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Keywords:

  • flared coaxial line;
  • reflection coefficient;
  • Hankel transform;
  • mode matching

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[1] 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.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[2] 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[3] 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

  • equation image
  • equation image

where βξ = ωequation image, equation imageξ is the permittivity in region (X), equation image, Rn(r) = J1nr)N0nb) − N1nr)J0nb), and Jn( · ) and Nn( · ) are the nth order Bessel and Neumann functions. Note that λn is the eigenvalue satisfying the characteristic equation

  • equation image

The normalization factors of TM0n modes of the coaxial line are

  • equation image
  • equation image

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 equation image modes as

  • equation image

where equation image, equation imagep is the permittivity in region (p), equation image, and equation image. Similarly, equation image is the eigenvalue of the coaxial line with the equation

  • equation image

The normalization factors are given as

  • equation image
  • equation image

In region (S) (0 < z < d), the field is represented in terms of the continuous mode in the spectral domain as

  • equation image

where equation image and equation image. Similarly in region (O), the field is given by

  • equation image

where equation image and equation image.

image

Figure 1. Longitudinal section of a flared coaxial line with a flange.

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image

Figure 2. Geometry of N-steps model for a flared coaxial line.

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3. Enforcement of Boundary Conditions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[4] The tangential E-field continuity at z = −t1 requires

  • equation image

Multiplying (10) by equation image and integrating from ρ1 to r1, we obtain

  • equation image

The boundary condition for tangential magnetic field at z = −t1 yields

  • equation image

Multiplying (12) by equation image and integrating from ρ1 to b yields

  • equation image

The tangential E-field continuity between region (p − 1) and region (p) at z = −tp(p ≥ 2) requires

  • equation image

Multiplying (14) by equation image and integrating from ρp to rp, we obtain

  • equation image

Similarly the magnetic field continuity between region (p − 1) and region (p) at z = −tp (p ≥ 2) leads to

  • equation image

Multiplying (16) by equation image and integrating from ρp to rp−1 gives

  • equation image

The tangential E-field and H-field continuities at z = d lead to

  • equation image
  • equation image

Applying the Hankel transform equation image to (18) and (19), respectively, yield

  • equation image
  • equation image

Applying the Hankel transform to the tangential E-field continuity at z = 0

  • equation image

and solving for equation image, we have

  • equation image

where

  • equation image
  • equation image

The tangential H-field continuity at z = 0 is given as

  • equation image

Substituting (20), (21), and (23) into (26) leads to

  • equation image

where

  • equation image
  • equation image
  • equation image

Multiplying (27) by equation image and integrating from ρN to rN gives

  • equation image

It is possible to solve a system of simultaneous equations (11), (13), (15), (17), and (30) for the modal coefficients cn, (equation image), (equation image), … , (equation image). The reflection coefficient for the TEM mode is given by c0.

4. Special Cases

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[5] 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

  • equation image

and g1(ζ) = g2(ζ). Therefore, (30) can be rewritten as

  • equation image

For a case of a dielectric slab backed by a metal plate instead of air, the field representation in region (S) is modified as

  • equation image

Equation (30) can be reduced using equation image,

  • equation image

It can be reexpressed as

  • equation image

where

  • equation image

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]

  • equation image
  • equation image
  • equation image
  • equation image

where equation image denotes equation image and

  • equation image
  • equation image
  • equation image
  • equation image

5. Numerical Computations and Measurements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[6] 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.

[7] 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, equation imager = 2.08equation image0), 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 (equation imager = 2.08 + i2.08 × 10−3) up to 10 GHz. The effect of a finite sample and finite flange is ignored in theoretical computation.

[8] 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.

image

Figure 3. Comparison between calculated and measured reflection coefficient.

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image

Figure 4. Comparison between calculated and measured reflection coefficient for teflon slab (d = 10) backed by metal plate.

