continuum properties from interdigital dielectrometry

8/17/2019 Continuum Properties from Interdigital Dielectrometry

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IEEE nan sa ctio ns on Electrical Insulation Vol. 23 No. , December 1988

897

Continuum Properties from

Inter digita1 Elect rode Dielec rome t ry

Ma rk C. Zaretsky, Lama Mouayad and

Ja me s

R. Melcher,

Lab.

E& E

Systems M.I.T. Cambridge, MA

A B S T R A C T

Using a modal approach,

a

model is derived tha t makes th e in-

terdigital electrode microdielectrometer developed by Senturia

and co-workers applicable to measuring continuum param eters

in a wide range of heterogeneous m edia. In this ‘imposed w -

C’

technique, the medium is excited at t he tempora l (ang ular) fre-

quency

w

by means of an interdigital electrode structure hav-

ing a spatial periodicity length X = 27r/IC and hence a dominant

wavenumber IC.

Given the surface capacitance density C(w,

C

of any linear system having property gradients perpendicular

to the plane of the electrodes, the model predicts the com-

plex gain, taking into account the properties, geometry, and

terminal configuration of the interdigital electrode structure.

This capability can then be used with an app ropriate parame-

ter estimation strategy to determ ine the continuum properties

an d/o r geometry of the medium. Specifically illustrated, us-

ing a secant method root searching routine for the parameter

estimation, are estimations of film thickness, film permittiv-

hickness known, and film surface conductivity with

I N T R O D U C T I O N

state of

a

medium can be obtained from the electrical

frequency response. The modal approach t o dielectrom-

etry described here not only extracts its information

from the control of the temporal frequency, but also from

the imposition

of a

spatial

periodicity as

well.

In this

IMPOSED w SENSING

LASS ICAL

dielectrometry illustrates how important

c nformation concerning the physical properties and

0018-9367/88/1200-897 1.00

@ 1988 IEEE


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898

Zaretsky

et al.:

Continuum Properties

from

Interdigital Electrode Dielectrometry

‘imposed w -

’

technique, the medium is excited at the

temporal (angular) frequency w by means of an inter-

digital electrode structure having a spatial periodicity

length X

=

2 a / k and hence a dominant wavenumber k .

+

Y

L--q 1

I

-

A

I “F

t -

I

I

I

J

1

Figure 1.

Representation of interdigital electrode struc-

ture and associated circuitry interacting with a

heterogeneous medium.

In Figure 1, the medium is above and the struc-

ture used to make the measurements is below. Because

the electrodes introduce higher spatial harmonics, field

quantities, such as the potential, are expressed as

where

&(z,y)= b,(z)cos(k,y) ;

k, =

2 n r / X

representing a superposition of standing waves of domi-

nant wavenumber k = k l in the plane of the electrodes.

n=O

The choice of the cosinusoidal representation exploits

the symmetry of the electrode structure and the pre-

sumed symmetry of the situation above (for example,

there is no motion parallel to the electrode plane).

The advantages of the

w

- k approach are many.

First, there is the coupling into the medium from

a

sin-

gle surface, allowing measurements to be more readily

made noninvasive, if desired. For example, dielectric

measurements of thin films can be performed without

having to vapor-deposit an upper metal electrode for

use with

a

capacitance bridge, and in many cases, can be

taken a t the s ite of the coating process without changing

the ambient environment.

Secondly, because the fields generated by the elec-

trodes in the medium (above in Figure 1) are quasistatic,

they tend to decay into the material exponentially with

a characteristic length that is at most a fraction of the

spatial wavelength. Consequently, the electrodes are

sensitive to dielectric materials located within a region

roughly 5

X/3

from the electrode plane. Thus, the spa-

tial sensitivity of the device can be tailored to individual

needs.

A third advantage comes with modern microfabri-

cation techniques that not only make micrometer-scale

electrodes possible, but make the integration with the

signal processing electronics feasible

as

well. Electrodes

with

a

very short spatial wavelength

X

can be deposited.

Thi s refines the spati al resolution. For example, given

a spatial wavelength of

50

pm, thin films of 5 20 pm

(perhaps coated on the electrodes) can be distinguished

and their dielectric dispersions examined.

Having short wavelengths also improves the sensi-

tivity t o measurements of surface conductivityQ,. Sens-

ing conduction on the surface of an insulator without

making electrical contact with the surface requires ca-

pacitive coupling. Thus , the frequency must be high

enough to make th e capacitive reactance on the order of

the resistance of the surface. Roughly, thi s requires that

to measure a surface conductivity

U ,

at the frequency

f w / 2 a , the ratio u,/f~X)ust be of the order of

unity. There is a practical lower limit on the frequency.

Therefore, the smaller the wavelength A, the smaller

the surface conductivi ty th at can be measured. Having

the electronics integrated with the electrode structure

makes it possible to use frequencies as low as 0.005 Hz

using the ‘chip’ described in the next Section.

Although the emphasis here will be on dielectrom-

etry, the imposed w-k approach can be used in systems


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Insulation

Vol. 23 No. 6 , December

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899

where the electrical response reflects electromechanical

and electrochemical phenomena. This is illustrated by

earlier studies where interdigital electrodes were used

to study the electromechanics of double layers [l],elec-

trohydrodynamic surface waves and instabilities

[2]

and

sensing of electrification effects in insulating fluids

[3,4]

IN

T IMAT

E

S EN

SIN G

Y

contrast with remote sensing, where electromag-

B netic fields described by the wave equation are of-

ten used to resolve properties an d st ate s of materia ls

at great distances, the imposed

w

- k approach exploits

fields th at are quasi static from the electromagnetic point

of view. In the electroquasistatic embodiment described

here, the dynamics result from charge migration and

diffusion, and polarization relaxation. In the magnet @

quasistatic analogue, the dynamics would involve cur-

rent diffusion an d magnetizat ion hysteresis. In either

of these latter cases, the fields are quasistatic and tend

to be confined to the immediate neighborhood of the

interdigital electrode structure. Because measurements

reflect the properties a nd s tate within a short distance,

the application of such structures to parameter estima-

tion might be dubbed ‘intimate sensing’. Strategies ex-

ploiting the wavelength controlled resolution inherent to

having fields that are confined to the proximity of the

electrode can utilize the model developed here.

With the combination of microfabrication technol-

ogy and the ‘imposed

w

-

k’

technique, a determina-

tion of electromechanical or electrochemical structures

and dynamics on a microscale becomes feasible. Fluid

boundary layers, electrochemical double layers and thin

film structures all fall in this microscale realm.

Intuitively, it makes more sense to determine the

structure of heterogeneous media by probing spatially,

rather than temporally. Wit h the usage of the approach

presented here,

a continuum model coupled with a pa-

rameter estimation routine, it becomes feasible to ex-

tract information at

a

particular temporal frequency,

varying the spatial frequency of the applied potential.

This method does not have the restriction regarding the

frequency dependence of parameters such as the com-

plex permittivity. Practically, this effort amounts to

having multiple electrode structures, each having a dif-

ferent spatial wavelength, monitoring the same medium,

or perhaps one array of electrodes with switchable ter-

minal connections to produce the various spatial wave-

lengths.

MICRODIELECTROMETRY

ICRODIELECTROMETRY , as

developed by Senturia

M

and co-workers

[5,6],

is a commercially available

technique

[8]

for measuring complex permittivity utiliz-

ing microfabrication technology to incorporate both the

sensing electrodes and associated circuitry on the same

microchip (Figure

1 ) .

A set of interdigital, planar elec-

trodes are deposited on an insulating oxide layer along

with two FET transistors [7].

One set of electrodes is

driven by a variable frequency (0.005 to 10000 Hz) ac

voltage. The other set of electrodes is allowed to float

by connecting it to the gate of one of the transistors.

The other transistor serves

as

a reference.

