Recei ed: 31 Ma ch 2020 Re ised: 21 Augus 2020 Accep ed: 24 Augus 2020 IET Powe Elec onics
DOI: 10.1049/pel2.12030
ORIGINAL RESEARCH PAPER
Fi s sel - esonan equency o powe induc o s based on
app oxima ed co ec ed s ay capaci ances
Ignacio Lope1,2Claudio Ca e e o1Jesus Ace o3
1Depa men o Applied Physics, Uni e si y o
Za agoza, Za agoza, Spain
2BSH Home Appliances G oup, Za agoza, Spain
3Depa men o Elec onic Enginee ing and
Communica ions, Uni e si y o Za agoza, Za agoza,
Spain
Co espondence
Claudio Ca e e o, Depa men o Applied Physics,
Uni e si y o Za agoza, 50012 Za agoza, Spain.
Email: cca @uniza .es
Funding in o ma ion
Minis e io de Ciencia e Inno ación, G an /Awa d
Numbe : RTC-2017-5965-6; BSH Home Appli-
ances G oup, G an /Awa d Numbe : -; DGA-FSE,
G an /Awa d Numbe : -; Minis e io de Economía,
Indus ia y Compe i i idad, Gobie no de España,
G an /Awa d Numbe : PID2019-103939RB-100
Abs ac
Induc i e de ices a e ex ensi ely employed in powe elec onic sys ems due o hei mag-
ne ic ene gy s o age and powe ans e capabili ies. The cu en end is owa ds inc easing
he equency o ope a ion in o de o educe he size o he magne ic componen s, bu he
main d awback is ha he pa asi ic capaci ance e ec can become signi ican , and deg ade
he pe o mance o he sys em. This wo k analyses he in luence o his s ay capaci ance,
and conside s how o imp o e he pe o mance o he de ice. In gene al, he impac o
he s ay capaci ance on a magne ic componen can be educed by wo me hods: educ-
ing he pa asi ic capaci ance be ween u ns o he winding o , al e na i ely, modi ying he
a angemen o he connec ion be ween u ns. To e alua e he las op ion, an app oxima ed
exp ession o he i s sel - esonan equency o he magne ic de ice is p oposed. This
gi es a apid assessmen o he pe o mance o di e en de ices main aining he o e all
equi alen induc ance. The p oposed exp ession accoun s o he in luence o he connec-
ion be ween u ns in he bandwid h o he componen . Finally, some nume ical esul s a e
e i ied wi h plana coils manu ac u ed on wo-laye p in ed ci cui boa ds.
1 INTRODUCTION
Plana coils a e o en selec ed o powe elec onics sys ems
because hese componen s a e well adap ed o he geome y
o many applica ions and a e con enien o in eg a ed appli-
ca ions. Some examples include induc i e powe ans e sys-
ems (IPTs) [1–7], de ices o medical applica ions [8–12]and
domes ic induc ion hea ing sys ems [13, 14]. Wi h espec o he
in eg a ed applica ions, he e is a endency owa ds using plana
cons uc ions o induc o s and ans o me s in swi ching-mode
powe supplies (SMPS) [15–17]. The mos common manu ac-
u ing echniques o plana coils a e con en ional wi ing, PCB
implemen a ions, as shown in Figu e 1, and mic o ab ica ion
[18].
The p esen endency owa ds highe ope a ion equencies
has also led o an in e es in modelling pa asi ic capaci i e
e ec s because, among o he easons, his s ay capaci ance can
be used as he esonan elemen in con e e s o de e mines
he bandwid h o he magne ic componen . Va ious p opos-
als ying o cap u e he capaci i e e ec s by means o equi -
alen ci cui s can be ound in he li e a u e. One app oach is a
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is
p ope ly ci ed.
© 2020 The Au ho s. IET Powe Elec onics published by John Wiley & Sons L d on behal o The Ins i u ion o Enginee ing and Technology
simple equi alen ci cui composed o an equi alen RL ci cui
connec ed in pa allel o an equi alen capaci ance [19]. In o he
app oaches, he alue o he equency-dependen impedance
has been ob ained by means o empi ical me hods based on
impedance measu emen s and cu e- i ing echniques [20–22].
Al hough he p eceding me hods a e aluable by hemsel es,
he applicabili y o he esul s is limi ed o he p o o ypes s ud-
ied. In hese simpli ied lumped equi alen ci cui s i is assumed
ha all u ns a e d i en by he same cu en and depending on
he de ice o he equency, he accu acy could be a ec ed and
he e o e some al e na i es ha e been p oposed. These al e na-
i es include dis ibu ed ci cui s [23–29] and ansmission line
app oaches [30], which a e mainly ocused on ob aining sel -
esonan equencies associa ed wi h he pa asi ic capaci ances
and induc ances o he sys em. These pa asi ic capaci o s could
appea , o example, be ween u ns, winding laye s o winding
and magne ic co es [31–34]. In gene al, s ay capaci ances a ise
om he elec ical coupling be ween elemen s o he de ice [35,
36]. Howe e , he connec ion o he winding’s u ns can be op i-
mised o imp o e he bandwid h o he de ice wi h a mino
impac in i s pe o mance [37–41].
