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United States Department of Commerce

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National Institute of Standards and Technology

N/ST Technical Note 1384

5753

0.1384

1996

Dielectric Measurements Using

a Reentrant Cavity:

Mode-Matching Analysis

James Baker-Jarvis

Bill F. Riddle

L

(

NIST Technical Note 1384

Dielectric IVIeasurements Using

a Reentrant Cavity:

IVIode-IVIatching Analysis

James Baker-Jarvis

Bill F. Riddle

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

November 1996

.^<^

'—^—7 * >;ss&:

*^4TES O^

U.S. DEPARTMENT OF COMMERCE, Michael Kantor, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1384, 16 pages (November 1996)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1996

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
Dielectric Measurements Using a Reentrant Cavity:

Mode-Matching Analysis

James Baker-Jarvis and Bill F. Riddle

Electromagnetic Fields Division, National Institute of Standards and Technology, Boulder, CO

80303-3328

Abstract

The coaxial reentrant cavity dielectric measurement technique is examined. A full-mode

model for the cavity is developed and solved numerically. Analytical expressions for wall losses are presented. The filling factor due to a partially-filled cavity is discussed. Dielectric results are presented and compare very closely to previous round-robin results.

Key words: cavity; coaxial; dielectric constant; loss factor; microwave measurements; per- meability measurement; permittivity measurement; reentrant; resonant

1. Introduction

In this report we develop the theory underlying the reentrant-cavity dielectric measurement technique. A full-mode model for the cavity is developed and solved numerically. Analytical expressions for reentrant- cavity characterization are presented.

Many radio-frequency applications require an accurate value of the component of the permittivity normal to the plane of the material. The reentrant cavity is unique since it is possible

to accurately measure materials at frequencies from 100 MHz to 1 GHz with axial electric field

[1, 2]. The reentrant cavity consists of a coaxial line or other transmission line with end walls,

and a gap in the inner electrode where a sample is positioned. If the gap is at either end of the

cavity, then the system is a singly reentrant cavity; otherwise it is a doubly reentrant cavity. The

doubly reentrant cavity has a stronger electric field in the gap than the singly reentrant cavity and
M 2b- H

1 2

2a

.3

2c

Z = dn

z = di

z =

z = -de

Figure 1: The reentrant cavity with regions 1, 2, and 3 denoted. The coaxial line inner and outer

diameters 2a and 2b, and sample diameter 2c are indicated in the figure.

is more useful for permittivity determination. Applications include measurements of substrates,

circuit boards, printed-wiring boards, and bulk materials.

Two approaches have been used for modeling the reentrant cavity. The first is a lumped-circuit

approximation [1,2]. The other approach, which we develop in detail, is a rigorous mode-matching

technique [3-5]. Our approach differs from previous work in that we develop the full mode-matched

solution for the doubly reentrant cavity and present the details of the calculations.

2. Mode-Matching Theory

The reentrant cavity method can be used to measure either permittivity or permeability. Per- meability is most accurately measured by placing the sample in the strong magnetic field near an end plate. Permittivity is most accurately measured in a region of strong electric field, such

as in a gap in the center conductor away from the end plates. We will restrict our analysis to permittivity determination; however, the equations are valid for arbitrary permeability. The material under test is assumed to have permittivity of e* = [e'^ - j^rj^o — (-*^o and permeability

/^s — [^^'r ~ JfJ'r]l^o = /^*/^o, where €o and //q are the permittivity and permeability of vacuum. The
dimensions of the cavity are shown in figure 1. The coaxial sections have inner and outer radii a and 6, respectively. The sample has radius c, which may be smaller than the inner conductor

radius. The sample thickness is d\. In the mode-matching technique the frequency

LJ = Ur + JLOr/2Q, ( 1)

where u>r is the measured resonant frequency and Q is the quality factor is complex.

We assume that only TMom modes are excited in the cavity. The TEM mode is excluded

because of the gap in the center conductor. The fields can be derived from either H^ or E^ in all

regions. The magnetic field satisfies the vector Helmholtz equation

V'^H + PH = 0. (2)

Symmetry requires that only an azimuthal magnetic field is excited. H^ satisfies

in regions where i = 1,2, or 3, kf = u>'^e*-fi*Jcl, and c^ is the speed of light relative to vacuum. The electric field components are Ez and Ep.

