ISEETRANSACTIONSONMICROWAVSTHSORYANDTSCHNIQUSS,VOL.MIT-28, NO.3, MARCH1980 225

High-Accuracy Wide-Range MeasurementMethod for Determination of Complex

Permittivity in Reentrant Cavity:Part A —Theoretical Analysis of the Method

ANDRZEJ IQCZKOWSKI AND ANDRZEJ MILEWSKI

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I. INTRODUCTION

,HE REENTRANT cavity has been used for theT measurement of t by many investigators, including

the authors [ 1]–[8], l%e method should be well proven

and reliable. Unfortunately, in the majority of available

publications, the accuracy with which the electrical prop-

erties of the material under test affect the resonator char-

acteristics is far from satisfactory. Usually, the error is

caused by inadequate accuracy of the approximation of

the electromagnetic field distribution within the sample

cavity.

Each of the simplifications can be a source of consider-

able error in the calculation of the equivalent circuit of

the resonator. The errors of resonant frequency can ex-

ceed 50 percent (as is shown by Uenakada [7], [8]).

IL ANALYSIS OF mm RIWNTRANT CAVITY

A few publications [3], [5] give evidence of attempts to

find an accurate field distribution in a reentrant cavity

with a dielectric sample. Both papers apply the following

assumptions.

1) The dielectric sample has a diameter equal to the

diameter of the resonator metal cores.

2) Only the condition of fundamental resonance (this of

the lowest resonant frequency) is considered.

However, the applied assumptions impose the following

limitations.

1) Since the amount of resonator off-tuning increases

with sample perrnittivity, the measurable values of { are in

practice limited to 20 for a sample diameter equal to the

core diameter [3], [5].

2) The high degree of resonator “filling” by the material

under the test in the area where the electric field is

Manuscript received August 14, 1978; revised October 3, 1979.The authors are with Instytut Tecbnologii Elektronowej, Politechnika

Warszawskaj ul. Koszykowa 75JJ0-662 Warsaw, Poland.

considerably high leads to a limitation of measurable tand

values to 10-2 maximum.

The results from the above show that to increase the

practical range of measurable c’ and tand values, samples

with a diameter smaller than that of the core should be

used for measurements,

Most encouraging for the solution of this system ap-

pears to be the “subarea bounding” method, as used for

some particular cases by Karpowa [3] and Milewslci [5].

The geometry of the resonator as considered in the

paper is shown Fig. 1.

When deriving the resonance condition, use has been

made of the following principles.

1) The fields and the resonator shape feature axial

symmetry.

2) Electric and magnetic fields are continuous so that

the tangent component of electric field at the metal walls

of the resonator is equal to zero.

3) Resonant frequency ~\27r is the complex value be-

cause of the dielectric lQSSin the sample.

Finding the ~ and @ fields in subareas A, B, and C

(Fig. 1) and using the assumptions presented above, the

resonant condition equation is obtained as follows:

det[~,,~]=O, forl,m=O,l, ”””, ce.< (1)

The terms of the matrix [&J are given by the formulas

i,,m=W ~:oanl?:[g(n,m)+g(n, -m)]

“ [ g(wl)+g(% -0] +&,m&where

(C?l,m= 1’ forl=m

o, for I+m

(

1/2, form=Oam=

1, form>O

0018-9480/80/0300-0225$00.75 01980 IEEE

226 rnm rRANSACI’IONS ON MICROWAVE‘msoRY ANDTscrnaQms, VOL.MTr-28,No. 3, MARCH1980

Fig. 1. Geometry of the resonator analyzed in the paper.

A

The coefficients B,, ~ and fi2, ~ meet the following system

of equations:

where

‘:=[(:)’-(3’]”2‘:=[(%(%’]1’2%=[&)’-(y)’]”2.

The resonance condition [1] has the following p~op~rties,

1) All resonances with field components (E,, E=, Hv) of

axial symmetry are taken into consideration.

2) Both real and imaginary parts of complex dielectric

parnittivity are connected with both components of com-

plex resonance frequency.

3) The diameter of the sample is optional, so that the

ranges of measurable values (e’) and (tan 8) are much

wider.

111. OPTIMAL ITERATIVE SOLUTION

To investigate the permittivity { it is necessary to com-

pute the value of the determinant as given by [1]. Figs. 2

and 3 shown the section shape of the surface determined

by the equation:

Fig. 2 shows the sections of surface (2) in planes which are

parallel to the e’ axis (Fig. 2(a)) and to the c“ axis (Fig.2(b)) for four determinant sizes: 1 X 1, 2X 2, 3 X 3, and

4x 4. The planes pass through the point which represents

the solution of (l). Fig. 3 shows the sections in these

planes, parallel to ~ for the determinant size of 3 x 3. It is

seen from Fig. 2 that the nonlinearity of the function (2)

increases with rising degree of the determinant. It should

be noted that for a single-term determinant (1= m = O),

the function is approximately linear and the solutions of

(1) are very close to each other, irrespective of the number

Fig. 2. Sections of surface given by fun@ion (2) by planes: a/c’-cons$

b/c’ = constant through the point det[A] = O. Numbers of curve meansthe degree of the determinant. Axis of ordinates is reduced to the

function value at point 2=5.5 –jO.04. Dimensions of resonator: ~ = ij=7.6 mm, T2=25.5 mm, L=20.O mm, d=5.1 mm, and L, =0.

b

1

0

-1

1 2 3 4~.