Using the

feedback circuit shown in Figure

1 ,

in conjunction with

the transistors, a very high impedance measurement of

the floating gate voltage can be obtained, even at very

low frequencies. Th e outp ut of the device, defined as

the complex gain

G ,

is th,e complex ratio of the floating

to driven gate voltage,

(VFIVD),

nd is obtained using

a correlation analysis

of

the two voltages.

Two specialized models have been developed for in-

terpreting dat a obtained with the microdielectrometer.

One model is for its application to the monitoring of

epoxy resin cures

[9,10].

This model, incorporated

in

the form of

a

lookup table supplied with the microdi-

electrometer system [7], makes it possible to relate the

measured complex gain to the complex permittivity of

a uniform infinite half space. For this case,

a

finite dif-

ference simul ation of Laplace’s equation was used to de-

termine the electric field distribution, treating the re-

gion above the electrode structure

as

a semi-infinite,

isotropic medium [ lo] .

The second model previously

developed is for the study of a very thin film

(<

1 pm)

[11,12].

Th e thin film-oxide layer was represen ted

as

an RC transmission line shunted by lumped capacitors.

This made it possible to represent the gate voltage as

the transmission line response to a driving voltage.

Th e work rep orted her e is an outgro wth of work be-

ing done to develop the microchip for monitoring trans-

former insulation [13]. There, the intent is to coat the

microchips either to:

( 1 )

passivate the chip to the ad-

sorption of moisture on the silicon dioxide interface so

that they can be used to measure the dispersion of the

oil and oil-impregnated system or

( 2 )

to utilize their

thin region sensitivity to monitor changes in dielectric

properties of finite thickness coat ings, distinguishing be-

tween bulk dispersion and effects of heterogeneity.


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Zaretsky et al.: Continuum Pr opertie s from Interdigital Electrode Dielectrometry

OBJECTIVES

HE first objective is t o develop a general and adap t-

T

ble model for determining the output of the inter-

digital electrode dielectrometer. The second is to es-

tablish a rapport for the relationship between predicted

frequency responses and several physical situations of

immediate interest. Finally,

a

method for parameter

estimation utilizing the continuum model and demon-

strating its practical application, will be outlined and

demonstrated.

CONTINUUM

MODEL

APPROACH

N Figure 2 is shown a schematic view summarizing the

I

tructure of the continuum model developed in the

next Section. One set of inputs t o the model, the lower

box on the left in the diagram, describes the electrode

array structure. Parameters such

as

interelectrode spac-

ing a and the spatial wavelength X are specified here.

In addition, for the specific electrode structure consid-

ered here, interdigital electrodes deposited on an oxide

layer with ground plane underneath, the electric field

distribution below the electrodes is completely specified

by the insulating oxide layer thickness

h

and permit-

tivity coz . The other input, the upper box on the left,

describes the half-space of media above the electrodes.

This description is in the form of a complex surface ca-

pacitance density

C,,

representing the response of the

half space to one Fourier component of a potential ap-

plied

at

the electrodes. As shown in the upper part of

the figure, this responseis the complex ratio of the nor-

mal displacement field D; measured at the interface to

the interfacial potential @E The A signifies a complex

quantity, the superscript indicates tha t t he function is to

be evaluated jus t above the interface and the subscript

indicates the nt h Fourier component. All the hetero-

geneity and structure of the medium is incapsulated in

C, .

The e box represents t he st ep of solving the elec-

trosta tic field problem. Field quantit ies in the oxide

layer and in the medium are represented in terms of their

Fourier components. Boundary conditions at t he elec-

trode interface are used to match up the field solutions

for the regions above and below the electrodes. A mixed

boundary value problem occurs at this interface, as the

potential is constrained along that part occupied by th e

electrodes while conservation of charge and

Gauss’

law

/ n

a r r a y

spec f c a t on

A

-,G

f

w

Figu re 2.

Schematic representation of continuum model.

combine to pu t a constraint on the potential and on the

jum p in i ts normal gradient along the interelectrode part

of th e surface. The solution involves discretizing the in-

terelectrode surface by introducing a grid of unknown

voltages 5 a t k collocation points. The potential is as-

sumed to vary linearly between these collocation points.

A system

of

equations is generated by maintaining a dis-

cretized conservation of charge along surface segments

bracketing these collocation points. These equations are

written in the form A.V=X where the elements of A are

the coefficients of the unknown voltages

VJ

and the ele-

ments of X represent the known voltage c ~ .nverting

the matrix A yields the voltage distribution along the

interelectrode surface, and thus, the electric field distri-

bution.

In the final box, the evaluation of the

yij’s

is tan-

tamount to a determination of the response with any

terminal configuration. Due to the symmetry of the

electrode structure, Y11 =

E>z.

All of the admittances

representing the electrode st ructu re are determined by

finding the electrical terminal currents,

iD

and i ~ ith

the floating gate electrode grounded. These currents

are obtained by integrating the current density over the


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Electrical Insulation

Vol. 23 No. 6, December

1988

901

surfaces of the electrodes. Using these admit tances, the

response of the microdielectrometer w ith an arbit rary

load is the complex gain, the ratio of the floating to

driven gate voltage

where Yl is the load capacitance of the floating gate

FET.

In the medium above the electrodes and in the ox-

ide substrate below, the potential is represented by

(1)

and the fields are taken

a s

electroquasistatic, th e elec-

tric field intensity is irrotational and hence represented

by the potential Cp .

E

Vip

(3 )

For completeness, the oxide layer will be given

a

complex

permittivity,

E; ..

ELECTRODE STRUCTURE BOUND ARY

CONDITIONS

HE electrode structure of Figure 1 is modeled as

T

hown in Figure 3. The electrodes are treated as

being one dimensional, one set driven sinusoidally with a

peak voltage VD and frequency w , the other set grounded.

It is possible to include the finite thickness of the elec-

trodes using the techniques outlined here, but the effort

is not judged t o be worthwhile a t present. For gener-

ality, an interelectrode spac ing of arbi tra ry width

a

is

allowed. Th us , in a half period, the endpoint of one

electrode is at y = yo and the beginning of the other

electrode is at

y

= yo+

a

yk+1.

I

A

D

...

i

F igure 3 .

Coordinate system used for electrode structure

of

Figure 1 .

amount of free surface charge density

o,, (6

is the unit

normal t o the interface)

+

6

D + V c .

D,=

(5)

where implies the difference in the function within

the brackets evaluated ju st above and ju st below the in-

terface and Vc. s the surface_ divergence. Th e surface

electric displacement vector D, represents the possibil-

ity of polarization phenomena concentrated within the

interface. Conservation of charge accounts for the accu-

mulation of via the discontinuity in normal conduc-

tion current

J ,

including conductivity within the_ nter-

face represented by the surface current density J,,

( 6 )

S . p - ~ + V c . ~ + -o U = o

At th e electrodes, the potential

is

constrained, whereas

a t

along the surface between the electrodes there is only

continuity of the potential

Substituting

( 5 )

into

( 6 ) ,

using the complex nota-

tion introduced by

(1)

and assuming sinusoidal steady

state

(-

-+

i w ) yields the following boundary condition

Re

{

~ D e i u t } for o

5

yo

for Yk+l

I

a

(4)

at

where complex bulk an d surface permittivi ties have been

introduced,

E * = E' E

and

E:

=

E:

-

E ; ,

and the con-

duction and displacement volume and surface current

On the interelectrode surfaces, Gauss' Law relates

the discontinuity in the normal displacement field to the


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Zaretsky

et al.:

Continuum Properties from Interdigital Electrode Dielectrometry

densities have been combined 2,s the component of

l? in the interfacial plane),

A . .

d .

A , .

iw~ ,?? = J+ wl? ; we:E , = J, +

w 5 s ;

(8)

For an ohmic conductor E' and E: are constant, and

6'' =

a / w ) and

E: =

a , / w ) , where

a

and 0 are con-

stant.

For uniform surface properties

(7 )

yields the second

interelectrode region jump condit ion,

(4c), and conservation of charge, (6)' is satisfied. The

stra tegy taken here is to represent the interface potential

by piece-wise continuous linear segments having the col-

location potential V at each collocation end-point yj.

