Analytic and numerically efficient scattering equations for an infinitely flanged coaxial line.
A flanged coaxial line has been extensively studied to obtain material characteristics for biological substances , non-destructive test , resonant dielectric absorption , permittivity determination , moisture layers , agricultural products , IC packages , and concrete . Even though the applications for material measurements are somewhat different, the basic idea is very similar one another . Measuring the reflection coefficients with a fixed probe is utilized to determine the complex permittivity of an unknown material. We usually perform iteration procedures to reduce errors between preobtained measurement results and simulations with various complex permittivity.
As a result, the analytic formulations of scattering phenomena for probe and material are very important to estimate material characteristics. To obtain precise scattering equations, a variety of numerical methods have been proposed [9-16]. The Wiener-Hopf method and unilateral Fourier transform  were applied to a problem of radiation from a coaxial line with an infinite flange and thick inner conductor. A quasi-static analysis  was proposed to get an approximate closed-form solution. The Hankel transform and mode-matching technique [11-13] allow us to formulate the scattered fields in analytic representations. However, the radiation integrals in [11-13] have singularities and oscillating behaviors, thus indicating that the proposed integrals are complicated to deduce and inefficient for numerical computations. In [14, 15], the authors used versatile numerical algorithms such as the boundary integral equation  and the two-dimensional finite-difference frequency-domain method , respectively. Recently, by using the Sommerfeld identity similar to the Hankel transform, an approximate but efficient exponential series solution was given in  based on the matrix pencil method. The analytic method based on the eigenfunctions satisfying the edge condition was also used in [18, 19]. Even though the method in  is analytic, the radiation integrals in  still have several singular points which make numerical integrations difficult.
The aim of this work is to show an alternative method based on the mode-matching technique and Green's function. The Green's function approach combined with virtual current cancellation  and integral path deforming also yields a fast-convergent radiation integral that is free of singularity and very efficient for numerical integration. In the following Section, we extend the methodology proposed in  to the problem of a flanged coaxial line.
[FIGURE 1 OMITTED]
2. FIELD REPRESENTATION AND ANALYSIS
Consider the [TM.sub.0s] mode propagates through a coaxial line shown in Figure 1 and is reflected at the end of a coaxial line (z = 0). Note that s represents the mode number of an incident wave. The coaxial line is truncated at z = 0 and connected to a perfectly conducting infinite flange at z = 0. Due to the geometric discontinuity at z = 0, the infinite number of guided waves are generated in region (I) (z [less than or equal to] 0) and some incident power is transmitted into the free-space that is denoted as region (II) (z > 0). We select the time convention as [e.sup.-iwt] which is suppressed throughout. The incident and reflected electric fields [E.sub.rho] are conveniently defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [rho] = [square root of [x.sup.2] + [y.sup.2]]], [[eta].sub.1] = [square root of [[mu].sub.1/[[epsilon].sub.1],[[xi].sub.m] = [square root of [k.sup.2.sub.1] -[k.sup.2.sub.m]] = w [square root of [[mu].sub.1] [[epsilon].sub.1]],
(x)' denotes differentiation with respect to the argument,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)
[J.sub.0](x) and [N.sub.0](x) are the zeroth-order Bessel functions of the first and second kinds, respectively, [k.sub.0] = 0, and [k.sub.m] is determined by equating [C.sub.0]([k.sub.m]a) = 0 (m = 1,2, ...). A coaxial function [C.sub.0]([k.sub.m][rho]) satisfies the Bessel's differential equation as
[1/[rho] d/d[rho]([rho] d/dp +[d.sup.2.sub.m]] [K.sub.0]([K.sub.m][rho]) = 0. (4)
When s = 0, (1) denotes a TEM mode. The incident [E.sup.i.sub.[rho] field is scattered at the boundary at z = 0. Based on the Fourier-Bessel series, the magnetic vector potential [A.sup.I.sub.z] in region (I) (z < 0) is formulated to represent the scattered fields as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)
where [p.sub.m] are the unknown modal coefficients to be determined with simultaneous equations that will be shown in (20). The electric field in region (I) [E.sup.I.sub.[rho]([rho], z) can be obtained with [E.sup.I.sub.[rho](p, z) = i/[w[[mu].sub.1][[epsilon].sub.1] [partial derivative].sup.2]/ [partial derivative].sup.z] [partial derivative].sup.[rho]][A.sup I.sub.z]([rho], z). In terms of the Sturm-Liouville theory, the eigenfunctions [C.sub.0]([k.sub.m][rho]) are orthogonal each other and constitute a complete basis set for region (I). Therefore, we can consider [p.sub.m] the expansion coefficients for the Fourier-Bessel series.
