Design and Evaluation of a Measurement
Procedure to obtain the Electric Permittivity and
the Magnetic Permeability
Dise
˜
no y evaluaci
´
on del procedimiento de medici
´
on para obtener la permitividad el
´
ectrica y la
permeabilidad magn
´
etica
Rittner I. L.
∗
, Fano W. G.
†
∗
Hamburg University of Technology
Am Schwarzenberg-Campus 1 - 21073 - Hamburg - Germany
i.rittner@tuhh.de
†
Universidad de Buenos Aires, Departamento de Electr
´
onica, Laboratorio de Radiaci
´
on Electromagn
´
etica
Av. Paseo Col
´
on 850 - C1063ACV - Buenos Aires - Argentina
gustavo.gf2005@gmail.com
Abstract—An implementation of the transmission/reflection
line method is presented to determine the intrinsic
electromagnetic properties of unknown materials with a
relative permeability not equal to one. A low-cost, easy
to manufacture sample holder is realised using a coaxial
transmission line with N-type female connectors. The reflection
and transmission coefficients are measured to simultaneously
extract the electric permittivity and magnetic permeability.
The classic Nicolson-Ross-Weir (NRW) extraction technique
in frequency domain is presented such that the equations can
be easily implemented with open-source tools such as GNU
Octave. The shift in phase reference planes and connector
calibration measurements are performed such that a low-
cost vector network analyser (VNA) without de-embedding
function can be used. The measurement procedure is valid
from 2 MHz to 6 GHz. The article also comments on the
sample preparation especially the thickness of the material
slab.
Keywords: transmission line measurements; parameter
extraction; calibration.
Resumen— Se presenta una implementaci
´
on del m
´
etodo
de l
´
ınea de transmisi
´
on / reflexi
´
on para determinar las
propiedades electromagn
´
eticas intr
´
ınsecas de materiales
desconocidos con una permeabilidad relativa distinta de uno.
Se construy
´
o un portamuestras de bajo costo y f
´
acil de fabricar
utilizando una l
´
ınea de transmisi
´
on coaxial con conectores
hembra tipo N. Los coeficientes de reflexi
´
on y transmisi
´
on
se miden para extraer simult
´
aneamente la permitividad
el
´
ectrica y la permeabilidad magn
´
etica. La t
´
ecnica cl
´
asica de
extracci
´
on de Nicolson-Ross-Weir (NRW) en el dominio de la
frecuencia se presenta de tal manera que las ecuaciones se
pueden implementar f
´
acilmente con herramientas de c
´
odigo
libre como GNU Octave. El desplazamiento en los planos
de referencia de fase y las mediciones de calibraci
´
on del
conector se realizan de tal manera que se puede usar un
analizador vectorial de (VNA) de bajo costo sin las funciones
incorporadas. El procedimiento de medici
´
on es v
´
alido desde
2 MHz hasta 6 GHz. El art
´
ıculo tambi
´
en comenta sobre la
preparaci
´
on de la muestra especialmente el espesor del bloque
de material.
Palabras clave: mediciones en l
´
ınea de transmisi
´
on; ex-
tracci
´
on de par
´
ametros; calibraci
´
on.
I. INTRODUCTION
In a technology dominated world there is a constant
growth and an increasing diversity of applications to pro-
cess and transmit microwave signals. Therefore it becomes
increasingly interesting to design materials with desired
behaviours in the presence of electromagnetic fields. Dif-
ferent techniques for the electromagnetic characterization
of unknown materials have been developed. All of them
have limitations based on the frequency range, the sample
material to be used, the parameters to be obtained etc. In
the following a brief overview of the procedures explained
in detail in [1], [2] and [3] is given.