Illustrated in Figure 4 are four collocation points , intro-

duced in the interelectrode region between yo and yk+l.

An initial effort divided the region into equal length line

segments [13].

A

cosinusoidal distribution has been uti-

lized here to give finer resolution in the regions of higher

electric stress (adjacent to the electrodes). Thus, the

collocation points are at

Y j

= (YO

+ -) -

(;)cos(") k + l (12 )

for

0

5

j

5

k + 1,

and the potential distribution is

( V D

for

0

I y

L Y o

BULK RELATIONS

&"(y)

=

( Y j - Y j - 1 )

I

j

for Yk+l

I

Y

L

i

V j ( Y - Y j - l ) - V j - ~ ( Y - Y j ~

for

y j - l

5

j=1,2, . . . k+l

A

(13)

s discussed earlier, the field distribution in the half-

A

space of material either above or below the elec-

trodes can be represented by the appropriate surface

capacitance density. This quant ity is the ratio of the

the complex Fourier amplitudes of normal electric dis-

placement to potential, both evaluated on the ( a )or ( b )

side

of

the interface.

Substituting for &"(y) into

( 1 1 )

yields the Fourier

series coefficients

. XVD

1

,h a;= [ C O S ( ~ n Y O )-

COS(knY1)l

t

zh

-

n. ~

C;,b (10) (n n I2

Y 1

-

Yo)

42b

Here, the subscript n refers to the order of the Fourier

modes. Given the distribut ion of a complex permittiv-

ity

E: z)

above the electrodes, with the coordinate

2:

perpendicular to the interface (including surface singu-

1

2b

j = 1

C O S ( ~ ~ Y ~ + I )cos(knyj-1)

cos(kn

j )

-

Yj+1 - Y j Y j Y j - 1

larities in that d*istribution), the complex surface capac-

itance density CE is found by determining the Fourier

amplitude of the normal

flux

density at the surface

a)

caused by the nth term of the imposed potential having

the form of

(1).

POTENTIALS

AT

COLLOCATION POINTS

The Fourier amplitudes of (1) are

The potential

&

is known only at the electrodes.

On the surface, in the interelectrode region, the poten-

tial assumes a distribution such that it is continuous,

The problem has been transformed to one of solving

for k unknowns, the $ 9, with the continuity of potential

satisfied by (4c). The second continuity condition at the

interface,

(9),

remains to be satisfied. By dividing the

interelectrode surface into k intervals, centered on the

collocation points and satisfying conservation of-charge

(9), within every interval, k equations for the k

VJ's

are

obtained. The intervals are demarcated by points y lo-

cated a t the edges of the electrodes and at the midpoints

of the line segments formed by consecutive collocation

points, yJ an d yj+l (Figure 4).

YO for

j =

1

Yj* =

{

i Y j + Y j - l )

for

2

5 j 5 k (15 )

Yk+l for

j = k + I


7/21

IEEE Transactions on Electrical Insulation

90

A

"I

"2

A

A

"3

v4

A

A

v

-[

k = 4

Figure 4 .

Voltage distribution along discretized interelec-

trode surface for k=4 collocation points. Un-

known voltages

Vj

are introduced at the points

yj . Conservation of charge is maintained for line

segments defined by the

y

Vol.

23

No.

6,

December 1988

903

Figure 5.

Distribution of tangential electric field,

h;(y),

along electrode-medium interface.

respective segments, extend to the electrode as shown

in Figure 5.

Integrating (9 ) over each interval] from

y j

to

y ~ + l l

and setting the result to zero yields k equations for k

unknowns,

With this choice of

y;,

the ambiguity in the value

of the tangential electric field at the collocation points

yj is avoided. Using (3 ) and ( 13 ) yields

for j = 1 , 2 , . . I C . The third term, reflecting contribu-

tions due t o surface effects, integrates t o

(18)

(16)

Y j - Y j - 1

and using (16) yields

E (y*)

=

-

Y J

As

shown in Figure

5, E;(y)

is a piecewise function

E:, [(

1 +

).;

Yj+1

- Y j

Y j -

Y j - 1

(19)

tha t is discontinuous a t the collocation points. Th e in-

tervals over which charge is conserved (between broken

lines in Figure

5 )

extend half a segment in either direc-

tion about

a

collocation point

y j

except for the first and

last intervals, which respectively have end points a t t he

electrode edges. In additi on, the values of the tangential

field used for the endpoints

y;

and are not aver-

aged even though they are located at a discontinuity. It

is more realistic to assume that these values, for their

b

v, l -

Y j + l

Y j

Y j

- Y j - 1

Integrating the remaining two terms of

(17) ,

using

the cosinusoidal dependence of the normal electric field

upon

o

implied by (1)and

( 3 ) ,

yields the set of equations


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Zaretsky et

al.:

Continuum Pr operti es from Interdigital Electrode Dielectrometry

" 1

kn

- ( E :&

-

~ik:) sin(k,yj+,) in(knyj)]

n = l

1

Yj+1

-

Y j

+ E : ,

[

(-

Y j

-

Y j - 1

] = o

t 1

5-1

Y j + 1

j

Y j

-

Y j - 1

1 1

~ 3 ( i , = E

[(- +

-) 6 ( i

-

Yi+l

-

Yi

Yi -

yi-1

or

j

= 1 ,2 , . . k.

(23)

(24)

( i - j

+ 1)

-

q - j

-

1)

Yi+l

-

Y i

Y i

-

yi-1

and

X ( i )

= Xl(i)

+ XZ(2)

+ X 3 2 )

gain substituting for the electric fields, this time

using

(10)

(remembering continui ty of the potential across

an interface) into (20) yields

where

(C:

e:)

A

n = l

kn(nT)'

X,(i) =

-

[sin(kny;+, in(kn~i')]

1

+-+

X , ( i )

= -

[e: e;

(591

y;+l yi')

1

Yj+l

-

Y j

Y j

-

Y j - 1

6 ( i -

1)

X3(i) = &;o-

Y 1

-Yo

G + l

Y j t l

-

Y j

Y j

Y j - 1

for

j

= 1,2, . ,k.

The set of equations described

by

(21) can be ex-

pressed in matrix form, A.V=X, with the elements of A

representing the coefficients of the unknown normalized

voltages pj (V,/Vo), and the elements of X repre-

senting the coefficient of the known drive. In order to

generate the matrix elements, (14) is substituted into

(21) and terms are rearranged such that all terms pro-

portional to the drive are moved to the right hand side.

Any expression involving Cz also goes on the right hand

side as it can only arise from an externally applied field

and th us, is also a driven signal. This yields

where yj and yj* have been previously defined and

6 ( i )

is defined as unity if the argument is zero and zero for

nonzero argument. Using this representation, the set of

k equations in k unknowns can be numerically inverted

to determine the $,;'S.

A

more rapidly convergent series

for (23a) and (25a) is derived in Appendix

A .

EQUIVALENT CIRCUIT PARAMETERS

HE

pi network of Figure 2 is helpful for formalizing

T he transition from the field analysis just under-

taken to the general situation where the detection elec-

trodes are not grounded but rather are free to assume

a potential consistent with th ? attached circuit. With

the driven gate at potential V, and the floating gate

grounded, Y11 an d Y1z ar e directly related to the elec-

trode currents, and hence calculable from the electric

fields just determined with the floating gate grounded.

where

The electrode currents are found by integrating (6),

conservation of charge, throughout a volume enclosing


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I E E E lkansactions

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Electrical Insulation

Vol. 23 No,

6,

December

1988

905

each electrode.

length of M c l , these currents are

Assuming a total electrode meander

+ 2 J A i Z

k + l

[E:E,n(y

Solving (26) for Y11 and Y1z in terms of ;D and ;F

- .

and incorporating

( l ) , 3 ) , (10)

and (16) yields

Yl l

= 2iwM,l

{ E : , (-

- 3, -

Y 1 - YO Yk+l Yk

n

(sin(knyo) - Sin(knyk+l))

+&:

[e: - e:] ( Y O +

x / 2

- Yk+l)}

Ylz

=

2iwM,1

{ E : , , (

U

)

1

-4: [Cz

-

e:]

sin(k,yk+l)

kn

Yk+l - Yk

+

-&: [e:

-

e;] ( x / 2

- yk+l)}

where

Uj

( / V D ) .

