3D-printers dielectric materials characterization at microwave frequencies


ACTA IMEKO 
ISSN: 2221-870X 
September 2020, Volume 9, Number 3, 26 - 32 

 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 26 

Characterisation of dielectric 3D-printing materials at 
microwave frequencies 

Andrea Alimenti1, Kostiantyn Torokhtii1, Nicola Pompeo1, Emanuele Piuzzi2, Enrico Silva1 

1 Dipartimento di Ingegneria, Università degli Studi Roma Tre, 00146 Roma, Italy 
2 Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni, Sapienza Università di Roma, Roma, Italy 

 

 

Section: RESEARCH PAPER  

Keywords: microwave characterisation; loss tangent; 3D-printer materials; dielectric resonator  

Citation: Andrea Alimenti, Kostiantyn Torokhtii, Nicola Pompeo, Emanuele Piuzzi, Enrico Silva, 3D-printers dielectric materials characterization at microwave 
frequencies, Acta IMEKO, vol. 9, no. 3, article 5, September 2020, identifier: IMEKO-ACTA-09 (2020)-03-05 

Editor: Jan Saliga, Technical University of Košice, Slovakia 

Received January 17, 2020; In final form April 14, 2020; Published September 2020 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Andrea Alimenti, e-mail: andrea.alimenti@uniroma3.it  

 

1. INTRODUCTION 

In recent years, the fast development and improvement of 
3D-printing technologies has had a significant impact on many 
areas of human activity [1]-[5]. Printed materials are now utilised 
in high-frequency applications such as telecommunication 
technologies and microwave frequencies [6]-[12]. Thus, a reliable 
and handy electromagnetic (e.m.) characterisation of these 
materials is increasingly required [13]. 

In this work, the microwave characterisation of plastic 3D-
printing materials is performed using a resonant perturbative 
technique. The physical quantity under investigation is 

permittivity , which is defined as the quantity (in a more general 
sense, the tensor) that describes the proportionality between the 

electric displacement vector 𝑫 and the electric field strength 
vector 𝑬 in a medium, 𝑫 = 0𝜺 ⋅ 𝑬, with 0 being the vacuum 
permittivity. In the case of the materials studied here, after one 
accounts for the isotropic, homogeneous and linear limits, then 

𝜺 = ̃  is a scalar quantity that does not depend on positioning. 
The scalar complex relative permittivity is defined as ̃ = ′ −
i ′′, where the real part ′ is a measure of the energy storage 
properties of the medium, while the imaginary part ′′  is related 

to the e.m. losses, and i = √−1. Since ̃ is a complex quantity, it 
is often represented on the complex plane, where the angle 𝛿 

between ̃ and the real axis is known as a loss angle. Thus, the 
ratio ′′ ′⁄ = tan𝛿 is called the loss tangent.  

In this work, a microwave (∼ 12.9 GHz) measurement 
method based on a resonant technique is proposed. The 
dielectric loaded resonator (DR) presented here was designed 
considering the e.m. properties of the dielectric materials usually 
used in 3D-printing and exploiting the possibility, given by 3D-
printers, of shaping the samples in almost every desired shape. 
In the paper, an in-depth uncertainty analysis shows how the 
designed resonator is optimized to provide the best accuracy for 
this kind of material [14]. However, it is clear that its use is not 
limited only to 3D-printing plastics but to every kind of material 
electromagnetically and mechanically similar to them.  

Moreover, to improve the repeatability of the measurement, 

the DR was specially designed to measure ̃ by placing the 
dielectric sample on one of its flat bases, removing the need to 
disassemble and reassemble the whole structure for each 
measurement and thus reducing the uncertainties involved. DRs 
are well known for their high sensitivity [15] but also for poor 
measurement repeatability [16], [17], particularly in relation to 
resonant frequency. Thus, a closed structure is especially useful 
for characterising samples as it reduces the systematic errors 
inevitably introduced by each mounting procedure. Moreover, 
since it is not necessary to reassemble the resonator for each 
measurement, the measurement time is reduced.  

ABSTRACT 
3D-printer materials are becoming increasingly appealing, especially for high frequency applications. As such, the electromagnetic 
characterisation of these materials is an important step in evaluating their applicability for new technological devices. We present a 
measurement method for complex permittivity evaluation based on a dielectric loaded resonator (DR). Comparing the quality factor Q 
of the DR with a disk-shaped sample placed on a DR base, with Q obtained when the sample is substituted with an air gap, allows a 
reliable determination of the loss tangent. 

mailto:andrea.alimenti@uniroma3.it


 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 27 

A part of the volume of the resonating structure is substituted 
with the dielectric material under study. Comparison of the 
changes induced by the insertion of the sample on the unloaded 

quality factor 𝑄 and the resonant frequency 𝑓0 can be used to 
evaluate the electric/magnetic properties of the sample. If the 
changes to the distribution of the e.m. field caused by the 
insertion of the sample are ‘small’, the resonant medium 
perturbation method [15] can be used.  

Dielectric printed materials are already used for high 
frequency applications, and some works have explored their 
dielectric permittivity. Of particular note is the result obtained in 
[18], where acrylonitrile butadiene styrene (ABS) doped with 

different quantities of BaTiO3 microparticles gave 2.6 <  
′ <

8.7 and 0.005 < tan 𝛿 < 0.027, thus opening up the possibility 
of engineering these materials for specific uses. The 
measurements were performed at 15 GHz with a split post 

dielectric resonator obtaining 𝑢(tan 𝛿)/ tan 𝛿 ∼ 0.4 %. The 
split post resonator is a very sensitive measurement instrument 
but does have some critical issues related to its assembly 
procedure [15]. The resonator presented in this work is tuned to 

a similar frequency (∼ 12.9 GHz) and has a somewhat reduced 
sensitivity compared with the split post resonator but a much 
improved ease of operation – a useful feature in view of the 
routine measurements required.  