IET Powe Elec on. 2021;14:257–267. wileyonlinelib a y.com/ie -pel 257
258 LOPE ET AL.
FIGURE 1 Plana PCB coils
In his pape , a simple exp ession enabling a apid calcula-
ion o he i s sel - esonan equency om he dis ibu ed
ci cui elemen s o he componen is de i ed. In he p oposed
exp ession, he ci cui elemen s a e in insic pa ame e s associ-
a ed wi h he geome y and physical p ope ies o he media,
bu , he i s sel - esonan equency is dependen on he u ns’
a angemen . F om his exp ession i is e iden ha he band-
wid h o he coil can be ex ended by selec ing he app op ia e
connec ion be ween he u ns as di e en connec ions p o ide
simila low equency cha ac e is ics.
The emainde o he pape is o ganised as ollows: in Sec-
ion 2, he dis ibu ed ci cui model o induc o s including s ay
capaci ances is desc ibed; nex , he beha iou o hese sys em
a e explained in Sec ion 3; in Sec ion 4, he p ocedu e o cal-
cula e he dis ibu ed ci cui elemen is gi en; expe imen al al-
ida ion o he p oposed model is d awn in Sec ion 5and Sec-
ion 6concludes.
2ELECTRICAL EQUIVALENT
OF THE INDUCTOR
2.1 Ideal coil
Gene ally speaking, a plana induc o consis s o he se ies con-
nec ion o nconcen ic u ns. Fo he sake o simplici y, spi al
u ns will be conside ed in his pape , bu he analysis is also
alid o u ns wi h al e na i e geome ies, o example, ellip ical
o squa e. The u ns o a plana coil a e placed concen ically o
inc ease he induc ance alue o he de ice while main aining a
educed size. The induc ance o his a angemen , Leq, is g ea e
han he sum o he alues o he sel -induc ances o he u ns,
Lii, due o he e ec o he mu ual coupling be ween hem,
Lij.
Elec ical cha ac e is ics o an ideal mul i- u n coil can be
desc ibed om he coupling induc ances, Lij, which ela e he
FIGURE 2 Dis ibu ed ci cui model o an ideal coil
alue o he induced ol age Viin each u n and sou ces o he
magne ic ields, which a e gi en by he cu en ampli ude, Ij,
ci cula ing h ough each u n. In his desc ip ion, he ha monic
app oach is assumed, ha is, ij( )=ℜe(Ij⋅ej𝜔 ), whe e 𝜔is
he angula equency o he exci a ion. The induced ol age in
he i h u n, Vi, is de e mined by he sum o he induced ol -
ages Vij =LijIj. The induced ol age Vij can be easily ca ied
ou by in eg a ing, along he ajec o y o he i h u n, he elec-
ic ield, Ej, c ea ed by he cu en lowing h ough he j h
u n, as s a ed in [42].
As is commen ed abo e, induc o s a e no mally buil by con-
nec ing he u ns in se ies. The e o e, he cu en o each u n,
Ij, is equal o he sou ce’s cu en , IS, injec ed h ough coil
e minals. In ha case, he ol age in each u n is he e o e
Vi=∑n
j=1j𝜔LijIS, and he o al ol age in he coil is VS=
∑n
i=1Vi=j𝜔IS∑n
i=1∑n
j=1Lij. Thus, he equi alen induc-
ance o he ideal induc o , Leq, is he well-known exp ession
Leq =∑n
i=1∑n
j=1Lij, which is independen o he exci a ion
equency, acco ding o he abo e assump ions. Mo eo e , he
equi alen induc ance, Leq, does no depend on he sequence o
he se ies connec ion among u ns.
Figu e 2 ep esen s he ci cui diag am o dis ibu ed ele-
men s co esponding o an ideal winding. Simila ly, a ci cui o
dis ibu ed elemen s can be cons uc ed by di iding he coils
in o se e al segmen s and applying he de ini ion o coupling
induc ance based on he in eg a ion o he elec ic ield along
he pa h o he elemen . This ex ended model does no p o ide
addi ional da a bu i can be he s a ing poin o a ci cui model
ha includes capaci i e e ec s.
2.2 Dis ibu ed model o he coil wi h s ay
capaci ances
The model desc ibed in he p eceding subsec ion only con-
side s induc i e e ec s a ising om cu en s. Howe e , a
comple e elec ical model o he componen also needs o
conside he inclusion o capaci i e e ec s. As a consequence,
an accu a e e alua ion o he componen conside s bo h
sel - and coupling induc ances as well as s ay capaci ances,
as shown in Figu e 3. The s ay capaci ances a e s ongly
dependen on he geome y and ela i e dis ances be ween
segmen s o he conduc o s. In pa icula , mul i-laye ed spi al
coils exhibi a highly a iable s ay capaci ance be ween acing
segmen s o di e en u ns associa ed wi h he apid a ia ion
LOPE ET AL.259
FIGURE 3 Coil wi h s ay capaci ances be ween pai o segmen s
FIGURE 4 Dis ibu ed ci cui model o a h ee- u n coil
on hei ela i e posi ions. S ay capaci ances can be ound
om analy ical models [43, 44] as well as om nume ical
simula ions [45–47]. The s ay capaci ances can also be ound
by conside ing segmen s o conduc o s a di e en elec ic
po en ials [48].
