Microsoft Word - Numerical Modelling of Dielectric Mixtures

Technical University of Cluj-Napoca

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1 REZUMAT 1.1 Introducere

Materialele compozite sunt tot mai folosite în cadrul aplicaţiilor mecanice, termice sau electrice, având o largă răspândire. Majoritatea materialelor folosite în viaţa de zi cu zi sunt compoziţi formaţi din unul sau mai mulţi constituenţi. Cercetătorii au fost atraşi de-a lungul timpului datorită proprietăţilor mecanice excelente ale materialelor compozite, în special datorită combinaţiei dintre densitatea scăzută şi rezistenţa ridicată. În general, compoziţii dielectrici sunt reprezentaţi de un sistem neomogem, iar în mod comun, compoziţii dielectrici bifazici sunt alcătuiţi dintr-un material gazdă cu o incluziune din alt material. Acest tip de dielectrici poate fi un amestec dintre dielectric-dielectric, dielectric-conductor sau dielectric-semiconductor. La momentul actual de producţie şi în dezvoltările ulterioare, dielectricii au avantajul preciziei procesului de fabricaţie care a devenit din ce în ce mai exact, putând fabrica materiale pentru scopuri speciale în condiţiile dorite. Chiar dacă studiul dielectricilor a început cu multe decenii în urmă, rezultând un număr mare de cărţi şi studii de specialitate, încă este o dorinţă crescută din partea comunităţii de cercetatori în a explora în continuare acest domeniu. Acest lucru poate fi atribuit industriei software şi implicit prin folosirea calculatoarelor cu ajutorul cărora putem obţine proprietăţile dorite ale sistemelor de izolaţie, sau putem estima şi corecta erorile prevăzute prin simulări software. 1.2 Aspecte Teoretice

1.2.1 Proprietăţile dielectricilor

Studiul proprietăţilor dielectricilor utilizaţi ca izolanţi electrici prezintă o deosebită importanţă pentru utilizarea corectă a acestora în diferite aplicaţii. Introducerea într-un agregat a unui material cu proprietăţi incompatibile cu mediul în care se află poate duce la scurtarea duratei lui de viaţă şi la prejudicierea instalaţiilor electrice aferente. Solicitările la care sunt supuse materialele dielectrice sunt diferite şi nu putem astfel să le clasificăm în proprietăţi principale şi secundare, fiind necesară cunoaşterea completă a tuturor proprietăţilor materialului.

Proprietăţile dielectricilor pot fi grupate în chimice, fizice generale, mecanice, termice şi electrice. Proprietăţile chimice definesc compatibilitatea materialului cu corpurile cu care vine în contact, cum ar fi coroziunea, solubilitatea sau indicele de aciditate (caracteristic dielectricilor lichizi). Printre proprietăţile fizice putem aminti: porozitatea, permeabilitatea faţă de aer, permeabilitatea la umiditate sau absorbţia de apă.

Proprietăţile mecanice ce trebuie cunoscute sunt: rezistenţa la tractiune, lungirea specifică, rezistenţa la compresiune sau duritatea. Proprietăţile termice prin care se caracterizează materialele dielectrice sunt: căldura specifică, conductivitatea termică, coeficientul de transmisie a căldurii, stabilitatea termică sau îmbătrânirea termică. 1.2.2 Polarizaţia electrică. Constanta dielectrică Michael Faraday a descris pentru prima dată conceptual de polarizaţie electrică în 1837, când a publicat şi primele măsurări numerice asupra dielectricilor. El a observat faptul că atunci când se introduce un material dielectric între armăturile unui condensator, valoarea capacităţii sale va creşte.


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Permitivitatea electrică relativă (constanta dielectrică) se defineşte ca fiind raportul dintre capacitatea C a unui condensator cu dielectric între armătură şi valoarea C0 a aceluiaşi condensator plasat în vid.

0

r

C

Cε = (1.1)

Cunoscând susceptivitatea electrică ca fiind 0

0 0e

C C C

C Cχ

− ∆= = putem defini următoarea

relaţie: 1

r eε χ= + (1.2)

Susceptivitatea electrică şi permitivitatea relativă sunt mărimi macoscopice, adimensionale. Creşterea capacităţii condensatorului este strâns legată de aptitudinea materialului de a se polariza în câmp prin rotirea moleculelor sau prin deplasarea în sens opus a sarcinilor pozitive şi negative. Mai exact, vectorul polarizaţie şi permitivitatea relativă sunt legate prin următoarea relaţie: ( 1) ( )

o e o r oP E E Eε χ ε ε ε ε= = − = − (1.3)

Figura 1-1 Mecanisme de polarizare: a) polarizaţie electronică b) polarizaţie ionică c)

polarizaţie de orientare d) polarizaţie interfacială

1.2.3 Tipuri de amestecuri dielectrice Dielectricii neomogeni sunt foarte mult utilizaţi în sistemele de izolaţie şi în alte aplicaţii practice, deoarece răspund mai bine la solicitarile de diferită natură cărora trebuie să le facă faţă (electrice, mecanice, termice etc.) Un dielectric neomogen este un amestec între doi sau mai mulţi dielectrici omogeni. Unii autori grupează amestecurile dielectrice după distribuţia componentelor lor astfel: statistice, matriciale şi compoziţii structurale.


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În cazul amestecurilor statistice distribuţia particulelor de natură diferită se face haotic, putând spune ca în acest fel nu se formează o structură regulată. În această categorie intră amestecurile de lichide electroizolante sau glazurile ceramice. La amestecurile matriceale se constată o matrice continuă la o orice concentraţie a fazei de dispersie. Particulele nu au contact fizic între ele. Uneori amestecul matriceal poate fi considerat un caz separat de amestec statistic de două componente, în care o fază este în proporţie foarte mică în amestec şi particulele acesteia nu au contact unul cu altul. Acest tip de amestec este specific materialelor composite. Compoziţiile structurale formează structuri ordonate după plane sau după volum. Principalele compoziţii structurale cunoscute sunt stratificatele si materialele fibroase. Amestecul omogen de dielectrici diferiţi prezintă aceleaşi proprietăţi în orice volum suficient de mic din materialul considerat. Amestecul sub formă de dispersie de particule este considerat ca fiind format dintr-un dielectric de tip gazdă în care sunt dispersate particule sferice de aceeaşi rază distribuite ordonat. Sistemul este asemănător cu cel al amestecului matriceal. Când concentraţia incluziunilor este foarte mică poate fi considerat şi amestec omogen de doi dielectrici. 1.2.3.1 Modelul dielectricului stratificat Dintre toate tipurile de dielectrici neomogeni acesta este cel mai simplu de studiat. Se consideră un stratificat format din n straturi paralele de dielectrici omogeni cu caracteristici diferite (grosime şi permitivitate), dispuse fie paralel fie perpendicular faţă de electrozii condensatorului plan în care este introdus startificatul. În funcţie de dispunerea paralelă sau perpendiculară a straturilor, se defineşte permitivitatea medie sau efectivă a amestecului ca fiind valoarea permitivitaţii unui dielectric omogen fictiv care dispus între armăturile condensatorului produce aceeaşi capacitate ca şi dielectricul neomogen pe care îl înlocuieşte:

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1 ni

ims i

q

ε ε=

=∑ ; 1

n

mp i i

i

qε ε=

=∑ (1.4)

În acelaşi mod Hasshin – Shtrikman au găsit relaţile de calcul pentru modele sferice în care o sferă cu permitivitatea ε1 este situată într-o sferă de permitivitate ε2, şi invers:

2 11 2

1 2

1 2 1 1 2 2

1 13 3

ef

q q

q qε ε ε

ε ε ε ε ε ε

+ ≤ ≤ +

+ +− −

(1.5)

1.2.3.2 Modelul amestecului dielectric sub formă de dispersie de particule Expresia permitivităţii medii (efective) pentru un amestec matriceal în care amestecul este un solid dielectric care încorporeaza incluziuni separate din al doilea dielectric a fost stabilită de Maxwell – Wagner. Se consideră N incluziuni repartizate uniform în interiorul unei sfere S de raza R>>r2 . Daca densitatea incluziunilor în sfera S nu este prea mare se poate considera potentialul P


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într-un punct P din mediul gazdă, ca o sumî a contribuţiilor fiecărei incluziuni luate separat. Dacă se consideră punctul P suficient de departe de sferă, astfel ca R<<r, atunci potenţialul în punctul P este:

3

2 1 22

1 2

cos2NP

rV NE

r

ε εθ

ε ε

−= ×

+ (1.6)

Figura 1-2 N particule sferice repartizate uniform intr-o sfera de raza R situata in mediul 1

Considerând sfera S ca find constituită dintr-un dielectric omogen, de permitivitate necunoscută εm , contribuţia acestei sfere la potenţialul P este:

3

12

1

cos2

mNP

m

RV E

r

ε εθ

ε ε

−= ×

+ (1.7)

Comparând aceste două relaţii şi notând 3

23

Nrq

R= , raport ce reprezintă concentraţia volumică

a dielectricului 2 în amestec, rezultă permitivitatea medie:

2 1 2 11

2 1 2 1

( 2 ) 2 ( )

2 ( )m

q

q

ε ε ε εε ε

ε ε ε ε

+ + −=

+ − − (1.8)

1.2.3.3 Modelul amestecului dielectric cu incluziuni plane (Zaidel)

Zaidel a stabilit o altă expresie a permitivităţii medii care a considerat incluziuni plane din dielectricul de permitivitate ε2 dispersate în dielectricul de permitivitate ε1. El a considerat aceste incluziuni sub forma unor mici placuţe cu grosimea mult mai mică in raport cu celelalte dimensiuni. Calculând valorile medii ale D si E şi înlocuind in expresia:

i m i

D Eε= , (1.9)

rezultă expresia permitivităţii medii care se poate generaliza pentru N tipuri de incluziuni plane:


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( )1 1

1

1

1

2

3

11 1

3

N

i i

im N

i

i i

q

q

ε ε ε

εε

ε

=

=

+ −

=

+ −

∑

∑ (1.10)

Dacă se notează cu q0 concentraţia substanţei de baza astfel încât, 0

1N

i

i

q=

=∑ , relaţia de mai

sus se poate scrie sub forma:

2 11

1 2

1 2( / )

2 ( / )m

ε εε ε

ε ε

+=

+ (1.11)

1.3 Metode numerice utilizate în cazul amestecurilor dielectrice În cazul amestecurilor dielectrice se disting două tipuri de probleme: calculul polarizării dielectricilor plasaţi într-un camp dat sau determinarea distribuţiei potenţialului pentru un sistem de conductoare încarcate în prezenţa dielectricilor. În cazul amestecurilor dielectrice problema de câmp este de primul tip. Pentru abordarea problemelor de calcul a câmpurilor electrice în medii neomogene ceea mai indicată metodă este metoda elementelor finite, atât datorită facilităţilor de introducere a condiţiilor la limită, a bunei sale precizii, cât şi datorită invariaţiei formulelor de calcul cu configuraţia geometrică analizată. 1.3.1 Metoda elementelor finite Principiul metodei elementelor finite este bazat pe formularea integrală a problemelor cu derivate parţiale. Domeniul se partitionează în elemente de mici dimensiuni. La partiţionarea domeniului se respectă următoarele două reguli:

• Două elemente distincte nu pot avea în comun decât puncte situate pe frontiera lor comună, dacă aceasta există. Aceasta condiţie exclude suprapunerile. Frontierele pot fi puncte, curbe, suprafeţe.

• Ansamblul tututor elementelor trebuie să constitue un domeniu cât mai apropiat de domeniul dat. Se exclude în special spaţiile între elemente.

