Determination of complex conductivity of thin strips with a transmission method Morteza Shahpari1, a) School of Engineering, Griffith University, QLD 4215, Australia (Dated: 27 March 2018)
Induced modes due to discontinuities inside the waveguide depend on the shape and material properties of the discontinuity. Reflection and transmission coefficients provide useful information about material properties of discontinuities inside the waveguide. A novel non-resonant procedure to measure the complex conductivity of narrow strips is proposed in this paper. The sample is placed inside a rectangular waveguide which is excited by its fundamental mode. Reflection and transmission coefficients are calculated by the assistance of the Green functions and enforcing the boundary conditions. We show that resistivity only impacts on one of the terms in the reflection coefficient. The competency of the method is demonstrated with a comparison of theoretic results and full wave modelling of method of moments and finite element methods. I.
INTRODUCTION
Developing new methods to measure various material parameters are of prominent importance as they enable one to perform precise measurements. Conventional methods to measure the conductivity of materials at microwave frequencies are the cavity, reflection and transmission methods1,2 . Cavity methods3–12 are generally based on the pertubation theory, where the sample under test (SUT) is placed inside a cavity and perturbs the natural modes of the cavity. Perturbations result in the shifts in the resonant frequency fr and quality factor Q. Material properties are extracted from the changes in fr and Q. In reflection methods13–16 , SUT is placed as the termination of the transmission line, and the material is characterized through investigation of the reflection coefficient. Transmission methods1,17–22 are based on the placing the object inside transmission line, and both transmission and reflection coefficients are used. Cavity methods3–11 are inherently narrowband, while reflection13,15,16 and transimssion methods18–21 are broad band methods. However, cavity methods are often considered as more accurate methods for material characterizations. In this paper, a method to measure the complex conductivity of the graphene and thin film materials is proposed. The method is based on the standard reflection/transmission methods, which are inherently broadband measurement method. The whole aperture of the waveguide had to be covered with the sample under test in previous transmission methods(e.g.20 ) for surface conductivity measurement. Af far as the authors are aware of, this is the first time a method for the conductivity of a thin strip is proposed, and it should be useful to measure the performance of materials that are synthesized in a strip shape. An example of such materials is graphene produced by laser depositions23,24 . Since the graphene is one atom thick material, the current state
a) Electronic
mail:
[email protected]
of the art methods for graphene fabrication need to support graphene on a substrate. For such cases, one needs to use the appropriate Green function while repeating the calculations. The outline of the paper is as follows: Initially, section II provides rigorous field theory for a conductive strip inside a rectangular waveguide. Simplified results for the impedance of the conductive strip is discussed in section III for uniform and cosinus hyperbolic distributions, and their corresponding fields on the aperture are reported in section IV. Numerical results from the analytic theory are validated in section V by modelling similar structure using full-wave commercial packages. Step by step procedure to measure the surface conductivity is also proposed (four steps).
II.
THEORY
This study follows the procedure described by Collin25 (Sec.8.5) to find reflection coefficient due to a narrow strip in a rectangular waveguide. The difference is that we assume a finite conductivity for the strip, while Collin assumed a perfect electric conductor (PEC) strip. We consider a waveguide with a cross section a×b in the x−y plane as shown in Fig. 1. An infinitely thin conductive strip with width 2t and conductivity of σ is placed in x = x0 and z = 0, and is stretched from y = 0 to y = b. It is assumed that a T E10 mode travels from −z. Electric and magnetic fields due to this mode are denoted by E i and H i . Presence of the strip makes discontinuity in the waveguide, induces currents J on the strips, and scatters the wave in both directions. One way to represent the scattered fields is to use Green functions which satisfy the waveguide boundary value problem. One can write Green function for the scattered electric field due to a current J = J (x0 )ˆ y inside rectangular waveguide as25 (Sec.5.6): Ge = −
∞ jωµ0 X 1 nπx nπx0 −γn |z| ˆ sin sin e y a n=1 γn a a
(1)
2 y
2t
Rcs
b
Zshunt x0
a
x Xa
FIG. 1. Geometry of the problem; thin strip of width 2t inside the rectangular waveguide.