In Sec. 2, the measurement method and system are presented. 
Then, in Sec. 3, a detailed uncertainties analysis is provided, while 
the results are shown in Sec. 4. Finally, conclusions are given in 
Sec. 5, comparing the obtained results to those given by a 
standard waveguide transmission/reflection method and to 
other relevant scientific works [13], [18], [19].  

2. DESCRIPTION OF THE METHOD 

A special configuration of a Hakki–Coleman dielectric loaded 
resonator [20] was designed to guarantee enhanced measurement 
repeatability at room temperature. The two physical quantities 
that characterise the response of the resonator are the unloaded 

quality factor 𝑄 and the resonant frequency 𝑓0. 𝑄 is defined as 
𝑄 = 𝜔0𝑊/𝑃, where 𝑊 is the energy stored into the resonator 
at the resonant angular frequency 𝜔0 = 2𝜋𝑓0 and 𝑃 is the power 
dissipated at the same frequency. Thus, as will be shown below, 
one can obtain information about the dielectric losses of the 

material under study (i.e. tan𝛿) from the 𝑄 measurement. 
𝑃 is the sum of all the power losses 𝑃 = 𝑃𝑆 + 𝑃Ω + 𝑃𝑉 , where 

the subscripts 𝑆, Ω, 𝑉 indicate the quantities related, respectively, 
to the sample, to the metal surfaces and to all the other dielectric 
materials inside the resonator volume. Hence: 

1

𝑄
=

𝑃𝑆 + 𝑃Ω + 𝑃𝑉
𝜔0𝑊

=
1

𝑄𝑆
+

1

𝑄Ω
+

1

𝑄𝑉
, (1) 

with: 

1

𝑄𝑆
=

∫ ε𝑆
′′ε0|𝐄|

2d𝑉
𝑉𝑆

2𝑊
= [

ε𝑆
′ ∫ ε0|𝐄|

2d𝑉
𝑉𝑆

2𝑊
]

ε𝑆
′′

ε𝑆
′ = 𝜂𝑆tan𝛿, (2) 

1

𝑄Ω
= ∑

∫ 𝑅𝑖|𝑯𝜏 |
2d𝑆

𝑆𝑖

2𝑊
𝑖

= ∑
𝑅𝑖
𝐺𝑖𝑖

 , (3) 

1

𝑄𝑉
=

∫ 𝑉
′′

0|𝑬|
2𝑑𝑉

𝑉𝑉

2𝑊
= 𝜂𝑉 tan 𝛿𝑉  , 

(4) 

where 𝑬 is the electric field and 𝑯𝜏  is the magnetic field 
tangential to the 𝑖-th metallic surface 𝑆𝑖  with surface resistance 
𝑅𝑖  and geometrical factor 𝐺𝑖 . 𝜂𝑆 and 𝜂𝑉  are the filling factors of 
the sample and of the dielectric elements inside the resonator, 
respectively. Thus: 

1

𝑄
= 𝜂𝑆 tan 𝛿𝑆 + ∑

𝑅𝑖
𝐺𝑖

+ 𝜂𝑉 tan 𝛿𝑉  

𝑖

. (5) 

Since 𝑊 and the field configuration depend on ′ of all the 
dielectric elements inside the resonator, both 𝜂 and 𝐺 are also 
functions of 𝑠

′ . Therefore, to evaluate tan 𝛿𝑠 of the dielectric 
sample placed in the resonator from Eq. (5), 𝑠

′  must be 

measured. To achieve this aim, one can exploit 𝑓0 of the 
resonator as a second measurand. However, it is known that the 

absolute value of 𝑓0 is strongly affected by many intrinsic (e.g. 
the electromagnetic properties of the elements inside the 
resonator) and extrinsic (e.g. temperature, pressure, humidity) 
factors, meaning it is very difficult to utilise it in practice. 

However, the variation Δ𝑓0/𝑓0,𝑟𝑒𝑓  of 𝑓0, with respect to the 

reference value 𝑓0,𝑟𝑒𝑓 , is much more reliable due to the change 

of one or more parameters [15]. In this case, Δ𝑓0, as caused by 
the insertion of the dielectric sample, is measured. Then, with 

electromagnetic simulations, Δ𝑓0 is evaluated as a function of 𝑆
′ . 

Thus, 𝑆
′  is determined as the value at which the simulated Δ𝑓0 

coincides with the measured one. Thus, 𝑆
′  is evaluated with the 

aid of e.m. simulations of the resonator. 

After 𝑆
′  is evaluated, the factors 𝜂 and 𝐺 in Eq. (5) can be 

analytically or numerically (with simulators) calculated. Following 

this, Eq. (5) can be inverted to obtain tan 𝛿𝑆 from the 
𝑄 measurements if all the 𝑅𝑖  and tan 𝛿𝑉  of the resonator are 
known from previous measurements or calibration procedures.  

It must be mentioned that the unloaded 𝑄 in Eq. (5) differs 
in principle from the measured 𝑄𝑙  because of the coupling of the 
resonator with the external lines. However, with very small 

coupling (i.e. 𝑃𝑒𝑥𝑡 /𝑃 < 0.01, with 𝑃𝑒𝑥𝑡  being the losses in the 
external transmission lines), as in our set up, one has 𝑄𝑙 ∼ 𝑄 and 
𝑢(𝑄) ∼ 𝑢(𝑄𝑙 ) [15]. 

