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工學碩士學位論文

평행평판 커패시터를 이용한 유전율 측정

Dielectric Measurements using a Parallel-Plate Capacitor

忠北大學校大學院

電波通信工學科電波通信工學專攻

바체른 스리삼부 (Battseren Sharavsambuu)

2010 年 2 月

工學碩士學位論文

평행평판 커패시터를 이용한 유전율 측정


指導敎授安炳哲

電波通信工學科電波通信工學專攻

바체른 스리삼부 (Battseren Sharavsambuu)

이 論文을 工學碩士學位論文으로 提出함.

2010 年 2月

本論文을 바체른 스리삼부의 工學碩士學位論文으로

認定함.

審査委員長 안 재 형 ㊞

審査委員 김 경 석 ㊞

審査委員 안 병 철 ㊞

忠北大學校大學院

2010 年 2月

-i-

CONTENTS

Abstract ······································································································· ii

List of Figures ··························································································· iv

List of Tables ···························································································· vi

I. Introduction ····························································································· 1

II. Theory ·································································································· 3

2.1 Dielectric Measurement by Parallel-Plate Capacitor ················· 3

2.2 Fringing Capacitance ···································································· 13

2.3 Effect of Electrode Thickness ··················································· 21

2.4 Effect of Surrounding Structures ·············································· 22

III. Measurements ····················································································· 24

3.1 Design of a Dielectric Test Fixture ··········································· 24

3.2 Calibration of Test Fixture ·························································· 26

3.3 Dielectric Measurements by One-Sample Method ···················· 29

3.4 Dielectric Measurements by Two-Sample Method ··················· 30

3.5 Dielectric Measurements using Large Material Sheet ·············· 32

IV. Conclusions ························································································ 34

References ································································································· 36

Acknowledgements ··················································································· 42

-ii-


Sharvsambuu Battseren

Department of Radio and Communications Engineering

Graduate School, Chungbuk National University

Cheongju, Korea

Supervised by Professor Ahn, Bierng-Chaerl

Abstract

Dielectric measurements play important roles in many fields of

engineering and science. A parallel-plate capacitor can be used in the

measurement of dielectric constants at low frequencies up to 100MHz. A

material under test is placed between two parallel electrodes and the

capacitance is measured, from which the dielectric constant of the material

is calculated. Electrodes of the capacitor are usually of circular shape.

Guard electrodes can be employed to remove the effect of the fringing

capacitance.

In this thesis, dielectric measurements using a parallel-plate capacitor

without guard electrodes are investigated. Electrodes of circular shape are

employed. First, the fringing capacitance of a parallel-plate capacitor with

zero-thickness circular disk electrodes is investigated. Accuracies of fringing

capacitance formulas published in the literature are compared. The fringing

capacitance is also evaluated using a commercial simulation software.

-iii-

Next a dielectric measurement fixture based on a parallel-plate capacitor

is designed. The fringing capacitance of the capacitor is measured and

compared with numerical simulation. Using the designed test fixture,

dielectric constants of some materials are measured by one-sample and

two-sample methods and compared with values given in the literature.

Experiments show that the low-frequency dielectric constant can be

measured accurately using the designed test fixture and that the accuracy of

the loss tangent needs improvements.

-iv-

List of Figures

Fig. 2.1 Parallel-plate capacitor ··········································································· 3

Fig. 2.2 Electric field distribution on a parallel-strip capacitor ······················ 4

Fig. 2.3 Parallel-plate capacitor filled with a dielectric materal ····················· 5

Fig. 2.4 Equivalent circuit of the material-filled capacitor ······························ 7

Fig. 2.5 Dielectric measurement with two samples ·········································· 8

Fig. 2.6 Parallel-plate capacitor filled with a larger-diameter material ·········· 9

Fig. 2.7 Equivalence between a parallel disk capacitor and a disk on a

grounded dielectric slab ···································································· 11

Fig. 2.8 Effect of making the one electrode infinitely large ························ 11

Fig. 2.9 Comparison of fringing capacitance formulas ·································· 16

Fig. 2.10 Normalized fringing capacitance of a parallel-plate disk capacitor

filled with an infinitely large dielectric sheet of ·············· 19


filled with an infinitely large dielectric sheet of ············ 19


filled with an infinitely large dielectric sheet of ············ 20


filled with an infinitely large dielectric sheet of ·········· 20

Fig. 2.14 Electric field around the edge of thick electrodes ························ 22

Fig. 2.15 Parallel-plate capacitor with a surrounding structure ····················· 22

Fig. 3.1 Designed parallel-plate diectric test fixture ······································ 25

Fig. 3.2 Photograph of the designed dielectric test fixture ··························· 26

Fig. 3.3 Photograph of the dielectric measurement setup ······························ 26

Fig. 3.4 Measured fringing capacitance of the designed test fixture ··········· 28

Fig. 3.5 Dielectric measurement by one-sample method ······························· 29

-v-

Fig. 3.6 Photograph of materials used in one-sample method ····················· 30

Fig. 3.7 Photograph of materials used in two-sample method ····················· 32

-vi-

List of Tables

Table 2.1 Example of a fringing field calculation ········································· 14

Table 2.2 Fringing capacitance of a circular disk capacitor with zero-

thickness electrode ············································································· 15

Table 2.3 Normalized fringing capacitance of a parallel-plate disk capacitor

filled with an infinitely large dielectric sheet ······························· 18

Table 2.4 Capacitance change due to the electrode thickness ······················ 21

Table 2.5 Dimensions of the structure shown in Fig. 2.19 ·························· 23

Table 2.6 Capacitance change due to a surrounding structure ····················· 23

Table 3.1 Dimensions of the designed dielectric test fixture ······················· 24

Table 3.2 Measured total capacitance of the desinged test fixture ·············· 27

Table 3.3 Measured fringing capacitance of the designed test fixture ········ 28

Table 3.4 Dielectric constant of some materials measured by one-sample

method ································································································ 30

Table 3.5 Dielectric constants of some materials measured by two-sample

method ································································································ 31

Table 3.6 Dielectric constants measured using large material sheets ··········· 33

-1-

I. Introduction

The measurement of the dielectric constant finds diverse applications in

such area as materials research and development of manufacturing processes

and electronic components, just to name a few. At low frequency ranges,

the capacitor cell is the most widely used method of dielectric

measurement, while at high frequencies various methods are employed,

among which are the open-ended probe, transmission line insertion method,

resonant cavity method, and free-space method [1].

