# Precision Permittivity Measurement for Low-Loss Thin Planar Materials Using Large Coaxial Probe from 1 to 400 MHz

^{*}

## Abstract

**:**

_{r}and tangent loss, tan δ via closed form capacitance model and lift-off calibration process. Average measurement error of dielectric constant, Δε

_{r}is less than 6% from 1 MHz to 400 MHz and the resolution of loss tangent, tan δ measurement is capable of achieving 10

^{−3}.

## 1. Introduction

_{r}measurements in most of the material processing industries are up to a few hundred MHz [1,2,3].

_{r}of a material. Coaxial probes have been commercialized and used commonly since 1990 [4]. Recently, several probes have produced by manufacturers such as SPEAG Inc. [5], KEYCOM [6], and APREL Inc. [7]. However, coaxial probes (N-type’s or SMA’s diameter size) are less sensitive to small changes in the MUT especially for thin and low-loss materials in permittivity, ε

_{r}measurement at MHz frequency. This causes the measurement results for low-loss materials at low frequencies to be highly scattered and less precise [8]. In fact, most coaxial probes are only suitable for half-space infinite lossy material with ε

_{r}′ > 5 and tan δ > 0.05 [4,5,6,7,8,9,10,11].

_{r}of low-loss materials having thickness of 1 mm precisely from 1 to 400 MHz. The designed large probe is heavy whereby it is capable of supplying stable reflection measurements. Additionally, it is sensitive to measurements at low frequencies due to its large size. Furthermore, large probe aperture (sensing area) can reduce the uncertainty of measurement, which mostly caused by the inequality of composite distribution in MUT, especially for inhomogeneous MUTs. The probe design of the probe and its performance were analyzed in detail. An explicit formulation for the prediction of ε

_{r}, which does not involve numerical inversion routines (iterative method) as in Refs. [8,9,10,11] is used. Since this study is focused on low frequencies measurement (up to 400 MHz), thus, a simple closed-form capacitance model [12,13,14,15,16,17,18] is implemented to predict the dielectric properties of MUT based on the measured reflection coefficient at probe aperture. The differences in this study compared with previous works are summarized in Table 1. In Refs. [8,9,10,11], the iterative methods (inverse methods) used to estimate the permittivity, ε

_{r}of lossy MUT in which the predicted values of ε

_{r}are obtained by minimizing the difference between the measured aperture reflection coefficient and the theoretical calculations. Those iterative methods are complicated solutions and less suitable for real-time ε

_{r}estimation.

## 2. Large Coaxial Probe Design

#### 2.1. Coaxial Probe Formulations

_{o}terms in which it is easily used to estimate the relative complex permittivity, ε

_{r}of MUT at low operating frequencies as follows:

_{o}is the characteristic admittance of coaxial line. In 1986, the modeling of open-ended coaxial probe that is terminated by layered media was studied by Ref. [22]. Subsequently, probe modeling studies which considered multi-layer MUT were rapidly developed during 1990 [23,24,25,26,27,28,29,30,31,32,33].

#### 2.2. Dimensions and Structure

_{o}= [60 × ln(b/a)/√ε

_{c}] = 50 Ω. The maximum limit of the operating frequency, f

_{max}propagating in the coaxial line of the probe is determined using TE

_{11}cut-off as: f

_{max}= (3 × 10

^{8})/[π(b + a)√ε

_{c}] ≈ 2.1 GHz. In fact, the limit of operation frequency of the practice coaxial line is always much lower than the TE

_{11}cutoff frequency based on the quality and precision of machining. The symbol ε

_{c}(Air: ε

_{c}= 1, Teflon: ε

_{c}= 2.06) represents the relative permittivity of material filling in the coaxial line in between the inner and outer conductor. The total weight of the coaxial probe is 2.6 kg. It has been divided into three sections: (I) N-type connector, which is used to connect the coaxial probe with the network analyzer via cable. (II) Transition section, which is a 50 mm of air-filled conical taper. The radius a and b of the conductors are increased along the transition length with a constant ratio, b/a = 2.3. Ratio, b/a = 2.3 is required to maintain Z

_{o}= 50 Ω along the transition length to achieve low return loss and the lowest standing wave ratio (SWR) during the transformation from small to large coaxial line. (III) Large coaxial line section, which is a 100 mm length of 50 Ω Teflon-filled coaxial line with b/a = 3.3. The Teflon isolation block is used to prevent the MUT from getting into the coaxial line. In addition, Teflon has high flexural strength, excellent chemical resistance, and high stability over a wide temperature range.