Figura 1-3 Elemente utilizate la partiţia domeniului de calcul


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1.4 Rezultate experimentale 1.4.1 Date generale Principalele aspecte studiate în cazul amestecurilor dielectrice sunt: determinarea permitivităţii amestecului cunoscând permitivitatea componentelor omogene şi concentraţia lor în amestec, răspunsul dielectricului în funcţie de forma şi poziţia incluziunilor, precum şi apariţia pierderilor maxime şi determinarea erorilor. Materialele folosite în sistemele de izolaţie sunt amestecuri dielectrice (dielectrici neomogeni ca stratificatele, materiale compozite, mixturi etc.) care rezistă mai bine la diferite solicitări la care pot fi supuse, mai mult sau mai puţin complexe. Se pune problema realizării unor materiale cu anumite proprietăţi îmbunătăţite care să poată fi utilizate în anumite condiţii. Asta se poate realiza proiectând un amestec care să satisfacă cerinţele aplicaţiei dorite. S-a studiat în literatura de specialitate că permitivitatea amestecului este dependentă şi de forma incluziunilor, dimensiunile lor, orientarea lor în câmpul electric aplicat şi distribuţia în amestec. Simulările computerizate au reprezentat un mare avantaj în rezolvarea tuturor acestor probleme, ţinându-se cont de toţi aceşti factori de influenţă. 1.4.2 Calculul numeric al permitivităţii în câmp electrostatic Având în vedere că în cazul amestecurilor dielectrice ne interesează în special comportarea în domeniul frecvenţelor joase s-a făcut calculul permitivităţii în regim electrostatic, considerându-se dielectrici neomogeni plasaţi în câmp electrostatic uniform. Astfel, parametrul cheie ce trebuie calculat pentru a determina proprietăţile dielectrice ale amestecurilor este distribuţia câmpului electric E(x,y,z) în domeniul de calcul. Aşa cum s-a descris în partea teoretica a acestui rezumat, cea mai indicată metodă pentru a realiza acest calcul este metoda elementelor finite. Utilizând produsul software Maxwell Ansoft care se bazează pe metoda elementelor finite, s-a putut calcula câmpul electric, respectiv inducţia electrică pentru a determina valoarea permitivităţii medii a amestecului folosind una dintre următoarele relaţii:

m

D

Eε = (1.12)

21 1 1

2 2mE E DdεΩ

= ⋅ ΩΩ ∫

, (1.13)

unde Ω reprezintă domeniul de calcul:

1

E EdΩ

= ΩΩ ∫

, 1

D EdεΩ

= ΩΩ ∫

(1.14)

Astfel prima metoda se bazează pe determinarea permitivităţii medii prin determinarea câmpului electric mediu şi a inductiei electrice medii. Cea de-a doua metodă implică calcularea energiei electrostatice disipată în întreg domeniul de calcul. S-a utilizat prima metodă iar ca domeniu de calcul s-a ales condensatorul plan în care există un câmp electric omogen. Condiţiile de frontieră aplicate în toate simulările efectuate sunt prezentate în figura urmatoare:


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Figura 1-4 Modelul de studiu utilizat si conditiile de frontiera

1.4.3 Modelarea amestecurilor dielectrice cu diferite forme de incluziuni Cunoscând faptul că permitivitatea depinde de forma şi marimea incluziunii, am studiat în acest scenariu amestecuri dielectrice cu diferite forme de incluziuni: cilindrice, dreptunghiulare, pătratice, triunghiulare, elipsoid orientat vertical sau orizontal. Pentru toate cazurile, modelarea s-a facut în 2D şi am considerat o secţiune printr-un cilindru, prismă, paraleliped sau elipsoid. Concentraţia amestecului a fost considerată q=0.3 %, determinându-se suprafaţa incluziunii în funcţie de q şi de suprafata materialului gazdă care a fost considerată aceeaşi pentru toate simulările. În mod normal, concentraţia depinde de raportul dintre volumele celor două materiale. Aşadar, am considerat concentraţia ca fiind raportul dintre cele două suprafeţe:

gazda/i incluziune material

q S S=

Figura 1-5 Amestecuri dielectrice cu diferite forme de incluziuni, unde 1ε reprezintă

permitivitatea mediului gazdă, iar 2ε reprezintă permitivitatea incluziunii

Suprafaţa materialului gazdă a fost acceaşi pentru toate simulările: un pătrat cu latura de 10

mµ , rezultând astfel din formula de mai sus o suprafaţă de 30 mµ pentru fiecare tip de incluziune. Având desenat modelul de studiat, putem trece la modelarea acestuia. S-au facut două tipuri de simulări pentru fiecare dintre incluziunile descrise mai sus. S-a considerat polietilena ca fiind materialul gazdă iar pentru incluziuni s-au considerat două situaţii: o dată, sticla cu permitivitatea ε =3.78 şi o dată titanat de bariu cu permitivitatea ε =1000. Polietilena are o permitivitate de ε =2.25.


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Tabel 1-1 Variatia permitivităţii în funcţie de forma şi materialul incluziunii, unde 1effε

reprezintă permitivitatea

efectivă (medie) a amestecului format din polietilena cu incluziuni de sticlă ( 1ε=2.25, 2ε

=3.78) , iar 2effε

reprezintă permitivitatea medie a amestecului format din polietilena cu incluziuni din titanat de bariu

( 1ε=2.25, 2ε

=1000). 1ε reprezintă permitivitatea materialului gazdă iar 2ε

permitivitatea incluziunilor.

ε /forma incluziunii 1eff

ε 2effε

Cilindrică 2.611893226 4.066342398 Dreptunghiulară 2.626417563 4.589550387 Pătratică 2.615193870 4.278775943 Triunghiulară 2.619990037 5.254818519 Elipsoid Orizontal 2.339934803 2.723327579 Elipsoid Vertical 2.328272277 2.519362426

Tabel 1-2 Variaţia erorii în funcţie de forma incluziunii şi tipul amestecului

ε /forma incluziunii Eroarea% / 1eff

ε Eroarea% / 2effε

Cilindrică 0.0595 0.4650 Dreptunghiulară 0.0993 0.7690 Pătratică 0.1010 0.0572 Triunghiulară 0.1000 0.3130 Elipsoid Orizontal 0.5470 0.0621 Elipsoid Vertical 0.6170 0.0543

1.4.4 Modelarea amestecurilor matriceale Pentru amestecurile matriceale s-au considerat două siutaţii. Pentru prima situatie am considerat o structură ordonată de incluziuni cilindrice, iar pentru a doua situaţie o structură dezordonată de incluziuni cilindrice. În cazul fiecărei dintre cele două situaţii am considerat polietilena ca şi material gazdă, cu o permitivitate de 2.25, iar pentru incluziunile cilindrice am luat din nou două situaţii: prima situaţie cu incluziune din sticlă de permitivitate 3.78, iar în a doua situaţie incluziuni din titanat de bariu cu permitivitatea de 1000.

a) structură ordonată b) structură dezordonată


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Figura 1-6 Amestecuri matriceale cu incluziuni: a) structură ordonată b) structură

dezordonată

Cunoscând concentraţia amestecului ca fiind de 0.4, şi latura unui patrat ca fiind de 10 microni am putut afla, astfel, dimensiunea unei incluziuni cilindrice din următoarea formulă:

2 2/i

q r lπ= [%] (1.15)

Tabel 1-3 Variaţia permitivităţii în funcţie de concentraţie şi tipul incluziunii. 1effε


medie (efectivă) a amestecului când materialul gazdă este polietilena iar incluziunile sunt din sticlă

( 1ε=2.25, 2ε

=3.78) , iar 2effε

reprezintă permitivitatea medie când materialul gazdă este polietilena iar

incluziunile sunt din titanat de bariu ( 1ε=2.25, 2ε

=1000). 1ε reprezintă permitivitatea mediului gazdă şi 2ε

a incluziunilor. Concentraţia q % ε /structură matriceală

1effε 2eff

ε

0.4 Ordonată 2.752042924 5.085069146 0.4 Dezordonată 2.754186525 5.216091304 0.099 Ordonată 2.364426288 2.725865857 0.099 Dezordonată 2.364502187 2.727342757 Table 1-4 Variaţia erorii în cazul unui amestec matriceal în funcţie de concentraţie şi tipul incluziunilor

Concentraţia q % ε /structură matriceală Eroarea % / 1eff

ε Eroarea % / 2effε

0.4 Ordonată 0.000605 0.002190 0.4 Dezordonată 0.000658 0.002290 0.099 Ordonată 0.000902 0.002200 0.099 Dezordonată 0.001160 0.002390

a) Structură Ordonată(Incluziuni din sticlă) b) Structură Ordonată(Incluziuni din BaTiO3)


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c) Structură dezordonată(Incluziuni din sticăa) d) Structură Dezordonată(Incluziuni din BaTiO3)

Figura 1-7 Variaţia erorii în funcţie de numărul de paşi, tipul incluziunii şi concentraţie

1.4.5 Modelarea amestecurilor dielectrice în funcţie de distanţa dintre incluziuni S-au considerat două incluziuni cilindrice plasate într-un material gazdă în formă de dreptunghi cu o lungime de 50 de microni şi o lăţime de 25 microni. Materialul gazdă a fost considerat polietilena cu o permitivitate de 2.25, iar pentru cele două situaţii de incluziuni s-a considerat sticla cu o permitivitate de 3.78, respectiv titanat de bariu cu o permitivitate mult mai mare, 1000. Am studiat variaţia permitivităţii în funcţie de distanţa dintre incluziuni, de la minimum de 0 microni până la o distanţă maximă de 12 ori raza unei incluziuni cilindrice. Raza a fost considerată de 2 microni.

a) distanţa d=0 microni b) distanţa d=24 microni

Figura 1-8 Amestec dielectric cu două incluziuni cilindrice aflate la o distanţă de: a) 0 microni

b) 24 microni

Tabel 1-5 Variaţia permitivităţii în funcţie de distanţa dintre incluziuni, unde 1effε


medie (efectivă) când materialul gazdă este polietilena iar incluziunile sunt din sticlă ( 1ε=2.25, 2ε

=3.78) , iar

2effε

reprezintă permitivitatea medie când materialul gazdă este polietilena iar incluziunile sunt din titanat de

bariu. ( 1ε=2.25, 2ε

=1000). 1ε reprezintă permitivitatea materialului gazdă şi 2ε

permitivitatea incluziunilor.


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ε /distanţa 1eff

ε 2effε

d=0 2.278093619 2.431498328

d=r 2.277027228 2.365913617

d=2r 2.276761658 2.359967690

d=3r 2.276762259 2.357939643

d=4r 2.276782985 2.357184891

d=5r 2.276878173 2.357129213

d=6r 2.277005645 2.357362583

d=7r 2.277203074 2.357950836

d=8r 2.277402008 2.358622804

d=9r 2.277670073 2.359595494

d=10r 2.277932512 2.360597417

d=11r 2.278259099 2.361886507

d=12r 2.278593454 2.363191827

1.4.6. Modelarea amestecurilor dielectrice în funcţie de concentraţie Cunoscând concentraţia amestecului am putut determina dimensiunile incluziunilor. S-a considerat o concentraţie de 0.05 a unui material gazdă cu 5 incluziuni cilindrice. Permitivitatea materialului gazdă a fost luată de 2.25, aceasta fiind permitivitatea polietilenei, iar incluziunile au fost alese drept sticla cu o permitivitate de 3.78, respectiv titanat de bariu cu o permitivitate de 1000. Formula de mai jos prezintă metoda folosită pentru determinarea razei unei incluziuni cilindrice:

2

2 2

5incluziune ii

gazda

N S qrq r

S l l

π• •= = ⇒ = (1.16)

Tabel 1-6 Variaţia permitivităţi în funcţie de concentraţia incluziunilor, unde 1effε

reprezintă permitivitatea medie (efectiva) a amestecului format din polietilena drept material gazdă cu incluziuni din sticla

( 1ε=2.25, 2ε

=3.78) , iar 2effε

reprezintă permitivitatea compozitului format din polietilenă cu incluziuni din

titanat de bariu ( 1ε=2.25, 2ε

=1000). 1ε reprezintă permitivitatea materialului gazdă şi 2ε

a incluziunilor.

Concentraţie/q %

1effε 2eff

ε

0.1 2.364464817 2.733126868 0.15 2.425284262 3.000804975 0.20 2.490959193 3.103327554 0.25 2.553108290 3.702588906 0.35 2.683325479 4.530542638 0.45 2.824619176 5.863116344


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a) incluziuni din sticlă b) incluziuni din BaTiO3

Figura 1-9 Variaţia potentialului electric φ a unui amestec dielectric cu cinci incluziuni

cilindirce: a) incluziuni din sticlă b)incluziuni din titanat de bariu

1.5 Concluzii Având în vedere faptul că majoritatea tipurilor de materiale utilizate în sistemele de izolaţie sunt amestecuri formate din mai mulţi dielectrici omogeni, trebuie să remarcăm faptul că fac faţă mai bine solicitarilor complexe la care sunt supuse (electrice, termice, mecanice etc.), avantajul lor fiind posibilitatea de a fi proiectate pentru anumite scopuri sau condiţii. Metodele experimentale presupun timp şi bani, dar printr-o modelare numerică folosind simularile computerizate se pot estima proprietăţile materialelor dielectrice, respectiv corectarea, din timp, a eventualelor erori. Folosind Maxwell Ansoft s-a facut în lucrarea de faţă o modelare a amestecurilor dielectrice plasate în câmp electrostatic omogen pentru determinarea permitivităţii medii sau efective a acestora. S-a studiat variaţia permitivităţii în funcţie de forma incluziunii, concentraţia incluziunilor, distanţa dintre incluziuni sau poziţionarea ordonată/dezordonată a incluziunilor într-un amestec de tip matriceal. S-au verficat rezultatele numerice cu ajutorul metodelor analitice de tipul Maxwell – Wagner. S-a observat o creştere a erorii în cazut unei concentraţii mai mici a incluziunilor, sau în cazul unui amestec ordonat faţă de cel ordonat. Pentru toate tipurile de simulari menţionate mai sus s-a calculat permitivitatea amestecului atât pentru incuziuni de permitivitate apropiată de cea a materialului gazda cât şi pentru incluziuni de permitivitate foarte mare.