The dielectric measurement fixture in the form of a parallel-plate

capacitor is useful for dielectric measurements at 100Hz-100MHz frequency

range [1]. The capacitor electrode is usually of circular shape and a guard

electrode is often employed to remove the effect of fringing fields [2].

Although useful in many applications, the parallel-plate capacitor test fixture

with guard electrodes does not offer a greater flexibility in the shape of the

material sample to be measured than a fixture with unguarded electrodes.



employed. First, the theory of dielectric measurements with the parallel-plate

capacitor is elaborated. Equations for the extraction of dielectric constant

and loss tangent of the material under test are presented for one-sample

method where the sample is smaller the electrode, for two-sample method

where two samples of the same material smaller than the electrode are

used, and for the sheet-material method where the material is larger than

the electrode.

For accurate dielectric constant measurements using a parallel-plated

capacitor with unguarded electrodes, the fringing field contribution to the

-2-

capacitance needs to be accurately assessed. The fringing capacitance of a

parallel-plate capacitor with zero-thickness circular disk electrodes is

investigated. A literature survey reveals that there are many formulas for

the fringing capacitance of the parallel-plate disc capacitor with varying

range of validity. Accuracies and regions of validity of fringing capacitance

formulas available in the literature are investigated. The fringing capacitance

is also evaluated using the commercial simulation software COMSOL

MultiphysicsTM. Based on the inter-comparison, formulas suitable for the

dielectric constant measurement are recommended.




dielectric constants of some materials are measured by three methods, i.e.,

one-sample method, two-sample methods, and sheet-material method.

Measured dielectric constants and loss tangents are compared with values

given in the literature.

In chapter II, the theory of dielectric measurements using a parallel-plate

capacitor is presented. In chapter III, the design of a parallel-plate capacitor

test fixture and dielectric measurements with the designed fixture are

described. In chapter IV, conclusions are drawn.

-3-

II. Theory

2.1 Dielectric Measurements by Parallel-Plate Capacitor

In chapter II, the theory of dielectric measurements using a parallel-plate

capacitor cell is presented. Fig. 2.1 shows a parallel-plate capacitor with

finite-thickness electrodes of infinite conductivity. Lines of the electric field

are drawn to illustrate the nature of the field distribution inside the parallel

plate region and in the region near the edge of the electrode. The vacant

space is filled with vacuum or air. In Fig. 2.1 one can see the fringing

field along edges of the electrode which contributes an additional fringing

capacitance to the total capacitance. Each electrode is a circular disk with a

finite thickness.

2a

0e d

t

t

Fig 2.1 Parallel-plate capacitor

The capacitance of the air-filled capacitor can be written

∆ (2.1)

where and ∆ are the capacitance without the fringing field contribution

and the capacitance due to the fringing field. The ideal-case capacitance C0

is given by the following elementary formula.

(2.2)

where S is the area of electrode surface facing each other, d is the gap

-4-

between facing surfaces of the electrode, a is the radius of the circular

electrode, and ×Fm is the permittivity of vacuum (or

approximately of air).

The capacitance without fringing field contribution is called the

geometric capacitance. In the case of circular disk electrodes, the effect of

the fringing capacitance is often expressed using the effective radius where

(2.3)

Thus the normalized effective radius is give by

∆

(2.4)

For a quantitive assessment of the extent of the fringing field, Fig. 2.2

shows the electric field distribution on the edge of a parallel-strip capacitor,

where the ratio of strip width to strip separation is high [3].

Fig. 2.2 Electric field distribution on a parallel-strip capacitor [3]

-5-

The electric field intensity rapidly decays outside the parallel plate

region. Electric flux lines are distorted from the ideal shape only up to

about one half of the electrode separation away from the edge into the

parallel-plate region. This suggests one important point useful in dielectric

measurements. When the plate diameter is large compared with the plate

separation, one can place the material anywhere inside the parallel plate

away from the edge by one-half of the plate separation without altering the

fringing capacitance.

A. Dielectric Measurements with One-Sample Method

Fig. 2.3 shows the same capacitor filled with a material with dielectric

constant of and loss tangent of tan. The size of the material is

sufficiently smaller than that of the electrode so that the fringing field does

not penetrate into the material when the material is placed at the center of

the electrode. Although a disk type is most common, the shape of the

material can be arbitrary as far as it is part of a cylinder, where the side

surface is perfectly in a right angle with top and bottom surfaces of the

material. When the material is placed in the parallel plate region, that there

will be no component of the electric field.

Sd

tandre 0e

Fig. 2.3 Parallel-plate capacitor filled with a dielectric materal

-6-

The capacitance with a material placed in the parallel plate is given by

∆

(2.5)

where and are the dielectric constant and the area of top and bottom

surfaces of the material under test. From (2.1) and (2.3), one can obtain

the equation for the dielectric constant of the material.

(2.6)

Equation (2.6) offers a simple method for dielectric measurements with a

parallel-plate capacitor. First, a material to be tested is fabricated in a

shape as cylindrical as possible with top and bottom surfaces parallel to

each other. The height of the material is d. The area of top and bottom

surfaces of the material is and should be sufficiently smaller than that

of the electrode. As a rule of thumb, side surfaces of the material should

be at least away from the electrode edge by d, the material height.