#### 2.3. Probe Characterization Test

_{AA′}= |Γ

_{AA′}|exp(jφ

_{AA′}) at plane AA′ for four MUTs were measured using Keysight E5071C network analyzer (Keysight Technologies, Santa Rosa, CA, USA) in the frequency ranging from 0.3 MHz to 650 MHz at 25 °C. Calibration was done at the AA′ plane, as shown in Figure 2, using Keysight 85032F kit (Keysight Technologies, Santa Rosa, CA, USA).

_{AA′}(rad) as shown in Figure 3a,b starts to become constant when the metallic plate is moved away from the probe aperture in air. Figure 4 shows the time-domain measurements of the coaxial probe, which is operated with minimum windowing and bandpass mode. Clearly, coaxial probe of 15 cm length is sufficient to avoid the interference between plane-AA′ and -BB′ for the frequency-domain Γ

_{AA′}measurement.

## 3. Calibrations

#### 3.1. Calibration Formulations

_{BB′}at the probe aperture (BB′ plane) and measured Γ

_{AA′}can be represented by an error network as shown in Figure 5, and its formulation is given as [34]:

_{12}e

_{21}terms in Equation (2) represents the values of tracking error, as well as the e

_{11}and e

_{22}are the values of the directivity error and the source match error, respectively. These errors are contributed by shifted phase, attenuate loss in the coaxial line (e

_{12}e

_{21}), and the radiate fringing effects at the probe aperture (e

_{11}and e

_{22}), which are mainly caused by the occurrence of standing waves in the coaxial line.

_{11}and e

_{22}are assumed to be zero. Finally, Equation (2) can be reduced as:

_{AA′_Air}is the reflection coefficient measurement, for air at plane AA′, and Γ

_{BB′_Air}is the standard value of the air reflection coefficient at plane BB′, obtained using the COMSOL simulator. Similarly, for MUT measurement, the Equation (2) can be expressed as:

_{BB′_MUT}of the MUT can be obtained as:

_{AA′_MUT}is the reflection coefficient measurement, for MUT at plane AA′. Equation (5) can be converted in terms of normalized admittance parameter as given in Equation (12). It should be noted that the errors of e

_{11}and e

_{22}are implicitly removed by effective permittivity calibration process as described in Section 3.3.

_{T}at BB′ plane can be written as:

_{eff}′ is the effective dielectric constant of finite layer MUT as:

_{r}

_{1}′ > ε

_{r}

_{2}′. For finite MUT with one layer thickness, d is backed by a conducting metallic plate (ε

_{r}

_{1}′ < ε

_{r}

_{2}′), the effective dielectric constant, ε

_{eff}′ can be expressed as:

_{r}

_{2}′ ≈ ∞), therefore, the 1/ε

_{r}

_{2}′ term in Equation (10) is neglected and subsequently yields

_{1}, a

_{2}, and a

_{3}(to be determined) in which the values of the constants are implicitly represented the e

_{11}and e

_{22}errors.

#### 3.2. Probe Aperture Calibration

_{eff}of the MUT can be estimated as [15]:

_{AA′}and ${\tilde{Y}}_{A{A}^{\prime}}$ is given as:

_{o}= 0.02 S, C = 2.38ε

_{o}(b − a) [36], C

_{f}= 0.0107 pF and ω are the characteristic admittance, aperture probe capacitance, fringing field capacitance, and the angular frequency, respectively.

#### 3.3. Effective Permittivity Calibration

_{eff}of the thin specimen will be measured, but not the actual permittivity, ε

_{r}of the material [10]. In this study, the relationship between the actual relative permittivity, ε

_{r}and effective relative permittivity, ε

_{eff}for a finite thickness planar specimen was empirically expressed as Ref. [35]:

_{1}, a

_{2}, and a

_{3}) values in Equation (15) were found by using three offset-short terminators (ε

_{r}= 1), yielding the following:

_{1}, h

_{2}, and h

_{3}are the known lift-off distances between the aperture probe from the shorted plate. On the other hand, ε

_{eff}

_{1}, ε

_{eff}

_{2}, and ε

_{eff}

_{3}are the corresponding effective permittivity of the three offset shorts, in which the values are obtained from (12). The unknown values of a

_{1}, a

_{2}, and a

_{3}in Equations (16)–(18) were explicitly determined using Cramer’s rule as:

_{1}, h

_{2}, and h

_{3}), three cylindrical rings have been printed using 3D-printer with polylactic acid (PLA) material in which the ring centers are concave rounded with depth, h

