Download Full Article in PDF - Advances in Radio Science

Adv. Radio Sci., 10, 221–225, 2012
www.adv-radio-sci.net/10/221/2012/
doi:10.5194/ars-10-221-2012
© Author(s) 2012. CC Attribution 3.0 License.
Advances in
Radio Science
Simplified susceptibility analysis of shielded multiconductor
transmission lines
B. Schetelig1 , S. Parr1 , R. Rambousky2 , and S. Dickmann1
1 Faculty
of Electrical Engineering, Helmut-Schmidt-University/University of the Federal Armed Forces Hamburg, Germany
Research Institute for Protective Technologies and NBC Protection (WIS) Munster, Germany
2 Bundeswehr
Correspondence to: B. Schetelig ([email protected])
Abstract. In this contribution, a new kind of coupling parameters for shielded multiconductor transmission lines is
presented. These parameters are easy to calculate and simplify the vulnerability analysis of shielded cables with multiple inner conductors. In this paper, the underlying theory is
derived in detail. The gained insight is demonstrated with a
sample measurement.
1
Introduction
If the susceptibility of complex structures in large volumes is
analyzed, the attention has to be drawn to a smart integration
of the transmission lines into the entire model of the structure. A numerical field analysis can be accelerated by separating the analysis into two parts. The field analysis is carried
out numerically and afterwards, the cables are added using
multiconductor transmission line theory. This approach is
particularly favourable for shielded transmission lines.
Cable shields can be integrated into transmission line theory by introducing the so-called transfer impedance and
transfer admittance. These quantities are well-defined for
single conductor transmission lines. Transfer parameters
for multiconductor transmission lines are determined in several ways in literature (e.g. Weitze, 1991; Kley, 1991; Degauque and Hamelin, 1993; Kasdepke, 1997). In this paper, a new definition of multiconductor coupling parameters
is introduced that uses some approximations to simplify the
description of the coupling process.
2
Theory of the transfer parameters
In this section, the theory of the transfer parameters is
briefly reviewed to the extent as it is applied in the
proposed procedure.
To describe the voltages and currents on shielded cables,
the well-known equivalent circuits from transmission line
theory are used. Because of the shield, which separates the
inner conductors from the outer world, one circuit is used for
each of these areas (Fig. 1). The inner system consists of the
inner conductor and the inside of the shield (reference conductor). The outer system is made up by the outside of the
shield and the remaining structure around the transmission
line. The shield is assumed to be a good one, which allows
the application of the “good shielding approximation” and
the concept of electromagnetic topology (Baum, 1980). This
permits to consider the influence of external electromagnetic
fields on the inner conductor primarily as an indirect coupling via the cable shield. In this concept, the field couples
to the shield and the resulting voltages and currents of the
shield then cause interferences inside.
The outer and the inner systems are linked via the transfer
impedance and the transfer admittance. They can be defined,
using a short section of a cable (Fig. 2), as (Vance, 1978):
1 dUi
0
ZT
=
·
; Ii = 0 ,
IS dz
1 dIi
Y 0T = −
·
; Ui = 0 .
US dz
(1)
(2)
These two parameters can be found in the definitions of the
distributed sources of the inner system (Fig. 1):
U 0 s,i = Z 0 T · IS ,
(3)
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
ckmann1
Fig. 2. Very small section of transmission line to define the transfer
University / University of the Federal Armed Forces Hamburg, Germany
impedance (left) and transfer admittance (right)
ologies and NBC
Munster,
Germany
222 Protection (WIS) B.
Schetelig
et al.: Simplified susceptibility analysis of shielded multiconductor transmission lines
s briefly
d proce-
0
Us,i
= ZT0 · IS
L0i
Ii
Ci0
G0i
inner system
Ui
0
Is,i
= −YT0
· US
∆z
Ra0
2
Fig. 3. Principle of using an equivalent cable description to define
0
Us,a
Fig.susceptibility
3. Principle
of using
an equivalent
cable
description
alternative
transfer
Schetelig et al.: Simplified
analysis
of parameter.
shielded
multiconductor
transmission
lines
IS
L0a
to define
alternative transfer parameter
0
l << λ
outer system
l
G0a
dz
Ca0
0
l << λ
l
0
10
US
wire A
wire B
wire C
wire D
dz
0
Is,a
I
impedance
and the transfer admittance. They can be defined,
10
U
Fig.
1.
