1
Modeling and Measu ing he Shielding
E ec i eness o Ca bon Fibe Composi es
Luis D. Angulo, Da id Esco Bocaneg a, Pa icia G´
omez de F ancisco, Bo ja Plaza Galla do, Da id Poya os
Ma inez, Miguel R. Cabello, Sal ado G. Ga cia, Senio Membe , IEEE
Abs ac —We p o ide a model able o p edic he shielding
e ec i eness (SE) o ca bon ibe composi e (CFC) panels made
o s acked laye s o conduc ing ibe s. This model pe mi s
us o ob ain simple o mulas in which he only pa ame e s
needed a e he shee squa e esis ance and he e ec i e panel
hickness. These ools le us o p edic a minimum SE, which
always inc eases wi h he equency and he e o e cons i u ing
he wo s case, om an elec omagne ic shielding pe spec i e.
Consequen ly, he measu emen o minimum SE equi emen s
can be simply measu ed wi h a mic o-ohmme e using an speci ic
expe imen al se up which is also desc ibed he e. Addi ionally,
his me hod allows o measu e e y high SE alling a beyond
he dynamic ange o he alues measu able wi h he mos
commonly used s anda d, he ASTM D4935. A e desc ibing
he modeling echnique and he di e en es se ups used, a
c oss- alida ion be ween heo e ical and expe imen al esul s is
made o ou di e en samples o CFC; wo designed o es he
modeling assump ions and wo which a e ep esen a i e o he
ones nowadays used in a eal ai c a .
Index Te ms—ca bon ibe composi es, elec omagne ic com-
pa ibili y, shielding e ec i eness, hin-laye modeling
I. INTRODUCTION
IN RECENT TIMES ca bon ibe composi e (CFC) panels
ha e ea ned a la ge sha e as design ma e ials in many in-
dus ies, which g ea ly app ecia e hei mechanical p ope ies
[1]. Howe e , om an elec omagne ic compa ibili y (EMC)
pe spec i e, CFCs ha e in e io capabili ies when compa ed
o he me als ha hey usually eplace [2]. Fo his eason,
i is con enien o adequa ely model hei elec omagne ic
shielding e ec i eness (SE) in o de o ensu e an adequa e
p o ec ion o he di e en sys ems in which hey a e used.
E en when hei mechanical cha ac e is ics a e well-known,
hei elec omagne ic beha io p esen s many sub le ies which
a ise om hei ilamen a y and s ock-piled geome y. On op
o his, hey pose an in insically mul i-scala p oblem when
hey mus be inco po a ed in o simula ions which in ol e
an en i e sys em. Such p oblem is no ye ully sol ed o
aniso opic hin laye s, bu some s eps ha e been made in ha
di ec ion [3], [4].
Luis D. Angulo, M. R. Cabello, and Sal ado G. Ga cia a e wi h he
Uni e si y o G anada, G anada 18071, Spain. (e-mail: lmdiazangulo@ug .es)
Da id Esco Bocaneg a, Pa icia G´
omez de F ancisco, Bo ja Plaza Galla do
and Da id Poya os Ma inez a e wi h he Na ional Ins i u e o Ae ospace
Technology (INTA), in Spain.
The wo k desc ibed in his pape and he esea ch leading o hese
esul s ha e been suppo ed by he Spanish MINECO and EU FEDER
unde P ojec TEC2016-79214-C3-1,2,3-R (MINECO, Spain), he EU PRACE
p ojec 2018184436 ESECELS, and he Alhamb a-LFT con ac wi h AIR-
BUS (Spain).
The heo e ical modeling o hin panels has been add essed
by many au ho s in he pas . A low equencies o when
he conduc ing ibe s in he panel a e e y loose, a simple
impedance model can be deduced [2], [5], [6]. Howe e ,
o ha e loose ibe s is no he mos common si ua ion o
CFCs. On he con a y, ibe s a e ypically packed e y close
and he main app oach consis s on de eloping an e ec i e
pe mi i i y model om he conduc ing cha ac e is ics o ibe
and he ma ix ma e ial [7], [8]. These implici ly assume he
exis ence o a p oximi y e ec be ween cu en s induced in he
ibe s, which domina es he SE a high equencies. Al hough
alid, hese app oaches equi e he use o nume ical ools
which make e y di icul o ack he ob ained SE o he
di e en design pa ame e s in he CFC, such as he adius and
ibe conduc i i y. Addi ionally, in hese wo ks, p edic ions
a e alida ed wi h nume ical simula ions p oducing e y good
ma ches, bu always lacking an expe imen al con i ma ion.