_{1}= (1.0 ± 0.1) mm, h

_{2}= (2.0 ± 0.1) mm, and h

_{3}= (3.0 ± 0.1) mm, respectively. It should be noted that the effective lift-off distance may be slightly shorter than the actual physical distance due to the strong coupling fringing field near to the probe aperture. The cylindrical rings backed by aluminum plate as shown in Figure 8, are used as calibration kits for this thin planar material measurements. As mentioned earlier, a large aperture probe has a strong and wide fringing field distribution compared to a small probe, thus non-metallic material such as polylactic acid (PLA) has been chosen to construct the cylindrical ring in order to reduce the field coupling effect from side wall in the area between aperture probe and aluminum plate.

## 4. Results and Discussion

#### 4.1. Reflection Coefficient, Γ_{AA′}

_{AA′_MUT}| and φ

_{AA′_MUT}data of four thin low-loss MUTs obtained from network analyzer. The |Γ

_{AA′_MUT}| and φ

_{AA′_MUT}distinction between four MUTs are less significant when it is below 50 MHz, in which mainly caused by the size limitation of the coaxial probe. For instance, various coaxial probes (in Figure 10) are tested in order to observe and compare the measurement sensitivity towards the size of coaxial probe as listed in Table 2 and Table 3.

_{AA′_MUT}| between the RF-4 (h = 1.0 mm) and the Teflon (h = 1.0 mm) is close to 0.004 in which is ten time greater to the existing resolution error (standard deviation: 10

^{−4}) as shown in Table 2. On the other hand, the phase differential, ∆φ

_{AA′_MUT}between FR-4 and Teflon at 50 MHz is capable of achieving 6° as illustrated in Table 3. Clearly, the larger the probe, the more stable and sensitive it can be achieved for reflection measurements at very low frequencies.

#### 4.2. Normalized Admittance, Ỹ_{BB′}

_{AA′_MUT}data using Equations (13) and (14). Typically, the lower the loss for the MUT, the lower the value for real part, ℜe$\left({\tilde{Y}}_{B{B}^{\prime}\_MUT}\right)$ (normalized conductance) as shown in Figure 11a. The case for the imaginary part, $\mathfrak{J}m\left({\tilde{Y}}_{B{B}^{\prime}\_MUT}\right)$ (normalized susceptance) indicates that the $\mathfrak{J}m\left({\tilde{Y}}_{B{B}^{\prime}\_MUT}\right)$ property is no longer linearly proportional to the operating frequency, f over than 400 MHz as shown in Figure 11b. Hence, the prediction of ε

_{r}value will be less accurate because the model (11) used in the conversion assumed the value of $\mathfrak{J}m\left({\tilde{Y}}_{B{B}^{\prime}\_MUT}\right)$ is proportional to the frequency.

#### 4.3. Effective Permittivity, ε_{eff}

_{eff}= ε

_{eff}′ − jε

_{eff}″ versus operating frequency, f.

_{eff}has been determined using Equation (12) and it is higher than the actual ε

_{r}value of the MUT in which the ε

_{eff}value is depended on the thickness, h of the MUT backed by metal plate. This is caused by the thinness of MUT at coaxial probe aperture, whereby the scattering of the wave from probe aperture will penetrate the MUT and coupling with the metal plate, as well as reflected by the metal plate on other side of the MUT [37].

#### 4.4. Actual Relative Permittivity, ε_{r}

_{r}′ and tan δ = ε

_{r}″/ε

_{r}′ of four thin low-loss MUTs which are in good agreement with expected values as tabulated in Table 2. The scattered ε

_{r}′ and tan δ data in Figure 13a,b have been smoothed by Local Polynomial Regression (Loess) algorithm, which is available in built-in MATLAB “smooth” command. The smoothed data are represented by the black solid lines in Figure 13a,b. As expected, the coaxial probe is very difficult to provide high stability reflection measurement for the low-loss MUT at very low operating frequencies (refer to Figure 9). Hence, indirectly, the uncertainty of the predicted ε

_{r}′ and tan δ (especially for small values of tan δ) are increased when the operating frequency decreases to below 50 MHz.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Cross-sectional side view and dimensions (in mm) of the coaxial sensor. (

**b**) Side view of the large coaxial sensor. (

**c**) Internal configuration of the coaxial sensor.

**Figure 3.**Variation in phase shift φ

_{AA′}with air thickness, h backed by metal plate at (

**a**) 50 MHz and (

**b**) 400 MHz.

**Figure 7.**Simulated ${\tilde{Y}}_{B{B}^{\prime}\_Air}$ = [ℜe(${\tilde{Y}}_{B{B}^{\prime}\_Air}$) + $j\mathfrak{J}m({\tilde{Y}}_{B{B}^{\prime}\_Air})$] for air at aperture probe (plane BB′).

