fr e q u e n c y is ω M = 2 ω s , t h er e ar e t w on ei g h b or c o m p o n e nts wit h t h e s a m e fr e q u e n c y, i. e., ω s a n dω − 1 = − ω s , m e a ni n g t h at t h er e will b e a v er y str o n g e ff e ctat t h e si g n al fr e q u e n c y ω s d u e t o t h e ti m e m o d ul ati o n of t h ei n d u ct or, w hi c h is t h e s a m e e ff e ct as us e d i n cl assi c al p ar a-m etri c a m pli fi ers [ 4 3 ]. L et us d eri v e t h e m ast er e q u ati o n oft his cir c uit.

As t h e R C c o m p o n e nts ar e dis p ersi v e b ut n ot ti m e m o d-

ul at e d, m atri c es ¯̄Z R a n d ¯̄Y C c a n b e o bt ai n e d b y r e pl a ci n gt h eir o ff- di a g o n al t er ms i n E qs. ( 3 9) a n d ( 3 7) wit h z er os.T h er ef or e, t h e m ast er e q u ati o n of t h e R L C cir c uit i n t h efr e q u e n c y d o m ai n is

v s = ( ¯̄Z R + ¯̄Z L + ¯̄Y − 1C ) · i, ( 4 4)

w h er e v s a n d i ar e t h e s o ur c e v olt a g e v e ct or a n d t h e c ur-r e nt v e ct or, r es p e cti v el y. F or t h e s o ur c e v olt a g e i n E q. ( 4 0),t h e v olt a g e v e ct or v s h as c o m p o n e nts V 0 / 2 at fr e q u e n ci es± ω s a n d 0 at all ot h er fr e q u e n ci es. T h er ef or e, w e c a n e asil y

0 1 4 0 1 7- 7


c al c ul at e t h e c urr e nt v e ct or fr o m t h e m ast er e q u ati o n ( 4 4)a n d t h e n o bt ai n t h e c urr e nt i n t h e ti m e d o m ai n b y t a ki n gt h e r e al p art of E q. ( 2 5). T h e v olt a g es a cr oss t h e el e m e ntsc a n als o b e o bt ai n e d si mil arl y.

H er e, w e c o n cl u d e b y f or m ul ati n g a si m pl e r e ci p e f orest a blis hi n g t h e m ast er e q u ati o n f or g e n er al li n e ar el e c-tri c al cir c uits c o nsisti n g of dis p ersi o nl ess ti m e- m o d ul at e del e m e nts: ( 1) writ e d o w n t h e V -I e q u ati o n i n t h e fr e q u e n c yd o m ai n, ass u mi n g t h at t h er e is n o ti m e- m o d ul at e d el e m e nt;( 2) e x p a n d t h e fr e q u e n c y s p e ctr u m b as e d o n E qs. ( 2 6)a n d ( 3 4); ( 3) f or e a c h el e ctri c al c o m p o n e nt i n cl u di n g t h eti m e- m o d ul at e d o n e, r e pl a c e t h e s c al ar i m p e d a n c e b y t h ec orr es p o n di n g m atri x i m p e d a n c e, as ill ustr at e d i n Fi g. 1 .N e xt, w e dis c uss t his p arti c ul ar pr o c e d ur e i n d et ail i n b ot ht h e ti m e a n d fr e q u e n c y d o m ai ns.

B. Ti m e- d o m ai n a n al ysis

L et us c o nsi d er a p arti c ul ar n u m eri c al e x a m pl e t o v erif yt h e a b o v e t h e or eti c al dis c ussi o n ( w e r ef er t o t h e s a m e R L Ccir c uit s h o w n i n Fi g. 1 ). I n s u bs e q u e nt n u m eri c al e x a m-pl es w e ass u m e t h at t h e ti m e- m o d ul at e d c o m p o n e nts ar edis p ersi o nl ess. I n t his e x a m pl e, w e ass u m e t h at t h e v olt a g es o ur c e is d e fi n e d b y E q. ( 4 0) at 1 0 0 k H z, i. e., ω s = 2 π ×1 0 0 kr a d/s, wit h m a g nit u d e V 0 = 1 V / m, a n d t h e i n d u ct oris m o d ul at e d as s h o w n i n E q. ( 4 1) at t h e d o u bl e fr e q u e n c yω M = 2 ω s ar o u n d t h e i n d u ct a n c e v al u e L 0 = 1 0 0 μ H ( n ot et h at i n t his e x a m pl e, L 0 i s n ot fr e q u e n c y d e p e n d e nt b e c a us ew e n e gl e ct fr e q u e n c y dis p ersi o n of its m at eri als). I n a d di-ti o n, w e c h o os e R 0 = 1 0 0 a n d C 0 = 1 /( ω 2

s L 0 ), s o t h att h e cir c uit is r es o n a nt at t h e si g n al fr e q u e n c y ω s w h e nt h e i n d u ct or is n ot m o d ul at e d. T h e n, t h e m o d ul ati o n d e pt hm M a n d m o d ul ati o n p h as e φ M ar e v ari e d t o i n v esti g at e t h ecir c uit p erf or m a n c e.

T h e d e v el o p e d a n al yti c al t o ol is v ali d at e d b y c o m p ar-i n g ti m e- d o m ai n s ol uti o ns. First, t h e R L C cir c uit c a n b en u m eri c all y st u di e d b y s ol vi n g t h e c orr es p o n di n g ti m e-d o m ai n di ff er e nti al e q u ati o ns. T his is d o n e usi n g M A T H-

E M A TI C A s oft w ar e, a n d t h e bl a c k o p e n s q u ar e m ar k ersi n Fi g. 2 s h o w t h e n u m eri c all y c al c ul at e d ti m e- d o m ai nst e a d y-st at e c urr e nt w a v ef or ms I (t). T h e n, w e a n al y z e

4. 9 8 4. 9 9 5. 0 0

– 1 0

0

1 0

I L (

mA)

– 1 0

0

1 0

I L (

mA)

t ( ms)4. 9 8 4. 9 9 5. 0 0

t ( ms)

N u m eri c al N = 0 N = 1 N = 2

( a) ( b)

0. 1 0. 6

FI G. 2. N u m eri c al si m ul ati o n of t h e c urr e nt w a v ef or m of I (t)f or ( a) m o d ul ati o n d e pt h m M = 0. 1 a n d ( b) m o d ul ati o n d e pt hm M = 0. 6. Si m ul ati o n is p erf or m e d f or φ M = 0, R 0 = 1 0 0 ,L 0 = 1 0 0 μ H, a n d ω s = 2 π × 1 0 0 kr a d / s.

t h e p erf or m a n c e wit h t h e pr o p os e d t o ol a n d c o m p ar e t h er es ults. I n t his c as e, t h e c h oi c e of t h e tr u n c ati o n i n d e x Nm att ers. F or e x a m pl e, w h e n t h e m o d ul ati o n d e pt h is s m all( wit h m M = 0. 1 a n d φ M = 0 r a d), a s m all tr u n c ati o n i n d e xis e n o u g h t o r e pr es e nt t h e cir c uit p erf or m a n c e, as s h o w nb y t h e r e d li n e f or N = 0 i n Fi g. 2( a) . T his is b e c a us et h e c o u pli n g b et w e e n t h e h ar m o ni cs is n e gli gi bl e d u e t ot h e r el ati v el y s m all l± . H o w e v er, w h e n t h e m o d ul ati o nb e c o m es str o n g er ( e. g., m M = 0. 6 wit h φ M = 0 r a d), t h ec o u pli n g b et w e e n h ar m o ni cs b e c o m e str o n g er as t h e c o ef-fi ci e nts l± ar e of t h e s a m e or d er as l0 , a n d w e s h o ul d us ea l ar g er tr u n c ati o n i n d e x. T his is ill ustr at e d i n Fi g. 2( b) ,w h er e w e s e e t h at t h e us e of a l ar g er i n d e x N = 2 gi v es a nal m ost i d e nti c al c urr e nt w a v ef or m t o t h e n u m eri c al o n e.Fr o m t h e ti m e- d o m ai n c urr e nt w a v ef or m, it is als o cl e art h at t h e d o u bl e-fr e q u e n c y m o d ul ati o n (i. e., ω M = 2 ω s ) c a nr es ult i n hi g h er- or d er- h ar m o ni c v olt a g es ( c urr e nts) a cr oss(t hr o u g h) t h e cir c uit d u e t o c o u pli n g b et w e e n di ff er e ntm o d es.

C. E ff e cti v e i m p e d a n c e of t h e ti m e- m o d ul at e d i n d u ct o rat t h e si g n al f r e q u e n c y

If w e k n o w t h e v olt a g e a cr oss a n d c urr e nt t hr o u g ha ti m e- m o d ul at e d i n d u ct or, d e n ot e d b y V L (t) a n d IL (t),r es p e cti v el y, t h e e ff e cti v e i m p e d a n c e of t his ti m e-m o d ul at e d i n d u ct or at t h e si g n al fr e q u e n c y ω s c a n b ed e fi n e d as

Z ω sL − e ff =

F [V L (t)]|ω = ω s

F [IL (t)]|ω = ω s

, ( 4 5)

w h er e F [·] d e n ot es t h e F o uri er tr a nsf or m. N ot e t h at t hisd e fi niti o n is f or a dis p ersi o nl ess c o m p o n e nt. H er e, t h eti m e- m o d ul at e d el e m e nt is n ot dir e ctl y c o n n e ct e d t o am o n o c hr o m ati c s o ur c e b ut t hr o u g h ot h er el e ctri c al el e-m e nts. T h er ef or e, t h e a b o v e d e fi niti o n is di ff er e nt fr o mt h e g e n er al d e fi niti o n i n S e c. II w h er e w e als o t o o k i nt oa c c o u nt fr e q u e n c y dis p ersi o n. I n t his c as e, if w e ass u m et h at t h e c urr e nt t hr o u g h t h e el e m e nt is si m pl y IL (t) =c os ( ωs t + θ ) wit h p h as e a n gl e θ s hift wit h r es p e ct t o t h esi g n al v olt a g e V s (t) t h e n, a c c or di n g t o E q. ( 3 1), t h e v olt-a g e a cr oss t h e ti m e- m o d ul at e d i n d u ct or c o nt ai ns m ulti pl efr e q u e n c y h ar m o ni cs. H o w e v er, t h e e ff e cti v e i m p e d a n c eZ ω s

