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Progress In Electromagnetics Research, PIER 79, 367–386, 2008

A NONDESTRUCTIVE TECHNIQUE FORDETERMINING COMPLEX PERMITTIVITY ANDPERMEABILITY OF MAGNETIC SHEET MATERIALSUSING TWO FLANGED RECTANGULARWAVEGUIDES

M. W. Hyde IV and M. J. Havrilla

Air Force Institute of TechnologyWright-Patterson AFBOH 45433-7765, USA

Abstract—In this paper, a nondestructive technique for determiningthe complex permittivity and permeability of magnetic sheet materialsusing two flanged rectangular waveguides is presented. Thetechnique extends existing single probe methods by its abilityto simultaneously measure reflection and transmission coefficientsimperative for extracting both permittivity and permeability over allfrequencies. Using Love’s Equivalence Principle, a system of coupledmagnetic field integral equations (MFIEs) is formed. Evaluationof one of the two resulting spectral domain integrals via complexplane integration is discussed. The system, solved via the Methodof Moments (MoM), yields theoretical values for the reflection andtransmission coefficients. These values are compared to measuredvalues and the error minimized using nonlinear least squares to findthe complex permittivity and permeability of a material. Measurementresults for two magnetic materials are presented and compared totraditional methods for the purpose of validating the new technique.The technique’s sensitivity to uncertainties in material thickness andwaveguide alignment is also examined.

1. INTRODUCTION

Waveguide probes, whether coaxial, rectangular, or circular, have beenextensively researched. Existing applications of waveguide probesinclude subsurface crack detection; electromagnetic characterizationof materials, especially liquids and biological tissue; and microwavehyperthermia treatment [1–19]. Nearly all published waveguide probe

368 Hyde IV and Havrilla

research focuses on obtaining the reflection coefficient, using a singleprobe, from an unknown material. While this arrangement is perfectfor nondestructively determining complex permittivity of an unknowndielectric, it fails when one wants to fully characterize a magneticmaterial. Several researchers have developed methods — two thicknessmethod, frequency varying method, and sample added method —to overcome this shortfall; however, these techniques are not alwaysapplicable [15–19]. A suitable alternative would be a system whichallows the measurement of reflection and transmission coefficients sincesuch measurements are independent over all wavelengths and are themost efficient means of simultaneously determining permittivity andpermeability [20]. The focus of this paper is the development of sucha system.

In order to fully characterize a planar layer of material, theoreticalexpressions for the reflection and transmission coefficients must befound. Figure 1 shows the structure analyzed in this paper. Using

x

x

y

z

y

, 0a

x

PEC Flange

,a x b y

z

,ε µ

10TE

TEmnTMmn

1M2M

10TE

TEmnTMmn

0z =

y

Waveguide

Region

Waveguide

Region

Parallel-Plate

Region

1n

2n

x∆

z d=

∆ ∆

∆∆

( )

( )

+ +

( )

( )

ε µ,

Figure 1. Side (top) and top (bottom) views of the flanged waveguidesmeasurement geometry.

Progress In Electromagnetics Research, PIER 79, 2008 369

Love’s Equivalence Principle, the waveguide apertures in Figure 1can be replaced with equivalent magnetic currents emanating in aparallel-plate environment [21, 22]. Making use of these currents andthe parallel-plate Green’s function, one can find an expression for themagnetic field in the parallel-plate region. Enforcing the continuityof tangential fields at the waveguide apertures produces a system ofcoupled MFIEs, which when solved via the MoM [22], yields theoreticalvalues for the reflection and transmission coefficients. These theoreticalvalues can then be compared to measured data and the systemoptimized via nonlinear least squares to yield complex permittivityand permeability values.

In this paper, a thorough analysis of Figure 1 will be presented,including the development of the system of coupled MFIEs andspectral domain integration via complex plane analysis. Lastly,measurement results of two magnetic materials comparing the flangedwaveguides technique to standard waveguide methods [23, 24] will beshown. Included in these measurement results will be the technique’ssensitivity to probe misalignment and sample thickness.

2. DERIVATION AND SOLUTION OF COUPLEDMAGNETIC FIELD INTEGRAL EQUATIONS

Material characterization requires one to solve forward and inverseproblems. The forward problem involves finding theoretical expressionsfor the reflection and transmission coefficients; whereas, the inverseproblem involves finding the complex permittivity and permeability,by some means, using the results of the forward problem. In very rareinstances closed-form expressions can be found that directly relate theforward and inverse problems, i.e., relate complex permittivity andpermeability to reflection and transmission coefficients [23, 24]. Inmost cases (including the one presented in this paper), the inverseproblem is solved using numerical techniques such as Newton’s methodor nonlinear least squares. The purpose of the next two sections willbe to solve the forward problem.