F om an expe imen al app oach, we can ind many wo ks
measu ing CFC panels wi h a ew laye s, ypically less han
ou , and SE below 100 dB [9], [10]. This is ypically accom-
plished using he ASTM me hod [11] o some a ian [12]. The
measu emen o highe SE has no been epo ed despi e he
ac ha CFCs o 10 o 30 laye s, wi h a bi a y o ien a ions,
a e ubiqui ously used in ehicles, and can a guably p esen
much highe shielding. The eason o his can be a ibu ed
o he lack o a dynamic ange in he ins umen s employed
which is ypically abou 100 dB. These wo ks also lack a
connec ion be ween he mic oscopic le el and he measu ed
SE. Mo eo e , he in e p e a ion o expe imen al esul s is
con ined o he i ing o an e ec i e conduc i i y, which
clea ly lacks he capabili y o explaining some o he obse ed
phenomena.
In his con ex , he aim o his wo k is o p opose and
alida e a model which allows o connec high equency (HF)
p ope ies wi h hei low equency (LF) ones, he la e much
easie o measu e and ha ing a much la ge dynamic ange.
Wi h he help o hese models, we de ise a new me hod
which can accu a ely p edic and measu e much highe SE
o CFCs, going beyond he ypical equency anges, han he
ones epo ed un il now.
II. MODELING METHOD
A CFC is usually composed o se e al laye s o s acked
ibe s wi h di e en o ien a ions, ypically o ming angles o
0, 45, o 90 deg ees among each o he (Fig. 1). These ibe s
can be wea ed o laid s aigh , he la e being he subjec o
2
Fig. 1. Mic oscopic image o a CFC panel cu . Elec omagne ic wa es a e
assumed o illumina e he panel coming om op o bo om o he image, o
ice- e sa. Laye s o ming 45 o 90 deg ees a e seen as ellipsoids o lines,
espec i ely.
his wo k. A non-conduc i e ma ix esin is usually used o
cu e he panels, gluing he ibe s oge he .
When a ha monic plane-wa e, ha ing an inciden elec ic
ield ampli ude (Ei) illumina es pe pendicula ly a la panel
(Fig. 2), pa o i can be e lec ed (E ), ansmi ed (E ),
o abso bed. The SE can be de ined as he in e se o he
ansmission coe icien ,
SE =1
|T|=|Ei|
|E |(1)
In o de o de i e an exp ession om i s geome ical and
physical pa ame e s, hin panels can be concep ualized as wo-
po ne wo ks. The ields on one side (E1,H1) and he o he
(E2,H2) a e ela ed by
E2
H2= [Φ] E1
H1=
Nl
Y
i=1
[Φi]E1
H1(2)
wi h Φand Φibeing he ansmission ma ices, o ABCD ma-
ices, o he whole hin panel and o each o he s acked laye s
composing i , espec i ely. Fo panels loca ed in ee-space,
cha ac e ized by an in insic impedance η0, he ansmission
coe icien (T) can be ob ained as [13],
T=2η0
Φ11η0+ Φ12 + Φ21η2
0+ Φ22η0
(3)
No e ha Eq. (1) is a scala , which is su icien o iso opic
media. Howe e , as CFCs a e in insically aniso opic media,
we would need o accoun o he di e en pola iza ion modes
ha illumina e he ibe in a ans e sal (TE) o pa allel (TM)
di ec ion. Fo his kind o aniso opic media, we can de ine an
a e age SE combining he ansmission coe icien s as
SEa =2
|TTM|+|TTE|(4)
Please no e ha (4) does no conside in e -mode con e sions
which would domina e i wo consecu i e laye s o conduc i e
ibe s a e no pe pendicula ly o ien ed. Howe e , o he sake
o simplici y, we do no conside hese cases in he emainde
o his wo k.
