Modified Nicolson-Ross-Weir (NRW) method to retrieve the constitutive parameters of low-loss materials Adriano Luiz de Paula Division of Materials, AMR Institute of Aeronautics and Space, IAE São José dos Campos, Brazil
[email protected] [email protected]
Joaquim José Barroso Associated Plasma Laboratory National Institute for Space Research, INPE São José dos Campos, Brazil
[email protected]
Mirabel Cerqueira Rezende Division of Materials, AMR Institute of Aeronautics and Space, IAE
[email protected] Abstract — The constitutive parameters of a Teflon sample of arbitrary thickness are retrieved from scattering parameters experimentally measured with a vector network analyzer and a waveguide system into which the test sample is inserted by filling the transverse section of the X-band waveguide. Based on the Nicolson-Ross-Weir (NRW) method and modified to eliminate the phase ambiguity inherent to the NRW procedure, the algorithm used is able to correctly retrieve over the frequency range considered (8.20 to 12.40 GHz) the permittivity and permeability without any divergence at half-wavelength resonant frequencies for either dielectric or magnetic materials. Index Terms—electric permittivity; magnetic permeability; radiation absorbing material; computational modeling
I.
INTRODUCTION
Efficient measurements of complex permittivity () and permeability (μ) of broadband materials are of great interest in scientific and industrial applications, in studies of the biological effects of electromagnetic radiation, in sintering of ceramics, plastic welding, communications systems and remote sensing [1]. Among the measurement methods [2], the technique of transmission / reflection (TR), established by Nicolson and Ross [3] is a non-interactive method, which retrieves the complex permittivity and complex permeability of isotropic materials from measurements of scattering parameters. While efficient and compact, the explicit Nicolson-Ross equations become unstable when the reflection scattering parameter S11 is close to zero and exhibits ill-conditioned behavior for the samples under test showing thicknesses corresponding to integer multiples of half the length wave in the sample. An experimental solution to overcome this deficiency is by using samples with a length less than half the wavelength. But this approach severely limits the application of the TR since shortlength samples make measurements increasingly uncertain. In fact, to minimize the uncertainties in low-loss materials, a relatively long sample thickness is preferred [2]. Moreover, the inverse problem produces multi-valued solutions that may result from ambiguity in determining the phase shift of the
field transmitted at frequencies of multiples of half wavelength in the material. To resolve this ambiguity, Weir [4] proposed a technique that compares the delay time calculated and measured signal propagating through the sample, although this method cannot be very valid for dispersive media (as in photonic structures), since the concept of group velocity is no longer useful in regions of anomalous dispersion. Advances in the technique of TR were made by BakerJarvis et al. [5] by obtaining expressions invariant with respect to the reference position of the plan and length of the sample. By setting the relative permeability μr = 1, this new procedure allows measurements to be performed on samples of arbitrary length and minimizes the instability of the equations of Nicolson Ross-Weir (NRW). For the combined complex permittivity and permeability, however, the equations also show themselves to be unstable at multiples of halfwavelengths in the material. By exploiting a variant of the method of NRW, Boughriet et al. [6] presented a formalism that is stable for low loss materials, without any disagreement or undesirable ripples that appear in the calculated permittivity spectrum. Chalapat et al. [7] combined the NRW and the Baker-Jarvis methods to obtain an explicit and reference-plane invariant method. This method, however, still produces sharp discontinuities in discrete frequency spectrum calculated from the real part of refractive index for the dielectric samples [7]. By redefining new variables in the inverse problem [2]-[6], a way to circumvent this instability is by using only the amplitudes of the scattering parameters, but supplemented by the phase of the transmission coefficient as discussed in [2]. Furthermore, including the determination of reference planes, the procedure NRW standard procedure was modified to retrieve the effective constitutive parameters of metamaterials for which and μ are simultaneously negative [8-11]. In the present report, however, we use a non-iterative method that eliminates the phase ambiguity and so determines unique constitutive parameters as recently presented in [12]. In using this method, complex scattering parameters are
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measured on a Teflon sample of arbitrrary length. The calibration procedure also establishes the reference planes for the measurement test ports, illustrated in Figs. 1 and 2.
