INSTITUTE OF PHYSICS PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 17 (2006) R55–R70 doi:10.1088/0957-0233/17/6/R01

REVIEW ARTICLE

Frequency domain complex permittivitymeasurements at microwave frequenciesJerzy Krupka

Department of Electronics and Information Technology, Institute of Microelectronics andOptoelectronics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa,Poland

E-mail: krupka@imio.pw.edu.pl

Received 16 May 2005, in final form 17 August 2005Published 26 April 2006Online at stacks.iop.org/MST/17/R55

AbstractOverview of frequency domain measurement techniques of the complexpermittivity at microwave frequencies is presented. The methods are dividedinto two categories: resonant and non-resonant ones. In the first categoryseveral methods are discussed such as cavity resonator techniques, dielectricresonator techniques, open resonator techniques and resonators fornon-destructive testing. The general theory of measurements of differentmaterials in resonant structures is presented showing mathematicalbackground, sources of uncertainties and theoretical and experimentallimits. Methods of measurement of anisotropic materials are presented. Inthe second category, transmission–reflection techniques are overviewedincluding transmission line cells as well as free-space techniques.

Keywords: frequency domain measurements, microwave measurements,permittivity, permeability, dielectric losses, magnetic losses

1. Introduction

One of the most widely known books on dielectricmeasurement is that of Von Hippel [1] published more than50 years ago. A number of review papers, books [2–4]and hundreds of journal papers on this topic have beenpublished since that time but still readers may find it difficultto choose an appropriate measurement technique to satisfytheir measurements needs. In this paper, an overviewof measurement techniques applicable for measurements ofthe complex permittivity of various materials at microwavefrequencies is presented.

In the frequency domain the complex permittivity ofany linear material is generally defined as a tensor quantitydescribing the relationship between the electric displacement( D) and the electric field ( E) vectors

D = ↔ε E. (1)

For passive reciprocal materials such as ionic dielectric singlecrystals the permittivity tensor is symmetric and can be

diagonalized which means that in a certain specific coordinatesystem it takes the diagonal form

↔ε =

ε11 0 0

0 ε22 00 0 ε33

. (2)

For polycrystalline materials, glasses, plastics and somecrystals (e.g., having cubic crystallographic structure) alldiagonal elements become identical and the complexpermittivity becomes a scalar quantity. The complexpermittivity of an isotropic material in general can be writtenas

ε = ε0εr = ε0

(ε′

r − jε′′rd − j

σ

ωε0

)= ε0ε

′r(1 − j tan δ) (3)

where tan δ is the total dielectric loss tangent given by

tan δ = tan δd +σ

ωε0ε′r

(4)

where εr is the relative complex permittivity, ω is the angularfrequency, σ is the conductivity, ε0 = 1/(c2µ0) ≈ 8.8542 ×10−12 (F m−1) denotes permittivity of vacuum and tan δd is the

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dielectric loss tangent associated with all other dielectric lossmechanisms except conductivity.

The dielectric loss tangent of any material describesquantitatively dissipation of the electric energy due to differentphysical processes such as electrical conduction, dielectricrelaxation, dielectric resonance and loss from nonlinearprocesses (such as hysteresis) [5]. When we measure theloss of a dielectric at a single frequency we cannot, in general,distinguish between them. Phenomenologically, they all giverise to just one measurable quantity, namely, the total measuredloss tangent. The origin of dielectric losses can also beconsidered as being related to the time delay between theelectric field and the electric displacement vectors.

For some materials like superconductors conductivityitself contains two terms: real (conductivity related to normalcurrent) and imaginary (conductivity related to supercurrent).It is easy to prove that conductivity related to the supercurrentcan be formally treated as a negative real permittivity.Similar phenomena take place in metals and semiconductorsat very high frequencies (infrared and optical). From ameasurement point of view materials such as dielectrics,metals and semiconductors can be characterized by thecomplex permittivity. The only differences between thesematerials at microwave frequencies are related to the valuesof real and imaginary parts of permittivity. For example, formetals the imaginary part of permittivity is many orders ofmagnitude larger than the real one while for dielectrics usuallythe real part is larger than the imaginary one.

Some materials commonly used at microwave frequenciessuch as ferrites exhibit magnetic properties that mustbe considered in measurements of their permittivity. Thepermeability tensor µ describes relationship between themagnetic induction B and magnetic field H vectors (1):

B = ↔µ H. (5)

The most important microwave applications of ferrites arerelated to their non-reciprocal properties. In the presence ofstatic magnetic field magnetizing ferrite material along z-axisof Cartesian or cylindrical coordinate system permeability offerrite material is represented by Polder’s tensor (6) [6]. Off-diagonal components of Polder’s tensor are purely imaginarybut they do not describe any magnetic losses since they appearwith opposite signs. If ferrite material is lossy then particulartensor components (µ, κ, µz) become complex:

↔µ = µ0

µ jκ 0

−jκ µ 00 0 µz

. (6)

In addition to the intrinsic material properties defined above,we must concern ourselves with the extrinsic quantitieswe encounter in measurements of samples in differentmeasurement cells. At microwave frequencies the size ofmeasurement cells can be smaller, comparable or largerthan the wavelength. In the first case, a measurement cellcontaining a sample under test can be treated as a lumpedimpedance circuit. In such a case the measured quantityis the complex impedance (or the complex admittance)and measurements are usually performed using impedanceanalysers. At higher frequencies external parameters arethe complex reflection and/or the complex transmissioncoefficients and their measurements are usually performed

Figure 1. Transmission line measurement cell.

using vector network analysers. The second group ofmeasurement techniques is often called wave techniques. Bothlumped impedance and wave techniques can be employed inresonators. Resonators are measurement cells with resonatingelectromagnetic fields inside them that are used to obtainhigh sensitivity for measuring the loss of low-loss dielectrics.Various wave techniques are used to measure the complexpermittivity. Differences between techniques are related toboth the cells used for measurements and mathematical modelsdescribing relationships between quantities that are directlymeasured (e.g., transmission and reflection coefficients) andthe complex permittivity. Choice of a specific techniquedepends on several parameters such as frequency, size andshape of available samples, range of dielectric losses andpresence of anisotropy.

2. Transmission-line and free-space methods

Transmission and reflection methods are based onmeasurements of transmitted and/or reflected electromagneticpower from a sample under test illuminated by a well-determined incident electromagnetic wave. If the materialof the sample is isotropic then determination of the complexpermittivity and the complex permeability is possible fromtwo measured parameters: the complex reflection (S11)and the complex transmission (S21) coefficients. For non-magnetic materials the complex permittivity can be derivedonly from one measured complex parameter (either S11 orS21). Transmission–reflection measurements can be performedemploying closed measurement cells, open-ended probes butcan also be undertaken in a free-space environment.

2.1. Coaxial and waveguide cells

At frequencies below millimetre waves the sample undertest is often situated in a measurement cell made as a shortsection of coaxial line, e.g., [7–9] or waveguide as shownin figure 1. There are published standard methods for thetransmission-line technique: ASTM D5568–01 [10] and UTE(Union Technique de l’Electricite) [11]. ASTM D5568–01 standard presents a full procedure for transmission-linemeasurements, including measurements on magnetic materialsand measurements in waveguide. The UTE standard [11]is for coaxial measurements on thin specimens in a specialcoaxial-line cell. Specimens are machined to the same lengthas the cell. To mitigate the air gap problem, the gaps arefilled using a low-melting-point alloy. In determination ofmaterial properties one has to know the relationship betweenmeasurable quantities such as the complex reflection and thecomplex transmission coefficients, dimensions of the celland sample and the complex permittivity of the sample.

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Figure 2. Transmission lines containing samples having dimensionssmaller than their cross section.

Figure 3. Ratio of the actual (real) permittivity of a sample to themeasured permittivity versus relative gap value between sample andinner conductor of coaxial cell.

This requires solutions of Maxwell’s equations for the cellcontaining the sample. Exact solutions are available for a fewsimple shapes of transmission lines when the sample undertest occupies their whole cross section between two planesperpendicular to the line (e.g., for the structure shown infigure 1). When this condition is not satisfied, only numericalsolutions of Maxwell’s equations are available. Computationsbased on exact solutions usually employ explicit Nicolson andRoss expressions for the complex permittivity [7], but theycan be unstable producing erroneous results when the lengthof the specimen is close to one or more half-wavelengths inthe medium of the dielectric. Paper [12] presents an explicitalgorithm which is believed to be more stable. An alternativeapproach uses iterative algorithms [13].

Transmission/reflection techniques are especially usefulfor broad frequency band measurements of lossy dielectricliquids [14]. Measurements of solid materials are moredifficult, especially those having large permittivities. Thisis associated with the presence of air gaps between sampleand metal parts of the measurement cells as shown infigure 2. The electric field for the TEM mode of coaxialcell is perpendicular to metal–dielectric interfaces and inthe presence of air gaps becomes discontinuous. In mostcases air gaps are non-uniform and very difficult to bemeasured and therefore to be accounted for. When the gapis neglected in electromagnetic analysis it causes substantialerrors in permittivity determination. In figure 3 quantitativepermittivity errors are shown versus relative gap valuesbetween inner conductor and a dielectric sample. One cannotice that relative permittivity errors increase with increasing

Figure 4. Coaxial and waveguide cells used for measurements ofhigh permittivity materials. To minimize air gap influence both endsof samples are metallized.

gap and permittivity values. In order to mitigate errorsassociated with air gaps solid samples are often metallizedor covered with conductive pastes or solders (as in [11]), butthis usually creates additional errors in dielectric loss tangentdetermination.