The use of Eq. (5) can give unacceptably large uncertainties 
since, at microwave frequencies, the accuracy with which all the 
quantities in Eq. (5) are known is poor when compared to dc or 
low frequency measurements. In fact, in our case, 

𝑅 = 92 mΩ with 𝑢(𝑅) 𝑅⁄ ∼15 % and tan 𝛿𝑉 = 4 ⋅ 10
−5 with 

𝑢( tan 𝛿𝑉 )/ tan 𝛿𝑉 ∼ 50 %.  
In order to reduce the effects of these uncertainties on 

tan 𝛿𝑠, a perturbative approach is proposed here. The difference 
Δ(𝑄−1) between the measured quality factors 𝑄𝑆

−1 and 𝑄𝐴
−1, 

obtained with the sample into the resonator (subscript S) or with 

a gap of air in its place (subscript A), respectively, can be written 

as: 

Δ(𝑄−1) = 𝜂𝑆 tan 𝛿𝑆 + ∑ 𝑅𝑖 Δ(𝐺𝑖
−1) + Δ(𝜂𝑉 ) tan 𝛿𝑉 ,

𝑖

 (6) 

 

where it is clear that the smaller Δ(𝐺𝑖
−1) and Δ(𝜂𝑉 ) are, then the 

smaller their contributions to the uncertainties of 𝑅𝑖  and tan 𝛿𝑉 
are. 

2.1. Measurement system and procedure 

The resonator used for this study is depicted in Figure 1. The 

single sapphire crystal is a cylinder (height ℎ = (5.0 ± 0.1) mm, 



 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 28 

diameter ⌀ = (8.0 ± 0.1) mm). A K-type coaxial transmission 
line, ending with coupling loops, is used to excite and sense (in 
transmission mode) the e.m. field configuration in the resonator. 
The dielectric samples are supported by a brass mask with a 

central hole ⌀ = (13.00 ± 0.01) mm and closed with a brass 
cap in order to prevent energy radiation, as depicted in Figure 1. 

The DR is excited in the TE011 mode, and thus, the 𝑬 field is 
oriented parallel to the bases of the resonator. It is important to 

underline the orientation of the 𝑬 field because the layered 
deposition techniques typical of most of the 3D-printers can 
generate anisotropic effects on the e.m. properties of the printed 
samples. A uniaxial anisotropy factor of almost 7 % was 

measured for ′ at 40 GHz for polylactide (PLA) samples using 
the waveguide reflection method [19]. In the method presented 

here, the 𝑬 field is almost parallel to the deposition layers of the 
sample; thus, our results probe the direction along the layer 
deposition without significant mixing of the perpendicular 
component. 

The resonator transmission complex scattering parameter 

𝑆12 , from which 𝑄 and 𝑓0 are evaluated, is measured with an 
Anritsu 37269D Vector Network Analyzer (VNA) in the 
following way: 

- The VNA is calibrated using the Short Open Load 
Through (SOLT) method, and the 12-errors parameters 
are applied to the frequency range in which the 
measurements are performed;  

- The transmission scattering parameter 𝑆12(𝑓) is 
acquired with 1601 points evenly distributed in a 

frequency range width 7Δ𝑓−3𝑑𝐵 , where Δ𝑓−3𝑑𝐵  is the 
width of the resonance curve at half power. Each data 
point is averaged with 10 acquisitions to reduce the noise 
contribution;  

- The absolute value of the acquired points |𝑆12(𝑓)|, 

which have the uncertainty 𝑢(𝑆21(𝑓)), given by the 
VNA after the calibration [21], is fitted to the Fano 
resonance curve [22], [23]: 

|𝑆12(𝑓)| = |
𝑆12(𝑓0)

1 + 2𝑖𝑄
𝑓 − 𝑓0

𝑓0

+ 𝑆𝑐 | , (7) 

where the complex constant 𝑆𝑐  represents the cross-
coupling contribution. For each resonance curve, 𝑄 and 
𝑓0 are evaluated with their uncertainties 𝑢(𝑄), 𝑢(𝑓0). 
The uncertainties of the fitted parameters are obtained 
by standard statistical methods starting with the fitting 

residuals variance 𝜎𝑅
2 [24];  

- For each mounting, 10 resonance curves are acquired. 

Then, the mean values of 𝑄 and 𝑓0 are evaluated with 
their standard deviation: 𝑢(𝑄) 𝑄⁄ ∼ 0.05 % and 
𝑢(𝑓0 )/  𝑓0  ∼ 1 ppm; 

- For each sample, 5 mountings are performed, 
disassembling and resetting the sample in its position. 

Then, the mean value of 𝑄 and 𝑓0 with their standard 
deviation are evaluated. The final uncertainties 

𝑢(𝑄) 𝑄⁄ ∼ 1 % and 𝑢(𝑓0 )/𝑓0  ∼ 20 ppm exist mainly 
due to the repeatability of the assembly.  

3. UNCERTAINTIES ANALYSIS 

In this section, the behaviour of the measurement technique 
is explored for the whole sample parameter space in order to 
establish the best working condition and its boundaries as a 

function of 𝑠 ′, tan 𝛿𝑠 and sample thickness 𝑡.  
First, the sensitivity of the resonator to tan 𝛿𝑠 variations is 

analysed. The sensitivity is evaluated from Eq. (5) as: 

𝑐 =
𝜕𝑄

𝜕 tan 𝛿𝑆 
= −

𝜂𝑆
(𝜂𝑆 tan 𝛿𝑆 + 𝑙𝑟 )

2
= −𝜂𝑆 𝑄

2, (8) 

with 𝑙𝑟 = ∑
𝑅𝑖

𝐺𝑖
+𝑖 𝜂𝑉 𝑡𝑎𝑛 𝛿𝑉, which, as a first approximation, in 

this analysis is assumed to be independent from the sample 
properties; in the small perturbation limit, the changes in the e.m. 
field configuration due to the sample are small and practically 
negligible, meaning the conduction/volume losses given by the 
resonator components do not change appreciably. In our case, 

𝑙𝑟 ∼ 5000 = 𝑄𝐴
−1. 

In Figure 2, |𝑐(tan 𝛿𝑠, 𝜂𝑠 )| is reported for 
10−5 < tan 𝛿𝑠 <    10

0 and 𝜂𝑠 = {10
−4, 10−3, 10−2, 10−1}. 