The boundary conditions on the fields are

Ez{a,z) = 0, {d2> z > di, 0> z > -ds),

E,{b,z) = 0, id2>Z>-d3),

Epip,d2) = Ep{p,-d3) = 0, ia<p<b),

Ep{p,di) = Ep{p,0) = 0, (0</9<a),

EAa^+\z) = E,{a^-\z), {0 < z < d^),

H^{c^+\z) = H^{c^-\z), (0<^<rfi),

/^^(a(+\^) = H^{a^-\z), (0<z<rfi),

E,{c(^\z) = E,{c^-\z), {0<z<di). (4)

The boundary-value problem defined in eqs (3) and (4) can be shown to be unique.

2.1. Fields

In this section we develop expressions for TMon field structure in the reentrant cavity. The electric

field has components Ep and Ez, and the magnetic field has component H^. The general solution can be written in the form

H4>(i)ip, Z) = Y1 ^nU)K(t)ip)fn(t){-)- (5)
The eigenfunctions satisfy

^^2 - KUz). (6)

Since we require Ep to vanish ^X z — —d-^ and z = (^2 we must have

f(. = .,) =f(.= -.3) = 0; (7)

therefore in region 1

!n{\){z) = cos (((/2 - z)ni^jL), (8)

where X = c?2 + c?3 and n = 0,1,2... In regions 2 and 3, the eigenfunctions satisfy eq (6) with

the same boundary conditions, but on the ends of the inner conductor instead of on the cavity

endplates. In regions 2 and 3,

/n(2)(^) = cos(n7rz/rfi). (9)

The magnetic field in region 1 is

-^<^(i)(p)^) =

m=o Jkl - {rmr/Ly

X cos{(d2 - z)m7r/L). (10)

Ji, ^h^ Jqi and A^o are Bessel functions of complex argument. Equation (10) is written in a form

to be consistent with previous publications [3]. The other field components can be generated by

jue* p dp

These are

Ep{i){p,z)

= E Ami^)-T-^^^^^^^ ^'^^^ A' - i^^/Lyp) + QxmN,{yJkl - (m;r/X)V)]

m=i yjk\ - {rmr/Ly

xsin{(d2 - z)m-K/L), (13)

Ez{i){p,z)

E A^^,po{^Jkl-{m7:/LYp) + Q^mNoiy/kj - (m7r/X)V)]

m=0

X cos {mTr{d2 - z)/L). (14)
The condition that Ep vanishes on the outer conductor requires

JoiJkj - {mn/Lm

Q\m =

/

• (15)

Noi^kl-^m-KlLfh)

In region 2, the fields are

E,{2){p,z) = -Y^ Am{2)[Mykj - {rmr/difp)

m=0

+Q2mNo{^Jkl - (mx/diyp)] cos {rmrz/di), (16)

E,i2){p.z) = f; A^^,)

^^^/^^^ [M^kl-imn/d,rp)

m=i Jkl - {mi: jdiY

-\-Q2mNiiyJkj - {rmr/diyp)]sm{rmrz/di), (17)

HmiP,^) - - E ^m(2) . ^""'^ [M^kl-imi^ld.fp)

m=0 Jk\ - {Vfl-K

I

diY

+<52miVi(^A;2 _ (m7r/di)2p)]cos(m7r^/di). (18)

In region 3, where the sample resides, the fields are

OO Ez(3){p,^) = - XI ^mi3)MyH - {rmr/difp)cos{Tmrz/di), (19)

m=0

E.isM^) = E A^(3)

^""""^^^-^ M^kl-{mn/d,yp)sm{m7rz/d,), (20)

m=i Y^l - {im: IdiY

H4>(3){p,z)

OO • # = - E -^m(3) / —^^

Ji{^kl-[mi:ldrYp)cos{mivzld^). (21)

m=o \lk\ - {mi: IdiY
2.2. Matching between Regions

If we equate expansions for i/^ between regions 1 and 2, at p = a in eqs (10) and (18), multiply

each side by an axial eigenfunction, and integrate over [0,(ii], we obtain the following system of

equations

^(1 + ^no)An(2)^

'^ [JiiJkl - {nir/d.ya) + Q2nNiiJkj - {nx/d^)^a)] (22)

