Solutions: Solution. d = 3.0g/cm we can calculate the number of Xe atoms per unit volume, Given m and the given values from Table 7.

Transcription

1 Tutial-09 Tutial - 09 Sectin6: Dielectic Mateials ECE:09 (Electnic and Electical Ppeties f Mateials) Electical and Cmpute Engineeing Depatment Univesity f Watel Tut: Hamid Slutins: 7.3 Electnic plaizatin in liquid xenn Liquid xenn has been used in adiatin detects. The density f the liquid is 3.0 g cm 3. What is the elative pemittivity f liquid xenn given its electnic plaizability in Table 7.1? (The expeimental is 1.96.) Slutin Given -3 d 3.0g/cm we can calculate the numbe f Xe atms pe unit vlume, N N A d M at cm With and 3 8 N m and the given values fm Table 7.1, CV 1 m 1, fm Eq 7.14, we have α e Fm Nαe But, if we use Clausius-Msstti Equatin, ( ( 1) + ) Nα 3 e Nαe Nαe 1 3 (Xe) Which is clse t the expeimental value Relative pemittivity, bnd stength, bandgap and efactive index Diamnd, silicn, and gemanium ae cvalent slids with the same cystal stuctue. Thei elative pemittivities ae shwn in Table 7..

2 a. Explain why inceases fm diamnd t gemanium. Tutial-09 b. Calculate the plaizability pe atm in each cystal and then plt plaizability against the elastic mdulus Y (Yung's mdulus). Shuld thee be a celatin? c. Plt the plaizability fm pat (b) against the bandgap enegy, E g. Is thee a elatinship? d. Shw that the efactive index n is. When des this elatinship hld and when des it fail? e. Wuld yu cnclusins apply t inic cystals such as NaCl? Slutin a. In diamnd, Si, and Ge, the plaizatin mechanism is electnic (bnd). Thee ae tw facts that incease the plaizatin. Fist is the numbe f electns available f displacement and the ease with which the field can displace the electns. The numbe f electns in the ce shells inceases fm diamnd t Ge. Secndly, and mst imptantly, the bnd stength pe atm deceases fm diamnd t Ge, making it easie f valence electns in the bnds t be displaced. b. F diamnd, atmic cncentatin N is: DN N M at A ( 3.5 kg/m )( 6.0 ml ) 3 ( 1 kg/ml) The plaizability can then be fund fm the Clausius-Msstti equatin: α e N 3 α e ( 1) ( + ) 1 ( F/m)( 5.8 1) 9 3 ( m )( ) α e F m N m -3 The plaizability f Si and Ge can be fund similaly, and ae summaized in Table 7Q-: N (m -3 ) α e (F m ) Diamnd m F m Si m F m

3 Tutial-09 Ge m F m Table 7Q4-1: Plaizability values f diamnd, Si and Ge Plaizability pe atm vesus Yung's Mdulus Plaizability pe atm (F m ). 6E-40 5E-40 4E-40 3E-40 E-40 1E-40 0 y -5E-43x + 5E-40 R Yung's Mdulus (GPa) Figue 7Q4-1: Plt f plaizability pe atm vesus Yung s mdulus. As the plaizatin mechanism in these cystals is due t electnic bnd plaizatin, the displacement f electns in the cvalent bnds depends n the flexibility elasticity f these bnds and hence als depends n the elastic mdulus. c.

4 Tutial-09 Plaizability pe atm vesus Bandgap Plaizability pe atm (F m ) 6E-40 5E-40 4E-40 3E-40 E-40 1E-40 0 y -8E-41x + 5E-40 R Bandgap Eg (ev) Figue 7Q4-: Plt f plaizability vesus bandgap enegy. Thee indeed seems t be a linea elatinship between plaizability and bandgap enegy. d. T facilitate this pf, we can plt a gaph f efactive index, n, vesus elative pemittivity,. Refactive Index vesus Relative Pemittivity Refactive Index (n). 1 y x R Realtive Pemittivity ( ) Figue 7Q4-3: Lgaithmic plt f efactive index vesus elative pemittivity.