Consider the measurement geometry shown in Figure 1. In orderto find theoretical expressions for the reflection, Γ, and transmission,T, coefficients, one must find expressions for the electric and magneticfields in the waveguide and parallel-plate regions. The transverse fieldsin the waveguide regions take the form of transverse electric (TE) and


transverse magnetic (TM) rectangular waveguide modes:⇀

Et= ⇀

e10e−γ10z +

∑m

∑n

Γmn⇀emne

γmnz

⇀

Ht=

⇀

h10e−γ10z −

∑m

∑n

Γmn

⇀

hmneγmnz

(1)

when z < 0 and⇀

Et=

∑m

∑n

Tmn⇀emne

−γmn(z−d)

⇀

Ht=

∑m

∑n

Tmn

⇀

hmne−γmn(z−d)

(2)

when z > d [21]. In (1) and (2), ⇀emn and

⇀

hmn are the transverse modaldistributions of the electric and magnetic field, respectively; they areshown in the appendix. If the waveguides in Figure 1 are perfectlyaligned, only higher order modes with symmetric distributions aboutthe center of the waveguides (m = 1, 3, 5, . . . ; n = 0, 2, 4, . . . ) will begenerated. This is due to the symmetry of the TE10 incident field andthe structure. On the other hand, if the waveguides in Figure 1 aremisaligned, all higher order modes, with the exception of TE0n modes,will be generated.

The transverse fields in the parallel-plate region of Figure 1 can befound by replacing the waveguide apertures with equivalent magneticcurrents,

⇀

M1 and⇀

M2 [21, 22]. Using the parallel-plate Green’s functionand the electric vector potential, the transverse magnetic field can befound from the following expression

⇀

Ht=

1jωµε

(k2 + ∇t∇·

) ⇀

F (3)

where the electric vector potential,⇀

F , is

⇀

F =

b∫0

a∫0

↔G

(x, y, z|x′, y′, 0

)· ε

⇀

M1

(x′, y′

)dx′dy′

+

b+∆y∫∆y

a+∆x∫∆x

↔G

(x, y, z|x′, y′, d

)· ε

⇀

M2

(x′, y′

)dx′dy′, (4)

k = ω√µε is the material’s complex-valued wavenumber, and ∇t is

the transverse gradient operator. In (4),↔G is the dyadic, parallel-plate


Green’s function derived in [25, 26]:↔G = xGtx + yGty + zGnz

Gtn =

1(2π)2

∞∫−∞

∞∫−∞

Gtn

(ξ, η, z|z′

)ejξ(x−x′)ejη(y−y′)dξdη (5)

Gtn =

cosh p (d− |z − z′|) ± cosh p (d− (z + z′))2p sinh pd

where Gt and Gn represent the transverse and normal componentsof the Green’s function, respectively, and p =

√ξ2 + η2 − k2 is the

spectral domain, z-directed wavenumber [4].With expressions for the fields in the waveguide and parallel-plate

regions of Figure 1, a system of coupled MFIEs can be formed byenforcing the continuity of transverse magnetic fields at the waveguideapertures. Setting the transverse magnetic fields of (1) and (2) equalto (3) at z = 0 and z = d, respectively, results in the desired coupledMFIEs:

⇀

h10 −∑m

∑n

Γmn

⇀

hmn =1

jωµ

(k2 + ∇t∇·

)∫

s1

↔G

(z = 0|z′ = 0

)·

⇀

M1ds′ +

∫s2

↔G

(z = 0|z′ = d

)·

⇀

M2ds′

∑m

∑n

Tmn

⇀

hmn =1

jωµ

(k2 + ∇t∇·

)∫

s1

↔G

(z = d|z′ = 0

)·

⇀

M1ds′ +

∫s2

↔G

(z = d|z′ = d

)·

⇀

M2ds′

(6)

where s1 and s2 have been introduced to represent the limits ofintegration in (4).

Solving (6) for Γ and T can be accomplished via the MoM. Thefirst step in the MoM is to choose suitable basis functions to representthe unknown currents.