One can notice a different log |𝑐| slope 𝑚 at high tan 𝛿𝑠 values 
(𝑚 = − 2) and at low tan 𝛿𝑠 (𝑚 = 0) in the log–log plot (Figure 
2). The 𝑚 = −2 behaviour is due to the losses inside the 
dielectric samples when the following hold: 𝜂𝑠 𝑡𝑎𝑛 𝛿𝑠 ≫ 𝑙𝑟 , and, 
from Eq. (8), 𝑐 → − 𝜂𝑠

−1(tan 𝛿𝑠)
−2. Conversely, when 

𝜂𝑠 tan 𝛿𝑠 ≪ 𝑙𝑟 , 𝑐 → − 𝜂𝑠 𝑙𝑟
−2

; as such, 𝑐 is no longer dependent 
on tan 𝛿𝑠. As expected, the bigger 𝜂𝑠, the higher |𝑐|, as long as 
the sample losses are small. Thus, for low tan 𝛿𝑠 samples, a 
higher 𝜂𝑠 is preferable, while at higher tan 𝛿𝑠, a lower 𝜂𝑠  gives 
better performances.  

At a fixed tan 𝛿𝑠 value, the maximum sensitivity is obtained 
at the crossover of |𝑐| (see Figure 2); thus, 𝜂𝑠 =    𝜂𝑠,𝑜𝑝𝑡 =   𝑙𝑟 /

tan 𝛿𝑠 and |𝑐|𝑚𝑎𝑥 = (4𝑙𝑟 tan 𝛿𝑠 )
−1. 𝜂𝑠,𝑜𝑝𝑡  is the optimum 

sample filling factor and gives the maximum sensitivity for 𝑄 
measurements. As such, the geometry of the samples under 

 

Figure 1. Sketch of the dielectric loaded resonator (not to scale). 

 

Figure 2. Solid lines: the absolute value of resonator sensitivity |𝑐| as a 
function of the sample 𝑡𝑎𝑛 𝛿𝑠 and filling factor 𝜂𝑠. Dotted line: the maximum 
sensitivity |𝑐|𝑚𝑎𝑥  reachable for every 𝑡𝑎𝑛 𝛿𝑠 value. 



 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 29 

investigation can be adjusted in order to fulfil the requirement 

for 𝜂𝑠,𝑜𝑝𝑡  . In our case, this is the expected tan 𝛿𝑠 ∼ 10
−2 . So, 

from Figure 2, 𝜂𝑠,𝑜𝑝𝑡 ∼ 10
−2. 

The loss tangent measurement uncertainty 𝑢(tan 𝛿𝑠) is 
evaluated as follows [20]: 

𝑢2(tan 𝛿𝑆)

=
1

𝜂𝑆
2

[(𝑢(Δ(𝑄−1)))
2

+ ∑(Δ(𝐺 −1)𝑢(𝑅𝑖))

𝑖

2

+ ∑ (𝑅𝑖 𝑢(Δ(𝐺
−1)))

2

+ (tan 𝛿𝑉 𝑢(Δ(𝜂𝑉 )))
2

𝑖

+ (Δ(𝜂𝑉 )𝑢(tan 𝛿𝑉 ))
2

+ (tan 𝛿𝑆𝑢(𝜂𝑆 ))
2], 

(9) 

with: 

𝑢2(Δ(𝑄−1)) = (
𝑢(𝑄𝑆 )

𝑄𝑆
2

)

2

+ (
𝑢(𝑄𝐴)

𝑄𝐴
2

)

2

,  (10) 

𝑢2(Δ(𝐺 −1))   = (
𝑢(𝐺𝑖,𝑆)

𝐺𝑖,𝑆
2

)

2

+ (
𝑢(𝐺𝑖,𝐴)

𝐺𝑖,𝐴
2

)

2

− 2𝑟𝐺
𝑢(𝐺𝑖,𝑆 )𝑢(𝐺𝑖,𝐴)

𝐺𝑖,𝑆
2 𝐺𝑖,𝐴 

2
, 

(11) 

𝑢2(Δ(𝜂𝑉 )) = 𝑢
2(𝜂𝑉,𝑆) + 𝑢

2(𝜂𝑉,𝐴) − 2𝑟𝜂 𝑢(𝜂𝑉,𝑆)𝑢(𝜂𝑉,𝐴). (12) 

The correlation factors 𝑟𝐺  and 𝑟𝜂  are supposed to be almost 1 

since the evaluation of 𝐺𝑖  and 𝜂𝑉  is performed with the same 
algorithm and with the same settings. Conversely, the 𝑄 
measurements are not strongly correlated, since the different 
mountings can introduce different uncorrelated error 
contributions. 

Next, 𝑢(tan 𝛿𝑠) is explored in the space (1 <
′ < 10, 

10−5 < tan 𝛿𝑠 < 10
0, 0.5 mm < 𝑡 < 2 mm) to establish the 

operative limits of this technique. 𝑢(tan 𝛿𝑠) is evaluated using 
Eq. (9) with geometrical 𝐺 factors and filling 𝜂 factors obtained 
through e.m. simulations. Using e.m. simulations, it has been 

verified that tan 𝛿𝑠 variations (in the studied space) do not alter 
the e.m. field configuration. Therefore, for the evaluation of 𝐺 
and 𝜂, tan 𝛿𝑠 is fixed (i.e. tan 𝛿𝑠 = 10

−2). Both 𝑢(𝐺) and 𝑢(𝜂) 
are obtained with Monte Carlo e.m. simulations randomly 
varying all the physical dimensions and the e.m. properties of the 
materials from which the resonator is made in their uncertainty 
space [25].  