[FIGURE 2 OMITTED]
The magnetic vector potential [A.sup.II.sub.z] in open half-space region (z > 0) can be obtained based on virtual current cancelation. Even though virtual currents induced on a virtual coaxial line at [rho] = a, b and z > 0 illustrated in Figure 2 are artificial, this concept facilitates the simplified yet precise field representations in open region. It should be noted that we assume the virtual coaxial line at [rho] = a, b and z > 0 to be made of PEC (Perfect Electric Conductor). Since a virtual coaxial line is introduced in open region (z > 0), we can easily match the potential [E.sub.m]([rho], z) (z > 0) for the virtual coaxial line and [A.sup.I.sub.z](p, z) (z < 0) by a standard mode-matching technique. In the next step, the component [R.sup.E.sub.m](p, z) is generated to remove the inevitable discontinuities at [rho] = a and b caused by a virtual coaxial line. By this basic concept of virtual current cancelation, we can formulate the magnetic vector potential [A.sup.II.sub.z] for open half-space region (z > 0) as
[A.sup.II.sub.z] ([rho],z) = [[mu].sub.2] [[infinity].summation over (m=0)] [r.sub.m] [E.sub.m]([rho], z) + [R.sup.E.sub.m]([rho], z)], (6)
where [r.sub.m] are the unknown modal coefficients for region (II),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
[[eta].sub.m] = [square root of [k.sup.2.sub.2] - [k.sup.2.sub.m], and [k.sub.2] w[square root of [[mu.sub.2][[epsilon].sub.2]. In order to determine the coefficients [r.sub.m], the normal electric field continuity [partial derivative][E.sub.z]/[partial derivative]z [[rho].sub.e]/[epsilon] at z = 0 should be utilized. The z-directed electric field [E.sub.z] is represented as [E.sub.z] ([[rho], z) = i/w[mu][epsilon]([partial derivative].sup.2]/[partial derivative][z.sup.2] + [k.sup.2])[A.sup.I.sub.z]([rho], z). Enforcing the normal electric field continuity at z = 0, we get
[r.sub.m] = -[[epsilon].sub.2]/[[epsilon].sub.1]i[xi].sub.m[p.sub.m]. (8)
where we assume [partial derivative]/[[partial derivative].sub.z][R.sup.E.sub.m]([rho], z)|z=o = 0. Note that the condition [partial derivative]/[[partial derivative].sub.z](p,z)\z=o = 0 will be verified later.
In terms of the Green's function relation, the component [R.sup.E.sub.m([rho]m(p,z) to cancel out Em(p, z) is formulated as
R.sup.E.sub.m]([rho], z) = -[integral][partial derivative]/[partial derivative]n'[E.sub.m]([bar.r')] [G.sup.zz.sub.A]([bar.r], [bar.r'], (9)
where n is an outward normal direction denoted in Figure 2 and [G.sup.zz.sub.A]([bar.r], [bar.r'] is the z-directional Green's function excited by the z-directed source in terms of a magnetic vector potential A. Substituting (7) into (9) yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Since (10) has singular points at [kappa] = [kappa].sub.m], (10) is not efficient for numerical computations. As such, we use the integral path deformation proposed in . According to , we replace [zeta] with [zeta] = [[kappa].sub.2v(v - i).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
Considering (12), we can prove [partial derivative]/ [partial derivative]z[R.sup.E.sub.m]([rho]|,z=o = 0. Although the integrand in (12) is oscillating when v [much greater than] 1 and z [much greater than] 1, the integrand does not have any singularities. Thus, we can integrate (12) very easily to get the magnetic vector potential [A.sup.II.sub.z] in (6) for the open region near z [approximately equal to] 0.