L − e ff at fr e q u e n c y ω s c a n b e f o u n d fr o m E q. ( 4 5) as

Z ω sL − e ff = j ω s L 0 1 + 1

2m M e j ( φM − 2 θ ) . ( 4 6)

T his is t h e first- or d er a p pr o xi m ati o n r es ult, as t h e m ulti-pl e v olt a g e h ar m o ni cs will i n d u c e m ulti pl e c urr e nt h ar-m o ni cs. B ut it alr e a d y gi v es us ef ul i nf or m ati o n o n t h es yst e m. Fr o m t his r es ult, w e c a n e asil y s e e t h at t h e ti m em o d ul ati o n of t h e i n d u ct or c o ntri b ut es a n a d diti o n al e ff e c-ti v e i m p e d a n c e of 1

2j ω s L 0 m M e j ( φM − 2 θ ) . T h e c orr es p o n di n g

first- or d er a p pr o xi m ati o n f or t h e e ff e cti v e r esist a n c e R ω sL − e ff

0 1 4 0 1 7- 8


a n d e ff e cti v e i n d u ct a n c e L ω sL − e ff at ω s r e a d

R ω sL − e ff = 1

2m M ω s L 0 si n (2 θ − φ M ), ( 4 7)

L ω sL − e ff = L 0 + 1

2m M L 0 c o s (2 θ − φ M ). ( 4 8)

We c a n s e e t h at, w h e n t h e m o d ul ati o n p h as e φ M (r el-ati v e t o a n gl e θ ) is v ar yi n g, t h e ti m e- m o d ul at e d i n d u c-t or e x hi bits di ff er e nt e ff e cti v e b e h a vi or: w h e n 2θ − φ M =n π ( n ∈ Z ), t h e ti m e m o d ul ati o n e ff e cti v el y c o ntri b ut esa d diti o n al i n d u ct a n c e t o L 0 ; w hil e w h e n 2θ − φ M = (n +12) π (n ∈ Z ), t h e ti m e m o d ul ati o n e ff e cti v el y a d ds p ositi v e

or n e g ati v e r esist a n c e; f or ot h er p h as e v al u es, t h e e ff e c-ti v e r esist a n c e v ari es b et w e e n ± 1

2m M ω s L 0 a n d t h e e ff e cti v e

i n d u ct a n c e v ari es b et w e e n (1 ± 12m M )L 0 , r es p e cti v el y.

T his is a n i nt er esti n g r es ult as c o m p ar e d t o t h e cl assi-c al lit er at ur e of p ar a m etri c a m pli fi ers [ 3 ,4 4 ,4 5 ] a n d r e c e nta d v a n c e m e nts i n Fl o q u et i m p e d a n c e m at c hi n g [ 3 1 ], w h er eti m e- m o d ul at e d el e m e nts ar e o nl y e x pl or e d as e ff e cti v en e g ati v e r esist ors. T his is b e c a us e us u all y o nl y a p arti c ul arp h as e of t h e m o d ul ati o n is us e d a n d t h er ef or e o nl y n e g ati v er esist a n c e is pr es e nt.

Vari ati o ns of t h e e ff e cti v e r esist a n c e a n d i n d u ct a n c e v er-s us φ M ar e e v al u at e d, a n d t h e r es ults ar e gi v e n i n Fi g. 3 .We c a n s e e t h at, w h e n m M = 0. 1, t h e first- or d er a p pr o x-i m ati o n r es ults (r e d li n es) fr o m E qs. ( 4 7) a n d ( 4 8) a gr e ewit h t h e n u m eri c al r es ults ( bl a c k o p e n s q u ar es) v er y w ell.T his is b e c a us e, f or l o w m o d ul ati o n d e pt hs, t h e cr oss-c o u pli n g b et w e e n n ei g h b ori n g h ar m o ni cs is n e gli gi bl e a n d

– 4

– 2

0

2

4

Rω

s

L–e

ff(Ω

)

0. 9 5

1. 0 0

1. 0 5

Lω

s

L–e

ff /L

0

– 2 0

– 1 0

0

1 0

2 0

Rω

s

L–e

ff(Ω

)

– π – π / 2 0 π / 2 π0. 6

0. 8

1. 0

1. 2

1. 4

Lω

s

L–e

ff /L

0

– π – π / 2 0 π / 2 π

– π – π / 2 0 π / 2 π– π – π / 2 0 π / 2 π2 θ – φ

M (r a d) 2 θ – φ

M (r a d)

2 θ – φM (r a d)2 θ – φ

M (r a d)

N u m eri c al A p pr o xi m ati o n M atri x m o d el (N = 2)

( a) ( b)

0. 1 0. 1

( c) ( d)

0. 6 0. 6

FI G. 3. T h e e ff e cti v e r esist a n c e a n d i n d u ct a n c e ( n or m ali z e d)of a ti m e- m o d ul at e d i n d u ct or v ers us 2 θ − φ M f or m o d ul ati o np h as es ( a), ( b) m M = 0. 1 a n d ( c), ( d) m M = 0. 6. N u m eri c alr es ults ar e c al c ul at e d b y n u m eri c all y s ol vi n g t h e ti m e- d o m ai ndi ff er e nti al e q u ati o ns, first- or d er a p pr o xi m ati o n r es ults ar e c al-c ul at e d usi n g E qs. ( 4 7) a n d ( 4 8), a n d m atri x m o d el r es ults ar ec al c ul at e d fr o m E qs. ( 4 4) a n d ( 4 5).

t h e first- or d er a p pr o xi m ati o n is e n o u g h f or c h ar a ct eri zi n gt h e el e m e nt. H o w e v er, f or l ar g e m o d ul ati o n d e pt hs, f ore x a m pl e, m M = 0. 6, d u e t o str o n g c o u pli n g b et w e e n cl os e-fr e q u e n c y m o d es, t h e c urr e nt at si g n al fr e q u e n c y i n d u c e db y ot h er v olt a g e h ar m o ni cs is missi n g i n t h e first- or d era p pr o xi m ati o n. T h er ef or e, t h e r es ults ar e s hift e d. N e v er-t h el ess, t h e s h a p e is r et ai n e d a n d t h e r a n g e gi v e n b y t h efirst- or d er a p pr o xi m ati o n is still v ali d f or l ar g e m o d ul ati o nd e pt hs. O n t h e ot h er h a n d, fr o m Fi g. 3 , w e c a n s e e t h att h e e ff e cti v e r esist a n c e a n d i n d u ct a n c e o bt ai n e d fr o m t h em atri x m o d el, i. e., c al c ul at e d fr o m E qs. ( 4 4) a n d ( 4 5) ( bl u ed as h e d li n es), a gr e e e x a ctl y wit h t h e n u m eri c al r es ults.T his m e a ns t h at t h e c orr e ct w a y f or m o d eli n g ti m e- v ar yi n gel e m e nts is t h e i m p e d a n c e m atri x m et h o d.

V. TI M E- M O D U L A T E D M U T U A L I N D U C T A N C EF O R E N H A N C E D WI R E L E S S P O W E R T R A N S F E R

I n t his s e cti o n, w e a p pl y t h e a b o v e- d e v el o p e d g e n er ala n al ysis t o ol t o ti m e- m o d ul at e d W P T s yst e ms. We st artwit h a dis c ussi o n of t h e t h e or eti c al li mits i n cl assi c al W P Ts yst e ms. N e xt, w e i ntr o d u c e a n d dis c uss a p ossi bilit y t oo v er c o m e t h os e f u n d a m e nt al li mits usi n g ti m e- m o d ul at e dm ut u al i n d u ct a n c e.

A. F u n d a m e nt al li mit ati o ns of wi r el ess p o w e r t r a nsf e r

H er e, w e c o nsi d er cl assi c al W P T s yst e ms wit h o ut a n yti m e- m o d ul at e d c o m p o n e nts. I n t h es e s yst e ms, t h e p o w ertr a nsf er c a p a bilit y a n d t h e s yst e m e ffi ci e n c y ar e t h e m osti m p ort a nt p erf or m a n c e i n di c at ors.

A cir c uit di a gr a m of a n i n d u cti v el y c o u pl e d W P T s ys-t e m is s h o w n i n Fi g. 4 , w h er e w e ass u m e t h at t h er e is n oti m e m o d ul ati o n. T h e d e vi c e c o nsists of a h ar m o ni c p o w ers o ur c e V s (t) [ E q. ( 4 0)] wit h i nt er n al i m p e d a n c e Z s , a tr a ns-mitti n g r es o n at or (ill ustr at e d as a n L C R cir c uit L 1 , C 1 , a n dR 1 ), a n d a r e c ei vi n g r es o n at or (L 2 , C 2 , a n d R 2 ) c o n n e ct e d t oa n el e ctri c al l o a d R l o a d. T h e tr a ns mitti n g r es o n at or i n d u c-ti v el y c o u pl es wit h t h e r e c ei vi n g r es o n at or wit h a m ut u ali n d u ct a n c e of M . T h e p erf or m a n c e i n di c es, t h e d eli v er e dp o w er a n d e ffi ci e n c y, will str o n gl y d e p e n d o n M a n dR l o a d. T h e i n p ut i m p e d a n c e Z i n s e e n b y t h e s o ur c e at t h e r es-o n a n c e fr e q u e n c y ω s i s Z i n = R 1 + ( ωs M )2 /( R 2 + R l o a d)[4 6 ]. T h e m a xi m u m p o w er tr a nsf er fr o m t h e s o ur c e o c c ursw h e n Z i n = Z s

∗ (t h e ast eris k d e n ot es c o m pl e x c o nj u g a-ti o n), w hi c h c orr es p o n ds t o a p arti c ul ar M v al u e f or a gi v e nR l o a d. W h e n e v er t h e m ut u al i n d u ct a n c e di ff ers fr o m t his

s ( )1 2

1 21 2

l o a d

s

( )

FI G. 4. Cir c uit di a gr a m of a n i n d u cti v el y c o u pl e d W P T s ys-t e m w h er e t h e m ut u al i n d u ct a n c e c a n b e m o d ul at e d i n ti m e.