Referenced to the surfaces of the planar slab, the reflection and transmission S-parameters are written as [2-8] S11 = Γ
S 21 = T
1−T 2
(1)
1 − Γ 2T 2
1− Γ2
(2)
1 − Γ 2T 2
where Z − Z0 Z + z0
Γ=
(3)
with Z= j
Fig. 1. Waveguide calibration set for X band .
ω μ0 μr γ
Z0 = j
ω μ0 γ0
(4)
denoting the reflection coefficcient between Z and Z 0 . The term T is the transmission coeffficient in the finite-length slab, namely, (5) T = exp(−γ d ) where γ= j
2π
λ0
γ0 = j
Fig. 2. Setup for measurements of S parameters.
II.
METHODS AND EQUATION
The problem of transmission / reflection to t be considered is depicted in Fig. 1, where a homogeneous, isotropic and flat slab of arbitrary thickness (d) and impeddance Z is placed inside a rectangular waveguide characterizzed by impedance (Z0). The cross section of the guide is complletely filled by the sample, onto which is launched from the left a TE10-mode electric field with the Poynting vector inn the longitudinal direction.
Fig. 3. Geometry of the retrieval problem: a loss planar slab completely filling the cross sectional area of a rectangular waveguidde characterized by the operating impedance.
2π
λ0
§λ · ε r μ r − ¨¨ 0 ¸¸ © λc ¹
§λ 1 − ¨¨ 0 © λc
· ¸ ¸ ¹
2
(6a)
2
(6b)
and 0 the free-space wavelenngth. We note that the complex propagation factor emboddies simultaneously the wave guiding medium (through the cutoff wavelength) and the material sample, through the relative r constitutive parameters , ,, , ,, ε r = ε r − jε r and μ r = μ r − jμ r . The coefficients Γ and T are expressed from the S parameters by [3, 4] Γ = K ± K 2 −1 T=
S11 + S 21 − Γ 1 − Γ( S11 + S 21 )
,
K=
2 2 S11 − S 21 +1 2 S11
(7) (8)
where the sign to be assigned in i (7) is such that Γ ≤ 1 , which is equivalent to the requiremennt Re Z >0 based on causality for passive materials [13]. Then noting n that Z / Z 0 = (1 + Γ ) /(1 − Γ ) and combining (4) with (6), the relative complex permeability is recast in the form λ0 g § 1 + Γ · (9) μr = ¸ ¨ Λ ©1− Γ ¹ where λ0 λ0 (10) , λ0 g = Λ= 2 1 − ( λ0 / λc ) ε r μ r − (λ0 / λc ) 2 are the guided wavelengths coorresponding to the empty and loaded waveguides, respectiveely. Using (10) to extract from (9) gives
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RESULTS AND DISCUSSION
III.
μr
(11)
Since the complex guided wavelength Λ is related to the wavenumber γ in the transmission coefficient
T = exp(−γ d ) through, γ = 2π / Λ then an explicit expression for Λ as function of T is obtained 1 j = ln(T ) (12). Λ 2π d Equally applied to rectangular and cylindrical wave guiding systems, the explicit NRW procedure is formulated by (1)(12), through which and are extracted from the reflection and transmission coefficients, Γ and T , determined in turn from the measurable parameters S11 and S21 according to (7) and (8). This calculation, however, has two main problems. The first arises from the term (1 + Γ) /(1 − Γ) in the right-hand side in (9), which can be expressed as 2 (1 + S11 ) 2 − S 21 1+ Γ =± 2 1− Γ (1 − S11 ) 2 − S 21
We utilize a sample of polytetrafluoro-ethylene (PTFE) in the shape of flat slab 0.4625-in thick, completely filling the transverse area of an X-band waveguide (0.40x0.90 in2) operating in the the TE10 propagation mode. The equipement used is an 8510C vector network analyzer and the WR-90 calibration kit from Agilent Technologies. The measured scattering parameters, S11 and S21, used to retrieve the permittivity and permeability are shown in Fig. 4, where the peak in the curve of S11 is an indicative measure of the first resonance ccurring at 10.07 GHz given the dimensions and constitutive properties of the sample. This resonance effect is also apparent in the phase plot in Fig. 4 (b), where an antisymmetric pair of curves intersects the horizontal axis, with a phase jump at the resonance frequency. In these conditions, the scattering parameters exhibit the values S11! 0 with S21! -1. 10
0 S Parameter in Magnitude in dB
εr =
§ 1 1 · + 2 ¸¸ 2 λc ¹ ©Λ
λ20 ¨¨
(13).
