Notes on Dielectric Characterization in Waveguide

R.Nesti, V. NataleIRA-INAF Arcetri Astrophysical Observatory

1. Theory

Let's suppose we have to characterize the electromagnetic properties of adielectric material, in terms of the relative dielectric constant and the loss tangent��tan . The loss tangent is given by�

tan (1)��� �

�� ��

�ZZ

� �

�

where is the angular frequency and F/m is the dielectric constant� ��c��� ����� � �

of vacuum. The contribution of the bulk conductivity ( ) have been here separated to��any other sort of losses ( ).�ZZ

The problem of the electromagnetic scattering due to a dielectric slab is brieflyintroduced. To this aim the model shown in Fig. 1 is considered: a unit amplitudewave is incident on a dielectric slab of thickness inside a waveguide, which isassumed air-filled in the remaining part.

Fig. 1: Multiple reflection model of the electromagnetic scattering of adielectric slab in a waveguide

Due to the material discontinuity at the two interfaces between air and dielectricthe electromagnetic field scattering can be modeled by multiple reflections and can betreated as an extension of the classical problem of the incidence of a plane wave at theplanar interface between two semi-infinite dielectrics.

We introduce the field impedance in the air and in the dielectric material,respectively, and whose expression is given by� �� �

� ���

�(2)

where is the wave propagation factor and is given by the following general�

expression

� � � � �� � � � � � � �� � �

!

b, , (3)

with the propagation constant , the eigenvalue , the wave constant and the� �! �

attenuation constant being given in Table I according to material, domain and mode�

of propagation. As regards the magnetic permeability, only standard materials(including vacuum) having H/m are here considered� � � � � � ��

c�

TABLE I

�

� � �

� � � � � � �

�

� � �

� �

�� ��

!

� � �

� �

�� � �tanCoaxial cable Rectangular waveguide Circular waveguide(Free space) ( , , ) (Radius )

Mode TEM TE TE�� �������

� � � �

� � � �

� �

�

� � �

� � � � � � �

Air Dielectric

tan

tan

Air Dielectric

cos

� � �

� �

� �

� �

�

�

c�

� �

!

� � �

�

�

� �

� � � �� � �� �

�� � � ��

� �

! !

� �

� � �

�

�

�

$

tan

� �� � �� � �

tan

0 sin tan

� �

� � ��

��

� � � � � � ��

� �

� � � � �

tan�$

It is worth to spend a few words about Table I where medium electromagneticproperties are organised in a quite strange manner which is however best suited toimplement later formulas. Basically Table I has to be seen as a two column table, thefirst column giving an electromagnetic property and the second one its expression interms of known quantities. The vacuum propagation constant is given first than a��relative propagation constant is defined, both used to derive further parameters.��

A first complication comes out in the definition of the eigenvalue as it is�!different for the three domain we desire to characterize: the coaxial cable supportingTEM propagation, the rectangular waveguide (TE mode propagation) and the��

circular waveguide (TE mode propagation). It has to be noticed that the coaxial��

cable and free space, as propagation domains, can be considered the same case.A second diversification is made as regards the medium of propagation and we

distinguish between air (which is assumed to have vacuum features) and dielectricwith the propagation constant � (that is the unbounded medium wave propagationfactor) given for both media. It is also useful to define here , and , mainly � �introduced to easily express further important parameters as they have no particularphysical meaning: in the air medium, represents the square of the wave constant

normalized to the vacuum propagation constant while in the dielectric medium, wherelosses are considered, it is used to compute both a sort of normalized wave constantamplitude ( ) and the wave constant phase ( ).� �

The above definitions allow a compact and simple form to express the wavepropagation factor (Eq. (3)) and to analytically obtain its real and imaginary part,�

respectively ( ) and ( ); about these last items a useful approximation is also given� �

in the case of small losses � �tan .� �Analogously, a simple and compact form for the Fresnel coefficient is defined at

the second interface as follows:

� �� � �

� � �� �

� �

(4)

We can now try to describe the multiple reflection process in order to give, aselementary processes combined together, one single reflection and one singletransmission coefficient accounting for, respectively, the reflected and transmittedwave.