where γn = αn +jβn is the complex propagation constant of mode n in the waveguide. The scattered electric field E s is Z E s (x, z) = Ge (x, x0 ) J (x0 ) dx0 . (2) S
and the total electric field inside the waveguide is E = E i + E s . Here, we assume the T E10 mode with the incident field πx −γ1 z ˆ e E i = sin y (3) a π πx γ πx 1 ˆ− ˆ (4) e−γ1 z x cos e−γ1 z z Hi = − sin jωµ0 a jωµ0 a a Because the strip is made of conductive material, so called standard impedance boundary conditions (SIBC)26 (Sec.2.4) are to be satisfied on the strip. ˆ × (E + + E − ) = η¯ · n ˆ ×n ˆ × (H + − H − ) n
(5)
where η¯ is the condition tensor of the sheet. In the following, we assume an isotropic non-magnetic sheet with conductivity of σs . Therefore, η¯ = (2σs )−1 I¯ where I¯ is the identity tensor. Plus and minus superscripts denote the field on z = 0+ or z = 0− , respectively. The tangential component of E has to be continuous at the junction. Therefore: Z πx ˆ + Ge (x, x0 )J (x0 ) dx0 y ˆ E + = E − = sin y (6) a We also find H s by inserting (1) in (2), and ∇ × E s = −jωµ0 H s . Tangential component of H s on each side of the boundary are: Z ∞ 1X nπx nπx0 ˆ sin sin J (x0 ) dx0 x a n=1 a a Z ∞ 1X nπx nπx0 ˆ =− sin sin J (x0 ) dx0 x a n=1 a a
Xcs
H s+ t =
(7)
H s− t
(8)
Difference in the sign is due to the fact that one mode is propagating to z > 0 while other propagates to z < 0.
FIG. 2. Equivalent circuit for a conductive strip inside waveguide
We juxtapose components into (5): −2 sin
Z ∞ jωµ0 X 1 nπx nπx0 πx −2 sin sin J (x0 ) dx0 a a n=1 γn a a Z ∞ 1 2X nπx nπx0 =− sin sin J (x0 ) dx0 (9) 2σs a n=1 a a
The coefficient of the non-evanescent part of the scattered E s traveling in the z < 0 would be the reflection coefficient Γ: Z jωµ0 πx0 Γ=− sin J (x0 ) dx0 . (10) aγ1 a The transmission coefficient T is also equal to sum of incident wave and non-evanescent part of scattered wave traveling towards z > 0. Z jωµ0 πx0 T =1− sin J (x0 ) dx (11) aγ1 a From (10) and (11), one finds T = Γ + 1 relationship. Therefore, we can consider the strip discontinuity as a shunt element across a transmission line (see Fig. 2). One can also show that Zshunt = − Γ+1 2Γ . As a result we to find expressions for (Γ + 1)/(2Γ) in some variational form. We can re-arrange (9) by taking the n = 1 term out of the series, and also moving all terms with n ≥ 2 to the right hand side: Z πx jωµ0 πx πx0 sin − sin sin J (x0 ) dx0 a aγ1 a a Z ∞ 1 X nπx nπx0 = sin sin J (x0 ) dx0 2aσs n=1 a a Z ∞ X jωµ0 nπx nπx0 + sin sin J (x0 ) dx0 (12) aγ a a n n=2 The left hand side of (12) can be factorized as (1 + Γ) sin πx a . Following the Collin’s procedure, we arrive at (Γ + 1)/2Γ form by multiplying both sides by J (x), integrating over x, and dividing both sides by 2Γ. Therefore,
3 the total shunt impedance is calculated as: 1 aγ1 2 × R πx j2ωµ0 sin a J (x) dx ( ∞ Z Z 1 X nπx0 nπx sin J (x0 )J (x) dx0 dx sin 2aσs n=1 a a ) Z Z ∞ X jωµ0 nπx nπx0 0 0 + sin sin J (x )J (x) dx dx aγn a a n=2 (13)
Zshunt =
By direct comparison with the analytic case solved by Collin, we can identify the terms corresponding to the equivalent circuit proposed in Fig. 2. The reactance jXa is independent of the material properties of the conductive strips. Interestingly, it appears in the same form as derived by Collin for a lossless strip inside a waveguide. On the other hand Zcs is the contribution due to the material properties of the conductive strip. Thus Zshunt = Zcs + jXa
FIG. 3. Currents on the conductive strip (x − y plane)
Collin also simplifies (17) for the case when the strip is exactly in the middle of the waveguide (x0 = a/2) jX =
∞ P
Zcs =
Zcs
γ1 n=1 = j4ωµ0 σs ∞ P
jXa =
γ1 n=2 2
1 γn
RR
0
nπx 0 0 sin nπx a sin a J (x )J (x) dx dx , R 2 sin πx J (x) dx a (15)