Resonant methods e.g. applying a cavity resonator can be
used to measure the permittivity and permeability of very
small samples of the material under test (MUT). The basic
idea is that the material properties can be inferred from
changes of the resonant frequency and the quality factor
of the resonator. The results are obtained with very high
accuracy but are limited to a narrow frequency band. With
nonresonant methods the electromagnetic properties of ma-
terials are deduced from their impedance and wave velocities
within the material. Reflection methods such as the open-
ended coaxial probe method collect the reflected energy after
it is directed toward a material. Therefore only the electric
permittivity
R
may be determined. The open-ended coaxial
probe method is a non-destructive testing method usually
used for biological specimens as the sample does not need to
be machined to fit a sample holder. The electric permittivity
R
may also be determined with the parallel plate capacitor
method as used in [4]. Here the measurement device is con-
structed as a parallel plate capacitor, which is filled with the
MUT alternating the capacitance. Voltage measurements are
used to then calculate
R
. Transmission/reflection methods
evaluate the reflected and transmitted energy directed toward
a material. All types of transmission lines such as free-space,
hollow metallic or dielectric waveguides may be used to
carry the electromagnetic wave impinging on the material
surface. The free-space measurement method can yield both
Revista elektron, Vol. 2, No. 1, pp. 30-38 (2018)
ISSN 2525-0159
30
Recibido: 05/05/18; Aceptado: 13/06/18
parameters
R
and µ
R
. It consists of an arrangement with
two directive antennas facing each other connected to a
VNA. The material is placed as a thin plate in the middle
of the set-up. This method can be used to determine the
electromagnetic parameters for a wide frequency range and
allows high frequency measurements. The free-space mea-
surements inherit the great disadvantage that the MUT has to
be large and thin. With coaxial lines or metallic waveguides
as sample holders smaller sample sizes may be used, which
results in a destructive technique because the sample must
be prepared to fit the sample holder. These transmission-
line techniques allow a broadband characterization of the
complex electromagnetic parameters
R
and µ
R
of unknown
materials.
The measurement technique presented is designed for
materials eventually being used for radio frequency ab-
sorbers. Therefore a wide-band method for the simultaneous
determination of
R
and µ
R
for solid materials is needed. A
two-port transmission-line technique is implemented using
a coaxial transmission-line as sample holder. This coaxial
transmission-line allows a transversal electromagnetic wave
(TEM) to propagate such that the energy density of the TEM
is tangential to the sample interface therefore increasing the
measurement accuracy. The idea of the general procedure is
to measure the forward- and back-scattered energy of elec-
tromagnetic waves in the frequency range 2 MHz −6 GHz.
The measurements are realised with a VNA at its connector
reference planes. The data is then post-processed yielding
the relationship of the forward- and back-scattered energy
S
11
and S
21
at the sample interfaces, from which the
intrinsic electromagnetic parameters
R
and µ
R
may be
calculated as functions of frequency. To extract the complex
parameters
R
=
0
R
− j
00
R
and µ
R
= µ
0
R
− jµ
00
R
from
the measured scattering parameters S
11
and S
21
the classic
NRW method is used [5], [6]. Later developed techniques
such as the NIST iterative or the new non-iterative extraction
or conversion technique [3] assume µ
R
= 1. This makes
these techniques not suitable for the proposed application
as the synthesized absorbing materials may have µ
R
6= 1.
The equations derived in the original papers are extend like
in [7] and implemented in the post-processing with the open-
source software GNU Octave, which is similar to MATLAB.
II. THEORY
First of all, a theoretical model of the wave propagation
inside the sample holder has to be derived in order to under-
stand the derivation of the Nicolson-Ross-Weir Method [5],
[6]. The original notation as in the first paper by Nicolson
and Ross [5] is used. The influence of the connectors and
the length of the coaxial line on the wave propagation is
modelled as a shift in reference plane explained in section
II-B.
A. TEM Wave Propagation in the Presence of a Sample
Figure 1 shows a sample slab of thickness d positioned
in the middle of the sample holder designed as an air-filled
coaxial transmission line of length L
L
. The screw holes for
the connectors are indicated by the four small circles on
each side.
In the presence of a current flow and a potential difference
between the inner and outer conductor the electric and the
thickness d
length L
L
r
1
r
2
diameters
Fig. 1: Model of a sample holder as a coaxial TEM trans-
mission line.
magnetic field are perpendicular over the entire coaxial
transmission line. This enables the propagation of a TEM
wave, which is represented by a voltage wave as indicated
in figure 2. Under TEM conditions the forward-travelling
voltage wave V
A
inc
incides perpendicularly on the sample
interface.
Z
0
Γ
−Γ
Z
A
A’
V
A
inc
R
, µ
R
Fig. 2: Schematic of an air-to-sample interface A-A’ for a
semi-infinite sample inside a coaxial TEM transmission line.
At the air-to-sample interface denoted as A-A’ in figure
2 a discontinuity occurs, when the characteristic impedance
changes from Z
0
to Z. For the case of a semi-infinite sample
part of the electromagnetic wave is reflected, while the
other part is transmitted and propagates within the MUT.
