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Indian Journal or Pure & A ppli ed Phys ics Vol. 37. September 1999. pp. 667-675
Decomposition of dielectric dispersion into debye domains
B Hemalatha * . V G R aul & Y P Singh2
Department of Electrical Engineer ing. Indian Institute or Technology. Kharagpur 721 302
Received 23 October I 99l:i: rev ised 29 December 1998: accepted 26 July 199LJ
Th is paper presents a Genera l M ode l. based on a systems approach for representing the dielectric response or a system
possessi ng two or more distinct absorpti on regions. A novel method for determining the minimum number of secti ons in the
model and the estimati on of the model parameters bas been proposed. First the minimum number of model secti ons which can
best descr ibe the response data is round either by using Levy ' s complex curve fitting technique or a M onte Carlo search procedure.
With thi s information, the model parameters are accurately es timated by the minimization of an error function. using the Neider
- M ead Simplex algorithm. The proposed methodology includes two important cases. namely. when both the real and imaginary
part or the response is kn own and whcn onl y the imaginnry part of thc response is known. Several illustra ti ve ex amples arc
presented.
1 Introduction The di e lectri c re laxati on spectrum of seve ral linear
d ie lectri cs , parti cul arl y the die lectri c liquids and liquid
mi xtures in the mi c rowave range are known to possess
di stinct absorpti on regions I . In such cases, the re laxati on
spectrum is decomposed into di stinct number of Debye
domains. When two absorption regio ns are known to
ex ist, g raphical me th ods like that due to Davidson and
Co le2.:l ha ve been used fo r separation of the loss peaks.
For three di spe rs ion regions. the use of Fl etcher-Powell
al go rithm for separatio n of the loss peaks was described
by Salefran4 The di sad vantage of thi s method is that it
requires both £, and £= to be kn own apri o ri . In additi on,
the nume rical methods';'(' proposed so far for deco mpo
siti on are hi ghl y sensiti ve to the initi a l values used . Based on a systems approach, thi s pape r proposes a
gene ral mode l fo r describin g the die lectric respon se of
a syste m hav ing 11 d isti nct absorpti on regions 7 Thi s
pape r desc ribes a syste matic procedure for decompos in g
the response data into appropriate number of Debye
do main s . The f irst step decides the numbe r of Debye
sections and the approx imate magnitudes of the parame
te r values. The second step performs the minimi zati on
whi ch es timates the magnitudes of the re fined di spe r
sion paramete rs. T wo impo rt ant cases have been cons id
e reel , th at is, when both £' ( real part of pe rmitti v ity) and
£" (i mag i nary pa rt of pe rmitti v ity) a re ava i lable and that
whe n onl y £" is a vailable.
2 Model for a Dielectric Possessing Multiple
Absorptions
For a linear die lectri c system in gene ra l, £*(s) can be
re presented in the Laplace domain as :
P ( S) £* ( 5) =--
Q (.I' ) ... ( I )
where pes) and Q (s ) are polynomial s of the complex
ope rator 5: 2 z· 1 z pes) = ao + al s + a 2S + ... . +az. ls + azs .
Q( .) - I l b .2 b jI·1 I JI .\ - + 'h~ + 2·\ + .. .. . + /1. 1.1 + )//.1
.. (2) .. .(3)
The roots of the denominator po lyno mi a l g i ve the
trans iti on frequenc ies and hence the re laxati on times.