Although in principle measurements in coaxial linesemploying TEM mode can be performed up to very highfrequencies, in practice, they are rarely performed atfrequencies higher than 10 GHz. Reasons for this are decreaseof measurement accuracy related to smaller size of samples(their length should be smaller than half the wavelength toavoid resonances in the sample), increase in parasitic lossesand increasing influence of imperfections in measurementsystems on determination of the complex permittivity. Moreinformation on this topic can be found in [4]. The waveguidetransmission line method has similar accuracy and resolutionas the coaxial transmission line method but it is typically usedat higher operating frequencies (lower frequency is alwayslimited by the cut-off frequency of a specific waveguide).One of its advantages over the coaxial transmission linemethod is lack of an inner conductor which makes the airgap problem less critical. On the other hand, the frequencycoverage for the waveguide transmission line method issmaller than for the coaxial counterpart and requires additionalequipment like coax-waveguide adapters when used withmodern network analysers. Measurements of high permittivitymaterials, such as ferroelectrics using samples that fullyoccupy cross section of coaxial line or waveguide becomeinaccurate due to high impedance mismatch between thesample and air filled sections of the transmission line. Formeasurements of ferroelectrics the cells shown in figure 4are used [15, 16]. For such cells exact solutions of Maxwell’sequations are not available so numerical techniques ofelectromagnetic analysis must be used to find the relationshipbetween transmission and reflection coefficients and thecomplex permittivity. The upper measurement frequencylimit for the coaxial cell shown in figure 4 is about1 GHz for materials having permittivities of the order of afew thousands.

The main features of the guided transmission linetechnique can be summarized as follows:

Advantages

• Relatively broadband frequency coverage;• Suitable for measurements of magnetic materials

(ferrites);• One of the best techniques at microwave frequencies for

high loss and medium loss samples. Uncertainties for realpermittivity better than ±1%.

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(a) (b)

Figure 5. Open transmission line measurement cells: (a) coaxial and (b) waveguide.

Figure 6. Schematic diagram of arrangements for free-space transmission/reflection or reflection measurements.

Disadvantages

• For solid specimens significant errors caused by air gaps;• Limited resolution of loss tangent measurements

(typically ±0.01).

2.2. Open-ended probes

One of the best techniques for measurements of lossydielectrics (especially biological tissues) is an open-endedtransmission line probe [17–24] schematically depicted infigure 5. This technique is applicable for measurements ofnon-magnetic materials. Since for open-ended transmissionline geometry exact solutions of Maxwell’s equations arenot available, the complex permittivity is determined frommeasurements of the complex reflection coefficient S11

employing one of the rigorous techniques of electromagneticanalysis such as mode matching, finite element of finitedifference. A single coaxial probe can typically operateover a frequency range of about 30:1 with uncertainty forreal permittivity order of ±3% for suitable materials. Thefrequency range depends on the diameter of the coaxialaperture and, e.g., a coaxial probe with 7 mm aperture canoperate from 200 MHz to 6 GHz. The best measurementsaccuracy and resolution is achieved around the centrefrequency of this band. Smaller probes can cover appropriatelyhigher frequencies but measurement uncertainties typicallyincrease with increasing frequency. In measurements of solidsthe air gap problem is important but it can be mitigated bypressing the probe against a flat surface of the sample undertest. It is only helpful if the sample is relatively soft.

Open-ended waveguide probes [20–24] are used lessoften than coaxial probes, partly because they are limited infrequency range and at lower frequencies they are physicallylarge. Waveguide probes offer two advantages over coaxialprobes for specific applications. First, such probes are bettermatched for measuring lower permittivity than coaxial probes

of similar size and at a similar frequency. Second, they canbe used for measurements of anisotropic materials since theelectric field for the dominant mode of rectangular waveguideis linearly polarized. Two permittivity tensor componentsof uniaxially anisotropic material can be derived by carryingout two measurements on the flat face of the specimen, withthe specimen orientated in two orthogonal directions: oneparallel to the anisotropy axis and the other perpendicular tothe anisotropy axis. A full numerical analysis that takes intoaccount permittivity as a tensor must be performed in order toevaluate properly the two permittivity components [24]. In allother respects waveguide probes are similar to coaxial probes.

Advantages of coaxial and waveguide open-ended probetechniques:

• Quick, easy and relatively cheap to use compared withother methods;

• A single probe can be used over a frequency range of 30:1with suitable samples;

• Well-suited for non-destructive testing.

Disadvantages of coaxial and waveguide open-endedprobe techniques:

• Air gaps between specimens and probes are difficult toavoid with hard solid specimens;

• Difficult calibration;• The methods are generally less accurate than techniques

described in the previous section with typical uncertaintiesof ±3% for real permittivity.

2.3. Free-space measurements of the complex permittivity

At millimetre wave frequencies instead of transmission linecells and open-ended probes, free-space measurements areused. Measured quantities are again the complex scatteringmatrix coefficients (transmission, reflection or both) measuredon a sample under test situated between two antennas or in frontof one antenna as shown in figure 6. One of the requirements

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for accurate free-space measurements is that the size ofthe samples under test in the direction perpendicular to theincident wave is larger than the electromagnetic beam width,so diffraction from the edges of samples can be neglected.

Free electromagnetic fields are typically radiatedas beams from antennas. In the unfocused-beammethods electromagnetic energy creates diverging beams, socorrections for attenuation of the beam between transmittingand receiving antennas have to be incorporated in the method.The beam typically extends beyond the aperture of thereceiving antenna at relatively short distance and may alsospread beyond the aperture of the sample. Such methods aretherefore prone to errors caused by diffraction at antennasand sample edges. For this reason focused-beam methodsare more often used in practice. Dielectric lenses or concavemirrors are used to focus the beam. The most commonlyused calculable beam is the Gaussian beam [24, 25] that canbe radiated from corrugated-horn antennas [26, 27]. For aGaussian beam, electromagnetism decays exponentially (inthe manner described by the Gaussian function) in the radialdirection with maximum magnitude at the axis of beampropagation, so diffraction problems may be made negligible.The practical ability to focus Gaussian beams improves as thefrequency increases up to the terahertz region. At sufficientlyhigh frequencies quasi-optical methods essentially becomeoptical methods. Normal-incidence unfocused reflection andtransmission have been used by a number of researchersfor measuring large-area laminar specimens (see, e.g., [28]for a review). The measurements can be easily performedby using matched waveguide horns attached via coaxialcables to a network analyser, although in the past accuratemeasurements were performed with less advanced equipmentsuch as waveguide bridges [29]. Computation of transmissionand reflection coefficients is simple, assuming a plane-wave approximation, and such an approximation becomesmore accurate as frequency increases. By rotating the sampleabout an axis perpendicular to the direction of propagationone can perform measurements at the Brewster angle ofincidence [31]. Measurements can also be performed as afunction of the angle of incidence [32]. Such measurementsare often more accurate than those performed at normalincidence. In recent years, focused reflection and transmissionmethods have been improved by full understanding of thetheory of corrugated-horn antennas and advances in theirmanufacturing. Complete measurement systems for laminarspecimens have been constructed either with mirrors [33] orlenses [34] for focussing. Such systems are commerciallyavailable at frequencies from 1.5 GHz up to the sub-millimetreregion of the spectrum. A typical measurement system isshown in figure 7.

The sample under test is situated in the narrowest part ofthe Gaussian beam where the beam is approximated by a planewave. Such approximation introduces some errors becausethe Gaussian beam wavelength is not exactly the same asthat of a plane wave at the same frequency. Free-spacetechniques are generally less accurate than their guided-waveequivalents and the focused methods are slightly more accuratethan the unfocused ones. It is difficult to assess uncertaintiesquantitatively but according to [4] they lie in the range±(1–10)% for real permittivity and from ±5% to over 20%for dielectric losses.

Figure 7. A quasi-optical focused beam technique for measuringthe dielectric properties of a laminar sample at normal incidence.

Major areas of application of transmission and reflectionmethods described in section 2 are measurements of medium tohigh loss isotropic materials as a function of frequency. Someof these techniques (those where the electric field is linearlypolarized) allow measurements of the complex permittivity foranisotropic materials.