S ay capaci ance can each high alues when he dis ance
be ween u ns is small o acing su aces a e la ge. In addi ion,
he in luence o he dielec ic p ope ies o he media, o
example subs a e o p in ed ci cui boa ds, play an impo an
ole in his cha ac e is ic. S ay capaci ance p o ides elec ical
connec ion be ween u ns a di e en ol age le els. The
dis ibu ed ci cui model o an ideal coil depic ed in Figu e 2
can be easily ex ended o include capaci i e e ec s by inse ing
s ay capaci ances, Cij, connec ed o he middle poin s o he
i h and j h u ns, as shown in Figu e 4. As i occu s wi h he
coupling induc ances, Lij, he numbe o s ay capaci ances
depends on he squa e o he numbe o u ns. Howe e , only
pa asi ic capaci ances be ween nea by u ns ha e a conside able
in luence in he equency esponse o he coil due o he
s ongly dependence o he s ay capaci ances wi h he ela i e
dis ance be ween segmen s o he u ns.
The model desc ibed in his pape does no conside he
e ec associa ed wi h powe losses in he winding, because i
would comp omise simplici y in desc ibing he beha iou o
he sys em. In o de o ex end he analysis o include winding
losses, on he one hand, i would be necessa y o include in he
FIGURE 5 Ideal single- u n coil wi h long e minals. (a) Schema ic o he
geome y. (b) Dis ibu ed elemen model
model ep esen ed in Figu e 4 esis i e ype elemen s connec ed
in se ies wi h he induc i e elemen s, in o de o conside bo h
he cu en conduc ion losses and he p oximi y losses associ-
a ed wi h he a iable magne ic ield in which he conduc o s
a e imme sed [49, 50]. On he o he hand, esis i e ype ele-
men s connec ed in pa allel wi h he pa asi ic capaci o s should
be inse ed o model he conduc ion and dielec ic losses in
he subs a e [51]. Howe e , he p eceding elemen s ha e a lim-
i ed impac in he equi alen impedance o he whole sys em,
hus, hei in luence in he i s sel - esonan equency alue is
mode a e.
3ELECTRICAL BEHAVIOUR OF COIL
WITH STRAY CAPACITANCES
3.1 LC-pa allel esonan ci cui
S ay capaci ances in coils in oduces addi ional ci cui pa hs
o he cu en which inc ease he complexi y o analysis o he
magne ic de ice’s elec ical beha iou . Fo illus a ion pu poses,
a simple e e ence sys em composed o a loss-less single-loop
coil wi h long pa allel e minals is selec ed, as shown in Fig-
u e 5a. The dis ibu ed elec ical equi alen o he e e ence sys-
ems is also ep esen ed in Figu e 5b. As can be seen in he
schema ic, he equi alen o he single loop is he induc ance,
Leq, and he long pa allel e minals a e ans o med in pa allel-
connec ed s ay capaci ances, Ci, whose addi ion p o ides he
equi alen capaci ance Ceq. A he endpoin s o he coil’s e -
minals, a ol age sou ce, S( ), is connec ed d i ing he cu -
en iS( ). The simpli ied elec ical equi alen o he p eceding
sys em is shown in Figu e 6a which co esponds o he LC-
pa allel ci cui ed by a ol age sou ce. The cu en , iS( )canbe
decomposed in o wo con ibu ions ep esen ed in Figu e 6b:
he capaci o ’s cu en o eci cula ing cu en , iC( ), and he
induc o ’s cu en , iL( ). Applying he Ki chho ’s cu en law
260 LOPE ET AL.
FIGURE 6 Cu en s in pa allel LC esonan ank. (a) Sou ce’s cu en low.
(b) Dis ibu ed elemen model
condi ion, he ela ionship iS( )=iC( )+iL( ) can be s aigh -
o wa dly es ablished, and, ea anging he p eceding exp es-
sion, iL( )=iS( )−iC( ). Equi alen ly, he cu en lowing
h ough he loop u n can also be decomposed in o a eci cu-
la ing cu en , iC( ), and a sou ce cu en iS( ), as can be seen in
Figu e 6. Mesh de ined by he eci cula ing cu en , iC( ), obeys
he Ki chho ’s ol age law, ha is, he ol age in he equi alen
capaci ance, C( ), is equal o he ol age in he equi alen induc-
ance, L( ). Vol age C( ) is o ally de e mined by he p od-
uc be ween he eci cula ing cu en iC( ) and he equi alen
capaci i e impedance, whe eas, ol age L( ) is gi en by bo h
he eci cula ing cu en , iC( ), as well as he sou ce cu en ,
iS( ).