Material surfaces should be as smooth as possible since air gaps and

surface voids contribute to errors in the measured dielectric constant.

With the material prepared, one measures the capacitance of the test

fixture without the material with the electrode distance same as the height

d of the material. Or one can pre-measure capacitances of the vacant

fixture for many values of electrode distances and curve-fit them to derive

the equation for the capacitance of the air-filled fixture. Next with the

prepared material placed at the center of the parallel plate region, the

capacitance of the fixture is measured. Finally (2.6) is used to calculate the

dielectric constant of the material. The low-frequency capacitance is easily

measured using a low-cost LCR meter often readily available in electronics

-7-

experiment benches.

The loss tangent of the material is calculated from the measured quality

factor or resistance. Fig. 2.4 shows the equivalent circuit of the test fixture

with the material placed in the capacitor. The input admittance i n of the

material-filled capacitor is given by

Yin

dRdC

Fig 2.4 Equivalent circuit of the material-filled capacitor

in

(2.7)

where is given by (2.5) and

tan

(2.8)

In (2.8) tan is the loss tangent of the material under test. It is

assumed that the loss is entirely due to the material under test and the

fixture itself is lossless. The finite loss of the test fixture can be calibrated

out using modern LCR meters with four-terminal test leads often called

Kelvin probe [4]-[6].

From (2.8), the following equation is obtained.

tan

(2.9)

The quality factor of the material-filled capacitor is given by

-8-

(2.10a)

or

(2.10b)

Now the loss tangent of the material can be obtained using the following

equation.

tan

(2.11)

The loss tangent is obtained from either (2.9) or (2.11).

B. Dielectric Measurements with Two-Sample Method

With one-sample method, one needs to know the fringing capacitance

either by theory or by experiment. With two-sample method, the fringing

capacitance needs not be known.

Fig. 2.5 shows two capacitors filled with same materials of equal height

but with different area. The capacitance is given by

(2.12a)

(2.12b)

S1 S2

Fig. 2.5 Dielectric measurement with two samples

-9-

From (2.12a) and (2.12b), one obtains

(2.13a)

or

(2.13b)

(2.13b) is the equation that one uses to extract the dielectric constant

from two-sample measurement. The loss tangent is obtained from the

measurement with a larger sample for better accuracy. Thus we use the

following equation for the extraction of the loss tangent.

tan

(2.14)

C. Dielectric Measurements with Material Sheet

Fig. 2.6 shows the case where the size of the dielectric material is

greatly larger than the electrode. Ideally the area of the material should be

infinite.

re

2a

d

0e

Fig. 2.6 Parallel-plate capacitor filled with a larger-diameter material

Capacitance formulas similar to (2.1)-(2.4) can be derived. The total

capacitance is given by

-10-

∆ (2.15)

where is the geometric capacitance given by

(2.16)

The effective radius of the circular electrode is defined by

(2.17)

Then the normalized effective radius is give by

∆

(2.18)

The equivalent dielectric constant is defined by

(2.19)

where given by (2.15) and given by (2.1) are capacitances with

infinite slab material, and with material replaced by air, respectively. The

dielectric filling factor is defined by

(2.20)

In the literature many authors analyzed the capacitance of a circular disk

on a grounded dielectric slab. One can apply the theory of image in

electrostatics to find the capacitance of a parallel-plate circular disk

capacitor from that of a circular disk on a grounded dielectric slab. Fig.

2.7 shows the equivalence of two problems.

The total capacitance in each case is given by

∆ (2.21a)

∆ (2.21b)

-11-

V

V

V

Q Q

Q

Q

d

Problem 1 Problem 2

0e

red/2

0e

re

0e

re

Fig. 2.7 Equivalence between a parallel disk capacitor and a disk on a

grounded dielectric slab

where are geometric capacitances and ∆ ∆ are fringing

capacitances of Problems 1 and 2, respectively. It is evident that

(2.22)

Therefore

∆

∆ (2.23)

Thus the total and fringing capacitances of a parallel-plate capacitor with

electrode separation d is one half of those of the plate placed on a

grounded dielectric slab of thickness d/2.

From the foregoing analysis, one can find the effect of making one of

the electrode infinitely large. The situation is depicted in Fig. 2.8.

2a2a

d0e

re0e

0ere d

(a) (b)

Fig. 2.8 Effect of making the one electrode infinitely large

-12-

If one makes the lower electrode of Fig. 2.8(a) infinitely large, the total

capacitance increases due to the increase in the fringing capacitance. The

geometric capacitance remains unchanged. Let the fringing capacitance of

the capacitor in Fig. 2.8(a) be expressed as

∆ (2.24)

Now the fringing capacitance ∆ ′ in the case of Fig. 2.28(b) can be

expressed as

∆ ′ (2.25)

An example is in order. For a case with mm mm ,

uising Chew-Kong formula for the fringing capacitance [10], one finds

pF ∆ pF pF

′ pF ∆ ′ pF ′ pF

where primed quantities are for the case of Fig. 2.28(b).

Dielectric measurements with a large sheet material require the

evaluation of the fringing capacitance in the presence of the material in the

edge region. The fringing capacitance is calculated using a closed-form

formula described in the next section. The calculation of the fringing

capacitance requires the value of the dielectric constant of the material

under test. Therefore one extracts the dielectric constant by iteratively

matching measured capacitance value while monotonically increasing the

trial value of the dielectric constant until measured and calculated

capacitances coincide.

-13-

2.2 Fringing Capacitance

A. Air-Filled Capacitor

The capacitance of the air-filled parallel-plate test fixture needs to be

measured in order to extract the dielectric constant of the material. The

capacitance of an air-filled capacitor includes the capacitance contributed by

the fringing field. Therefore knowledge on the fringing capacitance is

important is the dielectric measurement using the parallel-plate capacitor. In

fact, if the fringing capacitance is known, one can calculated the

capacitance of the air-filled capacitor using (2.1).

