A Novel Method for Determining the R-Card Sheet ...

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 7, JULY 2009

A Novel Method for Determining the R-Card Sheet Impedance Using the Transmission Coefficient Measured in Free-Space or Waveguide Systems Milo W. Hyde, IV, Michael J. Havrilla, Senior Member, IEEE, and Paul E. Crittenden

Abstract—Free-space and rectangular waveguide techniques for determining the effective complex permittivity and, ultimately, the effective sheet impedance of an R-card using the forward transmission coefficient are presented. The advantage of using a transmission coefficient method instead of a more traditional reflection-based technique is discussed. The exact transcendental expressions relating the transmission coefficient and effective complex permittivity are derived and approximated using the Maclaurin series for sine and cosine. It is shown that the Maclaurin series expansion leads to simple closed-form solutions to the effective complex permittivity and avoids the use of sensitive and often unstable root search algorithms, which are necessary to solve transcendental equations. The accuracy of the approximations is directly related to the R-card’s thickness and wavenumber. Free-space (4–16 GHz) and waveguide (8.2–12.4 GHz) measurements are made using two R-cards of differing thicknesses and impedances to demonstrate the method and regimes of validity. An uncertainty analysis is also performed to demonstrate the robustness of the technique. Index Terms—Material measurements, permittivity, R-card, sheet impedance, transmission coefficient.

I. I NTRODUCTION

R

ECENTLY, monostatic radar cross-section (RCS) measurements were made for a resistive sheet wrapped around a polystyrene foam cylinder at 7 GHz to provide experimental verification of an R-card physical optics (PO) approximation [1]. In both polarizations (only H-pol is shown in Fig. 1), the measured monostatic RCS was approximately 1.5 dBsm higher than that predicted by the PO technique and the modal solution despite using a vector background subtraction calibration. In the derivations of the PO approximation and modal solutions, the R-card’s impedance value was assumed to be purely resistive, being based on a DC four-point probe measurement. Since both the PO and modal solutions were nearly identical, the unfavorable RCS measurement results of Fig. 1 prompted the investigators to examine the validity of using DC R-card values Manuscript received September 18, 2007; revised June 20, 2008. First published February 10, 2009; current version published June 10, 2009. This work was supported in part by the Air Force Research Laboratory. The Associate Editor coordinating the review process for this paper was Dr. Sergey Kharkovsky. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government. M. W. Hyde, IV and M. J. Havrilla are with the Air Force Institute of Technology, Wright–Patterson Air Force Base, Dayton, OH 45433-7765 USA. P. E. Crittenden was with the Air Force Institute of Technology, Wright–Patterson Air Force Base, Dayton, OH 45433-7765 USA. He is now with Jacksonville University, Jacksonville, FL 32211 USA. Digital Object Identifier 10.1109/TIM.2009.2013673

Fig. 1. Theoretical and average measured monostatic RCS (7 GHz, H-pol) of an 892-Ω/sq R-card wrapped around a foam cylinder. Included are error bars showing a ±0.5 dBsm measurement uncertainty in RCS values.

for applications at microwave frequencies. It is common to fabricate an R-card via carbon deposition on a very thin Kapton film. Depending on the operational frequency regime, this film may contribute a significant capacitive reactance to the overall sheet impedance of the R-card. An AC expression relating the effective sheet impedance Zeff and the effective complex permittivity εeff has been developed [2]–[9] based on Ampere’s law and an Ohms-per-square formulation, i.e., Zeff =

1 jη0 c 1 = =− . σeff τ jω(εeff − ε0 )τ ωτ (εeff r − 1)

(1)

In this relation, σeff is the effective complex conductivity, τ is the sheet thickness, ω is the angular frequency, ε0 is the permittivity of free space, η0 is the intrinsic impedance of free space, c is the speed of light, and εeff r = εeff /ε0 = (ε + σ/jω)/ε0 is the relative effective complex permittivity. For σ jω(ε − ε0 ) Zeff ≈

1 . στ

(2)

Thus, so long as the conduction current significantly dominates over the net displacement current, a DC four-point probe resistance measurement can be used at microwave frequencies with little error. Otherwise, for cases in which the net displacement current becomes significant, (1) must be used.

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HYDE et al.: DETERMINING THE R-CARD SHEET IMPEDANCE USING THE TRANSMISSION COEFFICIENT

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electric and magnetic fields at the material interfaces. Solving the resulting algebraic expressions produces the desired result, i.e.,

S21

1 − Γ2 e−jkτ = 1 − Γ2 e−j2kτ

Γ=

η − η0 η + η0

(3)

where the wavenumber k of the material under test (MUT) is given by the relation

Fig. 2.

k=ω

Resistive sheet with thickness τ immersed in free space.