5 Tutial-09 The lg-lg plt exhibits a staight line thugh the thee pints. The best fit line is n A x (Celatin cefficient is ) whee x / and A exp(0.0070) 1. Thus n is. The efactive index n is an ptical ppety that epesents the speed f a light wave, an electmagnetic wave, thugh the mateial (v c/n). The light wave is a high fequency electmagnetic wave whee the fequency is f the de f 14 t 15 Hz (ƒ ptical ). n and plaizability ( ) will be elated if the plaizatin can fllw the field scillatins at this fequency (ƒ ptical ). This will be the case in electnic plaizatin because electns ae light and apidly espnd t the fast scillatins f the field. The elatinship between n and will nt hld if we take at a lw fequency (<< ƒ ptical ) whee the slw plaizatin cntibutins (such as inic plaizatin, dipla plaizatin, intefacial plaizatin) als cntibute t. e. n wuld apply t inic cystals if is taken at the cespnding ptical fequency athe than at fequencies belw ƒ ptical. Tabulated data f inic cystals typically qute that includes inic plaizatin and hence this data des NOT cnfm t n. 7.6 Dielectic cnstant f wate vap steam The islated wate mlecule has a pemanent diple p f C m. The electnic plaizability α e f the wate mlecule unde dc cnditins is abut 4-40 C m. What is the dielectic cnstant f steam at a pessue f atm ( 5 Pa) and at a tempeatue f 400 C? [Nte: The numbe f wate mlecules pe unit vlume N can be fund fm the simple gas law, P (N/N A )RT. The Clausius Msstti equatin des nt apply t ientatinal plaizatin. Since N is small, use Equatin 7.14.] Slutin Given the gas law, P NRT N A whee the gas cnstant, R J.ml 1.K 1 and 1 3 N A 6.03 ml, N At T 673 K and P atm we will have, m PN N RT A m 3 f H O mlecules. Since p Cm and k J K 1, the dipla plaizability will be, α d p 3 kt Fm Theefe, given α 4 40 Fm, fm Eq 7.14 e

6 Tutial-09 N(αe + αd ) (steam) at atm and T 673 K. 7.8 Inic and electnic plaizatin Cnside a CsB cystal that has the CsCl unit cell cystal stuctue (ne Cs + -B - pai pe unit cell) with a lattice paamete (a) f nm. The electnic plaizability f Cs + and B - ins ae F m and F m, espectively, and the mean inic plaizability pe in pai is F m. What is the lw fequency dielectic cnstant and that at ptical fequencies? Slutin The CsB stuctue has a lattice paamete given by a nm, and thee is ne CsB in pai pe unit cell. If n is the numbe f in pais in the unit cell, the numbe f in pais, individual ins, pe unit vlume (N) is n 1 N m -3 a 3 9 ( m) 3 At lw fequencies bth inic and electnic plaizability cntibute t the elative pemittivity. Thus, fm Equatin 7.1, (whee α i is the mean inic plaizability pe in pai, α ecs is the electnic plaizability f Cs + and α eb is the electnic plaizability f B - ): ( lw) ( lw) ( Nα + Nα + Nα ) i ecs eb Remembe that (Nα i + Nα ecs + Nα eb ) shuld be witten as (N i α i + N Cs α ecs + N B α ecl ), but since thee is a ne-t-ne ati between the numbe f mlecules and ins in CsB, we can take all the N s t be the same. 1 3 ( Nα + Nα + Nα )( + ) + 1 (lw) i ecs eb (lw) 1 3 ( Nα + Nα + Nα ) ( Nα + Nα + Nα ) + 1 (lw) Islate and simplify: (lw) i ecs eb ( αi + αecs + αeb ) + N( α + α + α ) N 3 3 i ecs eb (lw) 3 i ecs eb (lw) ( m )( 5.8 F m F m F m ) + 3( F/m) ( F/m) ( 1.58 m )( 5.8 F m F m F m )