A low equency, he eci cula ing cu en is negligible and
he cu en h ough he equi alen induc ance is almos o ally
equal o he sou ce cu en , whe eas a highe equencies,
he eci cula ing cu en inc eases. Conside ing he ha monic
sou ce’s ol age S( )=VSej𝜔 , he eci cula ing cu en ampli-
ude is IC=j𝜔CeqVS, and he induc ance’s cu en ampli ude
is IL=VS∕j𝜔Leq. The esonan equency is eached when he
induc ance’s ol age is o ally due o he eci cula ing cu en ,
gi en he sou ce cu en , iS( ), equal o ze o a his poin . The
esonan angula equency is gi en by he well-known exp es-
sion 𝜔0=1∕√LeqCeq, being in ini e he equi alen impedance
o he LC- ank ci cui .
The p eceding LC- ank ci cui is selec ed as he e e ence sys-
em o unde s and he in luence o he pa asi ic capaci ances
in n- u ns coils. Reduc ion o he dis ibu ed ci cui model ep-
esen ed in Figu e 4 o he concen a ed ci cui model shown
in Figu e 6b is a challenging ask because he o al cu en ca -
ied by each u n is he composi ion o mul iple eci cula ing
cu en s. As was men ioned, a low equencies, he eci cula -
ing cu en s a e negligible. Howe e , he cu en dis ibu ion
changes depending on he u n posi ion when he ope a ion e-
quency inc eases. A equencies close o esonance, he cu en
in he coil is almos o ally associa ed wi h he eci cula ing cu -
en s. The cu en eaches i s maximum alue in he in e medi-
a e u ns and decays o ze o in he u ns nea he coil e minals.
The imagina y pa o he cu en in o he e minals is nega i e
o posi i e, ha is induc i e o capaci i e, espec i ely, depend-
ing on whe he he exci a ion equency is abo e o below eso-
nance, espec i ely.
The esonan equency o he dis ibu ed ci cui model will
be es ima ed by weigh ing he con ibu ion o each pa asi ic
capaci ance o he o al cu en in he induc o . A he es-
onan equency, he ol age d op be ween he u ns can be
de e mined om he induc i e elemen s’ ol ages, whe eas he
e ec o he pa asi ic capaci o s comes om he c ea ion o
closed pa hs o he eci cula ing cu en s because he cu en
d i en by he ex e nal ol age sou ce ends o ze o a his ope -
a ing poin .
The exp ession o es ima e he i s esonance equency is
ob ained by pe o ming wo di e en s eps: i s , he app ox-
ima e alue o he eci cula ing cu en can be de e mined
assuming ha he same cu en ci cula es, IL h ough all he
u ns o he coil as well as eci cula ing cu en , iCij is p o-
po ional o he po en ial d op be ween he poin s ha p esen
he Cij pa asi ic capaci ance; second, om he eci cula ion cu -
en s, and conside ing ha he cu en o he sou ce eeding he
winding ends o ze o a he esonance equency, ha is, he
cu en s in each loop a e due o he sum o he eci cula ion cu -
en s, he dependence o he po en ial d op wi h espec o he
equency in he comple e coil is calcula ed, being he exp es-
sion o he app oxima ed sel - esonance equency s aigh o -
wa dly de i ed om he p eceding ela ionship.
3.2 App oxima ed eci cula ing
cu en s a esonance
To ob ain a simpli ied exp ession o he i s esonan e-
quency o a mul iple u n coil, he eci cula ing cu en s iCij will
be es ima ed. Fi s , he cu en s’ h ough he induc i e elemen s
a equencies close o he esonance a e app oxima ed o he
es ima ed mean cu en , Ii≅I′
L. The ol age d op ampli ude in
he i h u n, Vi, can be now gi en by he ollowing exp ession:
Vi=
n
∑
l=1
j𝜔Lil Il≅j𝜔I′
L
n
∑
l=1
Lil ,(1)
whe e Ilis he cu en ampli ude h ough he l h u n.
