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 Estimation of dielectric slab permittivity is considered by using a flared coaxial line. A problem of reflection from a flared coaxial line that radiates into a dielectric slab with a flange is solved. A flared coaxial line is modeled with multiply stepped coaxial lines with different inner and outer conductors. A set of simultaneous equations for the modal coefficients is constituted based on the boundary conditions. Computations are performed to illustrate the reflection behavior in terms of the coaxial line geometry, frequency, and permittivity of a dielectric slab. Nomograms are developed to estimate the permittivity from the measured reflection coefficients. The utility of a flared coaxial line for the determination of slab permittivity is discussed.
 The nondestructive estimation of material permittivity has been extensively studied by using microwave sensors, including rectangular waveguides, coaxial lines, cavities, etc. An open-ended coaxial line has been widely used to measure dielectric properties of a slab at microwave frequencies due to its easy application [Levine and Papas, 1951; Noh and Eom, 1999]. A large aperture size of an open-ended coaxial line is required to accurately measure the reflection coefficients of low permittivity materials at low frequencies [De Langhe et al., 1994]. To improve the measurement sensitivity, we will introduce a flared coaxial line consisting of a conical waveguide with a large aperture size. A conical waveguide is attached to the terminal of a coaxial line to increase an aperture size and reduce the reflection. In sections 2, 3, and 4, we shall solve the boundary-value problem of radiation from a flared coaxial line. For scattering analysis, a conical waveguide is divided into multiple steps of coaxial line sections with different radii. The Hankel transform and mode matching is used to obtain the modal coefficients for multiply stepped coaxial lines. In section 5, computations are performed to construct nomograms that allow us to estimate the permittivity from the measured reflection coefficient. Measurements are performed to convert the measured reflection coefficients into estimated permittivities. A brief summary is given in Conclusion.
2. Field Representations
 Consider a flared coaxial line that radiates into a dielectric slab backed by air. The problem geometry is shown in Figure 1. A flared coaxial line has an infinite conducting flange. A flared section of a coaxial line is modeled in terms of the N-step coaxial lines with constant radii ρ and r shown in Figure 2. Assume that an incident TEM mode exicites the coaxial line. In region (X) (a < r < b, z < −t1), the incident and reflected H fields are
where βξ = ω, ξ is the permittivity in region (X), , Rn(r) = J1(λnr)N0(λnb) − N1(λnr)J0(λnb), and Jn( · ) and Nn( · ) are the nth order Bessel and Neumann functions. Note that λn is the eigenvalue satisfying the characteristic equation
The normalization factors of TM0n modes of the coaxial line are
We divide the conical waveguide region into N coaxial line sections with different radii and the same length. In region (p) (ρp < r < rp, −tp < z < −tp+1, 1 ≤ p ≤ N), H field consists of the TEM mode and higher modes as
where , p is the permittivity in region (p), , and . Similarly, is the eigenvalue of the coaxial line with the equation
The normalization factors are given as
In region (S) (0 < z < d), the field is represented in terms of the continuous mode in the spectral domain as
where and . Similarly in region (O), the field is given by
where and .
3. Enforcement of Boundary Conditions
 The tangential E-field continuity at z = −t1 requires
Multiplying (10) by and integrating from ρ1 to r1, we obtain
The boundary condition for tangential magnetic field at z = −t1 yields
Multiplying (12) by and integrating from ρ1 to b yields
The tangential E-field continuity between region (p − 1) and region (p) at z = −tp(p ≥ 2) requires
Multiplying (14) by and integrating from ρp to rp, we obtain
Similarly the magnetic field continuity between region (p − 1) and region (p) at z = −tp (p ≥ 2) leads to
Multiplying (16) by and integrating from ρp to rp−1 gives
The tangential E-field and H-field continuities at z = d lead to
Applying the Hankel transform to (18) and (19), respectively, yield
Applying the Hankel transform to the tangential E-field continuity at z = 0
and solving for , we have
The tangential H-field continuity at z = 0 is given as
Multiplying (27) by and integrating from ρN to rN gives
It is possible to solve a system of simultaneous equations (11), (13), (15), (17), and (30) for the modal coefficients cn, (), (), … , (). The reflection coefficient for the TEM mode is given by c0.
4. Special Cases
 In this section, we shall consider two special cases - an open half space and a dielectric slab backed by a metal plate. For a case of open half space, we let βs = βo and d = 0 in Figure 2. Under this condition, (9) then reduces to
and g1(ζ) = g2(ζ). Therefore, (30) can be rewritten as
For a case of a dielectric slab backed by a metal plate instead of air, the field representation in region (S) is modified as
It is possible to evaluate A(m,n) in rapidly convergent series by utilizing residue calculus. As such, (36) can be rewritten as [Lee et al., 1996]
where denotes and
5. Numerical Computations and Measurements
 We shall consider a flared coaxial line with a uniform inner conductor (θi = 0). The dimensions of the structure are a = 0.815mm, b = 2.655mm, Ro = 30mm, H = 20mm, and Ri = 0.815mm. A flared region is modeled with 20 steps (N = 20) of coaxial line with different outer conductor radii. For computations, it is necessary to truncate the number of modes in the simultaneous equations (11), (13), (15), (17), and (30). Three higher order modes (n = 1, 2, 3) in each step are included in computation to achieve numerical accuracy to better than 1% error.
