R/2 L/2 i + di ---> + + v G C. v + dv - - <--- i. R, L, G, C per unit length

Size: px
Start display at page:

Download "R/2 L/2 i + di ---> + + v G C. v + dv - - <--- i. R, L, G, C per unit length"

Transcription

1 _03_EE394J_2_Sping2_Tansmissin_Linesdc Tansmissin Lines Inductance and capacitance calculatins f tansmissin lines GMR GM L and C matices effect f gund cnductivity Undegund cables Equivalent Cicuit f Tansmissin Lines (Including Ovehead and Undegund) The pwe system mdel f tansmissin lines is develped fm the cnventinal distibuted paamete mdel shwn in Figue i ---> R/2 L/2 i di ---> G C v v dv R/2 L/2 - - <--- i <--- i di < dz > R L G C pe unit length Figue istibuted Paamete Mdel f Tansmissin Line Once the values f distibuted paametes esistance R inductance L cnductance G and capacitance ae knwn (units given in pe unit length) then eithe "lng line" "sht line" mdels can be used depending n the electical length f the line Assuming f the mment that R L G and C ae knwn the elatinship between vltage and cuent n the line may be detemined by witing Kichhff's vltage law (KVL) aund the ute lp in Figue and by witing Kichhff's cuent law (KCL) at the ight-hand nde KVL yields Rdz Ldz i Rdz Ldz i v i v dv i t 2 2 t This yields the change in vltage pe unit length v z i Ri L t which in phas fm is V ~ ( R jωl)i ~ z Page f 30

2 _03_EE394J_2_Sping2_Tansmissin_Linesdc KCL at the ight-hand nde yields i i di Gdz ( v dv) Cdz ( v dv) t 0 If dv is small then the abve fmula can be appximated as v di ( Gdz) v Cdz t I ~ ( G jωc)v ~ z i v Gv C which in phas fm is z t Taking the patial deivative f the vltage phas equatin with espect t z yields 2 ~ V 2 z ~ I ( R jωl) z Cmbining the tw abve equatins yields 2 ~ V ~ 2 ~ ( R jωl)( G jωc) V γ V whee γ ( R jωl)( G jωc) α jβ and 2 z whee γ α and β ae the ppagatin attenuatin and phase cnstants espectively The slutin f V ~ is ~ V ( z) Ae γz γz Be A simila pcedue f slving I ~ yields ~ I ( z) γz γz Ae Be Z whee the chaacteistic "suge" impedance Z is defined as Z ( R jωl) ( G jωc) Cnstants A and B must be fund fm the bunday cnditins f the pblem This is usually accmplished by cnsideing the teminal cnditins f a tansmissin line segment that is d metes lng as shwn in Figue 2 Page 2 f 30

3 _03_EE394J_2_Sping2_Tansmissin_Linesdc Sending End Receiving End Is ---> I ---> Tansmissin Vs Line Segment - <--- Is <--- I z -d V - z 0 < d > Figue 2 Tansmissin Line Segment In de t slve f cnstants A and B the vltage and cuent n the eceiving end is assumed t be knwn s that a elatin between the vltages and cuents n bth sending and eceiving ends may be develped Substituting z 0 int the equatins f the vltage and cuent (at the eceiving end) yields ~ V R ~ A B I R Slving f A and B yields ( A B) ~ ~ VR ZI R VR ZI R A B 2 2 ~ ~ Substituting int the V ( z ) and I ( z) equatins yields Z ~ VS ~ I S ~ ~ VR csh( γd ) Z0I R sinh( γd ) ~ VR ~ sinh( γd ) I R csh( γd ) Z A pi equivalent mdel f the tansmissin line segment can nw be fund in a simila manne as it was f the ff-nminal tansfme The esults ae given in Figue 3 Page 3 f 30

4 _03_EE394J_2_Sping2_Tansmissin_Linesdc Sending End Receiving End Is ---> I ---> Ys Vs Ys Y V - <--- Is <--- I - z -d z 0 < d > Y S Y R d tanh γ 2 Z Y SR Z sinh ( γd ) Z ( R jωl) γ ( R jωl)( G jωc) ( G jωc) R L G C pe unit length Figue 3 Pi Equivalent Cicuit Mdel f istibuted Paamete Tansmissin Line Shunt cnductance G is usually neglected in vehead lines but it is nt negligible in undegund cables F electically "sht" vehead tansmissin lines the hypeblic pi equivalent mdel simplifies t a familia fm Electically sht implies that d < 005 λ whee wavelength 8 30 ( ) m / s λ Hz Hz Theefe electically sht f ε Hz vehead lines have d < Hz and d < Hz F undegund cables the cespnding distances ae less since cables have smewhat highe elative pemittivities (ie ε 2 5 ) Substituting small values f γd int the hypeblic equatins and assuming that the line lsses ae negligible s that G R 0 yields YS jωcd YR and 2 Y SR jωld Then including the seies esistance yields the cnventinal "sht" line mdel shwn in Figue 4 whee half f the capacitance f the line is lumped n each end Page 4 f 30