In dealing with a parallel-plate with circular disk electrodes, it is often

convenient to normalize the capacitance with . From (2.1) and (2.2)

one obtains

∆ (2.26)

(2.27)

As an example of the fringing capacitance calculation, we use following

case to simulate with COMSOL MultiphysicsTM [7].

mm mm

Table 2.1 shows the result of capacitance calculation where

: total capacitance

: theoretical capacitance neglecting fringing capacitance

∆ : fringing capacitance

The most important fact learned from Table 2.1 is that the ratio ∆

-14-

is rapidly decreased as increases. That is, the fraction of the fringing

capacitance o t of the total capacitance decreases as the electrode spacing

is reduced with the given size of electrodes. When is 20, the fringing

capacitance acco nts for abo t 9% of the total capacitance, while it takes

up 22.8% of the total capacitance when is 5.

Table 2.1 Example of a fringing field calculation

d

(mm)

(pF)

(pF)

∆

(pF)

∆

∆

∆

2 20 22.253 24.466 2.213 1.989 0.1107 0.0904

4 10 11.126 12.993 1.867 1.678 0.1867 0.14378 5 5.563 7.201 1.638 1.472 0.3276 0.2275

16 2.5 2.782 4.199 1.417 1.274 0.5668 0.337532 1.25 1.391 2.745 1.354 1.123 1.0832 0.4933

To find analytical means for evaluating the fringing capacitance, an

extensive literature survey has been carried out. There are several formulas

for a parallel-plate disk capacitor with zero-thickness electrodes.

Kirchoff [8]:

∆

ln ≫ (2.28)

Nishiyama and Nakamura [9]:

∆

≤≤

≤≤ (2.29)

Chew and Kong [10]:

∆

ln

≤ ≤ (2.30)

Wintle [11]:

-15-

∆

ln

(2.31)

≥

where (2.31) is obtained by curve-fitting Wintle's numerical results [11].

Fringing capacitances formulas (2.28)-(2.31) are compared in Table 2.2.

Table 2.2 Fringing capacitance of a circular disk capacitor with

zero-thickness electrode

∆

Kirchoff

[8]

Nishiyama-

Nakamura

[9]

Chew-Kong

[10]

Wintle

[11]

0.1

1.0

10

100

1000

10000

0.1

1.0

10

100

1000

10000

0.1956

0.9286

1.6615

2.3945

3.1274

3.8604

1.2453

1.2980

1.7631

2.3948

3.2529

4.4185

3.3318

1.3175

1.7757

2.4811

3.2113

3.9440

1.1508

1.3183

1.7559

2.4124

3.1305

3.8603

The formula (2.20) obtained by curve-fitting Wintle's numerical results

yields accurate capacitance for ≥ so that it can be used as a

reference for the ratios of in this range.

Fig. 2.9 is a graphical comparison of formulas (2.28) to (2.31). The

formula by Nishiyama and Nakamura gives accurate capacitance values for

≤≤. As approaches the infinity, the accurace of

Nishiyama-Nakamura formula is gradually decreased. The formula by

Kirchoff is accurate only for ≥. When becomes less than 20,

-16-

the accuracy of Kirchoff formula gradually decreases.

0.1 1 10 100 1000 100000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5D

C0

/pe 0

a

a/d

Kirchoff Wintle Nishiyama-Nakamura Chew-Kong

Fig 2.9 Comparison of fringing capacitance formulas

The Chew-Kong formula (2.30) is most versatile since it gives accurate

capacitances for the technically important range ≥, and it includes the

case where the diameter of the dielectric material is larger than that of the

electrode.

There are many other works in the literature on parallel-plate disk

capacitors and similar structures including still more fringing capacitance

formulas [12]-[19], analysis of fringing fields [20]-[22], theoretical [23]-[24]

and numerical [25] analyses.

B. Dielectric Slab-Filled Capacitor

For the circular disk capacitor filled with an infinite dielectric slab

shown in Fig. 2.6, numerous authors presented the fringing capacitance

-17-

formula [10], [26]-[43]. Also there are many theoretical research results on

a circular disc on a dielectric slab [44]. Here we investigate only some of

representative formulas.

Chew and Kong [10]:

∆

ln

for ≤ ≤ ≤≤

(2.32)

Scott and Curtis [26], ASTM D150-98 [27], and IEC 70727:1978 [28]:

∆

ln in mm (2.33)

Wintle [29]:

∆

ln

(2.34)Wheeler [30]:

∆

(2.35)

Formulas (2.32), (2.34) and (2.35) are compared in Figs. 2.10-2.13 and

Table 2.3. The Chew-Kong formula (2.10) gives accurate capacitance values

for ≤ ≤ so that we use it as a reference for the ratio of

in this range. From Table 2.3 and Figs. 2.7-2.10, the Chew-Kong formula

-18-

Dielectric

constant of

the material

∆

Chew-Kong

[10]

Wintle

[29]

Wheeler

[30]

5

0.1 1.367 0.545 0.729

1 0.657 0.584 0.65710 0.718 0.692 0.711

100 0.856 0.833 0.7851000 1.001 0.980 0.855

10000 1.148 1.126 0.925

10

0.1 1.121 0.498 0.6631 0.574 0.517 0.570

10 0.585 0.571 0.583100 0.652 0.641 0.621

1000 0.725 0.715 0.658

10000 0.798 0.788 0.694

50

0.1 0.924 0.459 0.611

1 0.508 0.463 0.50110 0.480 0.474 0.475

100 0.490 0.488 0.4811000 0.504 0.503 0.488

10000 0.519 0.517 0.496

100

0.1 0.900 0.454 0.6041 0.500 0.456 0.492

10 0.466 0.462 0.461100 0.470 0.469 0.463

1000 0.476 0.476 0.466

10000 0.484 0.483 0.470

gives a good accuracy when ≥ and for all vaues of .