⇀

M1 and⇀

M2 in (6) are related to the transverseelectric fields at the waveguide apertures by the following expressions

⇀

M1 = −n1 ×⇀

Eap

(z = 0) = −z ×⇀

Eap

(z = 0)⇀

M2 = −n2 ×⇀

Eap

(z = d) = z ×⇀

Eap

(z = d). (7)

Since the transverse electric fields must be equal across each apertureinterface, it is logical to substitute the electric field expressions of (1)


and (2) into (7) resulting in the following expressions for⇀

M1 and⇀

M2

⇀

M1 = −z ×[

⇀e10 +

∑m

∑n

Γmn⇀emn

]

⇀

M2 = z ×[∑

m

∑n

Tmn⇀emn

] . (8)

Substituting (8) into (6) results in a system of 2 equations and 2Nunknowns. To make the system well defined, testing functions mustbe chosen (the last step in the MoM) and applied to (6). The choice oftesting functions is somewhat subjective; however, in this case, it makessense to test with

⇀

hmn to take advantage of the orthogonal propertiesof the waveguide modes. Application of the testing function

⇀

hmn andsubsequent integration over the guide cross section results in a 2N by2N matrix equation, Ax = b, where the matrix A is the impedancematrix, x is a vector containing the unknown modal reflection andtransmission coefficients, and b is a vector containing the incident fieldexcitation. Inspection of A reveals the following form

A1110,10 · · · A11

10,mn A1210,10 · · · A12

10,mn...

. . ....

.... . .

...A11

mn,10 · · · A11mn,mn A12

mn,10 · · · A12mn,mn

A2110,10 · · · A21

10,mn A2210,10 · · · A22

10,mn...

. . ....

.... . .

...A21

mn,10 · · · A21mn,mn A22

mn,10 · · · A22mn,mn

(9)

where the submatrix A11 represents a “self” term, i.e., the source andobserver are at z = 0. Likewise, A22 is also a “self” term — sourceand observer are at z = d. The off-diagonal submatrices, A12 and A21,are “coupling” terms. They describe how a source at z = d influencesthe fields at z = 0, A12, and vice versa, A21. Submatrices A12 andA21 are found to be equal as expected by reciprocity. It is also foundthat A11 and A22 are equal, regardless of waveguide misalignment, asanticipated by symmetry. The subscript of each submatrix elementdescribes how a certain waveguide mode couples into another. Forexample, the mn,mn elements describe how the mnth source modecouples into the mnth field mode.

In the next section, it is shown how complex plane integrationcan be utilized in evaluating the matrix elements in (9). By using thespectral domain representation of the parallel-plate Green’s function,


closed-form solutions to all spatial integrals in (6) and an infinite seriesrepresentation of one of the spectral integrals in (5) are obtainedwhich greatly accelerate convergence and enhance physical insight.For the sake of brevity, only the evaluation of the A12

m0,m0 elementsis demonstrated.

3. SPECTRAL DOMAIN INTEGRATION

In the previous section, the forward problem for the measurementgeometry in Figure 1 was introduced. Analysis ultimately led to asystem of coupled MFIEs for the modal reflection and transmissioncoefficients. Applying the MoM to the system of MFIEs resulted in amatrix equation, Ax = b, where A was the impedance matrix, x wasa vector containing the unknown modal reflection and transmissioncoefficients, and b was a vector containing the incident field. Thissection will complete the forward problem of Figure 1 by calculatingthe A12

m0,m0 elements of the impedance matrix. This class of elementsprovides an excellent example of the role waveguide misalignment playsin the integration. The remaining elements of the impedance matrixcan be evaluated in an analogous manner.

The impedance matrix elements, A12m0,m0, have the following form

A12m0,m0 =

mπ/a

jωµ (2π)2

∞∫−∞

∞∫−∞

(k2 − ξ2

)p sinh pd

fm0(ξ, η)gm0(ξ, η)dηdξ (10)

where

fm0(ξ, η) =

a∫0

sin(mπx

a

)ejξxdx

b∫0

ejηydy

gm0(ξ, η) =

a+∆x∫∆x

sinmπ

a

(x′ − ∆x

)e−jξx′

dx′b+∆y∫∆y

e−jηy′dy′

. (11)

The spatial integrals in (11) can be evaluated in closed-form and areshown in the appendix. Substituting the results of (11) into (10)


produces the following expression

A12m0,m0 =

mm2π3/a3

jωµ (2π)2∞∫

−∞

(k2 − ξ2

) (1 + (−1)m+1 e−jξa

) (1 + (−1)m+1 ejξa

)e−jξ∆x(

ξ2 − (mπ/a)2) (

ξ2 − (mπ/a)2)

∞∫−∞

e−jη∆y

p sinh pd

(1 − e−jηb

) (1 − ejηb

)η2

dηdξ

. (12)

Inspection of the η integrand in (12) yields a double pole at η = 0

and simple poles at η = ±√

k2 − ξ2 − ("π/d)2 (" = 0, 1, 2, . . . ). Thedouble pole at η = 0 comes from a combination of the TEm0 basisand TEm0 testing functions. If the y-directed waveguide misalignmentis zero (∆y = 0), this double pole becomes a simple pole since thestructure in Figure 1 becomes perfectly symmetric in y. The simple

poles at η = ±√

k2 − ξ2 − ("π/d)2 are a result of the Green’s functionand physically represent the natural modes of the parallel-plate region.Because the η integrand is even in p =