 To check the validity of our analysis method, our numerical results are compared to the measured data. A flared coaxial line consists of a semirigid 50 Ω coaxial line (UT 250-A-TP type: a = 0.815mm, b = 2.655mm, r = 2.080), a SMA type connector, and a flare horn made of duralumin. A conducting rectangular flange (1m × 1m) is used. A cutoff frequency of the first higher-order mode in the coaxial line is about 55 GHz. All measurements were performed from 45 MHz to 6 GHz using a Wiltron 37225A network analyzer. A teflon slab (300mm × 300mm × 10mm) is used for a dielectric slab sample. Its permittivity is almost uniform (r = 2.08 + i2.08 × 10−3) up to 10 GHz. The effect of a finite sample and finite flange is ignored in theoretical computation.
 A comparison between the calculated and measured reflection coefficients is shown in Figure 3. Good agreements between the measured data and the theory are realized for air half space and teflon slab (d = 10mm) attached to the aperture. When a teflon slab is backed by a metal plate, the measured reflection coefficient is plotted in Figure 4. A good agreement is also observed except near higher frequencies. The difference between the calculation and measurement at higher frequencies (above 4.5 GHz) may be due to a possible measurement error. Also, we show the rate of convergence for |cn| for a metal-backed case in Table 1. It indicates that the series converges very fast and one or two higher order modes (n = 1, 2) are needed to achieve the accuracy of computations.
The parameters and the situation are the same as in Figure 4.
Figure 5 shows nomograms that provide the magnitude and phase information of reflection coefficient c0 as used in [Hirano et al., 1998] for an infinitely thick half space at 3GHz. The nomograms for open-ended coaxial line (a = 0.815mm, b = 2.655mm) and for the flared coaxial line are depicted in Figures 5a and 5b, respectively. A larger sensitivity ∂c0/∂r is obtained in Figure 5b, indicating that the flared coaxial line is less sensitive to an error in estimating from the measured c0.
Figure 6 illustrates the nomograms for a dielectric slab backed by air and by a metal plate, respectively. Using the measured data in Figures 3 and 4, we estimate the permittivity of a teflon slab from the nomogram in Figure 6. The estimated permittivities are shown in Table 2, indicating that error in r is less than 10% at 3 GHz with the true value (r = 2.08 + i2.08 × 10−3). This error reflects the difference of |c0| between the calculated and measured reflection coefficients at 3 GHz in Figures 3 and 4.
Table 2. Permittivity From Nomogram With Measured Data at 3GHz
Permittivity (Inverse Solution)
Real Part (Error, %)
Air half space
Teflon slab backed by air
Teflon slab backed by a metal plate
 It is possible to improve the bandwidth characteristics of permittivity estimation with an impedance matching technique. The impedance matching is realized by varying the radius of an inner conductor of the flared coaxial line. The characteristic impedance of the conical waveguide is given by [Marcuvitz, 1951]:
where r is permittivity of material that fills the interior of a conical waveguide and the angles, θi and θo are shown in Figure 1. Using other parameters (a = 0.815mm, b = 2.655mm, Ro = 30mm, H = 20mm), we obtain the parameter for impedance matching, θi = 25° or Ri = 9.3mm.
 In Figure 7, we show the behavior of when the flared coaxial line is impedance-matched (Ri = 9.3mm) and when it is unmatched (Ri = 0.815mm). The true reflection coefficient can be calculated with the true permittivity (true). The measured reflection coefficient is assumed to vary within ±5% in magnitude and ±10° in phase relative to the true reflection coefficient. The measured permittivity (measured) can be estimated from the nomograms using the measured reflection coefficient. The dimension of an unmatched flared coaxial line is designed to maximize ∂c0/∂r at 3 GHz, resulting in minimum (Δ/) at 3 GHz in the unmatched case. It is, however, seen that a use of the impedance matching reduces the relative error in the estimated permittivity (Δ/) substantially over a 1∼6 GHz range. This means that a flared coaxial line with a varying radius of inner conductor provides a simple microwave sensor for accurately estimating the slab permittivity.
 A problem of a flared coaxial line radiating into a dielectric slab with a flange is solved using the Hankel transform and mode matching technique. Computations are performed to illustrate the reflection behavior. Measurements are performed over 45 MHz ∼ 6 GHz and are compared favorably with the theory. Nomograms are given to determine the permittivity of a slab material from the measured reflection coefficient. A sensitivity of the reflection coefficient to a change in permittivity increases when the impedance matching is used by varying the radius of an inner conductor of a flared coaxial line. A flared coaxial line provides a simple means to accurately estimate the dielectric slab permittivity.