Reconstruction of annular bi-layered media in cylindrical waveguide section
- Anders Eriksson^{1},
- Larisa Beilina^{2}Email author and
- Truls Martin Larsen^{3}
https://doi.org/10.1186/s13362-017-0036-x
© The Author(s) 2017
Received: 6 October 2016
Accepted: 23 February 2017
Published: 11 April 2017
Abstract
We consider a radial transverse resonance model for a circular cylindrical waveguide composed into two layers with different frequency dependent complex dielectric constants. An inverse problem with four unknowns - three physical material parameters and one dimensional dielectric layer thickness parameter - is solved by employing TE110 and TE210 modes with different radial field distribution. First the resonance frequencies and quality factors are found fitting a Lorentzian function to the ‘measured’ data, using the method of least squares. Then found resonance frequencies and quality factors are used in a second inverse Newton-Raphson algorithm which solves four transverse resonance equations in order to get four unknown parameters. The use of TE110 and TE210 models offers one-dimensional radial tomographic capability. An open ended coaxial waveguide quarter-wave resonator is added to the sensor topology, and the effect on the convergence of numerical method is investigated.
Keywords
reconstruction of material parameters in a waveguide transverse resonance model open ended coaxial waveguide resonator least squares problem1 Introduction
Extraction of material parameters and/or dimensions based on distributed resonator measurements has been around for decades. Characterization of distributed microwave resonators dielectric material from resonance frequency and quality factor measurements is found in [1]. A comparison of inverse methods for extracting resonant frequency and quality factor is given in [2]. Typically, the dielectric filling of the resonators is homogeneous, but there are not really any restrictions for allowing inhomogeneous dielectric filling.
In this work we use the Bruggemans model for emulsion permittivity [3, 4]. This model is applied for saline water mixed with oil as well as for liquid droplets in gas. For the latter, the Bruggemans model is used twice - first for water mixed with oil in liquid droplets, and then again to calculate the effective permittivity of liquid droplets in gas. Several models for effective complex permittivity of metal powders in insulating dielectrics (e.g. teflon) are studied in [4], and one conclusion is that the Bruggemans model is relatively accurate for predicting of the real part of permittivity, while the well-known Maxwell-Garnett model [4] have higher accuracies for the imaginary part. This is relevant for the case of an oil continuous regime with water content of high salinity. In this case the electrically conducting saline water droplets are comparable to electrically conducting metal powder. Another popular model - the Gadanis model for saline water [5] - we use to determine a complex permittivity of the saline water. In this model the complex permittivity is a function of salinity (s), temperature (T) and frequency (f).
Thus, by applying these two permittivity models with corresponding fractions for each medium to our problem (assuming that the known chemical substances are presented but with unknown ratios), the parameters are reduced to four unknowns. In the case of using directly frequency-dependent complex permittivity for the liquid, and another frequency-dependent complex permittivity for the gas, four unknowns would be left to find just for one frequency point along with the liquid thickness. Then from these permittivities, the water liquid ratio (WLR) and droplet gas ratio (DGR) can be found.
2 Model equations for cylindrical waveguides
Let \(G\subset\mathbb{R}^{n}, n=1,2,3\) be a bounded domain with a piecewise smooth boundary ∂G and x be a point in this domain. Let \(\Omega \subset G\) be bounded subdomain representing our waveguide with a boundary ∂Ω. As a model problem we have used a full wave RF resonance model of [1] using transverse resonance method. This model is very computationally efficient and compact (equation wise). It can model infinite long cylindrical pipe or waveguide, filled with arbitrary concentric layers.
We note that often the solutions of (7) is efficient represent in the terms of Hankels functions \(H_{m}^{(1)},H_{m}^{(2)} \) of the first and second order, see [6], and these solutions are given in [7]. In Sections 4.1, 4.2 we use both Hankels and Bessels functions. Below we present solutions of (7) in a cylindrical waveguides for TE and TM modes expressed via Bessels functions.
