WO2011012767A1

WO2011012767A1 - Broadband reference-plane invariant method and algorithm for measuring electromagnetic parameters of materials

Info

Publication number: WO2011012767A1
Application number: PCT/FI2010/050591
Authority: WO
Inventors: Gheorge Sorin Paraoanu; Kari Sarvala; Khattiya Chalapat
Original assignee: Aalto-Korkeakoulusäätiö
Priority date: 2009-07-27
Filing date: 2010-07-09
Publication date: 2011-02-03
Also published as: FI20095811A0; FI122901B; FI20095811A

Abstract

The present invention describes a method for measuring the electromagnetic parameters of materials, such as the complex index of refraction, the complex wave impedance,the complex electric permittivity and the complex magnetic permeability. The method is explicit and does not require the transformation of the scattering parameters from the calibration reference- planes to the surfaces of the materials. The propagation factor and the square of the reflection coefficient are explicitly described in terms of reference- plane invariant quantities. The method can be used to characterize both dielectric and magnetic materials. The invention is easily implemented as a computer program run by a vector network analyzer.

Description

BROADBAND REFERENCE-PLANE INVARIANT METHOD AND ALGORITHM FOR MEASURING ELECTROMAGNETIC PARAMETERS OF MATERIALS BACKGROUND OF THE INVENTION

Field of the invention:

The present invention relates to the measurement of the electromagnetic parameters of materials, specifically the complex index of refraction, the com- plex wave impedance, the complex electric permittivity and the complex magnetic permeability.

Description of the related art: In recent times, fast and accurate knowledge of the electromagnetic properties of materials is increasingly required in the design and development process of a vast number of industries, spanning from food processing to communication systems. Also under- standing and measuring material parameters, such as the complex refractive index n and the complex electric permittivity ε, is fundamental to basic science research.

Non-resonant techniques such as transmission and reflection measurements are largely used nowadays for characterizing the electromagnetic properties of materials, see e.g. L. Chen, V. V. Varadan, C. K. Ong, C. P. Neo, Microwave Electronics: Measurement and Materials Characterization, John Wiley and Sons, 2004. The fundamentals of these techniques have been already established in the 1970 's by the seminal papers of Nicolson, Ross (A. M. Nicolson and G. F. Ross, Measurement of the intrinsic properties of materials by time-domain techniques, IEEE Trans. Instrum. Meas . , vol. IM-19, pp. 377-382, Nov. 1970) and Weir (William B. Weir, Automatic measurement of complex dielectric constant and permeability at microwave frequencies, Proc. IEEE, vol. 62, pp. 33-36, Jan. 1974). These techniques are relatively simple and accurate. They have the advantage of broadband characterization of materials and devices. For almost four decades they have been widely applied to measure the permittivity and permeability of various synthetic and natural materials .

A known drawback of the original Nicolson- Ross-Weir (NRW) method is that it requires the transformation of S-parameter measurements from the calibration reference planes to the surfaces of the material. The phases of the transmission and reflection signals are strongly dependent on the positions of the reference planes, so the uncertainties in the transformation of S-parameters can result in significant errors. The precision of this transformation can be enhanced in various ways, for example, by adding more steps to the calibration process or running extra cal- culation algorithms which complicate the measurement. Besides the Nicolson-Ross-Weir algorithm, other methods based on transmission/reflection measurements exhibit the same sort of difficulties.

An important step forward has been achieved in 1990, when Baker-Jarvis and collaborators showed that it is possible to derive S-parameter equations which are reference-plane invariant (see J. Baker- Jarvis, E. J. Vanzura, W. A. Kissick, Improved technique for determining complex permittivity with trans- mission/reflection method, IEEE Tran. Microw. Theory Tech., vol. 38, pp. 1096-1103, Aug. 1990). Using some of these equations, they showed that it is possible to extract the values of the material parameters by using an iterative algorithm. This algorithm requires some initial values for permittivity and permeability as input, and therein lies one of its limitations: a good guess is needed, otherwise the algorithm can produce wrong results.

A further deficiency in using the Baker- Jarvis method is that it cannot be used for determin- ing the permeability of magnetic materials.

