694
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 3, MARCH 2014
Circular Layered Waveguide Use for Wideband Complex Permittivity Measurement of Lossy Liquids Valery N. Skresanov, Member, IEEE, Zoya E. Eremenko, Senior Member, IEEE, Ekaterina S. Kuznetsova, Yun Wu, and Yusheng He Abstract— The electromagnetic characteristics of a circular metal waveguide with a coaxial cylindrical dielectric insert and a layer of absorbing liquid that fills the space between the insert and the waveguide wall have been studied. The transverse waveguide dimensions are comparable with the wavelength. Dependences of the attenuation and phase coefficients of waveguide modes on the structure size and material properties of the layers are found by solving the boundary value problem. It is shown that based on the proposed layered waveguide with the H E11 type of wave, a measuring cell can be designed to work at either fixed frequency with high differential sensitivity or in the frequency range of the single-mode waveguide operation. The cell is of class of cells with calculable geometry. In this case, a reference liquid with known dielectric properties is not required for absolute measurements of the complex permittivity (CP) of the absorbing liquid. The method of finding the CP of absorbing liquids are verified using electromagnetic modeling with CST Microwave Studio. Index Terms— Absorbing liquid, attenuation and phase coefficients, circular metal waveguide, complex permittivity (CP), electromagnetic characteristics, microwave frequencies, waveguide.
I. I NTRODUCTION
M
ETHODS for measuring the imaginary and real components of the complex permittivity (CP) for polar liquids are in demand in various fields of science and technology. For instance, numerous examples of the practical use of CP measurement results for absorbing liquids, which primarily include water and water solutions of various substances, are given in [1]. There is also a continuing interest in the study of the dielectric properties of biological fluids [2]–[4]. We have been engaged in studying the possibility of dielectrometry to address the control of natural wines and juices. In particular, it has been shown that the method of dielectrometry is more efficient in comparison with already known detection methods of water dilution in wines and juices [5], [6]. The CP measuring is often sufficient to be done at fixed frequencies, however, it can be more informative if performed in Manuscript received February 19, 2013; revised July 9, 2013; accepted July 29, 2013. Date of publication September 30, 2013; date of current version February 5, 2014. This work was supported by the scientific technology center in Ukraine through the STCU under Project 3870. The Associate Editor coordinating the review process was Dr. Wendy Van Moer. V. N. Skresanov, Z. E. Eremenko, and E. S. Kuznetsova are with O. Ya. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkiv 61085, Ukraine (e-mail:
[email protected]). Y. Wu is with the University of Science and Technology, Beijing 100190, China. Y. He is with the Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. Digital Object Identifier 10.1109/TIM.2013.2282003
a frequency range. Measurements in the dispersion range of the CP allow determining the relaxation time of the water solution. This parameter is sensitive to the concentration of a number of substances, and therefore is an indicator of changes in the composition of the solution [7]. As known, the first dispersion range of CP in water solutions for various substances belonging to the class of polar liquids is located in the microwaves and usually covers the band from 4 to 40 GHz [8], [9]. Methods and techniques of CP measurement for absorbing liquids in the microwave frequency range are now sufficiently developed. Their comparative analysis has been done in numerous reviews (see [1], [2], [8], [10], and [11]). The sections of coaxial transmission lines [10] or section of metal hollow waveguides [12] are used in the known broadband CP measuring cells operating in the mode of electromagnetic wave transmission. Metal waveguides are preferable for measurements in the high-frequency range. A test liquid fully fills the volume of the waveguide that, in the case of aqueous solutions, leads to a large attenuation of electromagnetic waves, and therefore to the need for reducing the length of the cell. Decreasing the length of the measuring cell leads to larger errors in the measurement. In this paper, we have proposed a specific waveguide structure for CP measurements of high loss liquids. The proposed measuring cell is a circular metal waveguide with a coaxial insert in the form of a circular cylinder with a small dielectric loss tangent. The tested liquid fills the space between the wall of the waveguide