√ξ2 + η2 − k2, the branch

points, η = ±√

k2 − ξ2, and associated branch cut contributions areremovable [4]. This result is intuitive since the infinite parallel-platestructure in Figure 1 is closed and consequently non-radiating [4].Expanding the product in the η integrand produces the followingintegral

∞∫−∞

e−jη∆y

p sinh pd

(1 − e−jηb

) (1 − ejηb

)η2

dη =

∞∫−∞

1p sinh pd

2e−jη∆y − e−jηbe−jη∆y − ejηbe−jη∆y

η2dη

. (13)

To ensure convergence of (13), proper integration contours must bechosen. Letting η = ηre + jηim, reveals that the magnitude and sign ofthe y-directed waveguide misalignment (∆y) are extremely important.For instance, if ∆y > b, then lower half plane (LHP) closure is requiredto ensure (13) converges; however, if 0 < ∆y < b, then a combinationof LHP and upper half plane (UHP) closure is required. Since theformer represents an extreme misalignment case, certainly noticeableto the unaided eye, it will not be investigated. The remainder of this


section will deal with the latter case, 0 < ∆y < b, since it presentsa much more realistic misalignment scenario. Under the assumptionthat 0 < ∆y < b, (13) should be split into the following two integrals

reC

C

C

reC

imη

x

x

reη

x

xx

C η−

C η

0C+

0C−

+

8

8−

+

"

"

Figure 2. Complex η-plane (branch cuts are removable).

∞∫−∞

1p sinh pd

2e−jη∆y − e−jηbe−jη∆y − ejηbe−jη∆y

η2dη =

∞∫−∞

1p sinh pd

−ejηbe−jη∆y

η2dη +

∞∫−∞

1p sinh pd

2 − e−jηb

η2e−jη∆ydη

(14)

where the first integral in (14) requires UHP closure and the secondLHP closure. Figure 2 shows the complex η-plane complete with theappropriate integration contours. Using Cauchy’s Integral Theoremand Jordan’s Lemma,

∫C±∞

= 0, (14) becomes

∞∫−∞

1p sinh pd

−ejηbe−jη∆y

η2dη =

∫Cre

= −∫

C+0

−∞∑

�=0

∮C−η�∞∫

−∞

1p sinh pd

2 − e−jηb

η2e−jη∆ydη =

∫Cre

= −∫

C−0

−∞∑

�=0

∮C+η�

(15)

where C±0 are semi-circular contours around the double pole at η =0 [27]. By utilizing Cauchy’s Integral Formula and the following residue


formulas [27]

a−1 =1

(m− 1)!∂m−1

∂zm−1[(z − z0)

m f (z)]z=z0

a−1 =f1 (z0)f ′2 (z0)

, (16)

(13) becomes

∞∫−∞

e−jη∆y

p sinh pd

(1 − e−jηb

) (1 − ejηb

)η2

dη = 2π (b− ∆y)1

p0 sinh p0d

−j4πd

∞∑�=0

e−jη�∆y − e−jη�b cos η�∆y

η3� (−1)� (1 + δ�,0)

p0 =√

ξ2 − k2, η� =√

k2 − ξ2 − ("π/d)2, δ�,0 ={

1 " = 00 " �= 0

. (17)

Note that the parallel-plate mode poles η� become branch pointsin the ξ-plane resulting in irremovable branch cut contributions [4]. Itis for this reason that the ξ integral is calculated numerically usingreal-axis integration instead of Cauchy’s Integral Theorem. It is worthnoting that the integral can be simplified for numerical computationby taking advantage of the evenness of the integrand. It can be easilyshown that when m + 1 and m + 1 sum to an even number, (12)simplifies to

A12m0,m0 =

mm2π3/a3

jωµ2π2

∞∫0

(k2 − ξ2

) (1 + (−1)m+1 e−jξa

) (1 + (−1)m+1 ejξa

)cos ξ∆x(

ξ2 − (mπ/a)2) (

ξ2 − (mπ/a)2)

[2π (b− ∆y)

1p0 sinh p0d

− j4πd

∞∑�=0


η3� (−1)� (1 + δ�,0)

]dξ

.