Once (28) is programmed, it is

only the surface capacitance densities,

Ct,

that are al-

tered when the medium above is changed.

DETERMINATION

OF

GAIN

T this point it is instructive to recap the major

A steps for calculating the

xj’s

and hence, the gain.

Solution of a s ystem of equatio ns of the form

A-U=X

with th e matr ix elements defined by (22)-(25) deter-

mines the Uj’s nd the electric field distribution. With

these Cj’s , the Fourier coefficients for the potential a t

the interface, E and i ; t using

(14),

nd hence, the cir-

cuit parameters Y11 and Y12, using (28), are eva luated.

Finally, given

a

load capacitance

x

he gain is then

calculated according to

(2) .

COLLOCATION POINTS, FOURIER MODES

AND CONVERGENCE

E V E R A L tradeoffs must be considered for deciding

S

he optim um number of collocation points and Four-

ier modes. Increasing the number of collocation points

improves the representation of the actual potential

dis-

tribution - a t the cost of greatly increasing the comput-

ing time required to calculate each element of the ma trix

A

and t o compute the matri x inverse. Increasing the

number of Fourier modes used to compute the voltage

distribution increases the accuracy - of representing this

distribution with a piecewise linear function. Beyond a

certain number of modes the piecewise representation

may differ significantly from the ac tua l one. However,

as previously mentioned, and elaborated in Appendix

A, a more rapidly convergent series is used to sum the

Fourier modes for the elements of the A matrix . This

series effectively sums up roughly

1000

modes. In this

case, it is quicker to compute either a small or a large

number of modes than to compute an intermediate num-

ber. Using the large number of modes, th e number of

Fourier modes summed in (28) when computing Y11 and

Y1z can be altered. Increasing the number of terms used

here has

a

much smaller, though still significant, effect

on the computing time.

Determination of convergence requires a specifica-

tion against which various results can be compared. For

the case of the micr odielectrometer, where comparisons

are made with experimental results, the accuracy of the

gain is given as 0.1 dB [7]. This accuracy will be used

as

a tolerance for measuring convergence. Modeling an

uncoated microchip in air, the high-frequency coupling

capacitances per unit length (normalized to the oxide

layer permittivity) and resulting gains were computed,

using the large number of Fourier modes, for a mat rix of

values for the number of collocation points and summa-

tion terms (used in (28)). It was observed that using 25

collocation points and 100 summation terms provided

sufficient convergence when compared to t he maximum

values of

35

and

1000,

respectively (as well as taking a

reasonable amount

of

computing time). For the rest of

this work, unless otherwise noted, 25 collocation points

and

100

summation terms will be standard. The spe-

cific values used for the oxide layer thickness

h

and load

admi ttan ce were supplied by Micromet Instru ments,

Inc., on a proprietary basis and were used to generate

the specific gain-phase responses in this paper.


10/21

906

Zaretsky

et al.:

Continuum Pr opertie s from Interdigital Electrode Dielectrometry

TYPICAL RESPONSES USING

CONTINUUM MODEL

PRED ICTED RESPONSES

s a first step toward developing a scheme for iden-

A tifying continuum parameters, this Section is de-

voted to forming insights concerning the general rela-

tionships between these parameters and the gaip-phase

response. *For all of the following examples E, = 0 ,

and thus C:

=

0. Using the form expressed in ( l ) , he

solution of Laplace's equation for the potential in the

insulating oxide layer below the electrodes gives

(29)

where the layer has a thickness h and is bounded from

below by a perfectly conducting' s ubstrate.

F i g u r e 6.

Electrode structure below a ) uniform medium

or

b)

variable thickness homogeneous layer and

uniform medium.

UNIFORM MEDIA

This situation, a semi-infinite half space of uniform

complex permittivity, is pictured in Figure 6a. The po-

tentia l satisfies Laplace's equation and decays to zero

as

x

goes to infinity. Thus,

(These expressions, ( 2 9 ) and ( 30 ) , illust rate how transfer

relations naturally fit into the spectral field description

~ 4 1 ) .

- 1 0

\

U

A'

i

I I

- 3 - I I 3

l o g f

Figure 7 .

Predicted responses for microchip

(X=50

mi-

crometers and

a

=

X / 4 )

in uniform medium

( B I = 2 x

lo-

F/m), vary conductivity

ul.

For ohmic media, typical responses have the fea-

tures of

a

linear system with one time constant - the

charge relaxation time, r, = ( E ~ / c T ~ )Figure 7 ) . With

the frequency high,

w r ,

>>

1,

the effect of dissipation

currents is small. Th e coupling between the two elec-

trodes is purely capacitive, explaining the low gain at


11/21

IEEE Transactionson Electrical Insulation

Vol. 23

No. ,

December

1988

90

7

are shorted together ( 0 dB gain and zero phase). It is ev-

ident from the equations th at the frequency enters only

- 4 0 -

-

I I 1

VARIABLE THICKNESS LAYER

with the bulk conductivity - altering the conductivity

0 .

merely shifts the frequency response without changing

the curve shapes (Figure 7). Changing the bulk per-

mittivity alters both the breakpoint (due to

a

change

in

r,

and the high frequency gain (due to

a

change in

the capacitive coupling),

as

shown in Figure

8.

When

the phase is plotted versus gain for the curves shown in

Figs. 7 and 8 (eliminating the frequency as a parame-

ter), the graph reproduces the lookup table developed

for the application of the microdielectrometer to homo-

- 2 0 -

W

V) - 40

,

a

-60-

geneous systems by Lee [IO].

-80-

A layer of thickness dz and characterized by a uni-

form complex bulk permittivity is placed immediately

above the electrodes (Figure 6b). Solution of Laplace's

equation in

a

piece-wise fashion, perhaps using transfer

relations (see Appendix B or [14]), gives

-

-

-

-

-

-

-

+

coth(k,dz)]-'}

As background for inferring the layer thickness from

a

measurement of the gain, the effect on the frequency

response of varying dz while holding the complex bulk

permittivities of each region constant is illustrated by

Figure

9.

In this case, the half space is taken t o be more

insulating than the layer and the surface properties of

the layer-half space interface are taken to be zero.

- 3

- I

3

l o g f

Figure

8.

Predicted responses for microchip in uniform

medium u1

= lo-'

S/m), vary permittivity

c l .

the thick layer for purely geometrical reasons. As the

layer thickness is decreased, the response is still

a

bulk

one, but now it also reflects the fact that the fields have

begun to penet rat e into the semi-infinite region, where

the permittivity is less (accounting for the decrease in

high-frequency gain) and the conductivity is zero. At

dz - O. l (A / 27 r ) , th e layer is so thin that it might well be

described by

a

surface conductivity on the interelectrode

interface.

This case is discussed in the next Section.

Further decreases in the layer thickness reduces the ef-

fective interelectrode surface conductivity. Thi s results

in a shift of the interelectrode surface type response to

lower frequencies. Generalization of th e

C,

to include

multiple layers is presented in Appendix B.

SINGULAR PROPERTIES AT

SUBSTRATE-MEDIUM INTERFACE

For

a

thick layer, dz >

(A /27r ) ,

the response is that

of

a

uniform half-space, determined solely by the prop-

erties of th e layer. T he electrodes do not 'see' beyond

This is again the case of a uniform half-space (Fig-

ure

sa)

with

C:

given by (30) , but with a surface con-

duct ivit y at t he SiOz-medium interface between the elec-


12/21

908

Zaretsky

e t

al.: Continuum Properties from Interdigital Electrode Dielectrometry

-

40 r

a

- * O F

-

IZ0L

3

el

=

c 0 F / m -

-

I I I

- I

I

3

I I

1

l o g f

l o g

f

Figure 9 . Figure 10.