3. Resonance methods

Resonators and cavities form a special class of measurementcells that are especially useful for measurements of verylow loss materials, but they also offer the highest possibleaccuracy of measurements of real permittivity. Resonantmethods can be divided into two categories. The first categoryincludes different kinds of resonant cavities (including re-entrant cavities, cylindrical and rectangular cavities), openresonators and resonators loaded with a dielectric (e.g., splitpost dielectric resonators). For the second category the sampleunder test, itself, can create a dielectric resonator. Resonantmethods employing cavities [35–44] and open resonators[45–52] operating at a single, dominant or a higher orderwell-established mode have been used for measurement ofdielectric materials for more than 60 years. In a resonantcavity the electric energy stored in the sample under test isusually small compared to the total electric energy storedin the whole cavity (except re-entrant cavities [43, 44]).In the second category, a cylindrical or spherical dielectricsample under test, enclosed in a metal shield or situated inan open space, constitutes a dielectric resonator, where theresonance frequencies predominantly depend on permittivityand dimensions of the sample. In the dielectric resonatortechniques, typically more than 90% of the total electricenergy is stored in the sample. Progress in measurementsof dielectrics employing resonant techniques during the lastdecades has been associated with two factors: the developmentof new low-loss dielectric materials and the advances inrigorous techniques of electromagnetic field computations.Using new low-loss dielectric materials it was possible toconstruct resonators filled with dielectrics that have higherQ-factors and better thermal stability than traditional all-metalcavities. Developments in electromagnetic field simulationshave made it possible to construct resonant structures withoutrestrictions on sample size or shape and also to obtain highaccuracy with measurements.

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(a) (b)

Figure 8. Cylindrical cavities: (a) containing a dielectric rod and(b) containing a dielectric disc.

(a) (b)

Figure 9. Spherical resonators: (a) a dielectric sphere in a freespace and (b) a spherical cavity containing a dielectric sphere.

Figure 10. Cylindrical dielectric resonator between conductingplanes.

3.1. General concepts of measurements by resonancemethods

Measured quantities in resonance techniques are the resonantfrequency and the Q-factor of a specific mode excited in theresonant structure containing the sample under test. Thecomplex permittivity of the sample can be evaluated fromthese two measured quantities provided all other parameters ofthe structure are known. These parameters include dimensionof the structure, surface resistance of metal parts, couplingcoefficients, radiation losses and the complex permittivitiesof dielectric supports (if present). Exact relations betweenpermittivity, sample and cavity dimensions, and measuredresonant frequency and the unloaded Q-factor can only bederived if resonant structures that permit theoretical analysisby separation of variables are used. Practically, this is onlypossible when the measurement system has simple cylindrical,spherical or rectangular geometry and where any permittivityinhomogeneity in the measured structure exists in only one ofthe principal coordinate directions and all conductive surfacesare made of perfect conductors. Examples of such geometriesare shown in figures 8–10. For these geometries transcendentalequations can be derived that represent relationship betweenthe complex permittivity and the complex angular frequenciesof a specific resonant structure. The general form of such anequation can be written as

F(εr, ω) = 0, (7)

where Re(ω) = 2πf, Im(ω) = Re(ω)/2/Qd and Qd is theQ-factor depending on dielectric losses and radiation losses inthe resonant structure.

Strictly transcendental equations can be derived if metalwall losses and coupling losses are assumed to be zero,but these extra losses are also taken into account in the

complex permittivity determination. This problem will bediscussed later on. Examples of transcendental equations thatare commonly used in measurements of dielectric propertiesand that are important for understanding the mathematicaldescription of different physical phenomena in resonantstructures are given below.

The transcendental equation for TE01p modes of acylindrical cavity [41] is shown in figure 8(b) for L1 = 0.

kz0 tan(kzh) + kz tan[kz0(L − h)] = 0, (8)

where k2z0 = (ω/c)2 −(u′

01/b)2, k2z = (ω/c)2εr −(u′

01/b)2 andu′

01 is the first root of the derivative of Bessel function of thefirst kind and zero order u′

01 = 3.831 71.The transcendental equation for all modes in a parallel

plate cylindrical dielectric resonator [53] is[εrJ

′m(kρ1a)

kρ1aJm(kρ1a)− H ′(2)

m (kρ2a)

kρ2aH(2)m (kρ2a)

]

×[

J ′m(kρ1a)

kρ1aJm(kρ1a)− H ′(2)

m (kρ2a)

kρ2aH(2)m (kρ2a)

]

= m2k2z

k22

[1

(kρ1a)2− 1

(kρ2a)2

], (9)

where

k21 = ω2

c2εr k2

2 = ω2

c2

k2ρ1 = k2

1 − k2z k2

ρ2 = k22 − k2

z

k2z = pπ

L.

The transcendental equation for TEn0p modes of a dielectricsphere in free space [54] is

ε1/2r Jn−1/2(ka)H

(2)n+1/2(k0a) − Jn+1/2(ka)H

(2)n−1/2(k0a) = 0,

(10)

where

k = ω

c

√εr, k0 = ω

c

and J and H(2) denote Bessel’s and Hankel’s functions.The transcendental equation for TEn0p modes of a

dielectric sphere in a spherical cavity [54] is

ε1/2r Jn−1/2(ka)[Jn+1/2(k0a)Yn+1/2(k0b)

− Yn+1/2(k0a)Jn+1/2(k0b)]

− Jn+1/2(ka)[Jn−1/2(k0a)Yn+1/2(k0b)

− Yn−1/2(k0a)Jn+1/2(k0b)] = 0. (11)

3.1.1. Q-factors and parasitic losses in resonant structures.The Q-factor of a resonant structure, for any mode of operation,is associated with different kinds of losses such as dielectriclosses (Qd), conductor losses (Qc), radiation losses (Qr) andcoupling losses (Qcoupling). Particular Q-factors are related bythe well-known formulae (12)–(16):

Q−1 = Q−1d + Q−1

c + Q−1r + Q−1

coupling (12)

Q−1d =

I∑i=1

pei tan δi (13)

Q−1c = RS/G (14)

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where Q is the total Q-factor of the resonant structure, RS isthe surface resistance of the metal shield at a given resonantfrequency, G is the geometric factor of the shield, pei is theelectric energy filling factor for the ith dielectric region.

The geometric factor G and electric energy filling factorspei are defined as follows:

G = ω

∫∫∫Vt

µ0|H|2 dv∫∫S|Hτ |2 ds

(15)

pei =∫∫∫

Vdεi |E|2 dv∫∫∫

Vtε(v)|E|2 dv

(16)

where Vd is the volume of dielectric resonator, Vt is the volumeof whole resonant structure, ε(v) is the spatially dependentpermittivity inside the whole resonant structure and εi is thepermittivity of ith dielectric region.

For low-loss materials electric energy filling factors canbe alternatively determined from incremental frequency rules,that require computations of the derivative of the resonantfrequency with respect to the real permittivities of dielectricregions [55]. For cylindrical TE0 mode resonant structureshaving axial symmetry an incremental frequency rule can alsobe used to determine geometric factor [56]. In this casederivatives are to be computed with respect to cavity wallperturbations.

Coupling losses are usually determined experimentally.For a resonator with two coupling ports the relationshipbetween the total Q-factor (or loaded Q-factor), the unloadedQ-factor (Qu) and coupling coefficients (β1 and β2) is givenby the formula

Q−1u = Q−1

d + Q−1c + Q−1

r = Q−1

1 + β1 + β2. (17)

The unloaded Q-factor can be determined either if couplingcoefficients are known from measurements or if they bothare at least two orders of magnitude smaller than unity. Inthe last case one can assume within experimental errors thatthe unloaded Q-factor is the same as the total measured Q-factor. The Q-factor due to conductor losses can be evaluatedfrom equations (14) and (15) providing that the magnetic fielddistribution for a specific mode of interest in the resonantstructure and the surface resistance of metal shield are known.For close resonant structures radiation losses are equal to zero,so the Q-factor due to dielectric losses can be determined fromthe measured unloaded Q-factor value subtracting the partdepending on conductor losses using equation (17). Then thecomplex angular frequency can be evaluated (as in formulae(7)) and finally transcendental equation can be solved directlyfor the complex permittivity.

3.2. Measurements in resonant cavities

Resonant cavities having axial symmetry are those most oftenused in the dielectric metrology. Cavities of this kind canoperate on different modes but in practice one of the few firstmodes of the frequency spectrum is used.

3.2.1. TE01n mode cavities. Some of the most frequentlyemployed modes are the TE01n ones that have been usedfor more than 60 years for measurements of the complexpermittivity of disc samples (as shown in figure 8(b)) madeof low-loss dielectrics [38–40] but also for measurements of

9.44

9.46

9.48

9.50

9.52

9.54

10-2 10-1 100 101 102 103 104 105 106

Re(εr)=10

Re(εr)=40

f0

f(GHz)

Im(εr)

Figure 11. Resonant frequency versus imaginary part of permittivityfor the TE011 mode of a cylindrical cavity containing a dielectricdisc sample. Parameters assumed in computations: L = 25 mm,b = 25 mm, L1 = 0 mm, h = 0.5 mm. The dotted line correspondsto the TE011 mode resonant frequency of the empty cavity.