The addi ion o he ol age d op ampli udes o e all he u ns
o he coil gi es he o al ol age d op in he coil, Vcoil which is
equal o he sou ce’s ol age ampli ude, VS:
VS=
n
∑
i=1
Vi≅j𝜔I′
L
n
∑
i=1
n
∑
l=1
Lil .(2)
LOPE ET AL.261
Rew i ing he p eceding equa ion, he cu en ampli ude I′
L
can be app oached by he ollowing equa ion:
I′
L≅VS
j𝜔
n
∑
i=1
n
∑
l=1
Lil
=VS
j𝜔Leq
.(3)
Simila ly, he ol age d op be ween he middle poin i′+1
2o
he i h u n and he middle poin o he nex u n is gi en by:
V(i′+1
2)(i′+1
2+1)≅j𝜔I′
L(n
∑
l=1
Lil
2+
n
∑
l=1
L(i+1)l
2).(4)
And he ol age d op be ween he e minals o he pa asi ic
capaci o , VCij =V(i′+1
2)(j′+1
2), can be app oached by:
VCij ≅j𝜔I′
L
j−1
∑
m=i(n
∑
l=1
Lml
2+
n
∑
l=1
L(m+1)l
2).(5)
Subs i u ing he alue I′
Lgi en by (3), we ob ain:
VCij ≅
j−1
∑
m=i
n
∑
l=1
Lml +L(m+1)l
2Leq
VS.(6)
The las exp ession es ablishes he ela ionship be ween he
ol age d op, VCij, in he middle poin s o he i h and j h u ns,
espec i ely, and he sou ce’s ol age, VS. To a ce ain ex en ,
exp ession (6) could be conside ed as he o mula o a ol -
age di ide . The ac o o p opo ionali y, F
Cij, de ined om he
iden i y VCij =F
CijVS, can be exp essed as:
F
Cij =
j−1
∑
m=i
n
∑
l=1
Lml +L(m+1)l
2Leq
.(7)
The ac o o p opo ionali y, F
Cij, anges om 0 o 1,
because i is posi i ely de ined and he maximum alue is
eached o he middle poin s o he u ns connec ed o he
coil’s e minals. Fo he maximum alue con igu a ion, he ol -
age d op VC1nis lowe o VSdue o he non-ze o ol age
d ops be ween he coil’s e minals and he middle poin s o
he ex eme u ns, V1′(1+1
2)′and V(n+1
2)′(n+1)′, espec i ely. The
ac o o p opo ionali y, F
Cij, is independen o he equency,
as i also occu s o he s ay capaci ances, bu p esen s depen-
dence on he sel - and mu ual induc ances o he u ns, ha is,
i depends on he geome y and magne ic p ope ies o he coil.
No e ha he ac o o p opo ionali y, FCij, can be modi ied by
changing he a angemen o connec ions be ween u ns bu he
equi alen induc ance o he coil, Leq, emains he same. Equa-
ion (6) can be ew i en as:
VCij ≅F
CijVS.(8)
Capaci i e eci cula ing cu en ICij is p opo ional o he
ol age d op be ween he s ay capaci o e minals placed in
he middle poin s o he i h and j h u ns, as i is gi en by
ICij =j𝜔CijVCij. Thus, conside ing Equa ion (8), he eci cu-
la ing cu en ICij can be exp essed as:
ICij ≅j𝜔F
CijCijVS.(9)
By de ining he equency-independen co ec ed capaci ance
be ween he i h and j h u ns, C∗
ij,as:
C∗
ij =F
CijCij.(10)
Thus, he eci cula ing cu en ampli ude can be ew i en
as:
ICij ≅j𝜔C∗
ijVS.(11)
As can be app ecia ed in he p eceding exp ession, he co -
ec ed capaci ance C∗
ij ep esen s he e ec i e capaci ance asso-
cia ed wi h he s ay capaci ance. As a consequence, he in lu-
ence o he physical s ay capaci ance can be al e ed by modi y-
ing he ac o o p opo ionali y, F
Cij, ha is. changing he u ns
be ween e minals o he s ay capaci ance.
3.3 App oxima ed i s sel - esonan
equency
Once he eci cula ing cu en ampli ude is es ima ed, he o e -
all cu en ca ied by he induc i e elemen s o he dis ibu ed
model a esonance can be calcula ed by he addi ion o he
eci cula ing cu en s because he sou ce’s cu en is ze o a
his ope a ing poin . The es ima ed cu en , I′
i′+1, a esonance
h ough he i′+1 h poin is:
I′
i′+1≅
i′
∑
k=1
n
∑
m=i′+1
−ICkm ,(12)
which co esponds o he cu en lowing h ough induc ance
elemen wi h e minals a he i′+1
2 h and i′+1 h endpoin s
as well as he elemen a he i′+1 h and i′+1
2+1 h end-
poin s. The es ima ed ol age d op in he i h u n, V′i,canbe
calcula ed as:
V′
i≅j𝜔0(n−1
∑
l=1
Lil
2I′
(l+1)′+
n−1
∑
l=1
Li(l+1)
2I′
(l+1)′).(13)
Including (12) in he p eceding exp ession, yields:
V′
i≅−j𝜔0
n−1
∑
l=1
l
∑
k=1
n
∑
m=l+1
Lil +Li(l+1)
2ICkm .(14)