For phys ical syste m p 2:::.. Since di e lec tri c systems have
finite £, and ~, the numerato r and denominator polyno
mia l have the same order. Thus, p = z = n and the
permitti vity functi on of the die lectri c is of the fo rm :
2 1/ - I /I
( ao + al s + a2S + ... + all- IS + (I"S
£ * s) = -------------- ... (4) 2 /1 - I /I
I+bls + b2s + .. .+ b,,_IS + b"s
Therefore, the paramete rs £, and £N. are writren as:
E, = lim £ * ( s ) = on ... (5) .1-->0
. ) a £~ = 11m £ * ( s =.....l!. .1'-)000 17"
... (6)
2.1 Block diagram representation
Writing the denominator po lynomi a l, Qls) of Eq. (3 )
as:
...(7)
66R INDIAN J PURE & APPL PHYS. VOL 37. SEPTEMBER 1 999
The complex permittivity £*(j0) can be expressed in partial fraction form as:
A I A2 An £ * ( j Ul ) = Eoo + + ---- + . . . + ----
I + j Ul 't I I + j Ul 't2 I + j Ul 'tn . . . (8)
where. £.,." (A I , A2 . . .An) and (1 1 , 12, . . . . , 111) are the
dispersion parameters, Aj = £,j - £.,." £,j and £.,., being the permittivities for the i
th section in the low and h igh
frequency regions respectively, and 1 1 • . . . . . . • 1n are the n discrete relax
·ation times. Therefore,
i=n
i= 1
I=n
E, = L £x i ;=1
. . . (9)
. . . ( 1 0)
, £, = £.,., + A I + A2 + A1 + . . . . . . + An . . . ( 1 1 ) The complex permittivity of die lectrics possessing
multiple absorption regions can thus be represented by the block diagram shown in Fig. �. This system comprises of one feed forward element and n closed loop first order (Debye) systems . Dill is the dielectric displacement due to induced polarization and Dor; is the dielectric displacement due to orientational polarization of the /h section.
3 Estimation of Dispersion Parameters
For a given £*(j0) , over a range of frequencies, the determination of dispersion parameters i nvolves the following two steps :
(a) The determination of the number of Debye sections and the approximate values of corresponding coefficients. In most cases the choice is between two or three Debye sections . Each section must, however, have a relation to the physical processes.
(b) Refining the approximate values of the dispersion parameters by the minimization of an appropriate error function.
From Eq. (8) it is seen that 2n+ I parameters are to be determined for a system with n relaxation time constants.
3.1 Case A : Only E" is available
From Eq. (8) £"(0) can be written as:
£ " ( E., ) __ A I (J)'t I + A2 ffi'try A (J)'t UJ + . . . + ---",,-' _tL." -
1 + j 0)2 1; I + j 0)2 1i I +.i 0)2 1� . . . ( 1 2)
In this case, 2n unknown parameters have to be determined for a system with n discrete relaxation t ime constants.
3. 1. 1 Estimation of Initial Values by Monte-Carlo method
Since, the number of Debye sections n i s not known
aoriori. the lowest value of n = I is chosen. Let £" (O)k)
o(s)
Fig. I - B lock diagram representation for a dielectric with mUlt iple absorption regions
'""' .
,. \
HEMALATHA l't al DECOMPOSITION OF DI ELECTRIC DISPERSION <169
1\
be the actual va lue and E " (Wk) be the estimated value of
E" (w), at a particular sampling frequency Wk Since:
~ " = A I Wk1 , + Ary wk 1~ + ... + AI! Wk1" ... ( 13) J I ") " ') ')
I + WZ 1; I + Wi "; I + Wi "~
The error, Ei (Wk), at the frequency Wk would be : 1\
e, wl• =E"(Wk )- E"(Wk ) ... ( 14)
" { A I w,." , A ry W, L~ A W/ I } = E W - . , . + " + ... + I! ,,, A,),) ') J ') .,
I + w; 1; I + Wi Tz I + Wi 1~ ... ( 15)
Summing up the square of the error Ei (Wk) over the entire range of M available 'frequ encies ei an object ive fun cti on J I is defined as :
M
... ( 16)
k~ 1
A random sea rch proced ure is employed to determine
. the va lues of the coefficients, (A I, A1 ... , An) and "1 ,11,
.. ·'tn, which yield a minimum value for J I •
In order [0 initiate the random search procedure, a region of search for eac h one of the coeffic ients must be defined. If the frequency response data covers a wide range offrequencies above and below the loss peak, then
the area of the E" versus In (w) curve gives the approxi
mate va lu e of E" since
E, oc f E" ( W ) d ( In W ) ... ( 17) ()
From Eq. (II ) it is c lear that , and each of the coeffi
cients A 1, 1\1, .. .. , All must be pos iti ve but less than En. A reg ion for searching the coeffic ients A I , A1, .... , An is thus defined. The region of search for the relaxation time constants is spec ified to cover the entire range of fre
quencies. With the limit s fo r E=, A and" spec ified , a Monte-Ca rl o search is initi ated. For each va ri ab le, a rand om vari ab le is pi cked from the co rres pond ing search region, and the obj ec ti ve functi on J I is co mputed. The lowcst va lue of J I , and the corresponding variab le va lues, ari sin g from WOO run s is determin ed.
