Scattering and Transfer Matrix of Screening Effectiveness - modeling of the triaxial setup

The scattering and transfer matrix of screening effectiveness,
modeling of the triaxial setup
Thomas Hähner
Nexans
Draveil, France
[phone], [e-mail]
Abstract
The method described in the literature to model the triaxial setup to
measure the screening attenuation in the case of different coupling
sections is based on the principle of (voltage) wave propagation and
(multiple) reflections. This approach leads to cumbersome
formulae. The following article describes a more versatile way
based on the theory of cascaded multi-ports and the introduction of
the scattering and transfer matrix for the electromagnetic coupling
through the screen. In that way also the impact of impedance
variations in the device under test can be investigated. Furthermore
a method is described to extend the frequency range for transfer
impedance and screening attenuation measurements.
Keywords: transfer impedance; screening attenuation; screening
effectiveness, scattering matrix, transfer matrix, cascaded multi-
ports, Mason’s gain formula, triaxial set-up, IEC 62153-4 series
1. Introduction
The screening effectiveness of screened cables, connectors and
cable assemblies is described by the (surface) transfer impedance
[1][2] and screening attenuation [1][3].
The relation between the transfer impedance and the screening
attenuation of the screen, for a uniform device under test (DUT) is
well described in the literature [4][5][6]. At IWCS 2017 the first
time a paper was presented dealing with non uniform devices like
cable assemblies or connecting hardware [7]. That paper presented
formulae which allow to take into account variations of the
impedance (and velocity) in the outer circuit (cable surrounding).
However the given formulae are limited to 4 segments and exclude
impedance variations in the inner circuit (DUT).
A new, more versatile and elegant approach is based on the theory
of cascaded multi-ports and the use of the scattering matrix S and
transmission matrix T. It allows to evaluate the impact of variations
of the impedance and velocity in the inner and outer circuit for any
number of segments.
Today the triaxial method (see Figure 1) is the preferred method to
measure the transfer impedance [2] and screening attenuation [3].
Other methods are the line injection method [9] for transfer
impedance and the absorbing clamp [10] or injection clamp [11]
method for screening attenuation.
The maximum frequency up to which the transfer impedance can be
measured depends on the load conditions of the test configuration
and the test length [2]. In general the frequency is limited to about
100MHz. The IEC standard 62153-4-16 [8] describes a method to
extrapolate the test results to higher frequencies for transfer
impedance or to lower frequencies for screening attenuation. But the
method is only applicable in the case of a matched inner circuit
(device under test). However transfer impedance measurements are
often made with a mismatched inner circuit. A new approach
applicable to any load conditions is presented.
Note that the symbols used in the following text are described in
paragraph 7.
Figure 1: triaxial setup
2. Scattering matrix S of screening
attenuation
2.1 Coupling matrix
The coupling between two lines (the inner and the outer circuit) can
be depicted as a four port coupling device (see Figure 2). Ports 1-3
represent the inner circuit (device under test) and Ports 2-4 represent
the outer circuit (surrounding).
Figure 2: 4-port coupling device
For passive and reciprocal devices like cables, cable assemblies or
connecting hardware the forward scattering parameters Sxy equal the
reverse scattering parameters Syx .
The coupling between the inner and outer circuit is loose therefore
the forward and reverse scattering parameters of the inner circuit
(S31=S13) and outer circuit (S42=S24) equal the (wave) attenuation of
the inner respectively outer circuit:
l
IC eSSS 
 1
1331
(1)
l
OC eSSS 
 2
2442
(2)
For matched circuits the reflection coefficients are zero:
0
44332211
SSSS (3)
In the case of matched conditions in the inner and outer circuit [12],
[4] describe the formulae for the voltages coupled from the inner to
the outer circuit at the near end and the far end. But scattering
parameters are ratios of (square root) of power waves. Therefore the
formulae for the voltage ratios need to be extended by introducing
the square root ratio of the impedances in the inner and outer circuit:
coupling length
a
pp
. 2m
terminatin
g
resistor R = Z
cable sheath cable screen
tube
U2
si
g
nal
g
enerato
r
cable under test
input voltage U111
calibrated receive
r
or network anal
y
zer
U
1
Port1
Port2
Port3
Port4
S
31
S
42
S
21
S
41

12 1
112 2
11
2
l
TF
N
Z
ZZ
Ce
Z
Z
(4)
12 21
112 2
11
2
ll
FT
F
Z
ZZ
Cee
Z
Z
 (5)
43342112
SSSSC
N
 (6)
23324114 SSSSCF (7)
The scattering matrix for the coupled four port device is then
expressed by:
0
0
0
0
NOCF
NFIC
OCFN
FICN
C
CSC
CCS
SCC
CSC
S (8)
It is important to note that the reference impedance of this scattering
matrix is not the reference impedance of the vector network
analyzer, in general 50 Ohm. But the reference impedance of the
inner circuit (port 1 to 3) equals the impedance of the inner circuit.