262 LOPE ET AL.
Thus, he o al ol age ampli ude in he coil is he addi ion o
he ol ages’ ampli udes, V′io e all he u ns:
V′
coil ≅−j𝜔0
n
∑
i=1
n−1
∑
l=1
l
∑
k=1
n
∑
m=l+1
Lil +Li(l+1)
2ICkm .(15)
The coil’s ol age ampli ude is VS=V′
coil, and he eci cu-
la ing cu en is gi en by (11), hus:
VS≅𝜔
2
0
n
∑
i=1
n−1
∑
l=1
l
∑
k=1
n
∑
m=l+1
Lil +Li(l+1)
2C∗
kmVS.(16)
Rea anging he p eceding exp ession:
𝜔2
0≅VS
n
∑
i=1
n−1
∑
l=1
l
∑
k=1
n
∑
m=l+1
Lil +Li(l+1)
2C∗
kmVS
.(17)
And, inally, he i s sel - esonan angula equency can be
app oxima ed by he ollowing ela ionship:
𝜔0≅1
√n
∑
i=1
n−1
∑
l=1
i
∑
k=1
n
∑
m=i+1
Lil +Li(l+1)
2C∗
km
.(18)
Equa ion (18) can also lead o an es ima ion o he pa allel
esonan capaci o o he lumped ci cui equi alen shown in
Figu e 6a. F om he de ini ion, 𝜔0=1∕√LeqCeq , he equi a-
len capaci ance is p o ided by:
Ceq ≅
n
∑
i=1
n−1
∑
l=1
l
∑
k=1
n
∑
m=l+1
Lil +Li(l+1)
2Leq
C∗
km.(19)
This means ha 𝜔0can be inc eased by educing he equi -
alen capaci ance applying he co ec a angemen o he u ns
which is use ul because i inc eases he bandwid h o he induc-
i e de ice.
4 CALCULATION OF THE
DISTRIBUTED CIRCUIT ELEMENTS
4.1 Induc i e pa ame e s
The analy ical exp ession o he induc ance o a coil made o
ili o m concen ic spi al loops is al eady ound in Maxwell’s
ea ise [52]. Despi e, analy ical exp essions o al e na i e
induc o s uc u es a e no easily a ailable, he sel - and mu ual
induc ance assuming linea condi ions can be easily calcula ed
wi h nume ical ools. Simpli ied geome ic s uc u es p o ide
FIGURE 7 Two se ies-connec ed spi al u ns placed in pa allel planes
(da k g ey: uppe laye u n, ligh g ey: lowe laye u n) connec ed in he inne
adius common poin
accu a e esul s due o he low sensi i i y o he mu ual cou-
pling be ween u ns. Consequen ly, mu ual induc ance can be
calcula ed by aking ad an age o an impe ec symme y, as i
occu s o spi al winding which can be accu a ely modelled wi h
2D axisymme ic models. Induc i e elemen s, Lij, can be easily
ex ac ed om he ela ionship be ween he induced ol age in
he j h u n, Vj, and he cu en ampli ude, Ii h ough he i h
u n, as is gi en in [42].
4.2 Capaci i e pa ame e s
The o al capaci ance be ween wo u ns co esponds o he
o al e ec o se e al dis ibu ed capaci ances. S ay capac-
i ances can be also e alua ed om analy ical exp essions in
simpli ied s uc u es [53], bu analy ical exp essions a e no
a ailable o common a angemen s o a e inaccu a e [54].
Howe e , he nume ical ex ac ion o he pa asi ic capaci ances
can be pe o med in an easy way because hey a e physically
decoupled om he induc i e e ec s. Pa asi ic capaci ance Cij
can be calcula ed by addi ion o he capaci ance o acing
segmen s o he u ns. The nume ical ex ac ion o s ay capac-
i ance can be pe o med by applying Cij =Qj∕Vi, whe e Qj
is he cha ge accumula ed in he j h u n when a ol age Viis
applied in he i u n, and he emaining u ns g ounded.
4.3 Example o pa ame e calcula ion
The elec ical pa ame e ex ac ion o he wo- u n coil ep e-
sen ed in Figu e 7is pe o med wi h COMSOL®. Compu a-
ional ex ac ion can de e mine bo h induc i e and capaci i e
elemen s o he de ice [55, 56]. Tu ns a e buil o s anda d PCB
coppe acks wi h hickness, h=70 µm, and wid h, w=500 µm,
placed in each side o he boa d wi h hickness, =1.55 mm,
espec i ely. The middle poin s o he in e nal e minals a e a
adial dis ance, in =30 mm, and he ex e nal e minals a e
a ex =31 mm, ha is, he sepa a ion be ween acks is d=
500 µm. Sel - and mu ual induc ances can be easily calcula ed
LOPE ET AL.263
FIGURE 8 Schema ic o he PCB-coil (side iew including axis o e olu-
ion)
by using he magne o-quasi-s a ic app oach. Tu ns a e modelled
by wo ec angula c oss-sec ion u ns wi h axial 2D symme y
o mean adius i= j=30.5 mm acing a a dis ance .Mag-
ne ic ield a ises om he cu en Iiuni o mly dis ibu ed in he
c oss-sec ion a ea o he i h u n. The induced ol ages Viand
Vjo he i h and j h u ns ha e been de i ed by in eg a ing he
elec ic ield. Finally, sel -induc ance, Lii =211 nH, and mu ual
induc ance Lij =115 nH, a e ob ained om nume ical simula-
ions, espec i ely, being a good app oach o he exac alues. In
case o inse ing magne ic ma e ials, pa ame e ex ac ion can
be pe o med in a simila ashion because nume ical ools easily
include he in luence o hese elemen s by conside ing an objec
wi h a high ela i e pe meabili y.