(18)

Similarly, when m+ 1 and m+ 1 sum to an odd number, (12) reduces


to

A12m0,m0 =

−mm2π3/a3

ωµ2π2

∞∫0

(k2 − ξ2

) (1 + (−1)m+1 e−jξa

) (1 + (−1)m+1 ejξa

)sin ξ∆x(

ξ2 − (mπ/a)2) (

ξ2 − (mπ/a)2)

[2π (b− ∆y)

1p0 sinh p0d

− j4πd

∞∑�=0


η3� (−1)� (1 + δ�,0)

]dξ

.

(19)

Figure 3. 6′′ × 6′′ × 0.25′′ aluminum flanges attached to X-bandrectangular waveguides.

4. MEASUREMENT RESULTS

In this section, measurement results will be shown for thenondestructive technique derived in the previous sections. The resultswill be compared with those returned from traditional destructivewaveguide techniques [23, 24]. Figure 3 shows a picture of themeasurement apparatus. The flanges were constructed from aluminumand measure approximately 6′′ × 6′′ × 0.25′′. They were attachedwith screws to X-band rectangular waveguides. The apparatuswas calibrated using a thru-reflect-line (TRL) calibration [3]. Forthis experiment, two lossy, magnetic materials (ECCOSORB r©FGM-40 and FGM-125) were measured using an HP 8510C vectornetwork analyzer (VNA). The relative complex permittivity, εr, and


permeability, µr, values for FGM-40 and FGM-125 were found bysolving the following system of equations,∣∣∣Sthy

11 (εr, µr) − Smeas11

∣∣∣ ≤ δ

∣∣∣Sthy12 (εr, µr) − Smeas

12

∣∣∣ ≤ δ


21

∣∣∣ ≤ δ


22

∣∣∣ ≤ δ

Sthy11 = Sthy

22 = Γ10, Sthy12 = Sthy

21 = T10

, (20)

within some tolerance, δ, using nonlinear least squares. It should benoted that while the development shown in the previous sections isapplicable to any material, the technique, in practice, should only beused to extract the parameters of lossy materials. This consideration isdue to the finite truncation of the flanges in order to make the methodpractical. Dominant mode propagation in the parallel-plate region ofFigure 1 is cylindrical in nature with propagation constant k, i.e.,∣∣∣⇀

E∣∣∣ , ∣∣∣⇀

H∣∣∣ ∼ ∣∣∣H(2)

0 (kρ)∣∣∣ (21)

where H(2)0 is a zeroth-order Hankel function of the second kind. It is

possible to introduce measurement error from detecting waves reflectedfrom the edges of the flanges. Depending on the dynamic range ofthe VNA used, the cross sectional dimensions of the flanges shouldbe large enough or the material lossy enough to ensure that wavesemanating from the waveguide apertures are sufficiently decayed as toapproximate infinite flanges. Figure 4 shows the two-way attenuationof the materials measured in this experiment as well as that of acrylic.The dynamic range of the HP 8510C is listed at 90 dB [28]. It isclear from Figure 4 that 6′′ × 6′′ flanges are sufficient to characterizeboth magnetic materials with negligible two-way reflection error. This,however, is not the case for acrylic. In order to negate any possibletwo-way reflection error for a low loss material like acrylic, one needsflanges on the order of 200′′ × 200′′.

Complex permittivity and permeability results for FGM-40are shown in Figure 5. In the figure, the flanged waveguidestechnique, when 1, 5, and 10 modes are modeled, is compared tostandard waveguide extraction methods [23, 24]. The error bars


8 8.5 9 9.5 10 10.5 11 11.5 12 12.5-1400

-1200

-1000

-800

-600

-400

-200

0

Freq (GHz)

Atte

ntua

tion

(dB

)

HP 8510C Dynamic RangeFGM-125FGM-40Acrylic

Figure 4. Attenuation of fields propagating to and from the edges of6′′ × 6′′ PEC flanges filled with FGM-125, FGM-40, and acrylic.

on the waveguide method traces show the effect of a ±0.002′′material thickness uncertainty; whereas, the error bars on theflanged waveguides technique traces show the combined effect ofa ±0.002′′ material thickness uncertainty and a ±0.01′′ waveguidealignment uncertainty. Although both sources of error are combinedin the flanged waveguides technique traces, the material thicknessuncertainty drives the overall uncertainty. The technique is quiteresistant to uncertainty in waveguide alignment even to a largedegree. For instance, when accounting for 10 modes, a waveguidealignment uncertainty of ±0.01′′ accounts for an average uncertaintyin permittivity and permeability of approximately ±0.0257. On theother hand, a material thickness uncertainty of ±0.002′′ accounts for anaverage uncertainty in permittivity and permeability of approximately±0.6646, or 26 times greater than that of the waveguide alignmentuncertainty. Overall, the flanged waveguides technique does very wellwhen compared to results returned by traditional waveguide methods.Since FGM-40 is a relatively thin material (≈ 0.040′′), the benefit of