The reference impedance of the outer circuit (port 2 to 4) equals the
impedance of the outer circuit. Hence the Sxx of the coupling matrix
equal zero (see formula (8))
2.2 Junction matrix
The coupling matrix is valid for matched circuits. To take into
account impedance variations between different cascaded
segments or the load conditions of the triaxial setup1 a junction
matrix is introduced.
The formulae given in [13] (page 61 ff) for a two port can be
extended to a 4 port device and leads at the near end to:
NOCNOC
NICNIC
NOCNOC
NICNIC
JN
rr
rr
rr
rr
S
,
2
,
,
2
,
2
,,
2
,,
010
001
100
010
(9)
And at the far end to
FOCFOC
FICFIC
FOCNOC
FICFIC
JF
rr
rr
rr
rr
S
,
2
,
,
2
,
2
,,
2
,,
010
001
100
010
(10)
Where the reflection coefficients r are obtained for a wave
propagating from the near end (generator side) to the far end
(receiver side):
11
,
11
n
IC n
n
Z
Z
r
Z
Z
 11
,
11
f
IC f
f
Z
Z
r
Z
Z
(11)
1 In the triaxial method the outer circuit is short circuited at the
near end and mismatched at the far end.
22
,
22
n
OC n
n
Z
Z
r
Z
Z
 22
,
22
f
OC f
f
Z
Z
r
Z
Z
(12)
2.3 Transfer matrix T
The conversion from the scattering matrix S to the transfer matrix
T (also known as cascade matrix or transfer scattering matrix) is
well known for two ports. Recent work extended the T-Matrix of
2-ports to multi-ports [14][15]. It is interesting to note that already
in 1961 Wilhelm Klein described the conversion between the
scattering matrix and transfer matrix for M x M multi-ports [16]
(page 90 ff). In the following the notations described by Wilhelm
Klein are used. Here the conversions for a 4-port are shown:
1
,,
1
,
1
,,,
1
,,,
IIIIIIIIII
IIIIIIIIIIIIIIIII
SSS
SSSSSS
T (13)
IIIIIIIIIII
IIIIIII
III
IIIIIIIII
TTT
TTTTTT
S
,
1
,
1
,
,
1
,
,
,
1
,, (14)
Where the S-Matrix and T-Matrix are split into 4 sub-matrices:
IIIIIII
IIIII
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
S
,,
,,
4443
3433
4241
3231
2423
1413
2221
1211
(15)
IIIIIII
IIIII
TT
TT
TT
TT
TT
TT
TT
TT
TT
TT
T
,,
,,
4443
3433
4241
3231
2423
1413
2221
1211
(16)
2.4 Triaxial set-up as a cascade of near end
junction, coupling and far end junction
The resulting transfer matrix T of the triaxial set-up and
consequently resulting scattering matrix is obtained from a simple
matrix multiplication:
JFCJNtriax TTTT
(17)
For the determination of the reflection factors [formulae (11)(12)]
of the junction matrices [formulae (9)(10)] one has to consider
that the outer circuit has a short circuit at the near end (Z2,n=0) and
a mismatch at the far end (Z2,f=Z0). For the inner circuit (device
under test) the load at the near is in general Z1,n=Z0 and at the far
end depending on the used method Z1,f=0 or Z1,f=Z1 or Z1,f=Z0.
It is to note, that the short circuit at the near end of the outer
circuit, i.e. rOC,N=1, leads to a sub-matrix SII,I of the near end
junction matrix which is singular. Thus the matrix inverse does
not exist and the conversion to a T-matrix is not possible (formula
(13). Therefore a value close to zero, e.g. 10-9 Ohm, shall be taken
for Z2,n.