SUMN[ARY OF THE INVENTION

The present invention introduces a method for measuring the electromagnetic parameters of materials.

The method combines the ideas from the NRW and the Baker-Jarvis techniques. More precisely, the scattering parameters are combined into a specific set of reference-plane invariant equations (similar to Baker-Jarvis) , and the equations are used together with group velocity data (similar to NRW) . This results in a simple, explicit, and reference-plane invariant methodology which can be used to characterize both dielectric and magnetic materials. With respect to NRW, the advantage of this method is the use of reference-plane invariant quantities, so it is easier to implement and also the errors due the reference plane positions are eliminated. With respect to the Baker-Jarvis algorithm, the improvement consists in the use of additional information about the sample, extracted from group velocity measurements. This removes the ambiguity in the determination of the phase. Also, since our results do not depend on choosing good initial values for ε and μ as in Baker-Jarvis, dielectric materials with unknown properties and also mate- rials with magnetic properties at high frequency can be characterized.

The method is characterized in that the method comprises the steps of:

determining the scattering parameters by measuring transmission and reflection signals, with and without a material sample placed between ports of a measurement device; calculating the reflection coefficient of the material by using an explicit expression which is reference-plane invariant;

calculating the propagation factor of a transverse electromagnetic wave propagating through the material by using an explicit expression which is reference-plane invariant; and

calculating the electromagnetic parameters of the material from the reflection coefficient and the propagation factor.

In one embodiment of the present invention, the length of the sample is measured.

In one embodiment of the present invention, the distance between the ports is determined.

In one embodiment of the present invention, the propagation factor P and the square of the reflection coefficient T _r which are reference-plane invariant, are calculated as

l +BT _r where ^²¹^¹²

c

ι4π(L_a,_r-L) β _ °21

B = e ^λ [S₂₁S₁₂ -S_nS₂₂) . _and S_{21 ^} wherein S₁₃ denotes the measured transmission and reflection signals from port j to port i of a measurement device, and λ denotes the wavelength of the signal in the region between the ports outside the material.

In one embodiment of the present invention, the method further comprises calculating the complex index of refraction or the complex wave impedance from the propagation factor and/or the reflection coefficient . In one embodiment of the present invention, the complex index of refraction from

l

and/or the complex wave impedance from 1+ F _are calculated explicitly.

In one embodiment of the present invention, the complex electric permittivity and the complex magnetic permeability are calculated from the propagation factor and/or the reflection coefficient.

In one embodiment of the present invention, the complex electric permittivity is calculated from n

ε_r=- z and the complex magnetic permeability from μ_r = nz _

In one embodiment of the present invention, the electric permittivity of a non-magnetic material

9

p — τ_η ^Δ

is calculated from ^r

In one embodiment of the present invention, the magnetic permeability of an unknown material is determined by checking whether the negative or positive root of the square of the reflection coefficient, is physically correct by comparing the sign of the calculated scattering parameters with the measured scattering parameters.

In one embodiment of the present invention, the electric permittivity and the magnetic permeabil- ity of a material are determined by the distinguisha- bility between the electric permittivity and the magnetic permeability, regardless of the sign of the reflection coefficient.

In one embodiment of the present invention, a group velocity or a group delay through the space between measurement ports is measured.

In one embodiment of the present invention, the physically correct solution for the calculated electromagnetic parameters is chosen by comparing the calculated group delay with the measured group delay.

According to the second aspect of the present invention, the invention comprises a computer program for measuring the electromagnetic parameters of a material. The computer program is characterized in that it controls a data-processing device which performs the following steps:

determining the scattering parameters by measuring transmission and reflection signals, with and without a material sample placed between ports of a measurement device;

calculating the reflection coefficient of the material by using an explicit expression which is ref- erence-plane invariant;

In one embodiment of the present invention, the computer program controls the data-processing de- vice to perform at least one of the method steps described earlier.

In one embodiment of the present invention, the computer program is implemented for use by a vector network analyzer.