and the insert. The attenuation per length in this cell is several orders less than in the known waveguide measuring cells mentioned above. As a result, for the CP measuring of water solutions, a few centimeters long cell can be used. Loss reduction in the cell can be also achieved using the capillaries located between the broad walls of a rectangular waveguide [13]. Such measurements require, however, the procedure for calibrating the cell by means of a liquid with known CP (the reference liquid) and are not suitable for absolute measurements. The reference liquid is also required in the widely used methods of measurement using the coaxial probes in reflected wave mode regime [14]. The sensitivity of these methods lower than the sensitivity for the methods with transmission type cuvette. In contrast, the proposed cell belongs to a class of cells with calculable geometry, i.e., CP measurement can be performed without the use of the reference liquid. It must be emphasized that the layered structure, which we propose to use for measuring CP of liquids, is used in microwave engineering for a long time. In particular, exact
0018-9456 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
SKRESANOV et al.: CIRCULAR LAYERED WAVEGUIDE USE FOR WIDEBAND CP MEASUREMENT
complex characteristic equation for such a structure is well known and can be solved by numerical methods [15], [16]. For example, in [17], the solution of this equation has been considered to optimize the characteristics of broadband attenuator with fixed attenuation, in [16], to clarify the calculation of optical waveguides. This paper of the optical [18] and the quasi-optical [19] waveguides under the approximation of a thick absorbing dielectric layer had been carried out for the first time. In the latter case, we have a special case of the equation. This approach was also used to study wave propagation in mines and tunnels [20]. Earlier, we have used a thick layer (the layer thickness is much more than an operating wavelength) of a fluid in the development of the methodology and the design of the cell for measuring the dielectric constants of strongly absorbing liquids at a fixed frequency [5], [21]. The cell of [5] can be regarded as a prototype of the cell described in this paper. The studied electromagnetic structure has a complex mode structure and a careful analysis of the spectrum of the waves, as well as ways of the working mode efficient excitation, is required before proceeding to the construction of the measuring cell. The solution of these problems is presented in this paper. II. G EOMETRY AND C HARACTERISTIC E QUATION OF E LECTROMAGNETIC S TRUCTURE We study the spectrum of modes in a circular metal waveguide of the radius b with two coaxial dielectric layers (Fig. 1). The first layer is a circular coaxial cylindrical insert of the radius a with CP, ε1 = ε1 + i ε1 (ε1 /ε1 1). The second layer is an absorbing liquid with CP ε2 = ε2 + i ε2 , filling the space a < r < b, where r is the radial coordinate. The permeability of two layers, μ1 and μ2 , is equal to one. All the calculations are performed for the insert made of silica (ε1 = 3.8 + i 0.0004). We have restricted ourselves to water and water solutions of ethanol, as typical representatives of absorbing liquids. As known, the dispersion relation of CP for such liquids up to K a frequency range is well described by the Debye formula with a single relaxation time τ εs − ε∞ (1) ε2 = ε∞ + 1 + i 2π f τ where εs and ε∞ are the static and the optical permittivities, respectively, and f is the operating frequency. The dependence of the parameters in (1) on the ethanol concentration was obtained by fitting the experimental data presented in [7], [22]. For example, for water at 23 °C at the frequency f = 31.82 GHz, we have εs = 83.06, ε∞ = 6.024 and τ = 9.69 ps, and for 40% ethanol solution (by the volume percentage), they are εs = 58.43, ε∞ = 5.28, and τ = 33.8 ps. The results of calculations of (1) are the initial data for the analysis and shown in Fig. 2 by Cole–Cole diagrams. In addition, Fig. 2 shows the results of the proposed procedure verification of the CP recovering from the measurement for these liquids. The verification procedure is described in the following. The electromagnetic problem for the structure of Fig. 1 has been solved by the separation of variables method in
695
Fig. 1. Longitudinal cross section of a circular waveguide with (1) cylindrical dielectric insert and (2) coaxial test liquid layer.
Fig. 2. Initial Cole–Cole diagrams for water (1), 40% (2) and 96% (3) of ethanol in water (solid lines). Dots and triangles: the values of CP recovered from the numerical experiments with the proposed cell for the frequencies 26, 30, 35, and 40 GHz for water and 40% ethanol in water, respectively.