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Table 1. Convergence Rate of |cn| Versus na
 1 GHz2 GHz4 GHz6 GHz
  • a

    The parameters and the situation are the same as in Figure 4.

n|cn||cn||cn||cn|
00.9978900.9621630.3827050.761784
10.0026260.0046620.0071350.014279
20.0008700.0015370.0023670.004708
30.0004710.0008310.0012830.002548
40.0003150.0005560.0008600.001705
50.0002470.0004350.0006740.001335
60.0002330.0004100.0006340.001257
70.0001540.0002720.0004200.000833

[9] 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/∂equation imager is obtained in Figure 5b, indicating that the flared coaxial line is less sensitive to an error in estimating equation image from the measured c0.

image

Figure 5. Nomogram of reflection coefficient for complex permittivity (equation image) of infinite thickness materials at 3GHz. (a) Open-ended coaxial line (a = 0.815, b = 2.655). (b) Ours (a = 0.815, b = 2.655, Ro = 30, H = 20, and Ri = 0.815).

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[10] 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 equation imager is less than 10% at 3 GHz with the true value (equation imager = 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.

image

Figure 6. Nomogram for finite thickness (d = 10mm) dielectric slab at 3GHz. (a) Case of dielectric slab backed by air. (b) Case of dielectric slab backed by metal plate. The computation uses the dimensions of the flared coaxial line (a = 0.815, b = 2.655, Ro = 30, H = 20, and Ri = 0.815).

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Table 2. Permittivity From Nomogram With Measured Data at 3GHz
 Permittivity (Inverse Solution)
Real Part (Error, %)Imaginary Part
Air half space1.1 (10)0.02
Teflon slab backed by air2.01 (3.4)0.004
Teflon slab backed by a metal plate1.88 (9.6)0.07

[11] 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]:

  • equation image

where equation imager 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.

[12] In Figure 7, we show the behavior of equation image 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 (equation imagetrue). 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 (equation imagemeasured) can be estimated from the nomograms using the measured reflection coefficient. The dimension of an unmatched flared coaxial line is designed to maximize ∂c0/∂equation imager at 3 GHz, resulting in minimum (Δequation image/equation image) 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 (Δequation image/equation image) 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.

image

Figure 7. Effect (Δequation image = equation imagemeasuredequation imagetrue) of impedance matching of flared coaxial line (I: infinitely thick teflon sample, II: teflon slab (d = 10mm) backed by air half space, III: teflon slab (d = 10mm) backed by metal plate).

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6. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information

[13] 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.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information
  • De Langhe, P., L. Martens, and D. De Zutter, Design rules for an experimental setup using an open-ended coaxial probe based on theoretical modeling, IEEE Trans. Instrum. Meas., 43(6), 810817, 1994.
  • Hirano, M., M. Takahashi, and M. Abe, A study on measurement of complex permittivity by using flanged rectangular waveguide (in Japanese), EMT-98–70, 1998.
  • Lee, J. H., H. J. Eom, and K. H. Jun, Reflection of a coaxial line radiating into a parallel plate, IEEE Trans. Microwave Guided Wave Lett., 6(3), 135137, 1996.
  • Levine, H., and C. H. Papas, Theory of the circular diffraction antenna, J. Appl. Phys., 22(1), 2943, 1951.
  • Marcuvitz, N., Waveguide Handbook, p. 99, McGraw-Hill, New York, 1951.
  • Noh, Y. C., and H. J. Eom, Radiation from a flanged coaxial line into a dielectric slab, IEEE Trans. Microwave Theory Tech., 47(11), 21582161, 1999.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Field Representations
  5. 3. Enforcement of Boundary Conditions
  6. 4. Special Cases
  7. 5. Numerical Computations and Measurements
  8. 6. Conclusion
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
rds4923-sup-0001-tab01.txtplain text document0KTab-delimited Table 1.
rds4923-sup-0002-tab02.txtplain text document0KTab-delimited Table 2.

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