The procedure is then repeated fo r II = 2 ,3, ... and the mi nimum va lue or J I is tabul ated in each case. The order 11 corres pondi ng to the lowest value oU I from thi s tab le i:-. reckoned as the Illodel order. and the corres ponding paramete r va lue~ are taken as the initi al va lues for subsequent minillli za ti on usin g the Ne ldcr- Mead Silll-
. x C)
plex algo rithm ' .
3.2 Case R: When both E' and EN are availahle
3.2. 1 NUlI1ber a/sections and illitial values 17.1' Lev,' '.I' l17ethod
Determination of the number of Debye sections and the approximate va lues of correspondin g coeffi cients, is accompli shed by usin g Levy 's complex curve fitting technique lO
, which is briefl y described in the Appendi x. The objecti ve functi on J I is computed for several values of II , and the number n corresponding to the minimum value of J I is reckoned as the system order and the corresponding model coefficients are determined and used as initi al va lues in subsequent minimi zat ion. For hi gher order systems, Levy's technique yie lds a poor fit at lower frequenci es, the coeffi cient va lues thus obtained have to be refined by a subsequent min imizati on tec hnique.
3.2.2 Objectivl'.ji.lllctioll fo r lIIinilll i:{/tioll
After ascertaining the number of Debye secti ons and the approximate va lues of the coeffic ients fo r each Debye secti on by Levy's method, a minimi zati on tec hnique is empl oyed for a more accurate est imation of the dispersion parameters. For a given order, 11, at each sam
pi i ng frequ ency, Wk, the error E( Wk) between the actual
value of permitti vity, E':', and its est imated counterpart
E* is given by : 1\
e wk = E* ( wk ) - E * ( wk )
and the rea I part of the error Er (Wk) is gi ven by:
er ( wk ) =
... ( 18)
... ( 19)
, ) [A I Wktl A? wk t ? A"wkt" 1 E (wk - £ - + + .. . + ~ 7 ? ? ? , ?
I + Wi; "C, I + Wi; "C2 I + (J)i; t~
.. . (20)
whil e the imaginary part of the error, E;(wd i:-. defin ed by Eq. ( 15). Thus the magn it ucle of squared error, E2( w;), .
is obtained as:
E\ Wk) = Er 2( Wk) + Ei "( w,) ... (2 1)
By summin g the squared error ove r the entire range of M ava ilable frequencies, an objec ti ve functi on J is defined as
, ~Al
J = I. e2 ( w, ) ... (22)
,=1
If in add ition to the compl ex response data, the value
of E~ is known apriori, then onl y the co mponent E;(wd
670 INDIAN J PURE & APPL PHYS, VOL 37, SEPTEMBER 1 999
i s considered for minimization of the objective function J 1 as defined in Eq. ( 1 6) by employing the NeIder-Mead
algorithm. Subsequently, £, is then determined using Eq. ( I I ) .
4 Illustrative Examples The methods described above have been appl ied to
data drawn from avai lable l i terature. A l l computations have been done using MA TLAB 4.2c (Ref. I I ) on a 75 MHz, Pentium machine. The functions FMINS avai lable in the opt imization toolbox!) of MAT LAB has been used for Neider Mead min imization. The minimization is terminated when the change in the value of the variables and a change in the value of the objective function between two iterations is less than 1 0-4.
The errors between the estimated and actual values are defi ned as:
k=M . . . (23)
. . . (24)
and final ly , the square of the absolute error, E is obtained as: E2 = E � + E ; . . . (25)
Thi s equation is employed for estimating the error, when only the i maginary component is avai lable E, = O.
The errors in the final values of Ai and 1:;' S depend upon the accuracy of the measured data'
-That the pro
posed methodology can yield acceptable estimates of Ai and 1:i i s i l lustrated i n the fol lowing example by considering a known system corrupted with measurement nOIse.
S ince the estimation of Ai and 1:i' S involves constrained opti mization of a set of non l i near equations the decomposition may not be mathematically unique, but for a given set of data under the above mentioned constraints the methodology yields the best solution which is physical ly admissible.