Proper application of (1) requires an accurate measurement of εeff r (as well as τ ). There are instances in the literature in which simple methods for extracting the relative effective complex permittivity are derived using the reflection coefficient in a waveguide system [4]. Since the quality of the reflection coefficient measurement heavily relies on the location of the sample with respect to the calibration plane and can be an order of magnitude less certain than the transmission coefficient in most operational ranges [10]–[12], using it to compute the relative effective complex permittivity of thin sheets can be problematic. The goal of this paper is to present a novel method for calculating the relative effective complex permittivity and, ultimately, the effective impedance of thin sheets based on a transmission measurement. The transmission coefficient has no dependence on the location of the sample relative to the calibration plane and can be a more certain measurement [10]. Consequently, the method proposed here can offer an improved measurement alternative as compared to more traditional reflection-based techniques. Section II presents a method for determining the relative effective complex permittivity (and, thus, the effective impedance) from the transmission coefficient in free-space and rectangular waveguide systems. The exact transcendental expressions and the N th-order approximations for computing εeff r are derived and discussed. Section III demonstrates the accuracy of the approximations compared with the exact transcendental solutions using measured data from two impedance sheets. An uncertainty analysis is performed to show that the technique is relatively robust when thickness ambiguities are present. Finally, Section IV concludes this paper by showing the dramatic effect that using the true microwave frequencybased sheet impedance in (1) can have on predicting the RCS of R-card structures.

ε0 μ0 εeff r = k0

The geometry of a free-space measurement is shown in Fig. 2. In region 1 of Fig. 2, a plane wave is normally incident on an infinitely planar medium of thickness τ . Since the materials investigated in this paper are isotropic, the transmission coefficient is independent of the polarization of the incident field. An expression for the forward transmission coefficient S21 can be found by enforcing the continuity of transverse

(4)

and η and η0 are the intrinsic impedances of the MUT and free space, respectively [13], [14]. This expression can further be reduced into an explicit function of k, i.e., S21 (k) =

2kk0 . jk 2 sin kτ + 2kk0 cos kτ + jk02 sin kτ

(5)

Since values of the material’s relative effective complex permittivity εeff r are sought [being related to Zeff via (1) and k by the use of (4)], one would like to manipulate (5) into an analytical expression of the wavenumber as a function of S21 . This, however, is not possible. To find values of εeff r using (5), numerical inversion techniques, such as Newton’s method or nonlinear least squares, must be used. While extremely accurate, these routines require an initial guess of the unknown parameter. If one has no prior knowledge of the MUT, the correct solution or even convergence of these techniques is not guaranteed. To get accurate values for εeff r without having prior knowledge of the MUT, N th-order approximations, having closed-form solutions of (5), can be made by utilizing the Maclaurin series for sine and cosine, namely

sin x =

∞ (−1)n 2n+1 x (2n+1)! n=0

cos x =

∞ (−1)n 2n x . (2n)! n=0

(6)

Substituting (6) into the denominator of (5) produces ∞ ∞ 2kk0 (−1)n τ 2n+1 2n+1 (−1)n τ 2n 2n k k = jk 2 + 2kk0 S21 (2n + 1)! (2n)! n=0 n=0

+jk02

∞ (−1)n τ 2n+1 n=0

II. F REE -S PACE AND W AVEGUIDE T RANSMISSION C OEFFICIENT M ETHOD

εeff r

(2n + 1)!

k 2n+1 . (7)

Expanding the summations and grouping like powers of k produces the N th-order approximation, i.e., n−1 2n−1 τ 2 2n j(−1) + ··· + k 0 = 2 + jk0 τ − S21 k0 (2n − 1)! jk0 (−1)n τ 2n+1 2(−1)n τ 2n + + + · · · . (8) (2n)! (2n + 1)!


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The N th-order approximation for a waveguide measurement is found in precisely the same manner and can directly be obtained from (8) merely by replacing the free-space wavenumbers with waveguide wavenumbers, i.e., n−1 2n−1 τ 2 2n j(−1) + · · · + kz 0 = 2 + jkz0 τ − S21 kz0 (2n − 1)! jkz0 (−1)n τ 2n+1 2(−1)n τ 2n + + + · · · (9) (2n)! (2n + 1)! where kz and kz0 are 2 kz = k02 εeff r − (π/a)

kz0 =

k02 − (π/a)2 .