From field analysis of transmission lines [8] follows that
the characteristic impedance depends on the factor
q
0
R
µ
0
µ
R
,
where
0
and µ
0
are the electric and magnetic constant and
Z
0
=
q
µ
0
0
= 377 Ω, yielding
Z =
r
µ
R
R
Z
0
. (1)
The propagation at the interface of a semi-infinite sample
can be described by the voltage reflection coefficient
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Γ =
Z − Z
0
Z + Z
0
=
q
µ
R
R
− 1
q
µ
R
R
+ 1
. (2)
and the transmission coefficient 1+Γ [8]. Note that in the
case of a wave propagating through the material and inciting
on an interface with air its propagation is described by the
reflection and transmission coefficients −Γ and 1 − Γ.
Rewriting equation 2 yields the quotient
µ
R
R
as function
c
1
of Γ:
µ
R
R
=
(1 + Γ)
2
(1 − Γ)
2
= c
1
(Γ) (3)
At this point the propagation model has to be extended
to include the wave propagation in the presence of a sample
slab of finite thickness d as depicted in figure 3.
Z
0
Γ
Z
0
−Γ
Z
Γ
A
V
A
inc
A’
B’
B
z
1 2
d
R
, µ
R
Fig. 3: Schematic of the two air-sample-interfaces when a
sample of finite thickness d is placed in the middle of a
coaxial TEM transmission line.
In this case, the amplitude and phase change due to the
propagation within the sample before the wave incides on
the second interface has to be considered. In general, for a
forward travelling wave in +x direction the wave propagation
is accounted for by the factor
z(x) = e
−γx
, (4)
where γ = α + jβ = jk
c
is the frequency-dependent,
complex propagation constant with wave number k
c
. The
wavenumber may be represented as a function of the product
√
µ
R
R
[9]:
k
c
= ω
√
µ =
ω
c
√
µ
R
R
, (5)
where c = (
√
0
µ
0
)
−1
is the speed of light and ω the angular
frequency.
The amplitude change and phase factor for a propagation
distance of x = d result in:
z = e
−j
ω
c
√
µ
R
R
d
(6)
In the following this is denoted as the propagation factor
z. With the propagtation factor z the wave propagation
within the sample can be described independent of the
direction of the travelling wave as indicated in figures 3
and 4. Rewriting equation 6 enables the description of the
product µ
R
R
as function of z:
µ
R
R
= −(
c
ωd
ln
1
z
)
2
= c
2
(z) (7)
From c
1
(Γ) and c
2
(z) the electromagnetic parameters
R
and µ
R
may easily be determined. At this point Γ and z
are needed as functions of the measured quantities S
0
11
and
S
0
21
in order to calculate the permittivity and permeability
of the material from the measurements being realised. The
extension of the calibration surface from ports 1’ and 2’
to ports 1 and 2 is explained in section II-B. Therefore,
first of all the relation between the scattering parameters
S
11
and S
21
at ports 1 and 2 and the complex permittivity
R
and permeability µ
R
is derived in the following. The
equations presented in [10] allow to rewrite equations 3 and
7 as functions of the cut-off wavelength λ
c
for a waveguide
system as sample holder.
z
+ x
A
Z
0
Γ
V
A
inc
A’
−Γ
Z
B’
B
Z
0
Γ
V
A
ref
V
B
ref
Fig. 4: Schematic of the incident signal V
A
inc
, reflected
signal V
A
ref
and transmitted signal V
B
ref
with multiple
reflections present inside the sample.
Figure 4 shows how a voltage wave, which incides at
the interface A-A’, propagates in the presence of a finite
material sample. A complete bounce diagram indicating the
time relations may be found in [11]. The incident voltage
wave V
A
inc
is both reflected and transmitted first at the air-
to-sample A-A’ interface, later at the sample-to-air interface
B’-B and then at the sample-to-air interface A’-A and so on.
Therefore multiple reflections occur within the MUT. The
reflected voltage waves at reference planes A and B, V
A
ref
and V
B
ref
, consist of an infinite number of contributions
with decreasing amplitude the more reflections within the
sample have occurred. This propagation process may also
be described by a signal flow graph. Figure 5 shows the
signal flow graph presented in [5], from which equations 8
and 9 can be inferred.