4.1 Ideal case corrupted with noise
In order to examine the efficacy of the proposed technique data were computed for the ideal case of a system having three d ispersion regions. S ince the experimental data are i nvariably affected by noise, the effectiveness of the method is examined by corrupting the s imulated data with 1 0% measurement noise, corre
sponding to the worst case. If £* is the complex permit
t i v i t y , then £n* , correspond i n g to 1 0% no i se at
frequency, 00, i s s imulated as :
£,,* = £* + O. I £*x . . . (26)
where x i s random number sampled from a uniform distribution i n the i nterval from - I to I .
Assuming £00 = 2, A I = 0.2, Az = 0.35, A3 = 7 .5 , 1:1 = 0. 1 85 ps, 1:2 = 1 .7 ps and 1:3 = 1 70 ps in the frequency ranrre 1 0 MHz- l OOO GHz. data are !!enerated with 4
7�--------�----------------�------------�-r o Experimental response
6 ---. Model response
5
I.
3
2
dis�rsion 0 2nd 0 10 lO00 GHz o MHz
0 2 3 I. 5 6 7 8 9 .1 0 1 1 0 E o) t s
€'
Fig. 2 - Complex p lane plot for an ideal system with 3 dispersion regions corrupted with 1 0% noise
HEMALATHA e/ al: DECOMPOSITION OF DIELECTRIC DISPERSION 671
points/decade, and corrupted with 10% noise. The corresponding complex plane plot is shown in Fig. 2. Since, both the real and the imaginary pat1S of the data are avai lable, the' number of Debye sections and the corresponding initial values are found using Levy 's technique . The error, E for the first four orders is shown in Table I .
From Table I it is seen that the third o rder model has the lowest error. It is thus, appropriate to decompose the system into three Debye domains. The initial values
corresponding to the third order are£.,., = 1.92, A 1 = 0 . 11 2,
A] = 0.235. A, = 10.0,11 = 0.51 ps, 1] = 0.7 ps and 1, = 203.22 ps. Using these initial values, minimization of equation (22) leads to AI = 0. 186 1, A2 = 0.3512, A, = 7.4517,11 = 0. 1494 ps, 12 = 1.63 ps, 1, = 170.64 ps and
= f.~ = 1.96. The model is thus : 1\ 0. 1861 0.3512 7.4517 E ( .\' ) = 1.96 + + ---.++----
1+0. I 494s 1.+1.63s 1+170.64s
... (27) The error E is 0.1194. The I % error in the estimation
of the final values of £.,." A I, A2, A" 11, 12 and 1, are 2, 6.95,0.3,0.6, 6,4 and 0.38 respectively. The method has thus succeeded not on ly in identifying the correct order but also in yielding good est imates of the model parameters . The actua l response and the model response, which consists of one large di spersion region and two small dispers ion regions are also shown in Fig. 2.
The following examples illustrate the use of the proposed methodology for decomposi ng the response spectrum of die lectric liquids and liquid mixtures. In all the examp les data has been drawn from li terature.
4.2 Example-I
The complex permittivity data for n-propyl alcohol at
200 e and 400 e (Ref. 12) are presented in Table 2(a). The aim is to determine the number of Debye sections, along with the corresponding coefficients, representing the best fit for the data at each temperature.
Levy's complex curve fitting is used for determining the model order and the initial values, at each of the temperatures. The error E for first, second , third and fourt h order i. shown in Table 2(b).
Thus, both at 200 e and 4()Oe the third order mode l has the lowest error. The coefficients at each tempe rature are shown in Table 2(c).