γ1 n=1 j4ωµ0 σs
Zcs =
0
nπx 0 0 sin nπx a sin a J (x)J (x ) dxdx . R 2 J (x) sin πx a dx (16)
(19)
∞ γ1 πt X 1 nπt . csc2 sin2 2 j4ωµ0 σs a n=1,3,... n a
(20)
This can be further simplified to (see Appendix)27 : γ1 π2 t πt csc2 . j4ωµ0 σs 4a a
(21)
CURRENT DISTRIBUTION J (x) ON THE STRIP
The formulations for Zcs and Xa in (15) and (16) are in the variational forms. This implies that choosing a testing function for J (x) does not make a huge influence on the results. Choosing an appropriate testing function, which is closer to the true current distribution, would undoubtedly improve the accuracy of the results. Nevertheless, trying different testing functions also assists to examine the numerical sensitivity to the choice of the testing functions.
A.
nπt 2 0 sin nπx a sin a , 2 sin πxa 0 sin πt a
1 n2
and for the case of the centered strip we have:
Zcs = III.
(18)
Similarly, we also find expressions for Zcs :
(14)
where ∞ R R P
∞ γ1 1 nπt πt X csc2 sin2 . 2 a n=3,5,... n2 γn a
Uniform Current
If the strip is reasonably thin t a, a rough approximation is to assume J (x) as uniform over the strip25 ∞ P
jX =
γ1 n=2 2
nπt 2 0 sin nπx a sin a . 2 sin πxa 0 sin πt a
1 n 2 γn
(17)
B.
Hyperbolic Cosine Distribution
A second current distribution of J (x) = J0 cosh[(x − x0 )/t]ˆ y was also studied for this problem. The motivation for such choice of testing function is to analytically model the singularity on the H fields, especially Hz component on the edges of the strip (x → x0 ± t). This effect significantly increases J near the edges (see Fig.3 for currents and fields near a PEC strip). By the assumption on the current J(x) as hyper cosine distribution, we get the Zcs and jXa as ∞ P
In2 γ1 n=1 Zcs = , j4ωµ0 σs I12 ∞ P 1 2 I γ1 n=2 γn n , jXa = 2 I12
(22)
(23)
4 where
FEM MoM
x+
In =
J(x) sin x−
=
nπx dx a
Uniform
(24)
nπx+ + − − A[sin nπx + sin nπx − cos nπx a a ] − B[cos a a ] . 2 exp(b)[(nπt)2 + (ab)2 ] (25)
−14 8
and A, B and x± are: 2
B = πat (exp(2b) + 1) x± = x0 ± t.
(27) (28)
It should be noted that I1 is found by setting n = 1 in (25).
10 11 f (GHz)
12
is in the following form for uniform and cosh where distributions, respectively: nπ γ1 Γ cos nπ a (x0 − t) − cos a (x0 + t) Ens = (30) nγn cos πa (x0 − t) − cos πa (x0 + t) γ1 Γ In Ens = (31) nγn I1 NUMERICAL RESULTS
Intuitive tests can be readily performed to validate the rationality of results. If the strip is assumed as PEC then σs → ∞, therefore, (13) would also reduces to (16). On the other hand, Zcs goes to infinity if we presume that the strip in the waveguide aperture has very low conductivity (σs → 0). This is equivalent to replacing Zcs with an open circuit in the equivalent circuit (see Fig.2). Therefore, no reflections occur at zero conductivity, which resembles no discontinuity on the aperture. To show the competency of the method, we compare our analytic method with modelling results. A general purpose programming tool28 was used to find analytic results from (17) and (21) or (22) and (23). We
MoM Uniform
2.4
cosh
2.35 2.3 2.25
8
9
10 11 f (GHz)
12
−1
FEM
|T |(dB)
MoM
−1.2
Uniform cosh
−1.4 −1.6 8
9
10 11 f (GHz)
12 FEM MoM
0.25 ∠T (deg)
Ens
FEM
2.45
FIELD ON THE CONDUCTIVE STRIP
It is instrumental to study the E and H fields on the discontinuity since they provide deeper insights into the problem. One can also check whether appropriate boundary conditions (here SIBC) were satisfied or not. In the previous section, we have assumed that induced ˆ or J (x) = J0 cosh[(x − current is in the form of J = J0 y x0 )/t]ˆ y where J0 is a complex number. One finds J0 from (10) after computing the LHS from the results of the section III The scattered field Exs is then found by using (2): X nπx Exs = Ens sin (29) a n
V.