Fig. 3. “Horizontrd” sections of the surface (2) for determinant of

degree – 3. The same case as in Fig. 2.

of terms in the determinant det {&J. It has been found

that the function shapes as shown in Figs. 2 and 3 are

typical for all reentrant cavities under consideration,

For the linearization of the function used for direct

iterative solution of ~ the authors have proposed a formal,

manipulation which involved the following notation:

F?(c’,E”)F(c’,d’)= —X( LE’,C”)

where

~=det[~,,~], forl, m= O,l,...

~=det[~l,~], forl, m=l,2, ~..

and

2(6’,6’’) =0.

In its mathematical essence, (4) is equivalent

(3)

(4)

IO (l),

because ~ equals zero when—and only ‘when— W (i.e...,the determinant of the complete matrix det [&J) is equal

to zero. Since (1) was the condition for a resonance, the

same can be said about (4).It has been shown that when computing the complex

permittivity, the mathematical manipulation (3) suggested

by the authors reduced the procedure of computing byapproximately ten times, simultaneously increasing the

accuracy due to the application of linear interpolation (or

extrapolation) in the successive approximations.

IV. ACCURACY OF COMPUTING CONVERGENCE OF

SERIES

As mentioned previously, the expression for resonance

condition (4) or (1) contains an infinite determinant of a

square matrix, and that each term of the matrix contains,

in turn, an infinite series. It is obvious that finite algo-

rithms should be used in practice.

ICA(ZICOWSXI AND MILBWSKI: HIGH-ACCURACY WIDE-RANGE MSASUREMSNT MKIHOD: PART A 227

The following notation has been applied.

1) A number of rows and columns limited to a natural

number M (i.e., 1,m = O, 1.. . M – 1) represents the number

of function development terms used to determine the

electromagnetic field in subareas B and C of the resonator

(Fig. 1).

2) The restriction of the maximum number of terms in

the series which are indexed by the letter n to the value of

N (i.e., n = 0,1 .0. N – 1) is equivalent to using the first

terms in the developments of fields in the subarea A (Fig.

1).

Now, a question arises: at what values of the numbers

M and N, as defined above, will the error of the method

be limited to a predetermined value, taking into account a

variety of proportions of the resonator and of the sample,

as well as a wide range of resonator off-tuning? To answer

the question, it has been checked how the values M and N

affect the permittivity value as obtained by the method.

Some examples of/ are illustrated in Table I.

The following conclusions can be derived.

1) The number of terms in the field function develop-

ments in the individual subareas A, B, and C should be

chosen according to the required accuracy of e’. It is seen

from the Table I that the errors of computing the im-

aginary part ~“, are considerably smaller.

2) The real part d is slightly undervalued when restrict-

ing the number of terms in the field function develop-

ments for subareas B and C.

3) The restriction of the number of terms in the field

function development for subarea. A results in values by

several to several tens percent higher.

4) Both functions c’ =j(M) and d =~(N) monotonically

approach the limit value. Thus, methodical errors can

increase by making the function base wider. It is an

obvious consequence of the base orthogonality.

To determine the value of computational error resulting

from a restriction of the number of terms in the series, it is

necessary to establish a reference level, i.e., to determine

the exact values of electric permittivity first. When check-

ing how < depends upon M and IV, the process of making

the function base wider was stopped each time when a

change of either M or N by ca. 25 percent did not affect

the value of d by more than 10-5 percent.

When investigating the effect of series size on the d for

various proportions of the resonator, it has been found

(Figs. 4, 5) that the proportions of the gap between the

resonator cores (d: rl) has the major effect on the error

resulting from the restriction of M. This conclusion seems

to acknowledge the intuitive suggestion that for a given

accuracy, fewer terms are required when the region of the

lumped capacitance approaches the shape of a flat capaci-

tor (d; r~O). In addition, a small effect of the resonatorlength (Z.: d) on the error magnitude is noted (Fig. 4.).

It is seen from Fig. 5 that in the range of great errors

&’> 1 percent, there is a considerable effect of the posi-

tion of the gap (~1: ~) on the accuracy of computing 2.