2.1 TE modes in a cylindrical waveguide
2.2 TM modes in a cylindrical waveguide
3 Statement of inverse problem
Parameters to be determined in IP
Parameter | Description |
---|---|
h | Thickness of liquid layer - liquid having arbitrary mixture ratio (Water Liquid Ration) of condensate and saline water. |
\(R_{\mathrm{WLR}}\) | Water Liquid Ratio (WLR) - water fraction in liquid. |
\(R_{\mathrm{DGR}}\) | Droplet Gas Ratio (DGR) - ratio of liquid immersed in droplet form in gas continuous volume (note that WLR and Salinity is the same in liquid film as well as in droplets). |
s | Salinity - salt concentration in water. |
Our inverse problem is following:
Inverse Problem (IP)
In our IP, the function \(\tilde{S}_{11} (r,\omega )|_{r=r_{\mathrm{obs}}}\) in (22) represents measurements at the point \(r=r_{\mathrm{obs}}\) for all physical frequencies on the frequency interval \([\underline{\omega}, \bar{\omega}]\).
4 Radial transverse resonance method
In [7] was shown that only for the angular index \(m=0\), pure TE or pure TM modes can accommodate in a two-layer dielectric cylindrical waveguide. However, it is trivial to show that if also the longitudinal wave-number \(k_{z}\) is zero, pure TE and TM modes can accommodate. We note that \(k_{z}=0\) is equivalent to pure radial resonance.
4.1 Impedance transformation for TE wave
4.2 Impedance transformation for cylindrical TM wave
For verification a test code (the same code as used for generating electromagnetic fields in [10]) based on a spectral domain Greens function for cylindrical geometry [11] was compared to the transverse resonance method for both TE and TM modes for the case \(k_{z} = 0\). In this code, an excitation current in z-direction renders TM modes, and an excitation in transverse angular ϕ direction renders TE modes.
4.3 TE110 and TE210 mode field distribution for the one-dimensional radial tomography
In the case when the angular index \(m\!\gg\!1\) the electric field is dominant near resonator radius, see the equivalent field distribution for parallel plate TM disc resonators in [12]. Even TE210 mode has electric field significantly more confined near pipe radius compared to TE110 mode, which has a more homogeneous electric field distribution. This can be exploited tomographically, since TE210 mode field pattern penetrates less radially inwards than the TE110 mode. Thus, the TE210 mode is more sensitive to the presence of an outer concentric dielectric layer than the TE110 mode.
Having the quality factor and resonance frequency for each TE110 and TE210 mode, a set of four unknown material and dimensional parameters described in Table 1 can in theory be extracted using the same transverse resonance technique described previously.
5 Open ended coaxial waveguide quarter-wave resonator probe
In this work, the response of the direct magnitude of the reflection coefficient \(S_{11}\), see Figure 2(left), is minimized with respect to the model of an open ended coaxial waveguide and its quarter-wave transmission-line circuitry without any intermediate resonance frequency and Q-factor calculations. We apply the full-wave Hankel model as in [13]. If the pipe diameter is significantly larger than the outer diameter b of the coaxial waveguide, the planar ground plane model used in [13] is assumed still to be valid. We note that the abrupt discontinuity from the coaxial waveguide section into a grounded plane excites higher order terms apart from the fundamental incident coaxial waveguide TEM mode. In [13], these higher modes are TM modes with only radial variations due to the angular symmetry. If the coaxial waveguide diameter b increases relative to the pipe diameter D, the angular variations of the basis functions/higher modes would be stronger. However, they will be not as large as the radial higher mode excitations. A compromise in accuracy can be to assume an incident ideal TEM wave, matching the tangential electric and magnetic fields at the curved open ended coaxial waveguide interface using a spectral domain approach [11], and using then basis functions/modes with angular dependency (in local open-ended coaxial waveguide coordinate system). For suitable coaxial waveguide dimensions a, b, a third characteristic, penetration depth which is smaller than in the TE210 mode, can be obtained. Thus, one can replace one of the four transverse resonance equations with an equation for the open-ended coaxial waveguide resonator.
As model for input impedance \(Z_{\mathrm{liq}}\), a full-wave model of [13] is employed; using fundamental TEM mode, and two lowest TM modes as basis functions, the input impedance is calculated from the obtained reflection coefficient. In Figure 2(left) the reflection coefficient \(S_{11}\) at the reference plane is obtained by the standard impedance transformations [8].
5.1 Coupling to the open-ended quarter wave coaxial waveguide resonator
By probing the resonator in the middle of the coaxial waveguide instead of at the left end, separation of the resonance frequencies is narrowed for the same physical resonator length. Practically (if not also theoretically) is impossible to select an optimal coupling to a quarter-wave open ended coaxial waveguide loaded with media when we have a large range of the real and imaginary parts of permittivity. We note that with salinities from 0 up to 25% the imaginary part of saline water changes with several orders of magnitude.