9
(26)
∠Γ(deg)
A = a2 bt(exp(2b) − 1)
IV.
cosh
−12
|Γ|(dB)
Z
Uniform cosh
0.2
8
9
10 11 f (GHz)
12
FIG. 4. Magnitude and phase of the reflection Γ and transmission coefficient T from a strip with σ = 0.01 − j0.01S m−1
used two commercial electromagnetic packages with finite element method (FEM) solver29 and method of moments (MoM) solver30 to simulate the structures. Analytic and modelling results for the following case were compared: The sample has a complex conductivity of σ = 0.01 − j0.01S m−1 which is chosen close to conductivity of the graphene at X-band.
5 FEM
0.8
Calibrate the aparatus
MoM
|Ey | V/m
Uniform
0.6
cosh
0.4
Measurement of S−parameters
0.2 0
0
5
10 15 x(mm)
20 Zsc FEM
6 |Hx | (mA)/m
Convert S11 to the shunt impedance Zshunt by 11 +1 − Xa = Rsc + Xsc = − S2S 11
MoM Uniform
Find ∆
cosh
4 2 σ
0
0
5
10 15 x(mm)
=
∆ Zsc
20 FEM
FIG. 6. The procedure to measure the conductivity by the proposed method
|Hz | (mA/m)
MoM Uniform
6
cosh
B.
4 2 0
5
10 15 x(mm)
20
FIG. 5. Fields on the aperture of the waveguide and discontinuity of conductive strip. E field at z = 0 and H fields at z = 0+ .
Fields on the Aperture
Aperture fields on the strip discontinuity are examined in this subsection. Filed depicted in Fig. 5 are the total fields (incident+scattered) for a strip with 1 mm width and the resistivity of 50 + j50 Ohm. The frequency is set to 10 GHz. It is interesting to see that there is a drop in E and a significant jump in H field to satisfy the imposed boundary conditions. The singular like behaviour of the Hz near the edges of the strip is to make a closed loop of H field around the strip (also see Fig. 3). The fields from both current approximations are close to the simulation results when |x−x0 | > 2t. Particularly, Ex and Hz components of the cosh and full-wave solver are almost identical even on the conductive strip.
A.
Reflection and Transmission Coefficients
Fig. 4 illustrates the magnitude and phase of the reflection and transmission coefficients caused by the conductive strip discontinuity. In the modelling, we deembedded the excitation ports to the plane of the discontinuity, to achieve the correct phase for Γ and T .
Modeling data of Hx component over the strip does not agree with none of our numerical results. Presumably, a more realistic choice of the basis function would improve agreement. It is also noteworthy to mention that full wave modeling tools are also subject to error.
C.
A good agreement between analytic and simulation results are obtained for both current distribution (see Fig. 4). However, it is evident that the Γ and T from cosh assumption are much closer to full wave modelling results. This is best illustrated on the magnitude of T and Γ coefficients. It is also observed that MoM simulations lie closer to the theoretical calculations.