Equivalent
circuit
of
a
shielded
transmission
line
a shortanalysis
section
of a cable
(Fig. 2),
as (Vance,
Equivalent circuit of a shielded
transmission
line. using
2
Schetelig
et al.: Simplified
susceptibility
of shielded
multiconductor
transmission
lines 1978):
Ui
i
−1
IS
0
cable shield
l << λ
S
inner conductor
l
0
dz
l << λ
l
dz
Fig.
dure.2. Very small section of transmission line to define the transfer
impedance
(left)
and
transfer
admittance
(right)
To describe the voltages and currents on shielded cables,
Ui
U2/U0
umes is
egration
e strucy sepacarried
d using
oach is
s.
ine thence and
ned for
ters for
several
gauque
, a new
s introdescrip-
Ri0
0
10 −2
10
Ii
1 dUi
; Ii = 0 ,
·
IS dz
1 dIi
YT0 = − ·
; Ui = 0 .
US dz
ZT0 =
wire A
wire B
wire C
wire D
(1)
the well-known equivalent circuits from transmission line
−1
IS
10 −3
US
(2)
theory are used. Because of the shield,
which separates the
10
inner conductors
from the
world, one circuit is used for
cable shield
innerouter
conductor
each of these areas (Fig. 1). The inner system consists of the
−2
These
two1010
parameters
can be found in the definitions of the
Very small
section
transmission
line
Fig.
smalland
section
of
transmission
linetotodefine
definethe
thetransfer
transfer
inner2.conductor
theof
inside
of the shield
(reference
con−4
0
0.5
1
1.5
impedanceThe
(left)
and
(left)
and transfer
transferadmittance
admittance
(right).
ductor).
outer
system
is made up(right)
by the outside ofdistributed
the
sources of the innerf system
(Fig.
1): x 1092
/ Hz
shield and the remaining structure around the transmission
−3
line. The shield is assumed to be a good one, which allows
10
0 3. Principle
0
0
Fig.
of using
an equivalent cable description to define
Ithe
=
−Y
·
U
.
(4)
Us,i
=inner
ZT0conductors
· IS , of a shielded LiYCY
(3)
application
of
the
“good
shielding
approximation”
and
T
s,i
S
Fig. 4. Coupling to the four
LiYCY
alternative transfer parameter
the
concept
of
electromagnetic
topology
(Baum,
1980).
This
0
transmission line
line.
This description of cable shields can be expanded to multiIs,i
= −YT0 · US .
(4)
permits to consider the influence of external electromagnetic
−4
conductor transmission lines. While the equivalent circuit of
10
0
0.5
1
1.5
2
fields on theand
inner
conductor
primarily They
as an can
indirect
cou9
f / Hz
impedance
theremains
transfer
admittance.
defined,
the outer system
the
same, the model
ofbethe
inner
x 10
pling via the cable shield. In this concept, the field couples
cable
be expanded
the border of aof
bundle
of shields
conductorscan
is rather
small. The to mulusing
sectionby
of the
a cable
(Fig. 2),circuit
as (Vance,
1978):This description
systema short
is replaced
equivalent
of a multiconsimilar, if the cable is not mechanically damaged. This is
to the shield and the resulting voltages and currents of the
difference
between
the
less
important
transfer
admittances
is
ductor
transmission
line.
In
this
model,
each
inner
conducFig. 3. Principle of using an equivalent cable description to define
ticonductor
lines.andWhile
the Other
equivalent
valid transmission
for the absolute values
the phases.
inves- circuit
shield then cause interferences
1 dUinside.
i
Fig.
4. Coupling
to
the four inner
conductors of a shielded LiYCY
slightly
larger
(Kasdepke,
1997).
0individual
alternative
transferby
parameter
tor
is stimulated
distributed
voltage
and
current
ZT = ·
; Ii = 0 ,
tigations
show
that even thethe
difference
between
the transfer
of (1)
the outer
system
same,
the model
of the inner
The outer
and
thebe
inner
linked via the transfer
transmission
line remains
IS systems
dz asare
These
observations
are of
course
only
for cables
sources.
They
can
described
follows:
impedances
of a conductor
in the
centre
andvalid
a conductor
at
1 dIi
without
mechanical
defects.
If shielded
multiconductor
casystem
is
replaced
by
the
equivalent
circuit
of
a
multicon0
0
0
the
border
of
a
bundle
of
conductors
is
rather
small.
The
Y
=
−
·
;
U
=
0
.
(2)
i They can be defined,
T
(5)
U
· ISthe
, transfer
impedance
s,i = Z Tand
bles
are
bent
by
force,
the
coupling
parameters
of
the
indiUadmittance.
S dz
difference between
the less
transfereach
admittances
ductor transmission
line.
Inimportant
this model,
inneris conduc0
0 section of a cable (Fig. 2), as (Vance, 1978):
short
vidual ifwires
can
become
different from
each other.
the cable
is not mechanically
damaged.
ThisAs
is it
=a −Y
· US .