8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

0

5

10

15

20

Freq (GHz)

Rea

l Par

t & Im

ag P

art o

f εr

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5-3

-2

-1

0

1

2

3

Freq (GHz)

Rea

l Par

t & Im

ag P

art o

f µr

Waveguide1 Mode5 Modes10 Modes


Figure 5. Complex permittivity and permeability results for FGM-40 comparing the flanged waveguides technique to standard waveguidemethods. Real parts are positive and imaginary parts are negative.

including higher order modes is not readily apparent. The link betweensample thickness and higher order mode contribution will be discussedlater.

Figure 6 shows the complex permittivity and permeability resultsfor FGM-125. As in Figure 5, the flanged waveguides technique,when 1, 5, and 10 modes are modeled, is compared to standardwaveguide extraction methods [23, 24]. The error bars represent theexact same uncertainties as they did in Figure 5. As before, the flangedwaveguide technique does very well when compared to results returnedby traditional waveguide methods. Since FGM-125 (≈ 0.125′′) isroughly three times thicker than FGM-40, the benefit of includinghigher order modes is apparent.

The link between sample thickness and higher order modecontribution, evident when one compares Figures 5 and 6, deservesattention. Consider once again the structure in Figure 1. As thesample thickness, d, goes to zero, the waveguide apertures move


8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

0

1

2

3

4

5

6

7

Freq (GHz)

Rea

l Par

t & Im

ag P

art o

f εr

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5-1

-0. 5

0

0.5

Freq (GHz)

Rea

l Par

t & Im

ag P

art o

f µr



Figure 6. Complex permittivity and permeability results for FGM-125 comparing the flanged waveguides technique to standard waveguidemethods. Real parts are positive and imaginary parts are negative.

closer and closer together. At d = 0, the apertures merge and thestructure behaves like a traditional rectangular waveguide supportingthe dominant TE10 mode only. Keeping this simple physical picturein mind, it makes sense that the permittivity and permeability ofthicker materials would benefit more from the inclusion of higher ordermodes than that of thinner materials. This is shown experimentallyin Figure 7. The figure shows the magnitudes of the first 10 modalreflection coefficients at 10 GHz when the waveguides are misaligned.While both FGM-40 and FGM-125 share a nearly equal reflectedTE10 mode, the magnitudes of most other higher order modes aresignificantly greater for FGM-125 than for FGM-40. The mostsignificant higher order mode, as it pertains to permittivity andpermeability results, is the TE12/TM12 mode pair. The magnitude ofFGM-125’s TE12/TM12 mode pair is approximately three times thatof FGM-40’s. This is why there is a noticeable difference in FGM-125’spermittivity and permeability results when 10 modes are modeled ascompared to the same FGM-40 traces.


TE10 TE20 TE11 TM11 TE30 TE21 TM21 TE31 TM31 TE40 TE41 TM41 TE12 TM12 TE22 TM2210

-6

10-5

10-4

10-3

10-2

10-1

100 |∆x| = 0.01" |∆y| = 0.01"

Rectangular Waveguide Modes

Mag

nitu

de o

f Ref

lect

ion

Coe

ffici

ent,

Γ

FGM-40FGM-125

Figure 7. Magnitudes of the first 10 modal reflection coefficients forFGM-40 and FGM-125 at 10 GHz when the waveguides are misaligned.TE and TM modes of the same index are counted as one mode.

5. CONCLUSION

In this paper, a nondestructive technique for determining both complexpermittivity and permeability of magnetic sheet materials using twoflanged rectangular waveguides was presented. This paper began witha theoretical analysis of the measurement geometry in Figure 1. Itwas shown that replacing the waveguide apertures with equivalentmagnetic currents, in accordance with Love’s Equivalence Principle,resulted in a system of coupled MFIEs. This system was thentransformed into a matrix equation using the MoM. It was also shownthat the spatial integrals of the matrix elements can be evaluatedin closed-form and one of the spectral integrals can be representedas an infinite series using complex plane integration. This resultinganalysis accelerated convergence and led to enhanced physical insight.Lastly, this paper concluded with experimental results of two magneticmaterials comparing the technique to traditional destructive waveguide


methods [23, 24].The main contribution of the flanged waveguides technique is

that it allows for the simultaneous measurement of reflection andtransmission coefficients. Since reflection and transmission coefficientsare independent over all wavelengths and are the most efficient meansof determining both permittivity and permeability of a material,the ability to simultaneously measure these parameters is a majorenhancement over existing single probe methods [1–19]. Anotherbeneficial aspect of the flanged waveguides technique is in samplepreparation. Compared with traditional waveguide methods, whichrequire samples to be machined, fully-fill the waveguide cross section,and be normal to the guiding axis, the flanged waveguides techniqueonly requires that samples be lossy enough, or the flanges large enough,to ensure negligible two-way reflection error. The flanged waveguidestechnique’s most promising application would be in the RF materialsindustry. Its ability to nondestructively measure permittivity andpermeability of large sheets of magnetic material would be invaluableas a quality assurance tool.