Predicted response for microchip in uniform

medium

(c1 = 2x10-

F/m, u1 =

lo-"

S/m)

with

surface

conductivity on interelectrode sur-

face

( u a o= 1 0 - l ~ S).

Predicted responses

for

microchip with variable

thickness layer (EZ = 2 x lo- F/m, uz

=

2.4 x

IO-

S/m) in air.

trodes E:,, = - i (ano /w) . A typical response is shown

in Figure 10 - clearly distinguishable from

a

bulk re-

sponse. There is an additional characteristic time,

r6e

=

( c 2 / k n a 6 , ) , epresenting surface charge diffusion along

the interface. Again the interplay between surface and

bulk conduction and displacement currents accounts for

the transition from low to high gain and a peak in phase.

However, the slope of the gain is steeper than -20 dB/

decade and the gain has an overshoot. The phase curve

is asymmetric and the phase peaks at a larger angle than

previously seen. All these are characteristic of the sur-

face charge diffusion from driven to floating electrodes

made possible by the combination of the distributed sur-

face conduction and the shunt capacitance between th e

interface and the highly conducting subs trate.

As the surface conductivity decreases (rnencreases),

the curves pass through a regime where surface and

bulk conduction are equally importan t to a point beyond

which

a

purely bulk response is observed (Figure 11).

The overshoot phenomena,

a

prediction of gains be-

low the purely capacitive high-frequency gain, is further

evidence of the surface charge diffusion process. This

can be explained by separat ing the coupling between the

electrodes into two components, one through the bulk

media and one through a distributed transmission line

composed of the interelectrode surface and the insulat-

ing oxide layer. The coupling through the bulk makes

a

smooth transition from purely conductive to purely

capacitive, just as in the bulk response, and therefore

involves a t most a 9 0 phase shift. However, the part

of the signal resulting from transmission along the dis-

tributed

RC

transmission line comprised of the inter-

face and the insulating oxide layer suffers larger phase

shifts and hence

a

contribution tha t tends to cancel that

due to the 'direct' coupling. The result is

a

frequency

range over which the gain is smaller than that at high

frequency and the phase shift larger than 90

* .

The


13/21

-

rn

c

c

.-

a

In

0

.c

a

I E E E

I'ransactions on Electrical Insulation

\\ \ I

- 3

- I I

3

l o g

f

Figure

11.

Predicted responses for microchip in uniform

vary surface conductivity on interelectrode

sur-

face ( o ~ ~ ) .

medium

(cl

= 2x lo-" F/m,

01

=

lo-"

S/m),

diminution of surface conductance coupled with an ex-

isting large bulk capacitance 'snaps' the response from

a

surface conduction coupling to

a

purely capacitive cou-

pling and accounts for the rapid changes exhibited in the

frequency response. This behavior provided the ra t i e

nale for the distributed parameter model,

as

developed

and implemented by Garverick [ll].

LAYER WITH SURFACE COND UCTIVIT Y

In practical applications such

as

the reduction of

electrostatic discharges

(ESD),

polymer films are often

coated with conducting films. With th e objective of

measuring the surface conductivity of these films with-

out electrical contact, the frequency response of a layer

(Figure 6b) with surface conductivity attributed to the

Vol. 23 No. 6, December 1088

000

.-

-30

- 3 - I I 3

l og

f

Figure 12 .

Predicted responses for microchip with a

layer

( d 2

=

5pm, c z =

ZxlO-"

F/m, u 2

=

lo- S / m )

having a variable

surface

conductivity ( 0 . 2 ) in

air.

layer-medium interface is of interest. Th e surface ca-

pacitance density is given by

(31)

with

a 2

having only

an imaginary component.

In Figure

1 2

is shown the frequency responses for

an electrode str ucture with a layer thickness dz = 0.1X

in air. By increasing the layer's surface conductivity,

aa shown, two relaxation phenomena are observed. The

higher frequency one results from the spectrum of times

associated with diffusion of surface charge along the up-

per surface of the layer. Th e lower frequency one is

from relaxation determined by the bulk properties of

the layer.

If c82


14/21

g10

Zaretsky et al.: Continuum Properties from Interdigital Electrode D ielectrometry

mid-frequency gain plateau and the higher frequency

phase peak.

SENSITIVITY T O ELECTRODE STRUCTURE

PAR A

M

E TE

RS

There are four parameters describing the electrode

structure, each with its individual effect on the gain-

phase response. The spatial wavelength X has been dis-

cussed previously with the application of intimate sens-

ing. Thus, this value should be consistent with the char-

acteristic length of the medium under measurement.

Two other parameters are the insulat ing layer thick-

ness h and permittivity, They are dependent on the

material used (for

.cox)

nd the fabrication method (for

h) . As mentioned earlier, varying h can significantly

affect the response to a surface conductivity a t the elec-

trode boundary. Thi s effect can be extended further

to include the response to thin 4 - O.l(X/27r)) lay-

ers having either bulk

or

surface conductivities. With

a relatively thick, high-permittivity oxide layer, there is

no surface-like response with overshoot gains and asym-

metric , large phase peaks when examining such thin lay-

ers [15]. As shown by Li, this can be altered so

as

to

optimize sensitivity to these type of phenomena by re-

ducing

h

and

This lack of sensitivity with thick, high-permittivity

insulating layers actually extends to measurements of

not only surface but bulk properties of thick and thin

layers. Referring to

(2),

the gain changes with the ma-

terial above due mainly to changes in

Y12.

The admit-

tance Yl1 is basically capacitive and much larger than

the capacitive part of

Y12.

As shown by Li [15],

Y12

can be thought of

as

representing two parallel couplings

between the floating an d driven gates , one above the in-

terface and through the medium, and the other below

the interface and through the insulating layer. With h

large, the couplings above and below are equal. Chang-

ing the permittivity of the medium above will change

the net coupling (the sum of the two couplings) by a

much smaller percentage. If the permittivi ty below is

much larger than above, this percentage change shrinks

even further. With

h

small , the coupling below is much

smaller than above. Most of the field below couples

to the ground plane, thus the contribution to Y12 from

below is greatly reduced. Now changes in the medium

above are more fully reflected in changes in Y12 and

the sensitivity is greatly enhanced. Thus , for maxi-

mum sensitivity, the oxide layer thickness and permit-

tivity should be minimized (subject t o fabrication con-

straints).

The fourth parameter is the interelectrode spacing

a. Using physical intuition, if

a

is decreased then the

electrode-ground plane structure approaches that of a

parallel plate capacitor. It is as if a is an ‘electric’ shut-

ter controlling the amount of electric field penetrating

into the medium. Sensitivity to changes above will be

reduced with small values of a. However, increasing

a, while improving sensitivity, decreases the net cou-

pling between the electrodes. This reduces the floating

gate voltage and requires the electronics to accurately

measure very small voltages with correspondingly small

signal-to-noise ratios . Thu s, it appears th at the (X /4 )

width for a is a valid middle ground.

PARAMETER

ESTIMATION

GENERALAPPROACH

N the previous Section, the problem was posed as

I

ne of determining a frequency response, given all

the pertinent complex permittivities, layer thicknesses,

and other relevant parameters describing the media. In

practice, the situat ion is usually reversed. The unknown

quantity is a material property such

as

complex permit-

tivity (bulk or surface, dis tribution if inhomogeneous)

or layer thickness. What is known is one or more ex-

perimentally measured responses, perhaps at different

temporal or spat ial frequencies. A parameter estimation

scheme is required in which all

a pr ior i

knowledge of the

physical situation is utilized, such as number of layers,

values of complex permitt ivities for each layer, and layer

thicknesses. Noise or other stochastic processes affect-

ing the experimental data call for more sophisticated

techniques, not considered here. Thus, here the estima-

tion routines may be viewed either as root searching or

function minimization routines [16].