According to the third aspect of the present invention, the invention comprises a measurement apparatus for measuring electromagnetic parameters of a material. The measurement apparatus is further characterized in that it comprises:

measurement means configured to determine the scattering parameters by measuring transmission and reflection signals, with and without a material sample placed between ports of a measurement device;

processing means configured to calculate the reflection coefficient of the material by using an ex- plicit expression which is reference-plane invariant;

the processing means configured to calculate the propagation factor of a transverse electromagnetic wave propagating through the material by using an explicit expression which is reference-plane invariant; and

the processing means configured to calculate the electromagnetic parameters of the material from the reflection coefficient and the propagation factor.

In one embodiment of the present invention, the apparatus further comprises means configured to perform at least one of the method steps described earlier .

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 shows a model of multiple reflections between two interfaces of different materials,

Figure 2 shows a model of a transmission line containing material of length L, where L₃ (j = 1, 2) represents the distance from the reference plane of the S-parameter measurement to the corresponding interface between air and the material under test,

Figure 3 shows a reference-plane invariant measurement model, where L represents the length of the sample and L_air represents the distance between the calibration planes of the S-parameter measurement,

Figure 4 shows the measurement setup used to measure the electromagnetic parameters of materials in microwave frequency range between 2 to 18 GHz,

Figure 5 shows a comparison of the complex permittivity obtained using the Nicholson-Ross-Weir method (NRW) and the reference-plane invariant method according to the invention (RPI) for a PVC sample of length 20 mm, and

Figure 6 shows a comparison of the complex permittivity obtained using the Nicholson-Ross-Weir method (NRW) and the reference-plane invariant method according to the invention (RPI) for a PTFE sample of length 20 mm.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Reference will now be made in detail to the embodiments of the present invention, examples of which are illustrated in the accompanying drawings.

At first, the measurement is modelled within the framework of classical electrodynamics. We present a new algorithm which is reference-plane invariant and show how it can be used to determine the complex refractive index and the complex permittivity and permeability.

We start by deriving the mathematical rela- tions between the S-parameters and the material parameters. We describe the scattering of transverse electromagnetic waves based on the multiple reflection model shown graphically in Figure 1. In this figure, a first material is present in sections 12 and 14 having the electric permittivity E₁ and the magnetic permeability μ_j . A second material is located in between the two sections in volume 13, with the length of L, having the electric permittivity ε₂ and the magnetic permeability μ₂. The interfaces between the two mate- rials are shown as 10 and 11.

Within this model, the total reflection and transmission coefficients can be calculated using the superposition principle. Since a transverse electromagnetic (TEM) wave propagating through a distance L picks up a phase change of 2πL/λ, where λ is the wavelength in that region, the propagation factor of the TEM wave traveling through the material of length L, as shown in Figure 1, is given by

P = e^^L , :D where J₂ = i(ϋ /v₂ = i(ύn₂ Ic = i(ϋ_Λ

The total reflection coefficient is

Similarly, the total transmission coefficient in terms of Y and P is

T = _^(i-r²) :5)

I-Γ²P² The standard model of a transmission/reflection measurement is described in Figure 2. The transmission line is divided into three regions 22, 23 and 24. The reference plane for the first port is shown as 20 while the reference plane for the sec- ond port is shown as 21. The locations of the reference planes determine the values of Li and L₂. Typically, the regions of lengths Li and L₂ are assumed to be filled with air and the middle region of length L is filled with a material with the relative permittiv- ε μ ity ε =— and relative permeability μ ^{= J}— . The com- ε₀ μ₀ plex refractive index of the material can be expressed as n

• If we assume that the permittivity, εi, and the permeability, μi, are equal to the permittiv- 5 ity of free space ε₀ and the permeability of free space μ₀, and we can rewrite Eq. (3) as r = ^i (6) z + l

10 where z = -yjμ_r /ε_r is the impedance relative to vacuum (The total impedance Z = yfμ/ε = Z_oz , where

Z₀

is the characteristic impedance of vacuum) .