Iusing
(6) similar,
T
s,i
slightly
larger
(Kasdepke,
1997).
These two parameters can be found in the definitions tor
of the
is stimulated
by
individual
distributed
voltage
and
is
usually
not
known
throughout
a
susceptibility
analysis,
if current
valid
for
the
absolute
values
and
the
phases.
Other
inves1 inner
dUparameter
i system (Fig.
distributed
sources
the
1):
0of
These formulas
use
transfer
vectors
of
dimension
These
observations
are
of
course
only
valid
for
cables
ZT = ·
; Ii = 0 ,
(1)
there
areshow
mechanically
stressed
cables,
these differences
tigations
that
the
difference
between
the transferbesources.
They
can
be even
described
as follows:
IS dz
N to describe the coupling
to 0a shielded N conductor trans- impedances
without
mechanical
defects.
If shielded
multiconductor
tween
the
various
wires
cannot
be
taken
into
consideration,
of
a
conductor
in
the
centre
and
a
conductor
atca0
Us,i 1= ZdI
(3)
T ·i IS ,
mission line.
bles
are
bent
by
force,
the
coupling
parameters
of
the
indi0
anyhow.
the
border of a bundle0 of conductors
is rather small. The
YT = −0 · 0 ; Ui = 0 .
(2)
dzT · US .
Ufollowing
Z 0Tit ·isISassumed
, each
(5)
Is,iU=
(4)
vidual
wires
can
become
different
from
As
S −Y
s,i
difference
between
the
less=important
transfer
admittances
is it
Therefore,
in the
thatother.
an external
is
usually
not
known
throughout
a
susceptibility
analysis,
if
0
0
slightly
larger (Kasdepke,
1997).
electromagnetic
field
quite ·similar
on
two
parameters
can
beshields
foundcoupling
in the
of multhe
3These
Concept
of simple
equivalent
parameters
I s,icauses
=
−Y
. interferences
(6)
This
description
of cable
can
be definitions
expanded
to
T USthese
there
are conductors.
mechanically
stressed
cables,
differences be4,
the
shield
of
a
four-wire-cable
the
inner
In
Fig.
distributed
sources
of
the
inner
system
(Fig.
1):
These observations are of course only valid for cables
for multiconductor
ticonductor
transmissiontransmission
lines. While lines
the equivalent circuit
tween
the at
various
wires
cannot
be
intoUconsideration,
is excited
one end
of the
cable
bytaken
a voltage
0 . The resultwithout
mechanical
defects.
If shielded
multiconductor
caof the outer system remains
the
inner formulas
0
0 same, the model of theThese
anyhow.
use
transfer
parameter
vectors
of are
dimension
Us,i = Z
(3)
between
each
inner
wire
and
the
shield
ing
voltages
U
T · IS , industrial multiconducPrevious
work
(Jung,
2003)
analyzed
bles are bent by2,i
force, the coupling parameters of the
indisystem is replaced by the
equivalent
circuit
of
a
multicon0
0
Therefore,
in
the
following
assumed
external
measured
atcan
the
other
enddifferent
(setup
analogous
Fig.an
5).
Is,iwires.
=
US .shown
tor cables
with stranded
ItT ·was
that
the coupling
N(4)to describe
the
coupling
to ait isshielded
N
conductor
transvidual
wires
become
from
eachtothat
other.
As it
ductor
transmission
line.
In−Y
this
model,
each
inner
conducelectromagnetic
field
causes
quite
similar
interferences
on
parameters
of
the
wires
within
such
a
cable
usually
are
quite
This
property
of
approximately
identical
voltages
and if
curis
usually
not
known
throughout
a
susceptibility
analysis,
tor is stimulated by individual distributed voltage and current
line.
This description
of cable
can be expanded tomission
multhe
inner
conductors.
fig. 4, cables,
thethe
shield
of adifferences
four-wire-cable
similar,
ifThey
the can
cable
not shields
mechanically
rents
shall
be used toIn
simplify
calculation
of thebecouare
mechanically
stressed
these
sources.
be isdescribed
as follows:damaged. This is there
ticonductor
transmission
lines.
While
the
equivalent
circuit
is
excited
at one end
of cannot
the cable
atransmission
voltage
U0 . The
the
various
wires
beby
taken
into consideration,
valid for the absolute values and the phases. Other inves- tween
pling
parameters
of multiconductor
lines.resultThat
of the outer
system
0 same, the model of the inner
ing
U2,imulticonductor
between each inner
wire andline
the will
shield
anyhow.
for, voltages
the shielded
transmission
be are
retigations
show
that remains
even
between the transfer
U 0s,ithe
=the
Zdifference
(5)
T · IS ,
system
is
replaced
by
the
equivalent
circuit
ofa aconductor
multicon-at
measured
at
the
other
end
(setup
analogous
to
fig.