The search is for

the root t o the equation

e(@)

= 0 ( 32 )

where

e(0)

is a set of error functions (usually the differ-

ence between one or more measured an d predicted gains)

and

0

is a list of parameters to be estimated (such as

complex permittivities). Of course, the particular phys-

ical phenomenon associated with the parameters to be

estimated must make a significant contribution to the

gain, and hence, the error.

A secant method of searching was employed for

most of the parameter estimations described here (171.


15/21

I E E E Transactions on Electrical Insulation

Vol.23

No. 6,

December

1988

-

5 x

l o - ' 1

3x

l o - 11

2

x

10-1'

-

T

E O -

I I I I I l l

1 1 9

I t 1

In this method, guesses 8; are update d by A8; using the

secant formed by the two most recent guesses,

( 33 )

The danger that the root does not necessarily re-

main bracketed by the two guesses is decreased by pro-

viding bounds on the minimum and maximum values

of the estimated parameter based upon physical con-

st raints.

For application to microdielectrometry, acceptable

tolerances for convergence are de termin ed by th e exper-

imenta l error in the response. Th e present version of the

device, when using a microchip sensor, has an accuracy

of

0.1

dB in gain and

0.1

in phase [7]. In the routines,

the gain tolerance is tightened to 0.05 dB. This trans-

lates t o a tolerance of

0.5%

in gain at

-40

dB and

0.1%

in phase at -90 *

.

The emphasis

w a s

on developing search routines

that converged and were somewhat robust, not on gen-

erating the optimal search method. Most of the val-

idation of these search algori thms was obtained using

experimental da ta on semi-insulating materials. Con-

sequently, there are areas of parameter space where the

convergence and robustness of these routines have not

been examined.

SINGLE PARAMETER ESTIMATION

For these estimations, the error is defined as the

difference between th e measured and th e predicted com-

plex gain,

where 8

iy

the complex parameter to be estimated. Here,

n and G,,are formed by evaluating the complex loga-

rithm of the complex voltage ratio,

I

"D

where the real part is the gain in dB divided by

20

and

the imaginary part is the phase in radians (closely re-

lated to the output of the microdielectrometer). If

G ,

as defined in (2), s a complex analytic function of the

911

complex variable 8, then so are and 6. Thus, for the

class of constitutive laws used here, it

is

meaningful to

apply directly one-dimensional root-finding techniques

to t he estimat ion of

a

complex parameter.

Note that although one parameter is being esti-

mated, this does not constrain the medium ,to be uni-

form. Using the surface capacit ance density,

C i ,

hetero-

geneous media can be represented as long as all but one

of the parameters necessary to characterize t he medium's

dielectric properties is known.

LAYER THICKNESS

This is th e case of a layer and

a

surroun ding uniform

medium, as pictur ed in Figure 6b. At high frequen-

cies, when the electrodes are purely capacitively cou-

pled through the material, the phase will be zero and

the gain will be a function of the layer thickness and the

permittivities of the layer and the surrounding medium.

Given any three of these four quantities, it is possible

to estimat e the fourth. Here the layer thickness dz is

unknown, thus 8 = dz and the search is for a purely real

root.

E I = e oF/m

- I O

m

2 - 2 0

0

.-

U - 3 0

- 4 0

t /

I x

10-10

Figure

13.

Predicted high hequency (10

kHe)

gains

for

mi-

crochip with variable thickness layer (&), vary

layer permittivity ( L Z ) , in air.

For the case of air as the surrounding medium, the

curves of gain versus layer thickness for various layer

permittivities are shown in Figure

13 .

Finding the root

of these functions is relatively straightforward. For ro-

bustness, the routine first checks whether the experi-

mental data makes sense by calculating the gain of a


16/21

912

Zaretsky

et al.:

Continuum Properties from Interdigital Electrode Dielectrometry

uniform medium having the lesser of the two permittiv-

ities. Thi s gain should be less than the experimental

gain if a layer thickness is to be estimated. After pass-

ing this check the routine uses the secant method to

find the root. The search is conducted using

25

collo-

cation points I C ) and

100

summation terms

( N ) .

Due

to the well-behaved nature of the curves, the thickness

is uniquely estimated within 3 to 6 itera tions. For a

50 pm wavelength electrode structure, the 0.1 dB gain

tolerance implies

a

sensitivity of better than 5% in the

thickness estimate within the range of

2

to

12

pm. The

lower bound may be determined by the experimenta l er-

ror in the measurement or by features of the electrode

that are not modeled such

as

electrode thickness. The

upper bound reflects the decrease in sensitivity due to

the exponential decay of the electrostatic fields.

COMPLEX PERMITTIVITY

(BULK

OR SURFACE)

A S

long as

$ ( e )

is an analytic function o fa complex

bulk or surface permittivity, the search is only slightly

more complicated. A complex bulk or surface permit-

tivity is estimated using a complex data point (gain and

phase) and specifying all other relevant parameters such

as the complex bulk permittivities of the other regions,

the complex surface permittivities a t the interfaces and

the layer thicknesses. Now, in

(33),

6 =

E’

-

id’ or

E: ie; and .^ e) ,n

(34),

is complex.

The search routine can be started either with a user

input guess or with i ts own ‘guess’as to the complex per-

mittiv ity. For bulk proper ties this ‘guess’ corresponds

to that of an insulating, nonpolar medium. The first up-

date only is calculated using the local derivative. For the

secant met hod the order of convergence is the ‘golden

ratio’ of

1.618..

.

[18].

The key here is to get in t he neigh-

borhood of the root where the convergence is rapid. A

substantial reduction in the time to convergence is ob-

tained by letting the routine s tart with its own ‘guess’,

and using a technique of varying the number of col-

location points used for the function evaluation. This

technique performs sequential searches, beginning with

a coarse discretization of

2

collocation points and the

routine’s initial guess, and culminating with a search

using 25 collocation points and an initial guess from the

previous coarse estimate. Generally, anywhere from 20

to 40 iterat ions are required for convergence with 2 col-

location points, resulting in a requirement of only 1 to 6

subsequent iterations at

25

collocation points. It takes

roughly

12

times longer to compute a gain-phase re-

sponse using 25 collocation points versus 2 collocation

100)

1

I

I I I- I20

- 4 0 - 30 - 20 - I 0

G a i n

( d B )

Figure

14.

Parameter space for complex bulk permittivity

estimation of

a

5 micrometer layer in a uni-

form medium

( E ;

= 2 x 1 0- ” F/m and

E:

=

lo-’’ F/m). € 4 and

&‘

are in units

of E”

F/m).

points. Thus, a substantial savings in computing time

can be realized.

A two-dimensional view of the space over which the

routine searches when estimating the complex bulk per-

mittivity of a ayer 5 pm thick surrounded by a medium

with a bulk permittivity of E: = 2 x 10- l ’ F/m and a

loss factor of

E:

=

1 lo-’ F/m is shown in Figure

14.

This space is constructed by computing the gain-phase

response for a matrix of complex bulk permittivities of

the layer at a

cons tant frequency. The contours are lines

of constant

E ;

and E” and are orthogonal because 6 is an

analytic function of

E * .

Reiterating, this orthogonality

justifies the computation of the complex slope used by

the one-dimensional secant search routine. As required

by

(35),

the gain is in dB divided by

20

and the phase in

radians multiplied by log e. This figure is analogous to

the lookup table given by Senturia,

e t

al.

[9]

for the case

of a uniform medium. With the gain defined by (35) the

contours are orthogonal for the uniform medium case

too.

In Figure

14, the values of

E;

that make the layer

more lossy than the upper medium are concentrated

near the origin. The dot in Figure 14 is where the layer

and surrounding medium have equal complex permit-

tivities. By contrast , in Figure

15

the upper medium

is sufficiently insulating that the point of equal com-

plex permittivities (the do t) is near the zero phase axis.