102

103

104

105

106

107

10-2 10-1 100 101 102 103 104 105 106

Re(εr)=10

Re(εr)=40

Qd

Im(εr )

low loss region

high loss region

conductorloss region

semiconductorloss region

Figure 12. Q-factor due to dielectric losses versus imaginary part ofpermittivity for the TE011 mode of a cylindrical cavity containing adielectric disc sample.

conductivity of semiconductors and metals [41, 42]. A typicaloperating frequency range for the TE01n mode cavities is8 GHz–40 GHz. One of the main advantages of the TE01n

mode cavities is a circumferential electric field distribution thatis tangential to a cylindrical sample inserted symmetrically inthe cavity. As a result the electric field is continuous acrossdielectric–air interfaces that allows the air gap to be omittedwithout degradation of measurement uncertainties. Also thesurface currents in the metal cavity walls are circumferential,so physical contact between the lateral surface and the cavitybottoms is not important, allowing easy construction of tunablecavities and easy sample insertion.

Let us now consider theoretical aspects of measurementsof various materials in the TE011 mode cylindrical cavity shownin figure 7(b) when the sample under test having arbitrarylosses is situated directly on the bottom of the cavity. Resultsof the TE011 mode resonant frequency and Q-factor due todielectric loss evaluation are shown in figures 11 and 12.

Both resonant frequency and Q-factor due to dielectriclosses exhibit very characteristic behaviour (common for anyother resonator) as a function of the imaginary part of the

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permittivity. In a low-loss dielectric region the resonantfrequency is smaller than that for the empty cavity andthe resonant frequency shift ( f0 − f ) depends on the realpermittivity and thickness of the sample, but practically doesnot depend on losses. The inverse of the Q-factor due todielectric losses in this region depends linearly on the dielectricloss tangent according to formulae (18) with a constant valueof the electric energy filling factor pe:

Q−1d = pe tan δ. (18)

For the low dielectric loss region the real part of the complexpermittivity can be determined on the basis of the measuredresonant frequency from a simplified transcendental equationwhere both the complex permittivity and the complex angularfrequency have imaginary parts equal to zero. Evaluation ofthe dielectric loss tangent from the measured unloaded Q-factor value requires determination of all parasitic losses andelectric energy filling factor in the sample. For a closed cavityconductor losses can be evaluated from formulae (14) and (15)assuming that the surface resistance of the cavity is known.

In fact the surface resistance in a closed cavity is alwaysdetermined (at the resonant frequency of the empty cavity f0)from the measured Q-factor of the empty cavity and then itis scaled to the frequency of the cavity containing the sampleaccording to the formula

Rs(ω) = Rs(ω0)

√ω

ω0. (19)

The simplified approach of a low-loss sample gives resultsthat are essentially the same as the complex angular frequencyapproach if the dielectric loss tangent of the sample issmaller than 0.1. If the losses become higher then theresonant frequency varies as a function of loss as seenin figure 11. As losses in the sample increase and thedielectric loss tangent becomes greater than 10 then boththe resonant frequency and Qd predominantly depend on theimaginary part of the permittivity. In this region only theimaginary part of the permittivity can be determined. Forrelatively thin samples situated on the cavity bottom theelectric energy filling factor in the sample is very small andthe Q-factor depending on dielectric losses is greater than100, so materials having arbitrary losses can be measured(at least in principle). On the other hand, for the imaginarypart of the permittivity smaller than 0.1 (or dielectric losstangent smaller than 0.01) the Q-factor due to dielectric lossesbecomes larger than 105. Since Q-factors due to parasiticlosses (conductor losses) in TE011 mode cavities are typicallyin the range 15 000–30 000 (depending on frequency), so insuch cases measurement uncertainties for low-loss dielectricssubstantially increase. Measurements of low-loss dielectricsin TE01n mode cavities are possible either by using thickersamples (optimum thickness is half the wavelength or itsmultiple) or by elevating the sample to the position where theelectric field approaches a maximum. In both cases electricenergy filling factors in the sample become relatively large andhigh resolution of loss tangent determination can be obtained(of the order of 5 × 10−5). Typical uncertainties of realpermittivity determination employing TE01n mode cavities arethe order of 0.5%. At frequencies lower than 8 GHz, thedimensions of TE01n mode cavities (and samples) become toolarge for practical applications so different resonators must beused.

Figure 13. Schematic diagram of re-entrant cavity.

3.2.2. TM0n0 mode and re-entrant cavities. In the past, themost commonly used cells operating in the frequency rangefrom 2 GHz to 10 GHz were TM010 mode cavities with rodsamples as shown in figure 8(a) [16, 35, 37] and for frequenciesfrom 50 MHz to 2 GHz were re-entrant cavities [16, 43, 44] asshown in figure 13. The electric field for both kinds of cells,the TM010 mode cavity and the re-entrant cavity, has dominantaxial component with maximum at the resonator axis while themagnetic field has only azimuthal component. As a result ofsuch an electromagnetic field distribution the surface currentshave a radial component on horizontal metal surfaces and anaxial component on vertical surfaces of the cavities. Such afield distribution creates practical measurement difficulties.Firstly, any air gaps between sample and metal surfacesintroduce significant errors in real permittivity determination(similar to those described for the coaxial transmission linemethod). Secondly, surface current distribution does notallow for easy disassembling of the resonant structure (becausecontact impedances reduce the Q-factor, and they tend not tobe reproducible). This makes it necessary to have a dooror lid in the cavity for sample insertion. In a re-entrantcavity the door is typically made in the lateral surface ofthe resonator and usually does not significantly change theQ-factor; for a TM010 mode cavity a small lid is usually madein the centre of one or two cavity bottoms. Lids in the TM010

mode cavity must be sufficiently small to get reproducibleQ-factor values. Another possibility is to make small holesthrough the cavity and measure a sample that is longer thanthe height of the cavity. However, in such a case analysisof the structure becomes a problem. For a TM010 mode cavitythe transcendental equation is known only if the height ofthe sample is the same as the height of the cavity otherwisenumerical analysis of the structure is necessary. For a re-entrant cavity the transcendental equation in a closed form doesnot exist for any case but rigorous mode-matching solutionsare available [44]. Q-factors due to conductor losses for re-entrant cavities are typically in the range 1000–5000 and forTM010 mode cavities in the range 2000–10 000. Since theelectric energy filling factor in a re-entrant cavity is close tounity, resolution of dielectric loss tangent measurements inthis cavity is also relatively high, i.e. of the order of 5 × 10−5.Similar resolutions of dielectric loss tangent measurementscan be obtained in TM010 mode cavities if samples under

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(a) (b)

Figure 14. Fabry–Perot resonators used for measurements oflaminar dielectrics.

test have sufficiently large diameter. Uncertainties of realpermittivity determination for the re-entrant cavity and theTM010 mode cavity might be strongly affected by the presenceof air gaps. Errors associated with gaps increase as the aspectratio of the sample (diameter to height) increases. If largepermittivity samples are to be measured or/and their aspectratio is large then the flat surfaces of samples must be metalizedto obtain reasonable results. In re-entrant cavities uncertaintiesmight also be affected by computational errors, as a rigoroustranscendental equation for this cavity is not available. Itshould also be mentioned that in the same measurement set-up higher order TM0n0 modes can be used for measurementsof permittivity and dielectric loss tangent as a function offrequency. The transcendental equation for TM0n0 modesremains the same as for the TM010 mode. Resonators of thiskind are commercially available with software for the complexpermittivity determination.

Typical measurement uncertainties for real permittivityare about 0.5%–2% for TM010 mode cavity and are about 1%–3% for re-entrant cavity.

3.2.3. Fabry–Perot resonators. Fabry–Perot resonators thatare also known as open resonators are commonly used in themillimetre frequency range [45–52]. Typical geometries ofFabry–Perot resonators used for measurements of the complexpermittivity of low-loss dielectrics are shown in figure 14. Theelectromagnetic field distribution for Fabry–Perot resonatorsis not accurately known but Gaussian beam approximationfor the dominant family of TEM00q modes allows quiteaccurate measurements at the frequency of one of these modeswhere sample thickness is approximately the same as halfthe wavelength. The sample size must be sufficiently largeto comprise full Gaussian beam energy; otherwise diffractionfrom sample edges may cause radiation losses. Also the sizeof the mirrors must be sufficiently large compared to the widthof the Gaussian beam. The Q-factor for an empty Fabry–Perotresonator operating at millimetre wave frequencies is typicallyin the range 100 000–200 000 and increases with increasingmode index. On the other hand loss tangent resolution doesnot increase with increasing mode indices because in such acase electric energy filling factors in the sample decrease.

For a low permittivity sample having half the wavelengththickness the uncertainty of real permittivity can be as low

as 0.5% but it can be larger for samples having arbitrarythickness. This is related to the approximate electromagneticmodelling of such structures. Loss tangent resolution inFabry–Perot resonators is of the order of 1 × 10−5 if themode of interest does not interfere with spurious modes andradiation losses are not present. Full wave electromagneticmodelling of Fabry–Perot resonators is very difficult and atpresent none of the commercially available electromagneticsimulators gives satisfactory results for analysis of modeswith large axial mode indices. The Q-factor of Fabry–Perotresonators can be increased at one single frequency by a factorof 5 or more employing so-called Bragg reflectors [57]. Dueto their extremely high Q-factor such resonators seem to beideal for precise measurements of the complex permittivity ofgases.

3.3. Dielectric resonator structures

For measurement techniques belonging to this category, thedominant part of the electromagnetic energy (usually morethan 90%) is concentrated in the sample under test. A lot ofspecific techniques have been described that employ differentmodes and different measurement cells.

