# Determination of the Dielectric Properties of Storage Materials for Exhaust Gas Aftertreatment Using the Microwave Cavity Perturbation Method

^{1}

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## Abstract

**:**

## 1. Introduction

_{x}) storage of a lean NO

_{x}trap [14,15], and the soot load of a diesel particulate filter (DPF) [16,17] can be evaluated with the RF system.

_{2}), the oxygen storage component of the TWC [21]. The effect of Pt on the reducibility of CeO

_{2}at low temperatures was also confirmed with the microwave technique.

_{1}and the dielectric loss ε

_{2}during operation. However, exact values cannot be determined with this method.

## 2. Laboratory Setup, Simulation Model and MCP-Fundamentals

#### 2.1. Resonator Setup

_{c}80 mm) is made of aluminum. Several quartz glass tubes are inserted along the resonator axis. The sample with a height h

_{s}is located in the center of the resonator cavity on a porous quartz glass frit in the inner tube (Ø 10 mm) and can be flushed vertically with different process gases. The inner sample tube is encased by a double tube (Ø 20 mm and Ø 38 mm). The space between the double tube is evacuated to minimize thermal losses. Around the inner sample tube, the sample is heated indirectly by flowing heating gas (hot air), which allows sample temperatures of up to 600 °C. To measure the sample temperature, the NiCr-Ni thermocouples (TC) are inserted into the sample tube from above and below the resonator. Thus, the sample temperature is determined by taking the arithmetic mean of both temperatures [18]. Additionally, the thermocouples penetrate the sample tube only so far that they do not affect the RF measurement. A circulated water cooling is used to maintain the room temperature of the aluminum resonator during the operation. Further details on the design (sealing system, temperature distribution, thermal considerations, etc.) and more technical specifications can be found in [18].

_{0n0}modes (f

_{TM010}= 1.18 GHz, f

_{TM020}= 2.62 GHz, f

_{TM030}= 4.19 GHz). Usually, these modes have a constant E-field maximum along the resonator axis. The sample is, therefore, always located in the electric field maximum, the area with the highest RF sensitivity. The field distribution of the TM

_{010}fundamental mode, which has exactly one field maximum along the resonator axis, is shown qualitatively in Figure 1a,b.

^{®}5.5), reflecting the detailed cavity resonator design is used as well (Figure 1b). Implemented are the resonator housing made of aluminum (conductivity σ

_{Al}= 3.8 10

^{7}S/m), the quartz glass tubes (dielectric constant ε

_{r,quartz}’ = 4.35, no losses), and antennas as well as the top and bottom connectors made of stainless steel (σ

_{Steel}= 4.0⋅10

^{6}S/m). For air as the filling medium, a dielectric constant ε

_{r,air}’ = 1 and magnetic constant µ

_{r,air}’ = 1 were assumed. Calculations for the electrical field distributions were carried out by a modal analysis (eigenvalue problem).

#### 2.2. Microwave Cavity Perturbation Theory

_{0}= 8.854 ⋅ 10

^{−12}As/(Vm). The dielectric losses ${\epsilon}^{\u2033}$ include the (relative) losses ${{\epsilon}_{\mathrm{r}}}^{\u2033}$ due to the sample polarization in the alternating electromagnetic field as well as the ohmic losses due to the electric conductivity σ. The ohmic loss is a function of the circular frequency ω = 2πf of the electromagnetic wave. The change in resonant properties of the cavity resonator due to the insertion of a sample can be derived from Maxwell’s equations. Generally, it has the from [22,23,24,25,26,27]:

_{C}, the resonant frequencies f

_{0}and f

_{S}, the (unloaded) quality factors Q

_{0}and Q

_{S}of the empty and filled resonator, the electric and magnetic fields E

_{0}and H

_{0}of the empty resonator, their complex conjugated fields ${E}_{0}^{\text{}\ast}$, ${H}_{0}^{\text{}\ast}$ and the corresponding fields E

_{1}and H

_{1}of the filled resonator. For the volume outside of the sample the dielectric permittivity ε

_{0}and the magnetic permeability µ

_{0}of vacuum are assumed. In many cases this relation can be simplified. According to the classical MCP theory, the following correlation applies to the resonant properties of the TM

_{0n0}modes of cylindrical resonators and the dielectric properties of an inserted sample, located in the central electric field maximum [22,23]:

_{S}and effective volume V

_{eff}of the resonant mode (also denominated as mode volume). The mode volume considers the electric field distribution inside the cavity and corresponds to a kind of (inverse) “RF sensitivity” for the measurement method [22,23]. The smaller V

_{eff}, the higher is the change in the resonant parameters due to the insertion of a sample. The size of V

_{eff}for ideal cylindrical resonators can be calculated [22,23,25]:

_{1}of the 1st kind and 1st order and the n-th zero p

_{0n}of the Bessel function of the 1st kind and 0th order for the TM

_{0n0}mode. In case of the TM

_{010}mode V

_{eff}is 26.95% of V

_{res}. In such cases, the dielectric properties of the sample can be calculated easily from the changes in the resonant properties. The validity of this approach is, however, subject to a number of basic pre-conditions that are particularly crucial for the investigation of powders for exhaust aftertreatment [22,23,24,25,26,27,28,29,30,31,32,33]:

- The sample volume is small compared to the resonator volume.
- The sample height and the resonator height are identical (h
_{s}= h_{c}) or the sample has the shape of a thin rod at least. - The sample material is homogeneous and isotropic,

#### 2.3. Evaluation for Investigations on Powders for Exhaust Gas Aftertreatment

_{0n0}modes, whose electric field maxima drop rapidly in radial direction, significant deviations are expected for a sample diameter of 10 mm. A better consideration of the electric field distribution inside the resonator can be achieved by a simulative approach.

_{s}to the height of the resonator. However, the perturbation of the electric field also increases with the amount of powder. Especially for samples with high permittivity or high dissipation factor (tanδ = ${\epsilon}^{\u2033}$/${\epsilon}^{\prime}$), the accuracy of the method quickly diminishes [34,35]. This fact is particularly crucial for the investigation of storage materials for exhaust gas aftertreatment (such as zeolites, ceria), since their dielectric losses increase rapidly with adsorption or reduction, and with temperature as well [19,20,21,36,37].

_{s}less than the height of the cavity h

_{C}generate a depolarization field that counteracts the excitation field and weakens the net field inside the sample [24,25,32,38,39,40]. In the literature, various approaches to describe the depolarization effect have been established using different approximations for the resonator type (rectangular, cylindrical, split-ring) and sample dimensions [32,38,39,40]. In this study, a method is presented based on theories to account for depolarization in a cylindrical resonator using the TM

_{0n0}modes for different cylindrical sample geometries.

## 3. Extension of the Microwave Cavity Perturbation Theory

#### 3.1. Electric Field Distribution Inside the Cavity Resonator

_{eff}plays a central role for the correct evaluation of the properties of the introduced sample. The magnitude of V

_{eff}depends mainly on the field distribution inside the cavity and, thus, also on the resonant mode. Figure 2 shows the simulated field distributions |E

_{TM0n0}| of the (a) TM

_{010}, (b) TM

_{020}, and (c) TM

_{030}modes. In the center of the resonator, the typical position and geometry of the sample is shown (h

_{s}= 5 mm), whose properties match those of the filling medium (air) in this case. As it can be expected for of the TM

_{0n0}resonant modes, the electric field maximum is located along the resonator axis. While the TM

_{010}has only one maximum in the center, the higher modes have ring-shaped secondary maxima around the central axis. The width of these maxima decreases with the increasing harmonic number.

_{eff}, respectively. Therefore, a significant deviation from the theoretical value (simplified approach) for ideal cylindrical resonators according to Equation (5) must be expected. Instead, a more precise value for V

_{eff}can be determined with an approach considering the electric field distribution [22,23]:

_{0}= E

_{1}can be assumed, since its field perturbation is negligible. The calculated values for V

_{eff}inside the sample volume are shown in Table 1. For comparison, the values for the simplified approach for thin rod-shaped samples according to Equation (5) are also given. For the TM

_{010}mode both results are obviously identical. The simplified approach may be used here, as the sample is well placed within the area of the maximum electric field (Figure 2a). The deviations between both methods (here <1%) could partially result from the accuracy of the model.