Experimentally we have found that using a pair of resonators as shown in Figure 2(right), a better sensitivity in amplitude and frequency shift is achieved. In this figure, the upper circuit represents a simply galvanic coupled quarter-wave resonator, while the lower figure shows capacitive coupled quarter-wave resonator. In this work we have used a coupling capacitance of 10 pF. We should note that a pair of capacitive and galvanic coupled resonators cannot render more information than a phase and magnitude reflection measurements taken directly at the open end of the coaxial waveguide. Rather, the resonator pair transforms the complex reflection data to the resonance type response, so that the benefit of both amplitude change as well as the frequency shift can be advantageous.
6 Methods of reconstruction
In this section we present methods of reconstruction of parameters presented in Table 1. We note that the four transverse resonance equations are minimized in the same manner regardless parameters. Since we have four equations at hand and four unknowns, it is suitable to apply Newton’s method.
Typically, resonance frequency \(f_{0}\) and quality-factor Q can be extracted by a simple peak search and a numerical direct extraction of the quality-factor from a ratio of band-width at half maximum peak value and resonance frequency. These extracted values are good initial guesses for \(f_{\mathrm{init}}^{\mathsf{TE}110}\), \(Q_{\mathrm{init}}^{\mathsf{TE}110}\) and \(f_{\mathrm{init}}^{\mathsf{TE}210}\), \(Q_{\mathrm{init}}^{\mathsf{TE}110}\). For one TE resonance, assuming that above conditions hold, we have four unknowns to be measured: amplitude \(\mathrm{Amp}_{0}^{\mathsf{TE}110}\), \(Q_{0}^{\mathsf{TE}110}\), \(f_{0}^{\mathsf{TE}110}\) and \(C_{0}^{\mathsf{TE}110}\). The same is valid for TE210 resonance. Only \(\mathrm{Amp}_{0}^{\mathsf{TE}210}\), \(f_{0}^{\mathsf{TE}210}\) and \(Q_{0}^{\mathsf{TE}210}\), \(C_{0}^{\mathsf{TE}210}\) are needed to be measured in order to extract the four unknowns: liquid thickness, salinity, WLR and droplet ratio. The measured quality factors \(Q_{0}^{\mathsf{TE}110}\) and \(Q_{0}^{\mathsf{TE}210}\) as well as the measured resonance frequencies \(f_{0}^{\mathsf{TE}110}\) and \(f_{0}^{\mathsf{TE}210}\) can be approximated to the unloaded corresponding entities as long as the coupling to the resonator is sufficiently weak - otherwise, the unloaded quality factors and resonant frequencies must be calculated by an analysis that includes the coupling circuitry influence. In this work for simplicity we assume that the coupling is weak and that the unloaded and measured loaded entities are the same.
7 Results of simulations
There is room for some optimization regarding the open ended coaxial waveguide resonator pair. For example, the coupling capacitance value could be further optimized. Number of frequency points and frequency range are other issues that may increase the benefits of having the open ended coaxial waveguide resonator pair. Also, for some combinations of salinity, WLR, DGR and liquid layer thickness, it may be more beneficial include either capacitive or galvanic coupled open ended coaxial waveguide resonator to increase convergence of the iterative procedure.
The table shows the relative frequency change, by adding 0.5 μm to the liquid film thickness h . Considering the \(\pmb{f_{0}}\) listed in Table 3 , these numbers represent frequency changes of the order 10 \(\pmb{^{2}}\) -10 \(\pmb{^{3}\mbox{ Hz}}\)
Mode | Relative tunability h = 1 mm | Relative tunability h = 2 mm |
---|---|---|
TE110 | 3.701⋅10^{−6} | 5.107⋅10^{−6} |
TE210 | 7.452⋅10^{−6} | 1.293⋅10^{−5} |
Quarter wave | 1.520⋅10^{−5} | 1.059⋅10^{−5} |
The table shows the frequencies of the sensitivity analysed in Table 2
Mode | Frequency, \(\boldsymbol {f_{0}}\) h = 1 mm | Frequency, \(\boldsymbol {f_{0}}\) h = 2 mm |
---|---|---|
TE110 | 1,041,453,582 | 1,032,491,153 |
TE210 | 1,717,407,914 | 1,684,125,223 |
Quarter wave, capacitive coupled | 112,188,018 | 110,530,685 |
8 Discussion
We have presented a full wave transverse resonance model for a circular cylindrical annular geometry. It was demonstrated numerically that four unknown physical parameters could be extracted. If we combine the transverse resonance model with the reflection data from open ended quarter-wave resonators, we may improve convergence and reduce the computational error by a factor of 100.