Again, with a

5

pm thick layer, the surrounding medium

now has a bulk permittivity of E: =

2 x 1 0 - l ’

F/m and a

loss factor of E:

= 1 ~ 1 0 - ~ ’

/m. The kink that appears

in the left side of the plot is

a

result of charge diffusion

along the layer,

as

discussed earlier in regards to the

thin film humidity sensor.


17/21

I E E E Transactions on Electrical Insulation Vol. 23

No.

6, December 1988

913

0

Simultaneous parameter estimation of more than

one complex parameter clearly requires several experi-

mental data points, varying the value of

a

known pa-

rameter. As pointed out above, there should be at least

one data point representative of each physical process of

- 2 0

-

5

p

- 4 0

'y

- 60

-80 L

interest.

n

-100

- 120

- 40 -30 -20

-10

0

Gain ( d B )

SUMMARY REMARKS

Figure 15.

Parameter space for complex

bulk

permittivity

estimation of a 5 micrometer layer in a uni-

form medium (e; = 2 x lo-" F/m and e;

=

IO-

F/m) . (e; and e: are in units of eo F/m) .

Given the well-behaved nature of the spaces repre-

sented by Figures

14

and 15, it is expected th at estima-

tion of unique complex permittivities is a straightfor-

ward process. T he sensitivity of the estimation routine

can be observed from the two Figures. Poor estim ates

occur in regions of high contour density. For example,

gain-phase da ta adjacent to the gain axis yield poor

loss factor estimates while data with gains near zero

yield poor perm ittivit y estimates. The la tter is a conse-

quence of the high-impedance mode of operation which

utilizes th e measurement of a floating voltage.

For the special case of ohmic media, E = a / w .

Thus, contours of constant

E;

are also the gain-phase

trajectories obtained if the temporal excitation frequency

is varied. This may also be seen by eliminating the in-

dependent parameter of frequency from the plots in Fig-

ure

8.

For the gain-phase spaces of Figures 14 and 15,

valid data must lie in the region having a lower bound

given by the

E'

= curve. Upper and lower bounds

can be placed on the parameter estimates to prevent

the search from wandering too far

off.

These bounds are

physically motivated

- no

bulk permittivities less than

and

no

conductivities less tha n zero. Upper bounds

are chosen based on reasonable guesses of the order of

magnitude of the dielectric properties of the materials

to be measured.

In addition to estimating bulk parameters, this rou-

tine works just as well for estimating complex surface

parameters.

H modal representation relating the gain-phase

T

esponse of the microdielectrometer to the contin-

uum properties of

a

medium has been formulated so as

to retain generality and flexibility in its application to

various linear systems. This approach greatly extends

the usefulness of the device by providing

a

framework

for interpreting data obtained for heterogeneous media

in terms of absolute continuum parameters rather than

'lumped equivalents'. Typically, the expressions sum-

marized here are programmed as a subrout ine which can

then be incorporated into a parameter-estimating main

program. Th e main program requires the surface capac-

itance densities for the specific medium thought to be

under investigation. The medium might be represented

by a system of discrete layers, with the computation of

the surface capacitance densities organized

as

in Appen-

dix B.

Although parameter estimation using the contin-

uum model described here is applicable t o multi-variable

systems, the key to this and other approaches is in iden-

tifying the types of data that can be mapped into the

contin uum properties and geometry. Illustrated here

have been one-dimensional schemes for the identifica-

tion of film thickness, the complex permittivity of finite

thickness films and th e surface conduct ivity of films. As

a practical matter, these are being used for interpreta-

tion of experimental dat a in an on-line manner. Also

developed, though not discussed here, is the estimation

of both th e thickness and permit tivity of a film from two

measurements of gain.

Closely related to the techniques described here are

those under development for nondestructive evaluation

and robotic sensing [19-211. Capaci tiv e electrode arrays

are again employed. However, a finite element model is

used to interpret experimental results.

The ability to keep track of basic continuum param-

eters has proven to be useful in several contexts. One

is the estimation of thickness and properties of plasma-

deposi ted layers and of sedimenting colloidal particles in


18/21

914

Zaretsky et al.: Continuum Properties from Interdigital Electrode Dielectrometry

liquids [22]. Work will be reported elsewhere in which

these techniques have been used to investigate trans-

former oil, with specialized coatings used to turn the

microdielectrometer into

a

means for sensing either the

complex bulk permitt ivity of th e oil [22] or the mois-

ture content [23]. The ability to infer from the fre-

quency response the continuum parameters needed to

quantif y surface adsorpt ion of anti-static agents and

trace amounts of water while keeping track of changes in

the bulk properties of coatings

is

essential in sorting out

heterogeneous phenomena. There are, of course, limits

on how much information can be gotten from varying

the t empor al frequency. However, by varying the fun-

damental spatial wavelength of the electrodes, a new

approach can be taken to sorting out distributions of

properties. Para mete r estimat ion schemes for inferring

the spati al distribution of properties by exploiting mea-

surements of gain made a t variable wavelength an d fixed

temporal frequency can use the numerical approach that

has been derived here and will be described elsewhere.

sin(na) cos(np)

c

n

n2

n=

1

To

reduce the computational time required for this

summation a more rapidly convergent series was used.

Expression (36) can be expanded to the form of

Assuming Hn --+

Hmin

as n -

0

the sum

W

sin(n7)

Hmin 2

n = l

(37)

will be computed and the difference between Hn and

Hmin will be accounted for lat er on. From Oberhet-

tinger [24] comes the expression

2

-1 In [2sin(O/2)] de

n = l (39)

\ I

ACKNOWLEDGEMENT

Using a series expansion [25]

HIS

work is being carried out in the Laboratory for

T

Electromagnetic and Electronic Systems

at

MIT,

with financial support from members of MIT's Electric

Utilities Program. The companies which are supp ort-

ing the work include Allegheny Power Co., American

Electric Power Service Corp., Boston Edison Co., Em-

pire State Electric Energy Research Corp., Northeast

Utilities Service Co., N.Y. Power Authority, Southern

California Edison Co. and Tokyo Electric Power Co.

Professor Stephen D. Sent uria provided useful com-

mentary on the intellectual and stylistic content of this

paper. Micromet Instrum ents , Inc. and in particular,

Huan Lee and David Day provided critical information

regarding the operation and performance characteristics

of the microdielectrometer.

APPENDIX A

R A PI

D Y

C

0 N

V

E

RGE

N

T S

ER ES

H E individual terms in (23a) and (25a) that are to

T e summed over the indices n have the general form

of

t 2 t 4 t 6

In[sin(t)]

=

ln[t]-

- -

6 180 2835

(-

l ) m + 1 2 2 m B 2 m t 2 m

(40)

- -

. . .

2m( 2m )

( -

l)m+122mB2,,p

= 1n[t1-

2m( 2m )

m = l

for

t 2

<

x 2

where

B;

re the Bernoulli numbers, and

substit uting in (39) yields

.-

n = l

Due to the factorials present in the denominator in

the summation term of (41 ) , only 5 to 10 terms need

to be computed to obtain

a

high degree of accuracy

(comparable to computing the first 1000 terms of the

original series). At this point, a calculation

of

the terms

for which Hn is significantly different from H,n;n is per-

formed and the difference is added to th e results of (41).


19/21

I E E E

Transactions on Electrical Insulation Vol.

23 No. 6 ,

December

1988 915

APPENDIX B

From these equations, it follows that

A$)Ac;i,,

(47)

URFACE CAPACITANCE DENSITY OF

MULTIPLE LAYERS

+ l )

=

Am

22

(Ad - Ac;i,)

+

k y j

HE

medium may actually be comprised of

a

finite

T

umber of layers, each layer described by a set of

parameters,

or

layers may be used t o approximate what

is actually

a

smoothly inhomogeneous material. Wit h

the objective of describing either of these situations,

suppose that the medium is composed of

P

layers, as

shown in Figure 16. The j t h layer has a thickness

d j

and an upper surface designated by j . The surface per-

mittivity of the P

+

1 surface, the electrode-medium

interface, is handled differently from other continuum

properties and designated

czo .