APPENDIX A.

The rectangular waveguide transverse electric, ⇀emn, and magnetic,

⇀

hmn, field distributions are[⇀eTEmn

⇀eTMmn

]= x

[ky

kx

]cos kxx sin kyy + y

[−kx

ky

]sin kxx cos kyy

⇀

hTEmn

⇀

hTMmn

= z ×

⇀eTEmn

/ZTE

mn

⇀eTMmn

/ZTM

mn

(A1)

when z < 0 and[⇀eTEmn

⇀eTMmn

]= x

[ky

kx

]cos kx (x− ∆x) sin ky (y − ∆y)

+y

[−kx

ky

]sin kx (x− ∆x) cos ky (y − ∆y)

⇀

hTEmn

⇀

hTMmn

= z ×

⇀eTEmn

/ZTE

mn

⇀eTMmn

/ZTM

mn

(A2)


when z > d. In (A1) and (A2),

kx = mπa , ky = nπ

b , γmn =√

k2x + k2

y − k20

ZTEmn = jωµ0

γmn, ZTM

mn = γmn

jωε0

. (A3)

The integrals fm0(ξ, η) and gm0 (ξ, η) evaluate to

fm0 (ξ, η) =

a∫0

sin(mπx

a

)ejξxdx

b∫0

ejηydy

=

[−kxm

(1 + (−1)m+1ejξa

)ξ2 − k2

xm

] [j(1 − ejηb

)η

]

gm0 (ξ, η) =

a+∆x∫∆x

sinmπ

a

(x′ − ∆x

)e−jξx′

dx′b+∆y∫∆y

e−jηy′dy′

=

[−kxme−jξ∆x

(1 + (−1)m+1e−jξa

)ξ2 − k2

xm

] [−je−jη∆y

(1 − e−jηb

)η

]. (A4)

REFERENCES

1. Li, C. and K. Chen, “Determination of electromagnetic propertiesof materials using flanged open-ended coaxial probe — Full waveanalysis,” IEEE Trans. Instrum. Meas., Vol. 44, No. 1, 19–27, Feb.1995.

2. Chang, C., K. Chen, and J. Qian, “Nondestructive determinationof electromagnetic parameters of dielectric materials at X-bandfrequencies using a waveguide probe system,” IEEE Trans.Instrum. Meas., Vol. 46, No. 5, 1084–1092, Oct. 1997.

3. Chen, L. F., C. K. Ong, C. P. Neo, V. V. Varadan,and V. K. Varadan, Microwave Electronics Measurement andMaterials Characterization, John Wiley & Sons, New York, 2004.

4. Stewart, J. W. and M. J. Havrilla, “Electromagnetic characteriza-tion of a magnetic material using an open-ended waveguide probeand a rigorous full-wave multimode model,” Journal of Electro-magnetic Waves and Applications, Vol. 20, No. 14, 2037–2052,2006.

5. Olmi, R., R. Nesti, G. Pelosi, and C. Riminesi, “Improvementof the permittivity measurement by a 3D full-wave analysis of a


finite flanged coaxial probe,” Journal of Electromagnetic Wavesand Applications, Vol. 18, No. 2, 217–232, 2004.

6. Bois, K. J., A. D. Benally, and R. Zoughi, “Multimode solution forthe reflection properties of an open-ended rectangular waveguideradiating into a dielectric half-space: the forward and inverseproblems,” IEEE Trans. Instrum. Meas., Vol. 48, No. 6, 1131–1140, Dec. 1999.

7. Mautz, J. R. and R. F. Harrington, “Transmission from arectangular waveguide into half-space through a rectangularaperture,” IEEE Trans. Microwave Theory Tech., Vol. MTT-26,No. 1, 44–45, Jan. 1978.

8. Teodoridis, V., T. Sphicopoulos, and F. E. Gardiol, “Thereflection from an open-ended rectangular waveguide terminatedby a layered dielectric medium,” IEEE Trans. Microwave TheoryTech., Vol. MTT-33, No. 5, 359–366, May 1985.