5).
0
0 centre
impedances of a conductor
in
the
and
placed
by
an
equivalent
model,
which
is
made
up
by
only
one
3 (6)Concept
of simple
equivalent
coupling
parameters for
Therefore,
in the following
it is assumed
that an external
I s,i In
= −Y
· US . each inner conducductor transmission line.
this Tmodel,
This
property
of
approximately
identical
voltages
andon
curelectromagnetic
field
causes
quite
similar
interferences
tor is stimulated by individual distributed voltage and currentmulticonductor transmission lines
These
formulas
use
transfer
parameter
vectors
of
dimension
the
inner
conductors.
In
fig.
4,
the
shield
of
a
four-wire-cable
rents
shall
be
used
to
simplify
the
calculation
of
the
couAdv.
Radio
221–225, as
2012
www.adv-radio-sci.net/10/221/2012/
sources.
TheySci.,
can10,
be described
follows:
N to describe the coupling to a shielded N conductor transispling
excited
at one endofofmulticonductor
the cable by a voltage
U0 . The
resultparameters
transmission
lines.
That
Previous
work
(Jung,
2003)each
analyzed
mission line.
ing
voltages
U2,i between
inner
wireindustrial
and the
arerefor,
the shielded
multiconductor
transmission
lineshield
willmulticonducbe
U 0s,i = Z 0T · IS ,
(5)
measured
theequivalent
other end
(setup
analogous
fig. 5).
placed
an
model,It
which
istomade
up
by the
onlycoupling
tor(6)cables
withbyatstranded
wires.
was shown
that
I 0 = −Y 0 · U .
U2/U0
ng paras is preimplify
tiple inderived
sample
s,i
T
S
4 multiconductor transmission
Schetelig
lded multiconductor
transmission
lines susceptibility analysis of shielded
3
B. Schetelig
et al.: Simplified
lines et al.: Simplified
223susceptibility
cable under test
Zein,a
(9)
(10)
the
(11)
(12)
and
(13)
(14)
(15)
(16)
inner conductor
Zi
inner conductor and the shield (Fig. 3). Nevertheless, that
This
means
that coupling
to theconductors
inner system
a multicondoes
not mean
that the inner
are of
simply
shortcircuited
or that allline
wires
one are neglected.
Instead,
ductor
transmission
canexcept
be described
by scalar coupling
the observation
of almost identical
parameters.
The consequence
is that currents
one can and
nowvoltages
use the
from
above
is
applied
the
following
way:
simple procedure for a coaxial line to describe the multiconFollowing
ductor
coupling(Andrieu,
process. 2006), the differential equations of
the inner multiconductor system,


 
U1
I1



d 
U
 2  = (R 0 + j ωL0 ) ·  I2  ,
(7)
4 −Measurement
setup
to
determine



 the coupling para...
...
dz
metersUN
IN
 


I1
U1
The application
coupling para of the conceptof equivalent

d 
 I2  = (G0 + j ωC 0 ) ·  U2  ,
−
(8)
meters
from the 
measurement
procedure used
... 
... 
dzisindependent
to determine
approach is the
IN the parameters. A well-known
UN
“triaxial
This
measurement
setup
be simplified
can be setup“.
simplified.
This
is done by using
thecan
assumptions
as shown in fig. 5. The measurement principle is, to excite
N
P
the shield via the source U0 and
to
Ui measure the voltage U2
n=1the shield. Using transmisbetween
the
inner
conductor
and
,
(9)
Ueq = U1 = U2 = ... = UN =
sion line theory, these quantities N
can be used to calculate the
N
X
transfer impedance
and the transfer admittance (Tiedemann,
I
=
In = N · I1 .
(10)
eq
2001).
n=1
There is no difference between the calculation of the transThen, the Eqs. (7) and (8) reduce from dimension N to the
ferdimension
parameters
of a physical coaxial line and of the substitute
1:
transmission
line
from above. The equivalent cable is ded
0
· Ieq
j ωL0 eq )of
− Uineqthe
= (R
scribed
usual
a single conductor with(11)
the
eq +notation
dz
0
0
0
0
and
the
equiva,
C
,
L
,
G
per unit
length
parameters
R
eq
eq
eq
eq
d
0
0
Ieq =U(G
j ωCequivalent
lent−voltage
If +
these
are
eq ) · Ueq . substitute quantities(12)
eq . eq
dz
used0 instead0 of 0physical ones,
any
measurement
setup
can
be
0
Req
results of parameters
merging theof
rows
eq , Geq and
eq are thecoupling
used
to, L
determine
the C
equivalent
eq.
and columns:
17 and eq. 18.