-10
-20
-30 S11 S21
-40
It is apparent that (13) is algebraically unstable when S11 approaches zero, and hence S21 goes to unity for low-loss samples. Also, the uncertainty in the phase of S11 greatly increases when ¨S11¨→0. The ill-conditioned behavior of the scattering parameters manifests itself at frequencies corresponding to multiples of one-half wavelength in the sample. To see this, we rewrite S11 and S21 in the form
S 21 =
( z − 1) sinh( γ d )
2z
(15) where z = Z / Z 0 is the normalized complex impedance of the
S 21 → 1. The second problem in the NRW procedure is posed by (12), since the imaginary part of ln T, is a many-valued function with multiple branches, which may lead to ambiguities in retrieving the expressions for ε and μ . But as will be shown in the next section, the NRW method modified in [12] is able to correct these two problems.
10.5
11
11.5
12
12.5
100 50 0 -50
S11 S21
-100 -150
( z + 1) 2 sinh( γ d ) + 2 z exp( − γ d )
,,
10
150
( z + 1) sinh( γ d ) + 2 z exp( −γ d )
,,
9.5
200
(14)
α → 0 , that is when ε r and μ r simultaneously vanish (a typical condition of low-loss materials), then S11 → 0 and
9
(a)
2
sample. Writing the propagation factor as γ =α +jβ, , where α is the attenuation constant and β (=2π/λg) the wavenumber of the incident wave, and allowing for d≈m(λg/2), , m integer, then βd ≈mπ, yielding S11 ~ sin(α d ) with S 21 ~ exp(α d ) . If
8.5
Frequency in GHz
S Parameter in Phase in Degree
S 11 =
2
-50 8
-200 8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Frequency in GHz
(b) Fig. 4. Scattering parameters S11 and S21 measured for a 0.4625-in-thick Teflon sample with assigned εr=2.03-j0.01 and μr=1.00-j0.001. (a) Magnitude in dB and (b) Phase in degrees.
The set of figures 5-7 compares ε, μ and the normalized impedance z calculated using both the NRW standard procedure and the correction method. We see that the difference between the calculated values of μ (Fig. 5) and ε (Fig. 6) from the two methods is impressive. While the curves ε and μ are retrieved by the stable model and reproduce the values previously assigned to ε and μ, sharp discontinuities arise from the curves following the NRW procedure. Moreover, the imaginary part of the impedance given by the NRW curve [Fig 7 (b)] shows a sharp dip and a non-physical negative dip at the resonant frequency 10.07 GHz. Instead, the
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impedance curve (Fig. 7) generated by the correction method presents a smoother behavior over the frequency band considered, in which the real part of normalized impedance continuously increases toward the upper limit in the regime of high frequency.