°° €* I I = ^ A^(i)^^,^= ' [M\lkl - (m7r/i)2a) + Q,mN,{^kj - (mir/D^a)]. (23)

m=o Jkj - (nnr/Ly

Equation 23 may be rearranged and expressed in vector form

A2 = UAi, (24)

where

Bmn ., ;* .^jMJki-{m7T/L)^a) + Q^mN^iJkl-imiTlLfa)]

T _ y/ K^-^m-Kjl^) V V /Ol^^ -LiSmn — ; /

/ —

•

1^3;

f(1 + ^no)^j=^^j^[.h{^Jkl - [ni^ld.fa) + Q2nN,{^kl - {ni^ld^fa)]

Similarly, if we equate eqs (14) and (16) between regions 1 and 2, multiply each side by an axial

eigenfunction, and integrate over [—d^,d2\, we obtain

^(1 + <5no)A„(i)[Jo(V'^^2-(n7r/i)2a) + Q^^No^^Jkl - [ni: I Lfa)]

oo = Y. A^^2)Bmn[MyJkl-[mixld^fa) + Q2mNo{yJkl - [mir ld^)^a% (26)

771=0

or Ai - L6-42, (27)

where

^6mn —

Bmn[Jo{\Jkl - (m7r/c/i)2a) + Q2mNo{yJkl - (m7r/di)^a)]

f(l + 6no)[JoiyJki-{niT/Lya) + QmMy/kj - {nir/ L)^a)]

(28)

Requiring this integration to be over d\ < z < d2 and — ^3 < ^ < incorporates the boundary

condition £"2 = on the inner conductor.
The coefficients Bmn can be calculated analytically. For n = m, Bmn = ^i , otherwise

Bmn = / COS (nirz/di) COS {Tmr{d2 — z)/L)dz

Jo _ rfiZsin(7r(^-

^

+ n)) rfil sin (7r(^ - ^ - n))

27r(rfim + Zn) 27r(dim - Ln)

-^^1 T^ V-r^ ^^^ (7rd2m/i). (29)

7r(aim — Ln)[d\m + Ln)

Since the modal fields decouple between regions 2 and 3 at p = c, we can write

Am(2)[M^Jkl-{mi:ld,fc) + g2miVo(y^i-(m7rM)2c)] = A^ojiofT^l^^Tw^c), (30)

and therefore,

e-^^kl-irmr/di)^ r-— —

^m(3) / =Ji{\/H - [mn/diyc)

€*^Jkl -{rmr/diy = Am(2)[M\/^-{m7r/d,)^c) + Q2mN,(^kl-{m7r/d,)^c)]. (31)

We can determine the coefficients Q2m from eqs (30) and (31)

ZJoi^Jkj - {m^/d,yc) - M^kj - (m7r/di)2c)

Q2m - / / , (32)

Niiyjkl - (m7r/di)2c) - ZNoiyJkl - {mir/dij^c)

where _ e*^k', - {mn/d.YM^kl - (m7r/di)2c)] - / / —

{^^)

^2\/H ~ irmr/di)'^.Jo{Jkl - {mTC / di)'^ c)

Equations (24) and (27) can be combined to form

A2 = L5L6I2. (34)

In order to have a nontrivial solution to this homogeneous system of equations, the determinant

of the coefficients

det(I-L5L6) = 0, (35)

must vanish. Equation (35) yields 63, given complex resonant frequency and quality factor. There

are many roots to the equation, but use of an accurate initial guess allows convergence to the

correct solution. Alternatively, given £3, the complex resonant frequency can be determined.

The matrices L5 and Lg have infinite dimension. In calculations, the matrices must be truncated. Numerical calculation shows that in regions 2 and 3, only 2 to 4 eigenfunctions are required
since the fields are strongly evanescent. This can be seen by examination of the argument of the

Bessel function in region 3, Jk^ — {mir/diy. Since di is usually only a few millimeters, this

quantity has a large imaginary component which produces a strong damping. In region 1, since

the dimensions are larger, there is less damping, and more eigenfunctions are required to obtain

a converged solution. In region 1, 20 to 90 eigenfunctions are required for accurate calculation of

permittivity.