Table 2.3 Normalized fringing capacitance of a parallel-plate disk capacitor

filled with an infinitely large dielectric sheet

-19-

0.1 1 10 100 1000 100000.0

0.5

1.0

1.5

DC

0/(e 0e rp

a)

a/d

Chew-Kong Wintle Wheeler


filled with an infinitely large dielectric sheet of

0.1 1 10 100 1000 100000.0

0.5

1.0

1.5

DC

0/(e 0e rp

a)

a/d




-20-

0.1 1 10 100 1000 100000.0

0.5

1.0

1.5

DC

0/e 0e rp

a

a/d




0.1 1 10 100 1000 100000.0

0.5

1.0

1.5

DC

0/e

0e rpa

a/d




-21-

2.3 Effect of Electrode Thickness

The capacitance of a parallel-plate capacitor with thick electrodes is one

of important research topics and many authors have investigated the

problem [45]-[46]. Fig. 2.1 shows the parallel-plate disk with electrodes of

finite thickness. In order to assess the effect of the electrode thickness, the

following case is simulated and the result is listed in Table 2.4.

mm mm r

From Table 2.4 one can see that the thickness of the electrode

appreciably increases the total capacitance. The effect of the electrode

thickness can be understood more realistically from the electric field around

the edge of a parallel-strip capacitor shown in Fig. 2.14 [25].

Table 2.4 Capacitance change due to the electrode thickness

t (mm) t / d C (pF)

0 0.0 9.712 ( = ) 1.000

1 0.2 10.605 1.0912

2 0.4 10.639 1.0954

4 0.8 10.762 1.1081

8 1.6 10.939 1.1263

16 3.2 11.567 1.1910

-22-

Fig 2.14 Electric field around the edge of thick electrodes

2.4 Effect of Surrounding Structures

In a parallel-plate capacitor test fixure, there are some surrounding

structures. One example is shown in Fig. 2.15 where the electrode

shape is designed so that it can easily be attached to non-conducting

plastic plates holding the electrode. Dimensions of the structure are

listed in Table 2.5.

D1

D2

D3

t1

Parallel Plate

d

Dielectric Material

D4

t2t3

Fig. 2.15 Parallel-plate capacitor with a surrounding structure

-23-

d (mm)

Capacitance of

without surrounding

structure (pF)

Capacitance of with

surrounding

structure (pF)

2 24.466 24.637

4 12.993 13.233

8 7.201 7.276

16 4.199 4.291

Parts Dimension (mm)

50

60

80

160

3

5

3

?

Table 2.5 Dimensions of the structure shown in Fig. 2.19

Table 2.6 shows the capacitance change due to the surrounding structure

of Fig. 2.15 simulated by COMSOL MultiphysicsTM. The capacitance is

increased slightly due to the presence of the surrounding structure.

Table 2.6 Capacitance change due to a surrounding structure

-24-

III. Measurements

3.1 Design of a Dielectric Test Fixture

Many authors have investigated the problem of dielectric measurement

by parallel-plate method and the fixture design associated with it [47]-[48].

A parallel-plate capacitor test fixture without a guard electrode is designed.

The designed fixture is shown in Fig. 3.1.

Electrodes are shaped so that they can easily be attached to thick plastic

plates (thickness ). The gap between two electrodes are maintained by

springs installed at four long screws (diameter ) that run through plastic

plates around the edge. The diameter and thickness of electrodes are

80mm and 3mm, respectively. To ensure an accurate alignment of two

electrodes, two guide pins (diameter ) are installed on plastic plates.

Table 3.1 shows the dimensions of the designed test fixture. Fig. 3.2 shows

a photograph of the designed test fixture.

Table 3.1 Dimensions of the designed dielectric test fixture

Parts Dimension (mm)

80

60

50

5

10.4

6

3

5

3

10

160

-25-

D1D2D3

D4

D5 D6

h

Æ Æ

D1D2D3

t1

D4

t2t3 t4

123

Fig. 3.1 Designed parallel-plate diectric test fixture

-26-

Fig. 3.2 Photograph of the designed dielectric test fixture

3.2 Calibration of Test Fixture

The fringing capacitance of the designed test fixture is calibrated. Since

the test fixture has no micrometer for measuring the electrode gap, a level

indicator and a height meter are employed to measure the electrode

distance. A photograph of the measurement setup is shown in Fig. 3.3

Fig. 3.3 Photograph of the dielectric measurement setup

-27-

d (mm)

Simulated

capacitance

(pF)

Measured

capacitance

(pF)

Theoretical capacitnace (pF)

Chew-Kon

g

[10]

Nishiyam-

Nakamura

[9]

Wintle

[11]

2 24.450 24.119 24.454 24.980 24.329

4 12.927 12.245 13.096 12.884 12.970

8 7.149 7.520 7.324 7.237 7.19516 4.168 4.222 4.367 4.368 4.224

32 2.674 2.263 2.670 2.890 2.67864 1.888 1.345 1.749 1.677 1.418

The total capacitance of the fixture is measured and the fringing

capacitance is extracted. The capacitance is measured using an LCR meter

(Goodwill LCR-821).

Table 3.3 shows the measured total capacitance of the fixture. It seems

that the measured capacitance contains a systematic error due to the

inability to measure the electrode distance accurately. The numerical

simulation suffers from the problem of convergence. It shows varying

degree of agreement with the theory as the electrode distance d changes.

Table 3.2 Measured total capacitance of the desinged test fixture

Table 3.3 and Fig. 3.4 shows the fringing capacitance of the designed

test fixture extracted from the total capacitance measurement. The measured

fringing capacitance is smaller than the theoretical values for all values of

the electrode spacing. This, again, is believed to be due to inaccuracies in

the electrode spacing measurement.