On he o he hand, s ay capaci ance, Cij, was ob ained
by applying he elec o-quasi-s a ic app oach, because he e-
quency dependence o he capaci ance associa ed wi h he skin
e ec can be neglec ed o conduc o s o small c oss-sec ional
a ea, as i occu s in he coils analysed in his wo k. The dielec-
ic cons an o he FR4 subs a e is 𝜖 =5. Fi s , he u ns
a e di ided in o Nangula segmen s. Pa ial axial symme y is
exploi ed o each pai o acing segmen s, bu can be also con-
side ed as s aigh segmen s wi h a simila ou come. Pa ial pa a-
si ic capaci ances anges om 1.04 o 1.20 pF pe adian, due o
he a ia ion o he dis ance be ween segmen s wi h espec o
he angula posi ion. The calcula ed wo- u n s ay capaci ance
is Cij =7.18 pF.
5EXPERIMENTAL RESULTS
Th ee p o o ypes o wo-laye PCB coils wi h n=30 equally
dis ibu ed u ns we e buil o alida e he nume ical alues
using s anda d coppe hickness, h=70 µm, and dielec ic
hickness, =1.55 mm, o FR4 ma e ial (𝜖 =5). The emain-
ing common cha ac e is ics o he p o o ypes, as shown in Fig-
u e 8, a e he ollowing: ex e nal adius, ex =60 mm, in e nal
adius, in =30 mm, and ace wid h, w=500 µm, espec i ely.
Fu he mo e, a dis ance om he coil o he e i e equal o
d =1 mm was se . The i s and he second p o o ypes we e
es ed wi h e minals placed a di e en u ns, bu in bo h cases
he connec ion is pe o med om ou e u n o inne u n in
he bo om laye and in he opposing sense o he op laye .
The i s con igu a ion is de ined by he coil’s e minals placed
a ou e u ns, ha is i s and n h u ns, as can be seen in
Figu e 9a, and he second one co esponds o he e minals
FIGURE 9 Re e ence winding a angemen s o n- u ns (side iew includ-
ing axis o e olu ion): (a) ou e connec ion; (b) in e media e connec ion; (c)
al e na ing-laye s connec ion
FIGURE 10 Expe imen al se -up o measu ing PCB coil wi h e i e
a se en h and eigh h u ns o he op laye s, as shown in
Figu e 9b. In addi ion, he las p o o ype was buil by se ies con-
nec ing he u ns om ou e - o-inne adius u ns, bu , sequen-
ially exchanging he laye by means o n-1 ias. Measu emen s
we e pe o med wi h a high p ecision LCR-me e HP 4285A
(measu emen equencies ange om 75 kHz o 30 MHz and
he basic accu acy o he measu emen s is 0.1 %), as shown in
Figu e 10.
Figu e 11a shows he eac ance- o-angula equency a io,
equi alen o he induc ance a low equencies, o he coil in
ai con igu a ions. Model-based alues a e calcula ed om he
same dis ibu ed ci cui pa ame e s, hus, he equi alen induc-
ance o bo h con igu a ions a e he same bu he equi alen
capaci ance di e s because di e en co ec ed capaci ances a e
ob ained. The change in he alues o he co ec ed capaci ances
o each con igu a ion is due o he a ia ion in he alues o he
p opo ionali y ac o s, FCij, de ined in (7) because he pa asi ic
capaci y be ween he i h and he j h u ns, Cij, emain cons an
in all con igu a ions. The e o e, when applying a di e en
connec ion o he u ns, he equi alen capaci ance, Ceq can be
264 LOPE ET AL.
FIGURE 11 Ra io be ween eac ance and angula equency expe imen-
al (ci cles) and model-based (solid lines) alues applying he ou e (black) and
in e media e ( ed) connec ion-modes. (a) Two-laye PCB coil in ai . (b) Two-
laye PCB coil wi h e i e
educed i he con igu a ion educes he p opo ionali y ac o s,
F
Cij, be ween he u ns wi h he highes pa asi ic capaci ance
be ween hem. Mo eo e , Figu e 11b shows he esul s when a
e i e laye o magne ic pe meabili y, 𝜇 , =2000, is inse ed a
a dis ance o d =1 mm below he coil, wi h a good ag eemen
be ween expe imen al and modelled da a because he misma ch
be ween hem a ise om a sligh di e ence on he i s sel -
esonan equencies in he p o o ype and he simula ion esul s
bu a high co ela ion in he equency dependence be ween
bo h ypes o alues is obse ed o he di e en con igu a-
ions. As can be seen in he g aphs, con igu a ions wi h he
in e media e u ns’ e minals ex end he bandwid h o he coil
compa ed o con igu a ions wi h he ou e u ns’ e minals
bu be e esul s a e achie ed when acing u ns loca ed a
di e en laye s a e connec ed be ween hem, as i occu s in he
las p o o ype. Howe e , he cons uc ion o he las con igu-
a ion exhibi s wo main d awbacks: including ias in he coil
would impac in he powe losses in he coil and he e minal
a e placed in o dis an poin s. A compa ison be ween he
simula ed alues and he measu ed i s esonan equencies
FIGURE 12 Nume ical esul s o coil’s e minals a di e en consecu i e
posi ions (black: coil in ai , ed: coil wi h e i e plane). (a) Fi s sel - esonan
equency, 0. (b) Equi alen induc ance, Leq. (c) Equi alen capaci ance, Ceq
o he es ed a angemen s is gi en in Table 1. Expe imen al
i s sel - esonan equencies, 0, a e ob ained by i ing he
measu ed impedances o he equency esponse o pa allel LC
esonan anks.