9. Encinar, J. A. and J. M. Rebollar, “Convergence of numericalsolutions of open-ended waveguide by modal analysis and hybridmodal-spectral techniques,” IEEE Trans. Microwave TheoryTech., Vol. MTT-34, No. 7, 809–814, July 1986.

10. Popovic, D., et al., “Precision open-ended coaxial probes for invivo and ex vivo dielectric spectroscopy of biological tissues atmicrowave frequencies,” IEEE Trans. Microwave Theory Tech.,Vol. 53, No. 5, 1713–1722, May 2005.

11. Bao, J., S. Lu, and W. D. Hurt, “Complex dielectric measurementsand analysis of brain tissues in the radio and microwavefrequencies,” IEEE Trans. Microwave Theory Tech., Vol. 45,No. 10, 1730–1741, Oct. 1997.

12. Mazlumi, F., S. Sadeghi, and R. Moini, “Interaction of rectangularopen-ended waveguides with surface tilted long cracks in metals,”IEEE Trans. Instrum. Meas., Vol. 55, No. 6, 2191–2197, Dec. 2006.

13. Huber, C., H. Abiri, S. I. Ganchev, and R. Zoughi, “Modelingof surface hairline-crack detection in metals under coatings usingan open-ended rectangular waveguide,” IEEE Trans. MicrowaveTheory Tech., Vol. 45, No. 11, 2049–2057, Nov. 1997.

14. Yeh, C. and R. Zoughi, “A novel microwave method for detectionof long surface cracks in metals,” IEEE Trans. Instrum. Meas.,Vol. 43, No. 5, 719–725, Oct. 1994.

15. Baker-Jarvis, J., M. D. Janezic, P. D. Domich, and R. G. Geyer,“Analysis of an open-ended coaxial probe with lift-off fornondestructive testing,” IEEE Trans. Instrum. Meas., Vol. 43,No. 5, 711–718, Oct. 1994.


16. Wang, S., M. Niu, and D. Xu, “A frequency-varying methodfor simultaneous measurement of complex permittivity andpermeability with an open-ended coaxial probe,” IEEE Trans.Microwave Theory Tech., Vol. 46, No. 12, 2145–2147, Dec. 1998.

17. Maode, N., S. Yong, Y. Jinkui, F. Chenpeng, and X. Deming,“An improved open-ended waveguide measurement techniqueon parameters εr and µr of high-loss materials,” IEEE Trans.Instrum. Meas., Vol. 47, No. 2, 476–481, Apr. 1999.

18. Chen, C., Z. Ma, T. Anada, and J. Hsu, “Further study on two-thickness-method for simultaneous measurement of complex EMparameters based on open-ended coaxial probe,” 2005 EuropeanMicrowave Conference, Paris, France, Oct. 4–6, 2005.

19. Tantot, O., M. Chatard-Moulin, and P. Guillon, “Measurementof complex permittivity and permeability and thickness ofmultilayered medium by an open-ended waveguide method,” IEEETrans. Instrum. Meas., Vol. 46, No. 2, 519–522, Apr. 1997.

20. Hyde, M. W. and M. J. Havrilla, “Measurement of complexpermittivity and permeability using two flanged rectangularwaveguides,” 2007 International Microwave Symposium, 531–534,Honolulu, HI., USA, June 3–8, 2007.

21. Collin, R. E., Field Theory of Guided Waves, 2nd edition, IEEEPress, New York, 1991.

22. Peterson, A. F., S. L. Ray, and R. Mittra, Computational Methodsfor Electromagnetics, IEEE Press, New York, 1998.

23. Weir, W. B., “Automatic measurement of complex dielectricconstant and permeability at microwave frequencies,” Proc. IEEE,Vol. 62, No. 1, 33–36, Jan. 1974.

24. Nicolson, A. M. and G. F. Ross, “Measurement of the intrinsicproperties of materials by time-domain techniques,” IEEE Trans.Instrum. Meas., Vol. IM-19, No. 4, 377–382, Nov. 1970.

25. Hanson, G. W. and A. B. Yakovlev, Operator Theory forElectromagnetics: An Introduction, Springer-Verlag, New York,2001.

26. Hanson, G. W., A. I. Nosich, and E. M. Kartchevski, “Green’sfunction expansions in dyadic root functions for shielded layeredwaveguides,” Progress In Electromagnetics Research, PIER 39,61–91, 2003.

27. Arfken, G. B. and H. J. Weber, Mathematical Methods forPhysicists, 5th ed., Harcourt/Academic Press, New York, 2001.

28. Agilent 8510C Network Analyzer Data Sheet, Agilent Technolo-gies, 2000.

anondestructivetechniquefor ... · waveguide probes,whether coaxial,rectangular,or circular,have...

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