A Dielectric Loaded Resonator for the Measurement of the Complex Permittivity of Dielectric Substrates

A new configuration of dielectric-loaded resonator (DR), particularly versatile for the complex permittivity measurement of substrates for microwave circuits, even in the presence of back metal plates, is shown here. To test this technique in a wide interval of the values of the complex permittivity, the versatility of 3-D printing is exploited to print samples with different densities, thus artificially changing the effective permittivity in the interval (1.7–3.1) for the real part and (0.02–0.06) for the imaginary part. The designed resonator, tuned at ~12 GHz, is experimentally validated by the comparison of measurements obtained on these samples with a split ring resonator (SRR) and standard transmission/reflection waveguide methods. Then, the versatility of the designed resonator is shown in the characterization of FR4-fiberglass and Kapton polyimide samples.


I. INTRODUCTION
D UE to the importance of accurate knowledge of the highfrequency electromagnetic (e.m.) properties of dielectric materials, in particular, in view of the development of information and communication technologies (ICT), several microwave measurement techniques have been developed since the 1950s to meet the different operative needs in terms of measurement frequencies, geometries, and dimensions of the investigated samples as well as accuracy and sensitivity levels [1], [2], [3]. These measurement methods are traditionally divided into two macrocategories: 1) broadband transmission/reflection methods and 2) resonant. The great advantage of transmission/reflection methods lies in their wide and continuous frequency band of operation. However, their poor sensitivity makes these methods not appropriate for precise and accurate characterizations of low-loss materials [2], [3], [4], [5]. On the contrary, resonant techniques, which operate only at discrete frequencies, take advantage of the high sensitivity of the quality factor Q and resonance Depending on the sample e.m. and geometrical properties, different kinds of resonant fixtures can be found in literature [1], [2], for example, cavities, dielectric-loaded resonators (DRs), dielectric resonators, open resonators, and planar resonators.
To achieve the best performances, the fixture choice depends on the characteristics of the sample (size and state of matter), e.m. properties (insulators/conductors), and operating conditions (frequency in the intended application).
In this work, we propose the use of a modified configuration of a DR for the measurement of the dielectric substrates relative complex permittivityε r = ε r − iε r , with ε r = Reε r , ε r = −Imε r , i 2 = −1. The ratio tan δ = ε r /ε r is known as the loss tangent. Thanks to their high sensitivity, DRs are widely used both for surface resistance measurement of normal conductors, surface impedance measurement of superconductors, and complex permittivity measurements of low-loss dielectrics [1], [2], [6], [7], [8], [9], [10], [11], [12]. Despite their common use with low-loss materials, in this work we show the possibility of using this measurement fixture also for the characterization of lossy dielectric materials without losing sensitivity. A new closed kind of configuration, which does not require completely unmounting the resonator for sample loading, is exploited. In particular, the presented geometry allows the samples to be loaded simply by placing them into the resonator through special openings in the metal cavity. In this way, the measurement procedure is simplified, and the fixture/sample assembling time is reduced. We focused on the design of a fixture optimized for the characterization of materials with 1.5 < ε r < 5 and high losses, tan δ ∼ 10 −2 . These values for ε r and tan δ typically include soft substrates and materials used in additive manufacturing (AM) for highfrequency applications.
Due to the high e.m. losses of the materials under investigation, the best measurement sensitivity to the materialε r is obtained by limiting the volume of the dielectric sample to avoid a detrimental lowering of the fixture quality factor Q. A preliminary study focused on the optimization of the sample volume as a function of both ε r and ε r of the sample itself was shown in [13].
The presented fixture allows forε r measurements of solid and relatively thick dielectric materials between 1 and 2 mm. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ In particular, the measurement specifications and constraints addressed in this work are: 1) the need to characterize samples without disassembling the fixture, to simplify and speed up the user's operations; 2) the need to characterize thick (>1 mm) dielectric samples even at high frequencies (>10 GHz); and 3) the possibility to measureε r of dielectric materials provided on planar metal backing plates. The possibility of characterizing thick samples is fundamental for theε r measurement of typically used electronic substrates (e.g., FR4-fiberglass) which are generally provided with this thickness and, when needed, directly deposited on conductive ground planes. To the best of our knowledge, currently existing resonant fixtures are not able to fulfill all these measurement specifications simultaneously. For example, split post resonators (SPRs), which guarantee the highestε r measurement accuracy [2], [14] for low-loss dielectric materials, are not suitable for thick samples and cannot work with dielectric samples grown on metal plates [14], [15], [16]. The metal plate also prevents the use of Cohn resonators [17], cylindrical-radial resonators [18], sheet resonators [19], standard configurations of the DRs [20], [21], and coaxial surface-wave resonators [22]. Thus, this work aims to fill the current lack of measurement instrumentation capable of satisfying the aforementioned requirements, in view of the importance that the characterization of dielectric materials is increasingly acquiring, thanks both to the development of new high-frequency technologies (e.g., 5G [23], 60-GHz radar sensors [24]) and also to the rapid diffusion of new materials, as those used in AM techniques, in radio frequency (RF) and higher-frequency applications [25].
To compare the techniques in a wide range ofε r values, we printed, with photopolymer materials through the PolyJet [26] deposition technique, different artificially porous samples with different air volumes. In this way, with a single material, we obtained samples with effective dielectric permittivity 1.7 < ε r < 3.1 and 0.02 < ε r < 0.06 [27]. The new configuration of the DR shown in this article is experimentally validated through the comparison ofε r measurements obtained on the same materials with the well-known split ring resonator (SRR) method [28], [29], [30], [31], [32]. In particular, the SRR used for this validation is tuned at ∼2.2 GHz. Since SRRs generally operate at lower frequencies than that used by volume resonators as the DR, before performing this comparison, we experimentally verified the negligible frequency dependence (between 2 and 12 GHz) of ε r of the material family under investigation, through the use of a standard reflection/transmission technique using WR430 (at ∼2 GHz) and WR90 (at ∼12 GHz) waveguides. In addition to this, ε r measurements performed both with the SRR and waveguides on low-density polyvinylchloride (LD-PVC), polytetrafluoroethylene (PTFE), polymethyl methacrylate (PMMA), and polycarbonate (PC) were compared to verify the goodness of the calibration of the SRR. Thus, the SRR has been used to validate the DR by measurement comparisons.
The article is organized as follows. In Section II the measured samples are described. In Section III, we describe the proposed DR configuration, the measurement technique and method, the used system, and the obtained results. Then, in Section IV, the DR experimental validation is discussed starting from the description of the used SRR and its calibration check through the waveguides measurements. Finally, in Section V, we comment on the obtained results, and a brief concluding summary is given in Section VI.