As regards the reflection coefficient, by using the Fresnel coefficient (4) and themultiple paths shown in Fig 1, we have a summation of infinite terms

� � � � � � � �� � � � � � � � � � � � � � � ������ �� � � �� �c�� � � c�� �� �� � (5)

At each the reflection coefficient gains a factor contribution rebound �� c�� �� ��

which can be highlighted as follows

� � �� �

�� � � �

� � � �� �� �� ��~�

B

� c�� � ��� (6)

Providing and , as in physical passive devices, the � �� �� � � � � �

� �arginfinite summation in (6) converges to , with , thus giving�� � � � � � �� � �� c�� ���

� �� � �

� �

�� � � �

� � � � �

� � � � � �

� � � �� �� � � �c�� �

� c�� � � c�� �

c�� ��

� �

��

� �

�

(7)

The same procedure holds analogously to the transmission coefficient, giving

� � � � �� � � � � � � � � � � � � ������ �� � � �� �c� � c� �� �� �2 3 (8)

� � �� � � � � � �� �� � � �c� � � c�� �

�~

B�� �� �

0

(9)

� �

��

� � � � �

� � �

� �� � c� �

� c�� �

�

�

�

�

(10)

Note that the infinite summation in (9) converges to 1 .� � � �� �2. Dielectric material characterization in rectangular waveguide

Although dielectric material characterization can in principle be done both infree space and in waveguide, this last domain has the big advantage to allow verygood accuracy with easy measurement setup and easy procedures for materialpreparation. On the contrary, in free space, setting-up testing facilities requires veryhigh attentions to prevent undesired effects, like spurious reflections and standingwaves, that have sensitive effects on the measurement results.

We now show how with only two different measurements giving the amplitudeof the S and S parameters of a two port waveguide, we can obtain accurate11 21

prediction of and tan .� ��

Let's suppose we have first, case , a measurement of the empty waveguide of� ��

length . If the instrument calibration is well done we should read ideally S for � ²�³

11the reflection coefficient; in practice, in the hypothesis of a perfectly fabricatedwaveguide, we read a very small value that can be interpreted as the zero of themeasure. Ideally the amplitude of S should be equal to unity if no losses were²�³

��

present. In practice we see, as a result of the measurement, a curve versus frequencywhich has a characteristic shape and which is always down unity, highlighting somesort of losses; we can associate this to the ohmic losses of the metallic wall of thewaveguide, a distributed effects obeying to an exponential law according to theformula

� �S (11)²�³

��

c �� � ��

From this we can estimate the attenuation constant . To evaluate accurately��

this parameter other effects, not only distributed but also localized like for exampleradiation losses due to the leakage from the flanges, should be taken into account.However the differential nature of the method here presented dramatically reduces theeffects of the non precise estimate of . In fact all the effects that are common to the��

two measurements are almost eliminated by differential comparison.As second and last measurement, case a dielectric material is inserted inside� ��

the waveguide and a new measurement is done. To get accurate information from themeasurement it is important a very precise dielectric fabrication that should be inlength and fill exactly the waveguide. In this case the measurement of both scatteringparameters, named S and S , is significant as they can be directly related to,²�³ ²�³

�11 1respectively, � and of Eq. (7) and (10).

If we observe (7) we can predict minima and maxima of the measured reflectioncoefficient. A first analytical approximation of such extreme points is given byminimizing or maximizing the numerator in (7). Thus putting

� � � �c� � c� � c �� � �� � � (12)

we have the following condition:

� � �� � � � � � �� � � � ��� !

"� � �

� � � �

� � minima (13)

� � �� � � � � � �� � � � � � ��� !