The determination of these two quantities, z 15 and n, from the experimental data will be the main focus of the remaining part of this section. Based on the multiple reflection model the S-parameters are expressed in terms of V and P as follows

^{Z U ύ}U — ^{e l} tot - ^e ₁ _-p2pT ⁽ '

C _ g-2YA_r - _e-2YA U^{1 "}^ ) / O )

°22 — V ^l tot — V _{1 r}2_p2 \°' a_nH c - c -_p-iΛh+Li)_τ _ -I₁ (L₁ +L₁) J³Jl-F j ana ύ_2l-ύ_l2-e i_m - e Γ₂P₂

25

where Y₁ = iθύn_γ I c ~iϋ)l c (c is the speed of light in the vacuum) .

When there is no sample inside the transmission line, r and therefore:

30

S° =

-jAL_mr) ₍₁₀₎ Equation (10) allows us to experimentally determine the airline length, L_air, by calibrating the vector network analyzer and then measuring the transmission through the empty air line to obtain the phase of S₂°_l .

In the next step, V and P are expressed in terms of the S-parameters . This is similar to the Nicolson-Ross-Weir algorithm, with the essential difference that neither V nor P depends on Li and L₂. In- deed, for airlines operating at relatively high frequencies, measurements of Li and L₂ are prone to relatively large uncertainties. These errors will further propagate in the phase factors of the S-parameters,

where ye { 1,2}, leading to the larger errors of the phase factors at higher frequencies.

Figure 3 shows the schematic model of a ref- erence-plane invariant measurement. The first material is placed in volumes 32 and 34 and is preferably air. In between the air volumes is the second material 33 whose length is L and permittivity and permeability are ε₂ and μ₂. The calibration planes are depicted as 30 and 31, while their mutual distance is L_air.

We start by defining two quantities, which are related to measurable quantities, namely

and

p²—r²

ϋ-e l^ύ21^ύ12 ^ύll^ύ ₂₂)-^~ _2p2 ' [-L^J) where Y₁= /2π , in which λ is the wavelength of the sig- λ,

nal in the region outside of the material.

Experimentally the S-parameters are measured by a vector network analyzer (VNA) , the airline length, L_air, is found via Eq. (10), and the length of the sample, L, is measured before inserting the sample into the airline.

Solving Eq. (13) for P² P²=^^_τ (14) and substituting back into Eq. (12), we obtain:

We now approach the actual results of the invention and in a first exemplary embodiment of the invention, it is described how the complex index of refraction n can be measured using the method at hand.

From Eq. (15) we can finally solve the square of the reflection coefficient (T²)

where the sign in this equation is chosen so that r≤l. These expressions for P² and Y² are manifestly reference-plane invariant.

A very useful, simpler expression for P can be obtained if we define another quantity, R, directly related to the scattering parameters,

where S_y and S° are the measured signals from port j to i with and without the material in between the ports .

We can get rid of the quantities Li and L₂ by

Eq. (17) : we solve Eq. for the propagation factor P and substituting P² from Eq. (14) into the denominator of Eq. (17), we obtain

1+r² 1+Γ² -—

P = R _oe^^L =R _oe ^λ . (18)

After these calculations, assuming free space on either side of the sample, the complex index of refraction can be determined by

or more expl icitly

The logarithmic function in Eq. (19) and Eq. (20) is a multi-valued function, which results in an infinite number of discrete values for n. The correct solution must be chosen from these values. One way to do so is to check whether a chosen solution gives correct values for another measurable quantity or not. Also, this measurable quantity should not depend on Li and L₂.

By definition, the group delay is a measure of a pulse signal transit time through a transmission line. The transit time of a wave packet is defined as x

( 2 1 ) where x is the transit length and v_g is the group velocity of the wave pulse. In this case, : o L

τ = air

( 22 ) and

)

where x° is the group delay through an empty line, and T is the group delay through the line with an inserted sample of length L. By comparing the calculated group delay with the measured group delay,

the correct refractive index n can be determined. Another quantity which can be used as an alternative to the group delay is the group delay relative to the empty air-line

which is derived by substracting Eq. (23) from Eq. (22) .

In the following exemplary embodiment of the invention, it is described how to extract the material parameters ε_r and μ_r (relative permittivity and relative permeability) . A situation of practical interest is the case in which the experimentalist already has some information about the material. For example, if chemical analysis provides additional proof that the material does not contain magnetic elements, one can assume μ_r = 1 and determine the permittivity simply from the equation ε_r = n . As we will show later, this leads to better accuracies than the more general method presented below, which requires the determina- tion of z.