The signal flow graph allows to relate the incident wave at
reference plane A to the reflected waves at reference planes
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(1 − Γ)
A’
(1 + Γ)z
V
A
inc
V
A
ref
V
B
ref
(1 − Γ)
Γ
−zΓ
+
A
B’
−zΓ
Fig. 5: Signal flow graph connecting the reference planes A,
A’ and B’.
A and B, from which the scattering parameters S
11
and
S
21
may be determined as functions of z and Γ. The graph
in figure 5 interrelates the reflections and transmissions of
V
A
inc
at the air-to-sample interface with reference plane A,
the sample-to-air interface with reference plane B’ and the
sample-to-air interface with reference plane A’. At A the
impinging wave is partly reflected, which is represented by
the factor Γ. The transmitted part also propagates through
the finite sample of thickness d, which is described by the
factor (1 + Γ)z. At B’ the reflection coefficient becomes
−Γ as indicated in figure 3 resulting in a factor of 1 −Γ for
the transmitted part. The reflected part at this sample-to-air
interface again propagates through the sample represented
by a total factor of −zΓ. When a fraction of V
A
inc
incides on
the sample-to-air interface A’ (1 −Γ) is transmitted and the
reflected part again propagates through the sample, which
is described by the factor −zΓ. Multiple reflection occur
between B’ and A’.
Γ
(1 − Γ)
A’
(1 + Γ)z
V
A
inc
V
A
ref
+
A
B’
−zΓ
b)
Γ
(1 − Γ)
A’
(1 + Γ)z
V
A
inc
V
A
ref
−zΓ
+
A
B’
−zΓ
a)
z
2
Γ
2
Fig. 6: Modified signal flow graph a) and simplification b)
for the determination of the scattering parameter S
11
.
Figure 6 shows the relation of V
A
inc
and V
A
ref
. From the
version in figure 6 b), which is simplified according to the
rules in [8], S
11
can be determined as
S
11
=
V
A
ref
V
A
inc
=
(1 + Γ)z(−zΓ)(1 − Γ)
(1 − Γ
2
z
2
)
+ Γ
=
(1 − z
2
)Γ
(1 − Γ
2
z
2
)
.
(8)
(1 + Γ)z
V
A
inc
V
B
ref
(1 − Γ)
A
B’
z
2
Γ
2
Fig. 7: Signal flow graph for the determination of the
scattering parameter S
21
.
The modified signal flow graph in figure 7 shows the
relation of V
A
inc
and V
B
ref
, which allows the determination
of S
21
according to [8] as
S
21
=
V
B
ref
V
A
inc
=
(1 + Γ)z(1 − Γ)
(1 − Γ
2
z
2
)
.
(9)
Nicolson and Ross in [5] combined equations 8 and 9
such that the system of equations decouples [7] resulting
in explicit equations for the reflection coefficient Γ and the
propagation factor z as functions of S
11
and S
21
:
Γ =
1 − (S
21
+ S
11
)(S
21
− S
11
)
2S
11
±
s
(
1 − (S
21
+ S
11
)(S
21
− S
11
)
2S
11
)
2
− 1,
(10)
with the passivity condition of the material such that |Γ|≤
1 and
z =
S
21
+ S
11
− Γ
1 − (S
21
+ S
11
)Γ
. (11)
Equations 3 and 7 can then be written as functions of
the scattering parameters enabling the determination of the
intrinsic electromagnetic parameters as
R
=
s
c
2
(S
11
, S
21
)
c
1
(S
11
, S
21
)
(12)
and
µ
R
=
p
c
1
(S
11
, S
21
)c
2
(S
11
, S
21
). (13)
B. Shift of the Reference Planes to the Sample Surfaces
In order to calculate S
11
and S
21
from the measured
quantities S
0
11
and S
0
21
the VNA reference planes at port 1’
and 2’ have to be shifted to the sample surface with ports 1
and 2. Figure 8 illustrates the different calibration reference
planes. For the coaxial sample holder classic transmission
line theory can be applied to shift the phase reference
planes. The sample holder may be modelled by the two line
model shown in figure 8, where the two lines represent the
Revista elektron, Vol. 2, No. 1, pp. 30-38 (2018)
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inner and outer conductor of the coaxial transmission line.
Since the sample holder may be approximated as a lossless
transmission line, the shift for a forward travelling wave is
done with the correcting phase factor
e
−jβL
= e
−j
2π
λ
0
L
, (14)
where λ
0
˜=
c
0
f
is the wavelength in air and β =
2π
λ
0
.