z
y
x=ξ
ζ
ψ
ϕ
ETE
ETM
a
P
Fig. 2. Geome ical cha ac e is ics o he ca bon ibe seen om he on :
diame e a, and pe iod P. The ep esen a ion wi h local (ξ, ψ, ζ) and global
(x, y, z)coo dina e sys ems
A. Single laye o CFC
As a i s app oxima ion, le us assume a CFC composed
o a single laye o homogeneously dis ibu ed ibe s wi h
diame e a, in ini ely la ge, and pe iodically epea ed in he
plane a e a dis ance P. The ma e ial which cons i u es hem
is modeled wi h a conduc i i y σ which causes hem o ha e
an e ec i e pe mi i i y o
ε (ω) = ε0−jσ
ω(5)
Fibe s a e embedded in a dielec ic ma ix o esin wi h pe -
mi i i y εmwi h a o al hickness da e cu a ion. The e o e,
ibe s occupy a ela i e olume space o
g=πa2
4Pd (6)
This mix u e o ma e ials can be homogenized using an e ec-
i e media app oach [7] which simpli ies hei elec omagne ic
p ope ies o
εTE =εm(1 −g)−1
εTM =ε g
µTE =µTM =µ0(7)
when ε εm. This simpli ica ion is jus i ied when we no e
ha e en a equencies as high as ∼100 GHz, he conduc i e
e m (σ/ω) will domina e o e he pe mi i i y in (5) e en o
conduc i i ies as low as ∼6 Ω/m. In [2] we made an s udy
on se e al models a ailable and ound ha o mos cases his
homogeniza ion is su icien .
F om hese assump ions, we can now deduce how TE and
TM modes a e ansmi ed using he concep o ansmission
ma ix in oduced in (2). F om [14],
[Φ] = cosh(γd)ηsinh(γd)
η−1sinh(γd) cosh(γd)(8)
3
wi h γ=jω√µε and η=pµ/ε being he ma e ial’s complex
p opaga ion cons an and in insic impedance, espec i ely.
Being aniso opic, he CFC suppo s wo p opaga ion
modes, TE and TM. Fo TE modes he e m γTEd=
jk0dqεmε−1
0is negligible o he ypical hicknesses o CFC
panels a equencies below se e al GHz and a pe mi i i y o
a ew ε0. In consequence, (8) educes o he iden i y ma ix
and om (3) we ob ain ha TTE = 1.
Wi h he TM mode he si ua ion is e y di e en as γTM
is domina ed by he ibe conduc i i y. This allows γTMd
o be also exp essed as a unc ion o he skin dep h δ=
(π µσe)−1/2wi h σe=gσ being he e ec i e conduc i i y,
as
γTMd= (1 + j)d
δ(9)
and he in insic impedance as
ηTM =2
1 + j
1
σeδ(10)
Subs i u ing (8) in (3)
|TTM|=
cosh(γTMd) + 1
2
η2
TM +η2
0
ηη0
sinh(γTMd)
−1
(11)
o
|TTM|=
(ηTM +η0)2
4ηTMη0
eγTMd−(ηTM −η0)2
4ηTMη0
e−γTMd
−1
(12)
Le us now explo e Eq. 12 in wo equency egimes. In he
LF egime, we can pe o m a i s o de Taylo expansion as
e±γTMd≃≃1±γTMd. Using (9), (10) on (12); and conside ing
ha o conduc i e panels |η||η0|below equencies up o
hund eds o GHz, we ob ain
|TTM,LF|=
η0σed
2
−1
=
1
2
η0
R
−1
(13)
wi h
R=1
σed(14)
being he DC squa e esis ance o he laye . No e also, ha eq.
(13) does no depend on he equency. In he HF egime, he
app oach e−γTMd≃0can be made, and (12) akes he o m
|TTM,HF|=
√2
8η0σeδ eγTMd
−1
=
4
√2
η0
R
δ
ded/δ
−1
(15)
which is domina ed by he exponen ial e m γTMd∝ω1/2.
To iden i y he egime in which one o o he app oach is
mo e pe inen , we can de ine wcas he co ne equency in
which e−γTMd=e−1(o equi alen ly δ=d√2) esul ing in
ωc=1
µ0σed2=R
µ0d(16)
An in e es ing esul which de i es om his equa ion is ha
he SE o a hicke panel will p esen a HF exponen ial
beha io a lowe equencies, e en i i has he same squa e
esis ance as a hinne panel.