0 1 4 0 1 7- 9


v al u e, p o w er tr a nsf er is n ot o pti m al d u e t o t h e i m p e d a n c emis m at c h i n t h e s o ur c e t er mi n al. I n or d er t o e v al u at e t h etr a nsf err e d p o w er, w e d e fi n e t h e tr a nsf err e d p o w er r ati oρ l o a d a s t h e r ati o b et w e e n t h e o ut p ut p o w er at t h e l o a d P l o a d

a n d t h e m a xi m u m a v ail a bl e p o w er fr o m t h e s o ur c e [ 4 6 ]:

ρ l o a d = P l o a d/ P 0 ( 4 9)

wit h t h e m a xi m u m a v ail a bl e p o w er P 0 = V 20 /( 8 R e[ Z s ])

us e d as a r ef er e n c e p o w er. O n t h e ot h er h a n d, p o w er tr a ns-f er e ffi ci e n c y (η ) is m ai nl y d et er mi n e d b y dissi p ati o n i nt h e l oss y c o m p o n e nts a n d t h e p o w er s o ur c e, w hi c h c a n b ed e fi n e d as

η =P l o a d

P R 1 + P R 2 + P s + P l o a d, ( 5 0)

w h er e P R 1 , P R 2 , a n d P s ar e t h e p o w er l oss es i n t h e c oilr esist a n c e (R 1 , R 2 ) a n d t h e s o ur c e r esist a n c e (R e[ Z s ]).W h e n t h e i n p ut i m p e d a n c e of t h e W P T s yst e m is m at c h e dt o t h e s o ur c e i m p e d a n c e (i. e., Z i n = Z s

∗ ), t h e m a xi m u ma v ail a bl e p o w er fr o m t h e s o ur c e is d eli v er e d t o t h e W P Ts yst e m; h o w e v er, h alf of t h e g e n er at e d p o w er is dissi p at e di nsi d e t h e s o ur c e, r es ulti n g i n t h e e ffi ci e n c y b ei n g l esst h a n 5 0 %. T h er ef or e, t h er e is al w a ys a tr a d e- o ff b et w e e nm a xi mi zi n g d eli v er e d p o w er a n d e ffi ci e n c y.

B. Ti m e- m o d ul at e d m ut u al i n d u ct a n c e

H er e, w e i ntr o d u c e a n alt er n at e p ossi bilit y t o usi n gti m e- m o d ul at e d m ut u al i n d u ct a n c e t o o v er c o m e t his f u n-d a m e nt al li mit ati o n of i n d u cti v e W P T s yst e ms. We st arto ur dis c ussi o n b y e xt e n di n g t h e m atri x m o d el of ti m e-m o d ul at e d i n d u ct ors t o t h e m o d eli n g of ti m e- m o d ul at e dm ut u al i n d u ct a n c es. As w e h a v e r e vi e w e d i n pr e vi o us s e c-ti o ns, a ti m e- m o d ul at e d i n d u ct or or c a p a cit or c a n e q ui v-al e ntl y a ct as a n e g ati v e r esist or. We c a n i ntr o d u c e ti m em o d ul ati o n i n t h e tr a ns mitti n g r es o n at or (i. e., L 1 or C 1 ) t om a k e t h e i n p ut i m p e d a n c e b e t h e n e g ati v e of t h e s o ur c ei m p e d a n c e Z i n = − Z s , s o t h at t h e t ot al i m p e d a n c e i n t h es o ur c e l o o p c a n b e n ulli fi e d. I n t his w a y, a pr a cti c al s o ur c ewit h i nt er n al i m p e d a n c e Z s i s e q ui v al e ntl y a cti n g as a ni d e al s o ur c e t h e or eti c all y c a p a bl e of pr o vi di n g i n fi nit ep o w er. T h us, w e c a n us e t h e ti m e m o d ul ati o n of L 1 or C 1 t oo v er c o m e t h e t h e or eti c al li mit of t h e tr a nsf err e d p o w er t h atis d u e t o t h e i nt er n al s o ur c e i m p e d a n c e. H o w e v er, b e c a us eof t h e p h ysi c al pr es e n c e of t h e s o ur c e i m p e d a n c e a n dwi n di n g r esist a n c es, e ffi ci e n c y of s u c h a s yst e m will stills u ff er fr o m t h e t h e or eti c al u p p er b o u n d dis c uss e d a b o v e.

O n t h e ot h er h a n d, ti m e m o d ul ati o n of m ut u al i n d u c-t a n c e all o ws us t o o v er c o m e p h ysi c al li mit ati o ns o n b ot ht h e e ffi ci e n c y a n d tr a nsf err e d p o w er, as w e c a n dir e ctl yp u m p t h e e n er g y t o t h e r e c ei vi n g r es o n at or wit h o ut s a c-ri fi ci n g t h e e ffi ci e n c y d u e t o l oss es i n t h e pri m ar y s o ur c e.

L et us c o nsi d er a ti m e- m o d ul at e d m ut u al i n d u ct a n c e i na n el e ctri c al cir c uit s u c h as a n i n d u cti v e W P T s yst e m. T h e

Port 2

Port 1

FI G. 5. Ti m e- m o d ul at e d m ut u al i n d u ct a n c e a n d its e q ui v al e ntcir c uit.

d eri v ati o ns ar e si mil ar t o t h e c as e of ti m e v ar yi n g L (t)i n S e c. I V, b ut n o w w e h a v e a t w o- p ort n et w or k m o d el,as ill ustr at e d i n Fi g. 5 . L et us ass u m e t h at t h e m ut u ali n d u ct a n c e M (t) is p eri o di c all y v ari e d as

M (t) = M 0 [ 1 + m M c os ( ωM t + φ M )], ( 5 1)

w h er e M 0 i s t h e n o mi n al m ut u al i n d u ct a n c e a n d m M i s t h em o d ul ati o n d e pt h. Si mil ar t o E q. ( 4 2), it is c o n v e ni e nt t oe x pr ess M (t) as

M (t) = m 0 + m − 1 e− j ω M t + m 1 e

j ω M t ( 5 2)

wit h m 0 = M 0 a n d m ± 1 = 12m M M 0 e

± j φ M .U si n g t h e i ntr o d u c e d m et h o d of m atri x cir c uit p ar a m e-

t ers, w e c a n b uil d a t w o- p ort m o d el of t h e ti m e- m o d ul at e dm ut u al i n d u ct a n c e as

v 1

v 2=

j L1¯̄W ¯̄Z M

¯̄Z M j L2¯̄W

i1i2

, ( 5 3)

w h er e v 1, 2 (i1, 2 ) ar e t h e c o m pl e x s p e ctr a of v olt a g e ( c ur-r e nt) a cr oss (t hr o u g h) p ort 1 ( p ort 2), as ill ustr at e d i n Fi g. 5 .Usi n g t h e s a m e a p pr o a c h as i n S e c. III, w e c a n fi n d t h e

i m p e d a n c e m atri x ¯̄Z M r e pr es e nti n g t h e m ut u al c o u pli n g i nt h e fr e q u e n c y d o m ai n as

¯̄Z M = j ¯̄W ·

⎛

⎜⎜⎜⎜⎜⎝

m 0 m − 1 0 · · · 0m 1 m 0 m − 1 · · · 0

0 m 1 m 0... 0

......

......

...0 0 0 · · · m 0

⎞

⎟⎟⎟⎟⎟⎠

. ( 5 4)

N o w w e c a n us e t h e m atri x m o d el t o a n al ys e a n y el e ctri c alcir c uit wit h t h e ti m e- m o d ul at e d m ut u al i n d u ct a n c e, i n cl u d-i n g W P T s yst e ms. O ur a n al ysis s h o ws t h at t h e m atri xm o d el a gr e es w ell wit h t h e si m ul ati o n r es ults w h e n pr o p erN is us e d.

C. W P T wit h ti m e- m o d ul at e d m ut u al i n d u ct a n c e

T o d e m o nstr at e t h e c a p a biliti es of ti m e- m o d ul at e dm ut u al i n d u ct a n c e, l et us c o nsi d er a n e x a m pl e W P T s ys-t e m, as ill ustr at e d i n Fi g. 4 . Ti m e m o d ul ati o n of m ut u ali n d u ct a n c e c a n b e r e alis e d b y usi n g n o nli n e ar m a g n eti cm at eri als, m e c h a ni c al m o v e m e nt, or wit h a di git al c o ntr ol

0 1 4 0 1 7- 1 0


0. 0 0. 1 0. 2 0. 30. 0

0. 2

0. 4

0. 6

0. 8

1. 0

ρ load a

nd η

k

ρl o a d

η

FI G. 6. Vari ati o ns of t h e tr a nsf err e d p o w er r ati o ρ l o a d a n dt h e p o w er tr a nsf er e ffi ci e n c y η wit h r es p e ct t o t h e c o u pli n gc o e ffi ci e nt k i n a cl assi c al u n m o d ul at e d W P T s yst e m.

s c h e m e. H o w e v er, as w e ar e i nt er est e d i n a n al ysi n g t h e t h e-or eti c al li mits of t h e s yst e m p erf or m a n c e, w e d o n ot f o c uso n p arti c ul ar m e a ns of i m pl e m e nt ati o ns. We c o nsi d er a ne x a m pl e W P T s yst e m wit h t h e f oll o wi n g s yst e m p ar a m e-t ers: L 1, 2 = 1 0 0 μ H, R 1, 2 = 1 0 0 m , ω s = 2 π 1 0 0 kr a d / s,C 1, 2 = 1 /( ω 2

s L 1, 2 ), Z s = 2 , a n d R l o a d = 2 . T h e m o d-ul ati o n fr e q u e n c y is e q u al t o d o u bl e t h e s o ur c e fr e q u e n c yω s , i. e., ω M = 2 ω s , t o e ns ur e str o n g p erf or m a n c e e n h a n c e-m e nt of t h e W P T s yst e m. First, w e pr es e nt e ffi ci e n c y ηa n d p o w er r ati o ρ l o a d v ari ati o ns i n a cl assi c al arr a n g e m e ntwit h a st ati c m ut u al i n d u ct a n c e ( m ut u al i n d u ct a n c e is n otti m e m o d ul at e d). T h e r es ults i n Fi g. 6 s h o w t h e v ari ati o n i ntr a nsf err e d p o w er r ati o a n d e ffi ci e n c y of a n u n m o d ul at e dW P T s yst e m wit h r es p e ct t o t h e c o u pli n g c o e ffi ci e nt k =M 0 /

√L 1 L 2 . We c a n o bs er v e t h at i n c o n v e nti o n al s yst e ms

e ffi ci e n c y i n cr e as es wit h i n cr e asi n g c o u pli n g str e n gt h;h o w e v er, t h e d eli v er e d p o w er h as a m a xi m u m at a p ar-ti c ul ar c o u pli n g l e v el (k ≈ 0. 0 3 3 i n t his e x a m pl e), a n d itd e cr e as es at hi g h er or l o w er c o u pli n g l e v els.