II. SAMPLES PREPARATION
The samples measured in this work were printed with a Stratasys 1 Objet30 PolyJet 3-D-printer using the hightemperature-resistant photopolymer RGD52 material and the SUP706 support [33].
The density of the samples, and thus their effective complex permittivity according to the theory for effective media [34], [35], was controlled by printing empty circular holes crossing the samples volume and arranged in various lattices with different filling factors. Hexagonal and square lattices were designed with different lattice parameters l p and hole diameters h . All samples were prepared with l p λ, with λ being the wavelength in the medium, so that the samples are electromagnetically homogeneous enabling the use of an effective permittivity.
In Table I, we report the filling factors and the lattice features of the prepared samples.
For each density, samples with different physical dimensions were printed to fulfill the needs of DR and SRR measurement fixtures. For the DR, small and thin square samples with nominal side L = 15.00 mm were printed with different nominal thicknesses: 1.50 mm, 1.75 mm, and 2.00 mm. A picture of the samples printed for the DR is shown in Fig. 1. The samples for the SRR must be large enough to cover the whole surface of the resonator and must be thick enough to avoid leakage of the e.m. field out of the sample. Therefore, parallelepiped samples of size 40 mm × 40 mm × 30 mm were prepared. The samples were made by printing several layers of smaller height (2 mm) to reach a total height of 30 mm. This path was chosen since, by printing these large samples as a whole, manufacturing errors emerged, as a progressive misalignment of the circular holes. To prepare all the samples in the same way, the solid sample was obtained as a stack too. The layers were then stacked in a polystyrene support (see Fig. 6).