-28-

Table 3.3 Measured fringing capacitance of the designed test fixture

d (mm)

Measured

fringing

capacitance

(pF)

Theoretical capacitnace (pF)

Chew-Kong

[10]

Nishiyama-

Nakamura

[9]

Wintle

[11]

1 1.920 2.441 2.358 2.379

2 1.747 2.204 2.151 2.158

4 1.643 1.975 1.961 1.952

6 1.540 1.849 1.858 1.841

8 1.326 1.764 1.788 1.764

10 1.170 1.702 1.736 1.715

12 0.905 1.654 1.695 1.673

1 2 3 4 5 6 7 8 9 10 11 120.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

DC

(p

F)

d (mm)

Measurement Wintle Nishiyama-Nakamura Chew-Kong

Fig. 3.4 Measured fringing capacitance of the designed test fixture

-29-

3.3 Dielectric Measurements by One-Sample Method

In one-sample method, a material with a diameter smaller than that of

the electrode is used as shown in Fig. 3.5. The pre-measured fringing

capacitance is used in the extraction of the dielectric constant from the

measured total capacitance of the fixture with the material placed in the

center of the parallel plate region.

D1D2D3

t1

D4

t2t3

re

Dielectric Material

Fig 3.5 Dielectric measurement by one-sample method

Table 3.4 shows the measured dielectric constants of a few sample

materials. Fig. 3.6 shows a photograph of materials used in one-sample

method. Measured dielectric constants agree well with values in the

literature. As explained in the above, there are some systematic errors in

the fringing capacitance calibration, but they do not significantly contribute

to the errors in the dielectric constant measurement. The measured loss

tangent is significantly larger than those available in the literature. This

-30-

MaterialsDimension

(mm)

Measurement

(pF)Q tan

Polycarbonate a = 20mm

d = 1.96mm34.193 184.7 2.71 0.005

Teflona = 20mm

d = 5mm13.009 241.6 2.10 0.0041

Aluminaa = 20mm

d = 9.81mm15.956 216.6 9.68 0.0046

Hard papera = 20mm

d = 0.49mm123.97 64.52 2.27 0.015

seems to be due to the inability to calibrate out the inherent losses in test

cables and the test fixture.

Table 3.4 Dielectric constant of some materials measured by one-sample

method

Fig. 3.6 Photograph of materials used in one-sample method

3.4 Dielectric Measurements by Two-Sample Method

The theory of dielectric measurements by two-sample method is

-31-

MaterialsDimension

(mm)

Measured

∆ (pF)

Measured

Qtan

Polycarbonate

=40mm

=0.98mm 7.31 2.75

162.7 0.0061

=17.5mm

=0.98mm206.8 0.0048

Teflon

=30mm

=3mm 30.064 1.99

243.3 0.0041

=10mm

=3mm214.2 0.0047

Hard Paper

=30mm

=0.49mm 39.41 2.35

49.65 0.0200

=20mm

=0.49mm72.18 0.0140

presented in Chapter II. Table 3.5 shows measured dielectric constants by

two-sample method. Dielectric constants measured by two-sample method

are slightly different from those by one-sample method. The magnitude of

the difference shows the accuracy level of the measurement method. Fig.

3.7 shows materials used in two-sample method.

Table 3.5 Dielectric constants of some materials measured by two-sample

method

-32-

(a)

Fig 3.7 Photograph of materials used in two-sample method

3.5 Dielectric Measurements using Large Material Sheet

Dielectric measurements using large material sheets are carried out

according the theory presented in Chapter II. Table 3.6 shows the result.

Dielectric constants and loss tangents agree well with those obtained by

one- and two-sample methods withing the accuracy of measurement method.

-33-

Table 3.6 Dielectric constants measured using large material sheets

MaterialsDimensions

(mm)

Measured

capacitance

(pF)

Measured

Qtan

Polycarbonatea=50

d=0.94128.93 2.65 141.1 0.00727

Window

glass

a=43

d=1.9153.96 6.36 81.02 0.01270

Hard papera=50

d=0.53202.87 2.38 46.76 0.01850

-34-

IV. Conclusions



employed. First, the theory of dielectric measurements with the parallel-plate

capacitor is elaborated. Equations for the extraction of dielectric constant

and loss tangent of the material under test are presented for one-sample

method where the sample is smaller the electrode, for two-sample method

where two samples of the same material smaller than the electrode are

used, and for the sheet-material method where the material is larger than

the electrode.

For accurate dielectric constant measurements using a parallel-plated

capacitor with unguarded electrodes, the fringing field contribution to the

capacitance needs to be accurately assessed. The fringing capacitance of a

parallel-plate capacitor with zero-thickness circular disk electrodes is

investigated. A literature survey reveals that there are many formulas for

the fringing capacitance of the parallel-plate disc capacitor with varying

range of validity. Accuracies and regions of validity of fringing capacitance

formulas available in the literature are investigated. The fringing capacitance

is also evaluated using the commercial simulation software COMSOL

MultiphysicsTM. Based on the inter-comparison, formulas suitable for the

dielectric constant measurement are recommended.




dielectric constants of some materials are measured by three methods, i.e.,

one-sample method, two-sample methods, and sheet-material method.

-35-

Measured dielectric constants and loss tangents are compared with values

given in the literature.

A review of measurements shows that a precision distance meter

integrated with the test fixture is required for accurate assessment of the

electrode spacing. The loss tangent can be measured only in the gualitative

sense due to the inability to calibrate small losses inherent in the test cable

and the test fixture. This thesis can be utilized in the further study on the

dielectric measurement using the parallel-plate capacitor.

-36-

References

[1] A. van Roggen, "An overview of dielectric measurements," IEEE Trans.

Electrical Insulation, vol. 25, no. 1, pp. 95-106, Feb. 1990.

[2] M. Winkeler, "A review of electrical tests for insulating resins," Proc.

19th IEEE Electrical & Electronics Insulation Conf., 25-28 Sept. 1989,

pp. 191-194.

