Note: Parameter extraction of samples without the

Note: Parameter extraction of samples without the direct application of the passivity principle from reference-plane-invariant measurements U. C. Hasar, and G. Ozturk

Citation: Review of Scientific Instruments 89, 076104 (2018); doi: 10.1063/1.5028193 View online: https://doi.org/10.1063/1.5028193 View Table of Contents: http://aip.scitation.org/toc/rsi/89/7 Published by the American Institute of Physics

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REVIEW OF SCIENTIFIC INSTRUMENTS 89, 076104 (2018)

Note: Parameter extraction of samples without the direct application of the passivity principle from reference-plane-invariant measurements U. C. Hasar1 and G. Ozturk2 1 Department 2 Department

of Electrical and Electronics Engineering, Gaziantep University, 27310 Gaziantep, Turkey of Electrical and Electronics Engineering, Ataturk University, 25240 Erzurum, Turkey

(Received 8 March 2018; accepted 26 June 2018; published online 13 July 2018) An extraction algorithm has been proposed for explicit complex permittivity and complex permeability determination of samples using reference-plane-invariant scattering parameters without resorting to the application of the passivity principle. The algorithm relies on the definition of new odd and even reflection coefficients which result in no-crossing of the negative real axis for the argument of the square-root functions. Permittivity measurements of two low-loss samples were performed for validation of our algorithm. Published by AIP Publishing. https://doi.org/10.1063/1.5028193

Interaction of electromagnetic signals with properties of materials turns into their electromagnetic response. This response can be correlated to many properties of the material including chemical composition, chemical reaction state, cure state, mechanical strength, etc. Measurement of electromagnetic properties of materials can be performed by resonant and non-resonant methods at microwave frequencies.1–4 Although resonator methods are the natural choice when highly accurate measurement results are needed, the narrow band nature as well as elaborate sample preparation requirement limits their application. On the other hand, due to their simplicity and relatively high accuracy, transmission-reflection non-resonant techniques are widespreadly considered in examining the electromagnetic response of isotropic conventional materials.3–6 The passivity principle should be implemented4–6 for ascertaining the correct sign before the square root expression of the intermediate quantity reflection coefficient Γ to extract physically accurate relative complex permittivity (ε r ) and relative complex permeability (µr ) of Foster-type passive (energy dissipating/absorbing) materials in the transmission-reflection techniques. In a recent study,7 it has been demonstrated that physically accurate ε r and µr can be extracted without resorting to determining the correct sign by using new formulas involving new even and odd reflection coefficients which take into account internally the passivity principle. Nonetheless, this study was restricted to reference-plane-dependent scattering (S-) parameters. In this note, we extend this methodology for sign-ambiguity-free extraction of ε r and µr of isotropic materials using reference-plane-invariant expressions. Figure 1(a) illustrates the schematic view of a sample with length L positioned arbitrarily (L 01 , L 02 ) into its sample holder with length L g = L 01 + L + L 02 . The sample has the propagation constant γ = α + j β and wave impedance Z. Assuming that the measurement system is calibrated to the reference planes in Fig. 1(a), then the wave cascading matrix (WCM)8 of the holder configuration between these planes can be written as 0034-6748/2018/89(7)/076104/3/$30.00

" # 1 S21 S12 − S11 S22 S11 M1 = −S22 1 S21    α α T 2 − Γ2  α1 2  Γ 1 − T 1 1 2 α2     =
 1 2 T 2  , 1 − Γ2 T  − αα21 Γ 1 − T 2 1 − Γ α1 α2  where S ab (a, b = 1, 2) are the measured S-parameters, Γ=

Z − Z0 , T = e−γL , α1 = e−γ0 L01 , α2 = e−γ0 L02 . Z + Z0

(1)

(2)

Here, γ and γ 0 are the propagation constants of the samplefilled and air-filled sample holder regions and Z 0 is the wave impedance of the air-filled q region. For a waveguide q with the dominant mode, γ = jk0 ε r µr − fc2 /f 2 , γ0 = jk0 1 − fc2 /f 2 , Z = jω µ0 µr /γ, Z 0 = jω µ0 /γ, and ω = 2πf, where fc and f are the cutoff and operating frequencies. For coaxial-line or free-space measurements, fc goes to zero. Our purpose is to extract ε r and µr without using the passivity principle from the measured S-parameters. Toward this end, in parallel with the study,7 we define the following new odd and even reflection coefficients: Γe1 = S11 + S12 , Γe2 = S22 + S21 , Γo1 = S11 − S12 , Γo2 = S22 − S21 .