13

2. STATE OF THE ART 3. THEORETICAL FUNDAMENTALS 3.1 Introduction

Nowadays, composite materials are widely used within mechanical, thermal or electrical

applications worldwide. Most of the materials often used in our daily life are actually composites made up of at least two constituents. Researchers have been attracted along the time because of the outstanding mechanical properties of composite materials, and especially the combination of low density with high strength. In general, composite dielectrics are represented by a heterogeneous system and commonly two-phase composite dielectrics consist of a host material with an inclusion of another material. This type of dielectrics could be a mixture of dielectric-dielectric, dielectric-conductor or dielectric-semiconductor. Composites have the advantage that during the current and future developments, the precision of the manufacturing has become more and more precise which implies the fact that we can fabric materials for very special purposes and desired conditions.

Even if the study of dielectrics has started many decades ago and resulted in a large body of work, there is still an increasing thrill from the research community in further exploring this field. This is attributed to the software industry and thus by using computers we can achieve the desired properties of insulation systems, we can estimate them and correct the predicted errors. 3.2 Properties of the Dielectrics Studying the properties of the dielectrics used as electrical insulators represents a particular importance for the correct use within various applications. Introducing a material into an aggregate having incompatible properties with the surrounding medium can shorten its life and affect the corresponding electric installations. The stresses to which the dielectrics are under are widely different and thus we cannot consider that dielectrics have principal or secondary properties, because most of the times these properties are the main cause when choosing a dielectric. Therefore, it is best to know all of their properties. We can largely group the properties of the dielectrics into physical, chemical, mechanical, thermal and electrical. The chemical properties define the compatibility of the material to interact with other bodies. We can define: corrosion, solubility or acidity index I (characteristic to liquid dielectrics). Physical properties which present the most interest are: porosity P, air permeability

Pa, moisture permeability Pu, humidity U, relative air humidity, water absorption Aa etc. Mechanical properties are the following: tensile strength σ, compressive strength σc, ductility,

Brinell hardness, Vickers, Mohs or Rockwell (depending on the method used for measuring it) Thermal properties defined for dielectrics can be the following: specific heat C, thermal

conductivity λ, thermal expansion α, flammability temperature, thermal shock resistance or thermal stability. Electrical properties of the dielectrics, as electrical insulators depend on the phenomena which take place when introducing them into an electric field: electric polarization, conductivity and insulation pierce. The object of interest of this thesis represents the electrical properties of the heterogeneous dielectrics, further implying the study of the above phenomena and corresponding


14

properties: relative electric permittivity εr, conductivity σ, volume and surface resistivity, ρv

and ρs, loss factor tgδ and dielectric strength Estr. 3.3 Electric Polarization and Dielectric Constant

The concept of electric polarization was described for the first time by Michael Faraday in 1837 when he published the first numerical measurements for dielectrics. He noticed the fact that introducing a dielectric material between the plates of a capacitor with C0 capacitance, its value will rise to C. The growth rate is called electric susceptibility and is computed in the following way:

0

0 0e

C C C

C Cχ

− ∆= = (3.1)

The relative electric permittivity (also known as dielectric constant) is defined as the ratio

between the capacitance C of a capacitor with a dielectric material between its plates and the C0 capacitance value of the same capacitor when placed in vacuum:

0

r

C

Cε = (3.2)

From relations 3.1 and 3.2 it results the expression 3.3 which defines the dielectric constant

and the electric susceptibility as being dimensionless measures: 1

r eε χ= + (3.3)

The capacitance value depends on the ability of the material to get polarized in a field by the moving of positive and negative charges in opposite directions. The polarization vector and permittivity are connected through the following relation: ( 1) ( )

o e o r oP E E Eε χ ε ε ε ε= = − = − (3.4)

Where

oε represents the permittivity of the vacuum,

rε represents the relative permittivity

and ε represents the absolute permittivity of the material used as a dielectric. The above relation is also known as the law of temporal polarization.

3.3.1 Electric Polarization Mechanisms We can consider as dielectrics the materials in which the electrons are tightly connected to their atoms such that they are not responsible for a significant electric current. Having the fact that the charges present in the dielectric materials are not free, it doesn’t mean at all that they cannot roam from each other under the action of an electric current, causing the occurrence of electric dipoles, thus an electric polarization of the matter.


15

Considering the atomic structure, all the matter consists of positive and negative charged particles. Their charges are macroscopically balanced; therefore the lack of an internal electric field leads to a neutral charge of the matter. When applying an electric field, the balance of the charges is disturbed by the temporal polarization mechanisms. In this case the electric dipoles occur because of the opposite direction movement of the positive and negative charges. In the case of heterogeneous materials we can observe an interfacial polarization as a consequence of the appearance of electric charges on the surfaces which separate the homogenous parts, causing a supplementary electric field which creates an extra temporal polarization.

The electronic polarization is due to the movement of atomic electrons against the nucleus of the atom. It is present in all materials, even if in some of them to such a small extent that it can be neglected (the case of metals or materials made of atoms that have lost their bonding electrons), it is also achieved in a very small amount of time and it maintains itself at very high frequencies (1015 Hz), being also known as optical polarization. If the atoms no longer have electrons on their outer electronic layer, the polarization that appears through the movement of the electrons from the inner layers is very low (the inner atoms have very strong bonds with the nucleus) and the polarization can maintain itself at even higher frequencies (1019 Hz).

Ionic polarization can only be met in materials with ionic bonds, it is due to the opposite movement of the positive and negative ions, it is achieved in a relatively longer period than the electronic one and it maintains itself at 1013 - 1014 Hz.

In polar materials, the field tends to orient the spontaneous dipolar moment Pp (the electric moment of molecules) to its direction, a situation in which the potential energy of the dipole is minimum. Thermal agitation opposes the orienting action of the field and thus, not all the electric moments of the molecules are oriented according to the direction of the field. This is called orientation polarization and it is achieved in a significantly longer period than the one of the movement polarizations. Beside this phenomenon of orientation, the electric field can modify the dipolar moment of the molecule through its own and her orbital deformation.

Interfacial polarization is important in the dielectric materials that contain load carriers that can migrate on a certain distance within the thickness of the material (for example, through diffusion, rapid ionic conduction or interlacing) and, in this way, local accumulations of genuine electrical load are achieved. Such a change is perceived by an observer as an increase in the capacity of the assay and can be identified through a real growth in electric permittivity. Interfacial polarization is the only type of polarization which is followed by a microscopic change of the load transport (and if the load carriers that migrate are ions, there appears a macroscopic body of transport. It is achieved in a long period which can be of a couple of minutes or even longer (days).

Movement and orientation polarizations can be studied in models. The diversity of the mechanisms for the achievement of interfacial polarization do not permit the design of a generally accepted model, a more profound study on this type of polarization being necessary. According to the external factor that produces it, the permanent polarization can be: through thermal effect, also known as spontaneous polarization or piezoelectric (through mechanical effects).

Spontaneous polarization is the dielectric organization that appears in certain temperature domains, in structures without a centre of symmetry but with a polar axis, in the absence of the outer field. Among the dielectrics that own this type of polarization, only the ferroelectric crystals have practical usage, because in their case the direction or the sense of the polarization can be changed under the action of an electric field.

Piezoelectric polarization appears in structures without a centre of symmetry for electric loads, being caused by applied mechanical tensions. Under the action of the electric field and other factors (temperature, luminous flux, radiations etc), quasi-permanent polarization states (of the


16

electrets type) with a limit period of existence can appear. Permanent and quasi-permanent polarizations are accompanied by temporary polarizations. 3.4 Ways to Calculate the Permittivity of Dielectric Mixtures 3.4.1 Types of dielectric mixtures

Heterogeneous dielectrics are widely used in isolation systems and other practical applications as they have a very positive reaction to various loads (electrical, mechanical, thermal, chemical etc) which they have to face. An inhomogeneous dielectric is a mixture of two or more homogenous dielectrics. Their classification is needed for a thorough study of this type of mixtures. Some authors group these mixtures according to the features of the components distribution in:

• Statistical • Matrix • Structural Compositions

The distribution of the particles is done in a chaotic way in the case of statistical mixtures

(according to a statistical repartition), and thus, they do not form a regular structure. This is the case of electro insulating liquid mixtures, ceramic glazes, compounds etc.

In the case of matrix mixtures, one phase forms a continuous matrix at each concentration of the dispersed phase. There is no physical contact among the granules or the particles of the dispersion phase. Sometimes, the matrix mixture can be considered a particular case of a statistical mixture of two components in which a phase is, to a very little extent, in the mixture and its granules do not come into contact. This type of mixture is known to be specific to composite materials but its model can be also applied in the case of those with polycrystalline structure, such as ferrites, ceramic oxides etc.

In the case of structural compositions the components form structures ordered according to chains, plane or volume. The most widely known structural compositions are the stratified and the fibrous materials.

The homogenous mixture of different dielectrics is the blend which exhibits the same properties in any small enough volume of the considered material. The mixture in the form of particles dispersion is considered as being formed from a host dielectric in which spherical particles of the same radius are dispersed, orderly distributed. The system is similar to the matrix mixture. When the concentration of the inclusions is low, it can be considered a homogenous mixture of two dielectrics.

The way to calculate (analytically or numerically) the average or actual permittivity of the mixture will be chosen according to the type of mixture, the analytical formulas suggested by various authors being different for statistical mixture or the matrix ones.


17

Figure 3-1 Different arrangements for combination of two media []

3.5 Analytical Calculation Methods for the Permittivity of Mixtures 3.5.1 Classical approaches

In classical approaches, the calculation of average or actual permittivity of the inhomogeneous dielectric is done by either using equivalent schemes (in simple treatments, available for the simple distributions of the phases) or theories of the average field (actual) or theories similar to these with molecular approaches or considering a regular ordering of the inclusions. It is considered a physical mixture, in which the components do not chemically interact. Moreover, the granules that contain a high enough number of molecules are taken into consideration (with the exception of molecular approaches) and which are also homogenous and isotropic, with constant characteristics in their entire volume.

The calculation of the permittivity of mixture on the basis of the permittivity of the components and their concentration is done by using various models.


18

3.5.1.1 The Model of the Stratified Dielectric

This is the most simple and the easiest model to study among the ones suggested for inhomogeneous dielectrics.

The following is taken into consideration: a stratified dielectric formed of n parallel layers of homogenous dielectrics with different characteristics (thickness di and permittivity εi), arranged against the electrodes of the plane condenser in which the stratified dielectric is introduced either parallel to them ( figure 3-2. a) or perpendicularly (figure 3-2. b) The schemes that correspond to the two types of arrangements contain: n condensers Ci=εiA/di, connected in series or, in the second case, n condensers of Ci=εiA/di capacity, connected in parallel. In the written relations, A(Ai) is the area of the plates of the condenser and d(di) is the thickness of the dielectric between the plates.

Figure 3-2 Stratified dielectric consisting of n layers. a – layers are parallel with the

plates of the capacitor. b – layers are perpendicular on the capacitor’s plates. The average or actual permittivity of the inhomogeneous dielectric can be defined as the

value of the permittivity of a fictive homogenous dielectric that, if placed between the plates of the condenser produces the same capacity as the inhomogeneous dielectric which it replaces.

The equivalent capacity that corresponds to a series connection of the condensers is:

1 1

1 1n ni

i is i i ms

d d

C C A Aε ε= =

= = =∑ ∑ , out of which the following results:

1

1 ni

ims i

q

ε ε=

=∑ (3.5)

In the above relation, qi is the volume proportion of the i dielectric in the mixture:

i ii

V dq

V d= =

Similarly, in the parallel connection of the condensers, we have:

1 1

n nmpi i

p i

i i

AAC C

d d

εε

= =

= = =∑ ∑ , out of which the following results:


19

1

n

mp i i

i

qε ε=

=∑ (3.6)

where the volume proportion is calculated with the ii

Aq

A= relation.