Making use of these coeffic ients as initial va lues for minim iza tion of the objecti ve fun cti on of Eq. (22), the
estimates of the tran sfer functions are : At 20oe:
Table I - Error for tirst four orders for the ideal system corrupted with IO"A, noise
Order 2 3
E(Error) 0.896 0.3378 O.22IX
Table 2(a) - Complex permittivity of n-propyl alcohol
I 20°C 40°C
(GHz) E' En E' En
O,CXII 21.10 0.0015 18.52 O.(lO I
0.01 21 .00 0.56 18.56 0.5 1
(1.025 21.00 1.14 18.50 0.97 .. 0.05 20.79 2.21 18.42 1.40
0.075 20.50 3.20 18.29 I.X6
0. 10 20,ClO 4.20 18. 12 2.33
0. 125 19.50 5.07 17.95 2.76
0.15 IS .80 5.83 17.62 2.93
0. 175 18. 10 6.53 17.71 3. 18
0.20 17.40 7.27 17.49 3.64 _/ 0.225 16.50 7.86 17.29 4.()()
0.25 15.70 8.42 170 I 4.5X
2.97 4.35 2.70 5.36 3.50
9.32 3.53 1.16 3.66 1. 64
24.01 3.20 0.72 3.28 0.95
136.36 2.48 0.57 2.53 0.57
Table 2(b) - Error E for varioll s orders for propyl alcohol
Temp. (0C) First order Second Third order FOLirt h order order
20 14.631 3.729 1.21 J 1.66H
40 7.958 2.11 7 1.5 15 5.258
~* ( s) = 2.24 + 0.978 + 0.8870 + 17.0537
1.+1.9326.1' 1+30.043s 1+429.3 12.1' .. . (28)
and at 406C:
~*( s ) = 2.2 1 + 1.0805 + 1.6778 + 13.565
1+1.7076\' 1+29.94Ss 1+239.02 1I s .. .(29)
where, s = jw 10- 12 Of the three d ispe rsion regions at
200 e and 40oe, two are very small and are located in
IN DI AN .J PURE & APPL PH YS. VOL 37. SEPTEMB ER 1999
4.3 Example-II the hi gh frequency region. The di spersion regions are thu s made up of one large re laxation time and two small re laxation times. The large re laxati on times decrease
with inc rease in te mperature. w hi Ie the short re laxati on times show very little temperature dependence .
T he dispersion curve of p-to luya lde in dilute solut ion
The actual respollse and the mode l response are com
pared in Figs . 3(a) and 3(b) w hich show very good
agreeme nt .
of cyc lohexane at 20°C at six di screte frequency po in ts('
is considered . T here is no ex plic it constra int in thi s case
as both £, and £00 are unknown. The a rea Of£"(CD) versus
In (CD) curve obta ined by Ilumeri cal integration is 14.3.
The indi vidua l values of A I and A:1 and the ir sum should
Temperature ( 0C)
2()
.:j()
[ =
2. 1 (,
2. 12
Table 2(c) - Initia l values for the Debye sections for /1- propy l alcohol
Ini ti al val ues
AI A:1 A3 'I ' 2 ps ps
I.O(iX5 1 0675 16.1\7 17 1.7891\ 32.6(,7
1.0:>7<) 2. 1261 1 :>.09') 2.002 :1 <) .0:>57
1~'-----------------------------------------=l r For .' d isp erSion ° £ x ptr imen l al ; espon ..
11 S econd d ispers ion - Hodel r espon ..
" 1 Th ir d d i s pe rsion
I 10
\
13 6 . 6 o GHz 1 MH z
15 20 25 o 111 5
8 I Fir" di<{ll?l' s iln II Second di~siln
6 III Third di~persion
2 .
° 2.91GH z
10 E '
w _ __
----o Expl. r.spon«
- Model rupon ..
136.0 GHz lMHz O~~~~P-~--~6~--~8----~'0~--~1~2 --~1~4----~'6~--~1~6~~20 o 2 €'
'3 ps
457. 74:>
274. X(,5
Fig. :1 - (a) Complex plane for II -propy l alcohol at 20°C along w ith their dispersion reg ions: (b) Complex plane fo r n-propy l
alcohol at 40°C along w ith thei r uispersion regions
HEMALATHA l'1 (1/ DECOMPOSITION OF DIELECTRIC DISPERSION 673
be less than 14.3. Cons idering the samplin g space for
the re lax at ion time constants to span one decade above
and one decade be low the loss peak frequenc y, the
minimum va lues of e rror E obtained from 1000 Monte
Carl o run s, for various mode l orders, are shown in Table
3. Since, the second order mode l has the lowest e rror,
the data are decomposed into two Debye syste ms. The
coeffici ent va lues corresponding to this order are A I =
5.6237, A2 = 6 .371 , 1:1 = 0.831 ns, 1:2 = 2.404 ns, and are
used as initial va lues for subsequent minimizati on of the
objective function of Eq . ( 16), which leads to final
values of A I = 1.4243, A2 = 9 .894, 1: 1 = 2.91 ns and 1:2 =
1702 ns. The mode l is thu s:
£ OJ = III + -1\,,( ) I [1.4243 9.984]1 1+2.9 Is 1+ 17.02s 1.1=;1"
... (30)
. -9 where , s =.Iw 10
Fig. 4 graphically compares the computed values and
the actual values. The two di spe rsion regions are also
indi cated.