Our numerical experiments demonstrate, that the combination of a galvanic and capacitive coupled open ended coaxial waveguide resonators renders higher sensitivity to determine WLR and liquid thickness. This is valid in low saline regime: for salinity \(< \sim 3\%\) in a water continuous liquid case or for any salinity where \(R_{\mathrm{WLR}} < 0.5\).
Using our experiments we can conclude that the improvement of the frequency sensitivity is due to capacitive coupled open ended coaxial waveguide resonators. For the high saline regime (water-continuous and salinity \(> \sim 3\%\)), better sensitivity (in amplitude change due to change in salinity) is obtained using the galvanic coupled coaxial waveguide resonator.
Declarations
Acknowledgements
The authors are grateful for discussions with PS Kildal, Y Yang, Z Sipus, P Slättman, H Merkel and SP Hanserud. The work of LB is supported by the sabbatical program at the Faculty of Science, University of Gothenburg.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Eriksson A, Deleniv A, Gevorgian S. Orientation and direct current field dependent dielectric properties of bulk single crystal SrTiO3 at microwave frequencies. J Appl Phys. 2003;93:2848. View ArticleGoogle Scholar
- Petersan PJ, Anlage SM. Measurement of resonant frequency and quality factor of microwave resonators: comparison of methods. J Appl Phys. 1998;84:339. View ArticleGoogle Scholar
- Bruggeman DAG. Calculation of various physical constants of heterogeneous substances. Ann Phys. 1935;32(12). Google Scholar
- Kiley EM, et al.Applicability study of classical and contemporary models for effective complex permittivity of metal powders. J. Microw. Power Electromagn Energy. 2012;46:26-38. View ArticleGoogle Scholar
- Gadani DH, et al.. Effect of salinity on the dielectric properties of water. Indian J Pure Appl Phys. 2012;50:405-10. Google Scholar
- Lebedev NN. Special functions and their applications. New York: Dover Publications; 1972. MATHGoogle Scholar
- Harrington RF. Time-harmonic electromagnetic fields. Wiley: IEEE Press; 2001. View ArticleGoogle Scholar
- Collin RE. Foundations for microwave engineering. 2nd ed. Wiley: IEEE Press; 2000. Google Scholar
- Eriksson A, Deleniv A, Gevorgian S. Resonant tunneling of microwave energy in thin film multilayer metal/dielectric structures. Microwave Symposium Digest, 2002 IEEE MTT-S International. 2002;3. Google Scholar
- Beilina L, Eriksson A. Reconstruction of dielectric constants in a cylindrical waveguide. Springer Proceedings in Mathematics & Statistics, Inverse Problems and Applications. 2015;120:97-109. MathSciNetView ArticleMATHGoogle Scholar
- Šipuš Z, Kildal P-S, Leijon R, Johansson M. An algorithm for calculating Green’s functions of planar, circular cylindrical, and spherical multilayer substrates. ACES Journal. 1998;13(3). Google Scholar
- Eriksson A, Linner P, Gevorgian S. Mode chart of electrically thin parallel-plate circular resonators. IEEE Proceedings - Microwaves Antennas and Propagation. 2001;1(148):51-5. View ArticleGoogle Scholar
- Baker-Jarvis J, et al. Analysis of an open-ended coaxial probe with lift-off for nondestructing testing. IEEE Transactions on Instrument and Measurement. 1994;43(5). Google Scholar
- Tikhonov AN, Goncharsky AV, Stepanov VV, Yagola AG. Numerical methods for the solution of ill-posed problems. London: Kluwer; 1995. View ArticleMATHGoogle Scholar
- Bakushinsky A, Kokurin M, Smirnova A. Iterative methods for ill-posed problems. vol. 54. Berlin: De Gruyter; 2011. MATHGoogle Scholar
- Malmberg JB, Beilina L. In: Iterative regularization and adaptivity for an electromagnetic coefficient inverse problem, AIP conference proceedings, ICNAAM2016. Rhodes, GREECE. 2016. p. 19.09-25.09. Google Scholar
- Klibanov MV, Bakushinsky AB, Beilina L. Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess. J Inverse Ill-Posed Probl. 2011;19:83-105. MathSciNetView ArticleMATHGoogle Scholar