III. MEASUREMENT METHOD AND SYSTEM
In this section, we describe the proposed DR measurement configuration and discuss the measurement method. The section is articulated as: A) measurement method, B) measuring system, and C) measurement procedure and uncertainty evaluation.

A. Measurement Method
Sinceε r of all the dielectric components loaded inside the resonator determines the amounts of the stored and dissipated energy, both the resonance frequency f 0 and the quality factor Q are sensitive toε r [2]. Hence,ε r can be determined through their measurement.
The volume perturbation approach is used [2], [9]: a part of the inner volume of the resonator is filled with the sample under measurement keeping all the other parameters unchanged. Thus, in the small perturbation limit, the variation of the relative permittivity ε r = ε r − iε r of a small volume of the resonator determines a change in the resonance frequency f 0 ∝ ε r and in the quality factor (Q −1 ) ∝ tan δ [2]. Here, x = x − x ref denotes the variation of the parameter x with respect to a reference value x ref .
In real cases, the required measurement sensitivity imposes a volume of the dielectric sample larger than the one compatible with the small perturbation limit. Thus, the change in ε r of the volume of the resonator filled with the sample changes the field configuration inside the resonator. Thus, even if the e.m. properties and size of all the other components of the resonator are kept fixed, their contribution to f 0 and Q changes due to the altered field configuration. Hence, the relationship between the sample properties (i.e.,ε r ) and the resonator response (i.e., Q and f 0 ) is no more linear. A full e.m. simulation is thus used to obtain the Knowing ε r , the e.m. field configuration inside the resonator can be computed, as well as all the geometrical and filling factors, either analytically or through e.m. simulations, as done in this work. Then, it is possible to demonstrate [2], [13], [36] that with R i and G i being the surface resistance and the geometrical factor, respectively, of the i th metal surface of the cavity, tan δ d the loss tangent and η d the filling factor of the dielectric parts of the DR distinct from the sample. In particular, if the reference has tan δ ref = 0 (e.g., vacuum) and the perturbation is small (i.e., (G −1 i ) and η v are negligible), the small perturbation limit approximation

B. Measurement System
The straight vertical section of the used DR is shown in Fig. 2. It is composed of an aluminum cylindrical enclosure (height h = 6.50(1) mm and diameter = 30.00(1) mm) loaded with a sapphire single crystal (h = 5.00(1) mm, = 8.00(1) mm). The lid of the resonator can be opened for the insertion of the dielectric sample. The sample is held by a brass sample holder with a central hole of diameter = 13.00(1) mm. The TE 011 resonating mode is magnetically coupled in the cavity at ∼12.9 GHz.
The transmission S 21 scattering parameter between the two ports of the DR is acquired with an Anritsu 37269D vector network analyzer (VNA) and the obtained resonance curve fit with the complex modified Lorentzian model [9], [37], [38], [39], [40] where S CC is a complex parameter which represents the crosscoupling between the resonator couplers and (α + β f ) is a phase delay term. Thus, this model takes into account typical non-idealities, which make the measured resonance curves asymmetric (Fig. 3). By using this generalized model, the accuracy and precision of the Q and f 0 measurements are increased [9], [37], [41].