However, in many situations, especially concerning materials under research which contain magnetic elements, magnetic impurities, or magnetic nanoparticles, it is not possible to know beforehand what the electromagnetic properties are. In these situations, one needs to use not only n but also the relative impedance z. This is a further embodiment of the invention regarding magnetic materials, where we can obtain the material properties ε_r and μ_r as μ_r =n- z and ε_r =— , (26) z

Where

l-r (27) I+Γ The reflection coefficient Y is determined from Eq. (16) . But this equation gives Y only up to a sign, since ±Y both satisfy Eq. (16) . To get the correct sign for Y, we have to go back to Eq. (7) or (8) and check which one of +Y or -Y satisfies them. Note that this does not bring in additional errors, since it is just a sign check.

In other words, for getting the correct sign for r, the S-parameters calculated from ±r are compared with the measured S-parameters. Then, via Eq. (26) and (27), the complex electric permittivity and magnetic permeability can be directly obtained, providing that the complex index of refraction is known from the method stated above.

It is useful to note that even without such a check, only minimal information about the properties of the material may be sufficient. Suppose that T is the correct solution leading to the correct set of material parameters ε_r and μ_r. The properties of the con- formal mapping, Eq. (27), imply that the oppositely signed solution -V corresponds to a relative impedance z^"1. The effect in the final result Eq. (26) is therefore simply to swap the values of permittivity and permeability. In many practical situations, an experienced experimentalist could recognize easily, given two complex numbers and minimal information about the chemical composition of the material, which one is the permittivity and which one is the permeability.

An example of the experiment, where the above method is verified, is presented in the following. In this example, electric permittivity of polytetra- fluoroethyline (PTFE) and polyvinylchloride (PVC) is measured within the frequency range between 2 to 18 GHz. It is however notable that although this test is performed over a limited frequency range the method is generally frequency-independent.

Referring to Figure 4, the present method is applied to measure the electromagnetic parameters of materials within the microwave frequency range. The 7- mm precision coaxial air line 44 is used as a sample holder to conduct transverse electromagnetic waves between the measurement ports; the sample 45 can be seen in the middle of the air line. The group delays and the scattering parameters were measured by using a vector network analyzer (VNA) 40. The air line set was connected to the VNA ports 41 by 7 mm-to-2.92 mm (K) adapters. In this example, the frequency range of the measurement is limited by the operating frequencies of the 7-mm air line, i.e. up to 18 GHz. In principle, the method can also be applied in other frequency range, providing that transverse electromagnetic waves are conducted and isotropically propagate through the region between the measurement ports in that frequency range .

Prior to the measurement, a 2-port calibration is performed. Then, the empty air line was meas- ured to obtain S° , from which the length L_mr between the calibration planes 42, 43 inferred to be 17.3193 cm. After that, a toroidal sample was inserted between the inner and the outer conductor of the air line, and measurements were repeated again.

Measurements on a 20.00 mm PVC and a 20.00 mm

PTFE sample give the complex permittivity spectra as shown graphically in Figures 5 and 6. We see that comparing to the Nicolson-Ross-Weir (NRW) algorithm, the method according to the invention (RPI) results in less errors and stable results even around the three Fabry-Perot resonance frequencies in the above mentioned frequency range. This is a practical proof that the present method has significantly less errors compared to the NRW algorithm when it is used to charac- terize non-magnetic materials.

Numerical analysis and experiments show that the method according to the invention has relatively low uncertainties at higher frequencies. This characteristic is an intrinsic property of the transmis- sion/reflection method because a significant change in the phase of the S-parameter due to the presence of the material requires that the sample length should be above a certain size, compared to the wavelength of the signal.

The present algorithm is preferably implemented as a computer program which can be run by a suitable processing device. The processing device may be a microprocessor, which is preferably implemented in a vector network analyzer.

Furthermore, the algorithm has been verified experimentally for many materials. It can be seam- lessly integrated with any vector network analyzer on the market.

It is obvious to a person skilled in the art that with the advancement of technology, the basic idea of the invention may be implemented in various ways. The invention and its embodiments are thus not limited to the examples described above; instead they may vary within the scope of the claims.