3.3.1. TE011 mode dielectric resonators. Initially,the dielectric resonator technique for measurements ofpermittivity and losses of low-loss dielectrics was proposedby Hakki–Coleman [58] in 1960 employing the TE011 modeof operation in a rod resonator terminated from both sidesby conducting planes as shown in figure 10. Since itsdiscovery, it has become one of the most accurate and the mostfrequently used techniques for measurements of permittivityand dielectric losses of solid materials. It is also known underdifferent names as the Courtney [59] or parallel plate holderand it is also proposed as one of International Standards IECtechniques [60] for measurements of the complex permittivityof low-loss solids. It benefits from a very simple measurementconfiguration and easy access for introducing and removingspecimens. For the TE011 mode the applied electric field iscontinuous across the sample boundary, so air gaps betweendielectric and metal planes do not play a significant role.As a result, high measurement accuracy of real permittivitycan be achieved. This technique was applied by Courtney tomeasure both the complex permittivity and the complex scalarpermeability of microwave ferrites. The main disadvantageof the originally described technique was that the surfaceresistance of metal plates could not be measured withoutthe sample as in cavity methods. This difficulty has beenovercome by measurements of two different TE01n modes ontwo dielectric samples made of the same material [61].

The TE011 mode is one of several modes in the so-called ‘trapped state’, which means that radiation lossesusually are not present for infinitely extended metal plates.Electromagnetic fields are evanescent in the air region (seefigure 10) if the distance between the plates is smaller thanhalf the wavelength corresponding to the resonant frequency.This happens if the aspect ratio of the dielectric sample islarger than a certain minimum value which depends on thepermittivity of the sample. The lower the permittivity ofthe sample, the larger must be the aspect ratio of the sampleto allow omitting the radiation losses. For example, for a

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Figure 15. Schematic diagram of a TE01δ mode dielectric resonator.

PTFE sample (with a permittivity of about 2.06) the minimumaspect ratio is about 2 but for alumina it is about 0.6. Toavoid radiation losses in practice when dimensions of metalplates are finite the aspect ratio of the sample must be largerthan those minimum values. It should be mentioned that inthe case when metal plates are separated by a distance largerthan half the wavelength, the TE011 mode dielectric resonatorshave very low Q-factors predominantly affected by radiation.For example, for a sample having permittivity about 30 theQ-factor due to radiation losses is about 40. Under suchconditions resonators cannot be used for measurements of lowdielectric losses.

If an additional metal cylinder is introduced (creating acylindrical cavity) in the parallel plate dielectric resonatoroperating in the trapped state then its resonant frequencies andQ-factors remain almost unaffected when the diameter of thecylinder is sufficiently large. Resonators of this kind may havesome advantages over the parallel plate structure (especiallyfor variable temperature measurements). Such resonatorshave been used for measurements of dielectric properties ofseveral dielectric materials at cryogenic temperatures [62]. Tominimize conductor losses high temperature superconductorswere used as the end plates. A transcendental equation wasderived for all modes in such a structure in the presence ofuniaxial anisotropy of the sample allowing measurements oftwo permittivity tensor components. TE011 mode dielectricresonators made of sapphire have been also used formeasurements of the surface resistance of high temperaturesuperconductors, e.g., [63]. Sapphire has dielectric losstangent <10−7 at cryogenic temperatures so the dielectriclosses in the structure are negligible, the lateral metal walllosses are small and calculable and the surface resistance canbe easily determined from equations (12)–(17) (see [63] formore details).

The uncertainty of the real permittivity measured in aHakki–Coleman cell operating at ambient temperatures isusually of the order of 0.3% with dielectric loss tangentresolution of the order of 10−5. The latter value is limitedby relatively large parasitic losses in the metal plates.

3.3.2. TE01δ mode dielectric resonators. The effect ofconductor losses on the loss tangent determination in thedielectric resonator structures can be mitigated if the sampleunder test is situated away from all conducting walls as shownin figure 15. Usually the TE01δ mode is employed. The TE01δ

mode technique is one of the most accurate techniques formeasurements of the dielectric loss tangent of isotropic low-loss dielectric materials but it can also be used for precisemeasurements of real permittivity [64, 65]. Manufacturers

101

102

103

104

105

2 4 6 8 10

G(Ω)

Dc/d

εr=50

εr=10

εr=2

Figure 16. Geometric factor versus size of metal enclosure forTE01δ mode dielectric resonators having different permittivities.Dc/Lc = 2, L1 = (Lc–h)/2, dielectric support neglected.

of low-loss dielectric ceramics usually use this technique tomeasure only dielectric losses (as the inverse of the unloadedQ-factor of the structure shown in figure 15) avoiding analysisof the structure. They assume that for a metal cavity which isabout three times larger than the dielectric sample conductorlosses can be entirely neglected. As will be shown later such anassumption is not always valid. There are a lot of advantagesassociated with using the TE01δ mode. The most importantare the easy mode identification, small parasitic losses,especially for high permittivity samples, and the lack of modedegeneracy. Measurements of real permittivity are possiblebut its determination is more difficult than for parallel platestructure because of the lack of exact solutions of Maxwell’sequation. However, if accurate numerical techniques are usedthe uncertainties of real permittivity determination associatedwith numerical modelling are smaller than those associatedwith dimensional uncertainties.

In figure 16 results of numerical computations ofgeometric factors versus size of metal enclosure are presented.It is seen that for an empty TE011 cavity geometric factor valueis about 660 (which is visible as the limit for Dc/d = 10 forlow permittivity samples) while for a high permittivity samplewith εr = 50 geometric factor values exceed 10 000 for 5 <

Dc/d < 8.5.This means that conductor losses in the cavity for

high permittivity samples have been reduced more than oneorder of magnitude compared to the empty cavity, allowingmeasurements of dielectric losses at least one order ofmagnitude smaller than in the TE01n mode cavities. Typicalresolution of loss tangent resolution employing TE01δ modedielectric resonators technique with optimised enclosure isabout 10−6 for high permittivity samples εr > 20 anduncertainty of real permittivity measurements of the orderof 0.3% (similar for low and high permittivities). Lowpermittivity materials are more difficult to measure becausefor such materials the TE01δ mode becomes one of the higherorder modes in the frequency spectrum [65]. Only recentlyhigher order quasi TE0 modes in the structure shown infigure 15 have been employed for measurements of realpermittivity and dielectric loss tangent as a function offrequency [66].

3.3.3. Whispering gallery mode resonators. To minimizefurther parasitic losses (conductor losses for closed structures

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1

100

104

106

108

1010

1012

1014

1016

1 10 100

Qr

ε r

n=1

n=2

n=4

n=12

n=6

n=8

n=10

TE modes

Figure 17. Q-factors due to radiation losses for TE0 modes ofisolated dielectric sphere.

Figure 18. Schematic diagram of shielded cylindrical whisperinggallery mode resonator. Supports can be made of metal.

or radiation losses for open structures) one can employ themodes having large azimuthal indices that can be excited incylindrical or spherical dielectric samples. They are calledwhispering gallery modes because their electromagnetic fieldenergy is concentrated near the lateral surface but inside thedielectric sample. The simplest structure of this kind isa spherical dielectric resonator in free space (figure 9(a)).Even for lossless dielectrics, transcendental equation (10) hassolutions only for certain sets of complex angular frequenciesdue to radiation losses. Formally this is the consequenceof properties of Hankel’s function (they are complex valuedfor real arguments). Using the complex frequencies from thesolution of (10), the Q-factor due to radiation can be computedas

Qr = Re(ω)

2 Im(ω). (20)

The results of the computations of radiation loss versuspermittivity for the TEn01 family of an open spherical dielectricresonator are shown in figure 17. As seen in figure 17, for largeindices ‘n’ and large values of permittivity, the Q-factor dueto radiation approaches a very large value, e.g., Qr 108 ifεr 10 and n 12. Even for low permittivity samples itis always possible to obtain Qr arbitrarily large by choosing asufficiently large index n. It means that even for extremely low-loss dielectrics the radiation losses for sufficiently high orderwhispering gallery modes are much smaller than dielectriclosses.

102

103

104

105

106

107

108

109

1 1.5 2 2.5

G(Ω)

b/a

m=12 m=10

m=8

m=6

m=4

m=0 (TE)

m=1

m=2

symmetricmodes

Figure 19. Geometric factors of symmetric modes versusnormalized radius of perfect conductor shield for cylindricalresonator having permittivity εr = 10. It was assumed that L/h = b/a.

For totally shielded resonators there are no radiation lossesbut conductor losses resulting from the surface resistance of themetal shield. The most popular shielded whispering gallerymode resonators are cylindrical ones situated in a cylindricalmetal enclosure as shown in figure 18. The measurement cellis similar to that for the TE01δ mode dielectric resonator butsupports are usually made of metal and the sample is situatedsymmetrically in the cavity.