Figure 3 shows simulation results for the configuration indicated
in Table 1:
Table 1: parameters for simulation of triaxial set-up
Z0=50Ω ; L=2m
Z1= Z0 ; Z1,n= Z0 ; Z1,f= Z0 ; εr1=2,25 ; α1=0
Z2=150Ω ; Z2,n=0Ω ; Z2,f=Z0 ; εr2=1,0 ; α2=0
RT=13,6mΩ/m ; MT=0,93nH/m ; CT=0,04pF/m

Figure 3: cascaded multi-port vs. classical formulae
The results of the resulting coupling Striax,41=Striax,14 are in perfect
concordance with the results obtained from the classical formulae
described in [5][6].
2.5 Cascading of different segments
A non uniform device under test like a cable assembly, with
connectors at the cable ends, or a cable junction, with connectors
between two cable segments, may have variations of the
impedance (and propagation constant) in the outer and inner
circuit. In that case the resulting coupling can be obtained by
cascading the different uniform segments. The impedance
variations between the different segments are taken into account
by the junction matrix. The resulting transfer matrix T of the
cascaded segments and consequently resulting scattering matrix S
is obtained from a simple matrix multiplication; see Figure 4 and
formula (18).
Figure 4: cascaded multiports
112 2 1
,, , ,, ,
... nn n
R
JN Cs Jss Cs Js s Cs JF
TTT T T T T T
(18)
Figure 5 shows simulation results for the triaxial configuration
indicated in Table 2:
Table 2: parameters for simulation in triaxial set-up
reference Z0=50Ω
segments 4 segments
# 1 2 3 4
Z2 150Ω 100Ω 50Ω 150Ω
L 1m 0,25m 0,5m 0,25m
inner circuit Z1= Z0 ; εr1=2,25 ; α1=0 ; equal for all segments
outer circuit εr2=1,0 ; α2=0 ; equal for all segments
screen RT=13,6mΩ/m ; MT=0,93nH/m ; CT=0,04pF/m ;
equal for all segments
The results of the resulting coupling SR,41=SR,14 are in perfect
concordance with the results obtained from the approach
presented at IWCS 2017 [7]. I.e. the results and conclusions
presented there are confirmed and valid!
Figure 5: cascaded multiport approach vs. wave
propagation and multiple reflections
3. Extrapolation of measurement results
to higher frequencies for transfer
impedance and lower frequencies for
screening attenuation
The triaxial set-up can be used to measure both the surface
transfer impedance [2] and the screening attenuation [3]. The
transfer impedance is in general measured with a coupling length
of 0,5 m resulting in an upper frequency limit of around 100 MHz
whereas the screening attenuation is in general measured with a
coupling length of 2 m to 3 m resulting in a upper frequency limit
for the transfer impedance of around 10 MHz and a lower
frequency limit for the screening attenuation of around 100 MHz
(see also [1] clause 8 and 9).
Figure 6 shows the grey zone between electrically short
(measurement range for the transfer impedance) and electrically
long (measurement range for the screening attenuation). In the
example, the transfer impedance can be measured up to around 30
MHz using a coupling length of 50 cm and the screening
attenuation can be measured starting from 150 MHz using a
coupling length of 200 cm.
Figure 6: simulation of the scattering parameter S21
(left hand scale) and the transfer impedance (right hand
scale) for a single braid screen
The IEC standard [8] describes a method to extrapolate the test
results of transfer impedance to higher frequencies and the test
results of screening attenuation to lower frequencies. But the
method is limited to a matched inner circuit, whereas transfer
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0
0 500 1000 1500 2000 2500 3000
couplingindB
frequencyinMHz
cascadedmulti‐port
classicalformulae
Segment
1
Segment
2
Segment
n
Segment
n‐1
Junction
Nearend
Junction
Farend
Junction
S
1,2
Junction
S
n‐1,n
…
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0
0 500 1000 1500 2000 2500 3000
couplingindB
frequencyinMHz
IWCS2018
IWCS2017
1
10
100
1000
10000
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0
0,01 0,1 1 10 100 1000
mOhm/m
dB
MHz
S21‐ 200cm
S21‐ 50cm
Zt greyzone

impedance measurements are often made with a mismatched inner
circuit (device under test)
The first idea was to use the described approach of cascaded
multi-ports to extract (de-embed) the coupling matrix (TC
respectively SC) from the measured resulting S-Matrix (SM) and
the corresponding resulting T-Matrix (TM):
11   JFMJNC TTTT (19)
However in the triaxial setup the scattering parameters can only
be measured from two ports, i.e. port 1 and port 4 (see Figure 2).