7 (lw) 6.48 Tutial-09 At ptical fequencies thee is n cntibutin fm inic plaizatin. We nly cnside electnic plaizatin f individual ins and theefe the elative pemittivity at ptical fequencies, (p), is: ( p) ( m )( 3.35 F m F m ) + 3( F/m) ( F/m) ( 1.58 m )( 3.35 F m F m ) (p) Electnic and inic plaizatin in KCl KCl has the same cystal stuctue as NaCl. KCL s lattice paamete is 0.69 nm. The inic plaizability pe in pai (pe K + -Cl - in) is F m. The electnic plaizability f K + is F m and that f Cl - is F m. Calculate the dielectic cnstant unde dc peatin and at ptical fequencies. Expeimental values ae 4.84 and.19. Slutin The KCl stuctue has a lattice paamete given by a 0.69 nm, and thee ae 4 KCl in pais pe unit cell (see Table 1.3). The numbe f in pais, individual ins, pe unit vlume (N) is theefe: 4 4 N m -3 a 3 9 ( 0.69 m) 3 The electnic plaizability f the K + in is given as α ek F m, and plaizability f the Cl - in is given as α ecl F m. Fm Equatin 7.1, the elative pemittivity at ptical fequencies, (p), can be fund (see slutin f questin 7.5 f deivatin): ( p) N 3 ( α ek + α ecl ) + 3 N( α + α ) ek ecl ( m )( 1.64 F m F m ) + 3( F/m) ( p) ( F/m) ( m )( 1.64 F m F m ) (p).18 This value is vey clse t the expeimental value f Student micwaves mashed ptates A micwave ven uses electmagnetic waves at.48 GHz t heat fd by dielectic lss, that is, making use f f the fd mateial, which nmally has substantial wate cntent. An undegaduate student micwaves cm 3 f mashed ptates in 60 secnds. The micwave geneates an ms field f E ms f 00 V cm -1 in mashed ptates. At.48 GHz, mashed ptates have 1. Calculate the aveage pwe dissipated pe cm 3, and als the ttal enegy dissipated heating the fd (Nte: Yu can use E ms instead f E in Equatin 7.3.) Slutin

8 Tutial-09 Using Eq7.3 and substituting Eq7.31 int it we have, W vl ωe " 0 Substituting the elevant paametes int this equatin we get Aveage pwe, W vl π W cm -3 Since, 1.16 k W cm -3 Enegy Pwe Time Aea Aveage 3 5 Enegy J 700 kj 7.14 Dielectic lss pe unit capacitance Cnside the thee dielectic mateials listed in Table 7Q8-1 with the eal and imaginay dielectic cnstants, ' and ''. At a given vltage, which dielectic will have the lwest pwe dissipatin pe unit capacitance at 1 khz and at an peating tempeatue f 50 C? Is this als tue at C?

9 Tutial 09 Slutin Since we ae meely cmpaing values, assume vltage V 1 V f calculatin pupses. Fm example 7.5, the pwe dissipated pe unit capacitance (W cap ) is given by: W cap V ω whee ω is the angula fequency (πf) and and epesent the eal and imaginay cmpnents f the elative pemittivity, espectively. As a sample calculatin, the pwe dissipated in plycabnate is: ( ) [ ( )] ( ) V π 00 Hz (.47) W 7.63 W/F cap Theefe, 7.63 W/F is dissipated at 50 C at 1 V. The values f the the mateials at bth 50 C and C ae listed belw in Table 7Q14-: Mateial 50 C C Pwe Dissipated (W pe F) Plycabnate PET PEEK Pwe Dissipated (W pe F) Table 7Q14-: Pwe dissipated at diffeent tempeatues f the given mateials. At 50 C, all thee ae cmpaable in magnitude, but PET has the lwest pwe dissipatin. At C, plycabnate has the lwest dissipatin, while PET is almst ten times wse Dielectic beakdwn f gases and Paschen cuves Dielectic beakdwn in gases typically invlves the avalanche inizatin f the gas mlecules by enegetic electns acceleated by the applied field. The mean fee path between cllisins must be sufficiently lng t allw the electns t gain sufficient enegy fm the field t impact-inize the gas mlecules. The beakdwn vltage, V b, between tw electdes depends n the distance, d, between the electdes as well as the gas pessue,

10 Tutial 09 P, as shwn in Figue V b vesus Pd plts ae called Paschen cuves. We cnside gaseus insulatin, ai and SF 6, in an HV switch. a. What is the beakdwn vltage between tw electdes f a switch sepaated by a 5 mm gap with ai at 1 atm when the gaseus insulatin is ai and when it is SF 6? b. What ae the beakdwn vltages in the tw cases when the pessue is times geate? What is yu cnclusin? c. At what pessue is the beakdwn vltage a minimum? d. What ai gap spacing d at 1 atm gives the minimum beakdwn vltage? e. What wuld be the easns f pefeing gaseus insulatin ve liquid slid insulatin? Slutin a. At pessue P 1 atm Pa and ai gap d 5 mm, P d ( Pa)(0.005 m) Pa m. Fm Figue 7Q19-1, the cespnding values f beakdwn vltage f ai (V ai ) and f SF 6 (V SF6 ) ae: V ai 00 V 1.0 kv V SF V 50.0 kv