C. Measurement Procedure and Uncertainty Evaluation
In this section, the measurement procedure, the uncertainty analysis, and the measurement results obtained with the DR are reported.
For each sample, ten different mountings were performed and for each mounting, ten acquisitions of the resonator scattering parameters were performed with 1601 points evenly spaced in frequency. During a frequency sweep, every point was averaged over five acquisitions. The frequency span was chosen to be ∼5 full-width half-maximum (FWHMs) of the resonance curve to keep the point density fixed in every acquisition and to reduce the measurement uncertainties on the resonating parameters, as extensively described in [39]. The selected frequency span improves the determination of the complex cross-coupling factor S CC and of the phase delay term, increasing in this way the accuracy of the Q and f 0 measurements in asymmetric resonance curves [39].
The relative standard deviation on f 0 obtained from the different mountings was u m ( f 0 )/ f 0 < 10 −6 including the measurement repeatability and the fit precision. Since we need to associate to each sample thickness the measured f 0 , we propagated on the overall u( f 0 ) and also the uncertainty u(t) on the sample thickness t. The uncertainty on f 0 caused by u(t) is u t ( f 0 ) = u(t)∂ f 0 /∂t. The sensitivity function ∂ f 0 /∂t was evaluated with e.m. simulations and u(t) was evaluated as the standard deviation of ten different t measurements performed with a micrometer on each sample. Thus, . To implement the perturbative approach, reference measurements must be performed as explained in Section III-A. For this purpose, the volume occupied by the sample under measurement must be substituted by the same volume of air, so thatε ref = ε air ≈ 1 and tan δ ref = tan δ air ≈ 0. The above approximations can be done since the sensitivity limits of the measurement method prevent distinguishing the dielectric properties of air from those of vacuum [13]. Hollow rings, printed with the same thickness as the samples and with inner The experimental points for f 0 were reported on the calibration curves (obtained through e.m. simulation of the DR) shown in Fig. 4: it can be noticed that samples of different thicknesses but of the same filling factor are placed, within a good approximation, on the same f 0 (t, ε r ) curve. For a better evaluation of ε r , the calibration curves shown in Fig. 4 were cut at fixed t values, thus obtaining the ε r ( f 0 ) curves shown in Fig. 5. In this figure, the data points for f 0 measured on the full sample are reported as an example: it can be noted that the 1σ probability intervals of the obtained ε r at different thicknesses overlap. The best estimation of ε r for the (full) samples of different thicknesses was obtained from the average of their ε r values, and the uncertainty was evaluated with the standard propagation procedure on the calculus of the averages [42].
The imaginary part ε r of the relative permittivity was then obtained as ε r = tan δ s ε r from Q measurements using (1), with R = 92(12) m and tan δ v = 4(2) × 10 −5 evaluated with a calibration procedure based on the round robin rotation technique [43]. The uncertainties u(ε r ) were finally obtained from (1) with the standard uncertainty propagation procedure [42]: u(Q)/Q = 1%, which includes the mounting repeatability; u(G) and u(η) were obtained through e.m. simulations with the Monte Carlo method by varying all the physical dimensions and e.m. properties of the components of the resonator in their variability intervals [44]; u(R) and u(tan δ v ) are those above reported.
The measurement results are shown in Fig. 7 and will be commented in Section V, together with the results of the Section IV.

IV. EXPERIMENTAL VALIDATION
In this section, we discuss the experimental validation of the here presented measurement fixture. Since at these frequencies, there are noε r measurement standards (neither solid nor liquid), and it was not possible to directly check the DR measurement performances through the use of known materials. For this reason, we compared the measurement results obtained with the DR with those provided on the same material by a SRR tuned at ∼2.2 GHz. In addition to this, as a further validation, the calibration of the used SRR was checked through the comparison of ε r measurements performed on LD-PVC, PTFE, PMMA, and PC both with the SRR and with the standard reflection/transmission method based on the NIST variation of the Nicolson-Ross-Weir method [45]. The reflection/transmission measurements were performed with WR430 and WR90 waveguides to show the frequency independence of the e.m. properties of this class of dielectrics in the operative frequency range of the DR and the SRR.
We first describe the used SRR, then we show the comparison with the waveguide measurements, and finally we show and compare the results obtained with the DR on the samples described in Section II.