(3)

Substituting Γe1 , Γe2 , Γo1 , and Γo2 from (3) into (1), we obtain " # 1 −∆Γ Γe1 + Γo1 M1 = , (4) 2 Γe2 − Γo2 −(Γe2 + Γo2 ) where ∆Γ = (Γe1 Γo2 + Γo1 Γe2 ).

FIG. 1. (a) Schematic view of the sample with length L asymmetrically (L 01 , L 02 ) positioned inside its sample holder and (b) photos of the X-band sample holder (L g = 60.36 mm), machined polypropylene sample (L = 5.10 mm—light gray), and polyvinyl chloride sample (L = 5.18 mm—black).

89, 076104-1

Published by AIP Publishing.

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U. C. Hasar and G. Ozturk

Rev. Sci. Instrum. 89, 076104 (2018)

Assuming that S-parameter measurements of an empty sample holder with length L g − L are measured, then the WCM matrix of this holder can be written as " # α1 α2 0 M2 = . (5) 1 0 α1 α2 Next, utilizing M 1 and M 2 in (1) and (5), we find    T 2 − Γ2 α12 Γ 1 − T 2  1 −1    . (6) M1 M2 =
 1 1 − Γ2 T 2  1 − Γ2 T − α2 Γ 1 − T 2  1

It is noted that the matrix M1 M2−1 in (6) can be written in another form as # " #" # " α1 0 1 Γ T 0 −1 X , X = . (7) M1 M2−1 = X 0 α11 Γ 1 0 T1 From (4), (5), and (7), we obtain  1 1 ∆Γ  T + = u1 = 2α1 α2 − , T Γe2 − Γo2 α1 α2 α1 α2 = e−γ0 (Lg −L) ,

T = e−γL =

u1 −

T −1 = e+γL =

q u12 − 4

, 2 q u1 + u12 − 4

(8a) (8b)

(9)

, (10) 2 where u1 = Tr(M1 M2−1 ) and Tr(?) denotes the trace of the square matrix “?.” It is noted that the argument of the squareroot functions in (9) and (10) does not cross the negative real axis of the complex plane7 because |Γe1 | ≤ 1, |Γe2 | ≤ 1, |Γo1 | ≤ 1, and |Γe2 | ≤ 1. Besides, it is observed that when α 1 = α 2 = 1 (reference planes are located at the right and left surfaces of the sample), the expression for T −1 in (10) reduces to that in (5) in the study,7 validating our formalism. Once T is found from (9) uniquely without resorting to the passivity condition, Γ2 can be found from (6) as Γ2 =

(u2 + T ) ∆Γ , T , u2 = 1 + u2 T α1 α2 (Γe2 − Γo2 )

where α 1 α 2 is given in (8b). After, Γ can be uniquely determined from v   t 2 1 − Γ2 T 2
2 S11 1 − Γ2 T 2 S11 2 Γ=
, R1 =
2 . R12 1 − T 2 Γ2 1 − T 2

FIG. 2. Measurement setup (network analyzer, coaxial cables, waveguide straights, and sample holder) used for validation of our proposed algorithm.

(11)

(12)

Finally, ε r and µr of the sample can be calculated from (1 + Γ) γ 1 1 , µr = , (13) γ = ln (1 − Γ) γ0 L T 1   γ  2  fc  2  εr = + . (14) µr jk0 f We note that on contrary to the method5 which uses S-parameter measurements of an empty sample holder with length L g , our formalism here utilises S-parameter measurements of an empty sample holder with length L g − L to determine unique ε r and µr because the trace operation would otherwise produce a quantity changing with Γ and T.