Although with the short circuit at the near end of the outer circuit
(port 2) we know S21=S12=S23=S32=S24=S42=0 and S22=1, we have
the problem that in the case of an inner circuit which is matched at
the far end (S33=0) we do not know S34=S43. In the case of an
inner circuit with a short circuit at the far end we know
S31=S13=S43=S34=0 and S33=1 and the measured matrix could be
written as:
11 14
41 44
00
0100
0010
00
M
SS
S
SS
(20)
But the sub-matrix SII,I is singular and the matrix inverse does not
exist and the conversion to a T-matrix is not possible. Simulations
revealed that it is not sufficient to replace the zeros by a value
close to zero.
Therefore a new approach is based on the signal flow graph of the
triaxial setup and Mason’s gain formula.
3.1 Signal flow graph of the triaxial setup and
Mason’s gain formula
The signal flow graph of the triaxial setup is illustrated in Figure 7.
The middle section corresponds to the coupling between both
circuits and the sections at the left and the right correspond to the
load conditions of the measurement setup. For screened devices the
coupling is loose (weak) between the inner circuit (DUT) and the
outer circuit (environment). Therefore it is reasonable, for the sake
of simplification, to neglect the feedback from the disturbed circuit
to the disturbing circuit.
Figure 7: signal flow graph of the triaxial setup
To obtain the resulting scattering parameter S41 one has to find in
the signal flow graph all paths (P) from port 1 to port 4 and all loops
(L).
It exist 4 paths to get from port 1 to port 4:
31 41 42
31 21 44 42 42
31 31 11 43 42
31 31 11 23 44 42 42
1
2
3
4
JN C JF
JN C JN C JF
JN C JF C JF
J
NCJFCJNCJF
PSSS
PSSSSS
PSSSSS
P
SSSSSSS
(21)
The signal flow graph contains 6 loops:
33 11
33 11
31 11 13 33
1
2
3
JN C
CJF
CJFCJN
LSS
LSS
LSSSS
44 22
44 22
42 22 24 44
4
5
6
JN C
CJF
CJFCJN
LSS
LSS
L
SS SS
(22)
Applying Mason’s formula and taking into account that in SC, the
coupling matrix, S11= S22= S33= S44=0 leads to the resulting S41 of
the triaxial setup:
41
1234
36 36
1
R
P
PPP
S
L
LLL
 (23)
After some further conversions and taking into account the short
circuit at the near end of the outer circuit one obtains:
41
12 12
2
12
12
22
12
11
2
22
11 2 11 2
11
11
1
Rnf
LL
L
Ff N f
L
LL
nf f nf f
Srr
Cre Ce re
rre r e rrr e
(24)
Figure 8 shows the comparison between the results obtained from
the approach of cascaded multiports (with the conversion between S
and T matrix and multiplication of the T matrices) with the results
obtained using Mason’s formula. The configuration of the triaxial
setup is shown in Table 3 with a mismatched inner circuit having a
short circuit at the far end.
Table 3: parameters for simulation of triaxial set-up
Z0=50Ω ; L=2m
Z1= 75Ω ; Z1,n= Z0 ; Z1,f= 0 ; εr1=2,25 ; α1=0
Z2=150Ω ; Z2,n=0Ω ; Z2,f=Z0 ; εr2=1,0 ; α2=0
RT=13,6mΩ/m ; MT=0,93nH/m ; CT=0,04pF/m
Figure 8: cascaded multi-port vs. Mason’s formula
The results of both approaches are in perfect concordance. Thus
formula (24) obtained from Mason’s gain formula can be used to
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0
0 500 1000 1500 2000 2500 3000
resultingS41indB
frequencyinMHz
Mason'sformula
cascadedmultiport

convert the measured scattering parameter SR41 to the transfer
impedance ZT.