It should be noted that 𝑢(𝑄)/𝑄 is ideally constant for every 
𝑄 value if the measurement frequency span is kept proportional 
to 𝑓0/𝑄 and the number of points constant [26]. Yet, because of 
the mounting repeatability limitation, the presence of other 
resonance modes and other unideal factors (e.g. a complex 
background signal on the transmission parameter and a cross-

coupling contribution), 𝑢(𝑄) is somehow limited even at low 𝑄.  
As such, its absolute value is assumed constant, i.e. 𝑢(𝑄) ∼ 40. 
In Figure 3, 𝑢(tan 𝛿𝑠)/ tan 𝛿𝑠 is evaluated in the plane 
( ′, tan 𝛿), and it is reported for samples with 𝑡 = 1.5 mm and 
𝑡 = 0.5 mm. 𝑢(tan 𝛿𝑠)/ tan 𝛿𝑠 strongly depends on the sample 
thickness 𝑡 and, thus, on 𝜂𝑠, as expected from Figure 2. 
𝑢(tan 𝛿𝑠) sharply increases with thinner samples, particularly at 
low tan 𝛿𝑠 values: with tan 𝛿𝑠 ∼ 10

−2 and 𝑠
′ ∼ 2, 

𝑢(tan 𝛿𝑠) tan 𝛿𝑠⁄ ∼ 100 % for a 0.5 mm thick sample, while 
𝑢(tan 𝛿𝑠) tan 𝛿𝑠⁄ ∼ 10 % in the same conditions but 𝑡 =1.5 
mm. In Figure 3, the 𝑠 ′(tan 𝛿𝑠) curve corresponding to |𝑐|𝑚𝑎𝑥  
is reported; it agrees to a fair extent with the lowest 

𝑢(tan 𝛿𝑠) tan 𝛿𝑠⁄  level.  
Once the maximum 𝑢(tan 𝛿𝑠) threshold level is fixed, taking 

what is shown in Figure 3, the space ( ′, tan 𝛿) where the 
proposed technique can be reliably used is defined. However, 
two other limiting factors must be considered. The first is related 

to the impossibility to discriminate small Δ𝑄 ∼ 𝑄𝐴 − 𝑄𝑆 
variations because of the measurement noise. Thus, where Δ𝑄 <
Δ𝑄𝑚𝑖𝑛 , this technique is no more sensitive. In our case, with 
𝑄𝐴 ∼ 5000, Δ𝑄𝑚𝑖𝑛 ∼ 40. In Figure 3, the Δ𝑄𝑚𝑖𝑛  curve is 
reported; however, one can notice that it crosses the ( ′, tan 𝛿) 
plane where 𝑢(tan 𝛿𝑠 )/  tan 𝛿𝑠 > 100 %. The second limit is set 
where the losses of the sample are so high as to make 𝑄𝑠  too 
small to be reliably measured. Due to the 𝑆12(𝑓) background, we 
set this minimum value min(𝑄𝑠 ) ∼ 100. In Figure 3, the dotted 
line represents this limit: above this curve no tan 𝛿𝑠 
measurements are possible. 

The presented technique can be easily optimized for the 
characterization of these materials. In fact, the position of the 

minimum of 𝑢(tan 𝛿𝑠)/ tan 𝛿𝑠 shown in Figure 3 arises in the 
region of the expected values of  𝑠

′  and tan 𝛿𝑠 of 3D-printer 
materials, and it can be tuned with 𝜂𝑠. 

 

Figure 3. Relative loss tangent uncertainty 𝑢(𝑡𝑎𝑛 𝛿𝑆 )/ 𝑡𝑎𝑛 𝛿𝑠 contour plot in 
the plane ( ′, tan 𝛿)  for samples of thickness 𝑡 = 0.5 mm (upper panel) and 
𝑡 = 1.5 mm (lower panel). The thicker solid line represents the points of 
maximum sensitivity |𝑐|𝑚𝑎𝑥  as evaluated from Figure 2. The dashed line 
corresponds to the minimum quality factor appreciable variation 𝛥𝑄𝑚𝑖𝑛 =
𝑄𝐴 − 𝑄𝑆 ∼ 40 and the dotted line to the minimum evaluable 𝑄𝑆,𝑚𝑖𝑛 ∼ 100. 



 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 30 

4. RESULTS AND DISCUSSION 

The measurements were performed on four dielectric 

samples, which were of different thickness values 𝑡 as reported 
in Table 1, made of a photopolymer material printed using the 
PolyJetTM deposition technique. The thicknesses and flatnesses 
of the samples were checked with a micrometre. The mean values 

𝑡̅ and their standard deviation 𝜎𝑡 = √∑ (𝑡 − 𝑡̅)
2𝑁

𝑖=1 (𝑁 − 1)⁄  
have been obtained by 𝑁 = 10 different measurements of the 
thickness by probing the surfaces of the samples. Thus, 𝜎𝑡  can 
be read as a measure of the flatness of the samples.  

To achieve a sufficiently sensitive method of evaluating ′, the 
frequency repeatability of the setup needs to be reliable to allow 

the accurate measurement of the differences between 𝑓0,𝑆  

(measured with the dielectric samples mounted) and 𝑓0,𝐴 
(measured without the sample and leaving an air gap of the same 
sample thickness). This is achieved using 3D-printed rings 
prepared in the same way and of the same thickness as the 
dielectric samples. The data are presented in Figure 4, from 

which one can then evaluate ε'S = 2.9 ± 0.2. 
Once 𝑠

′  is estimated with its uncertainty, the geometrical and 
filling factors of the resonator components (with their 
uncertainties) are evaluated from the simulations as shown in the 
previous section. 

Then, tan 𝛿𝑠 is evaluated through Eq. (6) from the measured 
𝛥(𝑄−1) = 𝑄𝑆

−1 − 𝑄𝐴
−1. The quality factors 𝑄 are reported in 

Table 2. 