The 2.1. relation is the one used in the calculation of the average permittivity of most stratified dielectrics in isolation systems: pertinax, pressboard, polymer paper, solid dielectrics impregnated with dielectric liquids etc. The model of the stratified dielectric with two components is also known as the Maxwell-Wagner model because they were the ones to use it for the first time for the study of relaxation and the models shown in figure 3-2 are also known as the Wiener models.

Similarly, Hashin-Shtrikman found the calculation relation in the case of the spherical models in figures 3-3, a and b:

2 11 2

1 2

2 1 1 2 1 2

,1 1

3 3

ef ef

q qand

q qε ε ε ε

ε ε ε ε ε ε

= + = +

+ +− −

(3.7)

Figure 3-3 The models of Hashin-Shtrikman: a – dielectric sphere with permittivity ε1

situated in a sphere with permittivity ε2 ; b – dielectric sphere with permittivity ε2 situated in a

sphere with permittivity ε1.

The above mentioned models (for two components with a different arrangement against the electrodes-figure 3-2 a and b) have the following limits:

• Wienner’s inequalities:

1 21 1 2 2

2 1 1 2ef

q qq q

ε εε ε ε

ε ε≤ ≤ +

+ (3.8)

• Hashin-Shtrikman’s inequalities

2 11 2

1 2

1 2 1 1 2 2

1 13 3

ef

q q

q qε ε ε

ε ε ε ε ε ε

+ ≤ ≤ +

+ +− −

(3.9)


20

3.5.1.2 Theories of the Average Electric Field

The theories based in the concept of ‘actual electric field’ or ‘main electric field’ are widely used to interpret the experimental results on the permittivity of dielectric mixtures.

These approaches of the average field mean that out of the two phases, the inclusion one is dispersed inside the other one which constitutes the matrix (the hosting medium). Thus, it is included in the other phase in such a way that they are both imbedded in the same actual medium. The average value of the electric filed in such a medium, supposedly organized, represents the average field, on the basis of which the average value of the polarization in the dielectric mixture is determined. The method of calculation implies the determination of the average value of the field and of the electric induction in the inhomogeneous dielectric and the average permittivity of the mixture is determined form the law of the binding between D, E and P, written for the supposedly homogenous dielectric.

It is obvious that taken into account these suppositions, the resulted relations are thoroughly experimentally verified for low values of the concentration of the inclusion phase or for the close values of the dielectric constants of the two mediums. The relations obtained from this approach of the problem of actual permittivity of the mixture are various, each researchers taking into consideration other simplifying hypotheses.

3.5.1.3 The Model of the Homogenous Mixture of Two Dieletrics

A. Spherical Inclusions

Böttcher established a relation to calculate the average permittivity of such a mixture in the hypothesis that the two mediums are formed from small spheres sunk in the permittivity dielectric εm. He also considered that the second dielectric has a smaller concentration and the difference between the permittivities of the two components is also small.

By calculating the average polarization in the homogenous mixture, with the following relation:

0 0( )m e m

P E Eε χ ε ε= = − (3.10)

we come to the conclusion that the former is the mean value of the polarizations in the two

mediums:

2 1(1 )m

P qP q P= + − (3.11)

By solving the problem of the dielectric sphere of permittivity ε1 or ε2, placed in the

dielectric with εm permittivity in an initially uniform field of E intensity, and by integrating the Laplace equation, the followings are obtained for the field inside the spheres:

1 21 2

3 3, ,

2 2m m

m m

E E and E Eε ε

ε ε ε ε= =

+ + (3.12)


21

By calculating the P1 and P2 polarizations of the two components and replacing in (3.11), the followings result:

0 2 0 1 02 1

3 3( ) ( ) (1 )( )

2 2m m

m

m m

E q E q Eε ε

ε ε ε ε ε εε ε ε ε

− = − + − −+ +

(3.13)

thus obtaining the Böttcher relation:

1 2 1

23 2m

m m

qε ε ε ε

ε ε ε

− −=

+ (3.14)

B. Ellipsoidal Inclusions

Starting form Böttcher’s hypotheses, an expression of the permittivity of the mixture with

inclusions of ellipsoidal form has been established. Identical ellipsoids with the big axis parallel to the direction of the applied field have been considered. By integrating Laplace’s equation, the expression of the electric field of ellipsoids is obtained:

1 21 2

1 1

, ,1 (0) 1 (0)m m

m m

E EE and E

A Aε ε ε ε

ε ε

= =− −

+ +

(3.15)

The polarizations in the two mediums are calculated with the following relations:

1 01 1 0 1

11

( )( )

1 (0)m

m

EP E

A

ε εε ε

ε ε

ε

−= − =

−+

(3.16)

2 02 2 0 2

21

( )( )

1 (0)m

m

EP E

A

ε εε ε

ε ε

ε

−= − =

−+

(3.17)

where 1 20

(0)2 ( )

abc dA

a Rλ

λ

λ

∞

=+∫ and a, b, c are the dimensions of the three axes of the ellipsoid.

The average polarization of the dielectric mixture is calculated with the (3.11) relation and the expression of the permittivity results as the following:

1 2 1

2( ) (0)m

m m m

qA

ε ε ε ε

ε ε ε ε

− −=

+ − (3.18)


22

3.5.1.4 The Model of the Mixture in the Form of Particle Dispersion

Maxwell-Wagner have established the expression of the average permittivity of a matrix mixture in the situation of a mixture that is a solid dielectric incorporating the separate inclusions (without physical contact) of the second dielectric. The hosting medium was considered to be of ε1 permittivity and the dispersion one is formed of spherical particles of ε2 permittivity.

Starting from the influence of a single particle on the dielectric medium, evaluated in terms of a perturbation of the electric potential, the permittivity of the mixture is calculated by summing up the effects produced by all the dispersed particles.

The case of the spherical inclusion in a uniform field of E intensity, a field established in the hosting medium – considered homogenous – in the absence of the inclusion, is a problem of an electrostatic field, a problem which can be solved by separating the variables.

Figure 3-4 N spherical particles with permittivity ε2 uniformly distributed in a sphere

with radius R, situated in the medium with permittivity ε1.

When the permittivity of the hosting medium is bigger than the one of the inclusion, the

intensity of the electric filed in it is higher than in the matrix medium. This is of utmost importance in the case of the gas bubbles (especially air) which are present in the dielectrics, because in these, due to the intensifying of the field and the lower rigidity of the gases, partial discharges may appear which, in time, lead to perforation.

Further on, the following is taken into consideration: N inclusions uniformly distributed inside a S sphere with a R>>r2 radius, situated in the medium of dispersion 1 (figure 3-4)

If the density of the S inclusions is not too big, the potential in a P point, inside medium 1 can be expressed, as a sum of the contributions of each inclusion taken separately. If the P point is considered as far enough from the big sphere, in such a way that R<<r, then the potential in P is:

3

2 1 22

1 2

cos2NP

rV NE

r

ε εθ

ε ε

−= ×

+ (3.19)


23

Considering the S sphere as incorporating a homogenous dielectric of an unknown εm

permittivity, the contribution of this sphere to the potential in P is:

3

12

1

cos2

mNP

m

RV E

r

ε εθ

ε ε

−= ×

+ (3.20)

By comparing the previous two relations, the following results:

3 3 12 12

2 1 12 2m

m

Nr Rε εε ε

ε ε ε ε

−−=

+ + (3.21)

With the 3

23

Nrq

R= notation, a ratio that represents the volume concentration of dielectric two

in the mixture, the following results:

2 1 2 11

2 1 2 1

( 2 ) 2 ( )

2 ( )m

q

q

ε ε ε εε ε

ε ε ε ε

+ + −=

+ − − (3.22)

The closer the dielectric constants of the components of the mixture are, the better the

experimental verification of the (3.20) equation, known as the Maxwell-Wagner relation becomes, because in these conditions, the overlapping of the effects is better verified. 3.5.1.5 The Model of the Mixture with Fine Dispersion of the Inclusions (The Landau-Lifchitz model)

L. Landau and E. Lifchitz took into consideration a homogenous mixture with fine dispersion (emulsion, powder mixture etc) in which the electric field can be taken as a mean in high volumes in comparison with the dimensions of the inhomogeneities. In comparison with the considered average field, the mixture presents itself as a homogenous and isotropic medium, characterized by an average εm permittivity. If E and D are the average values of the field and the induction, the following can be written:

mD Eε= (3.23)

Considering all the components of the mixture as isotropic and with small differences

between their permittivities and against the average permittivity, the latter can be calculated in a general form, with a precision that reaches rank 2 terms in relation to the mentioned differences.

The local value of the intensity of the electric filed can be written under the following

form: E E Eδ= + , and the local value of the electric permittivity under the following form: ε ε δε= + , where ε can be defined as the following relation:

1

dvv

ε ε= ∫ (3.24)


24

The (3.24) relation represents a mediation on volume v. In these conditions, the average

value of the induction can be obtained from the following expression:

( )( )D E E E Eε δε δ ε δεδ= + + = + (3.25)

because, according to the definitions ofδε and Eδ , their mediated values are null. In the rank 0

approximation m

ε ε= , the first corrective non-zero term is of rank 2 in relation to δε , as it is shown

in relation (3.25) By writing the unmediated equation of the local form of the law of electric induction

0divD = , for a precision that reaches rank 1 terms, the following can be obtained: ( )( ) 0div E E div E Eε δε δ ε δ δε+ + = + ∇ = , or by replacing E grad Vδ δ= − , it results that:

V Eε δε∆ = ∇ (3.26)

By applying to the last relation the gradient operation, the following results:

1

( )E Eδ δεε

∆ = − ∇ ∇ (3.27)

The mediation of the Eδεδ product in the relation (3.25) can be done in two phases: a mediation according to the volume of the particles of the same substance (taking a constant value for δε ) and then a mediation according to all the components of the mixture. The mediated value of

Eδ can be easily obtained from the relation (3.27), because of the isotropy of the mixture in general.

After the mediation of both terms of relation (3.27), the following is obtained:

1

3 3

EE E Eδ δε δ δε

ε ε∆ = − ∆ ⇒ = −

Multiplying with δε and mediating according to the components of the mixture, the

following results:

( )

2

3

EEδεδ δε

ε= −

Replacing in relation (3.25) and comparing with the expression in (3.23), the following

expression is obtained:

( )

2

3m

δεε ε

ε= − (3.28)

The expression in (3.28) can be presented in a different form if it is noted that for a precision

that reaches rank 2 terms:


25

1/3 1/3

1

n

m i i

i

qε ε=

=∑ (3.29)

The Landau-Lifchitz relation was established without taking into consideration the form of the inclusions and it is valid in the conditions of small concentration of the dispersed dielectric. 3.5.1.6 The Model of the Dielectric Mixture with Plane Inclusions (The Zaidel Model)

Another expression of the average permittivity was obtained by R.M. Zaidel who took into consideration plane inclusions from the dielectric of permittivity ε2 dispersed in the dielectric of permittivity ε1. He considered these inclusions under the form of small plates whose thickness was far smaller than the other dimensions so that in their interior the electric field is identical to that from the infinite plane stratum. In this approximation, the presence of the plane particles does not disturb

the homogenous field 0E in the basic dielectric. Noting with n the unit normal at the surface of

separation, the E1 field in the interior of the plane particle (which satisfies the conditions at the limit on this surface) can be written as follows:

11 0 0

2

( 1) ( )E E n E nε

ε= + − ⋅ ⋅ ⋅

or in Cartesian system:

11 0

2

( 1) ,i i i k ok

E E n n Eε

ε= + − ⋅ ⋅ ⋅

where i=1,2,3. The k index means the 1 to 3 summation. The components of the induction vector in various fields are: 1i oi

D Eε= 1 2 1 2( )i oi i k ok

D E n n Eε ε ε= + −

In the mean calculation according to values i

E andi

D , the n vectors for various planes can

be considered as probably equally distributed in any direction. In this situation, the dielectric can be considered isotropic, characterized by εm average permittivity, determined by the following relation:

i m i

D Eε= (3.30)

The integration for separate domains which reunite comes to multiplying D or E with the

volume of the given. For a q concentration of the dielectric with plane inclusions, the following is obtained:


26

1

2

1 13i oi

qE E

ε

ε

= + −

(3.31)

( )1 2 1

2

3i oi

qD Eε ε ε

= + −

(3.32)

Replacing in (3.30) the expressions of the average field and of the average induction the

expression of the permittivity is obtained under the following form:

( )1 2 1

1

2

2

3

1 13

m

q

q

ε ε εε

ε

ε

+ −=

+ −

(3.33)

The (3.33) formula can be generalized for N types of plane impurities:

( )1 1

1

1

1

2

3

11 1

3

N

i i

im N

i

i i

q

q

ε ε ε

εε

ε

=

=

+ −

=

+ −

∑

∑ (3.34)

where εi , qi are the permittivity and respectively, the concentration of the i component of the dielectric mixture.