11 Tutial SF 6 Ai Pessue Spacing (Pa m) 5 5 Figue 7Q19-1: Beakdwn vltage vesus (pessue electde spacing) (Paschen cuves) b. At P atm Pa and d 5 mm, P d ( Pa)(0.005 m) 5065 Pa m. Using linea extaplatin n Figue 7Q19-1: V ai V 500 kv V SF V 0 kv Pessue inceases by times, but the beakdwn vltages incease by a fact f abut 5 times - this is a gd impvement. c. With a gap length f 5 mm, we need t knw the pessue at which the beakdwn vltage is a minimum. Fm the gaph, the minimum beakdwn vltage f ai is abut V ai 50 V, and the minimum f SF 6 is abut V SF6 40 V. The cespnding values f P d ae (P d) ai 0.6 Pa m and (P d) SF6 0. Pa m. Fm these we can detemine the values f pessue needed f minimum beakdwn vltage: P d 0.6 Pa m PSF6d ai 0. Pa m P 0.6 Pa m P m ai SF6 P ai 0. Pa m m Pa atm P 40.0 Pa 14 SF atm A lw pessue is needed f minimum beakdwn which explains why dischage tubes peate at a lw pessue. d. At a set pessue P 1 atm Pa, the ai gap spacing d f minimum beakdwn vltage can be fund in a simila manne t the ne abve, using the same values f P d:

12 Tutial 09 Pd ai 0.6 Pa m d ai 0.6 Pa m Pa d ai m This value cespnds t a beakdwn vltage f 50 V. Theefe a gap f abut 6 µm will nly need 50 V f beakdwn. e. HV and high cuent switches elays that have mving pats cannt be pactically insulated using slid dielectics. Liquid dielectics ae nt as efficient as gaseus dielectics because sme undeg chemical changes unde patial dischages. Futhe, they have a highe viscsity than gases that may affect the efficiency f the mving pats. Gas natually pemeates all the necessay space lcatins whee insulatin is citical. *7.0 Capacit design Cnside a nnplaized 0 nf capacit design at 60 Hz peatin. Nte that thee ae thee candidate dielectics, as listed in Table a. Calculate the vlume f the 0 nf capacit f each dielectic, given that they ae t be used unde lw vltages and each dielectic has its minimum fabicatin thickness. Which ne has the smallest vlume? b. Hw is the vlume affected if the capacit is t be used at a 500 V applicatin and the maximum field in the dielectic must be a fact f less than the dielectic stength? Which ne has the smallest vlume? c. At a 500 V applicatin, what is the pwe dissipated in each capacit at 60 Hz peatin? Which ne has the lwest dissipatin? Slutin

13 Tutial 09 Nte: All sample calculatins ae f Plyme film (PET). All methds f calculatin f the the mateials ae identical, and the btained values ae summaized in Table 7Q0-1. a. T find the vlume needed f C 0 nf given that the dielectic has the minimum pactical thickness, d (Table 7.13), find the capacitance pe unit vlume (C vl ): C d 1 (.854 F/m)( 3.) 8 3 vl 8.33 F/m 6 ( 1 m) The vlume V can nw be fund as fllws: V C / C vl (0-9 F) / (8.33 F/m 3 ) m 3 This is the vlume at lw vltage peatin based n the minimum pactical thickness. b. Suppse that d is the minimum thickness (in m) which gives a maximum field f half f E b at 500 V. Then: d V E max b 500 V V/m 5 m Nw the capacitance pe unit vlume can be fund: C d 1 (.854 F/m)( 3.) 8 3 vl F/m 5 ( m) V C / C vl (0-9 F) / ( F/m 3 ) m 3 This is the dielectic vlume at 500 V. c. The pwe dissipatin in the capacit at 500 V (60 Hz peatin) can be fund by fist btaining the pwe lst pe unit vlume W vl. It is given by: b W vl E ω tanδ η whee η is the safety fact (assumed t be equal t ) and ω πf is the angula fequency. Evaluating: W 7 (.5 V/m) 1 vl π ( ) W vl 3004 W/m ( ( 60 Hz) )( F/m)( 3.)( 5 )

14 Tutial 09 The pwe dissipated (W) is theefe: W W vl V (3004 W/m 3 )( m 3 ) W W Plyme Film PET Ceamic TiO High-K Ceamic (BaTiO 3 based) A Lw vltage vlume (m 3 ) B High vltage vlume (m 3 ) C Pwe dissipated (W) Table 7Q0-1: Summaized values f vlume and pwe f given capacits. Upn inspectin we see that f pat a and pat b, high-k ceamic has the smallest vlumes, and f pat c, ceamic has the lwest pwe dissipatin.