**Figure 9.**Variation in (

**a**) measured |Γ

_{AA′_MUT}| and (

**b**) measured φ

_{AA′_MUT}with frequency for the four materials under test (MUTs) at room temperature.

**Figure 11.**Comparison of (

**a**) real part, ℜe$\left({\tilde{Y}}_{B{B}^{\prime}\_MUT}\right)$, and (

**b**) imaginary part, $\mathfrak{J}m\left({\tilde{Y}}_{B{B}^{\prime}\_MUT}\right)$, of the normalized admittance at probe aperture for four MUTs.

**Figure 12.**Comparison of (

**a**) real part, ε

_{eff}′ and (

**b**) imaginary part, ε

_{eff}″, of the effective relative permittivity, ε

_{eff}for four MUTs.

**Figure 13.**Variation in (

**a**) predicted ε

_{r}′ and (

**b**) predicted tan δ with frequency for the MUTs at room temperature.

Ref. | Probe Size (cm) | f (MHz) | Transition Section | Sample Contact | Sample Size/Shape | Calibration Standards | Measured ε_{r}′ Range | Inverse Method |
---|---|---|---|---|---|---|---|---|

[8] | 2a = 1.00 2b = 3.25 L = 5.00 | 100–900 | without | Aperture probe | Half-space infinite | Air, short, NaCl solution. | ~5–80 | Iterative |

[9] | 2a = 1.00 2b = 3.25 L = 5.00 | 1–10 or 10–3000 | without | Aperture probe | Half-space infinite | Short cavity orAir, short, short cavity | ~30–80 (Lossy) | Iterative |

[10] | 2a = 1.18 2b = 4.00 L ≈ 13.0 | 200–1500 | with | Aperture probe | Half-space infinite | Air, copper plate, Teflon plate. | ~2–35 (Lossy) | Iterative |

[11] | 2a = 4.50 2b = 10.3 L ≈ 41.0 | 50–1000 | with | Filled in coaxial line | Toroid-shaped | 3 positions short-circuit along the coaxial line | ~7–80 (Lossy) | Iterative |

[12] | 2a = 2.35 2b = 5.4 L ≈ 4.4 | 100–250 | with | Filled in coaxial line | Toroid-shaped | Based on specified specimen under test | 1–50 (Low lossy) | Non-iterative |

This study | 2a = 1.50 2b = 4.80 L = 15.0 | 1–400 | with | Aperture Probe | Thin planar backed by metal plate | Air, 3 offset shorts | 1–20 (Lossless) | Non-iterative |

**Table 2.**Measured |Γ

_{AA′}| for Teflon and FR-4 sheets with thickness of 1 mm backed by metal plate at 50 MHz.

Probe Size | |Γ_{AA′_MUT}| | Differentiation of ∆|Γ_{AA′_MUT}| between Teflon and FR-4 | |||
---|---|---|---|---|---|

Teflon | FR-4 | ||||

This study 2a = 1.5 cm 2b = 4.8 cm | 1.000213 | Average: 1.0001019 Standard deviation: 0.0001784 | 0.9954108 | Average: 0.9959747 Standard deviation: 0.0004091 | 0.0041272 |

1.000345 | 0.9961075 | ||||

0.9999458 | 0.9957603 | ||||

0.9999258 | 0.9964970 | ||||

1.00008 | 0.9960977 | ||||

Probe 1 2a = 0.3 cm 2b = 1.0 cm | 1.001418 | Average: 1.001394 Standard deviation: 0.000155902 | 1.000593 | Average: 1.0005194 Standard deviation: 0.00010485 | 0.0008746 |

1.001521 | 1.000355 | ||||

1.001302 | 1.000626 | ||||

1.001177 | 1.000507 | ||||

1.001552 | 1.000516 | ||||

Probe 2 2a = 0.24 cm 2b = 0.8 cm | 0.9998745 | Average: 0.9998884 Standard deviation: 0.000131850 | 0.9995621 | Average: 0.99951490 Standard deviation: 0.000038824 | 0.0003735 |

1.000060 | 0.9994846 | ||||

0.9998878 | 0.9995518 | ||||

0.9999274 | 0.9994819 | ||||

0.9996925 | 0.9994941 | ||||

Probe 3 2a = 0.13 cm 2b = 0.42 cm | 0.9996894 | Average: 0.99972804 Standard deviation: 0.000077999 | 0.9995004 | Average: 0.99967564 Standard deviation: 0.00014613 | 0.0000524 |