In Figure 5, the measured tan 𝛿𝑠 is shown with the error bars 
evaluated with Eq. (9) using the uncertainties for the measured 
quantities and simulated parameter shown previously. Figure 5 

shows the best estimate: tan 𝛿𝑆 = (1.8 ± 0.2) ⋅ 10
−2 as 

calculated with ′𝑆 = 2.9 ± 0.2. The evaluation of tan 𝛿𝑆 is 
performed taking as best value the centre point in the common 
confidence interval of all the experimental points (the light blue 

band in Figure 5). Then, the uncertainty is the half width of that 
common interval.  One can note that the uncertainty bars rapidly 
increase in size when the sample thickness becomes small due to 
a lack of sensitivity, as expected from the analysis presented in 
Sec. 3. This effect is present also in Figure 4, where the simulated 
curves for small thickness values tend to coalesce. Conversely, 
samples with a large thickness can cause e.m. field radiation from 
the structure, thereby significantly changing the resonant mode 
and adding further losses. The method presented here, in our 
geometry, is then most suitable for samples of thickness between 
1 and 2 mm. 

5. CONCLUSIONS 

In this work, a dielectric loaded-resonator-based technique 
was developed and presented for the measurement of the 

complex permittivity ̃ of 3D-printing materials. We exploited 
the possibility of shaping the sample into appropriate shapes 
(disks, in our case) and the excellent frequency repeatability of 

our setup to reliably measure the quality factor 𝑄 and the 
resonance frequency 𝑓0 both with and without the sample loaded 
into the cavity. From the variations of 𝑄 and 𝑓0 given by the 
sample insertion, ̃ is obtained using the perturbation approach. 
The measurement technique performances were deeply analysed 
in terms of sensitivity and accuracy in the whole parameter space 
in order to establish the sample geometry, as a function of its 
e.m. properties, for the best measurement accuracy. 

The technique was tested by measuring photopolymer 
material printed using PolyJetTM deposition. With the presented 

technique here, it was obtained that ε'S = 2.9 ± 0.2 and 
tan 𝛿𝑠 = 0.018 ± 0.002.  

The obtained result agrees well with other studies. In 
particular, using a combined technique, a broad band 

(1 MHz÷11 GHz) characterisation of 3D-printer materials was 
presented in [13]. The high frequency range (8.2 GHz÷11 GHz) 

 

Figure 4. Differences between the resonant frequencies 𝑓0,𝑆 (with the 

samples mounted) and 𝑓0,𝐴 (with the sample substituted by an air gap of the 
same thickness). The red dots are the experimental data and the blue lines 
the simulations results. 

Table 1. The mean thickness 𝑡̅ of the samples and its standard deviation 𝜎𝑡 . 

 
 

𝑺𝟏 𝑺𝟐 𝑺𝟑 𝑺𝟒 

𝑡̅ (mm) 0.522 1.002 1.512 2.063 

𝜎𝑡  (mm) 0.003 0.004 0.005 0.004 

Table 2. The inverse of the measured quality factors when the four samples, 
S1-S4, are inserted. 𝑄𝑆

−1and 𝑄𝐴
−1 refer to the measurement with the sample 

or an air gap of equal thickness, respectively. 

 
 

𝑺𝟏 𝑺𝟐 𝑺𝟑 𝑺𝟒 

𝑄𝑆
−1 ⋅ 104 2.15 2.48 3.06 4.25 

𝑄𝐴
−1 ⋅ 104 1.95 1.96 1.98 2.00 

 

Figure 5. The measured loss tangent of the dielectric 3D-printed material 
𝑡𝑎𝑛 𝛿𝑆. The error bars are evaluated with Eq. (9) and the green area 
represents the confidence interval. 

0

0.02

0.04

0 0.5 1 1.5 2

f
0
-measure

'=2.0

'=2.2

'=2.4

'=2.6

'=2.8

'=3.0

'=3.2

'=3.4


f 0

 (
G

H
z
)

t (mm)
0

0.02

0.04

0.06

0 0.5 1 1.5 2 2.5

ta
n


t (mm)

tan
S
=0.018(2)



 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 31 

was studied through a waveguide in reflection mode, although an 
uncertainty study was lacking in this frequency range. The 

reported values at 11 GHz were 2.5 < ′ < 3.29 and 
0.005 <  tan 𝛿 < 0.037, perfectly in agreement with our results. 

In order to check the accuracy of the DR technique shown in 

this paper, ̃ of the studied materials was also measured with a 
standard reflection/transmission method. A WR90 waveguide 
was used with a PNA Network Analyzer (model E8363C, Agilent 
Technologies) with the Agilent 85071E software and the ‘NIST 
precision’ method [27]. To perform this measurement, 

parallelepipeds of  22.8 ⋅ 10.1 mm2 and different thicknesses (4, 
5, 6, 7, 8 mm) were printed. 𝑆

′ ∼ 3.1 was measured by 
extrapolating the value at 12.9 GHz, and no significative sample 
variations can be seen (Figure 6). This value compares well with 

the one obtained using the proposed DR technique and the 𝑓0 
variation.  

Then, the imaginary part 𝑠
′′ ∼ 0.23 is obtained with the 

waveguide method, which yields tan 𝛿𝑠 ∼ 0.074 with a 
significative inter-sample scattering. This value is about 4 times 
higher than the one obtained with the DR method. However, it 
must be mentioned that the ‘NIST precision’ method was 
developed to solve the accuracy problems of the 
Nicholson–Ross–Weir technique (NRW) near the sample 
resonances [15], [27]. The ‘NIST precision’ and the NRW give 
comparable results out of the resonances. It is well known that 

the NRW is not a reliable method for the ′′ evaluation of low-
loss materials. In fact, the same measurement fixture was tested 

with a Polytetrafluoroethylene sample obtaining 𝑃𝑇𝐹𝐸
′′ ∼ 1.5 ×

10−2. However, from the literature [28], this is expected to be 
about two orders of magnitude smaller. Also, for materials with 
higher losses, NRW accuracy is limited; the comparison 
presented in [29] between the NRW and other methods shows 

some discrepancies, even at tan 𝛿 ∼ 10−2 values (e.g. with nylon 
samples), as in our case. Thus, the disagreement between the DR 

tan 𝛿 measurement and that obtained in waveguide, which was 
somewhat expected, will be studied in further works. 