3.2 Conversion between resulting (measured)
scattering parameter SR41 and transfer
impedance ZT
In most cases the capacitive coupling through the cable screen can
be neglected. This is in particular true for screens composed of
metal foils, dense single braids or double braids. In this case
formula (24) can be rearranged to obtain the transfer impedance ZT:
41
12
12
12 12
12 12
2
12
022
12
2
22
11 2 11 2
11
12 12
2
11
1
11
11
F
TR
Z
nf
L
LL
nf f nf f
LL
LL
L
ff
ZZ
ZS
rr
rre r e rrr e
ee
ere re
(25)
For low frequencies (L<<1) formula (25) can be rewritten as:
41
21 1
12
2
f
nf
TR
nf
Z
ZZ
ZS LZZ  (26)
Figure 9 shows the comparison between the results of transfer
impedance obtained from formula (25) and the commonly used
conversion formula between the measured forward transfer
scattering parameter and transfer impedance as described in IEC
62153-4-3 [2]. The configuration of the triaxial setup is shown in
Table 3 with a mismatched inner circuit having a short circuit at the
far end.
Table 4: parameters for simulation of triaxial set-up
Z0=50Ω ; L=0,5m
Z1= 75Ω ; Z1,n= Z0 ; Z1,f= 0 ; εr1=2,25 ; α1=0
Z2=150Ω ; Z2,n=0Ω ; Z2,f=Z0 ; εr2=1,0 ; α2=0
RT=13,6mΩ/m ; MT=0,93nH/m ; CT=0
The transfer impedance obtained from formula (25) corresponds, as
expected, to the transfer impedance obtained from the screen
parameter (RT, MT, and CT). But using the formula described in IEC
62153-4-3 to convert the measured forward transfer scattering
parameter to transfer impedance limits the upper frequency for the
transfer impedance to about 30MHz.
Figure 9: comparison of formulae for conversion
between forward transfer scattering parameter and
transfer impedance
In formula (25) it is necessary to know the propagation constant of
the inner and outer circuit. A method to obtain them from
measurement is described in IEC 62153-4-16 [8].
4. Measurement versus simulation results
Figure 10 shows the results of the conversion from the measured
scattering parameter SR41 to the transfer impedance for a RG59 type
cable. The DUT was matched at the near end (generator side) and
far end. The conversion was done using a relative dielectric
permittivity of the inner circuit (DUT) of 2,3 (PE dielectric) and 1,1
of the outer circuit (PVC jacket in tube). The blue line and the green
line show the transfer impedance when measured with a 2m
coupling length respectively 0,5m and using the standard
conversion formula as described in [2] and formula (26). One can
see that the upper frequency limit for the validity of the transfer
impedance is 10MHz for the 2m length and 40MHz for the 0,5m
length. The red line shows the conversion to transfer impedance of
the 2m length using formula (25). Here the upper frequency limit
for the validity of the transfer impedance is 200MHz. The observed
peaks at higher frequencies are due to the capacitive coupling
impedance ZF which is not exactly zero (single braid with “low”
coverage) and due to uncertainties in the dielectric permittivity used
in the conversion formula (25).
Figure 10: Example for the conversion of measured
scattering parameter SR41 to the transfer impedance of a
RG59 type cable; inner circuit matched; assuming
relative dielectric permittivity of 2,3 and 1,1 for the
inner, respectively outer circuit
Figure 11 shows the test results of transfer impedance for a single
braided coaxial cable with 50Ω impedance. The DUT is short
circuited at the far end (Z1f=0). The results are shown for 3
different coupling lengths 35cm, 100cm and 200cm. The
conversion from the measured scattering parameter SR41 to the
transfer impedance was done according to IEC 62153-4-3 [2]. The
results show that the cut-off frequency for the transfer impedance
is decreasing with increasing length from 60MHz for 35cm to
10MHz for 200cm.
The IEC standard 62153-4-16 [8] to extend the frequency range is
only applicable for a matched DUT and cannot be used for a DUT
with a short circuit at the far end. The newly introduced equation
(25) however can be used for any load conditions. This is shown
for the above cable in Figure 12. The results for the coupling
length of 100cm and 200cm have been extrapolated using
equation (25). The conversion was done using a relative dielectric
permittivity of the inner circuit (DUT) of 2,3 (PE dielectric) and
1,0 of the outer circuit (cable jacket was removed). The cut-off
0
1
10
100
1000
10000
100000
0,01 0,1 1 10 100 1000
transferimpedanceinmilli‐Ohm/m
frequencyinMHz
ZTofscreen
ZTfromS41
ZTfromIEC62153‐4‐3