Summa izing, om he p e ious discussion we ha e educed
he p oblem o de e mining he SE o a CFC o knowing only
wo a iables, i.e. he shee esis ance Rand he hickness
d. Bo h, much mo e easy o ob ain expe imen ally han he
exclusi ely geome ic ones.
B. CFCs wi h an a bi a y numbe o s acked laye s
Le us conside a simpli ied case in which we ha e a s ack
o CFC laye s as he ones desc ibed in sec ion II-A o ming
angles ϕo 0 o 90 deg ees angles wi h espec o he zaxis.
We es ic ou discussion o jus hese angles o he simplici y
o a oiding he in e -mode e ms which would appea in Φ
a e applying he Moh ans o ma ions o enso o a ions.
Fo a wa e a eling pe pendicula o he CFC (xaxis) he
ields a bo h sides can be desc ibed wi h a gene alized e sion
o (2),




Ey,2
Hz,2
Ez,2
−Hy,2




=
Nl
Y
i=1 [Φϕi,i]
[Φϕi−90◦,i]



Ey,1
Hz,1
Ez,1
−Hy,1




(17)
whe e [Φϕi,i]co esponds o he i- h laye wi h ibe o ming
an angle ϕiwi h espec o he zaxis. Fo ins ance, le us
assume a wa e pola ized in he ydi ec ion. This implies
ha [Φ0◦]co esponds o a TM ansmission and [Φ90◦]
co esponds o a TE ansmission.
III. EXPERIMENTAL SETUPS
A. ASTM D4935 s anda d es
The ASTM S anda d D4935 [11] allows he measu emen
o he SE o a plana ma e ial unde he ac ion o TEM mode
elec omagne ic ields in a coaxial line. The use o his mode
only allows o measu e he SE a e aged o all di ec ions,
i.e. he measu emen o pu e TE o TM modes is impossible
wi h his me hod. The s anda d ensu es ha he me hod is
alid o e a equency ange o 30 MHz o 1.5GHz o a
no mally inciden wa e. I also p o ides he dimensions o
he specimen holde which is indeed an enla ged, coaxial
ansmission line wi h special ape sec ions and no ched
ma ching g oo es o main ain a cha ac e is ic impedance o
50 Ω h oughou he en i e leng h o he holde . Ins ead o
ab ica ing an exac copy o his langed-coaxial sample holde
(FCSH), we ha e de eloped an al e na i e se up based on
comme cial connec o s and which has been used o his
wo k. This is based on wo 3 1/8” EIA o N emale adap e s
om SPINNER wi h modi ied inne connec o s (Fig. 3). The
diame e s o he ou e and inne conduc o s o his adap e s
a e 76.9mm and 33.4mm espec i ely, wha yields a cu -o
equency o 1.73 GHz be o e o e moding, in ag eemen wi h
he bounds o he s anda d.
The signal gene a o and he ecei e desc ibed in he
s anda d we e subs i u ed o a ec o ne wo k analyze (VNA)
whose po s 1 and 2 a e di ec ly connec ed h ough coaxial
cables o each o he N connec o s o he adap e s. The SE
o he ma e ial unde es (MUT) agains a la TEM wa e
wi h no mal incidence is hen ob ained om he measu emen
o he inse ion loss (IL) in he FCSH o med by he wo
adap e s connec ed o each o he . Then, he expe imen al SE
is e alua ed as
SE = 20 log10 
S21,L
S21,R 
(18)
whe e S21,L and S21,R a e he measu ed sca e ing pa ame e s
ela ed o he elec omagne ic powe ans e ed om he VNA
4
Fig. 3. The wo 3 1/8” EIA o N emale adap e s used in his wo k as sample
holde s.
po 1 o he VNA po 2 when he FCSH is illed wi h he load
and e e ence samples o he specimen, espec i ely. Thus,
wo samples a e needed o a ce ain MUT and hey mus
be iden ical in hickness o ge he bes epea abili y o SE
measu emen s. The s anda d speci ies he dimensions o he
e e ence and load samples o he a o emen ioned FCSH. I
is wo h no ing ha he s anda d does no men ion any need o
u he manipula ion o he samples so hey we e no modi ied
o imp o e he elec ical con ac be ween hem and he coaxial
holde , as opposed o [9].