[3] R. Bansevicius and J. A. Virbalis, "Distribution of electric field in the

round hole of plane capacitor," J. Electrostatics, vol. 64, pp. 226-233,

2006.

[4] Agilent Technologies Inc., Impedance Measurement Handbook - A

Guide to Measurement Technology and Techniques, 4th Ed., Doc. No.

5950-3000, 2009.

[5] Agilent Technologies Inc., Application Note AN 1369-1: Solutions for

Measuring Permittivity and Permeability with LCR Meters and

Impedance Analyzers, Oct. 2008.

[6] K. Suzuki, "A new universal calibration method for four-terminal-pair

admittance standards," IEEE Trans. Instrum. Meas., vol. 40, no. 2, pp.

420-422, Apr. 1991.

[7] COMSOL MultiphysicsTM homepage. www.comsol.com

[8] G. Kirchoff, "Zur theorie des kondensators," Monastb. Akad. Wiss.

Berlin, pp. 144-162, 1877.

[9] H. Nishiyama and M. Nakamura, "Capacitance of disk capacitors," IEEE

Trans. Comp. Hybrids, Manufac. Tech., vol. 16, no. 3, pp. 360-366,

May 1993.

[10] W. C. Chew and J. A. Kong, "Effects of fringing fields on the

capacitance of circular microstrip disk," IEEE Trans. Microwave Theory

-37-

Tech., vol. 28, no. 2, pp. 98-104, Feb. 1980.

[11] H. J. Wintle and S. Kurylowicz, "Edge corrections for strip and disc

capacitors," IEEE Trans. Instru. Meas., vol. 34, no. 1, pp. 41-47, Mar.

1985.

[12] H. J. Wintle, "Capacitance correction for the guard, edge and corner

structure," Proc. IEEE Int. Symp. Electric. Insul., 3-6 June 1990, pp.

435-438.

[13] F. Thompson and J. K. Gagon, "Fringe capacitance of a parallel-plate

capacitor," Phys. Edu., vol. 17, pp. 80-82, 1982.

[14] K. Y. Kung, H. J. Wintl, A. Blaszcyk and P. L. Levin, "On

capacitace correction for unguarded disc electrodes," Jour. Electrostatics,

vol. 33, no. 3, pp. 371-378, Oct. 1994.

[15] W. C. Chew and J. A. Kong, "Asymptotic formula for the capacitance

of two oppositely charged discs," Math. Proc. Camb. Phil. Soc., vol. 89,

pp. 373-384, 1981.

[16] D. P. Hale, "Fringe capacitance of a parallel-plate capacitor," Phys.

Edu., vol. 13, pp. 446-448, Nov. 1978.

[17] H. Nishiyama and M. Nakamura, "Form and capacitance of

parallel-plate capacitors," IEEE Trans. Comp. Pack. Manufac. Tech. Pt.

A, vol. 17, no. 3, pp. 477-484, Sept. 1994.

[18] M. Hosseini, G. Zhu and Y.-A. Peter, "A new formula of fringing

capacitance and its application to the control of parallel-plate

electrostractic micro actuators," Analog Integrated Circuits and Signal

Processing, vol. 53, no. 2-3, pp. 119-128, 2007.

[19] P. F. Stefens and D. M. van Dommelen, "The influence of edge

effects on measurements of dielectric material characteristics," Proc. Int.

Conf. Microwave High Freq. Heating, 17-21 Sept. 1995, pp. 1-4.

-38-

[20] M. C. Hegg and A. V. Mamishev, "Influence of variable plate

separation on fringing electric fields in parallel-plate capacitors," Proc.

IEEE Int. Symp. Electrical Insul., 19-22 Sept. 2004, pp. 384-387.

[21] A. Naini and M. Green, "Fringing fields in a parallel-plate capacitor,"

Am. J. Phys., vol. 45, no. 9, pp. 877-879, 1977.

[22] G. J. Sloggett, N. G. Barton and S. J. Spencer, "Fringing fields in

disc capacitors," J. Phys. A: Math. Gen., vol. 19, no. 13, pp.

2725-2736, Oct. 1986.

[23] G. T. Carlson and B. L. Illman, "The circular disk parallel-plate

capacitor," Am. J. Phys., vol. 62, pp. 1099-1105, 1994. (theory)

[24] M. Norgren and B. L. G. Jonsson, "The capacitance of the circular

parallel plate capacitor obtained by solving the Love integral equation

using an analytic expansion of the kernel," Progress in Electromagnetic

Research, PIER 97, pp. 357-372, 2009.

[25] S. Catalan-Izquierdo, J.-M. Bueno-Barrachina, C.-S. Canas-Penuelas, F.

Cavalle-Sese, "Capacitance evaluation of parallel-plate capacitors by

means of finite element analysis," Proc. Int. Conf. Renew. Energy Power

Qual. (ICREPQ '09), 15-17 Apr. 2009, (page numbers unlisted).

[26] A. H. Scott and H. L. J. Curtis, "Edge correction in the determination

of the dielectric constant," Jour. Res. Natl. Bur. Standards, vol. 22, pp.

747-775, 1939.

[27] ASTM Standard D150-98 1998, Test Methods for AC Loss

Characteristics and Permittivity (Dielectric Constant) of Solid Electrical

Insulation.

[28] British Standard BS5737:1979 / IEC70727:1978, Method for the

Measurement of Relative Permittivity, Dielectric Dissipation Factor and

DC Resistivity of Insulating Liguids.

-39-

[29] H. J. Wintle and D. G. Goad, "Capacitance corrections for disc

electrodes on sheet dielectrics," J. Phys. D: Appl. Phys., vol. 22, pp.

1620-1626, 1989.

[30] H. A. Wheeler, "A simple formula for the capacitance of a disc on

dielectric on a plane," IEEE Trans. Microwave Theory Tech., vol. 30,

no. 11, pp. 2050-2054, Nov. 1982.