N P
N
P
the
hey
(17)
(18)
R 0 eq =
R 0 ij
i=1 j =1
N2
,
(13)
5 Susceptibility
analysis with equivalent coupling paraN P
N
P
meters
L0 ij
L0 eq =
R0 ,L0 ,
G0 ,C 0
10
U2
Simplifiedsetup
setuptotodetermine
determinethe
the transfer
transfer parameters
parameters of
Fig.Fig.
5. 5.Simplified
ofaa
shielded
single
conductortransmission
transmissionline
line.
shielded
single
conductor
ling
anstute
paetry
U0
Z2
10
i=1 j =1
N2
,
(14)
The overview of the procedure (Fig. 6) shows that the first
N X
N
three0 stepsX
are completed
so far. The fourth step is to do the
G eq =
G0 ij ,
(15)
susceptibility
analysis
of
the
structure. This includes calcui=1 j =1
lating the distributed
sources of the inner system using the
N X
N
X
0
0
0
0
quantities
Z
and
Y
C eq = T,eq C ij ,T,eq . Common single conductor trans(16)
mission line
can then be used to calculate the interi=1theory
j =1
ferences Ueq , Ieq at the end of the lines (Tesche et al., 1997).
After
that, eq. 9 and eq. 10 can be used to derive the interwww.adv-radio-sci.net/10/221/2012/
ferences U , I, of the original transmission line.
10
calculation of the
p.u.l. parameters of the
substitute transmission line
Determination of the
transfer parameters of the
substitute transmission line
Vulnerability analysis of the
substitute transmission line
Derivation of the
vulnerability of the
original transmission line
0
,L0eq ,
Req
0
0
Geq ,Ceq
Z’T / Ω/m
(8)
Z1
one equivalent conductor
(7)
calculation of the
p.u.l. parameters of the
original transmission line
10
10
0
ZT,eq
,
0
YT,eq
10
Ueq ,Ieq
10
U,I
Fig. 6.
Fig.
6. Summary
Summary of
of the
theconcept
conceptofofalternative
alternativecoupling
couplingparameters.
parameters
with ij : indexation of the matrix elements of the prevailing
6original
Conversion
of the parameters.
equivalent coupling parameters to
per unit length
the
conventional
ones
Equations (11) and (12) describe a shielded equivalent
transmission line with only one inner conductor. This substitute
system iscoupling
solely defined
by its
equivalent per
unit
The
equivalent
parameters
in combination
with
the
length
parameters.
The
corresponding
and
exact
substitute
equivalent single conductor notation of the inner system can
geometry
known
nor the
of interest
for this application.
be
used tois neither
determine
easily
interferences
on the inner
This
substitute
cable
can
now
be
used
conductors. Nevertheless, it is possible to
to calculate
transformthe
the
equivalent transfer
admittance.descripThey
equivalent
transferimpedance
parametersand
to transfer
the conventional
are defined
as: be useful, if there is already a multi conduction.
This might
Ueq line implementation for the inner system.
tor0 transmission
Z T,eq =
,
(17)
IS
The relationship
can be drawn from the definitions of the
parameters:
IeqUsing the assumption of approximately identiY 0 T,eq
=
. on the inner conductors, the conventional
(18)
cal
interferences
US
transfer impedance can be written as:
This means that coupling to the inner system of a multiconductor transmission
can be described by scalar couline
U1 
pling parameters. The consequence is that one can now
 IS 
use the simple procedure
line to describe the
 fora coaxial
 ...  U1 N ×1
multiconductor coupling
ZT0 =  process.
·1
.
(19)
=

 IS
U 
N
Measurement setup ItoS determine the coupling
parameters
Using the definition of the equivalent transfer impedance
0 application of the concept of equivalent coupling paThe
Z
T,eq (Eq. 17) and eq. 9, the demanded link can be estabrameters
lished: is independent from the measurement procedure
used to determine the parameters. A well-known approach
is the “triaxial setup“. This measurement setup can be sim0
N ×1
ZT0 =5.ZT,eq
.