N o w, l et us i ntr o d u c e ti m e m o d ul ati o n t o t h e m ut u ali n d u ct a n c e a n d a n al ys e its e ff e cts o n t h e tr a nsf err e d p o w err ati o a n d e ffi ci e n c y. We c h o os e M 0 = 0. 1

√L 1 L 2 , i. e.,

k = 0. 1, as a n u m eri c al e x a m pl e f or t h e a n al ysis. First,t h e e ff e ct of t h e m o d ul ati o n p h as e φ M i s a n al ys e d a n dt h e r es ults ar e s h o w n i n Fi g. 7 . It s h o ul d b e hi g hli g ht e dt h at, wit h ti m e m o d ul ati o n of t h e m ut u al i n d u ct a n c e, e xtr ae n er g y is p u m p e d t o t h e W P T s yst e m d u e t o t h e a cti o nof t h e ti m e- m o d ul ati o n p u m p. T h er ef or e, p o w er r ati os ar ec al c ul at e d c o nsi d eri n g t h e p o w er d eli v er e d t o t h e l o a db y t w o m e a ns: fr o m t h e m ai n s o ur c e V s (t) a n d fr o m t h ee n er g y p u m p t h at m o d ul at es t h e m ut u al i n d u ct a n c e. T h es o ur c e p o w er r ati o a n d p u m p p o w er r ati o ar e d e fi n e das ρ s o ur c e = P s o ur c e / P 0 a n d ρ p u m p = P p u m p / P 0 , r es p e cti v el y,w h er e P s o ur c e i s t h e p o w er d eli v er e d fr o m t h e m ai n p o w ers o ur c e a n d P p u m p i s t h e p o w er p u m p e d fr o m t h e ti m e-m o d ul ati o n p u m p.

I m p ort a ntl y, t h e e n er g y t h at c o m es fr o m t h e p u m p d u et o t h e ti m e m o d ul ati o n of t h e m ut u al i n d u ct a n c e c a n b edir e ctl y d eli v er e d t o t h e r e c ei v er wit h o ut i n c urri n g l oss esd u e t o t h e i nt er n al s o ur c e r esist a n c e or dissi p ati o n i n t h etr a ns mitt er c oil. T h er ef or e, t h e tr a nsf err e d p o w er t o t h el o a d c a n b e i n cr e as e d wit h o ut r e d u ci n g e ffi ci e n c y d u e t o

– π – π / 2 0 π / 2 π– 0. 20. 00. 20. 40. 60. 81. 01. 2

Powe

r ra

tios

φ M (r a d)

0. 8 0

0. 8 4

0. 8 8

0. 9 2

η

ρl o a d

ρs o ur c e

ρp u m p

η

FI G. 7. Vari ati o ns of p o w er r ati os ρ l o a d, ρ s o ur c e , a n d ρ p u m p , a n dt h e tr a nsf er e ffi ci e n c y η of t h e ti m e- m o d ul at e d W P T s yst e m wit hr es p e ct t o t h e m o d ul ati o n a n gl e φ M , wit h m M = 0. 5.

t h e i nt er n al r esist a n c e of t h e s o ur c e. T his is t h e f u n d a-m e nt al r ati o n al e b e hi n d t h e p ossi bilit y t o o v er c o m e t h et h e or eti c al li mit of e ffi ci e n c y.

D e p e n di n g o n t h e m o d ul ati o n p h as e φ M , t h e p o w er fl o wfr o m t h e p u m p c a n b e eit h er p ositi v e or n e g ati v e, as s e e nfr o m ρ p u m p i n Fi g. 7 . P ositi v e p u m p p o w er m e a ns t h att h e e n er g y is i nj e ct e d i nt o t h e W P T s yst e m, w hil e n e g a-ti v e p u m p p o w er m e a ns t h at e n er g y is e xtr a ct e d fr o m t h em ai n s o ur c e t o t h e p u m p. T h er ef or e, t h e m o d ul ati o n p h as es h o ul d b e t u n e d pr o p erl y. It c a n b e o bs er v e d t h at t h e p o w err ati o ρ l o a d r e a c h es its m a xi m u m w h e n t h e m o d ul ati o n of t h em ut u al i n d u ct a n c e is i n- p h as e wit h t h e s o ur c e v olt a g e.

T h e v ari ati o ns of p o w er r ati os a n d e ffi ci e n c y v ers us t h em o d ul ati o n d e pt h is pr es e nt e d i n Fi g. 8 . I nt er esti n gl y, w ec a n i n cr e as e b ot h t h e e ffi ci e n c y a n d tr a nsf err e d p o w er r ati ob y i n cr e asi n g t h e m o d ul ati o n d e pt h. F or e x a m pl e, w h e nm M = 0. 5, t h e tr a nsf err e d p o w er r ati o i n cr e as es fr o m 3 2 %t o 6 0 %, w hil e e ffi ci e n c y i n cr e as es fr o m 8 5 % t o 8 9 % c o m-p ar e d t o t h e c as e wit h o ut ti m e m o d ul ati o n. T his is n ota c hi e v a bl e b y cl assi c al m e a ns of W P T o pti mi z ati o ns s u c has i m p e d a n c e m at c hi n g or fr e q u e n c y t u ni n g b e c a us e t h el oss es i nsi d e t h e s o ur c e will al w a ys b e d o mi n a nt u n d er t h ei m p e d a n c e- m at c h e d s c e n ari o. F or i nst a n c e, if w e i n cr e as et h e c o u pli n g c o e ffi ci e nt t o 0. 1 5 i n a st ati c W P T s yst e m( w hi c h c orr es p o n ds t o t h e m a xi m u m c o u pli n g i n t h e ti m e-m o d ul at e d W P T wit h m M = 0. 5), tr a nsf err e d p o w er willb e r e d u c e d t o 1 6 % e v e n if t h e e ffi ci e n c y c a n b e i n cr e as e d

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0– 0. 20. 00. 20. 40. 60. 81. 01. 2

Powe

r ra

tios

mM

0. 8 0

0. 8 4

0. 8 8

0. 9 2

η

ρl o a d

ρs o ur c e

ρp u m p

η

FI G. 8. Vari ati o ns of p o w er r ati os ρ l o a d, ρ s o ur c e , a n d ρ p u m p ,a n d t h e tr a nsf er e ffi ci e n c y η of t h e ti m e- m o d ul at e d W P T s ys-t e m wit h r es p e ct t o m o d ul ati o n d e pt h m M , wit h φ M = 0 a n dM 0 = 0. 1

√L 1 L 2 .

0 1 4 0 1 7- 1 1


t o 9 1 %. We n ot e t h at t h e p erf or m a n c e will h a v e a si mi-l ar s h a p e wit h a v al u e s hift w h e n t h er e is a n o ffs et of M 0

( c h a n g e of t h e c o u pli n g c o e ffi ci e nt k ). We ar e a bl e t o o v er-c o m e t his t h e or eti c al li mit ati o n wit h ti m e m o d ul ati o n oft h e m ut u al i n d u ct a n c e b e c a us e w e dir e ctl y d eli v er a d di-ti o n al e n er g y t o t h e r e c ei v er fr o m t h e m o d ul ati o n p u m p.It s h o ul d b e hi g hli g ht e d t h at l oss es i nsi d e t h e p u m p t h atm o d ul at es t h e m ut u al i n d u ct a n c e h a v e n ot b e e n t a k e n i nt oa c c o u nt i n t his a n al ysis. T h es e l oss es will i n e vit a bl y r e d u c et h e a d diti o n all y d eli v er e d p o w er. H o w e v er, i n pri n ci pl e,t h e l oss es i n t h e m o d ul ati o n p u m p c a n b e m a d e v er y s m all,w hi c h is d e p e n d e nt o n t h e p arti c ul ar r e ali z ati o n m et h o d.I m p ort a ntl y, b y p assi n g t h e l oss y cir c uits of t h e pri m ar ys o ur c e a n d t h e p ar asiti c r esist a n c e of t h e tr a ns mitti n g c oilall o ws us t o o v er c o m e f u n d a m e nt al li mit ati o ns o n p er-f or m a n c e. We n ot e t h at, i n t his p arti c ul ar e x a m pl e wit hk = 0. 1, t h e d eli v er e d p o w er is m ai nl y tr a nsf err e d vi a t h ef u n d a m e nt al h ar m o ni c, a n d t h e c o ntri b uti o ns fr o m ot h erh ar m o ni cs ar e n e gli gi bl e. We e x p e ct t h at f urt h er e x pl o-r ati o ns of t h e c a p a biliti es of ti m e- m o d ul at e d W P T s u c h ast h e us e of m ulti pl e- h ar m o ni c p o w er tr a nsf er will o p e n u pm or e o p p ort u niti es f or W P T d e v el o p m e nts.