In general electromagnetic fields in cylindrical dielectricresonators are of hybrid nature, i.e. they have all six non-zerocomponents (except axially symmetric modes). The modesin the structure shown in figure 18 can be divided into twocategories: symmetric (with respect to the resonator plane ofsymmetry) for which the symmetry plane constitutes a perfectmagnetic wall (S-modes) and antisymmetric for which thesymmetry plane constitutes a perfect electric wall (N-modes).Such nomenclature is only valid if the shielded resonatorpossesses a plane of symmetry. For each category only twomode subscripts are used: the first one corresponding to theazimuthal index (m) and the second index ‘i’ correspondingto the sequence of the particular mode on the frequency axis.Such nomenclature is very useful since it does not requirecomplicated field plot analysis in order to distinguish betweentwo unknown indices n (radial) and p (axial). Modes belongingto the primary WGMR families are therefore labelled as Sm,1

or Nm,1 and all other modes as Sm,i or Nm,i. Modes withazimuthal index m > 0 of cylindrical resonators are alwaysdoubly degenerate.

The results of the numerical computations of thegeometric factor employing a mode-matching technique arepresented in figure 19. As seen in this figure the geometricfactors increase rapidly with an increase of azimuthal modeindex m, so it is always possible to choose this index soas to make the conductor losses negligible compared tothe dielectric losses. One of the first papers describingthe use of whispering gallery modes for measurements of

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dielectric losses was published by Braginsky et al [69]. Theauthors of this paper measured the modes having the largestQ-factors on sapphire at cryogenic temperatures. Dielectricloss tangents were evaluated as the inverse of measuredQ-factors. In later papers full numerical electromagneticanalysis of the structure was undertaken [68] and thewhispering gallery mode technique used for measurementsof two permittivity tensor components of several uniaxiallyanisotropic crystals [70]. This was done by employing twomodes: one belonging to the quasi TE mode family (subscriptH) and the other belonging to the quasi TM mode family(subscript E).

Real permittivities were evaluated as the solutions of thesystem of two nonlinear equations

F1(f(H), ε⊥, ε||) = 0

F2(f(E), ε⊥, ε||) = 0

. (21)

Once permittivities had been evaluated dielectric loss tangentswere determined from the system of two linear equations

Q−1(E) = p(E)

e⊥ tan δ⊥ + p(E)e|| tan δ|| + RS/G(E)

Q−1(H) = p(H)

e⊥ tan δ⊥ + p(H)e|| tan δ|| + RS/G(H)

(22)

where tan δ⊥ and tan δ|| are the dielectric loss tangentsperpendicular and parallel to the anisotropy axis;p

(H)e⊥ , p

(H)e|| , p

(E)e⊥ , p

(E)e|| are the electric energy filling factors

perpendicular (subscript ⊥) and parallel (subscript ‖) tothe anisotropy axis of the resonant structure, for quasi-TM whispering gallery modes (superscript E) and quasi-TE whispering gallery modes (superscript H), and G(E) andG(H) are the geometric factors for quasi-TM and quasi-TEwhispering gallery modes.

The main advantages of using whispering gallery modesare: practically unlimited resolution for dielectric loss tangentand the possibility of measurements of real permittivities foruniaxially anisotropic materials with uncertainties down to0.1%. The main disadvantage is difficult mode identificationand limited ability for measurements of materials having losseslarger than 10−4.

4. Measurements of anisotropic materials withinduced anisotropy

If samples under test are anisotropic they have to be orientedand cut with respect to their anisotropy axes. Naturalanisotropy might be present in materials that do not possesssymmetric crystallographic structure, but it might also beinduced by mechanical stress or biasing electric or magneticfield. Measurement techniques for uniaxially anisotropiccrystals have already been described in the former paragraphs.They employ two different modes excited in an orientedsingle-crystal sample. One of the most important materialsbelonging to the second group is microwave ferrites thatexhibit gyromagnetic properties in the presence of biasingstatic magnetic field. As already mentioned the magneticproperties of ferrite under uniform bias along the z-axis canbe described by Polder’s tensor (5). The complex permittivityof polycrystalline ferrites is a scalar quantity, which does notdepend on the static magnetic bias. At frequencies higher thanferromagnetic resonance magnetic losses in ferrites are usually

Figure 20. Measurement cells used for measurements of thepermeability tensor of microwave ferrites versus static magneticbias.

small so measurements of their properties requires applicationof resonant measurement cells. Several techniques havebeen used to measure ferrite properties that were describedin journal papers, e.g., [35], and summarized in overviewbooks on microwave ferrites (Gurevich [6], Baden-Fuller[71]). More recently a dielectric resonator technique wasproposed that allows measurements of all three permeabilitytensor components on one rod-shaped ferrite sample [72].Measurement cells for this technique are shown in figure 20.This technique uses three different modes in two dielectricresonators having the same height and internal diameter butdifferent external diameters. One (larger) dielectric resonatoroperates on the TE011 mode while the other on the (smaller)HE111 mode. The HE111 mode is the first mode in thetrapped state for a parallel plate dielectric resonator. Theexternal diameter of the smaller resonator is chosen suchthat its HE111 mode resonant frequency (with the sample butwithout any bias) is approximately the same as the TE011 moderesonant frequency of the larger resonator (with the samplebut without any bias). Two resonators were used in order tomitigate the influence of frequency on the measurement resultssince permeability components are frequency dependent.Without any bias the HE111 mode is doubly degenerate, whichmeans that the two modes corresponding to the oppositecircular polarizations, namely HE+

111 and HE−111, have identical

resonant frequencies. In practice however due to imperfectaxial symmetry the two modes are split by a few MHz whichgives rise to the measurement uncertainties. In the presenceof a biasing field mode degeneracy is removed due to theappearance of an off-diagonal component of the permeabilitytensor. As this component increases the frequency differencebetween the two modes also increases as shown in figure 21.Resonant frequencies also depend on the other twopermeability tensor components. At zero bias the resonantfrequencies of the degenerate HE111 mode and TE011 modedepend on two unknowns, namely the scalar permeabilityand the scalar permittivity of the ferrite sample, so thetwo unknowns can be determined from the system of twotranscendental equations

F10(µd, ε, fHE) = 0F30(µd, ε, fHE) = 0

. (23)

When the permittivity is known the three permeability tensorcomponents at arbitrary fixed bias can be determined from thesystem of three transcendental equations

F1(µ, κ, µz, ε, fHE+) = 0F2(µ, κ, µz, ε, fHE+) = 0F3(µ, κ, µz, ε, fTE) = 0

. (24)

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9000

9200

9400

9600

9800

10000

10200

0 20 40 60 80 100 120 140 160

f(MHz)

Hext

(kA/m)

HE+

111

HE-

111

H011

empty HE111

empty H011

Figure 21. Measured frequencies of three different modes of twocylindrical resonators containing a YIG sample versus staticmagnetic field bias.

0.000

0.200

0.400

0.600

0.800

1.000

0 20 40 60 80 100 120 140 160

Ms=140 kA/m

κµµ

z

µ'

Hext

(kA/m)

Figure 22. Real parts of three permeability tensor components of aYIG sample versus static magnetic field bias.

Finally three magnetic loss tangent components can be foundon the basis of measured unloaded Q-factors of three differentmodes as solutions of the system of linear equations

Q−11 = p1ε tan δε + p1µ tan δµ + p1κ tan δκ

+p1µz tan δµz + Q−11c + Q−1

1d

Q−12 = p2ε tan δε + p2µ tan δµ + p2κ tan δκ

+p2µz tan δµzz + Q−12c + Q−1

2d

Q−13 = p3ε tan δε + p3µ tan δµ + p3κ tan δκ

+p3µz tan δµzz + Q−13c + Q−1

3d

. (25)

Results of the determination of the real parts of thepermeability tensor versus bias for yttrium iron garnet (YIG)are shown in figure 22. More details about this technique(frequency corrections for individual tensor components)can be found in [72]. Measurement uncertainties for realpermeability components using this technique are typically ofthe order of 1% and the magnetic loss tangent resolution isabout 5 × 10−5.

D dielectricresonators

support

sample metalenclosure

coupling loop

L

z

r

h

dr

L

h

r1

g εd

εr

Figure 23. Schematic diagram of a split dielectric resonator fixture.

5. Resonators for measurements of laminardielectric materials and thin films deposited ondielectric substrates

Advances in numerical computations of electromagnetic fields[73] have enabled the implementation of resonant structureswith more complicated geometry. Hence the geometry of aresonant cavity can be chosen to suit materials having specificshape and dimensions. The split post dielectric resonator(SPDR) is a good example of such an approach as it has beendeveloped specifically for measurement of laminar dielectrics[74–76]. The geometry of a split dielectric resonator is shownin figure 23. Currently the SPDR is one of the most convenientand accurate techniques for determination of permittivity andloss tangent of PWB and LTCC materials and ferrite substrates[78] in the frequency range from 1 GHz to 30 GHz.

The main advantages of split post dielectric resonatorsare: arbitrary shape of samples under test (they only needto have uniform thickness), smaller dimensions than formetal cavity resonators, and possibility of measurement ofvarious materials including thin film ferroelectrics [79]. Withproperly chosen sample thickness it is possible to resolvedielectric loss tangents to approximately 2 × 10−5 andQ-factor measurements with an accuracy of 1%. For well-machined laminar specimens the uncertainty in permittivitymeasurements using the SPDR is about 0.3% [77].