The· 1measurement
principle (20)
is,
plified as shown in Fig.
to excite the shield via the source U0 and to measure the
4
Under the same conditions, the conventional transfer admitAdv. Radio Sci., 10, 221–225, 2012
10
Fig. 7
LiYC
tance
For th
The s
7 S
The p
shield
ture w
equiv
age U
scalar
admit
If t
bility
at the
lated
the m
Schetelig
et al.: of
Simplified
susceptibility
of shielded
transmission
lines
fied susceptibility
shielded
multiconductor
transmission
lines multiconductor
224 analysis
B. Schetelig
et al.:analysis
Simplified
susceptibility
analysis of
shielded multiconductor
transmission lines5
04
1
10
10
L0 ,
C0
10
calculation with ZT and YT of cable 2
3
10
−1
10
measurement cable 1, wire A
measurement cable 1, wire B
measurement cable 1, wire C
measurement cable 1, wire D
0
10
2
10
−2
T
U2 / V
10 1
10
Y’T // Ω/m
S/m
Z’
L0eq ,
0
Ceq
0
10−3
−1
10
10
−1
10
,
−2
10
−4
10
−2
10
−5
−3
10
10 4 4
10
10
Ieq
−3
6
6
10
10
10
8
f f/ /Hz
Hz
8
10
10
4
6
10
8
10
10
f / Hz
Fig. 9. Calculation
Fig.8.7. Absolute
Absolutevalue
valueof
of
the
equivalent
transfer
impedance
Fig.
Calculation of
of the
the induced
inducedvoltage
voltageatatan
aninner
innerconductor
conductorofof
Fig.
ofthe
theequivalent
equivalenttransfer
transferadmittance
impedancefor
foraa
aa shielded
cable.
Schetelig
et al.: Simplified
susceptibility
analysis
transmission
lines
5
LiYCYmulticonductor
transmission
line
(4
inner
wires).
LiYCY
transmission
line
wires)
shielded LiYCY
LiYCY
cable
LiYCY
multiconductor
transmission
line(4
(4inner
inner
wires)of shielded multiconductor
ith the
em can
e inner
m the
escriponducem.
of the
identintional
(19)
edance
estab-
(20)
admit-
Y’T / S/m
ters to
1
10
istance
applied
is: on a susceptibility analysis of the ”cable piece no.
1”.
I 
−1
1
10
Because
of eq. 9, the values
of the induced voltages of the

U
Sas
original cable are the same
the
voltage of the sub  induced
I1 N ×1
further
 calculation
0
...
stitute. −2That means,
no
has
YT =   =
·1
. to be carried
(21)
10
  US
out.
I 
N
−3
10
US
8ForConclusions
the equivalent transfer admittance, the following is valid:
−4
N
The 10
procedure, which is presented
here, is a considerable
P
In coupling parameters of
simplification when calculating
the
I
I1
eq
n=1
0
YT,eq
=
=
= Nlines
·
.with stranded
(22)
industrial
multiconductor
transmission
−5
US
US
US
10 4
6
8
wires. The
10 deviations are quite
10 acceptable. The10application
f / Hz
toThe
thesecond
susceptibility
analysis
of the
cables under test results in
demanded
link then
becomes:
a reduced complexity, since only
a
simple single conductor
0
Yconsidered.
T,eq
0 be
N ×1
transmission
line
has
to
procedure
Fig. 8. Absolute
value
of
the
equivalent
admittance
for
aa
· 1 transfer
.The total
(23)
Absolute valueYof
the equivalent
transfer
admittance
for
T =
N
isLiYCY
summarized
in
fig.
6.
LiYCY
multiconductor
transmission
line
(4
inner
wires).
multiconductor transmission line (4 inner wires)
voltage
U2on
between
the inner analysis
conductor
the shield.
References
7is
Sample
measurements
applied
a susceptibility
of and
the ”cable
pieceUsno.
ing
transmission
line
theory,
these
quantities
can
be
used to
1”.
Andrieu,
G.:
et presented
application
d’une
méthode
de faiscalculate
theElaboration
transfer
impedance
and the
transfer
admittance
The
procedure
which
was
was
applied
Because
of eq.
9, the
values of thebefore
induced
voltages
ofon
thea
ceau
équivalent
pour
l’étude
des
couplages
électromagnetiques
(Tiedemann,
2001).
shielded
LiYCY
cable
with
four
inner
conductors.
The
strucoriginal
cable
are
the same as the induced
voltage of the subsur
réseaux
câblages automobiles,
Ph.D.
thesis, Université
de
There
is
nodedifference
the
calculation
of the
transture
wasThat
condensed
tonoa between
shieldedcalculation
single
conductor
line.
The
stitute.
means,
further
has to be
carried
Lille,
2006.
fer parameters of
a physical
coaxial lineand
andthe
of the
substitute
equivalent
unit
length parameters
induced
voltout. C. E.:per
Baum,
Electromagnetic
topology:
A
formal approach
the
transmission
line
fromThis
above.
The
equivalent
cable
isto deage
U
were
derived.
data
then
is
used
to
calculate
the
eq and design of complex electronic systems, Tech. rep., Inanalysis
scribed
in
the
usual
notation
of
a
single
conductor
with
the
scalars
equivalent
transfer
impedance
(Fig.