For better accuracies the computational errors for all

TABLE I

ExmPLm OF mm R.RLATIONS ~=fiAf) ANO ~=j(N) PROPORTIONS OFXESONATOR DIMENSIONS ARR: d: r, = 0.679, L: d= 3.915: rl =0.615

1 9.5918- j 0.00130 1

2 9.5950 - j 0.00130 2

3 9.5950 - j 0.00131 4

5 9.6027 - j 0-00132 6

7 9.6063 - j 0.00132 13

10 9.6085 . j 0.00132 24

14 9.6086 - j 0.00132 58

10.9601 - j 0.00147

10.2687 - j 0.oo140

9*7417 - j 0.00134

9.6524 - j 0.00133

9.6173 - j 0.00132

9.6106 - j 0.00132

9.6086 - j 0.00132

Fig. 4. Dependence of computing relative error of c’ on number of

terms in development of eli%tromaguetic field in subareas 1? and C.

1 10 WJ lrm N

Fig. 5. Computing relative error of c’ as a function of munber of termsin development of field in subarea A.

proportions (~1: ~) should be comparable to each other if

the ratio L: d remains constant.

The result of the analysis of computational errors of the

method enables (Figs. 4 and 5) an evaluation of the

minimum length of function series necessary to achieve

the required accuracy.

v. SUMMARY

The test method developed by the authors provides

means for the determination of t of a wide range of

materials used in electronics. The paper presents accurate

solution on the resonance condition and reports the re-

sults of a careful investigation of the effect of function

base size on the magnitude of error M. The analysis of the

remaining errors iX resulting from errors in determining

228 EBB TRANSACTIONS ON MICROWAVE THEORY AND TSCHNIQUJ3, VOL. Nrrr-28, NO. 3, MARCH 1980

the dimensions of the sample and of the resonator, the

resonator resonant frequency and Q-factor, as well as [3]those resulting from assymetry of various kinds, is pre-

sented by the authors in [4], together with methods for a

minimalization of these errors.[4]

[1]

[2]

I?EFERENC13S [5]

1. I. Eldumiati and G. L Haddar& “Cavity perturbation techniquesfor measurement of the microwave conductivity and dielectric eon-

staat of a bulk semiconductor material: IEEE Trans. Mcrowaoe [6]

Theoty Tech., vol. MTT-20, PP. 126-132, Feb. 1972.L 1. Eldumiati and G. 1. 13adda@“Microwave properties of n-type

InSb in a magnetic field betwean 4 and 300 KV J. App. Plgw., vol.

44, pp. 395-4Q5, Jan. 1973.0. V. Karpow% “Absolute measurement method for determinationof c in reentrant cavityfl (in Russian) Fir. Twerd. Teia, vol. 1, pp.

246-255, Feb. 1959.

A. K?c*owski ad A. Milewski, “High-accuracy wide rangemeasurement method for determination of complex permittivity inreentrant cavity-Part B-experimental analysis of measuringerrors,” This issue, pp. 228–23 1.

A. Nfilewski, “Coaxiaf lumped capacitance resonators for tbe in-vestigations of dielectrics,” Electron TechnoL vol. 10, pp. 71–98, Jan.1977.

J. V. Parry, “The Measurement of permittivity and power factor ofdielectrics at frequencies from 300 to 600 Mc/s~ Proc. Inst. EIec.

Eng., vol. 98, no. 54, pp. 303–311, 1951.

High-Accuracy Wide-Range MeasurementMethod for Determination of Complex

Permittivity in Reentrant Cavity:Part B— Experimental Analysis of’

Measurement Errors

ANDRZEJ K&ZKOWSKI AND ANDRZEJ MILEWSKI

Ak?ra@--In Part ~ the measurement metbd of i was presentedfromthe M*- vfewpint. Experiments smdertalwnfn tfds secttonwere*@h*to~**titim_tikofPti& aaweUasto Vdy the -ti ealcldation mew d amqse measurementem

1. MEASUREMENT APPARATUS

A. Resonator

I

N THE EXPERIMENTAL part of this work, a reso-nator tunable by capacitance and inductance adjust-

ment was used (Fig. 1) with functional dimensions: rl =

14 mm, rz=48 mm, L1 = 32–42 ~ L=57–500 mm,

d= 0– 10 mm. As a result of the aforementioned adjust-

ments, measurements at a wide range of frequencies

200– 1000 MHz were possible.

Manuscript rcecived August 14, 1978; revised October 3, 1979.The authors are with the Instytut Technolo@i Elektronowej, Poli-

tcchnika Warszawska, ul. Koszykowa 7500-662 Warsaw, Poland.

Fig. 1. Tunable reentrant cavity.

B. Q Factor and Resonant Frequency Measurement System

Measurements of the Q factor and resonant frequency

were carried out by utilizing a Wobbulator, whose operat-

ing principle is shown in the block diagram (Fig. 2)

This apparatus makes it possible to obtain an accuracy

of 1 percent on the Q-factor measurement, and 10–3

0018-9480/80/0300-0228$00.75 @1980 IEEE