A. Split-Ring Resonator
The SRR consists of metal tracks etched on the top of a dielectric substrate and a metal ground, placed on the bottom part (Fig. 6). The tracks consist of two concentric rings, known as circular crowns, that present two diametrically opposed cuts, hence the name of the resonator, that is, "split" ring resonator [46]. On the ring sides, there are two microstrip-feeding lines of larger thickness with respect to the rings, while another smaller microstrip line, aligned with the feeds, is placed along the diameter of the innermost circular crown. The chosen substrate is Duroid 1 RT-5870, with nominal thickness t s = 1.19 mm, relative permittivity ε sub = 2.33(2), loss tangent tan δ 0.0012, nominal copper thickness t c = 35 μm, and a total substrate dimension of 68.000(5) mm × 100.000(5) mm. The SRR is accessible through two coaxial connectors, coupled to the feeding lines, which are located on the bottom part of the structure.
The resonance frequency f 0 depends on the geometry of the SRR; modeling the structure as an LC circuit [47], its f 0 is given by f 0 = 1/(2π √ LC). The overall geometry was simulated and optimized by the e.m. software CST Microwave Studio 1 . Due to the well-defined geometry of the couplers in the SRR, the simulated quantity is the directly loaded Q-factor Q l of the resonator. The final simulated resonance frequency in the air is approximately 2.22 GHz with Q l ∼ 140. Once the SRR was realized and tested in air, the size and the e.m. parameters of the simulation model were tuned, within the size uncertainties of the components of the SRR, through a multidimensional optimization, to minimize the differences between the simulated f 0 and Q l and the measured ones.
If the SRR works with a dielectric sample having complex relative permittivityε r in contact with the resonator, as shown in Fig. 6, its resonance frequency and quality factor are affected. Performing a series of CST 1 simulations with ε r varying between 1 and 10, it was possible to evaluate the relationship between resonance frequency shift ( f 0 ) and the sample permittivity (ε r ). The numerically obtained data were found to be best fit through a second-order polynomial yielding the following calibration equation: where f 0 is in GHz. Therefore, by performing f 0 measurements in the presence of a sample, it is possible to evaluate its relative permittivity ε r . For what concerns tan δ of the sample, However, it must be noticed that the calibration curve tan δ(Q −1 l ) depends on ε r of the sample. Thus, through e.m. simulations, the dependencies of the coefficients p 1 (ε r ) and p 2 (ε r ) were obtained by varying ε r in the interval 1-10 in steps of 1. The obtained points fit with the following functions: The determination coefficients of (5) and (6) are r 2 = 0.998 and r 2 = 0.974, respectively. Thus, once the sample ε r is determined using the measured f 0 with (3), p 1 and p 2 are determined and then tan δ obtained through the measured Q l with (4).
The measurement system is shown in Fig. 6. The SRR is placed in contact with and on top of the sample to be measured, which is in turn kept in place by a holder. This holder is made of polystyrene, so as not to affect the resonance frequency of the system. To force the SRR to adhere to the sample, exercising always the same force, a certified mass of 1 kg was used. This enhances the measurement repeatability [48]. The transmission scattering parameter S 21 is acquired through a PNA network analyzer, model E8363C, Agilent Technologies, with a 100-kHz sampling step, a 3-kHz resolution bandwidth, and −20 dBm source power. A vector error correction procedure is preliminarily performed, employing the N4691B Ecal kit.