_{020}and TM

_{030}modes ((b) and (c)). Especially for the TM

_{030}, the simplified approach leads to significant other RF sensitivities, or V

_{eff}, respectively. The effect is correspondingly large for the derivation of the dielectric parameters. In the case of the relative permittivity ${{\epsilon}_{\mathrm{r}}}^{\prime}$, the relative deviation Δε

_{rel}between the simplified approach and the method based on the integration of electric field squares can be expressed by considering Equation (3):

_{rel}as a function of the dielectric constant ${\epsilon}_{\mathrm{r}}{}^{\prime}$ of the sample:

_{010}mode. For higher modes, the simplified calculation leads to significant errors. Already with smaller dielectric constants the deviations rapidly increase and rise (for ${\epsilon}_{\mathrm{r}}{}^{\prime}$ > 10) up to 15% in case of TM

_{020}and exceed 40% for the TM

_{030}mode. Hence, consideration of the real field distribution in the resonator is crucial for the correct determination of the dielectric properties in this case. For the investigation of a solid ceria sample (sintered pellet) with ${\epsilon}_{\mathrm{r}}{}^{\prime}$ = 23 [44,45,46] the approach of Equation (6) should be used. Nevertheless, even for powders whose effective dielectric constants are usually much smaller (depending on the bulk density of the powder), the error in the calculation using the simplified approach cannot be neglected. The extended approach provides a significant improvement for the evaluation of RF sensitivity when measuring a sample in the excitation field of the resonator.

_{eff}does not depend on the sample height, since the electric field along the resonator axis remains constant for the TM

_{0n0}modes. However, the simulation (Figure 2) shows that the electric field of all modes decreases towards the two openings at the resonator bottom and top plane. For samples with a larger h

_{s}it is, therefore, expected that the modal volumes V

_{eff}will increasingly differ from the ideal theoretical value. For longer samples, the method using the E-field squares is therefore already recommended for the TM

_{010}mode. For smaller sample heights, the result for the modal volume remains widely stable, because the electric field changes along the symmetry axis can be neglected. The results shown in Table 1 and Figure 3 are therefore valid for samples with similar heights (h

_{s}≈ 5 mm).

_{030}mode (Figure 2c), the magnitude of the electric field strength also decreases from the center along the resonator axis. Furthermore, it is noticeable that the propagation of the main maximum is not limited to the cylindrical resonant cavity, but parts of the electric field are excited outside of it along the quartz tubes. Here, an approach that considers the electric field distribution provides more precise values for the modal volume V

_{eff}. However, the accuracy of approaches for cylindrical resonators is questionable in this case.

#### 3.2. Depolarization Field of the Sample

_{s}< h

_{c}a depolarization effect is observed within their volume. The sample material causes a depolarization field against the excitation field of the resonator. It weakens the total electric field within the sample [24,25,32,38,39,40]. For cylindrical samples, the extent of the depolarization field increases with their diameter/height ratio. The effect is, thus, more significant for flat samples than for elongated [32,40,47]. In Figure 4 the effect of depolarization is demonstrated by a simulation. The weakening of the electric field within the sample volume (h

_{s}= 5 mm, ${\epsilon}_{\mathrm{r},\mathrm{eff}}{}^{\prime}$ = 2.59, ${\epsilon}_{\mathrm{r}}{}^{\u2033}$ = 0) is clearly visible (Figure 4a). The chosen dielectric constant is typical for a loosely packed ceria powder, which will be investigated later to validate this study.

_{1}(black) with the excitation field of the empty resonator E

_{0}(grey dashed line) clearly shows the importance of the effect for the MCP method. Generally, the weakening of the electric field is also locally different. The depolarization is especially prominent in the vicinity of the upper and lower plane of the sample and decreases towards its center [40,47]. However, due to the small sample height, this local effect can hardly be observed for the selected sample shape. It should be noted that the permittivity of the sample also plays a role for the degree of depolarization. Materials with high dielectric constants yield a stronger depolarization field [40,47,48,49].

_{1}, the excitation field E

_{0}of the undisturbed resonator, the geometry-dependent depolarization factor N, and the polarization field P due to the introduction of the sample. The correlation between the polarization field P and a unidirectional excitation field E

_{0}(here: z-axis) [38,46,49] is given by:

_{z}of the sample in z-direction. The depolarization factor N allows values between 0 and 1 and serves as a weighting factor for depolarization. Its magnitude depends on both the sample shape and the orientation of the sample in the excitation field. If the sample is polarized along all three spatial directions, the individual depolarization factors add up to a value of 1:

_{cyl}of the cylindrical sample, the polarizability α

_{sph}of an ellipsoid with identical axial ratio and volume and the correction term for polarizability Δα. The correction term becomes small especially for very flat and very long samples. Values for Δα are, for example, tabulated in the numerical study [49]. However, the described method for determining the depolarization factor does not take into account conductive surfaces near the sample. For a sample inside the cavity, a change of the depolarization factor and, therefore, of the depolarization field is expected. Parkash et al. describe the effect on the depolarization factor of the sample using the mirror charge method [25]:

_{e}, which still depends strongly on the sample shape and dimensions. In the resonator, however, it is also a function of the height ratio of sample h

_{s}and resonator h

_{C}. As mentioned above, the depolarization of the sample also affects the RF sensitivity. The approach according to Equation (8) must therefore be considered in (2). The correlation between the dielectric properties of the sample and the resonant properties of the TM

_{0n0}modes of a cylindrical resonator (filling medium: ε

_{0}, µ

_{0}) with account for sample depolarization can be described by the following equations [25]:

_{s}< h

_{c}can be determined much more precisely with the extended method.

_{s}= 5 mm, ${\epsilon}_{\mathrm{r},\mathrm{eff}}{}^{\prime}$ = 2.59) are determined with the model shown in Figure 4. For the sample geometry, an effective depolarization factor of 0.4354 is calculated, which meets the expectations for a slightly flat cylinder. For the modal volumes V

_{eff}of the TM

_{0n0}modes the simulative results of Table 1 were used.

#### 3.3. Bulk Properties of the Powder Samples

_{eff}, the complex permittivity ε

_{i}and the volume fraction ν

_{i}of the i-th constituent and the number of constituents n. For Looyenga’s model the exponent k is 1/3 [53]. Since the effective medium approach according to Looyenga does not take into account the shape of the particles, the formula can be used especially well for homogeneous mixtures, such as powders in this case [53]. Other studies have also shown that compared to other mixing models, Looyenga’s model gives significantly better results for volume fractions <30%, which is typical for loose powder fillings, and for strongly dissipative particles, which could be important in the chemical reduction of ceria [54,55,56]. Theoretical reflections and findings of previous studies therefore indicate that the effective medium approach according to Looyenga might be suitable for investigations on ceria powders.

_{a}5 mm, Ø

_{i}3 mm) and fills the resonator (Ø 45 mm, h

_{c}= 40 mm) from the bottom to the top plane. To avoid water interferences, the sample was dried for 48 h at 120 °C. The volumetric proportion ν

_{CeO2}of ceria in the bulk is 20.6%. For the validation of a suitable mixture model, the TM

_{010}mode is used, because of its superior reliability over the higher modes. For this resonator geometry, it occurs at around 2.5 GHz. The modal volume of the TM

_{010}can be assumed to meet the theoretical value (26.95% of the cavity volume) in this case, since the diameter ratio between the sample and the cavity is even smaller than for the geometry of the original resonator presented in Figure 1. And since h

_{s}= h

_{c}, obviously, any sample depolarization is avoided, and the simplified approach according to Equations (3) and (4) is valid for the given configuration.

## 4. Validation of the Measurement Method, Transferability and Alternative Approaches

_{2}) in nitrogen (N

_{2}) and hydrogen-water mixture in N

_{2}with p

_{O2}≈ 10

^{−20}bar, respectively). Due to the formed free electrons in the reduced state and the thermally activated mobility due to the small-polaron conduction mechanism, the conductivity of the material is strongly increased. This behavior can be well described by the defect chemistry. Details on the defect chemistry of ceria and its effects on the electrical material properties, are provided in the literature [36,37], but is beyond the scope of this work. The results for the dielectric constant and the electrical conductivity according to the TM

_{010}, TM

_{020}, and TM

_{030}signals are shown in Table 4a–c.

_{e}= 0.402) (Section 3.2), and the Looyenga mixing model (Section 3.3). The results for the measurement at room temperature are given in Table 4a. For the three modes, a dielectric constant of 22.6–23.8 was determined. The calculated properties are therefore consistent with findings from literature (${\epsilon}_{\mathrm{r}}{}^{\prime}$ = 23) [8,44,45,46]. However, the losses inside the material at room temperature are too low to be determined by the measurement method. To highlight the effects of the new approach on the MCP measurement, Table 4a also shows alternative calculations without considering the accurate electric field distribution or without considering the depolarization field. If the simplified approach according to Equation (5) is used for the field calibration, the dielectric constant can still be quite accurately determined with the TM