L denotes the equivalent electrical length of an air-filled
transmission line, by which the phase reference plane is
extended. Note that equation 14 is a simplification of the
general propagation factor represented in equation 4. For
a symmetrically placed sample the scattering parameters at
ports 1 and 2 become
S
11
= S
0
11
e
j2βL
= S
0
11
e
j
4π
λ
0
L
(15)
and
S
21
= S
0
21
e
j2βL
= S
0
21
e
j
4π
λ
0
L
[8]. (16)
The scattering parameters relate the amplitudes of two
waves, a forward and a backward travelling one. That
is why a factor of 2 occurs when the reference plane
is shifted by the length of L. As shown in figure 8 the
forward travelling wave propagates over a transmission line
of length L until it incides on the sample surface. When
the electrical length of the connector L
C
is included L
results in L =
1
2
(L
L
− d) + L
C
. [7] provides equations for
any reference plane position, therefore applicable when the
sample is not exactly placed in the middle.
Z
0
L
C
L
C
L
2
=
1
2
(L
L
− d) + L
C
VNA
L
1
=
1
2
(L
L
− d)
plane of symmetry
1’
1
2 2’
d
L
L
Z
0
Z
0
Z
Z
0
Fig. 8: Transmission line model with different reference
planes corresponding to three phase factor arguments.
The results in figure 9 demonstrate how important it is
to include L
C
in the reference plane shift. The blue, dotted
line shows the results for L = 0 meaning that no phase
adjustment of the measured scattering parameters has been
1 2 3 4 5 6
f [GHz]
0
5
10
15
20
25
30
35
40
0
L1
L2
Electrical length
Fig. 9: Calculation results for the real part of
R
for L = 0,
L = L
1
and L = L
2
corresponding to the three reference
planes of figure 8.
done. The results improve when L = L
1
=
1
2
(L
L
− d) is
chosen as indicated by the yellow curve. When addition-
ally the wave propagation in the Teflon-filled connector is
calibrated, which results in L = L
2
=
1
2
(L
L
− d) + L
C
,
the permittivity approaches the known value
R
= 2, 1 [12]
independent of frequency.
The Teflon-filled connector is modelled as a transmission
line with characteristic impedance Z
0
for the determination
of the equivalent electrical length L
C
. As electromagnetic
waves propagate differently in air and Teflon, L
C
is not
equal to the physical length of the real Teflon-filled con-
nector. The employed transmission line model assumes a
wave propagation in air like in the coaxial sample holder,
in order to apply the phase correction with equations 15
and 16. L
C
may then be determined with the help of
an impedance transformation. The empty sample holder is
terminated and as the voltage amplitude for a mismatched
line varies with position, L
C
can be inferred from input
impedance calculations.
Z
0
L
C
L
Z
L
Z(z
0
)
z
0
Fig. 10: Transmission line model for the connector calibra-
tion measurements.
Figure 10 shows a transmission line model of the termi-
Revista elektron, Vol. 2, No. 1, pp. 30-38 (2018)
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nated sample holder with one connector screwed on. For
this lossless transmission line terminated with Z
L
the input
impedance as function of z
0
becomes
Z(z
0
) = Z
0
Z
L
+ jZ
0
tan(βz
0
)
Z
0
+ jZ
L
tan(βz
0
)
[8]. (17)
This result is referred to as the transmission line
impedance equation. If the transmission line is short-
circuited with Z
L
= 0 equation 17 simplifies to
Z(z
0
) = jZ
0
tan(βz
0
) [8]. (18)
At z
0
= L
L
+ L
C
the impedance of a short-circuited
line can be calculated from frequency dependent reflection
coefficient measurements using equation 2. The absolute
value of Z(L
L
+ L
C
) converges to infinity, if βz
0
=
β(L
L
+ L
C
) = n
π
2
with n = 1, 2, 3, ... Therefore for
the first frequency f
1
, for which |Z(f
1
)|→ ∞, holds that
β(L
L
+ L
C
) =
2π
c
0
f
1
(L
L
+ L
C
) =
π
2
and
L
C
=
c
0
4f
1
− L
L
. (19)
For the coaxial air-filled line the electrical length coin-
cides with the physical length L
L
.