B. Comparison With Transmission/Reflection Methods
The calibration curves obtained through the e.m. simulation of the SRR are experimentally verified by testing the SRR on the following materials: LD-PVC, PTFE, PMMA, and PC. These samples were characterized with the SRR and, for comparison, with the standard reflection/transmission technique, based on the NIST method [45], through WR430 (at ∼2 GHz) and WR90 (at ∼12 GHz) waveguides [49]. The results are shown in Table II.
The SRR and the WR430 systems provide comparable ε r on all the tested materials at the same working frequencies.
The results obtained on PTFE, PMMA, and PC are also well in agreement with [20]. The agreement of the measurements obtained with the WR430 and the WR90 waveguides experimentally demonstrates the negligible ε r frequency dependence of this class of materials up to ∼12 GHz. This validates directly the SRR method (at least for ε r measurements) and legitimates the comparison between the SRR and the DR despite the different operative frequencies.

C. Comparison With the DR
Once the calibration of the SRR is experimentally verified, the SRR can be used to validate the DR by a comparison ofε r measurement performed on the same materials. In this work, to compare the techniques in a sufficiently wideε r values space, test samples are printed with different infill percentages as discussed in Section II.
For each sample, with the SRR, ten repetitions were performed, each time removing the SRR and the sample from the polystyrene holder. A preliminary measurement in the air was performed, with the empty holder, to provide the reference for the evaluation of the resonance frequency shift due to the sample. The type-A standard uncertainty related to the measurement repeatability was then combined with the uncertainty given by the curve fitting algorithm (u(Q l )/Q l < 10 −4 and u( f 0 )/ f 0 < 6 × 10 −7 ) and then straightforwardly propagated to the measured permittivity using (3) and (4). A further uncertainty contribution related to the fitting of the calibration function to the numerically simulated data must also be taken into account. From the residuals of the fitting on the ten simulation points, u(ε r ) ∼ 0.1 can be assessed through (3). The same kind of source of uncertainty is also taken into account for the evaluation of tan δ from (4) through (5) and (6). We note that the above-reported uncertainty is the dominant term as compared to the contribution obtained from the propagation of the standard uncertainty of the measured resonance frequency shift.
The final results in terms of measured permittivity for the different samples are reported in Fig. 7.