I is impo an o men ion ha du ing he es s, he hal es
o he cell a e connec ed by means o nylon sc ews, in o de
o 1) minimize s ay ield coupling, 2) educe he con ac
esis ance wi h he sample, and 3) imp o e he capaci i e
coupling be ween he wo FCSH langes desc ibed in [12]. In
o de o be su e ha he o que gi en o he sc ews is always
he same, a calib a ed dynamome ic o que w ench was used.
B. DC esis ance es
In he ollowing se up, we aim o measu e he DC squa e
esis ance (14) o he samples. This is he esis ance be ween
wo opposi e sides o a squa e and i is independen o i s
size. To ca y ou his es he ma e ials equi ed we e: a
specimen suppo , a mic o-ohmme e , and a pneuma ic p ess
(Fig. 4). The specimen suppo consis s o wo me allic holde s
sepa a ed by wo polyme s uc u es ab ica ed wi h a 3D
p in e . This suppo makes possible ha he cu en d i es
only in one di ec ion h ough he sample ( e ically in Fig.
4). To ensu e a good con ac be ween he me allic holde s
and he sample, sil e pain was used on he sides o he
sample in con ac wi h he holde s, p e iously cleaned wi h
isop opyl alcohol. The pain ed sample is hen placed in o he
pneuma ic p ess, MEGA PRP20 model [15] wi h a nominal
o ce o 20 ons in o de o wa an y enough p essu e o
an adequa e ohmic con ac wi h he ibe s and epea abili y
o measu emen s. To measu e he elec ical esis ance o he
samples, a Kei hley 580 mic o-ohmme e [16] was used. This
Fig. 4. Tes se up o DC esis ance measu emen s. The ed pa s belong o
he hyd aulic p ess used o make p essu e on he specimen holded by he blue
pa s. A mic o-ohmme e was used o measu e he squa e esis ance o he
sample.
uni is designed o esis ance measu emen equi emen s om
10 µΩ o 200 kΩ.
The samples used in his se up we e 200 mm long and
200 mm wid h, while he hickness depends on he numbe
o CFC laye s.
IV. THEORETICAL MODEL AND EXPERIMENTAL
CROSS-VALIDATION
In o de o alida e he model in oduced in sec ion II and
he expe imen al se ups desc ibed in III we ha e selec ed
ou di e en samples o CFC. Two samples, Monolaye and
Bilaye , ha e been designed and manu ac u ed wi h he aim o
being as simple as possible in o de o simpli y hei modeling.
A second se ies o wo samples, codenamed Red and Blue,
is ep esen a i e o he CFCs used in he MILANO ai c a .
The MILANO is a emo ely pilo ed ai c a sys em de eloped
by INTA and cu en ly in ope a ion. All samples we e made
wi h a Cy ec MTM 45-1 epoxy ma ix and HexTow IM7-
12K ca bon ibe . The pa icula numbe o laye s and he
di e en o ien a ions used du ing he s acking sequence o he
di e en samples is summa ized in Table I. E e y laye is
made o unidi ec ional ows composed o 12000 ibe s each,
wi h diame e a= 5.2µm and esis i i y σ−1
= 1.5mΩ·cm.
Ha ing a ow densi y o h ee ows pe cen ime e his adds o
3.6×105 ibe s/m. A e cu a ion, each laye has a o al phys-
ical hickness dp= 128 µm. Howe e , we mus dis inguish
be ween he physical hickness and he e ec i e hickness,d.
We de ine das he dis ance o ibe ma e ial a e sed by
he wa e in which he ibe is aligned wi h he elec ic ield
(TM mode). Using Eq. (14) we ob ain a p edic ed alue o
R,l = 196 mΩ o each o hese laye s.