VI. TI M E- M O D U L A T E D R E SI S T O R S F O RA D V A N C E D A N T E N N A S

N e xt, w e pr es e nt a n e x a m pl e of a p ossi bl e us e ofti m e- m o d ul at e d r esist ors f or s e v er al p ot e nti al a p pli c ati o ns,i n cl u di n g br o a d b a n d m at c hi n g of a nt e n n as a n d e n h a n c e-m e nt of t h e b a n d wi dt h of a bs or b ers. F or e x a m pl e, w ec a n c o nsi d er t h e sit u ati o n w h er e w e m o d ul at e t h e i nt er n alr esist a n c e of a s o ur c e t h at f e e ds a tr a ns mitti n g a nt e n n a, t h er a di ati o n r esist a n c e of a n a nt e n n a, or t h e l o a d of a r e c ei v-i n g a nt e n n a. Alt h o u g h ti m e- v ar yi n g r esist ors h a v e b e e ndis c uss e d i n e arl y lit er at ur e o n m o d ul at ors, it a p p e ars t h att h eir c a p a biliti es of g e n er ati n g virt u al r e a ct a n c es f or br o a d-b a n d m at c hi n g or a bs or pti o n h as n ot b e e n i d e nti fi e d b ef or e.K n o wi n g t h at ti m e m o d ul ati o n of r esist a n c e c a n e m ul at er e a ct a n c e, w e e x pl or e t his p ossi bilit y t o m at c h a n a nt e n n ai n a wi d e fr e q u e n c y r a n g e. R e c e ntl y, t h e us e of ti m e-m o d ul at e d r e a cti v e el e m e nts w as c o nsi d er e d i n R ef. [ 3 1 ],w h er e e ff e cti v e n e g ati v e r esist a n c e w as us e d f or p ar a m et-ri c a m pli fi c ati o n of t h e a nt e n n a c urr e nt. T h e f u n d a m e nt ala d v a nt a g e of m o d ul ati n g r esist a n c e is t h at t h er e is n o n e e dt o i ntr o d u c e a d diti o n al r e a cti v e el e m e nts t h at i n e vit a bl yst or e r e a cti v e e n er g y a n d i n cr e as e t h e a nt e n n a q u alit y f a c-t or. I n t his c o n c e pt u al st u d y w e f o c us o n o p p ort u niti es t h ato p e n u p d u e t o ti m e m o d ul ati o ns of r esist a n c es, n ot g oi n gi nt o d et ails of s p e ci fi c i m pl e m e nt ati o ns.

A. E ff e cti v e i m p e d a n c e of t h e ti m e- m o d ul at e d r esist o r

First, l et us c o nsi d er a ti m e- m o d ul at e d r esist or R (t) i n a nel e ctri c al cir c uit wit h r esist a n c e d e p e n di n g o n ti m e as

R (t) = R 0 [ 1 + m M c os ( ωM t + φ M )], ( 5 5)

w h er e R 0 i s t h e n o mi n al r esist a n c e, m M , ω M (= 2 ω s ), a n dφ M ar e t h e m o d ul ati o n d e pt h, m o d ul ati o n fr e q u e n c y, a n dm o d ul ati o n p h as e, r es p e cti v el y. If w e ass u m e t h at t h e c o u-pli n g b et w e e n hi g h er- or d er h ar m o ni cs is n e gli gi bl e, w ec a n d eri v e t h e e ff e cti v e i m p e d a n c e of t h e ti m e- m o d ul at e dr esist or usi n g t h e m et h o d pr es e nt e d i n S e c. I V. T h e first-or d er a p pr o xi m ati o n r es ult r e a ds

R ω sR − e ff ≈ R 0 1 + 1

2m M c o s (2 θ − φ M ) , ( 5 6)

X ω sR − e ff ≈ − 1

2m M R 0 si n (2 θ − φ M ). ( 5 7)

Fr o m E q. ( 5 7), w e n ot e a n i nt er esti n g e ff e ct t h at t h e ti m e-v ar yi n g r esist a n c e c a n g e n er at e a n e ff e cti v e r e a ct a n c e t h ats h o ws a si n us oi d al v ari ati o n wit h r es p e ct t o t h e m o d ul a-ti o n a n gl e wit h a n a m plit u d e of 1

2m M R 0 . It a p p e ars t h at t his

e ff e ct h as n ot b e e n n ot e d b ef or e. It c a n b e r e g ar d e d as ad u al p h e n o m e n o n of t h e ti m e- v ar yi n g r e a ct a n c e ( e ff e cti v er esist a n c e c a n b e g e n er at e d b y t h e m o d ul ati o n of c a p a c-it a n c e or i n d u ct a n c e, a n d its v al u e is d e p e n d e nt o n t h em o d ul ati o n p h as e [ 4 0 ]). U n d er t h e ass u m pti o n of n e gli gi-bl e c o u pli n g b et w e e n hi g h er- or d er h ar m o ni cs, it is p ossi bl et o r e ali z e a n y r e a ct a n c e v al u e wit hi n t h e r a n g e ± 1

2m M R 0 b y

pr o p erl y t u ni n g t h e m o d ul ati o n a n gl e. T his is a n i nt er esti n gp ossi bilit y, as it all o ws us t o r e ali z e a t u n a bl e m at c hi n g el e-m e nt wit h o ut a d di n g a n y a d diti o n al r e a cti v e c o m p o n e nts.T his is t h e r e as o n f or c h o osi n g t h e m o d ul ati o n fr e q u e n c yt o b e d o u bl e t h e si g n al fr e q u e n c y, w hi c h c a n pr o vi d e t hisa d diti o n al virt u al r e a ct a n c e. N ot e t h at i n c o ntr ast t o r e a c-t a n c es of a n y cir c uit f or m e d b y c a p a cit ors a n d i n d u ct ors,t his e ff e cti v e r e a ct a n c e d o es n ot d e p e n d o n t h e fr e q u e n c y,o ff eri n g a p ossi bilit y t o o v er c o m e t h e f u n d a m e nt al li mit a-ti o ns o n t h e b a n d wi dt h of m at c hi n g n et w or ks t h at f oll o wfr o m t h e F ost er t h e or e m.

L et us n o w c o nsi d er a n u m eri c al e x a m pl e of a R L Ccir c uit wit h a ti m e- m o d ul at e d r esist or (s a m e as t h at i nFi g. 1 b ut n o w t h e ti m e- m o d ul ati n g el e m e nt is t h e r esis-t or) i n or d er t o f urt h er i n v esti g at e t h e e ff e cts of c o u pli n gb et w e e n hi g h er- or d er m o d es. We us e t h e f oll o wi n g p ar a m-et ers f or t h e n u m eri c al st u d y: L 0 = 1 0 0 μ H, R 0 = 1 0 0 ,C 0 = 1 / ω 2

s L 0 , a n d ω s = 2 π 1 0 0 kr a d / s. Vari ati o ns of e ff e c-ti v e r esisti v e a n d r e a cti v e i m p e d a n c es ( n or m ali z e d t o R 0 )wit h r es p e ct t o t h e m o d ul ati o n a n gl e φ M ar e s h o w n i nFi g. 9 f or di ff er e nt m o d ul ati o n d e pt hs. We c a n s e e fr o mFi g. 9 t h at t h e a n al yti c al first- or d er a p pr o xi m ati o n a gr e esw ell wit h n u m eri c al r es ults f or s m all m o d ul ati o n d e pt hs(m M = 0. 0 1 a n d = 0. 1), b ut d e vi at es f or hi g h er m o d ul a-ti o n d e pt hs. T his r es ult is e x p e ct e d b e c a us e t h e a n al yt-i c al a p pr o xi m ati o n n e gl e cts hi g h er- or d er h ar m o ni cs t h atb e c o m e str o n g at hi g h m o d ul ati o n d e pt h, a n d t h e p h as ea n gl e of t h e c urr e nt θ is n ot n e gli gi bl e f or l ar g e m M .H o w e v er, t h es e r es ults s h o w t h at, e v e n f or a n e xtr e m el yhi g h m o d ul ati o n d e pt h, t h e mi ni m u m a n d m a xi m u m v al-u es of t h e e ff e cti v e r e a ct a n c e d o n ot d e vi at e m u c h fr o mt h e a n al yti c al a p pr o xi m ati o n. T h er ef or e, o n e c a n still us e

0 1 4 0 1 7- 1 2


mM

= 0. 1 0. 2 0. 6

N u m eri c alA p pr o xi m ati o n

– π – π / 2 0 π / 2 π– π – π / 2 0 π / 2 π– 0. 4

– 0. 2

0. 0

0. 2

0. 4

Xω

s

R–e

ff /R

0

φM (r a d)

0. 6

0. 8

1. 0

1. 2

1. 4

Rω

s

R–e

ff /R

0

φM (r a d)

( a) ( b)

FI G. 9. ( a) E ff e cti v e r esist a n c e a n d ( b) e ff e cti v e r e a ct a n c e oft h e ti m e- m o d ul at e d r esist or v ers us t h e m o d ul ati o n p h as e φ M .

ti m e- m o d ul at e d r esist ors t o r e ali z e ar bitr ar y r e a ct a n c esa p pr o xi m at el y wit hi n ± 1

2m M R 0 b y j ust m o d ul ati n g t h e m

at t h e pr o p er p h as e a n gl e φ M .

B. B r o a d b a n d m at c hi n g of a nt e n n a wit hti m e- m o d ul at e d r esist o r

L et us c o nsi d er a s m all l o o p a nt e n n a, r e pr es e nt e d b y aL R cir c uit wit h a n i n d u ct a n c e L 0 = 1 n H c o n n e ct e d t o as o ur c e wit h t h e i nt er n al r esist a n c e of 1 0 0 (R 0 = 1 0 0 ).L et us ass u m e t h at w e ti m e m o d ul at e t h e i nt er n al s o ur c er esist a n c e as i n E q. ( 5 5). A c c or di n g t o t h e a b o v e dis-c ussi o n, w e c a n f ull y c o m p e ns at e t h e r e a ct a n c e of t h ei n d u ct or ω L 0 i n a wi d e fr e q u e n c y r a n g e b y usi n g m o d u-l ati o n wit h t h e c orr e ct p h as e a n gl e. T his m e a ns t h at t h eti m e- m o d ul at e d r esist or c a n b e pr es e nt e d as a n arti fi ci aln e g ati v e r e a ct a n c e. H er e, t h e ti m e m o d ul ati o n of t h e s o ur c er esist a n c e is at t h e d o u bl e fr e q u e n c y of t h e e x cit ati o n wit ht h e c orr e ct p h as e a n gl e s u c h t h at t h e e ff e cti v e virt u al r e a c-t a n c e i ntr o d u c e d b y t h e m o d ul at e d r esist or is e q u al t o− ω s L 0 . T his m o d ul ati o n c a n b e e asil y a ut o m at e d wit h asi m pl e f e e d b a c k c o ntr ol s yst e m t o e ns ur e t h at t h e c urr e ntt hr o u g h t h e i n d u ct or is i n p h as e wit h t h e v olt a g e a cr oss it.