At frequencies higher than 20 GHz the size of split postdielectric resonators becomes very small and their Q-factorsalso become smaller than at lower frequencies (about 7000 at20 GHz). At millimetre wave frequencies split metal cavitytechniques can be alternatively used for non-destructive testingof laminar specimens [80–82].

6. Summary

This paper has overviewed only a small fraction of theavailable techniques for measurement of material propertiesat microwave frequencies. A lot of techniques exist that werenot mentioned in this paper that utilize microstrip, stripline andcoplanar waveguide cells (both transmission line and resonantones). General properties of such structures are similar tothose described in this paper. Typically transmission/reflectiontechniques are useful for characterization of high and mediumloss materials and resonant techniques can be in principleused for measurements of materials having arbitrary losses.On the other hand, resonant techniques are in most cases

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limited to one fixed frequency (although sometimes a fewdifferent modes can be used in one measurement cell) whiletransmission/reflection techniques can typically operate atbroad frequency bands. One of the most important issuesfor all techniques is their sensitivity to the presence of airgaps between the sample and other parts of the measurementcell. Resolution of loss tangent measurements for arbitrarytechnique is associated with the presence of parasitic losses inmeasuring cells. They must be calculable and relatively smallwith respect to the losses in the sample in order to measureprecisely losses in the material under test. Determinationof both permittivity and permeability for magnetic materialsand measurements of anisotropic materials always requiresmeasurements of at least the same number of parameters as thenumber of unknowns. Precise determination of permittivityat microwave frequencies also requires full electromagneticmodelling of measurement cells, which is still one of the mostchallenging problems in microwave materials metrology.

References

[1] Von Hippel A 1954 Dielectric Materials and Applications(New York: Wiley/The Technology Press of MIT) (newedition published 1995) (Dedham, MA: Artech House)

[2] Birch J R and Clarke R N 1982 Dielectric and opticalmeasurements from 30 to 1000 GHz Radio Electron. Eng.52 565–84

[3] Anderson J C 1964 Dielectrics (Science Paperbacks, ModernElectrical Studies) (London: Chapman and Hall)

[4] Clarke R N 2004 A Guide to the Characterisation of DielectricMaterials at RF and Microwave Frequencies (Teddington:NPL)

[5] Burfoot J C 1967 Ferroelectrics: An Introduction to thePhysical Principles (London: Van Nostrand-Reinhold)

[6] Gurevich A G 1963 Ferrites at Microwave Frequencies (NewYork: Consultants Bureau Enterprises Inc.)

[7] Nicolson A M and Ross G F 1970 Measurement of theintrinsic properties of materials by time domain techniquesIEEE Trans. Instrum. Meas. 19 377–82

[8] Weir W B 1974 Automatic measurement of complex dielectricconstant and permeability at microwave frequenciesProc. IEEE

[9] Baker-Jarvis J, Vanzura E J and Kissick W A 1990 Improvedtechnique for determining complex permittivity with thetransmission/reflection method IEEE Trans. Microw.Theory Tech. 38 1096–103

[10] ASTM D 5568.01 1995 Standard test method for measuringrelative complex permittivity and relative magneticpermeability of solid materials at microwave frequencies(American Society for the Testing of Materials)(http://www.astm.org)

[11] UTE Standard 26-295 1999 Mesure de la permittivite et de lapermeabilite de materiaux homogenes et isotropes a pertesdans le domaine des micro-ondes. Methode de mesure enguide coaxial circulaire (France: UTE (Union Techniquede l’Electricite et de la Communication)) C26-295(http://www.ute-fr.com)

[12] Boughriet A H, Legrand C and Chapoton A 1997 Noniterativestable transmission/reflection method for low-loss materialcomplex permittivity determination IEEE Trans. Microw.Theory Tech. 45 52–7

[13] Jenkins S, Hodgetts T E, Clarke R N and Preece A W 1990Dielectric measurements on reference liquids usingautomatic network analysers and calculable geometriesMeas. Sci. Technol. 1 691–702

[14] Gregory A P and Clarke R N 2001 Tables of complexpermittivity of dielectric reference liquids at frequencies upto 5 GHz NPL Report CETM 33 (Teddington: NPL)

[15] Grigas J 1996 Spectroscopy of Ferroelectrics and RelatedMaterials (Amsterdam: Gordon and Breach)

[16] Brandt A 1963 Investigations of Dielectrics at Ultra HighFrequencies (Moscow: GIFML) (in Russian)

[17] Stuchly M A and Stuchly S S 1980 Coaxial line reflectionmethods for measuring dielectric properties of biologicalsubstances at radio and microwave frequencies—a reviewIEEE Trans. Instrum. Meas. 29 176–83

[18] Mosig J R, Besson J-C E, Gex-Fabry M and Gardiol F E 1981Reflection of an open-ended coaxial line and application tonon-destructive measurement of materials IEEE Trans.Instrum. Meas. 30 46–51

[19] Anderson L S, Gajda G B and Stuchly S S 1986 Analysis ofopen-ended coaxial line sensor in layered dielectricIEEE Trans. Instrum. Meas. 35 13–8

[20] Hodgetts T E 1989 The open-ended coaxial line: a rigorousvariational treatment Royal Signals and RadarEstablishment Memorandum, Royal Signals and RadarEstablishment, No 4331

[21] Baker-Jarvis J, Janezic M D, Domich P D and Geyer R G1994 Analysis of an open-ended coaxial probe with lift-offfor nondestructive testing IEEE Trans. Instrum. Meas. 45711–8

[22] Sphicopoulos T, Teodoridis V and Gardiol F E 1985 Simplenondestructive method for the measurement of materialpermittivity J. Microw. Power 20 165–72

[23] Bakhtiari S, Ganchev S and Zoughi R 1993 Open-endedrectangular waveguide for nondestructive thicknessmeasurement and variation detection of lossy dielectric slabbacked by a conducting plate IEEE Trans. Instrum.Meas. 42 19–24

[24] Clarke R N, Gregory A P, Hodgetts T E, Symm G T andBrown N 1995 Microwave measurements upon anisotropicdielectrics—theory and practice Proc. 7th Int. BritishElectromagnetic Measurements Conf. (BEMC) Paper 57(Teddington: NPL)

[25] Kogelnik H and Li T 1966 Laser beams and resonatorsProc. IEEE 54 1312–29

[26] Clarricoats P J B and Olver A D 1984 Corrugated Horns forMicrowave Antennas (IEE Electromagnetic Waves Series18) (London: Peter Peregrinus)

[27] Wylde R J 1984 Millimetre-wave Gaussian beam modes opticsand corrugated feed horns Proc. IEE H 131 258–62

[28] Afsar M N, Birch J R and Clarke R N 1986 The measurementof the properties of materials Proc. IEEE 74 183–98

[29] Cook R J and Rosenberg C B 1979 Measurements of thecomplex refractive index of isotropic and anisotropicmaterials at 35 GHz using a free-space microwave bridgeJ. Phys. D: Appl. Phys. 12 1643–52

[30] Stone N W B, Harries J E, Fuller D W E, Edwards J G,Costley A E, Chamberlain J, Blaney T G, Birch J R andBailey A E 1975 Electrical standards of measurement:Part 3. Submillimetre-wave measurements and standardsProc. IEEE 122 1054–70

[31] Campbell C K 1978 Free space permittivity measurements ondielectric materials at millimeter wavelengths IEEE Trans.Instrum. Meas. 27 54–8

[32] Shimbukuro F I, Lazar L, Chernick M R and Dyson H B 1984A quasi-optical method for measuring the complexpermittivity of materials IEEE Trans. Microw. TheoryTech. 32 659–65

[33] Qureshi W M A, Hill L D, Scott M and Lewis R A 2003 Useof a Gaussian beam range and reflectivity arch forcharacterisation of random panels for a naval applicationProc. Int. Conf. on Antennas and Propagation (ICAP)(University of Exeter) (London: IEE) pp 405–8

[34] Gagnon N, Shaker J, Berini P, Roy L and Petosa A 2003Material characterization using a quasi-optical measurementsystem IEEE Trans. Instrum. Meas. 52 333–36

[35] Bussey H E and Steinert L A 1958 Exact solution for agyromagnetic sample and measurements on a ferriteIRE Trans. Microw. Theory Tech. 6 72–6

R68

Review Article

[36] Li S, Akyel C and Bosisio R G 1981 Precise calculations andmeasurement on the complex dielectric constant of lossymaterials using TM010 perturbation techniques IEEE Trans.Microw. Theory Tech. 29 1041–8

[37] Risman P O and Ohlsson T 1975 Theory for and experimentswith a TM020 applicator J. Microw. Power 10 271–80

[38] Horner F, Taylor T A, Dunsmuir R, Lamb J and Jackson W1946 Resonance methods of dielectric measurement atcentimetre wavelengths J. IEE 93 (Part III) 53–68

[39] Cook R J, Jones R G and Rosenberg C B 1974 Comparison ofcavity and open-resonator measurements of permittivity andloss angle at 35 GHz IEEE Trans. Instrum. Meas. 23 438–42