The Monolaye sample is composed o a single laye as he
one desc ibed p e iously. In his case, he R alue could no
be measu ed wi h he DC esis ance es because being so hin
and weak, he laye could no be wa an ed an a achmen o
he holde wi hou being damaged. To compu e i s SE, we
5
TABLE I
MEASURED SAMPLES
Name S acking Sequence Laye s
Monolaye 0 1
Bilaye 0/90 2
Red 45/-45/0/-45/45/90/90/45/-45/0/-45/45 12
Blue 45/-45/-45/0/45/90/45/0/-45/-45/0/45/90/45/0/45/-45/45 18
TABLE II
MEASURED AND PREDICTED VALUES
Name d[µm]P ed. R[mΩ] Meas. R[mΩ] P ed. SELF [dB]Meas. SELF [dB]P ed. c[GHz]
Monolaye 128/0 196/∞ − 6.0 5.7±0.3 1.220
Bilaye 128 196 −59.6 57.1±0.5 1.220
Red 768 98 99.6±6.8 75.2−0.033
Blue 1152 49 46.4±0.4 78.7−0.015
Fig. 5. SE expe imen al esul s ob ained wi h he applica ion o he modi ied e sion o ASTM D4935 s anda d desc ibed in sec ion III-A.
ealize ha TTE = 1 is much la ge han TTM. The e o e
Eq. (4) gi es a alue o SEa = 2 = 6 dB (Table II). In
Fig. 5 we obse e he expe imen al esul s o he SE o
his sample and we can ealize ha a e ∼1.5GHz hey
end o he p edic ed alued o 6dB. The de ia ion om
he cons an alue in he ange om 0.1GHz o 1.5GHz
can be a ibu ed o he aniso opic na u e o he sample.
This e ec was co obo a ed by a simula ion using he HFSS
so wa e [17], a comme cial Fini e Elemen Me hod nume ical
sol e . Below 0.1GHz he de ia ion co esponds o he los o
capaci i e coupling be ween he samples and he FCSH which
makes impossible any ansmission [11], possibly mixed wi h
he capaci ance s ay e ec s epo ed in [12].
The Bilaye sample combines wo laye s pe pendicula ly
laid. Using (16) we ind ha non-cons an e ec s should be
obse ed beyond 1.220 GHz, as he da a in Fig. 5 s a s o
ain ly sugges . In his case we ind ha TTE =TTM a e
calcula ing he ansmission ma ix in (17). In consequence a
LF, he SE is simply
SELF =2R
η0
= 59.6dB (19)
which g ea ly coincides wi h he measu ed alue o 57.1±
0.5dB. We can also obse e ha , again, and mainly below
0.1GHz, he ASTM s anda d ails o p o ide eliable mea-
su emen s and ends o o e es ima e he SE, as men ioned in
i s in oduc ion [11].
The Red and Blue samples a e made o 12 and 18 laye s,
s acked pe pendicula ly wi h hei neighbo s in mos cases.
The cases in which he ansi ions a e no pe pendicula
would lead us o expec some in e -modal con e sions which
may modi y he assump ions made in sec ion II bu which
we will igno e as hey all beyond wha we can measu e
in he scope o his wo k. The squa e esis ance o hese
samples was measu ed using he se up desc ibed in sec ion
III-B inding alues ha co obo a e he squa e esis ance
p edic ions made om pu e geome ical conside a ions (Table
II). The same assump ions as wi h he Bilaye case we e made
o p edic SELF alues o 75.2dB and 78.7dB o he Red and
Blue samples, espec i ely. Howe e , when we compu e hei
co ne equencies using (16), we p edic ha his LF egime
ends a 33 MHz and 15 MHz, espec i ely. This means ha
abo e hose equencies we mus obse e a SE which g ows
exponen ially wi h ω1/2. This is wha we obse e in he ange

6
up o 0.4GHz jus be o e he alues escape he dynamic ange
o he VNA. As in he p e ious cases, he ASTM s anda d does
no p o ided eliable esul s below 0.1GHz. This, oge he
wi h he e y low expec ed alue o cmakes impossible o
p o ide a measu emen o SELF and we a e le only wi h
he heo e ical p edic ions which se an in e io limi o he
samples.
V. CONCLUSIONS
In his wo k we ha e p o ided a model able o p edic
he SE o some CFC panels ep esen a i e o he ones used
in eal ai c a s (Sec. II). This modeling pe mi ed us o
ob ain simple o mulas in which he only pa ame e s needed
o make p edic ions a e he shee squa e esis ance and he
e ec i e panel hickness; bo h easy o ob ain wi h inexpensi e
equipmen . These ools le us o p edic a minimum, o wo s
case, SE. This minimum alue is always a LF and always
inc eases wi h he equency. In consequence, he measu emen
o minimum SE equi emen s can be deduced simply wi h a
mic o-ohmme e using he expe imen al se -up desc ibed in
sec ion III-B.
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