E x a m pl e n u m eri c al r es ults of r e ali zi n g t h e br o a d b a n dm at c hi n g a nt e n n a wit h ti m e- m o d ul at e d r esist a n c e ar es h o w n i n Fi g. 1 0 , w h er e t h e r e q uir e d o pti m al m o d ul ati o np h as e is c al c ul at e d t o a c hi e v e i m p e d a n c e m at c hi n g at dif-f er e nt fr e q u e n ci es. T h e r es ults s h o w t h at t his n o n- F ost erb e h a vi or c a n b e a c hi e v e d at fr e q u e n ci es u p t o ar o u n d 8 0 0M H z. T h e u p p er c ut o ff fr e q u e n c y of t h e r a n g e w h er e w ec a n f ull y c o m p e ns at e t h e l o o p i n d u ct a n c e c a n b e a p pr o x-i m at e d as fc ut = 1

2m M R 0 /( 2 π L 0 ). Si mil ar c o nsi d er ati o ns

a p pl y t o s m all el e ctri c di p ol e a nt e n n as b y usi n g a C Rcir c uit.

T h e a b o v e a n al ysis h o w e v er is o nl y v ali d f or a si n gl e-fr e q u e n c y s o ur c e. W h e n w e w a nt t o us e a ti m e- m o d ul at e dr esist or t o i m pr o v e c o m m u ni c ati o n a nt e n n as, w hi c h s u p er-p os e a m ess a g e si g n al o n t h e c arri er w a v e, w e n e e d t os e n d m ulti pl e fr e q u e n ci es t hr o u g h t h e a nt e n n a, a n d t h ea nt e n n a b a n d wi dt h is a n i m p ort a nt p erf or m a n c e i n di c at or.T h er ef or e, w e n e e d t o st u d y t h e sit u ati o n w h e n t h e s o ur c e

XR – eff

LR – eff

O pti m al M f

c ut

X R

–eff (

Ω)

L R

–eff (

nH)

Opti

mal

M (

rad)

FI G. 1 0. T h e e ff e cti v e r e a ct a n c e, i n d u ct a n c e, a n d t h e o pti m alm o d ul ati o n p h as e of t h e ti m e- m o d ul at e d r esist or wit h r es p e ct t ot h e si g n al fr e q u e n c y, f or a c hi e vi n g br o a d b a n d m at c hi n g of t h ea nt e n n a ( L 0 = 1 n H, R 0 = 1 0 0 , m M = 0. 1).

h as m ulti pl e fr e q u e n ci es. T o t his e n d, l et us c o nsi d er a na m plit u d e- m o d ul at e d si g n al

V s (t) = V 0 c os ( ωs t)[ 1 + d c os ( ωd t + θ d )], ( 5 8)

w h er e ω s = 2 π fs i s t h e c arri er fr e q u e n c y, ω d = 2 π fd , θ d ,a n d d ar e t h e a n g ul ar fr e q u e n c y, p h as e, a n d a m plit u d e oft h e m ess a g e si g n al. I n t his c as e, t h e s o ur c e si g n al c o m-pris es t hr e e fr e q u e n c y c o m p o n e nts, ω s a n d ω s ± ω d , a n dit is i m p ort a nt t o fi n d t h e e ff e cti v e i m p e d a n c es at all t hr e efr e q u e n ci es.

L et us st u d y t h e i m p e d a n c e c h ar a ct eristi cs of t h e ti m e-m o d ul at e d r esist or i ntr o d u c e d i n t h e pr e vi o us e x a m pl e[i. e., t h e ti m e d e p e n d e n c e of t h e m o d ul at e d r esist or isgi v e n b y E q. ( 5 5)] t h at is e x cit e d b y t h e c o nsi d er e da m plit u d e- m o d ul at e d si g n al V s (t) i n E q. ( 5 8) wit h d = 0. 1a n d θ d = 0. Fi g ur e 1 1 s h o ws t h e e ff e cti v e r e a ct a n c e a n di n d u ct a n c e at fr e q u e n ci es fs , fs + fd , a n d fs − fd . We c a n s e et h at t h e e ff e cti v e r e a ct a n c e of t h e ti m e- m o d ul at e d r esist orc o m p e ns at es t h e a nt e n n a r e a ct a n c e at t h e c arri er fr e q u e n c yfs ; h o w e v er, t h e a nt e n n a r e a ct a n c e at t h e ot h er t w o fr e q u e n-ci es (i. e., at fs ± fd ) is n ot f ull y c o m p e ns at e d. T h er ef or e, i nor d er t o m a k e t h e ti m e- m o d ul at e d a nt e n n a r es o n a nt at allt h e fr e q u e n ci es, i n cl u di n g t h e c arri er fr e q u e n c y as w ell as

At fs At f

s+ f

d At f

s– f

d

0 2 0 4 0 6 0 8 0 1 0 0– 6 3 0

– 6 2 9

– 6 2 8

– 6 2 7

– 6 2 6

XR

–eff (

mΩ)

fd ( M H z)

0 2 0 4 0 6 0 8 0 1 0 0– 5

– 4

– 3

– 2

– 1

0

LR

–eff (

nH)

fd ( M H z)

( a) ( b)

FI G. 1 1. T h e e ff e cti v e r e a ct a n c e a n d i n d u ct a n c e of t h e ti m e-m o d ul at e d r esist or i n E q. ( 5 5) at t hr e e di ff er e nt si g n al fr e q u e n ci es(L 0 = 1 n H, R 0 = 1 0 0 , m M = 0. 1, fs = 1 0 0 M H z).

0 1 4 0 1 7- 1 3


φ φ φ 0

M +

M –

M

0 2 0 4 0 6 0fd ( M H z)

0 2 0 4 0 6 0– π / 2

– π / 4

0

π / 4

Opti

mal

φ M (ra

d)

fd ( k H z)

( a) ( b)π / 2

– π / 2

– π / 4

0

π / 4

Opti

mal

φ M (ra

d)

π / 2

FI G. 1 2. T h e o pti m al m o d ul ati o n p h as es r e ali z e p erf e ctm at c hi n g f or c o m m u ni c ati o n a nt e n n as, wit h ( a) c arri er fr e q u e n c yfs at 1 0 0 k H z ( wit h m 0

M = 0. 1 a n d m +M = m −

M = 0. 0 0 0 0 1) a n d( b) c arri er fr e q u e n c y fs at 1 0 0 M H z ( wit h m 0

M = 0. 1 a n d m +M =

m −M = 0. 0 1).

t h e si d e b a n d fr e q u e n ci es, w e i ntr o d u c e ti m e m o d ul ati o n oft h e a nt e n n a r esist a n c e at m ulti pl e fr e q u e n ci es as

R (t) = R 0 {1 + m 0M c o s (2 ω s t + φ 0

M )

+ m +M c o s[ 2 ( ωs + ω d )t + φ +

M ]

+ m −M c o s[ 2 ( ωs − ω d )t + φ −

M ]}, ( 5 9)

w h er e m 0, + ,−M a n d φ 0, + ,−

M ar e t h e m o d ul ati o n d e pt hs a n dm o d ul ati o n p h as es at fr e q u e n ci es 2 ω s , 2( ωs ± ω d ), r es p e c-ti v el y. T h at is, w e h a v e ti m e m o d ul ati o n of t h e r esist or atd o u bl e t h e t hr e e si g n al fr e q u e n ci es. B asi c all y, t h e r esis-t a n c e is ti m e m o d ul at e d b y a n e xt er n al f or c e t h at is s y n-c hr o ni z e d wit h t h e si g n al its elf. It is i nt er esti n g t o n ot et h at, usi n g t his m o d ul ati o n l a w, w e c a n f ull y c o m p e ns at et h e r e a ct a n c e of t h e a nt e n n a at all t hr e e fr e q u e n ci es of t h esi g n al s p e ctr u m.

L et us c o nsi d er t h e s a m e l o o p a nt e n n a e x a m pl e, m o d-el e d as a L R cir c uit as st u di e d a b o v e ( L 0 = 1 n H a n d R 0 =1 0 0 ), f e d b y a n a m plit u d e- m o d ul at e d si g n al V s (t) as i nE q. ( 5 8), wit h d = 0. 1 a n d θ d = 0. T h e r es ults i n Fi g. 1 2s h o w t h e m o d ul ati o n p h as es r e q uir e d f or p erf e ct m at c h-i n g of t h e r e a ct a n c e at t h e t hr e e fr e q u e n ci es fs , fs + fd , a n dfs − fd , w h e n w e r a n g e t h e fr e q u e n c y fd of t h e m ess a g esi g n al. T h e n u m eri c al c al c ul ati o ns ar e pr es e nt e d f or t w oc as es: wit h fs = 1 0 0 k H z a n d fs = 1 0 0 M H z. It is cl e arfr o m t h e r es ults i n Fi g. 1 2 t h at t h e r e q uir e d e ff e cti v e n e g a-ti v e i n d u ct a n c e c a n b e r e ali z e d i n a wi d e r a n g e of fd v al u es.Si n c e t h e s yst e m is li n e ar, t his a p pr o a c h is als o v ali d f org e n er al ( m ultit o n e) si g n als.