In summary, a new measurement technique was presented for 
the e.m. characterisation of dielectric materials, with interesting 
potential applicability to industrial fields due to its simple 
conceptual approach and high accuracy. 

REFERENCES 

[1] N. Hopkinson, P. Dickens, Emerging rapid manufacturing 
processes, in: Rapid Manufacturing; an Industrial Revolution for 
the Digital Age. N. Hopkinson, R. Hague, P. Dickens (editors). 
Wiley & Sons Ltd, Chichester, 2006, ISBN: 0-470-01613-2, 
pp. 55-80.  

[2] M. J. Werkheiser, J. Dunn, M. P. Snyder, J. Edmunson, K. Cooper, 
M. M. Johnston, 3D printing in zero-g ISS technology 
demonstration, AIAA SPACE 2014 Conference and Exposition, 
San Diego, California, 2014, pp 2054-2078.  

[3] J. Y. Wong, Ultra-portable solar-powered 3D printers for onsite 
manufacturing of medical resources, Aerospace Medicine and 
Human Performance 89 (2015) pp. 830-834.   
DOI: https://doi.org/10.3357/AMHP.4308.2015  

[4] C. Schubert, M. C. Van Langeveld, L. A. Donoso, Innovations in 
3D printing: a 3D overview from optics to organs, British Journal 
of Ophthalmology  98 (2014) pp. 159-161.  
DOI: https://doi.org/10.1136/bjophthalmol-2013-304446  

[5] F. Rengier, A. Mehndiratta, H. Von Tengg-Kobligk, 
C. M. Zechmann, R. Unterhinninghofen, H. Kauczor, 
F. L. Giesel, 3D printing based on imaging data: review of medical 
applications, International Journal of Computer Assisted 
Radiology and Surgery  5 (2010) pp. 335-341.   
DOI: https://doi.org/10.1007/s11548-010-0476-x 

[6] M. J. Burfeindt, T. J. Colgan, R. O. Mays, J. D. Shea, N. Behdad, 
B. D. Van Veen, S. C. Hagness, MRI-derived 3-D-printed breast 
phantom for microwave breast imaging validation, IEEE 
Antennas and Wireless Propagation Letters 11 (2012) pp. 1610-
1613. 
DOI: https://doi.org/10.1109/LAWP.2012.2236293  

[7] A. T. Mobashsher, A. M. Abbosh, Three-dimensional human head 
phantom with realistic electrical properties and anatomy, IEEE 
Antennas and Wireless Propagation Letters  13 (2014) pp. 1401-
1404.  
DOI: https://doi.org/10.1109/LAWP.2014.2340409  

[8] A. Bisognin, A. Cihangir, C. Luxey, G. Jacquemod, R. Pilard, 
F. Gianesello, W. G. Whittow, Ball grid array-module with 
integrated shaped lens for WiGig applications in eyewear devices, 
IEEE Transactions on Antennas and Propagation  64 (2016) 
pp. 872-882.   
DOI: https://doi.org/10.1109/TAP.2016.2517667  

[9] G. P. Le Sage, 3D printed waveguide slot array antennas, IEEE 
Access  4 (2016) pp. 1258-1265.   
DOI: https://doi.org/10.1109/ACCESS.2016.2544278   

[10] A. Rashidian, L. Shafai, M. Sobocinski, J. Peräntie, J. Juuti, 
H. Jantunen, Printable planar dielectric antennas, IEEE 
Transactions on Antennas and Propagation 64 (2016) pp. 403-413.  
DOI: https://doi.org/10.1109/TAP.2015.2502265  

[11] P. Nayeri, M Liang, R. A. Sabory-Garcı, M. Tuo, F. Yang, 
M. Gehm, A. Z. Elsherbeni, 3D printed dielectric reflectarrays: 
low-cost high-gain antennas at sub-millimeter waves, IEEE 
Transactions on Antennas and Propagation  62 (2014) pp. 2000-
2008.   
DOI: https://doi.org/10.1109/TAP.2014.2303195  

[12] C. R. Garcia, R. C. Rumpf, H. H. Tsang, J. H. Barton, Effects of 
extreme surface roughness on 3D printed horn antenna, 
Electronics Letters  49 (2013) pp. 734-736.   
DOI: https://doi.org/10.1049/el.2013.1528  

[13] P. I. Deffenbaugh, R. C. Rumpf, K. H. Church, Broadband 
microwave frequency characterisation of 3-D printed materials, 
IEEE Transactions on Components, Packaging and 
Manufacturing Technology  3 (2013) pp. 2147-2155.   
DOI: https://doi.org/10.1109/TCPMT.2013.2273306  

[14] T. D. Ngo, A. Kashani, G. Imbalzano, K. T. Nguyen, D. Hui, 
Additive manufacturing (3D printing): A review of materials, 
methods, applications and challenges, Composites Part B: 
Engineering  143 (2018) pp. 172-196.   
DOI: https://doi.org/10.1016/j.compositesb.2018.02.012 

 

Figure 6. Real part of the complex permittivity ℜ{ ̃} = ′ measured on 
dielectric samples of different thicknesses using the ‘NIST precision’ 
transmission/reflection method [26] through a WR90 waveguide. 

3.05

3.1

3.15

3.2

3.25

3.3

8 9 10 11 12 13

4 mm
5 mm
6 mm
7 mm
8 mm


'

f (GHz)

https://doi.org/10.3357/AMHP.4308.2015
https://doi.org/10.1136/bjophthalmol-2013-304446
https://doi.org/10.1007/s11548-010-0476-x
https://doi.org/10.1109/LAWP.2012.2236293
https://doi.org/10.1109/LAWP.2014.2340409
https://doi.org/10.1109/TAP.2016.2517667
https://doi.org/10.1109/ACCESS.2016.2544278
https://doi.org/10.1109/TAP.2015.2502265
https://doi.org/10.1109/TAP.2014.2303195
https://doi.org/10.1049/el.2013.1528
https://doi.org/10.1109/TCPMT.2013.2273306
https://doi.org/10.1016/j.compositesb.2018.02.012


 

ACTA IMEKO | www.imeko.org September 2020 | Volume 9 | Number 3 | 32 

[15] L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan, V. K. Varadan, 
Microwave Electronics: Measurement and Materials 
Characterisation., John Wiley & Sons, Chichester, 2004, 
ISBN: 0-470-84492-2. 