An X-band (8.2-12.4 GHz) measurement setup9 was used to test the accuracy of our proposed method, as shown in Fig. 2. Here, a portable vector network analyzer (VNA) from Keysight Instruments with model N9928A was functioned to generate electromagnetic signals and detect S-parameters. Two onemeter long 3.5 mm phase-stable cables were applied to secure the connection between the VNA and the coax-to-waveguide adapters which input/extract electromagnetic signals within waveguide straights. The sample holder with different lengths L g and L g − L houses the sample under test with length L. The well-known Thru-Reflect-Line calibration was applied to calibrate the setup up to the reference planes in Fig. 1(a). A waveguide section with length 10.16 mm was used as the line standard which produces a maximum ∓70◦ offset from 90◦ . A short-termination (with higher reflectivity) was used as the reflect standard. Then, we monitored the magnitudes of S 11 and S 21 of the thru connection with/without short termination. It is noted that |S 11 | < −40 dB and |S 21 | < −0.03 dB. S-parameters at 1001 frequency points in X-band were measured. Two different low-loss dielectric samples (polypropylene with L = 5.10 mm and polyvinyl chloride with L = 5.18 mm) were used for validation of our formalism. In the measurements, both samples were first arbitrarily positioned into an X-band sample holder (a waveguide straight) with L g = 60.36 mm. Figure 1(b) displays the photos of the machined dielectric samples and sample holder. Then, calibrated S-parameters were recorded for processing without knowing L 01 and L 02 values. Finally, we applied our formalism by first calculating T and Γ2 from (9)–(11) and Γ and ε r (µr = 1.0) of both samples from (13) and (14). For validation of our formalism, we also extracted ε r of both samples using the method in Ref. 5 by enforcing the passivity principle. Figures 3 and 4 illustrate the real and imaginary parts of the measured ε r and µr of both samples using our proposed method (denoted by “PM”) and the method in Ref. 5 (denoted by “Ref. 5”). It is seen from Figs. 3 and 4 that the extracted ε r and µr by our proposed method (shown by a dashed curve in red) are in good agreement with the extracted ε r and µr

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U. C. Hasar and G. Ozturk

Rev. Sci. Instrum. 89, 076104 (2018)

FIG. 3. Measured (a) ε r and (b) µr of the polypropylene sample (L = 5.10 mm) by the proposed method (denoted by “PM” and shown by a red solid curve) and the method in Ref. 5 (denoted by “Ref. 5” and shown by a blue solid curve).

FIG. 4. Measured (a) ε r and (b) µr of the polyvinyl chloride sample (L = 5.18 mm) by the proposed method (denoted by “PM” and shown by a red solid curve) and the method in Ref. 5 (denoted by “Ref. 5” and shown by a blue solid curve).

by the method in Ref. 5 (shown by a solid curve in blue). In addition, the extracted ε r by our method and the method in Ref. 5 are also consistent with the measured ε r with a dielectric resonator 10 (ε r  2.26 − i0.0004 around 9.4 GHz for the polypropylene sample and ε r  2.71 − i0.008 around 11.0 GHz for the polyvinyl chloride sample). Furthermore, the extracted µr values by our method and the method in Ref. 5 are in good agreement with the expected µr values of dielectric samples (µr  1.00 − i0.0). These results indicate that our signambiguity-free algorithm could be applied for electromagnetic property extraction of solid materials using reference-planeinvariant expressions. We have proposed a new extraction algorithm for sign-ambiguity-free determination of the complex permittivity and complex permeability of the sample arbitrarily located into its sample holder using reference-planeinvariant S-parameter measurements. Our proposed method, which uses new even and odd reflection coefficients, can be used for accurate extraction of electromagnetic properties of materials by using reference-plane-invariant

expressions without requiring the direct application of the passivity condition. This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Project No. 114E495. 1 L.

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