VII. C O N C L U SI O N

T his st u d y of ti m e- v ar yi n g el e ctri c el e m e nts ( c a p a ci-t or, i n d u ct or, a n d r esist or) b as e d o n t h e i ntr o d u c e d m o d elof m atri x cir c uit p ar a m et ers h as r e v e al e d i m p ort a nt f e a-t ur es t h at a p p e ar d u e t o t e m p or al m o d ul ati o n of t h os eel e m e nts. D e v el o pi n g t h es e r e as o n a bl y si m pl e m o d els f orn o nst ati o n ar y el e m e nts is p ossi bl e b e c a us e t h es e el e m e ntsar e li n e ar a n d c a us al. Ass u mi n g t h at t h e c o m p o n e nts ar e

el e ctri c all y s m all at all r el e v a nt fr e q u e n ci es all o ws t h e us eof b ul k- c o m p o n e nt m o d els, a n d if t h e si g n als a n d m o d ul a-ti o ns ar e p eri o di c al f u n cti o ns of ti m e, t h e m o d els f urt h ersi m plif y t o m atri x r el ati o ns b et w e e n v e ct ors f or m e d b yt h e fr e q u e n c y h ar m o ni cs of v olt a g es a n d c urr e nts. I m p or-t a ntl y, t h e d e v el o p e d a n al yti c al m o d els f ull y a c c o u nt f orfr e q u e n c y dis p ersi o n of ti m e- m o d ul at e d c o m p o n e nts.

B y a p pl yi n g t h es e m o d els t o ti m e- m o d ul at e d m ut u ali m p e d a n c e b et w e e n tr a ns mitti n g a n d r e c ei vi n g c oils i ni n d u cti v e wir el ess p o w er tr a nsf er s yst e ms, w e fi n d t h at itis p ossi bl e t o i m pr o v e t h e tr a nsf er p o w er r ati o a n d p o w ertr a nsf er e ffi ci e n c y b e y o n d t h e k n o w n p ossi biliti es b as e d o ni m p e d a n c e m at c hi n g i n r es o n a nt s c e n ari os.

C o nsi d eri n g t h e e ff e cts of ti m e- m o d ul at e d r esist a n c e, w efi n d a p ossi bilit y t o virt u all y m at c h t h e l o a d (f or e x a m pl e, as m all r es o n a nt a nt e n n a) t o a s o ur c e wit h o ut t h e n e e d t o a d da n y r e a cti v e m at c hi n g n et w or ks or ti m e- v ar yi n g r e a cti v ec o m p o n e nts. T his a p pr o a c h c a n als o b e us e d f or b a n d wi dt he n h a n c e m e nt of r e c ei vi n g a nt e n n as a n d a bs or b ers.

T o s u m m ari z e, w e str ess t h e k e y c o ntri b uti o ns oft his w or k: ( 1) cir c uit-t h e or y m o d els a n d ri g or o us c h ar-a ct eri z ati o n of dis p ersi v e ti m e- v ar yi n g cir c uit el e m e nts;( 2) i ntr o d u cti o n a n d a n al ysis of t e m p or all y m o d ul at e dr esist a n c e a n d m ut u al i n d u ct a n c e; ( 3) r e v e ali n g alt er n at ep ossi bl e a p pli c ati o ns of ti m e- v ar yi n g el e m e nts i n wir el essp o w er tr a nsf er a n d a nt e n n a s yst e ms. We e n visi o n t h at t h es er es ults m a y h a v e si g ni fi c a nt i m p a ct o n t h e f ut ur e of t hisr es e ar c h ar e a.

A C K N O W L E D G M E N T S

T his w or k w as s u p p ort e d b y t h e A c a d e m y of Fi nl a n du n d er t h e Gr a nts N o. 3 3 0 2 6 0 ( A c a d e m y Pr oj e ct f u n d-i n g) a n d N o. 3 3 3 4 7 9 ( P ost d o ct or al R es e ar c h er f u n di n g),a n d t h e E ur o p e a n U ni o n’s H ori z o n 2 0 2 0 F ut ur e E m er gi n gT e c h n ol o gi es c all ( F E T O P E N- RI A) u n d er Gr a nt A gr e e-m e nt N o. 7 3 6 8 7 6 ( pr oj e ct VI S O R S U R F). M. S. M. wis h est o a c k n o wl e d g e t h e s u p p ort of Ull a T u o mi n e n F o u n d ati o n.

[ 1] M. F ar a d a y, O n a p e c uli ar cl ass of a c o usti c al fi g ur es; a n d o nc ert ai n f or ms ass u m e d b y gr o u ps of p arti cl es u p o n vi br ati n gel asti c s urf a c es, P hil. Tr a ns. R. S o c. L o n d. 1 2 1 , 2 9 9 ( 1 8 3 1).

[ 2] A. L. C ull e n, A tr a v elli n g- w a v e p ar a m etri c a m pli fi er,N at ur e 1 8 1 , 3 3 2 ( 1 9 5 8).

[ 3] P. K. Ti e n, P ar a m etri c a m pli fi c ati o n a n d fr e q u e n c y mi xi n gi n pr o p a g ati n g cir c uits, J. A p pl. P h ys. 2 9 , 1 3 4 7 ( 1 9 5 8).

[ 4] F. R. M or g e nt h al er, Vel o cit y m o d ul ati o n of el e ctr o m a g n eti cw a v es, I R E Tr a ns. Mi cr o w a v e T h e or y T e c h. 6 , 1 6 7 ( 1 9 5 8).

[ 5] A. K. K a m al, A p ar a m etri c d e vi c e as a n o nr e ci pr o c alel e m e nt, Pr o c. I R E 4 8 , 1 4 2 4 ( 1 9 6 0).

[ 6] M. R. C urri e a n d R. W. G o ul d, C o u pl e d- c a vit y tr a v eli n gw a v e p ar a m etri c a m pli fi ers: P art I- a n al ysis, Pr o c. I R E 4 8 ,1 9 6 0 ( 1 9 6 0).

[ 7] J. C. Si m o n, A cti o n of a pr o gr essi v e dist ur b a n c e o n ag ui d e d el e ctr o m a g n eti c w a v e, I R E Tr a ns. Mi cr o w a v e T h e-or y T e c h. 8 , 1 8 ( 1 9 6 0).

0 1 4 0 1 7- 1 4

https://www.jstor.org/stable/107936

https://doi.org/10.1038/181332a0

https://doi.org/10.1063/1.1723440

https://doi.org/10.1109/TMTT.1958.1124533

https://doi.org/10.1109/JRPROC.1960.287569




[ 8] J. R. M a c d o n al d a n d D. E. E d m o n ds o n, E x a ct s ol uti o nof a ti m e- v ar yi n g c a p a cit a n c e pr o bl e m, Pr o c. I R E 4 9 , 4 5 3( 1 9 6 1).

[ 9] A. H ess el a n d A. A. Oli n er, Wa v e pr o p a g ati o n i n am e di u m wit h a pr o gr essi v e si n us oi d al dist ur b a n c e, I R ETr a ns. Mi cr o w a v e T h e or y T e c h. 9 , 3 3 7 ( 1 9 6 1).

[ 1 0] B. D. O. A n d ers o n a n d R. W. N e w c o m b, O n r e ci pr o cit y a n dti m e- v ari a bl e n et w or ks, Pr o c. I E E E 5 3 , 1 6 7 4 ( 1 9 6 5).

[ 1 1] D. E. H ol b er g a n d K. S. K u n z, P ar a m etri c pr o p erti es offi el ds i n a sl a b of ti m e- v ar yi n g p er mitti vit y, I E E E Tr a ns.A nt e n n as Pr o p a g. 1 4 , 1 8 3 ( 1 9 6 6).

[ 1 2] D. G. T u c k er, Cir c uits wit h ti m e- v ar yi n g p ar a m et ers ( m o d-ul at ors, fr e q u e n c y- c h a n g ers a n d p ar a m etri c a m pli fi ers),R a di o El e ctr o ni c E n gi n e er 2 5 , 2 6 3 ( 1 9 6 3).

[ 1 3] J. R. Z urit a- S á n c h e z, P. H al e vi, a n d J. C. C er v a nt es-G o n z ál e z, R e fl e cti o n a n d tr a ns missi o n of a w a v e i n ci d e nto n a sl a b wit h a ti m e- p eri o di c di el e ctri c f u n cti o n (t),P h ys. R e v. A 7 9 , 0 5 3 8 2 1 ( 2 0 0 9).

[ 1 4] J. S. M artí n e z- R o m er o, O. M. B e c err a- F u e nt es, a n d P.H al e vi, T e m p or al p h ot o ni c cr yst als wit h m o d ul ati o ns ofb ot h p er mitti vit y a n d p er m e a bilit y, P h ys. R e v. A 9 3 ,0 6 3 8 1 3 ( 2 0 1 6).

[ 1 5] M. S. Mir m o os a, T. T. K o uts eri m p as, G. A. Ptit c y n, S. A.Tr et y a k o v, a n d R. Fl e ur y, Di p ol e p ol ari z a bilit y of ti m e-v ar yi n g p arti cl es, ar Xi v: 2 0 0 2. 1 2 2 9 7 ( 2 0 2 0).

[ 1 6] D. M. S olis a n d N. E n g h et a, A g e n er ali z ati o n of t h eKr a m ers- Kr o ni g r el ati o ns f or li n e ar ti m e- v ar yi n g m e di a,ar Xi v: 2 0 0 8. 0 4 3 0 4 ( 2 0 2 0).

[ 1 7] T. T. K o uts eri m p as a n d R. Fl e ur y, El e ctr o m a g n eti c fi el dsi n a ti m e- v ar yi n g m e di u m: E x c e pti o n al p oi nts a n d o p er-at or s y m m etri es, I E E E Tr a ns. A nt e n n as Pr o p a g. 6 8 , 6 7 1 7( 2 0 2 0).

[ 1 8] G. Ptit c y n, M. S. Mir m o os a, a n d S. A. Tr et y a k o v, Ti m e-m o d ul at e d m et a- at o ms, P h ys. R e v. R es. 1 , 0 2 3 0 1 4 ( 2 0 1 9).

[ 1 9] C. C al o z a n d Z.- L. D e c k- L é g er, S p a c eti m e m et a m at eri-als – P art I: G e n er al c o n c e pts, I E E E Tr a ns. A nt e n n as Pr o p a g.6 8 , 1 5 6 9 ( 2 0 2 0).

[ 2 0] C. C al o z a n d Z.- L. D e c k- L é g er, S p a c eti m e m et a m at eri-als – P art II: T h e or y a n d a p pli c ati o ns, I E E E Tr a ns. A nt e n n asPr o p a g. 6 8 , 1 5 8 3 ( 2 0 2 0).