There is no cZ1 as this

surface property is associated with the j

=

1surface and

is normally a t infinity. Solution of the field laws within

the layer gives the transfer relations [14],

This expression can be used repeatedly, starting

from the top layer ( j

= 1)

and working down to the

P t h layer. BY definition, the surface capacitance den-

sity called for in evaluating the complex gain is

en

ep1)

(48)

- ' t h s u r f a c e

t h l a y e r

where DC l) and 62 ) are respectively the complex

amplitudes of the nth Fourier components of the dielec-

tric flux density and potential, evaluated just above the

lower interface of the layer and 6;')' and &;I are re-

spectively these quantities evaluated just below the up-

per interface. In the case of a layer having a uniform

complex permittivity [14].

@;ti)

j n ( j + O

f b +l)'

Dn

G P

At the jt h interface, the potential is continuous, but

there is a complex surface permittivity i jnd hence a

discontinuity in the dielectric flux.

(44)

(i)

= &(j)'

Figure

16.

Medium above electrodes represented by

a

mul-

tilayered structure of

P

homogeneou s layers.

In the case where the first surface is at infinity,

the terms

in

(47)

are

Ier0

so tha t cp'

be

evaluated without A1). For the finite thickness layer

bounded by an infinite half space, P = 2 and

(31)

fol-

lows from evaluation

of

(47) with CA2)- A(1)

The surface capacitance density of the jt h interface

is defined in terms of the quantities evaluated just above

that interface.

p

j )

(46)

-

2 .


20/21

916

Zaretsky

e t al.:

Continuum Properties from Interdigital Electrode Dielectrornetry

REFERENCES

on Electron Devices, Vol.

ED-29,

No. 1, pp. 90-94,

1982.

[12] T. M. Davidson and S. D. Senturia, “The Mois-

ture Dependence of the Electrical Sheet Resistance

of Aluminum Oxide Thin Films with Application to

Integrated Moisture Sensors ”, in Proc. IEE E, Int’l.

Reliability Physics Symp., San Diego, CA, pp. 249-

252, 1982.

[l] J. C. Ziircher and J . R . Melcher, “Double-Layer

Trans duction At A Mercury-Electrolyte Interface

With Imposed Temporal And Spatial Periodicity”,

J . of Electrostatic s, Vol. 5, pp. 21-31, 1978.

[2] J. Melcher, “Electro hydro dynami c Surface Waves”,

in Wa ves On Fluid Interfaces, R. E. Meyer, ed., Aca-

demic Press, NY, pp. 167-200, 1983.

[13] L. Mouayad, Mon itoring of Tra nsform er Oil using

Microdielectrometric Sensors, SM Thesis, EE, MIT,

Cambridge, MA, Feb., 1985.

[3] J . R. Melcher, “C harge Relaxation on a Moving Liq-

uid Inte rface,” P hys . Fluids, NO. 10, pp. 325-331,

1967.

[14]

J.

R. Melcher, Contin uum Electromechanics, MIT

Press, MA, p. 2. 33, 1981.

[4] S. M. Gasworth, J . R. Melcher and M. Zahn, “In-

duction Sensing of Electrokinetic Streaming Cur-

rent”, in Conf. on Interfacial Phenomena in Practi-

cal Insulating Systems, Nat’l Bureau of Standards,

Gaithersb urg, MD, Sept. 19-20, 1983.

[5] N. F. Sheppa rd, Jr., D. R. Day, H . L. Lee and S. D.

Senturia, “Microdielectrometry” , Sensors and Actu-

ators, Vol. 2, pp.

263-274, 1982.

[6] S. D. Senturia and S. L. Garverick, “Method and Ap-

paratus for Microdielectrornetry”, U. S. Patent No.

4,423,371, Dec. 27, 1983, assigned to MIT, licensed

to Micromet Instruments, Inc.

[7] Eumetric SYSTEM

I1

Microdielectrometer manufac-

tured by Micromet Instruments Inc., 2 1 Erie St.,

Suite # 22, Cambridge, MA 02139.

[8] Low Conduc tivit y Sensor, Model S-20, available from

Micromet Instruments Inc., 2 1 Erie St., Suite

#

22,

Cambridge, MA 02139.

[15] P. Li,

Low

Frequency, M illimeter Wavelength, In-

terdigital Dielectrom etry of Insulating Media in a

Tbansformer Environment,

SM Thesis, EE, MIT,

Cambridge, MA, May, 1987.

[16] L. Ljung, System Identification: Theory for the

User, Prentice-Hall, Inc., NJ, 1987.

1171 T. R. Cuthbert, Jr., Optimization Using Personal

Computers , Wiley, NY, 1987.

[18] W. H. Press, B. P. Flannery, S. A. Teukolsky and W.

T. Vetterling, Numerical Recipes, The Art o Scien-

tific Computing, Cambridge University Press, NY,

1986.

[19] B. A. Auld, J . Kenney and

T.

Lookabaugh, “Elec-

tromagnetic Sensor Arrays

-

Theoretical Studies”

Review of Progress in Quantitative Nondestructive

Evaluation, Vol. 5A, (Ed. by D. 0. Thompson and

D. E. Chimenti), Plenum Press, NY, pp.

681-690,

1986.

[91 s. D. Senturia, N. F- ShePPard, Jr.1 H. L. Lee and D.

R. Day, “In-situ Measurem ent of the Prope rtie s of

Curing Systems with Microdielectrometry”, J. Ad-

hesion , Vol. 15, pp. 69-90, 1982.

[20] P. R. Heyliger, J . C. Moulder, P. J. Shull, M. Gim-

ple, and

B.

A. Auld, “Numerical Modeling of Ca-

pacitive Array Sensors Using the Finite Element

Method”, Review of Progress in Quantitative Non-

destructive Evaluation”, Vol. 7A, (Ed. by D. 0 .

Thompson and D. E. Chimen ti), Plenum Press, NY,

lo ] H. L.

Lee, Optim ization of a Resin Cure Sensor, EE

Thesis, MIT, Cambridge, MA, Aug., 1982.

pp.

501-508, 1987.

[ l l ]

S. L. Garverick and S. D. Senturia, “An MOS De-

vice for AC Measurement of Surface Imped ance with

Application to Moisture Monitoring”, IEEE Trans.

[21] M. Gimple and B. A. Auld, “Position and Sam-

ple Feature Sensing with Capacitive Array Probes”,

Review of Progress in Quantitative Nondestructive


21/21

I E E E Transactions on Electrical Insulation

Vol.

23

No.

6 , December 1988

Evaluation”, Vol. 7A, (Ed. by

D.

. Thompson and

D. E. Chimenti), Plenum Press, NY, pp. 509-515,

1987.

[22] M. C. Zaretsky, P. Li and J. R. Melcher, “Estima-

tion of Thickness, Complex Bulk Permittivity and

Surface Conductivity Using Interdigital Dielectrom-

etry ”, IEEE Int’l. Symp. on Electrical Insulation,

Cambr idge, MA, pps. 162-166, 1988.

[23] M . C. Zaretsky, J. R. Melcher and C. M. Cooke,

“Moisture Sensing in Transformer Oil Using Thin

Film Microdielectromet ry”, IEE E Int’l. Symp. on

Electrical Insulatio n, Cambridge, MA, pps. 12-17,

1988.

I241

F.

Oberhettinger, Fourier Ezpansions, A Collection

of Formulas, Academic Press,

NY,

p. 9, formula 1.

25, 1973.

[25] W .

H.

Beyer, CR C Standard Math Tables, CRC

Press, FL, p. 350, 1981.

917

Manuscript was received

on

1 4 Mar 1987, in final form

5

Sep

1988.

continuum properties from interdigital dielectrometry

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