(10)

The benefit of this analysis is that by using (8) or (9), one can obtain accurate εeff r values without prior knowledge of the MUT. One only needs to find the roots of a polynomial of which only one is physically realizable. For typical R-cards, i.e., |kτ | 1, one or two orders of the approximation will be sufficient to be within ±0.01 of numerical inversion values. This implies that accurate values of εeff r can be found for most R-cards by solving either a linear or a quadratic equation. Note: The free-space N th-order approximation (8) can be put into a polynomial expression of εeff r rather than k by a simple substitution stipulated by (4). The waveguide N th-order approximation (9), on the other hand, cannot without a great deal of simplification because of the more complicated relationship between the wavenumber and εeff r in (10). It is for this reason and to show the similarity between the two expansions that both approximations were presented as polynomials of the wavenumber. III. R ESULTS Two R-cards having DC four-point probe sheet resistivities of 892- and 64-Ω/sq were measured at microwave frequencies to verify the accuracy of the N th-order approximation introduced in Section II. To demonstrate convergence of the method, the approximations were compared to the root search values obtained using a stopping criterion of 10−6 . The first resistive sheet investigated in this paper was an 892-Ω/sq, τ = 25.4 μm (0.001 in), R-card. Because this R-card is extremely thin, it is relatively difficult to get the material to support itself in a waveguide fixture. A free-space measurement system, which has a sample support holder, is much more applicable in this scenario. It is for this reason that only free-space measurement results are presented for this R-card. Fig. 3 shows the relative effective complex permittivity of the 892-Ω/sq R-card measured in a free-space system. It compares the first-, second-, and third-order approximations to the values returned by a root search of (5). Note that convergence of the approximation only requires three orders. This translates to finding the roots of a quadratic equation. Error bars in the figure show the acute sensitivity of the relative effective complex permittivity measurement to a ±0.025τ uncertainty. This is contrasted with the relative insensitivity of Zeff to the same uncertainty (see Fig. 4 with exceedingly small error bars). This is

Fig. 3. (a) Real and (b) imaginary effective complex permittivity values for the 892-Ω/sq R-card comparing free-space first-, second-, and third-order approximations with root search values.

easily explained by examining the first-order approximation for εeff r , i.e., εeff r =

2c − 2cS21 − jωτ S21 . jωτ S21

(11)

The denominator of (11) demonstrates why εeff r suffers from even small uncertainties in τ . Contrast this with the first-order approximation for Zeff , which is found by substituting (11) into (1), i.e., Zeff =

η0 cS21 . 2c − 2cS21 − jωτ S21

(12)

Since this R-card is electrically thin, the last term in the denominator of (12) is roughly two orders of magnitude smaller than the first term. It therefore contributes little to the value of Zeff , thereby making Zeff nearly independent of τ . This is an important finding of the Maclaurin series analysis. Standard commercial micrometers are accurate to within



Fig. 4. (a) Real and (b) imaginary effective sheet impedance values for the 892-Ω/sq R-card comparing free-space first-, second-, and third-order approximations with root search values.

±12.7 μm (±0.0005 in). For the 892-Ω/sq R-card examined in this paper, that represents a ±50% uncertainty. At only a ±2.5%τ uncertainty, Fig. 3 demonstrates the deficiency of to characterize thin R-cards. The effective sheet using εeff r impedance, being nearly independent of τ , is a very good alternative for such scenarios. As the thickness of the R-card increases, Zeff does begin to show dependence on τ . However, the τ of thicker R-cards is easier to measure, and εeff r becomes less sensitive to uncertainties in its value. The last resistive sheet investigated in this paper was a 64-Ω/sq, τ = 762 μm (0.030 in), R-card. This R-card, being 30 times thicker than the former, presented no measurement difficulties, and both free-space and waveguide data were collected. For the sake of brevity and since Figs. 3 and 4 demonstrate the N th-order approximation’s accuracy in a free-space system, only waveguide measurement results are shown for the 64-Ω/sq R-card. Fig. 5 shows the relative permittivity of the 64-Ω/sq R-card measured in a waveguide system. It compares the first-, third-, and sixth-order approximations to the values returned by a root search of (5). As compared with the thinner

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Fig. 5. (a) Real and (b) imaginary effective complex permittivity values for the 64-Ω/sq R-card comparing waveguide first-, third-, and sixth-order approximations with root search values.