If we use q0 to note the concentration of the basic substance, so that 0

1N

i

i

q=

=∑ and we

consider the following function: 1

( ) ( )N

i i

i

f f qε ε=

=∑ the (3.34) relation can be expressed under the

following form:

2 11

1 2

1 2( / )

2 ( / )m

ε εε ε

ε ε

+=

+ (3.35)

The obtained results can be applied even when the particles of impurity are not plane but

their thickness is small in comparison to the radial curves. For the flat dielectric ellipsoid, if εi>> ε1, the obtained formulae will be correct if the dimensions of the i species plan respect the following

conditions: 1iH

i

i

Lε

ε<<


27

3.6 Numerical methods for the calculation of the electric field in the dielectric mixtures 3.6.1 A general presentation of the calculation methods for the dielectric fields

Two types of problems can be taken into consideration for the configuration of conductors which are in electrostatic balance in a dielectric in which the load distribution is known. The first one is a determination of the potential in any point of the dielectric and on the surface of the conductors, the loads that charge them being known and the second one is the determination of the loads of the conductors when their potentials are known. The methods that can be employed to solve these problems of dielectric field can be classified in: analytical, numerical, graphical, and respectively graphical-analytical and analogical methods.

The main analytical methods are: the direct method, the one of integrating the Poisson-Laplace equations through the separation of the variables, the one of electrical images, of geometrical inversion (Kelvin), of functions of complex variable, of conform transformations and the one of Green functions.

The direct method is mainly applied for load repartitions in the isotropic dielectric, linear and homogenous, of constant permittivity and which occupies the infinite space or finite domains edged by equipotential surfaces. It is based on Coulomb’s experiments and uses the principle of action and reaction and the one of superposition of the effects.

In the method of electrical images, the massive conductors are replaced with image loads which are distributed inside their surfaces so that the field remains unchanged in the exterior of the conductors.

The method of geometrical inversion uses the properties of geometrical inversion in plane or space against a circle or sphere in order to determine the electrostatic field of some configurations of conductors when the distribution of the potential in the reversed configuration is known.

The method of complex functions and conform transformations is applied when solving problems of plane field – the parallels, in domains where the Laplace equation is respected. Separating the real and imaginary functions of a complex function and equating them with a constant, the possible field configurations are determined. A conform transformation allows the solving of relative problems at a girth which limits a certain domain through a transformation which makes either the superior semi plane, or a circle to correspond to this domain. In particular, the Schwartz-Cristoffel transformation gives correspondence between the superior semi plane and the interior or the exterior of a polygon. The method of the Green functions uses the principle of superposition in order to calculate the potential of a load repartition through the superposition of the elementary potentials established by elementary loads.

The most widely used of the numerical methods are: the method of the finite differences and the ones based on variational principles (Ritz-Galerkin, the method of finite elements etc). These methods can be applied to any field configuration, the obtained precision varying according to the method applied, the step of discretization and the capacity of the used calculator.

A clear distinction between the numerical and the analytical methods cannot be made. Through the method of separating the variables differential equations are obtained whose solutioning is possible only numerically or series of functions whose calculation needs the calculator use. Solving integral equations can, often be made only numerically


28

3.6.2 Numerical methods used in the case of dielectric mixtures

There are two types of problems in the study of the field in dielectrics: the calculation of the polarization of the dielectrics placed in a given field or the determination of the distribution of the potential for a system of loaded conductors in the presence of dielectrics. The latter is more complicated because the dielectric acts on the initial distribution of loads on conductors.

In the case of dielectric mixtures there is a first type of field problem. If the components of the mixture are considered to be linear, homogenous dielectrics, of constant permittivity, placed in a given field, the electric potential will satisfy a Poisson or Laplace equation in any sub-domain according to whether the dielectric is or is not electrically loaded.

The following models can be used in order to solve a field problem: the separation of the variables, the complex functions, the conform transformation, the plane or geometrical inversion, the numerical ones.

Except for some simple configurations (cylinder, sphere, dielectric ellipsoid in homogenous field), the analytical solutions are difficult to obtain or even impossible. In the case of dielectric mixtures – the configuration being extremely complex – the only way is the use of numerical methods of calculation which permit the determination of the potentials in any point of the domain and on their basis, the calculation of the electric field and of the polarization of the dielectrics.

Out of these, the method of finite differences is not really applicable to the modeling of systems of complex form because of the difficult writing of the conditions on the interface.

Establishing the equations with finite differences for the points on the interface becomes difficult in the case of curves and especially, of complex surfaces. Often, each point constitutes a particular case which needs a separate analysis. These equations were written for the points on the surface of a sphere by using an approximation of Poisson equation in spherical coordination and then writing the equations with finite differences for both the regular points of the network of division and those on the border.

The method of finite elements is preferable when approaching problems of calculation of the electric fields in inhomogeneous mediums, thus in the case of dielectric mixtures too, due not only to the facility of introducing limit conditions, its good precision but also to the invariation of the calculation formulae with the analyzed geometrical configuration. 3.6.3 The method of finite elements 3.6.31 The principle of the method

This method, firstly employed in mechanics, where O. C. Zienkiewics and his team obtained important results, was later introduced in electromagnetism too by P. Silvester and M. V. K. Chari in 1970. It then significantly developed due to the successive improvements brought by the university teams led by Mc. Gill in Canada, Rutherford in Great Britain and some other brought by some industrial research labs.

The principle of the method of finite elements is based on the integral formulation of the problems with partial derivates. This formulation can be of a variation type when an equivalence between the solution of the differential problem and the function that optimizes a functional, usually a Lagrange function of the system, is done. When such a functional cannot be found, the so-called Galerkin method is used, also known as the method of ‘ponderous residua’ , which involves the


29

minimization within the studied domain of the error integral which is obtained by replacing the exact solution through a function of attempt, conveniently chosen.

The domain places itself in small-sized but finite elements, on which the known function is approximated with the help of a function of linear attempt (for elements of first rank), polynomial (elements of second rank) or more complex (macroelements).

The F functional for the whole domain is equal to the sum of contributions brought through F(e) functional, particular to every finite element. If on each element where the potential V(x,y,z) is defined according to the V potentials of the nodes of the elements:

1

( , , ) ( , , )k

i i

i

V x y z N x y z V−

= ⋅∑ (3.36)

The defined integral associated to the functional is calculated, the latter becoming a function of Vi potentials:

( ) ( )1 2( , ,..., )e e

kF F V V V=

When the summing up of the elementary functions is done by taking into consideration the common sums to several elements, the global F functional becomes a function:

( )1 2, ,..., ,nF F V V V=

Of potentials in all the nodes of the division in finite elements. It is, then, written that the

complex of 1 2, ,...,n

V V V values which optimizes the functional has to verify the equations: 0

i

F

V

∂=

∂ where i=1,2,..n, which represents a system of algebraic equations. The solving of this system through an appropriate method allows us to know Vi values in every point of the domain. The method of finite elements are generally characterized through the order and nature of the used elements and through the formulation of variation used. In general, the finite element method is characterized by the following process.

One chooses a grid for Ω. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons. Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions. A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are merely continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial differential equations, one must use smoother basis functions. For instance, for a fourth order problem such as uxxxx + uyyyy = f, one may use piecewise quadratic basis functions that are C1.

Another consideration is the relation of the finite dimensional space V to its infinite

dimensional counterpart, in the examples above10H . A conforming element method is one in which

the space V is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite

dimensional space is not a subspace of the original10H .


30

3.6.3.2 The partition of the domain in finite elements. Triadic functions of approximation

The following two rules are respected at the partition of the v∑ domain in disjunctive sub

domains, called finite elements (vl) • Two distinct elements can only have in common points which are situated on their common

border, if the latter exists. This condition rules out overlapping. The borders can be points, curves, surfaces.

• The complex of all 1

v−

elements must constitute a domain as close as possible to the given one. The spaces between elements are particularly excluded.

When the border of the domain is represented by curves or surfaces which are more complex than the ones that define the borders of elements, an error is inevitable. This error is called error of geometrical syncretizing and can be reduced by diminishing the size of the elements (fig. 3.5., b) or using more complex elements (fig. 3.5, c)

Figure 3.5. Elements used in the partition of the calculation domain

The finite elements are characterized through their geometrical form and the number of the

degrees of freedom equal to the number of the nodes belonging to the element in which the values of the triadic function are prescribed.

The one-dimensional elements are especially used on the border of the domain. The most widely used elements in the analysis of the bi-dimensional problems are the linear triangular elements and the curvilinear square ones in which the geometry is approximated through the same square functions as the unknown size. In the tri-dimensional problem, the linear tetrads are widely used because of their high practicability; it seems that the hexahedral curvilinear elements have the advantage of a better representation of the phenomena in many problems.

They also have the advantage that the division of the domain can be more easily done and the errors of numbering the nodes are unlikely. First rank elements have the advantage of a higher simplicity of the concept and permit the usage of formulae of analytical integration for the calculation of functionals. However, they need a very careful division in order to obtain an appropriate precision.

The second rank curvilinear elements need an important work process and the use of methods of numerical integration for the calculation of functionals, a thing that can be, however, programmed without difficulties. These elements have the advantage that at an equal precision, the discretization is far less smooth than when using first rank elements.

Moreover, the curvilinear nature of the elements allows for a well adapted presentation of the geometrical forms which are to be found in the conception of electrical tools and machines.


31

The triadic functions of approximation in the method of finite elements are Lagrange, Hermite etc polynomials of interpolation. In the case of approximation with Lagrange polynomials of interpolation, the precision increases through the increase in the number of nodes per element, thus, a higher order of the element. When approximating through Hermite polynomials with the same purpose, the number of modal variables attached to each node is increased.

We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. P1 is a one-dimensional problem:

( ) ( ) ( )

( ) ( )

" in 0,1 ,

0 1 0,

u x f x

u u

=

= =

where f is given and u is an unknown function of x, and u'' is the second derivative of u with respect to x. The two-dimensional sample problem is the Dirichlet problem:

( ) ( ) ( ), , , in

=0 on ,

xx yyu x y u x y f x y

u

+ = Ω

∂Ω where Ω is a connected open region in the (x,y) plane whose boundary is ,∂Ω (e.g., a smooth manifold or a polygon), and uxx and uyy denote the second derivatives with respect to x and y, respectively.

The problem P1 can be solved "directly" by computing antiderivatives. However, this method of solving the boundary value problem works only when there is only one spatial dimension and does not generalize to higher-dimensional problems or to problems like u + u'' = f. For this reason, we will develop the finite element method for P1 and outline its generalization to P2. Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem using the finite element method.

In the first step, one rephrases the original boundary value problem in its weak, or variational form. Little to no computation is usually required for this step. The transformation is done by hand on paper.The second step is the discretization, where the weak form is discretized in a finite dimensional space.

After this second step, we have concrete formulae for a large but finite dimensional linear problem whose solution will approximately solve the original boundary value problem. This finite dimensional problem is then implemented on a computer.

3.6.3.3 Variational formulation

The first step is to convert P1 and P2 into their variational equivalents, or weak formulation. If u solves P1, then for any smooth function v that satisfies the displacement boundary conditions, i.e. v = 0 at x = 0 and x = 1, we have

( ) ( ) ( ) ( )

1 1 ''

0 0f x v x dx u x v x dx=∫ ∫

Conversely, if u with u(0) = u(1) = 0 satisfies (1) for every smooth function v(x) then one may show that this u will solve P1. The proof is easier for twice continuously differentiable u (mean value theorem), but may be proved in a distributional sense as well. By using integration by parts on the right-hand-side of (1), we obtain


32

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 ''

0 0

11' ' '

0 0

1 ' '

0

'

,

f x v x dx u x v x dx

u x v x u x v x dx

u x v x dx u vφ

=

= −

= − = −

∫ ∫

∫

∫ where we have used the assumption that v(0) = v(1) = 0.