6.----
5
4
E II 3
' 2
1
4.4 Example-III
The variation of £" with frequency for n-butyl alco-
hol 12 is considered. The area of £"(w) versus In (OJ) is
26.7, therefore AI + A2 + A, < 26.7. Using the Monte
Carlo method, it is seen from Table 4 that decomposing
the data into three Debye syste ms y ie lds the lowest error. The coefficient values corres.ponding to third order
areAl = 0.46, A2 = 1.25,A , = 9. 16, 1: 1 = 2.23 PS.1:2 = 19.0
ps, and 1:, = 654.46 ps. Minimi zation of objective func
tion of Eq . ( 16) leads to A I = 0 .362, A2 = 0.5759, A, =
16.3033 , 1:1 = I .0534ps, 1:2 = 5.5939 ps, and 1:, = 550.9 14 1
ps . Thus the mode l is:
Tablc 3 - Minimum valuc of /~. from 1000 Montc-ell-Io run~ for !I-toluya ldc
Firsl order Second ordcr
0.7434 0.6765
10 1
• Actual respor,s€' t.1od~1 response Dis~rsion rfgion 1
_ . - Disp~rsion region 2
Th ird order
09296
W x 10 7 (rad isec )
Fig . 4 - Deco l1lposit ioll 01" loss characteristics of /Ho luyalde in lI-cycJohexane into two Debye domains
674 INDIAN J PURE & APPL PHYS. VOL 37, SEPTEMBER 1999
9~------~----------------------------------------·
It
7
6
2 . . 0 16' 10 7 10i 10'
W ( rod /s)
O.lIctliDI rtlpons. - Hodel rlspons~
1. Oisparsion region 1 2. Dispersion fltian 2 1. Dispel'tiGn region "3
10 10 1011
---
1012
--_._.----- ---- ----
fi g . ."i - Decomposition of loss characteri sti cs o f II-butyl alcohol into 3 dispersion regions
f {J) = m + 1 -1\,,( ) I [ <U62 0.5759 16.3033] I
1+ 1.0534.1' 1+5.5939.1' 1+550.914h Is =) O)
... (30)
T;lhlc 4 - Minimuill value or I :' rroll! 1000 Monte-Carl o runs ror II-hutyl alcohol
First order Second order Th ird order Fourth order
1.434 0732
where, s =jw 10- 12
Fig. 5 depicts the c lose agreement between the mode l response and actual response .
5 Conclusions A General Model based on the systems approach has
been proposed fo r representing the response characteristics of a die lectri c having di stinct absorption reg ions. Us ing thi s mode l as the basis , a systematic procedure for deco mpos ing the relaxation spectrum into a number of e le mentary De bye domains has been pro
posed . Two important cases, when £' and £0 are known
and when only £0 is known to have been dealt with . Whe n both the real and imaginary parts of the spectrum are availabie. Levy's technique is useful in determining
the number of Debye sectio ns and the corresponding initial values. Whe n onl y the imaginary part of the dielectric response is ava il ab le. the random search procedure using Monte-Carlo technique can be successfull y used to determine the number of Debye sec ti ons and the approximate values of the coefficients. These initi a l va lues invariably lead to convergence of Nelder- Mead
si mpl ex procedure. The successfu l use of the method has been illustrated by applying it to practical syste ms.
Acknowlcdgments The authors thank the Directo r, Indian In st itute of
Technology, Kharagpur, for providing faci I it ies to carry out (his work , One of the authors (BH) gratefull y acknowledges the financial assistance prov ided by the CSIR, New Delhi, during the course o f this work .