[31] S. S. Bedair, "Closed-form expression for the capacitance of the

shielded microstrip circular disc capacitor by dividing the capacitance

approach," Int. Jour. Electronics, vol. 58, no. 3, pp. 509-511, Mar. 1985.

[32] A. K. Verma, R. Kumar and M. Sharma, "Closed-form expression for

accurate determination of capacitance and effective radius of microstrip

circular disk," Microw. Opt. Tech. Lett., vol. 14, no. 4, pp. 217-221,

Mar. 1997.

[33] C. H. Seguin, "Fringe field corrections for capacitors on thin dielectric

layers," Solid-State Electron., vol. 14, no. 5, pp. 417-420, May 1971.

[34] S. Coen and G. M. L. Gladwell, "A Legendre approximation method

for the circular microstrip disk problem," IEEE Trans. Microwave Theory

Tech., vol. 25, no. 1, pp. 1-7, Jan. 1977.

[35] I. Wolff and N. Knoppik, "Rectangular and circular microstrip disk

capacitors and resonators," IEEE Trans. Microwave Theory Tech., vol.

22, no. 10, pp. 857-864, Oct. 1974.

[36] J. Helszajn and R. W. Lyon, "Approximate determination of effective

radius and effective dielectric constant of microstrip disc resonators,"

Electon. Lett., vol. 17, no. 1, pp. 46-48, Jan. 1981.

[37] N. Kumprasert and W. Kiranon, "Simple and accurate formula for the

resonant frequency of the circular microstrip disk antenna," IEEE Trans.

Antennas Propagat., vol. 43, no. 11, pp. 1331-1332, Nov. 1995.

-40-

[38] B. Gelmont, M. S. Shur and R. J. Mattauch, "Disk and stripe

capacitances," Solid-State Electron., vol. 38, no. 3, pp. 731-734, 1995.

[39] C. Donolato, "Analytical approximation to the capacitance of the

microstrip disk capacitor," Solid-State Electon., vol. 39, no. 2, pp.

314-317, 1996.

[40] T. Gűnel, "Continous hybrid approach to the modified resonant

frequency calculation for circular microstrip antennas with and without

air gaps," Microw. Opt. Tech. Lett., vol. 40, no. 5, pp. 423-426, Mar.

2004.

[41] D. Guha, "Resonant frequency of circular microstrip antennas with and

without air gaps," IEEE Trans. Antennas Propagat., vol. 49, no. 1, pp.

55-59, Jan. 2001.

[42] A. Akdagli and K. Gűney, "Effective patch radius expression obtained

using a genetic algorithm for the resonant frequency of electrically thin

and thick circular microstrip antennas," IEE Proc. Microw. Antennas

Propagat., vol. 17, no. 2, pp. 156-159, Apr. 2000.

[43] A. Akdagli, "A novel expression for effective radius in calculating the

resonant frequency of circular microstrip patch antennas," Microw. Opt.

Tech. Lett., vol. 49, no. 10, pp. 2395-2398, Oct. 2007.

[44] K. Hongo and M. Takahashi, "Evaluation of integrals occurring in the

study of circular microstrip disk," IEEE Trans. Microwave Theor Tech.,

vol. 30, no. 8, pp. 1279-1282, Aug. 1982.

[45] K. Shah, J. Singh and A. Zayegh, "Modelling and analysis of fringing

and metal thickness effects in MEMS parallel-plate capacitors," Proc.

Microelectron.: Design Tech. Packaging II, SPIE Proc. Vol. 6035, 12

Dec. 2005.

[46] H. El Kamchouchi and A. A. Zaky, "A direct method for the

-41-

calculation of the edge capacitance of thick electrodes," J. Phys. D:

Appl. Phys., vol. 8, pp. 1365-1371, 1975.

[47] J. J. Yoho, Design and Calibration of a RF Capacitance Probe for

Non-Destructive Evaluation of Civil Structures, M. S. Thesis, Virginia

Polytechnic Institute and State University, 1998.

[48] S. Straulino and A. Cartacci, "An absolute electrometer for the physics

laboratory," Phy. Edu., vol. 40, no. 3, pp. 301-305, May 2009.

-42-

Acknowledgements

My foremost thank goes to my professor Dr. Biering-Cherl Ahn who

gave me a big chance to study in the Master's Course while providing the

financial supporting. I would like to thank members of the thesis revew

committee, Dr. Jae-Hyung Ahn and Dr. Kim Kyung-Seok.

I must express my thanks to all of my lab members at Chungbuk

National University. I thank my best Mongolian friend Bat-Ochir Chinzorig,

who has been living with me for three years, for his great help on the life

in Korea as well as studying in the lab.

Finally, I would like to thanks my family: Fathers Ayursed Ya,

Sharavsambuu L., and my wife Damdinsuren BYAMBAJARGAL.

2009-12-24 in lab

Sharavsambuu BATTSEREN

I. IntroductionII. Theory 2.1 Dielectric Measurement by Parallel-Plate Capacitor 2.2 Fringing Capacitance2.3 Effect of Electrode Thickness 2.4 Effect of Surrounding Structures

III. Measurements3.1 Design of a Dielectric Test Fixture 24 3.2 Calibration of Test Fixture3.3 Dielectric Measurements by One-Sample Method3.4 Dielectric Measurements by Two-Sample Method3.5 Dielectric Measurements using Large Material Sheet

IV. ConclusionsReferences

11I. Introduction1II. Theory 32.1 Dielectric Measurement by Parallel-Plate Capacitor 32.2 Fringing Capacitance132.3 Effect of Electrode Thickness 212.4 Effect of Surrounding Structures 22III. Measurements243.1 Design of a Dielectric Test Fixture243.2 Calibration of Test Fixture263.3 Dielectric Measurements by One-Sample Method293.4 Dielectric Measurements by Two-Sample Method303.5 Dielectric Measurements using Large Material Sheet32IV. Conclusions34References36

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