_{010}mode (${\epsilon}_{\mathrm{r}}{}^{\prime}$ = 23.0). This finding is in line with the results from Section 3.1. However, for the higher modes, the RF sensitivity is assumed to be higher, yielding significantly underestimated dielectric constants (${\epsilon}_{\mathrm{r}}{}^{\prime}$ = 16.0 for the TM

_{020}mode and ${\epsilon}_{\mathrm{r}}{}^{\prime}$ = 7.91 for the TM

_{030}mode). This evidences that a precise consideration of the field distribution (Equation (6)) is essential for providing more accurate results for the higher modes. Similar observations can be reported if the depolarization of the sample is neglected (Ne = 0). In this case, the weakening of the net field within the sample is not considered, which effectively reduces the signal amplitudes in the RF measurement. Therefore, the determined dielectric constants are too low (${\epsilon}_{\mathrm{r}}{}^{\prime}$ = 11.2–11.5). Since depolarization depends mainly on the shape of the sample, the phenomenon affects all three modes equally. The additional calculations show obviously that the field distribution, the occurring depolarization and the powder filling are correctly considered and a successful determination of the dielectric properties of the sample is only possible if all three effects are combined in the calculation.

_{030}mode, which shows a distinct stronger attenuation compared to the other two modes. This consequently leads to smaller quality factors (<1000) and restricts the precision of the measurement. The fact that the TM

_{010}and TM

_{020}modes are more reliable must, therefore, be attributed to the properties of the resonator itself and not to the presented approach. Due to the small signal amplitudes for oxidized ceria, reasonable conductivities can only be calculated from the TM

_{010}and TM

_{020}signals.

^{−5}S/cm can be determined for the oxidized ceria at 600 °C, which again corresponds well with values measured in the literature [36]. Thus, the approach also provides plausible values for the dielectric losses of ceria.

^{−5}S/cm) and are close to the values reported in literature [36]. The conductivity can be determined even from the TM

_{030}, since the signal amplitudes are significantly higher due to the reduction of the sample. The results show that the presented method allows to successfully determine the dielectric properties of the ceria sample. The polarization and dielectric loss of the sample under typical operating conditions can be evaluated properly. The presented method now takes into account the field distribution in the resonator, the depolarization of the sample and the filling of the powder. The approach extends the application range of MCP measurements significantly and provides reliable information about the dielectric properties of the inserted sample.

_{s}or a high (effective) dielectric constant, the disturbed field has to be taken into account. This could be necessary, for example, if the bulk densities or dielectric constants of the powder particles are significantly higher. Due to the low package density of powders, the assumption should be justified in most cases. For example, the measured dielectric constant ${\epsilon}_{\mathrm{r},\mathrm{eff}}{}^{\prime}$ for ceria was only 2.59 at a ceria volume fraction of 20.6%.

_{s}. On the other hand, errors in the estimated sample height lead to larger changes in the depolarization factor for smaller samples. In addition, Parkash et al. have already indicated that the approach according to Equations (14) and (15) can deliver inconsistent results in case of strong sample depolarization, or very flat samples, respectively [25]. In this study, the effect of the porous quartz frit on the sample depolarization was also neglected. Therefore, the height of the investigated material should preferably be as large as possible without violating other MCP conditions. Applying the method to very flat structures or materials (layers) is not recommended. It is also worth mentioning that the depolarization factor may also be determined by finite element simulation. However, this would require an individual simulation model for each investigated sample, because the simulation model must be specifically adapted to the shape and properties of the sample. An analytical approach can be implemented more quickly and also provides reasonably accurate results for the depolarization factor according to acknowledged methods. Venermo et al. quantified the deviation of their method for determining depolarization factors for cylindrical samples to be less than 1% [49]. From this perspective, the determination of the depolarization factor by simulations is also possible, but not more beneficial, since the cylindrical approximation of the sample geometry alone causes larger deviations. Alternatively, the sample depolarization can also be considered by insertion of a sample with known dielectric properties and geometry. Such calibration sample must have an identical shape as the powder sample in order to correctly determine the field weakening. However, for this method, a calibration measurement is required before each investigation, which only can be applied to the specific sample shape. An analytical approach is, hence, more flexible and can be easily adapted to the individual geometry of a sample.