5 _03_EE394J_2_Sping2_Tansmissin_Linesdc Rd Ld Cd 2 Cd 2 < d > R L C pe unit length Figue 4 Standad Sht Line Pi Equivalent Mdel f a Tansmissin Line 2 Capacitance f Ovehead Tansmissin Lines Ovehead tansmissin lines cnsist f wies that ae paallel t the suface f the eath T detemine the capacitance f a tansmissin line fist cnside the capacitance f a single wie ve the eath Wies ve the eath ae typically mdeled as line chages ρ l Culmbs pe mete f length and the elatinship between the applied vltage and the line chage is the capacitance A line chage in space has a adially utwad electic field descibed as E ql aˆ Vlts pe mete ε This electic field causes a vltage dp between tw pints at distances a and b away fm the line chage The vltage is fund by integating electic field Vab b E aˆ a ql ε b a V If the wie is abve the eath it is custmay t teat the eath's suface as a pefect cnducting plane which can be mdeled as an equivalent image line chage ql lying at an equal distance belw the suface as shwn in Figue 5 Page 5 f 30

6 _03_EE394J_2_Sping2_Tansmissin_Linesdc Cnduct with adius mdeled electically as a line chage ql at the cente b B A a h Suface f Eath bi ai h Image cnduct at an equal distance belw the Eath and with negative line chage -ql Figue 5 Line Chage q l at Cente f Cnduct Lcated h Metes Abve the Eath Fm supepsitin the vltage diffeence between pints A and B is Vab b bi ql b bi ql b ai Eρ aˆ Eρi aˆ ε a ai a bi a ai ε If pint B lies n the eath's suface then fm symmety b bi and the vltage f pint A with espect t gund becmes Vag ql ε ai a The vltage at the suface f the wie detemines the wie's capacitance This vltage is fund by mving pint A t the wie's suface cespnding t setting a s that Vg ql ε 2h f h >> The exact expessin which accunts f the fact that the equivalent line chage dps slightly belw the cente f the wie but still emains within the wie is V g q l ε h 2 h 2 Page 6 f 30

7 _03_EE394J_2_Sping2_Tansmissin_Linesdc The capacitance f the wie is defined as fmula abve becmes ql C which using the appximate vltage V g ε C Faads pe mete f length 2h When seveal cnducts ae pesent then the capacitance f the cnfiguatin must be given in matix fm Cnside phase a-b-c wies abve the eath as shwn in Figue 6 Thee Cnducts Repesented by Thei Equivalent Line Chages a ab b ac Cnduct adii a b c c aai aci Suface f Eath ai abi bi ci Images Figue 6 Thee Cnducts Abve the Eath Supepsing the cntibutins fm all thee line chages and thei images the vltage at the suface f cnduct a is given by Vag ε aai qa a abi qb ab aci qc ac The vltages f all thee cnducts can be witten in genealized matix fm as V V V ag bg cg ε p p p aa ba ca p p p ab bb cb p p p ac bc cc q q q a b c Vabc PabcQabc ε whee Page 7 f 30

8 _03_EE394J_2_Sping2_Tansmissin_Linesdc paa aai a pab abi ab etc and a is the adius f cnduct a aai is the distance fm cnduct a t its wn image (ie twice the height f cnduct a abve gund) ab is the distance fm cnduct a t cnduct b abi bai is the distance between cnduct a and the image f cnduct b (which is the same as the distance between cnduct b and the image f cnduct a) etc A Matix Appach f Finding C Fm the definitin f capacitance invesin Q CV then the capacitance matix can be btained via abc 2 Pabc C πε If gund wies ae pesent the dimensin f the pblem inceases pptinally F example in a thee-phase system with tw gund wies the dimensin f the P matix is 5 x 5 Hweve given the fact that the line-t-gund vltage f the gund wies is ze equivalent 3 x 3 P and C matices can be fund by using matix patitining and a pcess knwn as Kn eductin Fist wite the V PQ equatin as fllws: Vag Vbg Vcg V vg 0 Vwg 0 ε P P abc vw abc (3x3) (2x3) P abc vw P vw qa qb (3x2) qc (2x2) q v qw Vabc Vvw ε Pabc Pvw abc Pabc vw Qabc Pvw Qvw whee subscipts v and w efe t gund wies w and v and whee the individual P matices ae fmed as befe Since the gund wies have ze ptential then Page 8 f 30