Using the Green's second integral identity, the magnetic vector potential [A.sup.II.sub.z]/([rho], z) in (6) is also simplified to a finite integral as
[E.sub.m]([rho], z) + [R.sup.E.sub.m]([rho]p, z) = - [[integral].sup.b.sub.a] [C.sub.o]([[kappa]].sub.m][rho]')[K.sub.[rho]z]([rho]')[rho]'d[rho]' +[[DELTA].sub.0]([rho], z), (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)
and [[delta].sub.ml] is the Kronecker delta. When we adopt the spherical coordinate system (r, [theta], [phi]) and r [right arrow] [infinity], the far-field for the magnetic vector potential [A.sup.II.sub.z]([rho],z) in (6) is asymptotically given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)
Similarly, the [E.sub.[theta]] field in the far-field is represented as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)
Differentiating (5) and (6) with respect to the [[rho]-axis and applying the [H.sub.[phi]] field continuity at z = 0 gives the final simultaneous equations for [p.sub.m as
[[infinity].summation over (m=0)][p.sub.m][I.sub.E](m,l) = [S.sub.E,l], (20)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
[f.sup.E.sub.l](a; k) = kb[H.sup.(1).sub.0](kb)[C'.sub.0]([k.sub.l]b) [ka[H.sup.(1).sub.0](ka)[C'.sub.0]([k].sub.l]a). (25)
[f.sup.E.sub.l](b;k) = kb[J.sub.0](kb)[C'.sub.0](k.sub.l]b) - ka[J.sub.0](ka) [C'.sub.0]([k.sub.l]a). (26)
3. NUMERICAL COMPUTATIONS
We solved the simultaneous equations given by (20) and compared with [2,16]. Figure 3 illustrates the behaviors of reflection coefficients when a TEM mode (s = 0) impinges on the flanged open end of a coaxial line. In this figure, M is the number of truncated modes. Our computational results based on (20) agree very well with [2, 16] when M [greater than or equal to] 2. A dominant mode solution (M = 1) is only valid for low frequency.
In Figure 4, we can observe the magnitude variations of the integrand for [I.sup.E.sub.ml] in (24). The radiation integral with deformed integral path [I.sup.E.sub.ml] does not have any singularity: this is desirable for numerical integration. Furthermore, the integrand shown in Figure 4 has fast-convergent characteristics when the integration variable v becomes large. Our computational experience indicates that the maximum value of v for numerical integration can be five to ten in order to obtain numerical convergence for most of cases.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Figure 5 shows the behaviors of radiation patterns of a flanged coaxial line. Contrary to the reflection characteristics in Figure 3, a dominant mode solution (M = 1) is almost the same as the higher-mode solutions (M [greater than or equal to] 2). In addition, the radiation pattern of a flanged coaxial line has peak antenna gain at [theta] = 90[degrees] which is similar to a flanged monopole antenna. This indicates that the reflection characteristics are mainly determined near the open end and flange of a coaxial line.
Using the Green's function and mode-matching technique, an analytic and numerically efficient analytic approach is proposed for an infinitely flanged coaxial line. The reflection behaviors of a flanged coaxial line were compared with other results and showed good agreements. The permittivity measurement and estimation with a flanged coaxial line can be more precisely performed with the proposed analytic solutions with fast-convergent radiation integrals.
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Y. H. Cho *
School of Information and Communication Engineering, Mokwon University, 21 Mokwon St., Seo-gu, Daejeon 302-729, Republic of Korea
Received 9 November 2011, Accepted 16 December 2011, Scheduled 20 December 2011
* Corresponding author: Yong Heui Cho (email@example.com).
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|Publication:||Progress In Electromagnetics Research Letters|
|Date:||Jan 1, 2012|
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