We can loosely think of ( )10 0,1H

to be the absolutely continuous functions of (0,1) that are 0 at x = 0 and x = 1.

If we integrate by parts, we see that if u solves P2, then for any v,

( ),fvds u vds u vφΩ Ω

= − ∇ ⋅∇ = −∫ ∫

where ∇ denotes the gradient and denotes the dot product in the two-dimensional plane. Once more

φ can be turned into an inner product on a suitable space ( )10H Ω

of "once differentiable" functions

of Ω that are zero on. ∂Ω . We have also assumed that ( )10v H∈ Ω

. Existence and uniqueness of the solution can also be shown. 3.6.3.4 Discretization

Figure 3-6 A function in 1

0H , with zero values at the endpoints (blue), and a piecewise

linear approximation (red).

The basic idea is to replace the infinite dimensional linear problem:

Find 10u H∈ such that

( )10 , ,v H u v fvφ∀ ∈ − = ∫

with a finite dimensional version:

Find u V∈ such that


33

( ), ,v V u v fvφ∀ ∈ − = ∫

where V is a finite dimensional subspace of10H . There are many possible choices for V (one

possibility leads to the spectral method). However, for the finite element method we take V to be a space of piecewise linear functions.

For problem P1, we take the interval (0,1), choose n values of x with 0 = x0 < x1 < ... < xn < xn + 1 = 1 and we define V by

[ ] [ ] ( ) ( ) 1: 0,1 : is continuous, is linear for 0,... , and 0 1 0

k kx xV v v v k n v v

+= → = = =

where we define x0 = 0 and xn + 1 = 1. Observe that functions in V are not differentiable according to the elementary definition of calculus. Indeed, if then the derivative is typically not defined at any x = xk, k = 1,...,n. However, the derivative exists at every other value of x and one can use this derivative for the purpose of integration by parts.

Figure 3-7 A piecewise linear function in two dimensions.

For problem P2, we need V to be a set of functions of Ω. In the above figure, we have illustrated a triangulation of a 15 sided polygonal region Ω in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space V would consist of functions that are linear on each triangle of the chosen triangulation.

One often reads Vh instead of V in the literature. The reason is that one hopes that as the underlying triangular grid becomes finer and finer, the solution of the discrete problem (3) will in some sense converge to the solution of the original boundary value problem P2. The triangulation is then indexed by a real valued parameter h > 0 which one takes to be very small. This parameter will be related to the size of the largest or average triangle in the triangulation. As we refine the triangulation, the space of piecewise linear functions V must also change with h, hence the notation Vh. Since we do not perform such an analysis, we will not use this notation. 3.6.3.5 Comparison to the finite difference method

The finite difference method is an alternative way of approximating solutions of partial derivative equations. The differences between finite element method and finite difference method are:

• The most attractive feature of the finite element method is its ability to handle complicated geometries (and boundaries) with relative ease. While finite difference method in its basic


34

form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in finite element method is theoretically straightforward.

• The most attractive feature of finite differences is that it can be very easy to implement. • There are several ways one could consider the finite difference method a special case of the

finite element method approach. One might choose basis functions as either piecewise constant functions or Dirac delta functions. In both approaches, the approximations are defined on the entire domain, but need not be continuous. Alternatively, one might define the function on a discrete domain, with the result that the continuous differential operator no longer makes sense, however this approach is not finite element method.

• There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in finite difference method.

• The quality of a finite element method approximation is often higher than in the corresponding finite difference method approach, but this is extremely problem dependent and several examples to the contrary can be provided.


35

4. DESIGN AND EXPERIMENTAL RESULTS 4.1 General Data This section presents software simulations for various dielectric mixtures. The main aspects to be studied in the case of dielectric mixtures are: the determination of the permittivity of the mixture knowing the permittivity of the homogenous components and their concentration in the mixture, the feedback of the dielectric regarding the shape and position of the inclusions or when the maximal losses appear. The materials used in isolation systems are usually, dielectric mixtures (inhomogeneous dielectrics such as stratified, composite materials, mixtures etc) which have a high resistance to solicitations they are subjected to. The problem of creating materials with improved properties which could be used in certain conditions has also been suggested. This can be achieved by creating a mixture which would meet all the requirements of the target application. A computerized simulation would obviously represent a big advantage in solving this problem. However, what should be set as priority is the checking of the correctness of the numerical modeling which is not always easily done due to the fact that there are not analytical solutions which would permit the verification of the results’ accuracy. It has also been noted that the permittivity of the mixture depends on the form of the inclusions, on their size, orientation within the applied electric field and their distribution in the mixture. The simulations on statistical and matrix mixtures were done by taking into consideration all these factors of influence. 4.2 Software Presentation and Choice Reasoning For all the simulations and solving the equations presented in Chapter 3 we have chosen Maxwell 2D Student Version (SV), version 9. Maxwell is designed by Ansoft, an US based company and it’s a software package used for analyzing electromagnetic fields in cross-section structures. Given the fact that are a lot of solutions out there on the Internet market, choosing Maxwell has proven to be a smart decision due the fact that the Student Version is free with no trial or other costs involved. Plus that it supplied all the needed results or graphics for solving electrostatic problems. Maxwell SV uses finite element analysis (FEA) to solve two-dimensional (2D) electromagnetic problems. To analyze a problem, you need to specify the appropriate geometry, material properties, and excitations for a device or system of devices. The Maxwell software then does the following:

• automatically creates the required finite element mesh. • calculates the desired electrostatic or magnetostatic field solution and special quantities of

interest, including force, torque, inductance, capacitance, and power loss. Provides the ability to analyze, manipulate, and display field solutions.

• provides the ability to analyze, manipulate and display field solutions.


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4.3 Steps to be Performed for Solving an Electrostatic Problem

Our goal is to simulate the behavior of the dielectric mixtures in low frequency domain. We have computed the dielectric permittivity by considering inhomogeneous dielectrics placed in an uniform electrostatic field. In order to solve the desired problem the following steps have to be completed:

Use the Solver command to specify which of the following electric or magnetic field quantities to compute:electrostatic, magnetostatic, DC conduction, AC conduction etc.

Use the Drawing command to select one of the following model types: XY Plane or RZ Plane.

Use the Define Model command to access the following options: Draw Model, Group Object, Couple Model.

Use the Setup Materials command to assign materials to all the objects in the geometric model

Use the Setup Boundaries/Sources command to define the boundaries and sources for the problem.

Use the Setup Executive Parameters command to instruct the simulator to compute the following special quantities: matrix, force, torque, current flow

Use the Setup Solution Options command to specify how the solution is computed. Use the Solve command to solve for the appropriate field quantities. For electrostatic

problems, the simulator computes, the electric potential, from which it derives E and D. Use the Post Process command to analyze the solution, as follows: plot the field solution and

then use the calculator.

Figure 4-1 Maxwell Control Panel In the following flowchart there are presented the necessary steps for solving an electrostatic

problem. The commands shown on the Executive Commands menu must be chosen in the sequence in which they appear. For example, you must first create a geometric model with the Define Model command before you specify material characteristics for objects with the Setup Materials command. A check mark appears on the menu next to the completed steps:

:


37

Figure 4-2 Flowchart for solving a Maxwell 2D electrostatic problem 4.4 Modeling of Dielectric Mixtures with Various Inclusion Shapes

Knowing that the permittivity of the mixture depends on the shape and size of the inclusions, we have studied at first dielectric mixtures with various shapes of the inclusions: cylindrical, rectangular, squared, triangular, horizontal ellipsoid and vertical oriented ellipsoid (Figure 4-3). For all the cases, the modeling was done in 2D and we considered a cross section through a cylinder, a prism, parallelepiped, cuboid or ellipsoid. For all the mixtures studied in this section we have considered a plane capacitor in which we have a homogenous electric field. The concentration of the mixture was considered q=0.3, and we have determined the surface of the inclusion based on q and the surface of our host model which was the same for all simulations (a square with a side l=10 mµ ). Normally the concentration depends on the ratio between the volumes of the two materials. In our case, Maxwell provides data for when a cross section through the dielectric is considered, and which represents the object of our study. Hence, we will consider the concentration as a ratio between the areas of the two surfaces. The formula used is:

Inclusion Surface / Host Surface

iq = ,


38

where the host surface represents the area of the square with the side of 10 mµ , resulting an inclusion area of 30 mµ for every type of inclusion. Knowing the area and using exiting geometrical formulas we can compute the radius of the cylindrical inclusion, the side of the triangular inclusion or the axis of the ellipsoid. Having drawn our model we can proceed to the modeling of the desired dielectric.

We have considered polyethylene as the host material with a permittivity of ε =2.25 and for the inclusions we considered two situations: once glass with a permittivity of ε =3.78 and once barium titanate which has a permittivity of ε =1000.

Figure 4-3 Dielectric models with various shapes of the inclusions, where 1ε represents the

permittivity of the host medium and 2ε represents the permittivity of the inclusion

Table 4-1 Permittivity variation with respect to inclusion shape and material, where 1effε

represents the effective

(average) permittivity when having polyethylene as host material and glass inclusions ( 1ε=2.25, 2ε

=3.78) , and

2effε

represents the average permittivity when having polyethylene as host material and barium titanate

inclusions ( 1ε=2.25, 2ε

=1000)

ε /shape 1eff

ε 2effε

Circle 2.611893226 4.066342398 Rectangle 2.626417563 4.589550387 Square 2.615193870 4.278775943 Triangle 2.619990037 5.254818519 Ellipsoid H 2.339934803 2.723327579 Ellipsoid V 2.328272277 2.519362426

In Figure 4-4 we can observe the error variation depending on the number of steps, when

having polyethylene as host material and glass forming up the inclusions. We can notice that for the vertical oriented ellipsoid we have the highest error in the first two steps and the cylindrical inclusion has the most obvious error variation with respect to the number of steps. In Table 4-2 we can find the errors for each type of dielectric composite. In the case of glass inclusions the highest error is obtained for squared inclusion and in the case of barium titanate inclusions, the highest error


39

is obtained for the rectangular inclusion. It can be easily observed how the error varies depending on the type of inclusion and its permittivity.

Table 4-2 Error variation with respect to the shape of the inclusion, for each average permittivity

ε /shape Error% / 1eff

ε Error% / 2effε

Circle 0.0595 0.4650 Rectangle 0.0993 0.7690 Square 0.1010 0.0572 Triangle 0.1000 0.3130 Ellipsoid H 0.5470 0.0621 Ellipsoid V 0.6170 0.0543

a) Cylindrical b) Rectangular

c) Squared d) Triangular

e) Horizontal Oriented Ellipsoid f) Vertical Oriented Ellipsoid

Figure 4-4 Error dropdown with respect to the number of steps when having polyethylene as

the host material and quartz glass as the inclusion. Inclusions have the following shape: a)


40

cylindrical, b) rectangular, c) squared, d) triangular, e) horizontal oriented ellipsoid and f)

vertical oriented ellipsoid

a) Cylindrical b) Rectangular

c) Squared d) Triangular

e) Horizontal Oriented Ellipsoid f) Vertical Oriented Ellipsoid Figure 4-5 Error dropdown with respect to the number of steps when having polyethylene as

the host material and barium titanate (BaTiO3) as the inclusion. Inclusions have the following

shape: a) cylindrical, b) rectangular, c) squared, d) triangular, e) horizontal oriented ellipsoid

and f) vertical oriented ellipsoid


41

a) Triangle(quartz glass) b) Triangle(barium titanate)

Figure 4-6 Electric potential φ variation when the inclusion is triangular shaped: a) quartz

glass and b) barium titanate BaTiO3

a) Square (quartz glass) b) Square (barium titanate)

Figure 4-7 Electric potential φ variation when the inclusion is squared shaped: a) quartz glass

and b) barium titanate BaTiO3

a) Rectangle (quartz glass) b) Rectangle (barium titanate)

Figure 4-8 Electric potential φ variation when the inclusion is rectangular shaped: a) quartz



42

a) Horizontal oriented ellipsoid (quartz glass) b) Horizontal oriented ellipsoid (barium titanate)

Figure 4-9 Electric potential φ variation when the inclusion has the shape of a horizontal

ellipsoid: a) quartz glass and b) barium titanate BaTiO3

a) Vertical oriented ellipsoid (quartz glass) b) Vertical oriented ellipsoid (barium titanate)

Figure 4-10 Electric potential φ variation when the inclusion has the shape of a vertical

ellipsoid: a) quartz glass and b) barium titanate BaTiO3

a) Circle (quartz glass) b) Circle (barium titanate)

Figure 4-11 Electric potential φ variation when the inclusion is circular shaped: a) quartz



43

4.5 Modeling of Matrix Structured Dielectric Mixtures For this scenario we have considered a matrix structured dielectric with cylindrical inclusions arranged once as an ordered structure and once as a disordered structure. It can be observed in the first figure below that the geometrical center of an inclusion is the same with center of the square in which the inclusion is situated. If the centers of 4 neighboring inclusions are connected with each other, they will make up a square, which makes our structure an ordered one. The second part of this scenario involves the modeling of the same matrix structured dielectric but with its cylindrical inclusions randomly placed within their corresponding host squares. For each of the two models there were made a pair of two simulations. The first case was considered having polyethylene as the host material and glass as the inclusions, and the second case involved polyethylene as host material and barium titanate (BaTiO3) as the inclusion type. Simulations were also made for different concentrations of our dielectric composite.

a) ordered structure b) disordered structure

Figure 4-12 Matrix structured dielectric mixture: a) ordered structure b) disordered structure The concentration for these two models was calculated based on the ratio between the area of the circle which actually represents a cross section through a cylindrical inclusion and the area of the host square from the matrix structure. Therefore, knowing the side of the square with a side of and the radius of the cylindrical inclusion, the concentration q resulted by applying the following formula:

2 2/i

q r lπ= [%]

Table 4-3 Permittivity variation with respect to concentration and incusions type, where 1effε

represents the

effective (average) permittivity when having polyethylene as host material and glass inclusions ( 1ε=2.25, 2ε

=3.78)

, and 2effε



=1000). 1ε represents the permittivity of the host medium and 2ε

of the inclusions.