0.9996297 | 0.9995625 | ||||

0.9998415 | 0.9998173 | ||||

0.9997365 | 0.9998234 | ||||

0.9997431 | 0.9996746 | ||||

Probe 4 2a = 0.09 cm 2b = 0.3 cm | 1.000269 | Average: 1.000275 Standard deviation: 0.000159385 | 1.000199 | Average: 1.0002248 Standard deviation: 0.000084138 | 0.0000502 |

1.000183 | 1.000161 | ||||

1.000358 | 1.000366 | ||||

1.000075 | 1.000233 | ||||

1.000490 | 1.000165 |

**Table 3.**Measured φ

_{AA′_MUT}for Teflon and FR-4 sheets with h = 1 mm backed by metal plate at 50 MHz.

Probe Size | φ_{AA′_MUT} (°) | Differentiation of ∆φ_{AA′_MUT} between Teflon and FR-4 | |||
---|---|---|---|---|---|

Teflon | FR-4 | ||||

This study 2a = 1.5 cm 2b = 4.8 cm | −32.3000 | Average: −32.21717 Standard deviation: 0.134307 | −38.56207 | Average: −38.34329 Standard deviation: 0.495519 | 6.12612° |

−32.13884 | −38.64224 | ||||

−32.25762 | −38.43801 | ||||

−32.02661 | −37.46768 | ||||

−32.3628 | −38.60646 | ||||

Probe 1 2a = 0.3 cm 2b = 1.0 cm | −1.943045 | Average: −1.93933 Standard deviation: 0.00692831 | −2.522396 | Average: −2.542644 Standard deviation: 0.0144045 | 0.603314° |

−1.930690 | −2.554037 | ||||

−1.934963 | −2.535734 | ||||

−1.939431 | −2.542843 | ||||

−1.948521 | −2.558213 | ||||

Probe 2 2a = 0.24 cm 2b = 0.8 cm | −13.88491 | Average: −13.884262 Standard deviation: 0.0098386975 | −14.31968 | Average: −14.315222 Standard deviation: 0.00953097686 | 0.43096° |

−13.89140 | −14.31454 | ||||

−13.86970 | −14.32456 | ||||

−13.89475 | −14.31788 | ||||

−13.88055 | −14.29945 | ||||

Probe 3 2a = 0.13 cm 2b = 0.42 cm | −1.426506 | Average: −1.4218634 Standard deviation: 0.00444004 | −1.589263 | Average: −1.585402 Standard deviation: 0.00375146 | 0.163539° |

−1.417838 | −1.579990 | ||||

−1.419884 | −1.588442 | ||||

−1.418287 | −1.583723 | ||||

−1.426802 | −1.585592 | ||||

Probe 4 2a = 0.09 cm 2b = 0.3 cm | −2.60528 | Average: −2.6106474 Standard deviation: 0.00985552 | −2.71289 | Average: −2.688339 Standard deviation: 0.01568456 | 0.077692° |

−2.60139 | −2.679226 | ||||

−2.618748 | −2.694912 | ||||

−2.623582 | −2.674806 | ||||

−2.604237 | −2.679861 |

MUT | ε_{r}′ (Typ) | tan δ (Max) |
---|---|---|

Teflon | 2.06 | 0.0004 |

Acrylic | 2.75 | 0.019 |

FR-4 | 4.4 | 0.022 |

Glass | 6.1 | 0.0036 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

You, K.Y.; Sim, M.S. Precision Permittivity Measurement for Low-Loss Thin Planar Materials Using Large Coaxial Probe from 1 to 400 MHz. *J. Manuf. Mater. Process.* **2018**, *2*, 81.
https://doi.org/10.3390/jmmp2040081

**AMA Style**

You KY, Sim MS. Precision Permittivity Measurement for Low-Loss Thin Planar Materials Using Large Coaxial Probe from 1 to 400 MHz. *Journal of Manufacturing and Materials Processing*. 2018; 2(4):81.
https://doi.org/10.3390/jmmp2040081

**Chicago/Turabian Style**

You, Kok Yeow, and Man Seng Sim. 2018. "Precision Permittivity Measurement for Low-Loss Thin Planar Materials Using Large Coaxial Probe from 1 to 400 MHz" *Journal of Manufacturing and Materials Processing* 2, no. 4: 81.
https://doi.org/10.3390/jmmp2040081