The VNA is first calibrated at the connector calibration
planes. In case the VNA used does not have a de-embedding
function as mentioned in [3], the extension of the reference
planes and the connector and line calibration have to be
implemented in the post-processing. Therefore the calibra-
tion procedure explained above provides a simple method
to significantly upgrade a low-cost VNA.
III. DESIGN
A. Sample Holder
Fig. 11: Sample holder with the inner and outer cylindrical
conductor, a Teflon sample with d = 8 mm and two
connectors.
The sample holder is designed as a coaxial transmission
line of length L
L
= 50.4 mm. The conductors of the coaxial
line are made out of a solid bronze cylinder. Bronze is
chosen because it provides high conductivity and may be
machined easily. The geometry is designed to match the
50 Ω Amphenol N type female connectors. Figure 11 shows
the parts of the coaxial sample holder. The diameters r
1
and
r
2
of the two cylindrical conductors indicated in figure 1 are
determined according to:
Z
0
=
r
µ
0
µ
R
0
R
1
2π
ln
r
2
r
1
[8]. (20)
For a coaxial transmission line with air as dielectric the
geometry dependent characteristic impedance Z
0
yields
Z
0
≈ 60 ln
r
1
r
2
Ω. (21)
r
1
= 5, 4 mm and r
2
= 12, 5 mm are chosen, which yields
Z
0
≈ 50, 3 Ω, which has been confirmed by impedance
measurements. Since the diameters of the 50 Ω Teflon-filled
connectors do not coincide with the chosen values for r
1
and
r
2
, adjustments have to be done. Figure 12 shows the bronze
ring and the solder used to adjust the connectors. Note that
inaccuracies in the manufacturing process of the ring could
produce discontinuities at the connector transitions.
Fig. 12: Adjusted connector and original reference connec-
tor.
The connector calibration measurements as explained in
section II-B are realised with the equipment shown in figure
13. The sample holder was manufactured in the mechanic’s
workshop of the Facultad de Ingenier
´
ıa de la Universidad
de Buenos Aires with a tolerance of ±0.05 mm.
B. Sample Preparation
To evaluate the measurement procedure different Teflon
slab thickness as shown in figure 14 have been investigated.
Also the results of three slabs of an unknown nickel ferrite
(FNI) have been calculated. The samples have been manu-
factured with a tolerance of ±0.1 mm. Figures 17b and 17a
show that the results of the measurement procedure used
depend on the sample slab thickness. However, this should
not be the case, as the parameters are intrinsic. Therefore,
criteria for suitable slab thicknesses need to be established
investigating the behaviour of the known Teflon slabs. [3]
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Fig. 13: Utensils used to measure the transmission line in
short-circuit.
Fig. 14: Teflon slabs of thicknesses d = 4 mm, d = 8 mm
and d = 16 mm.
proposes that d ≈
λ
g
4
should be chosen, where λ
g
is the
wavelength in the sample. For f
max
= 6 GHz
λ
g
min
=
v
f
=
c
0
f
max
√
R
(22)
can be determined. For Teflon
λ
g
min
4
= 8, 6 mm and
the sample with d = 8 mm should yield the best results.
The error analysis of figures 17b and 17a proves that this
is indeed the case. Therefore also for FNI a sample with
d ≈
λ
g
min
4
should be chosen. To determine λ
g
of unknown
samples such as FNI a posteriori knowledge of
R
has to
be used. The results of figure 15b justify to set
R
= 3 −5
resulting
λ
g
min
4
= 7, 2 −5, 6 mm. The imaginary part of the
permittivity can be determined to be negligible. It follows
that the FNI sample with d = 6, 5 yields the most accurate
results.
For both MUT, Teflon and FNI, a divergence phenomenon
occurs for the largest sample thickness d. As explained in [7]
the classic NRW method diverges, when the sample thick-
ness approaches multiples of
λ
g
2
. Using equation 22 results
in
λ
g
2
≈ 17, 3 mm for Teflon and
λ
g
2
≈ 14, 4 − 12, 5mm
for FNI, when a permittivity
R
≈ 3 − 4 is a posteriori
assumed. This explains the divergence of the Teflon sample
with d = 16 mm and the FNI sample with d = 15 mm. [7]
proposes an iterative procedure based on the NRW method
to determine the complex permittivity independent on d. It
is only suitable for nonmagnetic materials as µ
R
is set to 1.