44

Concentration q % ε /matrix structure 1eff

ε 2effε

0.4 Ordered 2.752042924 5.085069146 0.4 Disordered 2.754186525 5.216091304 0.099 Ordered 2.364426288 2.725865857 0.099 Disordered 2.364502187 2.727342757 Table 4-4 Average permittivity calculated with Maxwell-Wagner formula for the two types of dielectric mixtures

and concentration modification: 1effε

represents the effective (average) permittivity when having polyethylene as

host material and glass inclusions ( 1ε=2.25, 2ε

=3.78) , and 2effε

represents the average permittivity when having

polyethylene as host material and barium titanate inclusions ( 1ε=2.25, 2ε

=1000)

Concentration q% ε /matrix structure

1effε 2eff

ε

0.4 Maxwell-Wagner model 2.589201877 5.051742649 0.099 Maxwell-Wagner model 2.321874655 2.667206000 Table 4-5 Error variation with respect to the dielectric structure and concentration

Concentration q % ε /matrix structure Error % / 1eff

ε Error % / 2effε

0.4 Ordered 0.000605 0.002190 0.4 Disordered 0.000658 0.002290 0.099 Ordered 0.000902 0.002200 0.099 Disordered 0.001160 0.002390

a) triangular mesh b) triangular mesh – close up

Figure 4-13 Triangular mesh for an ordered matrix structured dielectric mixture a) general

view b) close up view We can visualize the elements (triangles) of the mesh in the figure above, observing that the density of triangles is much higher in the interest area. When simulating this scenario the elements of the network reached to a number of 22200 triangles, depending on the structure deploy (ordered/disordered) on the permittivity of the inclusions and concentration of the mixture. We


45

obtained the highest number of triangles (22975) when having a disordered matrix structured dielectric mixture, barium titanate inclusions and a concentration of 0.4%.

a) Ordered structure(Glass Inclusions) b) Ordered structure(BaTiO3 Inclusions)

c) Disordered structure(Glass Inclusions) d) Disordered structure(BaTiO3 Inclusions)

Figure 4-14 Error dropdown with respect to the number of steps, inclusions type and

concentration When simulating with a concentration of 0.4% of the dielectric composite we obtained the error variation graphs from above. The error drops down under 0.001% or varies around this value, which was our initial target value for this scenario, but differs, depending on the permittivity of the inclusions, the number of triangles of the network and in our case, if the composite is an ordered or disordered one. The initial error, between the first three iterations of a simulation is much higher in the case of the ordered composites with glass inclusions, but at the end the error is much higher in the case of disordered mixtures and it can be observed in Table 4-5. In the following pages of this scenario we can observe the variation of the electric potential and electric field along our mixture. Regarding the electric potential, the changes are not very significant when the inclusions have low permittivity (Figure 4-15, Figure 4-17), but we can notice significant differences in the case of inclusions with higher permittivity (Figure 4-16, Figure 4-18), especially in the case of disordered placement of the inclusions. The electric field records the highest values between the inclusions and much lower values within the inclusions. In the case of disordered placement of the inclusions we can notice the value of E significantly increases when the inclusions are very close to each other (Figure 4-20).


46

Figure 4-15 Electric potential φ variation for an ordered structured dielectric mixture with

glass inclusions and polyethylene as host material

Figure 4-16 Electric potential φ variation for an ordered structured dielectric mixture with

BaTiO3 inclusions and polyethylene as host material


47

Figure 4-17 Electric potential φ variation for a disordered structured dielectric mixture with

glass inclusions and polyethylene as host material

Figure 4-18 Electric potential φ variation for a disordered structured dielectric mixture with

BaTiO3 inclusions and polyethylene as host material


48

Figure 4-19 Electric field plot for an ordered matrix structured dielectric composite having

polyethylene as host material and inclusions made up of barium titanate

Figure 4-20 Electric field plot for a disordered matrix structured dielectric mixture having

polyethylene as host material and inclusions made up of barium titanate


49

4.6 Modeling of Dielectric Mixtures Based on the Distance Between the Inclusions

In this chapter we considered two cylindrical inclusions placed in a rectangle with a length of 50 microns and a width of 25 microns. We considered the host material as polyethylene with a permittivity of 2.25 and for the inclusions we considered two separate situations: once glass inclusions with a permittivity of 3.78 and once barium titanate inclusions (BaTiO3) with a permittivity of 1000. The radius of an inclusion was considered to be r=2 microns. We studied the permittivity variation by varying the distance between the two inclusions, from the minimum of d=0 to the maximum of d=12r (24 microns).

a) distance d=0 microns b) distance d=24 microns

Figure 4-21 Dielectric composite with two cylindrical inclusions spaced at: a) 0 microns b) 24

microns

Table 4-6 Permittivity variation with respect to distance between the inclusions, where 1effε

represents the

effective (average) permittivity when having polyethylene as host material and glass inclusions ( 1ε=2.25, 2ε

=3.78)

, and 2effε




of the inclusions.

ε /distance

1effε 2eff

ε

d=0 2.278093619 2.431498328

d=r 2.277027228 2.365913617

d=2r 2.276761658 2.359967690

d=3r 2.276762259 2.357939643

d=4r 2.276782985 2.357184891

d=5r 2.276878173 2.357129213

d=6r 2.277005645 2.357362583

d=7r 2.277203074 2.357950836

d=8r 2.277402008 2.358622804

d=9r 2.277670073 2.359595494

d=10r 2.277932512 2.360597417

d=11r 2.278259099 2.361886507

d=12r 2.278593454 2.363191827


50

For both of the calculated average (effective) permittivities in Table 4-5 we considered polyethylene as the host material. When the inclusions consisted of glass, we noticed a maximum average permittivity when the inclusions were spaced at 0 microns apart. However, the average permittivity increased when the distance between the inclusions reached to d=11r, and obtained the maximum for d=12r. In the case of barium titanate inclusions the maximum average permittivity is obtained for d=0 microns, after which the permittivity has a significant decrease until d=5r. When reaching the maximum distance of d=12r, the permittivity increases only with 0.01 units, noticing that the average permittivity has a significant decrease with respect to distance in the case of inclusions with very high permittivity.

a) distance d= 0 microns b) distance d=12 x r = 24 microns

Figure 4-22 Triangular mesh for a dielectric composite with inclusions spaced at: a) 0 microns

b) 24 microns

a) distance d= 0 microns b) distance d=12 x r = 24 microns

Figure 4-23 Distribution network elements increase with respect to the number of steps

increase: a) Inclusions are spaced at 0 microns b) Inclusions are spaced at 24 microns


51

The number of the network elements (triangles) increases in the interest area, even when the inclusions are widely spaced apart as can be seen in Figure 4-22. Judging from the graphs in Figure 4-23, the network has approximately the same number of elements when the inclusions are spaced at 0 microns and when at 24 microns. Verifying the actual results from Maxwell, we recorded a number of 1548 triangles when the distance is minimum and a number of 1571 when the distance is maximum, in case of glass inclusions. When having BaTiO3 inclusions the number of triangles is 1582 for the minimum distance and 1591 for the maximum distance. Therefore, the number of network elements is much higher when the permittivity is much higher, but not significantly higher.

a) Glass inclusions spaced at d= 0r b) BaTiO3 inclusions spaced at d=0r

Figure 4-24 Error dropdown with respect to distance and inclusions type: a) glass inclusions b)

barium titanate inclusions

a) Glass inclusions spaced at d= 12r a) BaTiO3 inclusions spaced at d= 12r Figure 4-25 Error dropdown with respect to distance and inclusions type: a) glass inclusions b)

barium titanate inclusions The targeted error was 0.01%, and we obtained error values of 0.0135% and 0.0141% for the minimum and maximum distances in case of glass inclusions, and 0.0129% and 0.0170% for the minimum and maximum distances in case of barium titanate inclusions. Thus, the error increases with the distance and with the permittivity of the inclusions.


52

a) Glass inclusions spaced at d= 0r b) BaTiO3 inclusions spaced at d=0r

Figure 4-26 Electric potential φ variation when the inclusions are spaced at 0 microns apart:

a) glass inclusions b) barium titanate inclusions

a) Glass inclusions spaced at d= 12r b) BaTiO3 inclusions spaced at d= 12r

Figure 4-27 Electric potential φ variation when the inclusions are spaced at 24 microns apart:

a) glass inclusions b) barium titanate inclusions The electric potential varies based on the voltage applied on the plates of the capacitor. For our model we have considered 10V on the left plate and 0V on the right plate and, as can be seen in Figures 4-26, 4-27, the electric potential varies from 10V represented by dark red to 0V represented by dark blue. The distance between the inclusions does not affect the values of the electric potential along our model. The permittivity of the inclusions modifies the values of the electric potential φ

across the model, in the case of barium titanate inclusions. The change is noticeable in both cases: when the inclusions have no space between them, and when they are spaced at 24 microns. In the first case the two inclusions have the same effect that a bigger cylindrical inclusion would have on the electric potential. In the second case the value of the electric potential along the inclusion extends towards the neighboring values narrowing them.


53

a) Inclusions with low permittivity

b) Inclusions with high permittivity

Figure 4-28 Permittivity variation with respect to distance and average permittivity: a) glass

inclusions b) barium titanate inclusions


54

4.7 Modeling of Dielectric Mixtures Based on the Concentration of the Inclusions

Figure 4-29 Dielectric composite with five

cylindrical inclusions Figure 4-30 Dielectric composite with five

cylindrical inclusions – distribution mesh The last scenario involved the modeling of a dielectric composite with five cylindrical inclusions based on varying the concentration of the inclusions. We considered polyethylene as the host material with a permittivity of 2.25. We implemented two types of simulations: once for glass inclusions with a permittivity of 3.78 and once for barium titanate inclusions with a permittivity of 1000. The inclusions were placed in a square with a side of 10 microns. Having known the concentration, the radius of the cylindrical inclusions was computed with following formula:

2

2 2

5inclusion ii

host

N S qrq r

S l l

π• •= = ⇒ =

Table 4-7 Permittivity variation with respect to inclusions` concentration, where 1effε

represents the effective

(average) permittivity when having polyethylene as host material and glass inclusions ( 1ε=2.25, 2ε

=3.78) , and

2effε




of the inclusions.