cylindrical coordinates, (r, ϕ, z) [15], [16]. The types of waves propagating along the axis of the structure can be characterized with the propagation constant h = h + i h , where h and h are the phase and attenuation coefficients, respectively. The electromagnetic field depends on the azimuth ϕ as cos(nϕ), where n = 0, 1, 2, . . . is the azimuth index of the wave type. Along the radius r , the field is represented by a combination of Bessel Jn (·) and Neuman Nn (·) functions of orders n. As a result, the complex amplitudes of the electric and magnetic Hertz vectors are as follows in the dielectric cylinder and the absorbing layer, respectively: U1e = Ae Jn (k1r ) sin(nϕ) exp (i hz) U1m = Am Jn (k1r ) cos(nϕ) exp (i hz) U2e = [B e Jn (k2r ) + C e Nn (k2r )] cos(nϕ) exp (i hz) U2m = [B m Jn (k2r ) + C m Nn (k2r )] sin(nϕ) exp (i hz). (2) Here, the time dependence is as assumed as exp(−i ωt), and Ae , Am , B e , B m , C e , C m are the unknown coefficients. They are can be found by the standard way from a homogeneous system of linear algebraic equations, after solving the following characteristic equation for the propagation constant h = h + i h 2 n 2 h 2 k12 − k22 e m (ε2 f − ε1 F)(μ2 f − μ1 F) = (3) k02 k14 k24 a 4 where Nn (k2 a)Jn (k2 b) − Jn (k2 a)Nn (k2 b) k2 a[Nn (k2 a)Jn (k2 b) − Jn (k2 a)Nn (k2 b)] Jn (k2 a)Nn (k2 b) − Nn (k2 a)Jn (k2 b) fm = k2 a[ Jn (k2 a)Nn (k2 b) − Nn (k2 a)Jn (k2 b)] Jn (k1 a) F = k1 a · Jn (k1 a) fe =
696
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 3, MARCH 2014
where the primes denote the derivatives with respect to the arguments of functions Jn (·) and Nn (·) ki = εi μi k02 − h 2 are the transverse wavenumbers in the dielectric cylinder (i = 1) and the surrounding liquid (i = 2). For n = 0 (azimuthally homogeneous field), (3) separates into two independent equations that are equivalent to the presence, in the structure under study, of the guided waves of electric E 0m and magnetic H0m types. Here, the index m = 1, 2, . . . corresponds to the number of field variations along the radius of the cylinder. For all other modes of the structure, the continuity of the tangential components of electric and magnetic fields at the dielectric cylinder boundary can be achieved only in the class of hybrid waves of the H E nm or E Hnm types, with all six components of the electromagnetic field not equal to zero. In the limiting case of absence of the absorbing layer, when b = a, we have a circular metallic waveguide filled with dielectric of permittivity ε1 . As known, the propagation constants of the electric E nm and magnetic Hnm waves in such a waveguide are solutions of the equations Jn (k1 a) = 0 and Jn (k1 a) = 0, respectively. In the other limiting case, if b → ∞, the investigated structure transforms into a cylindrical dielectric channel in an infinite absorbing medium. Using the known asymptotic expressions for the Bessel and Neumann functions of large arguments, it is easy to show that (3) in this case leads to the familiar form [18]– [21] Hn(1)(k2 a) Jn (k1 a) − ε1 ε2 (1) k1 a · Jn (k1 a) k2 a · Hn (k2 a) (1) Hn (k2 a) Jn (k1 a) − μ1 × μ2 (1) k1 a · Jn (k1 a) k2 a · Hn (k2 a) =
n 2 h 2 (k12 − k22 )2 k02 k14 k24 a 4
.
(4)
Equation (3) describes, for example, the wave propagation in quasi-optical or optical waveguides representing a metal pipe of the radius b covered inside with a layer of the thickness (b − a) made of absorbing dielectric. Well established in practice is the fact that the working mode H E 11 attenuation coefficient monotonically decreases to acceptable values (of the fraction of decibel/kilometer) in oversized quasi-optical waveguides (a λ, where λ is the wavelength) with the waveguide diameter increase. It happens despite the large loss tangent of the dielectric layer. Qualitatively, the same results should be expected for a waveguide with a layer of absorbing liquid. Fig. 3 shows the results of the phase and attenuation coefficients calculation for the four modes of the structure (Fig. 1) with a fixed thickness layer of 40% ethanol in water at 40 GHz as a function of the quartz cylinder radius a. We observe that for all the modes, attenuation coefficients decrease monotonically, and the phase velocities tend to the phase velocity of the plane wave in quartz. In contrast to transmission lines, the attenuation of the operating mode in the measuring cell is a positive factor. This is because at the appropriate choice of length of the cell, it is possible to neglect the resonance effects in the
Fig. 3. Dependence of the normalized phase (i), h /(k0 ε1 ), and attenuation coefficients (ii), h , on the radius of the dielectric cylinder in the waveguide with a fixed thickness [(b−a) = 2.5 mm] of the layer of 40% ethanol solution in water at the frequency of 40 GHz.
cell associated with reflection from the edges. On the other hand, it is necessary to ensure the propagation of only one working mode in the measuring cell. However, other modes with the same azimuth index can be excited at the entrance of the cell at the working mode excitation. As the difference in attenuation of waves with the same azimuth symmetry is rather large (Fig. 3), there is an effective spatial filtering of higher modes. Thus, to build a measuring cell of interest, the waveguides with transverse dimensions comparable with the wavelength can be considered. Electromagnetic characteristics of such waveguides have not been studied in detail so far, in contrast to quasi-optical oversized waveguides. III. M ODES OF A C IRCULAR T WO -L AYER WAVEGUIDE W ITH A L IQUID A BSORBING L AYER Complex roots, h nm = h nm + i h nm , of the characteristic equation (3) determine the mode composition of the radially layered waveguide (Fig. 1). Along the z-axis, electromagnetic waves with phase h nm and attenuation h nm coefficients can propagate. For each azimuthal index n = 0, 1, 2, . . ., there is a countable set of roots, m = 1, 2, 3, . . ., for different values of complex propagation constants, where m is the radial index of the wave. We have restricted our calculations to the roots of the index n ≤ 2, for which h nm