Figu e 12 p esen s he expec ed i s sel - esonan equency
calcula ed o he equi alen induc ance, Leq and equi alen
LOPE ET AL.265
TABLE 1 Sel - esonan equency and elec ical pa ame e s o 30- u n coil
Addi ional Connec ion Measu ed pa ame e s Simula ed pa ame e s
media mode 0(MHz) Leq (µH) Ceq (pF) 0(MHz) Leq (µH) Ceq (pF)
Ai Ou e u ns 2.09 103.84 55.84 2.01 103.19 60.75
In e media e u ns 3.21 103.84 23.64 3.44 103.19 20.74
Al e na ing u ns 6.31 102.70 6.19 6.77 103.19 5.36
Fe i e plane Ou e u ns 1.44 189.30 64.89 1.48 189.20 61.30
In e media e u ns 2.54 187.40 21.03 2.39 189.20 23.36
Al e na ing u ns 3.63 180.90 10.60 3.33 189.20 12.07
TABLE 2 Sel - esonan equency o wo-laye PCB coil
Tu ns Measu ed 0(MHz) Simula ed 0(MHz)
18 6.17 5.70
26 3.61 3.43
34 2.56 2.47
capaci ance, Ceq o he es ed a angemen s bu a ying he
posi ion o he coil’s e minals, s a ing om he ou e con-
nec ion, ha is 1s and 30 h u ns, and a ying he connec ion
sequen ially. Ou e connec ion and inne connec ion exhibi a
simila pe o mance because u ns wi h la ge s ay capaci ances
a e connec ed h ough a high numbe o in e media e u ns,
implying a high alue o co ec ed s ay capaci ances. Maxi-
mum sel - esonan equency is achie ed o coils wi h e mi-
nals a in e media e u ns, bu he uppe -laye connec ion is
weakly p e e ed o he coil wi h e i e plane. Al e na i e com-
plex connec ions can inc ease he bandwid h o he induc o ,
ha is, i s , up- o-down o down- o-up u ns’ connec ion and,
nex , ou e - o-inne connec ion gi es esonan equencies a
0=6.77 MHz o coil in ai and 0=3.33 MHz o coil wi h
e i e plane. As can be seen in Figu e 12c, he equi alen capac-
i ance is clea ly dependen on he a angemen o he u ns, bu
he inclusion o he e i e plane sligh ly inc eases i s alue.
Addi ionally, h ee p o o ypes o wo-laye PCB coils wi h
n=18, n=26 and n=34 equally dis ibu ed u ns we e also
buil using coppe hickness, h=400 µm, and dielec ic hick-
ness, =1.55 mm, o FR4 ma e ial (𝜖 =5). The emaining
common cha ac e is ics o he p o o ypes a e he ollowing:
ex e nal adius, ex =60 mm, in e nal adius, in =12 mm, and
ace wid h, w=2000 µm, espec i ely. A compa ison be ween
he measu emen da a and he model esul s a e shown in Fig-
u e 13, and he sel - esonan equency ex ac ed om he mea-
su emen s and he calcula ed sel - esonan equency, 0,a e
lis ed in Table 2whe e he good ag eemen be ween hem can
be seen. Calcula ed sel - esonan equencies a e ob ained om
sel -induc ances o 18.20, 38.01 and 65.21 µH, and co ec ed
capaci ances o 42.82, 56.57 and 65.54 pF, espec i ely.
Finally, an a angemen composed o wo PCB’s was made
o double he numbe o u ns by applying a se ies connec ion
be ween hem in o de o es he p oposed model o mul i-
FIGURE 13 Ra io be ween eac ance and angula equency expe imen al
measu emen s (ci cula symbols) and model-based alues (solid lines) o wo-
laye PCB coils o n=18 u ns, n=26 u ns and n=34 u ns
FIGURE 14 Ra io be ween eac ance and angula equency expe imen al
measu emen s (ci cula symbols) and model-based alues (solid lines) o ou -
laye PCB coils o n=36 u ns, n=52 u ns and n=68 u ns
laye ed coil con igu a ions. The new con igu a ion was buil by
using wo PCBs wi h he same cha ac e is ic o he p eceding
a angemen . The dis ance be ween PCB’s in ha case is ixed
o 1.55 mm. These con igu a ions e i y he gene ali y o
he p oposed model. In ha case, he compa ison be ween he
expe imen al and model esul s a e shown in Figu e 14 e i ying
he co ec ness o he p oposed model because, as can be seen