[16] J. Mazierska, C. Wilker, Accuracy issues in surface resistance 
measurements of high temperature superconductors using 
dielectric resonators (corrected), IEEE Transactions on Applied 
Superconductivity  11 (2001) pp. 4140-4147.   
DOI: https://doi.org/10.1109/77.979858  

[17] J. Mazierska, M. V. Jacob, How accurately can the surface 
resistance of various superconducting films be measured with the 
sapphire Hakki–Coleman dielectric resonator technique?, Journal 
of Superconductivity and Novel Magnetism  19 (2006) pp. 649-
655.   
DOI: https://doi.org/10.1007/s10948-006-0129-z 

[18] F. Castles, D. Isakov, A. Lui, Q. Lei, C. E. J. Dancer, Y. Wang, 
J. M. Janurudin, S. C. Speller, C. R. M. Grovenor, P. S. Grant, 
Microwave dielectric characterisation of 3D-printed BaTiO3/ABS 
polymer composites, Scientific Reports  6 (2016) art. no. 22714 
DOI: https://doi.org/10.1038/srep22714 

[19] J. M. Felício, C. A. Fernandes, J. R. Costa, Complex permittivity 
and anisotropy measurement of 3D-printed PLA at microwaves 
and millimeter-waves, 22nd International Conference on Applied 
Electromagnetics and Communications (ICECOM), Dubrovnik, 
Croatia, 2016, pp. 54-60.   

[20] B. W. Hakki, P. D. Coleman, A dielectric resonator method of 
measuring inductive capacities in the millimeter range, IRE 
Transactions on Microwave Theory and Techniques  8 (1960) pp. 
402-410.   
DOI: https://doi.org/10.1109/TMTT.1960.1124749  

[21] K. Torokhtii, A. Alimenti, N. Pompeo, F. Leccese, F. Orsini, 
A. Scorza, S. A. Sciuto, E. Silva, Q-factor of microwave 
resonators: calibrated vs. uncalibrated measurements, Journal of 
Physics: Conference Series  1065 (2018) art. no. 052027.  
DOI: https://doi.org/10.1088/1742-6596/1065/5/052027 

[22] K. Leong, J. Mazierska, Precise measurements of the Q factor of 
dielectric resonators in the transmission mode-accounting for 

noise, crosstalk, delay of uncalibrated lines, coupling loss, and 
coupling reactance, IEEE Transactions on Microwave Theory and 
Techniques 50 (2002) pp. 2115-2127.   
DOI: https://doi.org/10.1109/TMTT.2002.802324  

[23] N. Pompeo, K. Torokhtii, F. Leccese, A. Scorza, A. S. Sciuto, 
E. Silva, Fitting strategy of resonance curves from microwave 
resonators with non-idealities, IEEE International 
Instrumentation and Measurement Technology Conference 
(I2MTC), Torino, Italy, 2017, pp. 1-6.   
DOI: https://doi.org/10.1109/I2MTC.2017.7969903  

[24] ISO/IEC GUIDE 98-3:2008 - Guide to the expression of 
uncertainty in measurement (GUM:1995). 

[25] ISO/IEC GUIDE 98-3/Suppl.1:2008 - Propagation of 
distributions using a Monte Carlo method. 

[26] K. Torokhtii, A. Alimenti, N. Pompeo, E. Silva, Uncertainty in 
uncalibrated microwave resonant measurements, Proc. of the 23rd 
IMEKO TC4 International Symposium Electrical & Electronic 
Measurements Promote Industry 4.0, Xi’An, China, 2019, pp. 98-
101. Online [Accessed 24 September 2020]  
https://www.imeko.org/publications/tc4-2019/IMEKO-TC4-
2019-022.pdf  

[27] J. Baker-Jarvis, E. J. Vanzura, W. A. Kissick, Improved technique 
for determining complex permittivity with the 
transmission/reflection method, IEEE Transactions on 
microwave theory and techniques 38 (1990) pp. 1096-1103.  
DOI: https://doi.org/10.1109/22.57336  

[28] J. Krupka, Measurements of the complex permittivity of low loss 
polymers at frequency range from 5 GHz to 50 GHz, IEEE 
Microwave and Wireless Components Letters  26 (2016) pp. 464-
466.   
DOI: https://doi.org/10.1109/LMWC.2016.2562640  

[29] Z. Abbas, R. D. Pollard, R. W. Kelsall, Complex permittivity 
measurements at Ka-band using rectangular dielectric waveguide, 
IEEE Transactions on Instrumentation and Measurement 50, 
(2001) pp. 1334-1342.   
DOI: https://doi.org/10.1109/19.963207   
 

 

https://doi.org/10.1109/77.979858
https://doi.org/10.1007/s10948-006-0129-z
https://doi.org/10.1038/srep22714
https://doi.org/10.1109/TMTT.1960.1124749
https://doi.org/10.1088/1742-6596/1065/5/052027
https://doi.org/10.1109/TMTT.2002.802324
https://doi.org/10.1109/I2MTC.2017.7969903
https://www.imeko.org/publications/tc4-2019/IMEKO-TC4-2019-022.pdf
https://www.imeko.org/publications/tc4-2019/IMEKO-TC4-2019-022.pdf
https://doi.org/10.1109/22.57336
https://doi.org/10.1109/LMWC.2016.2562640
https://doi.org/10.1109/19.963207