3. Cavity Losses

In this section we summarize the theory for calculation of dielectric losses in the reentrant cavity

[4]. Losses in cavities originate from imperfectly conducting walls, sample loss, and coupling port

losses. The analysis of cavity loss is more complicated than the analysis of the empty cavity when

the cavity is partially filled with material.

In the time domain, the electric fields in a lossy cavity are of the form

E^ exp {jujt - at)Eo, (36)

where a is an attenuation factor due to dielectric, coupling port, and cavity wall losses. The stored electric and magnetic energy in a cavity is

Use

^sm

^-Re I D-EdV, (37)

jRe f B -HdV. (38)

4

Since the stored electric and magnetic energies are equal at resonance,

f/,e = / '-^\E\'dV = J

^\H\'dV = Usm. (39)

Therefore the total stored energy in a cavity is

Us = 2Use = Wsm. (40)

The total stored energy dissipated in a cavity evolves in time as

or

Usit) = Usoex-p{-2at). (42)
In the forthcoming analysis it is necessary to study both the cases where the cavity is completely filled with material and where the cavity is partially filled. The power dissipated in a material that completely fills a cavity is

Wd ^l^\E\'dV^j^\H\'dV, (43)

where a^ is the effective conductivity of the dielectric, a^ = ujc" + a, and a is defined by J = aE.

The quality factor due to a dielectric that completely fills the cavity is used to define the loss tangent of the material under test

ijj (average stored energy) i^Usf (^^^o 1 /, ,x power loss per cycle W^f Cd tan S

'

where the subscript / refers to filled cavity. Generally, the materials under test will only partially

fill the cavity. Since the dielectric does not completely fill the cavity, a correction must be applied. In what follows we outline the procedure for calculating the correction due to partial dielectric

filling. The correction is obtained by considering losses in both the air-filled and dielectric-filled

sections of the cavity separately.

The power dissipated in walls and end plates is given by integration of the Poynting vector over the surfaces [5] W^ = ^l\H^\'dS, (45)

where the surface resistance is Rs = a/cJ/772^ — Ifdoyj, a^ is the conductivity of the walls, and

d is the effective penetration depth into the walls.

The total quality factor in the cavity is the ratio of energy stored in cavity and dielectric to power dissipated in walls, end plates, dielectric, and coupling ports

where Usd is the energy stored in the dielectric, Uga is energy stored in air section of the cavity

that contains a sample, and Wcp is the loss due to coupling ports. The coupling-port losses can be obtained by circle-fit procedures [6]. The quality factor without sample loss is Q[

'' W T '' Cp

In the nonmagnetic case, if we write Wd = ptanS, where p is a parameter identified after

performing the integration in eq (43) over the sample volume, then the net loss tangent can be
calculated using eqs (46) and (47) given expressions for Ug-, Usa-, ^w, ^nd Wcp and eq (47). It can be put in the form

Q[ is not directly measurable, however, a theoretical value for Q[ can be calculated from eq (47),

if the wall penetration depth is known. An estimate of wall penetration depth can be calculated

from quality-factor measurement on the empty cavity:

d = „ ';V' ,•:,

.

. (50)

therefore, the penetration depth is

The calculated penetration depth can then be used in eq (47) for use in eq (48).

3.1. Wall Losses

Equation (45) yields the wall losses given the surface conductivity. The integrals can be calculated

analytically on the outer walls, end plates, and inner conductor. The radial integration is slightly

more complicated since the radial functions are not orthogonal.

The total wall loss is P^=^^ j\H^\'dS

= irRslYl BlJA^^,) -—^'"'' IM^kj - {mi^lLYb) + Qi^7Vi(v/^f-(m7r/Z)26)]|

m=o Jk\ - [m-KJ LY

oo • ,

m-o Jkj - {tti-kI ly

/ I Y, A^(i)^=4=i=[.A( J^-2 - (m7r/I)2a) + Q,mN,{Jkl - {mn/Lya)]

Jo ^0 ^k1-{mixlLY

X cos ((^2 — z)rm: I L)^dz

+2 ' / Pl E ^rnix) I ^""'^ =\h{^k\ - [mi^ilfp) + Q,mN,{^kl - [mi: I Lfp)]\^ dp

Jo m=o,2,.. yJk^-imTr/L)^

+2/%| J2 -4^(2) / -^"^'^ [M^kl-{m7r/L)^p) + Q2mN,{^kl-{m7r/Lrp)]\\lp

•^'^ m=o,2,.. ^Jk] - {rmr/LY

+2/%| ^

^^3) / •^'^'' JA^kl-{m7:/L)^p)]\'dp]. (51)