R-card, convergence now requires six orders. This translates to finding the roots of a cubic equation. As before, error bars in the figure show the sensitivity of the relative effective complex permittivity measurement to a ±0.025τ uncertainty. Note that since this R-card is thicker, it is not as sensitive to uncertainties in τ . Fig. 6 shows the Zeff values for this R-card. Although this R-card is 30 times thicker than the 892-Ω/sq R-card, its Zeff still behaves as if it were relatively insensitive to thickness. Since the N th-order approximation (8) was derived using the Maclaurin series for sine and cosine, one can determine the number of orders required for accuracy by examining the value of the next higher order terms in the expansions for sine and cosine, i.e., 2n+2 (kτ )2n+3 and (kτ ) (13) (2n + 2)! (2n + 3)! respectively. The order of the expansions n should be chosen such that the remainder terms in (13) are less than a specified tolerance.


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Fig. 7. Theoretical and average measured monostatic RCS (7 GHz, H-pol) of an 892-Ω/sq R-card wrapped around a foam cylinder. Measurements are based on improved R-card values using the novel free-space and waveguide methods.

exact transcendental expressions via a root search. Since most R-cards will converge using less than sixth-order expansion terms, the resulting expressions for εeff r will be either second- or third-order (quadratic or cubic) equations. The gain in runtime is approximately a factor of ten when using the N th-order approximation versus a root search. Even if the application demands very high accuracy, the technique can still be used to supply a superior initial guess for a root search routine, resulting in much faster convergence and reliability. R EFERENCES

Fig. 6. (a) Real and (b) imaginary effective sheet impedance values for the 64-Ω/sq R-card comparing waveguide first-, third-, and sixth-order approximations with root search values.

IV. C ONCLUSION After analyzing the material measurement results and noting the significant reactive component of the 892-Ω/sq R-card, the investigators realized the limitations of using (2) as an approximation to the sheet impedance. Making use of the newly obtained values of the R-card’s effective sheet impedance based on (1), the theoretical RCS originally shown in Fig. 1 now agrees significantly better with the measurement (Fig. 7). The technique presented in this paper provides closed-form expressions to the effective complex permittivity (or the sheet impedance) of an R-card in a free-space or waveguide material measurement system using a Maclaurin series expansion. It was shown in the course of this investigation that convergence of these expressions is fast, typically requiring no more than a few orders. It was also found that Zeff is nearly independent of the thickness of the R-card, making it a superior parameter to εeff r in characterizing thin R-cards. Depending on the level of accuracy required, the technique presented in this paper can serve as a substitute to solving the

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Milo W. Hyde, IV received the B.S. degree in computer engineering from Georgia Institute of Technology, Atlanta, in 2001 and the M.S. degree in electrical engineering from the Air Force Institute of Technology, Wright–Patterson Air Force Base, Dayton, OH, in 2006. He is currently working toward the Ph.D. degree in electrical engineering with the Air Force Institute of Technology. From 2001 to 2004, he was a Maintenance Officer with the F-117A Nighthawk, Holloman Air Force Base, Alamogordo, NM. From 2006 to 2007, he was a Government Researcher with the Air Force Research Laboratory, Wright–Patterson Air Force Base. His current research interests include electromagnetic material characterization, guided-wave theory, scattering, and adaptive optics.

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Michael J. Havrilla (S’85–M’86–SM’05) received the B.S. degree in physics and mathematics and the M.S. and the Ph.D. degrees in electrical engineering from Michigan State University, East Lansing, in 1987, 1989, and 2001, respectively. From 1990 to 1995, he was with General Electric Aircraft Engines, Cincinnati, OH, and Lockheed Skunk Works, Palmdale, CA, where he worked as an Electrical Engineer. He joined the Department of Electrical and Computer Engineering, Air Force Institute of Technology, Wright–Patterson Air Force Base, Dayton, OH, as an Assistant Professor in 2002 and is currently an Associate Professor. His current research interests include guided-wave theory, electromagnetic materials characterization, microwave engineering, and electromagnetic radiation and scattering.

Paul E. Crittenden received the B.S. degree in mechanical engineering, the M.S. degree in engineering mechanics, and the Ph.D. degree in mathematics with a minor in electrical engineering from the University of Nebraska-Lincoln in 1992, 1995, and 2002, respectively. From 2002 to 2004, he was an Instructor of mathematics and electrical engineering and a Research Engineer in mechanical engineering with the University of Nebraska-Lincoln. From 2004 to 2007, he was a Visiting Assistant Professor of mathematics with the Air Force Institute of Technology, Wright–Patterson Air Force Base, Dayton, OH. He is currently an Assistant Professor with Jacksonville University, Jacksonville, FL. His current research interests include guided-wave theory, electromagnetic and thermal material characterizations, electromagnetic radiation and scattering, Green’s functions, and integral transforms.