7) 0and transfer
teraction
Notes - Note
400, http://socorro.eece.unm.edu/summa/
0
0
0
per unit length
admittance
(Fig.parameters
8).1980. R eq , L eq , G eq , C eq and the
8 notes/In/0400.pdf,
Conclusions
equivalent
Ueq
. If these equivalent
substitute
quantiIf theseP.voltage
coupling
parameters
are applied
to the susceptiDegauque,
and Hamelin,
J.: Electromagnetic
Compatibility,
Oxties
are
used
instead
of
physical
ones,
any
measurement
setup
fordprocedure,
Science
bility
analysisPublications,
ofwhich
the cable
under test,
theisinduced
voltage
The
is 1993.
presented
here,
a considerable
can be
used
to determine
the equivalent coupling
parameters
Jung,
Einfluß
von Schirminhomogenitäten
bei
Mehrleiterkabeln
at
theL.:inner
conductor
of the equivalent
cable
can be calcusimplification
when
calculating
the coupling
parameters
of
ofauf
Eqs.
(17)
and
(18).
die
komplexe
Ph.D.lines
thesis,
Technische
lated
(Fig.
9). TheTransferimpedanz,
transfer
parameters
are
calculated
with
industrial
multiconductor
transmission
with
stranded
Universität
Hamburg-Harburg,
2003.
the
measured
data of the
no. 2“.The
This
data then
wires.
The deviations
are”cable
quite piece
acceptable.
application
to
the
susceptibility
analysis
of
the
cables
under
test
results in
Adv. Radio Sci., 10, 221–225, 2012
a reduced complexity, since only a simple single conductor
transmission line has to be considered. The total procedure
is summarized in fig. 6.
10
5Kasdepke,
Susceptibility
analysis with
equivalent coupling
T.: Simulation
von Störströmen
auf geschirmten
calculation
with ZTthesis,
and YT of
cable 2
mehradrigen
Ph.D.
Technische
Universität
parametersKabeln,
Hamburg-Harburg,
1997.
measurement
cable 1, wire A
0
measurement
cable
1, wire
Kley,overview
T.:
Optimierte
Kabelschirme
Theorie
und Messung,
10
The
of the
procedure
(Fig.
6)Bshows
that the Ph.D.
first
measurement
cable 1,
wire C
thesis,
Eidgenössische
Technische
Hochschule
Zürich,
three steps are completed so far. The fourth step
is to1991.
do the
measurement
cableKarlsson,
1, wire D
Tesche, F. M., analysis
Ianoz,
M.
EMC Analysis
susceptibility
of V.,
the and
structure. ThisT.:includes
calcuMethods
and
Computational
Methods,
Wiley,
1997.
−1
lating10the
distributed sources of the inner system using the
Tiedemann, R.: Schirmwirkung koaxialer Geflechtsstrukturen,
quantities Z 0 T,eq and Y 0 T,eq . Common single conductor
Ph.D. thesis, Technische Universität Dresden, 2001.
transmission line theory can then be used to calculate the inVance, E. F.: Coupling to shielded cables, John Wiley & Sons, Inc.,
terferences
Ueq , Ieq at the end of the lines (Tesche et al.,
−2
1978.
10 After that, Eqs. (9) and (10) can be used to derive the
1997).
Weitze, F.:
Ausbreitung eingekoppelter Störströme auf
interferences
U , Leitungen,
I , of the original
line. Univerabgeschirmte
Ph.D. transmission
thesis, Technische
U2 / V
0
meters
sität Hamburg-Harburg, 1991.
−3
10
6
4
6
10
10
8
10
Conversion of the equivalent
coupling parameters
f / Hz
to the conventional ones
Fig. equivalent
9. Calculation
of the induced
voltageinatcombination
an inner conductor
The
coupling
parameters
with of
a shielded
LiYCY
cableconductor notation of the inner systhe
equivalent
single
tem can be used to determine easily the interferences on
the inner conductors. Nevertheless, it is possible to transKasdepke, T.: Simulation von Störströmen auf geschirmten
form the equivalent transfer parameters to the conventional
mehradrigen Kabeln, Ph.D. thesis, Technische Universität
description.
This might
Hamburg-Harburg,
1997. be useful, if there is already a
multi
conductor
transmission
lineTheorie
implementation
for the
Kley, T.: Optimierte Kabelschirme
und Messung,
Ph.D.
inner
system.
thesis, Eidgenössische Technische Hochschule Zürich, 1991.