[40] Stumper U 1973 A TE01n cavity resonator method todetermine the complex permittivity of low loss liquids atmillimetre wavelengths Rev. Sci. Instrum. 44 165–9

[41] Krupka J and Milewski A 1978 Accurate method of themeasurement of complex permittivity of semiconductors inH01n cylindrical cavity Electron. Technol. 11 11–31

[42] Dmowski S, Krupka J and Milewski A 1980 Contactlessmeasurement of silicon resistivity in cylindrical TE01n modecavities IEEE Trans. Instrum. Meas. 29 67–70

[43] Parry J V 1951 The measurement of permittivity and powerfactor of dielectrics at frequencies from 300 to 600 Mc/sProc. Inst. Electr. Eng. 98 303–11

[44] Kaczkowski A and Milewski 1980 A high-accuracywide-range measurement method for determination ofcomplex permittivity in reentrant cavity: Part A.Theoretical analysis of the method IEEE Trans. Microw.Theory Tech. 28 225–8

[45] Cullen A L and Yu P K 1971 The accurate measurement ofpermittivity by means of an open resonator Proc. R. Soc.Lond. A 325 493–509

[46] Jones R G 1976 Precise dielectric measurements at 35 GHzusing a microwave open-resonator Proc. IEE 123 285–90

[47] Clarke R N and Rosenberg C B 1982 Fabry–Perot and openresonators at microwave and millimeter wave frequencies,2–300 GHz J. Phys. E: Sci. Instrum. 15 9–24

[48] Hirvonen T M, Vainikainen P, Łozowski A and Raisanen A V1996 Measurement of dielectrics at 100 GHz with an openresonator connected to a network analyser IEEE Trans.Microw. Theory Tech. 45 780–6

[49] Lynch A C and Clarke R N 1992 Open resonators:improvement of confidence in measurement of lossIEEE Proc. A 139 221–5

[50] Heidinger R, Schwab R and Koniger F 1998 A fast sweepablebroad-band system for dielectric measurements at 90–100 GHz 23rd Int. Conf. on Infrared and Millimeter Waves(Colchester, UK) ed T J Parker pp 353–4

[51] Heidinger R, Dammertz G, Meiera A and Thumm M K 2002CVD diamond windows studied with low- and high-powermillimeter waves IEEE Trans. Plasma Sci. 30 800–7

[52] Danilov I and Heidinger R 2003 New approach for openresonator analysis for dielectric measurements at millimeterwavelengths J. Eur. Ceram. Soc. 23 2623–6

[53] Kobayashi Y, Fukuoka N and Yoshida S 1981 Resonant modesfor a shielded dielectric rod resonator Electron. Commun.Japan 64-B 46–51

[54] Julien A and Guillon P 1986 Electromagnetic analysis ofspherical dielectric shielded resonators IEEE Trans.Microw. Theory Tech. 34 723–29

[55] Kobayashi Y, Aoki Y and Kabe Y 1985 Influence of conductorshields on the Q-factors of a TE0 dielectric resonatorIEEE Trans. Microw. Theory Tech. 33 1361–6

[56] Kajfez D and Guillon P (ed) 1990 Dielectric Resonators(Oxford, MS: Vector Fields)

[57] Krupka J, Cwikla A, Mrozowski M, Clarke R N and Tobar M E2005 High Q-factor microwave Fabry–Perot resonator withdistributed Bragg reflectors IEEE Trans. Ultrason.Ferroelectr. Freq. Control 52 1443–51

[58] Hakki B W and Coleman P D 1960 A dielectric resonatormethod of measuring inductive capacities in the millimeterrange IEEE Trans. Microw. Theory Tech. 8 402–10

[59] Courtney W E 1970 Analysis and evaluation of a method ofmeasuring the complex permittivity and permeability ofmicrowave insulators IEEE Trans. Microw. Theory Tech.18 476–85

[60] IEC 61338, Sections 1-1, 1-2 and 1-3 (1996–2003)Waveguide type dielectric resonators: Part 1. Generalinformation and test conditions—Section 3. Measurementmethod of complex permittivity for dielectric resonatormaterials at microwave frequency (www.iec.ch)

[61] Kobayashi Y and Katoh M 1985 Microwave measurement ofdielectric properties of low-loss materials by the dielectricrod resonator method IEEE Trans. Microw. Theory Tech.33 586–92

[62] Krupka J, Geyer R G, Kuhn M and Hinken J 1994 Dielectricproperties of single crystal Al2O3, LaAlO3, NdGaO3,SrTiO3, and MgO at cryogenic temperatures and microwavefrequencies IEEE Trans. Microw. Theory Tech.42 1886–90

[63] Krupka J, Klinger M, Kuhn M, Baranyak A, Stiller M,Hinken J and Modelski J 1993 Surface resistancemeasurements of HTS films by means of sapphire dielectricresonators IEEE Trans. Appl. Supercond. 3 3043–8

[64] Takamura H, Matsumoto H and Wakino K 1989 Lowtemperature properties of microwave dielectrics Proc. 7thMeeting on Ferroelectric Materials and Their Applications,Japan. J. Appl. Phys. 28 (Suppl. 28-2) 21–3

[65] Krupka J, Derzakowski K, Riddle B and Baker-Jarvis J 1998 Adielectric resonator for measurements of complexpermittivity of low loss dielectric materials as a function oftemperature Meas. Sci. Technol. 9 1751–6

[66] Krupka J, Huang W-T and Tung M-J 2005 Complexpermittivity measurements of low loss microwave ceramicsemploying higher order quasi TE0np modes excited in acylindrical dielectric sample Meas. Sci. Technol.16 1014–20

[67] Krupka J, Cros D, Aubourg M and Guillon P 1994 Study ofwhispering gallery modes in anisotropic single crystaldielectric resonators IEEE Trans. Microw. Theory Tech.42 56–61

[68] Krupka J, Derzakowski K, Abramowicz A, Tobar M E andGeyer R G 1999 Whispering gallery modes for complexpermittivity measurements of ultra-low loss dielectricmaterials IEEE Trans. Microw. Theory Tech.47 752–9

[69] Braginsky V, Ilchenko V S and Bagdassarov Kh S 1987Experimental observation of fundamental microwaveabsorption in high quality dielectric crystals Phys. Lett.A 120 300–5

[70] Krupka J, Derzakowski K, Tobar M E, Hartnett J andGeyer R G 1999 Complex permittivity of some ultralowloss dielectric crystals at cryogenic temperatures Meas. Sci.Technol. 10 387–92

[71] Baden-Fuller A J 1987 Ferrites at Microwave Frequencies(IEE Electromagnetic Wave Series No 23) (London: PeterPeregrinus)

[72] Krupka J 1991 Measurements of all permeability tensorcomponents and the effective linewidth of microwaveferrites using dielectric ring resonators IEEE Trans. Microw.Theory Tech. 39 1148–57

[73] Maj Sz and Pospieszalski M 1984 A composite multilayeredcylindrical dielectric resonator IEEE MTT-S Int. MicrowaveSymp. Dig. pp 190–2

[74] Krupka J and Maj S 1986 Application of TE01δ mode dielectricresonator for the complex permittivity measurements ofsemiconductors CPEM’86 Conf. Dig. pp 154–5

[75] Nishikawa T, Wakino K, Tanaka H and Ishikawa Y 1988Precise measurement method for complex permittivity ofmicrowave substrate CPEM’88 Conf. Dig. pp 154–5

[76] Krupka J, Geyer R G, Baker-Jarvis J and Ceremuga J 1996Measurements of the complex permittivity of microwavecircuit board substrates using split dielectric resonator andreentrant cavity techniques DMMA’96 Conf. Dig. pp 21–4

R69

Review Article

[77] Krupka J, Gregory A P, Rochard O C, Clarke R N, Riddle Band Baker-Jarvis J 2001 Uncertainty of complexpermittivity measurements by split-post dielectric resonatortechnique J. Eur. Ceram. Soc. 21 2673–6

[78] Krupka J, Gabelich S A, Derzakowski K and Pierce B M 1999Comparison of split-post dielectric resonator and ferritedisk resonator techniques for microwave permittivitymeasurements of polycrystalline yttrium iron garnet Meas.Sci. Technol. 10 1004–8

[79] Krupka J, Huang W T and Tung M J 2005 Complexpermittivity measurements of thin ferroelectric filmsemploying split post dielectric resonator 11th Int.

Meeting on Ferroelectricity (Foz do Iguacu, Brazil,5–9 Sept. 2005)

[80] Kent G 1996 Nondestructive permittivity measurements ofsubstrates IEEE Trans. Instrum. Meas. 45 102–6

[81] Janezic M D and Baker-Jarvis J 1999 Full-wave analysis of asplit-cylinder resonator for nondestructive permittivitymeasurements IEEE Trans. Microw. Theory Tech.47 2014–20

[82] Janezic M D, Krupka J and Baker-Jarvis J 2001Non-destructive permittivity measurements of dielectricsubstrates using split-cylinder and split-post resonatorsDig. of Int. Conf. APTADM’2001 pp 116–8

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