"� � �

� � � �

� � � � maxima (14)

where is the light velocity in vacuum. Both (13) and (14) can be solved for ." !Considering for example a rectangular waveguide with largest dimension the�

above equations lead to following extreme points:

! � � � "

� �min�

�

�

�

� � ��(15)

! � � � �� "

� � �max�

� �

�

� � � ��(16)

Fixed , depending on the length , obviously not all the points given above are�� meaningful since the first points can fall under the cutoff region.#$%

However a numerical model based on (7), (10) and (11) can be implemented topredict the behavior of a waveguide filled in with dielectric, using for the propagationfactor an expression of the form:

� � � � � � �� � � � �b� � � � �, (17)

where and , as defined in Table I, analytically depends on geometrical and� �� �

electrical features and is obtained from the ( ) measurement according to (11).�� �Considering , tan and parameters (and thus , tan and , tan )� � � � � � � � �� � � � � �� � � �

family of curves ( ) can be generated and compared with S and S� � � � �! ! ²�³ ²�³

�11 1measurement data. To solve the problem of dielectric characterization a distance can1

be defined in the space of real functions of real variable

�� � � �! & � !$'% � &$'% (18)

and search in the space of the allowed value for and tan the one minimizing the� ��

distances S and S . The above considerations suggest� � � � � � � � � � �� � � �! ! ²�³ ²�³

11 21

to choose a dielectric length to be few wavelengths so that two or more minima andmaxima fall in the bandwidth of interest thus making easier to 'measure' the distancebetween the curves. In practice, to achieve about a few parts per thousand accuracy for� �� and a few percent accuracy for tan , simply an eye inspection to compare the plotsis sufficient if three minima (or maxima) or more fall inside the bandwidth.

1 An example of distance is the root mean square value + + ! ! l�²%³ c �²%³ ~ � % c � % �%�� c� �

� � �� � �

� �

3. Example results

Some examples are here given to show applications of the theory previouslydescribed.

First we consider a foam like material, that is Styrodur of BASF. A 50.3mmlength sample have been machined to fill in an aluminum WR42 standard waveguideand measurements have been done in the 18-26GHz band. First the empty waveguidehas been measured in order to have an estimate of and the results are given in��

Fig. 2.In rectangular waveguide the TE fundamental mode attenuation constant due to��

wall ohmic losses may be modeled by the conductivity through:�

� � �

�� �

! �� ��� � ��

� �

� ! ��� � � (19)

where , are the waveguide section dimensions ( ), and the surface resistance is� � � � �given by

� �� !

�� � �

�(20)

Fig. 2: Transmission of an empty 50.3mm WR42 aluminum waveguide. In thesimulation model a finite conductivity of 5 S/m is used.� � � �

Once has been estimated the dielectric parameters can be determined by��

comparing simulation curves with reflection and transmission measurements byinserting the dielectric in the waveguide. As final result in the Styrodur case it wasfound a dielectric relative constant and tan . As regards , the� � �� �

c� ��( � ) � �results have been determined by best matching between the two plots of Fig. 3.

Fig. 3: Reflection coefficient of a Styrodur filled 50.3mm WR42 aluminumwaveguide. In the simulation model (Eq. 7) , tan� ��

c� ��( � ) � �� � � �5 S/m are used.

In the case of tan the plots in Fig. 4 has been obtained as a result of best�

matching simulations and measurements. Here the plots in Fig. 2 has been consideredto remove common effects which has not been taken into account by using only wallohmic losses to characterize the attenuation constant in (19). In fact, like ohmic��

losses, these not considered common effects produce the same offset in Fig. 2 and 4(here mainly in the lower part of the band) between simulations and measurements.