## 5. Conclusions

_{x}storage materials or soot, with only a few modifications and, thus, has the potential to provide further important insights into microwave characteristics and the functioning of typical exhaust aftertreatment materials. The findings contribute to improving the microwave-based state diagnosis of automotive catalytic converters.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Illustration of the resonator setup for MCP measurements with the electric field (qualitatively) of the TM

_{010}mode: (

**a**) schematic sectional view of the setup; (

**b**) simulation model in COMSOL Multiphysics

^{®}5.5.

**Figure 2.**Simulated electric field inside the empty resonator (sectional view): (

**a**) TM

_{010}; (

**b**) TM

_{020}; (

**c**) TM

_{030}.

**Figure 3.**Relative deviation Δε

_{rel}between both methods for the calculation of the dielectric constant as a function of the dielectric constant of the sample ${\epsilon}_{\mathrm{r}}{}^{\prime}$

**Figure 4.**Depolarization of a ceria powder (${\epsilon}_{\mathrm{r},\mathrm{eff}}{}^{\prime}$ = 2.59) in the excitation field of the resonator: (

**a**) electric field distribution (sectional view) in the resonator and within the sample; (

**b**) electric field along the x-axis with pronounced weakening at the location of the sample (x = 0).

**Figure 5.**Effective dielectric constant of ceria powder, ${\epsilon}_{\mathrm{r},\mathrm{eff}}{}^{\prime}$, as it depends on volume fraction ${\nu}_{\mathrm{CeO}2}$ of ceria, according to some dielectric mixing models from literature.

**Figure 6.**Setup for the characterization of the dielectric properties of ceria powder at room temperature.

**Table 1.**Comparison of the modal volumes of the TM

_{010}, TM

_{020}, and TM

_{030}modes of the resonator.

Mode | TM_{010} | TM_{020} | TM_{030} |
---|---|---|---|

V_{eff}/V_{C} (simulation) | 26.78% | 13.73% | 12.84% |

V_{eff, th}/V_{C} (simplified approach) | 26.95% | 11.58% | 7.37% |

**Table 2.**Calculated dielectric constant of a ceria powder from simulation data with and without consideration of depolarization.

Mode | f_{s}/GHz | f_{0}/GHz | Q_{S} | Q_{0} | N_{e} | ${\mathbf{\epsilon}}_{\mathbf{r},\mathbf{eff}}{}^{\prime}$ (Equation (14)) | ${\mathbf{\epsilon}}_{\mathbf{r},\mathbf{eff}}{}^{\prime}$ (Equation (13)) |
---|---|---|---|---|---|---|---|

TM_{010} | 1.179563 | 1.179973 | 19,787 | 19,781 | 0.435 | 2.66 | 1.97 |

TM_{020} | 2.624186 | 2.625898 | 30,183 | 30,818 | 0.435 | 2.56 | 1.93 |

TM_{030} | 4.237119 | 4.240028 | 25,821 | 26,628 | 0.435 | 2.52 | 1.91 |

**Table 3.**Results of the investigation of the ceria powder sample in the simplified resonator geometry.

f_{s}/GHz | f_{0}/GHz | ${\mathbf{\epsilon}}_{{r},\mathbf{eff}}{}^{\prime}$ | ${\mathbf{\epsilon}}_{{r}}{}^{\prime}$ (Looyenga) | ${\mathbf{\epsilon}}_{{r}}{}^{\prime}$(Birchak) | ${\mathbf{\epsilon}}_{{r}}{}^{\prime}$(Lichtenecker) |

2.470882 | 2.478675 | 2.59 | 22.4 | 15.7 | 101 |

**Table 4.**Results for the dielectric properties of the ceria sample inside the resonator cavity: (a) measured at 25 °C and 21% O

_{2}; (b) at 600 °C and 21% O

_{2}; (c) at 600 °C and p

_{O2}= 10

^{−20}bar. The derived values for the material permittivity are highlighted. Please note the lower permittivity for higher modes when applying simplifications. The permittivity for the TM

_{030}mode in (b) is written in brackets, due to its uncertainty.