Conductivity Estimation
In the following, we demonstrate how one can use the analytic results of section III to measure the real and imaginary part of the conductivity of a thin strip. The measurement apparatus should basically include a vector network analyser and waveguides. Diagram in Fig.6 shows the required steps in a typical apparatus. For the
6
< σs (mS m−1 )
10.2 10 9.8 9.6 9.4 9.2
= σS (mS m−1 )
FEM Uniform FEM cosh MoM Uniform MoM cosh
8
9
10 f(GHz)
11
12
FEM Uniform FEM cosh MoM Uniform MoM cosh
−9
−9.5 −10 8
9
10 f(GHz)
11
tion and transmission of the T E modes in rectangular waveguide. Other transmission-reflection methods need that sample under test cover the aperture of the waveguide, however, our method needs that SUT only cover a small portion of the cross section. An equivalent circuit of the problem is proposed which is of assistance for intuitive understanding. We provide analytic formulas for the reflection Γ and transmission T coefficients, and derive terms related to each component in the equivalent circuit. Distribution of the fields and currents over the SUT are also reported. The reflection and transmission coefficients from a resistive sheet are compared with Sparameters from the commercial software with FEM and MoM solvers. A reasonable agreement is observed for Γ and T coefficients, as well as fields and currents over the aperture.
12
ACKNOWLEDGMENTS FIG. 7. Estimated conductivity for a strip with surface conductivity of (0.01 − j0.01)S m−1 . Solid black line shows the expected value.
uniform current approximation, ∆ is found from ∞ X γ1 1 nπt 2 πt ∆= , csc sin2 j4ωµ0 a n=1,3,... n2 a
(32)
This work was partly supported by Australian Research Council grant DP130102098. The author is grateful to Prof. Thiel of Griffith University for pointing this problem as well as fruitful discussions. The manuscript was also greatly benefited by suggestions of Dr Bird of Antengenuity, P as well 2as his 2simplified closed form formula for n=1,3,... sin nx/n . Finally, author thanks Dr Albooyeh of UC Irvine and Dr Sadatgol of MTU Houghton for quickly reviewing a draft of the manuscript.
while ∆ for cosh distribution is: ∆=
∞ γ1 X 2 I . 4ωµ0 I12 n=1 n
(33)
We used the modelling results of the full-wave simulators (FEM and MoM) as the input to examine the measurement procedure. Fig. 7 illustrates the computed conductivity by our method for a conductive strip of 0.01 − j0.01S m−1 . It is seen that the theoretical results are closer to the expected value of 0.01 S m−1 for the real part of the conductivity rather its imaginary part. This is observed when both MoM and FEM modelling data are used as the data sources. Generally, the theory presented in this paper is valid for the various positions of the conductive strip inside the waveguide and along its long side. When the strip is placed at the center of the guide, the maximum reflection from the strip occurs. The interaction with the fundamental mode is stronger with the conductive strip in the middle which improves dynamic range of the measurement method. Further investigations are needed to explore the accuracy and sensitivity of this method to various parameters in the apparatus. VI.
CONCLUSION
In this paper, a method to measure the conductivity of thin layers is proposed which is based on the reflec-
APPENDIX THE SIMPLIFICATION OF SERIES IN (??)
In (20), we have a series in the form of: S2 (x) =
∞ X
1 sin2 nx 2 n n=1,3,...
(34)
where x = (πt/a). This series can be slow to converge under some conditions and it is useful to obtain a closed form solution. It is recognized that this series is the difference of two infinite series containing all terms. Thus: S2 (x) =
∞ X
1 1 sin2 nx − 2 n 4 n=1,2,...
∞ X k=1,2,...
1 sin2 2kx n2
1 =S∞ (x) − S∞ (2x) 4
(35)
where S∞ (x) is: S∞ (x) =
∞ X
1 sin2 nx. 2 n n=1,2,...
(36)
Using 2 sin2 z = 1 − cos 2z, we find S∞ (x): S∞ (x) =
∞ ∞ 1 X 1 1 X 1 − cos 2nx 2 n=1,2,... n2 2 n=1,2,... n2
(37)
7 where the first series is known as the Euler’s result31 ∞ X
π2 1 = . n2 6 n=1,2,...
(38)
The second series is looked up from the tables32 (Sec.1.443) with a change of variable of x → 2x with a restriction of 0 ≤ x ≤ π/2. ∞ X
1 π2 cos 2nx = − πx + x2 2 n 6 n=1,2,...
(39)
Therefore: S∞ =
1 πx − x2 2
(40)
Finally, using (40) twice in (35) results in S2 (x) =
∞ X
1 πx sin2 nx = 2 n 4 n=1,3,...
(41)
where 0 ≤ x ≤ π/2. 1 J.
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