10
The loss characteristics of various metals used in cavity construction are given in table 1.

Table 1: Conductivities of metals

Material Conductivity, a Skin Depth, d (10 GHz)

(xlO^S/m) (xlO-^)m

6.60

8.26

6.42

7.85

12.70

Cu 5.80

Al 3.72

Ag 6.17

Au 4.10

70-30 brass 1.57

4. Measurements

Permittivity measurements were performed on a number of well-characterized materials. The

results are given in table 2. Uncertainties were calculated by comparison with previous measure- ments of these materials. Numerical experimentation was performed on the convergence of the

solution as a function of the number of TMom modes used in the expansions. 20 to 90 axial

eigenfunctions in region 1 and 2 eigenfunctions in regions 2 and 3 were required for these measurements. Higher permittivity requires more expansion functions in region 3. Uncertainties are produced from a number of sources and will be analyzed in more detail in a future publication.

Uncertainties for the measurements were calculated by a sensitivity analysis of sample thickness,

resonant frequency, and quality factor. The small air gap between sample and electrodes produces

a slightly lower measured permittivity. Corrections for the gap are summarized in [7]. The gap

effects on the real part of the permittivity can be mitigated by electroplating sample surfaces with

copper by applying conducting pastes, or by using tin foil with petroleum jelly as adhesive [8]. In the measurement of loss factor, gap mitigation methods will produce extraneous loss. Therefore,

Table 2: Dielectric measurements

Material Frequency Calculated (f'^, c'/)

Crosslinked polystyrene 915 MHz (2.524± 0.020, 0.003± 0.002)

Polytetrafluoroethylene 905 MHz (2.072± 0.020, 0.0025± 0.002)

1723 Glass 840 MHz (6.253± 0.045 ,0.0217± 0.003)

11
the loss measurement should be performed before the permittivity measurement.

5. Conclusions

The goal of this report was the development of the theory for the mode-matching solution for resonance in a reentrant cavity and the presentation of numerical results. The mode matching

results in a determinantal equation. The roots of this equation yield the permittivity of the

sample. A full-mode model for the cavity was developed and solved numerically. The solution

allows accurate measurement of permittivity. Analytical expressions for wall losses were presented.

The theoretical approach to characterization of the reentrant cavity was presented.

6. Acknowledgments

We thank Michael D. Janezic for help with software and David Hill for discussions.

References

[1] R.D. Harrington, R.C. Powell, and P.H. Haas, "A re-entrant cavity for measurement of complex permeability in the very-high frequency region," J. Res. Nat. Bur. Stand. (U.S.), vol. 56, no. 3, pp. 129-133, 1956.

[2] C.N. Works, T.W. Dakin, and F.W. Boggs, "A resonant cavity method for measuring dielectric

properties at ultra-high frequencies," AIEE Trans., vol. 63, pp. 1092-1098, 1944.

[3] W. Xi, W.R. Tinga, W.A.G. Voss, and B.Q. Tian, "New results for coaxial re-entrant cavity

with partially dielectric filled gap," IEEE Trans. Microwave Theory Tech., vol. MTT-40,

pp. 747-753, April 1992.

[4] R..]. Cook, Microwave cavity methods, pp. 12-27. IPC Science and Technology Press, 1973.

[5] J.D. Jackson, Classical Electrodynamics. New York: John Wiley and Sons, 1974.

[6] E.J. Vanzura, R.G. Geyer, and M.D. Janezic, "The NIST 60-millimeter diameter cylindrical

cavity resonator: Performance evaluation for permittivity measurements," Natl. Inst. Stand.

TechnoL, Tech. Note 1354, August 1993.

[7] J. Baker- Jarvis, "Transmission/reflection and short-circuit line permittivity measurements,"

Natl. Inst. Stand. TechnoL, Tech. Note 1341, July 1990.

12
[8] J.V.L. Parry, "Measurement of permittivity and power factor of dielectrics at frequencies from

300 to 600 Mc/s," Proc. IRE, vol. 98, pp. 303-311, 1951.

13

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