1.3 1.2
|Z| in Magnitude
1.1
1.4
1 0.9 0.8
|Z| calculated |Z| corrected
0.7 0.6
1.2
0.5 1
0.4 8
8.5
9
9.5
10.5
11
11.5
12
12.5
μ'
(a)
0.6
μ ' measured μ' calculated μ' corrected
0.4
0.2
0 8
10
Frequency in GHz
0.8
1
0.5 8.5
9
9.5
10
10.5
11
11.5
12
12.5 Z in Phase
Frequency in GHz
(a)
0
-0.5 Zº calculated Zº corrected
1.5
-1 1
-1.5 8
0.5
8.5
9
9.5
10
10.5
11
11.5
12
12.5
μ"
Frequency in GHz
0
(b) μ" measured μ" calculated μ" corrected
-0.5
-1
-1.5 8
8.5
9
9.5
10
10.5
11
11.5
12
Fig 7. Complex impedance for a sample of Teflon 0.4625-in-thick. (a) real and (b) imaginary part recovered by the procedure NRW (dotted line) and corrected (dashed line). 12.5
Frequency in GHz
(b) Fig.5. Complex permeability for a sample of Teflon 0.4625-in-thick, r = 1.00 to 0.001 j (a) real and (b) imaginary part of measured (solid line), recovered by the procedure NRW (dotted line) and corrected (dashed line). 2.5
2
ε'
1.5
1
ε ' measured ε ' calculated ε ' corrected
0.5
0 8
8.5
9
9.5
10
10.5
11
11.5
12
Since the TE10 mode begins to propagate at a cutoff frequency of 6.562 GHz, scattering parameters measured below this frequency exhibit unphysical behavior like the magnitude of S21 greater than unity, as apparent in the 2.0-6.5 GHz range in Fig. 8. To verify the behavior of the scattering parameters of the materials in a wider frequency range (2.00-26.50 GHz). Extrapolate the frequency range, using an interactive model NRW, changing the mode of spread for TEM. With this we calculate the scattering parameters using the equation 14:15 in amplitude (Fig. 9) and phase (Fig. 9b) in the range of desired frequency (2.00 -26.50 GHz). 20
12.5
Frequency in GHz
10 S Parameter in Magnitude in dB
(a) 2
1.5
ε"
1 ε" measured ε" calculated ε" corrected
0.5
0 -10 -20 -30 -40
S11 S21
-50 -60 0
0
5
10
15
20
25
30
Frequency in GHz
-0.5
-1 8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Frequency in GHz
Fig. 8. Scattering parameters S11 and S21 measured for a 0.4625-in-thick Teflon sample with assigned εr=2.03-j0.01and μr=1.00-j0.001in magnitude in dB (TE10).
(b) Fig 6. Complex permittivity for a sample of Teflon 0.4625-in-thick, with ε r = 2.03 to 0.001 j (a) real and (b) imaginary part of measured (solid line), recovered by the procedure NRW (dotted line) and corrected (dashed line).
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ACKNOWLEDGMENT 10
This work has been supported by FINEP (project no.1757/03), and CNPq (project no. 301583/06-3).
S Parameter in Magnitude in dB
0 -10 -20
REFERENCES
-30
[1]
-40
S11 S21
-50 -60 0
[2] 5
10
15
20
25
30
Frequency in GHz
(a)
[3]
200
[4]
S Parameter in Phase in Degree
150 100
[5]
50 0 -50
S11 S21
-100
[6]
-150 -200 0
5
10
15
20
25
30
Frequency in GHz
[7]
(b) Fig.9. Scattering parameters S11 and S21 calculated for a 0.4625-in-thick Teflon sample with assigned εr=2.03-j0.01 and μr=1.00-j0.001 (TEM). (a) Magnitude in dB and (b) Phase in degrees.
IV.
CONCLUSION
It is well known that the NRW method diverges for lowloss materials at frequencies corresponding to integer multiples of one half wavelength in the sample. At this particular frequency, the magnitude of the measured S11 parameter is particularly small (thickness resonance) and the S11 phase uncertainty becomes large. This leads to the appearance of inaccuracy peaks on the permittivity and permeability curves. This feature was verified in the present comparative study of the constitutive parameters of a Teflon slab for which the complex permittivity and permeability were retrieved using the Nicolson-Ross-Weir procedure (NRW), the most commonly used method to perform this calculation. The constitutive parameters of the test sample were also retrieved by using a correction method, which proved to be robust and no anomalies or divergences in the retrieved parameters were noticed in 2.00-26.50 GHz frequency range.
[8]
[9]
[10]
[11]
[12]
[13]
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