Claims

1. A method for measuring electromagnetic parameters of a material, cha r a c t e r i z e d in that the method comprises the steps of:

calculating the reflection coefficient of the ma- terial by using an explicit expression which is reference-plane invariant;

calculating the propagation factor of a transverse electromagnetic wave propagating through the material by using an explicit expression which is reference- plane invariant; and

2. The method according to claim 1, c h a r - a c t e r i z e d in that the method further comprises the step of:

measuring the length of the sample.

3. The method according to claim 1, cha r a c t e r i z e d in that the method further comprises the step of:

determining the distance between the ports.

4. The method according to claim 1, cha r a c t e r i z e d in that the method further comprises the step of:

calculating the propagation factor P and the square of the reflection coefficient F² , which are reference-plane invariant and given by

and

where _A=±^- ; B = e ^λ (S₂₁S₁₂-S₁₁S₂₂) and R=^L,

^21^12 ^21 wherein S_XJ denotes the measured transmission and reflection signals from port j to port i of a measure- ment device, and λ denotes the wavelength of the signal in the region between the ports outside the material .

5. The method according to claims 1-4, cha r a c t e r i z e d in that the method further com- prises the step of:

calculating the complex index of refraction or the complex wave impedance from the propagation factor and/or the reflection coefficient.

6. The method according to claim 5, cha r - a c t e r i z e d in that the method further comprises the step of:

calculating explicitly the complex index of re- fraction from n = c In(—Iλ and/or the complex wave im- ilπfL [P)

pedance from z = .

l+r

7. The method according to claims 1-4, cha r a c t e r i z e d in that the method further comprises the step of:

calculating the complex electric permittivity and the complex magnetic permeability from the propagation factor and/or the reflection coefficient.

8. The method according to claim 6, cha r a c t e r i z e d in that the method further comprises the step of:

calculating the complex electric permittivity from ε_r=— and the complex magnetic permeability from z

μ_r=nz .

9. The method according to claim 6, cha r a c t e r i z e d in that the electric permittivity of a non-magnetic material is calculated from Z_r=n .

10. The method according to claims 1-9, cha r a c t e r i z e d in that the magnetic permeability of an unknown material is determined by checking whether the negative or positive root of the square of the reflection coefficient, is physically correct by comparing the sign of the calculated scattering pa- rameters with the measured scattering parameters.

11. The method according to claims 1-9, cha r a c t e r i z e d in that the electric permittivity and the magnetic permeability of a material are determined by the distinguishability between the elec- trie permittivity and the magnetic permeability, regardless of the sign of the reflection coefficient.

12. The method according to claim 1, cha r a c t e r i z e d in that the method further comprises the step of:

measuring a group velocity or a group delay through the space between measurement ports.

13. The method according to claims 1-12, cha r a c t e r i z e d in that the physically correct solution for the calculated electromagnetic parameters is chosen by comparing the calculated group delay with the measured group delay.

14. A computer program for measuring electromagnetic parameters of a material, cha r a c t e r i z e d in that the computer program controls a data- processing device to perform the steps of:

calculating the reflection coefficient of the material by using an explicit expression which is reference-plane invariant; calculating the propagation factor of a transverse electromagnetic wave propagating through the material by using an explicit expression which is reference- plane invariant; and

15. The computer program according to claim 14, cha r a c t e r i z e d in that the computer pro- gram controls the data-processing device to perform the method steps according to claims 2-13.

16. The computer program according to claims 14-15, cha r a c t e r i z e d in that the computer program is implemented for use by a vector network analyzer.

17. A measurement apparatus for measuring electromagnetic parameters of a material, cha r a c t e r i z e d in that the apparatus comprises:

processing means configured to calculate the reflection coefficient of the material by using an ex- plicit expression which is reference-plane invariant; the processing means configured to calculate the propagation factor of a transverse electromagnetic wave propagating through the material by using an explicit expression which is reference-plane invariant; and

18. The measurement apparatus according to claim 17, cha r a c t e r i z e d in that the apparatus further comprises means configured to perform the method steps according to claims 2-13.