V. RESULTS AND DISCUSSION
In Fig. 7, the data forε r measured with both resonators are reported as a function of the percentage-filling factor of the sample ϕ. The experimental data for ε r are well fit by the linear function ε r (ϕ) = (ε r (100%) − 1)ϕ/100 + 1, while the imaginary part with ε r (ϕ) = ε r (100%)ϕ/100. The points ε r (0%) = 1, ε r (0%) = 0 are constrained by the values of air. The linear dependence of the measured points is well in agreement with the upper limit of the Wiener model for the effective medium theory [34], [35] as already observed in [50] and [51] for the effective values of ε r , and also for ε r [52]. From the slopes of the linear fits, we obtaiñ ε r (100%) = 2.80(5) − i0.057(2) for the full sample. The standard complex uncertainty u(ε r (100%)) is evaluated from the linear fit with the Monte Carlo method assuming all the u(ε r (ϕ)) and u(ϕ) to be Gaussian and centered [44]. u(ϕ) is obtained starting from the linear printing precision declared by the manufacturer of the 3-D-printer (i.e., ∼0.01 mm) [33] and propagating this uncertainty on ϕ taking into account the geometry of the samples. We notice thatε r (100%) measurement is in good agreement with literature [53], [54]. For what concerns the measurement uncertainty comparison of the new DR configuration with other measurement techniques, we can assess that the technique proposed in this work is aligned with the typical uncertainty levels of this kind of measurement. In [54], the complexε r of acrylonitrile butadiene styrene (ABS) samples doped with different amounts of BaTiO 3 microparticles was measured by means of a split-post dielectric resonator (SPDR) with a nominal resonance frequency of 15 GHz. SPDR is well known to be one of the most accurate methods for the measurement of planar dielectric samples [2], [14]. In [54] uncertainties u(ε r )/ε r ∼ 1% (including systematic contributions) and u(tan δ)/ tan δ ∼ 0.2% (evaluated as the standard deviation of six measurements) were reported. In [20], bulk ABS samples were measured at 10 GHz with a dielectric resonator, giving at room temperature u(ε r )/ε r ∼ 1% and u(tan δ)/ tan δ ∼ 20%. Broadband techniques were also used although, as previously stated, they are usually less sensitive and accurate than resonant methods [2]. In [53], a Nicholson-Ross-Weir waveguide method was used in the 8.2-11-GHz band on a wide variety of 3-D printing materials obtaining a maximum standard deviation for the set of ABS samples u(ε r )/ε r ≤ 5.8% and an approximate range of estimated errors 10%-300% on tan δ, although a complete uncertainty analysis in this frequency band was not presented. Therefore, the new configuration of the showed DR allows the characterization of samples with geometries and sizes difficult to characterize in other ways and with uncertainties aligned with the current state of the art.
The obtained results from the DR and the SRR are almost the same despite the different working frequencies. Even the uncertainties given by both techniques are the same despite the SRR being much more sensitive to ε r of the material under investigation than the DR. The sensitivity of the SRR f 0 to ε r variations is estimated to be ∂ f 0 /∂ε r ∼ 112 MHz at ε r = 2.4 while that of DR, from Fig. 4, ∼20 MHz in the same conditions and with the thickest sample (t = 2 mm). This is an expected result since both SRR and DR are based on volume perturbation, so that the SRR is favored by the use of thicker (bulk) dielectric samples. However, the typical final uncertainty is the same for both techniques, since u( f 0 )/ f 0 is much smaller for the DR thanks to its higher quality factor 2500 < Q < 5000 in the conditions of the measurements here presented. A final remark should be made about a possible anisotropy of printed samples. Using a waveguide reflection method, a uniaxial anisotropy factor of ∼7% was measured on ε r of 3-D-printed polylactide acid (PLA) samples at 40 GHz [55]. The samples were produced with the standard printing procedure, based on the extrusion and deposition of fused filaments. Such printing technique generates anisotropic structures due to the discrete and weakly connected printed layers [55], as also shown in [56]. On the contrary, for the realization of the samples here studied, a PolyJet printer was used: the material is deposited in liquid form and then polymerized with UV-rays. Thus, one expects a tighter connection between the printed layers, with a reduced amount of defects and an overall reduced anisotropy. The resulting anisotropy level is then believed to be well below u(ε r ) and u(ε r ). In addition, the DR is excited in a quasi-TE 011 mode and thus the electric field vector is, within a good approximation, directed along the printing layers. In the SRR, instead, the electric field direction is not directed along some preferred e.m. direction of the sample. Since no evident difference was measured onε r with both resonators, we conclude that the small anisotropy of the samples, including the contribution given by the aligned holes, does not affect the results presented.
At the end of the calibration and validation procedure of this new configuration of DR, we can reliably use this fixture for the characterization of other substrates materials. We measured ε r of a FR-4 fiberglass substrate with copper ground plane obtaining ε r = 4.76(8) and tan δ = 1.73(6) × 10 −2 fairly in agreement with [57] and [58]. We also tested the DR with Kapton 1 polyimide stacking ten layers of 127-μm-thick films and obtaining ε r = 4.01(5) and tan δ = 9.9(2) × 10 −3 in good agreement with [59]. These comparisons with the literature further validate the proposed fixture.

VI. SUMMARY
We have demonstrated the application of a DR to the measurement of the complex relative permittivityε r of dielectric samples printed with a PolyJet 3-D-printer. The 3-D printing technique allowed us to obtain samples with different filling factors, to vary the effective relative permittivityε r in a geometrically controlled way. The dependence ofε r on the density of the sample was found to be well described by the Wiener upper bound in agreement with [50], [51], and [52]. The measurement technique was validated by comparing thẽ ε r measurements performed with an SRR.
In conclusion, the obtained results demonstrated the possibility to employ a DR for microwaveε r characterization of 3-D printing materials, and soft substrates for electronic circuits with similar e.m. characteristics even in the presence of back metal ground plates.