(a) Conditions: 25 °C, p_{O2} = 0.21 bar | Literature:${\mathbf{\epsilon}}_{{r}}{}^{\prime}$= 23 [44,45,46] | |||||||

Mode | f_{s}/GHz | f_{0}/GHz | Q_{S} | Q_{0} | N_{e} | V_{eff}/V_{C} | ${\epsilon}_{{r}}{}^{\prime}$ | σ/(S/cm) |

TM_{010} | 1.180624 | 1.180870 | 13,890 | 13,930 | 0.402 | 26.78% | 22.6 | -- |

TM_{020} | 2.629937 | 2.631025 | 13015 | 12,756 | 0.402 | 13.73% | 23.6 | -- |

TM_{030} | 4.200389 | 4.203252 | 893.34 | 898.65 | 0.402 | 12.84% | 23.8 | -- |

For comparison: Calculation with simplified electric field calibration (V_{eff}): | ||||||||

TM_{010} | 1.180624 | 1.180870 | 13,890 | 13,930 | 0.402 | 26.95% | 23.0 | -- |

TM_{020} | 2.629937 | 2.631025 | 13015 | 12,756 | 0.402 | 11.58% | 16.0 | -- |

TM_{030} | 4.200389 | 4.203252 | 893.34 | 898.65 | 0.402 | 7.37% | 7.91 | -- |

For comparison: Calculation without considering depolarization (N_{e} = 0): | ||||||||

TM_{010} | 1.180624 | 1.180870 | 13,890 | 13930 | 0 | 26.78% | 11.2 | -- |

TM_{020} | 2.629937 | 2.631025 | 13,015 | 12756 | 0 | 13.73% | 11.5 | -- |

TM_{030} | 4.200389 | 4.203252 | 893.34 | 898.65 | 0 | 12.84% | 11.5 | -- |

(b) Conditions: 600 °C, p_{O2} = 0.21 bar | Literature:${\mathbf{\epsilon}}_{{r}}{}^{\prime}$= 23 [8,43,44,45,46], σ = 3.9 10^{−5} S/cm [36] | |||||||

Mode | f_{s}/GHz | f_{0}/GHz | Q_{S} | Q_{0} | N_{e} | V_{eff}/V_{C} | ${\mathbf{\epsilon}}_{{r}}{}^{\prime}$ | σ/(S/cm) |

TM_{010} | 1.179082 | 1.179328 | 12,327 | 12826 | 0.402 | 26.78% | 22.7 | 1.79·10^{−5} |

TM_{020} | 2.625354 | 2.626425 | 9572.3 | 10341 | 0.402 | 13.73% | 22.8 | 1.58·10^{−5} |

TM_{030} | 4.192808 | 4.194524 | 943.23 | 1210.1 | 0.402 | 12.84% | <17.6> | -- |

(c) Conditions: 600 °C, p_{O2} = 10^{−20} bar | Literature: σ = 2.0 10^{−2} S/cm [36] | |||||||

Mode | f_{s}/GHz | f_{0}/GHz | Q_{S} | Q_{0} | N_{e} | V_{eff}/V_{C} | ${\mathbf{\epsilon}}_{{r}}{}^{\prime}$ | σ/(S/cm) |

TM_{010} | 1.178967 | 1.179341 | 3441.6 | 12,878 | 0.402 | 26.78% | 42.8 | 4.22 10^{−2} |

TM_{020} | 2.625034 | 2.626473 | 2488.3 | 10,335 | 0.402 | 13.73% | 39.0 | 5.50 10^{−2} |

TM_{030} | 4.192956 | 4.194998 | 806.22 | 1210.8 | 0.402 | 12.84% | 23.5 | 5.48 10^{−2} |

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**MDPI and ACS Style**

Steiner, C.; Walter, S.; Malashchuk, V.; Hagen, G.; Kogut, I.; Fritze, H.; Moos, R.
Determination of the Dielectric Properties of Storage Materials for Exhaust Gas Aftertreatment Using the Microwave Cavity Perturbation Method. *Sensors* **2020**, *20*, 6024.
https://doi.org/10.3390/s20216024

**AMA Style**

Steiner C, Walter S, Malashchuk V, Hagen G, Kogut I, Fritze H, Moos R.
Determination of the Dielectric Properties of Storage Materials for Exhaust Gas Aftertreatment Using the Microwave Cavity Perturbation Method. *Sensors*. 2020; 20(21):6024.
https://doi.org/10.3390/s20216024

**Chicago/Turabian Style**

Steiner, Carsten, Stefanie Walter, Vladimir Malashchuk, Gunter Hagen, Iurii Kogut, Holger Fritze, and Ralf Moos.
2020. "Determination of the Dielectric Properties of Storage Materials for Exhaust Gas Aftertreatment Using the Microwave Cavity Perturbation Method" *Sensors* 20, no. 21: 6024.
https://doi.org/10.3390/s20216024