Rcfcl'cnccs I Bottcher C .I F & Bordewijk P. Th eory o/Electric Pol{/ri~{/t ioll
- Dielectrics ill tilJl e ticlIl'lulellt ./ieltls. Vol 2 (Ebevier Seiell Ii ri c. Amsterdam). 19711.
2 Davidson D W & Co le R I-I . .I Cllell! Pln·s. 19 ( 195 I ) 14114-14!)O.
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4 Salcl'ran J L . ChclII Pllys L"I/ , 45 ( 1977) 124- 129.
5 Shepparci R J . .1 PhI'S f): AJlII! PhI'S. 6 ( 1973) 790-794.
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II MATLAB. Yersion4.2c. (The Math Work Inc. USA). 1996.
12 Garg S K & Smyth C P . .I Ph\'s Chelll . (1) (1986) 1294-130 , .
Appendix Levy's technique
The dielectric permitti vity, E*(S) can be represented in the Laplace domain as a transfer function:
p (s ) E * ( .1 ) = -- ... (1)
Q (.1)
where, pes) and Q(s) are polynomial s of the complex operator s.
P( ) 1 1.- 1 z (11 ) s = On + OIS + 0 2.1' + .. .. +0 /. . 1.1' + ({z.l· . . .
Q( .) - I I . 1 .2 I }I -I I ]I (Ill ) .1 - + )1.1 + 72.1 + .. ... + ) ,,_1.1 + )1" \ .. .
The roots of the denominator po lynomial give the transition frequencies and hence the relaxat ion times .
For phys ical systems p ~ :::. Si nce, dielectric systems
ha ve a finite E., and £.." the order of the numerator and denominator pol ynomial is same, that is p = ::: = n. Therefore
E, = an .. . (TV)
(f
... (V) E.C>'> =~ hll
If E':' (jOJ) is known over a wiele range of frequencies, the problem is to find the order of the numerator and the denominator polynomials and the corresponding coeffi cients which fit the data with a minimum error. In the frequency domain s =jOJ and E*(jOJ) ca n be written as:
. N? + iN I £* (.JOJ) = - . ...(VI ) D2 +jD I
where
N 1 5 1= (/ 1 OJ - (Il OJ + 0 , OJ + ...
:2 4 Nc = (to - {/ 2 OJ + (f~ OJ + ...
... (V II )
...(VlIl )
... (IX)
:2 . 4 D2 = I - b2 OJ + h4 CJ.) +.. . ...(X)
Therefore, the real and imaginary parts of E* (jOJ) are
E' ( OJ) = [Nl DI + N2 D?] D~+D~
E" ( CJ.) = (NI D~ + N: DI] DI+D"2
. ... (XI)
...(XII)
The error of fit at any specific frequency (0" is :
e( OJ" ) = ['E1 ( OJ" ) _ NI" D~" + N??" D? ,, ]
Dlk + D 2"
. ["(' ) N I" D?k +N?k Dl kJ .- .1 'E OJ" + ? 0
D1" + D 2" ... (X lII )
MUltiplying the above equation by D" (OJ,,) = DCIA +
D22k, and resolving into real and imaginary parts yields:
D\ OJk)e «(0,,) = A( OJ,,) - j B( OJ,,) . ..(XIV)
where , ??
A( OJ,,) = E ( OJ,, ) (D;A + D ; k ) - (N lk D I " + N2A D1k )
. .. (XV)
1/ 2 ~ B ( OJ,, ) = E (OJ,, ) (D " + D 2k ) + ( N ib D2" - N:'A DI A)
...( XVI)
By squaring the magnitude of the weighted error function, and summing it over the entire range of sampling frequencies, an objecti ve function} is obta ined as:
} = L.. ID" ( OJ. ) e ( OJ. ) I:'
A ~ I
M
"~ I
.. . (XVII )
Minimization oU with res pect to the unkn own pa-rameters would lead to an appropriat model representing a good overall fit to the experi menta l data. Differentiating} with respect to each of the unk nown parameters and equating them to zero results in a set of linear simultaneous e(}u <1 ti ons, whose solution yields the va lues of the coe ffi cient s. S ince the system order is not known apri ori , the procedure is carri ed out for several assumed values of system order, and the oreler which corresponds to a minimum va lue of } is reckoned as the system order.
Recommended