For the symmetric placement of the samples the in-
strument shown in figure 16 has been manufactured. The
1 2 3 4 5 6
f [GHz]
0
1
2
3
4
5
16 mm
8 mm
4 mm
(a) Teflon.
1 2 3 4 5 6
f [GHz]
0
2
4
6
8
10
15,0 mm
6,5 mm
3,6 mm
(b) FNI.
Fig. 15: Real part of
R
for different slab thicknesses.
cylindrical element is hollow and has a diameter in between
the diameters r
1
and r
2
.
Fig. 16: Instrument to place the samples.
IV. RESULTS
The accuracy of the proposed measurement procedure is
assessed with the help of a well characterized material.
Therefore, the GNU Octave [13] calculation results of a
known Teflon sample are analysed. As discussed in section
III-B slab thickness d = 8 mm is used. The complex per-
mittivity and permeability are calculated from the scattering
parameters measured with an Agilent Field Fox N9923A
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1 2 3 4 5 6
f [GHz]
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
8 mm
4 mm
(a) Relative error of |
R
|.
1 2 3 4 5 6
f [GHz]
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
8 mm
4 mm
(b) Relative error of |µ
R
|.
Fig. 17: Relative error for Teflon with d = 4 mm and d =
8 mm.
vector network analyser, which works up to 6 GHz and is
calibrated by a standard two port full calibration with the
provided T tool. The preliminary results are presented in
the frequency range 1 − 6 GHz. Figures 18a and 18b show
the most accurate results for the appropriate slab thickness.
Figures 17a and 17b illustrates that indeed d = 8 mm
yields the smallest deviation from the known parameters
for the dielectric. It can be seen that the accuracy of the
measurement procedure can be characterized by a relative
error below ±0.17 for
R
and below ±0.33 for µ
R
.
The results presented in figures 18a and 18b show small
and overall fluctuations, which may partly be attributed to
the NRW algorithm itself. Previously published results of the
complex permittivity and permeability determined according
to the NRW method show similar fluctuations [7]. Further
measurement uncertainties may be due to sources of possible
impedance discontinuities for example at the transition to the
connectors as mentioned in section III-A. Future designs of
the sample holder focus on a smooth transition between the
coaxial line and the connectors. Air gaps between the sample
and the coaxial line are mentioned in [3] as a possible
reason for measurement deviations. Also the calibration of
the transmission line itself may be revised.
1 2 3 4 5 6
f [GHz]
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Real part
Imaginary part
(a)
R
.
1 2 3 4 5 6
f [GHz]
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Real part
Imaginary part
(b) µ
R
.
Fig. 18: Results for Teflon with d = 8 mm.
V. CONCLUSIONS
In the article a signal flow graph approach to derive the
classic NRW equations is explained in detail. Equations are
presented to shift the phase reference plane of the measured
scattering parameters in order to calculate the complex
permittivity and permeability. An approach to determine
the preferable sample slab thickness is derived. Finally
measurements of the known dielectric Teflon show that
the classic NRW method may be easily implemented up
to 6 GHz with a coaxial transmission line made of bronze
with N type female connectors. The line has been designed
to yield a characteristic impedance of Z
0
= 50 Ω. An
GNU Octave script has been written to compute and plot
the results of the electric permittivity and the magnetic
permeability. This post-processing also includes connector
and line calibrations. The results of
R
and µ
R
as func-
tions of frequency show small fluctuations attributed to the
NRW algorithm itself, which have been reported by other
researchers. This problem will be discussed in future papers
of the authors.
Future works also consist of the design and implemen-
tation of a new sample holder with the aim to reduce
discontinuities at the transition from the coaxial transmission
line to the N type female connectors. An error analysis will
also be provided. Furthermore an alternative post-processing
Revista elektron, Vol. 2, No. 1, pp. 30-38 (2018)
ISSN 2525-0159
37
http://elektron.fi.uba.ar
algorithm as presented in [10] may be implemented to yield
accurate results independent of the slab thickness d.
ACKNOWLEDGEMENTS
The authors acknowledge the fruitful discussions with
Silvia Jacobo and Carlos Herme, and the help of Eriel
Fernandez Galvan and his team manufacturing and adjusting
the sample holders. The work reported in this article was
realised by the kind collaboration and assistance of Carlos
Herme in preparing and measuring the samples. Finally the
authors would like to especially thank Ramiro Alonso and
Valentino Trainotti for their constant support and motivation.
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