Concentration/q %

1effε 2eff

ε

0.1 2.364464817 2.733126868 0.15 2.425284262 3.000804975 0.20 2.490959193 3.103327554 0.25 2.553108290 3.702588906 0.35 2.683325479 4.530542638 0.45 2.824619176 5.863116344


55

a) glass inclusions with q=0.1% b) BaTiO3 inclusions with q=0.1%

Figure 4-31 Error dropdown with respect to concentration and inclusions` type: a) glass

inclusions b) barium titanate inclusions

a) glass inclusions with q=0.45% b) BaTiO3 inclusions with q=0.45%

Figure 4-32 Error dropdown with respect to concentration and inclusions` type: a) glass

inclusions b) barium titanate inclusions Table 4-8 Error variation with respect to concentration and inclusions` type

Concentration/q % Error/ 1effε % Error/ 2eff

ε %

0.1 0.00962 0.01190 0.15 0.01020 0.01060 0.20 0.00887 0.01180 0.25 0.00739 0.01110 0.35 0.00741 0.01070 0.45 0.00785 0.00923

The error drops down with the growth of the concentration. This is due to the fact that the number of triangles (the elements of the mesh) is higher when the concentration is maximum,


56

leading to a lower error. Therefore the more elements the network has, the smaller the energy error will be (see Table 4-8). The error is much higher for the composite with the barium titanate inclusions, especially in the first two iteration steps, ranging between 7-8% when having a concentration of 0.1%, and between 10-12% when having a concentration of 0.45% (Figure 4-31b, Figure 4-32b). In case of glass inclusions the error for the first two iterations ranges between 4-4.5% for both type of displayed concentrations: 0.1% and 0.45%. The minimum error is 0.00785% for the dielectric composite with glass inclusions and 0.00923 for the dielectric composite with barium titanate inclusions.

The following plots display the electric potential variation across the dielectric composite. The value of the electric potential is much higher in the left part of the mixture, where a voltage of 10V was applied on the plate of the capacitor. The value of the electric potential continues to drop down from left to right, reaching 0V in the area represented with dark blue. In Figure 4-34b we can observe how inclusions with high permittivity (BaTiO3) and concentration (0.45%) affect the value of the electric potential along the mixture. These modifications are less significant with the dropdown of the concentration as can be seen in Figure 4-33b.

a) glass inclusions with q=0.1% b) BaTiO3 inclusions with q=0.1% Figure 4-33 Electric potential φ variation for a dielectric composite with five inclusions

having a concentration of 0.1%: a) glass inclusions b)barium titanate inclusions

a) glass inclusions with q=0.45% b) BaTiO3 inclusions with q=0.45% Figure 4-34 Electric potential φ variation for a dielectric composite with five inclusions

having a concentration of 0.45%: a) glass inclusions b)barium titanate inclusions


57

a) glass inclusions a) glass inclusions Figure 4-35 Dielectric composite with five glass cylindrical inclusions having q=0.45%: a)

Electric field plot b)Electric Induction plot

a) BaTiO3 inclusions b) BaTiO3 inclusions

Figure 4-36 Dielectric composite with five BaTiO3 cylindrical inclusions having q=0.45%: a)

Electric field plot b)Electric Induction plot The plots from above present the variation of the electric field E and induction D along the dielectric composite with five cylindrical inclusions. The values of E and D are much higher when the concentration of the mixture is higher. The values of E are much smaller within the inclusions, growing between the inclusions and in the area where the inclusions are close to the edges of the host material. However, the opposite thing happens in the case of electric induction. The value of D is much higher within the inclusions and on the area where the inclusions are very close to the edge of the host medium. The value continues to drop on the area between the inclusions (across the diagonals of the host medium if we can say so), reaching the lower values where the inclusions are further from each other. (see Figure 4-36b)


58

Figure 4-37 Dielectric composite with five BaTiO3 cylindrical inclusions having q=0.45%:

Electric Induction plot – 3D view

Figure 4-38 Dielectric composite with five BaTiO3 cylindrical inclusions having q=0.45%:

Electric field plot – 3D view


59

5. CONCLUSIONS AND FUTURE WORK 5.1 Conclusions 5.2 Future work

6. REFERENCES [1] Brendan Kevin Patrick Scaife, “Monographs on the physics and chemistry of materials – Principles of Dielectrics”, Oxford University Press 1989 [2] Steven A. Boggs, Chathan M. Cooke, R.J. Densley, Edward Sacher, James E. West, Committee on Dielectrics, Assembly of Engineering, National Research Council, Annual Report, “Conference on Electrical Insulation and Dielectric Phenomena”, National Academy of Sciences Washington D.C. 1979 [] Robert, P. “Materiaux de l’electrotechnique” , Laussannes, Press Polytechniques Romandes, 1989 [3] Ari Sihvola, “Electromagnetic mixing formulas and applications”, The Institution of Electrical Engineers, London, United Kingdom 1999 [4] Perambur S. Neelakanta, “Handbook of Electromagnetic Materials”, CRC Press LLC 1995 [5] Enis Tuncer, Yuiry V. Serdyuk and Stanislaw M. Gubanski, “Dielectric mixtures – electrical properties and modeling”, Chalmers University of Technology, 412 96 Gothenburg Sweden [] Bottcher, C.J.F. “Theory of Electrical Polarization”, Amsterdam, Elsevier, 1978 [] [6] Mailadil T. Sebastian, “Dielectric Materials for Wireless Communication”, Elsevier B.V. 2008 [7] Hermann A. Haus, James R. Melcher, “Electromagnetic Fields and Energy”, Sep 1989 [8] Gorur G. Raju, “Dielectrics in Electric Fields”, Marcel Dekker, Inc 2003 [9] Enis Tuncer, “A formula for dielectric mixtures”, March 30 2005 [10] Enis Tuncer, “The Landau-Lifshitz/Looyenga dielectric mixtures expression and its self-similar fractal nature”, March 31 2005 [11] Enis Tuncer, “Analogy between dielectric relaxation and dielectric mixtures: Application of the spectral density representation”, February 24 2005 [12] Enis Tuncer, “On micro-structural effects in dielectric mixtures”, IEEE Trans Diel Elec Insul, February 2nd 2008 [13] David A. De Wolf, “Essentials of Electromagnetics for Engineering”, Cambridge University Press 2001


60

APPENDIX A

15 15 15

0 120

10 10 10

8.854187817 10r r

D D D D

E E E Eε ε ε ε

ε

− − −

−

⋅ ⋅ ⋅= ⇒ ⋅ = ⇒ = =

⋅ ⋅ ⋅

310 / 8.854187817 ......eff D Eε −= ⋅ ⋅ ⇒

I. INCLUSION SHAPE Cylindrical – glass

effε = 2.3188370965801 310−⋅ / 0.0008877993452=2.6118932269067

Cylindrical – BaTiO3

effε = 3.757104838470646 310−⋅ / 0.000923951913003=4.06634239898

Rectangular – glass

effε = 2.33051820298678 310−⋅ / 0.00010021667635228=2.6264175631

Rectangular – BaTiO3

effε = 4.30500414972669 310−⋅ / 0.00093800128248907=4.589550387

Ellipsoid V – glass

effε = 2.06717606453431 310−⋅ / 0.00088785838500054=2.328272277

Ellipsoid V – BaTiO3

effε = 2.25248582939458 310−⋅ / 0.00089406978771589=2.519362426

Ellipsoid H – glass

effε = 2.07794149088439 310−⋅ / 0.0008880339263369=2.339934803

Ellipsoid H – BaTiO3

effε = 2.47883811533557 310−⋅ / 0.00091022399728328=2.723327579

Triangular – glass

effε = 2.32621525789178 310−⋅ / 0.000887871795080=2.619990037

Triangular – BaTiO3

effε = 5.01208477335928 310−⋅ / 0.000953807396991=5.254818519

Squared – glass

effε = 2.32033350113327 310−⋅ / 0.00088725164743568=2.615193870


61

Squared – BaTiO3

effε = 3.97053635136942 310−⋅ / 0.0009279608008757875=4.278775943

II. MATRIX MIXTURES 25 ordered – glass/0.4%

effε = 1.221188326390147 210−⋅ / 0.00443738836819546=2.752042924

25 ordered – BaTiO3/0.4%

effε = 2.366177637481008 210−⋅ / 0.00465318674998849444=5.085069146

25 disordered – glass/0.4%

effε = 1.222784036804622 210−⋅ / 0.0044397284836772735=2.754186525

25 disordered – BaTiO3/0.4%

effε = 2.492232813743497 210−⋅ / 0.004777970070604490=5.216091304

25 ordered – glass/0.099%

effε = 1.048488665258468 210−⋅ / 0.004434431601109071=2.364426288

25 ordered – BaTiO3/0.099%

effε = 1.233041658575903 210−⋅ / 0.00452348698214127=2.7258653853

25 disordered – glass/0.099%

effε = 1.048546168223899 210−⋅ / 0.00443453245176056=2.3645021873

25 disordered – BaTiO3/0.099%

effε = 1.234413414890217 210−⋅ / 0.00452606630171551=2.7273427577

III. DISTANCE BETWEEN INCLUSIONS

0r – glass

effε = 4.04747075999408 310−⋅ / 0.001776692022735=2.2780936190

1r-glass

effε = 4.04546911441329 310−⋅ / 0.001776645032436=2.2770272285

2r – glass

effε = 4.04506745896210 310−⋅ / 0.001776675851883=2.2767616583


62

3r – glass

effε = 4.04513717552922 310−⋅ / 0.001776706004019=2.2767622591

4r – glass

effε = 4.04520501646790 310−⋅ / 0.001776719626862=2.2767829855

5r – glass

effε = 4.04542341110569 310−⋅ / 0.001776741266862=2.2768781738

6r – glass

effε = 4.04566836975588 310−⋅ / 0.001776749380457=2.2770056450

7r – glass

effε = 4.04608080742898 310−⋅ / 0.001776776455843=2.2772030742

8r – glass

effε = 4.04648946380284 310−⋅ / 0.00177680069141689=2.2774020087

9r – glass

effε = 4.04705735279930 310−⋅ / 0.00177684090451992=2.2776700730

10r – glass

effε = 4.04760357269992 310−⋅ / 0.00177687598333170=2.2779325122

11r – glass

effε = 4.04839455678819 310−⋅ / 0.00177690845725213=2.2782590992

12r – glass

effε = 4.04912343798827 310−⋅ / 0.00177702759150242=2.2785934542

0r – BaTiO3

effε = 4.36076515912331 310−⋅ / 0.00179344773087430=2.4314983280

1r – BaTiO3

effε = 4.22167157631382 310−⋅ / 0.00178443734149670=2.3659126172

2r – BaTiO3

effε = 4.21093784539389 310−⋅ / 0.00178432012570200=2.3599676900

3r – BaTiO3

effε = 4.20780265551438 310−⋅ / 0.00178452517511900=2.3579396436


63

4r – BaTiO3

effε = 4.20685328863310 310−⋅ / 0.00178469381159700=2.3571848914

5r – BaTiO3

effε = 4.20716331456986 310−⋅ / 0.00178486749468000=2.3571292138

6r – BaTiO3

effε = 4.20795669578505 310−⋅ / 0.00178502735424500=2.3573625837

7r – BaTiO3

effε = 4.20950331678200 310−⋅ / 0.00178523794954300=2.3599508366

8r – BaTiO3

effε = 4.21108712515542 310−⋅ / 0.00178540083502603=2.3586228047

9r – BaTiO3

effε = 4.21347948066819 310−⋅ / 0.00178567872774000=2.3595954953

10r – BaTiO3

effε = 4.21576477924430 310−⋅ / 0.00178588867754280=2.3605977417

11r – BaTiO3

effε = 4.21892124573502 310−⋅ / 0.00178625516067190=2.3618805

12r – BaTiO3

effε = 4.22210725047182 310−⋅ / 0.00178661215988699=2.3631918

IV. CONCENTRATION

0.1% - glass

effε = 2.10209970447112 310−⋅ / 0.000889038267152=2.364464817

0.1% - BaTiO3

effε = 2.48337742560883 310−⋅ / 0.000908621350311=2.733126868

0.15% - glass

effε = 2.15726025093500 310−⋅ / 0.000889487589014=2.425284262

0.15% - BaTiO3


64

effε = 2.76822298265124 310−⋅ / 0.000920271674869=3.000804975

0.2% - glass

effε = 2.21735249396401 310−⋅ / 0.000890160103508=2.490959193

0.2% - BaTiO3

effε = 3.10332755428766 310−⋅ / 0.000932033466726=3.103327554

0.25% - glass

effε = 2.27270278253110 310−⋅ / 0.000890170930326=2.553108290

0.25% - BaTiO3

effε = 3.49185476184248 310−⋅ / 0.000943084649680=3.702588906

0.35% - glass

effε = 2.3899553575059 310−⋅ / 0.0008906691997180=2.683325479

0.35% - BaTiO3

effε = 4.3552389312603 310−⋅ / 0.0009613062448920=4.530542638

0.45% - glass

effε = 2.5164213054786 310−⋅ / 0.0008908886998239=2.824619176

0.45% - BaTiO3

effε = 5.7427166780152 310−⋅ / 0.0009794570344117=5863163442

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