Fig. 4: Transmission coefficient of a Styrodur filled 50.3mm WR42 aluminumwaveguide. In the simulation model (Eq. 10) ,�� � ��(tan , 5 S/m are used.� �� ) � � � � �c

As a second example we consider high density Polyethylene (HDPE). A 98.6mmlength sample material have been machined to fill a copper WR28 standardwaveguide and measurements have been done in the 26-40GHz band. In this case weobtained S/m, 2 278, tan 1 as the plots in Figs. 5-7 show.� � �� ��* � � � � � � �� c

�4

Fig. 5: Transmission of an empty 98.6mm WR28 copper waveguide. In thesimulation model a finite conductivity of 2.6 S/m is used.� � � �7

Fig. 6: Reflection coefficient of a HDPE filled 98.6mm WR28 copperwaveguide. In the simulation model (Eq. 7) 2 278,�� � �tan 1 2.6 S/m are used.� �� � � � � �c4 7

Fig. 7: Transmission coefficient of a HDPE filled 98.6mm WR28 copperwaveguide. In the simulation model (Eq. 10) 2 278,�� � �tan 1 2.6 S/m are used.� �� � � � � �c4 7

Also if the second material is quite different from the previous one the sameresults have been obtained in terms of and tan accurate estimation. In both cases it� ��

has to be noticed the very good agreement between simulations and measurements.Since in the HDPE case the length of the dielectric is about 7-8 wavelengths a

larger number of minima appears in the plots of S and S (Figs. 6, 7) with respect²�³ ²�³

�11 1to the Styrodur case (Figs. 3, 4), exhibiting about 2.5-3 wavelengths dielectric length.

As a third example we consider Plexiglas. A 66.2mm length sample materialhave been machined to fill a copper WR42 standard wave guide and measurementshave been done in the 18-26GHz band. In this case we obtained S/m,� � * � �6

� ��c� � � � �2 54, tan 4.35 as the plots in Figs. 8-10 show.3

Once again it has to be noticed the impressive agreement between simulationsand measurements. In this case the dielectric length is about 4 wavelengths.

Fig. 8: Transmission of an empty 66.2mm WR42 aluminum waveguide. In thesimulation model a finite conductivity of 6.0 S/m is used.� � � �6

Fig. 9: Reflection coefficient of a Plexiglas filled 66.2mm WR42 copperwaveguide. In the simulation model (Eq. 7) 2 54,�� � �tan 4.53 6 S/m are used.� �� � � � � �c 3

Fig. 10: Transmission coefficient of a Plexiglas filled 66.2mm WR42 copperwaveguide. In the simulation model (Eq. 10) 2 54,�� � �tan 4.3 6 S/m are used.� �� � � � � �c 3

As a general and practical consideration we can say that a larger helps forbetter characterization of the dielectric constant, since it is associated essentially to thespacing between minima (or maxima) of the and curves. As regards the tan� �

characterization the choice of is less critical also if we can say that smaller lengths(2 or 3 wavelengths) are slightly preferable.

For the sake of completness we collected the above presented results in thefollowing Table I together with some data already published. The data withoutreference are from this work.

TABLE IMaterial f(GHz) tan x RefStyrodur 4000CS (BASF) 19 - 26 1.07 0.3 " 320 - 0.030* 4

� ��c� 4

" 3.75 - 6.25 1.034 <1 8 " 7.5 - 12.5 1.040 <1 8 " 1000 1.042 - 5HDPE 0.1 2.34 - 6 " 26.5 - 40 2.34 - 6 " 26.5 - 40 2.278 1 " 35 2.316 1.34 7Plexiglass 19 - 26 2.54 45.3 " 50 2.557 32.7 7 " 3.75 - 6.25 2.50 41.0 8 " 6.25 - 12.5 2.55 41.0 8

* in Np/cm

References

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York (NY), 1989.

[2] R.E. Collin, , McGraw-Hill: New YorkFoundations for Microwave Engineering

(NY), 1992.

[3] F.E. Gardiol, , Artech House: Dedham (MA), 1984.Introduction to Microwaves

[4] J.W. Lamb, n.10, 1997.Int J. Infrared and Millimeter Waves, 18,

[5] Guozhong Zhao, et al., , , n.6, June 2002. J.Opt. Soc.Am. B 19

[6] K. Seeger, , n.2, Feb. 1991. IEEE Trans. MTT 39

[7] Afsar and Button, , n.1, Jan. 1985.Proc. IEEE 73

[8] S. Mariotti, Marzo 2005.Nota Tecnica IRA,