The relationship
canM.
be V.,
drawn
the definitions
the
Tesche,
F. M., Ianoz,
and from
Karlsson,
T.: EMC of
Analysis
parameters:
Using
the
assumption
of
approximately
identiMethods and Computational Methods, Wiley, 1997.
Tiedemann,
R.: on
Schirmwirkung
koaxialer the
Geflechtsstrukturen,
cal
interferences
the inner conductors,
conventional
Ph.D.impedance
thesis, Technische
transfer
can be Universität
written as: Dresden, 2001.
Vance, E. F.: Coupling to shielded cables, John Wiley & Sons, Inc.,
 U1 
1978.
I
Weitze,
U1 N×1 eingekoppelter Störströme auf
 SF.: Ausbreitung
Z 0 Tabgeschirmte
=  ...  = Leitungen,
·1
. Ph.D. thesis, Technische Univer(19)
IS
UN
sität Hamburg-Harburg,
1991.
I
S
www.adv-radio-sci.net/10/221/2012/
B. Schetelig et al.: Simplified susceptibility analysis of shielded multiconductor transmission lines
Using the definition of the equivalent transfer impedance
0
ZT,eq
(Eqs. 17 and 9), the demanded link can be established:
N×1
0
0
Z T = Z T,eq · 1
.
(20)
Under the same conditions, the conventional transfer admittance is:
 I1 
U
 S  I1 N×1
·1
.
(21)
Y 0 T =  ...  =
US
IN
U
S
For the equivalent transfer admittance, the following is valid:
N
P
In
Ieq n=1
I1
0
YT,eq
=
=
=N·
.
US
US
US
(22)
The second demanded link then becomes:
Y 0T =
7
Y 0 T,eq
N
· 1N ×1 .
(23)
Sample measurements
The procedure which was presented before was applied on a
shielded LiYCY cable with four inner conductors. The structure was condensed to a shielded single conductor line. The
equivalent per unit length parameters and the induced voltage Ueq were derived. This data then is used to calculate the
scalars equivalent transfer impedance (Fig. 7) and transfer
admittance (Fig. 8).
If these coupling parameters are applied to the susceptibility analysis of the cable under test, the induced voltage
at the inner conductor of the equivalent cable can be calculated (Fig. 9). The transfer parameters are calculated with the
measured data of the ”cable piece no. 2“. This data then is
applied on a susceptibility analysis of the ”cable piece no. 1”.
Because of Eq. (9), the values of the induced voltages
of the original cable are the same as the induced voltage
of the substitute. That means, no further calculation has to
be carried out.
www.adv-radio-sci.net/10/221/2012/
8
225
Conclusions
The procedure, which is presented here, is a considerable
simplification when calculating the coupling parameters of
industrial multiconductor transmission lines with stranded
wires. The deviations are quite acceptable. The application
to the susceptibility analysis of the cables under test results in
a reduced complexity, since only a simple single conductor
transmission line has to be considered. The total procedure
is summarized in Fig. 6.
References
Andrieu, G.: Elaboration et application d’une méthode de faisceau équivalent pour l’étude des couplages électromagnetiques
sur réseaux de câblages automobiles, Ph.D. thesis, Université de
Lille, 2006.
Baum, C. E.: Electromagnetic topology: A formal approach to the
analysis and design of complex electronic systems, Tech. Rep.,
Interaction Notes – Note 400, available at: http://www.ece.unm.
edu/summa/notes/In/0400.pdf, 1980.
Degauque, P. and Hamelin, J.: Electromagnetic Compatibility, Oxford Science Publications, 1993.
Jung, L.: Einfluß von Schirminhomogenitäten bei Mehrleiterkabeln
auf die komplexe Transferimpedanz, Ph.D. thesis, Technische
Universität Hamburg-Harburg, 2003.
Kasdepke, T.: Simulation von Störströmen auf geschirmten
mehradrigen Kabeln, Ph.D. thesis, Technische Universität
Hamburg-Harburg, 1997.
Kley, T.: Optimierte Kabelschirme Theorie und Messung, Ph.D.
thesis, Eidgenössische Technische Hochschule Zürich, 1991.
Tesche, F. M., Ianoz, M. V., and Karlsson, T.: EMC Analysis Methods and Computational Methods, Wiley, 1997.
Tiedemann, R.: Schirmwirkung koaxialer Geflechtsstrukturen,
Ph.D. thesis, Technische Universität Dresden, 2001.
Vance, E. F.: Coupling to shielded cables, John Wiley & Sons, Inc.,
1978.
Weitze, F.:
Ausbreitung eingekoppelter Störströme auf
abgeschirmte Leitungen, Ph.D. thesis, Technische Universität Hamburg-Harburg, 1991.
Adv. Radio Sci., 10, 221–225, 2012