[ 2 1] V. P a c h e c o- P e ñ a a n d N. E n g h et a, E ff e cti v e m e di u m c o n-c e pt i n t e m p or al m et a m at eri als, N a n o p h ot o ni cs 9 , 3 7 9( 2 0 2 0).

[ 2 2] Z. Y u a n d S. F a n, C o m pl et e o pti c al is ol ati o n cr e at e db y i n dir e ct i nt er b a n d p h ot o ni c tr a nsiti o ns, N at. P h ot o ni cs3 , 9 1 ( 2 0 0 9).

[ 2 3] D. L. S o u n as a n d A. Al ù, A n g ul ar- m o m e nt u m- bi as e dn a n ori n gs t o r e ali z e m a g n eti c-fr e e i nt e gr at e d o pti c al is ol a-ti o n, A C S P h ot o ni cs 1 , 1 9 8 ( 2 0 1 4).

[ 2 4] Y. H a d a d, D. L. S o u n as, a n d A. Al ù, S p a c e-ti m e gr a di e ntm et as urf a c es, P h ys. R e v. B 9 2 , 1 0 0 3 0 4( R) ( 2 0 1 5).

[ 2 5] D. L. S o u n as a n d A. Al ù, N o n-r e ci pr o c al p h ot o ni cs b as e do n ti m e m o d ul ati o n, N at. P h ot o ni cs 1 1 , 7 7 4 ( 2 0 1 7).

[ 2 6] S. T ar a v ati, N. C h a m a n ar a, a n d C. C al o z, N o nr e ci pr o c alel e ctr o m a g n eti c s c att eri n g fr o m a p eri o di c all y s p a c e-ti m em o d ul at e d sl a b a n d a p pli c ati o n t o a q u asis o ni c is ol at or,P h ys. R e v. B 9 6 , 1 6 5 1 4 4 ( 2 0 1 7).

[ 2 7] T. T. K o uts eri m p as a n d R. Fl e ur y, N o nr e ci pr o c al G ai ni n n o n- H er miti a n Ti m e- Fl o q u et S yst e ms, P h ys. R e v. L ett.1 2 0 , 0 8 7 4 0 1 ( 2 0 1 8).

[ 2 8] X. Wa n g, A. Di a z- R u bi o, H. Li, S. A. Tr et y a k o v, a n dA. Al ù, T h e or y a n d D esi g n of M ultif u n cti o n al S p a c e- Ti m eM et as urf a c es, P h ys. R e v. A p pl. 1 3 , 0 4 4 0 4 0 ( 2 0 2 0).

[ 2 9] X. Wa n g, G. Ptit c y n, V. S. As a d c h y, A. Di a z- R u bi o,M. S. Mir m o os a, S. F a n, a n d S. A. Tr et y a k o v,N o nr e ci pr o cit y i n Bi a nis otr o pi c S yst e ms wit h U nif or mTi m e M o d ul ati o n, P h ys. R e v. L ett. 1 2 5 , 2 6 6 1 0 2 ( 2 0 2 0).

[ 3 0] M. S. Mir m o os a, G. A. Ptit c y n, V. S. As a d c h y, a n d S. A.Tr et y a k o v, Ti m e- Var yi n g R e a cti v e El e m e nts f or E xtr e m eA c c u m ul ati o n of El e ctr o m a g n eti c E n er g y, P h ys. R e v. A p pl.1 1 , 0 1 4 0 2 4 ( 2 0 1 9).

[ 3 1] H. Li, A. M e k a w y, a n d A. Al ù, B e y o n d C h u’s Li mit wit hFl o q u et I m p e d a n c e M at c hi n g, P h ys. R e v. L ett. 1 2 3 , 1 6 4 1 0 2( 2 0 1 9).

[ 3 2] A. S hli vi ns ki a n d Y. H a d a d, B e y o n d t h e B o d e- F a n o B o u n d:Wi d e b a n d I m p e d a n c e M at c hi n g f or S h ort P uls es Usi n gT e m p or al S wit c hi n g of Tr a ns missi o n- Li n e P ar a m et ers,P h ys. R e v. L ett. 1 2 1 , 2 0 4 3 0 1 ( 2 0 1 8).

[ 3 3] M. Li u, D. P o w ell, Y. Z ar at e, a n d I. S h a dri v o v, H u y g e ns’M et a d e vi c es f or P ar a m etri c Wa v es, P h ys. R e v. X 8 , 0 3 1 0 7 7( 2 0 1 8).

[ 3 4] M. M. S al ar y, S. J af ar- Z a nj a ni, a n d H. M os all a ei, El e c-tri c all y t u n a bl e h ar m o ni cs i n ti m e- m o d ul at e d m et as urf a c esf or w a v efr o nt e n gi n e eri n g, N e w J. P h ys. 2 0 , 1 2 3 0 2 3 ( 2 0 1 8).

[ 3 5] N. C h a m a n ar a, Y. Va h a b z a d e h, a n d C. C al o z, Si m ult a n e-o us c o ntr ol of t h e s p ati al a n d t e m p or al s p e ctr a of li g ht wit hs p a c e-ti m e v ar yi n g m et as urf a c es, I E E E Tr a ns. A nt e n n asPr o p a g. 6 7 , 2 4 3 0 ( 2 0 1 9).

[ 3 6] S. T ar a v ati a n d A. A. Kis h k, D y n a mi c m o d ul ati o n yi el dso n e- w a y b e a m s plitti n g, P h ys. R e v. B 9 9 , 0 7 5 1 0 1 ( 2 0 1 9).

[ 3 7] T. T. K o uts eri m p as, A. Al ù, a n d R. Fl e ur y, P ar a m etri ca m pli fi c ati o n a n d bi dir e cti o n al i n visi bilit y i n P T-s y m m etri cti m e- Fl o q u et s yst e ms, P h ys. R e v. A 9 7 , 0 1 3 8 3 9 ( 2 0 1 8).

[ 3 8] T. T. K o uts eri m p as a n d R. Fl e ur y, El e ctr o m a g n eti c w a v esi n a ti m e p eri o di c m e di u m wit h st e p- v ar yi n g r efr a cti v ei n d e x, I E E E Tr a ns. A nt e n n as Pr o p a g. 6 6 , 5 3 0 0 ( 2 0 1 8).

[ 3 9] M. S. Mir m o os a, G. A. Ptit c y n, R. Fl e ur y, a n d S. A.Tr et y a k o v, I nst a nt a n e o us r a di ati o n fr o m ti m e- v ar yi n g el e c-tri c a n d m a g n eti c di p ol es, P h ys. R e v. A 1 0 2 , 0 1 3 5 0 3( 2 0 2 0).

[ 4 0] I. S. G o n or o vs k y, R a di o Cir c uits a n d Si g n als ( Mir P u blis h-ers, M os c o w, U S S R, 1 9 8 1), c h a p. 1 0.

[ 4 1] X. Wa n g, S urf a c e-I m p e d a n c e E n gi n e eri n g f or A d v a n c e dW a v e Tr a nsf or m ati o ns ( A alt o U ni v ersit y P u bli c ati o nS eri es, H elsi n ki, 2 0 2 0), D o ct or al Diss ert ati o ns.

[ 4 2] S. Y. El n a g g ar a n d G. N. Milf or d, M o d eli n g s p a c e-ti m ep eri o di c str u ct ur es wit h ar bitr ar y u nit c ells usi n g ti m e p eri-o di c cir c uit t h e or y, I E E E Tr a ns. A nt e n n as Pr o p a g. 6 8 , 6 6 3 6( 2 0 2 0).

[ 4 3] H. S u hl, T h e or y of t h e f err o m a g n eti c mi cr o w a v e a m pli fi er,J. A p pl. P h ys. 2 8 , 1 2 2 5 ( 1 9 5 7).

[ 4 4] L. R a yl ei g h, O n t h e m ai nt e n a n c e of vi br ati o ns b y f or c es ofd o u bl e fr e q u e n c y, a n d o n t h e pr o p a g ati o n of w a v es t hr o u g ha m e di u m e n d o w e d wit h a p eri o di c str u ct ur e, L o n d o nE di n b ur g h P hil os. M a g. J. S ci. 2 4 , 1 4 5 ( 1 8 8 7).

[ 4 5] R. E. C olli n, F o u n d ati o ns f or Mi cr o w a v e E n gi n e eri n g(J o h n Wil e y & S o ns, N e w Y or k, 2 0 0 7), c h a p. 1 1.

[ 4 6] S. Y. R. H ui, W. Z h o n g, a n d C. K. L e e, A criti c al r e vi e w ofr e c e nt pr o gr ess i n mi d-r a n g e wir el ess p o w er tr a nsf er, I E E ETr a ns. P o w er El e ctr o n. 2 9 , 4 5 0 0 ( 2 0 1 4).

0 1 4 0 1 7- 1 5



https://doi.org/10.1109/PROC.1965.4321

https://doi.org/10.1109/TAP.1966.1138637

https://doi.org/10.1049/ree.1963.0035

https://doi.org/10.1103/PhysRevA.79.053821


https://arxiv.org/abs/2002.12297

https://arxiv.org/abs/2008.04304

https://doi.org/10.1109/TAP.2020.2996822

https://doi.org/10.1103/PhysRevResearch.1.023014

https://doi.org/10.1109/TAP.2019.2944225

https://doi.org/10.1109/TAP.2019.2944216

https://doi.org/10.1515/nanoph-2019-0305

https://doi.org/10.1038/nphoton.2008.273

https://doi.org/10.1021/ph400058y

https://doi.org/10.1103/PhysRevB.92.100304

https://doi.org/10.1038/s41566-017-0051-x


https://doi.org/10.1103/PhysRevLett.120.087401






https://doi.org/10.1103/PhysRevX.8.031077

https://doi.org/10.1088/1367-2630/aaf47a

https://doi.org/10.1109/TAP.2019.2891706



https://doi.org/10.1109/TAP.2018.2858200


https://doi.org/10.1109/TAP.2020.2985712

https://doi.org/10.1063/1.1722623

https://doi.org/10.1080/14786448708628074

https://doi.org/10.1109/TPEL.2013.2249670

time-varying components for enhancing wireless transfer of

Documents