9 _03_EE394J_2_Sping2_Tansmissin_Linesdc 0 0 ε [ P Q P Q ] vw abc abc vw vw s that [ P Q ] Q vw Pvw vw abc abc Substituting int the V abc equatin abve and cmbining tems yields V abc [ PabcQabc Pabc vwpvw Pvw abcqabc ] [ P abc P abc vw P vw P vw abc ] Q abc ε ε V Q abc abc ' [ P abc ] Q abc s that ε ' ' C V whee C ε [ P ] ' abc abc abc abc Theefe the effect f the gund wies can be included int a 3 x 3 equivalent capacitance matix An altenative way t find the equivalent 3 x 3 capacitance matix ' C abc is t Gaussian eliminate ws 32 using w 5 and then w 4 Aftewad ws 32 ' will have zes in clumns 4 and 5 P abc is the tp-left 3 x 3 submatix Invet 3 by 3 ' P abc t btain ' C abc Cmputing 02 Capacitances fm Matices ' Once the 3 x 3 C abc matix is fund by eithe f the abve tw methds 02 capacitances can ' be detemined by aveaging the diagnal tems and aveaging the ff-diagnal tems f C abc t pduce CS CM CM avg C abc CM CS CS CM CM CS avg C abc has the special symmetic fm f diagnalizatin int 02 cmpnents which yields Page 9 f 30

10 _03_EE394J_2_Sping2_Tansmissin_Linesdc CS 2CM 0 0 avg C 02 0 CS CM CS CM The Appximate Fmulas f 02 Capacitances Asymmeties in tansmissin lines pevent the P and C matices fm having the special fm that allws thei diagnalizatin int decupled psitive negative and ze sequence impedances Tanspsitin f cnducts can be used t nealy achieve the special symmetic fm and hence impve the level f decupling Cnducts ae tanspsed s that each ne ccupies each phase psitin f ne-thid f the lines ttal distance An example is given belw in Figue 7 whee the adii f all thee phases ae assumed t be identical a b c a c then b then b a c then b c a c a b then then c b a whee each cnfiguatin ccupies ne-sixth f the ttal distance Figue 7 Tanspsitin f A-B-C Phase Cnducts F this mde f cnstuctin the aveage P matix ( Kn educed P matix if gund wies ae pesent) has the fllwing fm: avg Pabc 6 paa pab pac pbb pbc pcc 6 paa pac pab pcc pbc pbb 6 pbb pab pbc paa pac pcc pcc pac pbc paa pab pbb 6 pbb pbc pab pcc pac paa 6 pcc pbc pac pbb pab paa whee the individual p tems ae descibed peviusly Nte that these individual P matices ae symmetic since ab ba pab pba etc Since the sum f natual lgaithms is the same as the lgaithm f the pduct P becmes ps pm pm avg P abc pm ps pm pm pm ps Page 0 f 30

11 _03_EE394J_2_Sping2_Tansmissin_Linesdc whee s 3 bb Pcc aai bbi 3 3 a b c Paa P cci p and M 3 ac Pbc abi aci bci 3 3 abac bc Pab P p avg Since P abc has the special ppety f diagnalizatin in symmetical cmpnents then tansfming it yields whee Inveting C p0 0 0 ps 2 pm 0 0 avg P 02 0 p 0 0 ps pm p2 0 0 ps pm aai bbi cci abiacibci aai bbi cci abacbc p s pm ab c abacbc a b c abi acibci C 0 0 avg P 02 and multiplying by 0 0 avg 02 C 0 πε 0 2 πε yields the cespnding 02 capacitance matix p0 ps 2 pm ε p ps pm C p2 ps pm When the a-b-c cnducts ae clse t each the than they ae t the gund then aai bbi cci abi aci bci yielding the cnventinal appximatin 3 abacbc p2 ps pm 3 a b c GM2 p GMR 2 Page f 30

12 _03_EE394J_2_Sping2_Tansmissin_Linesdc whee GM 2 and GMR 2 ae the gemetic mean distance (between cnducts) and gemetic mean adius espectively f bth psitive and negative sequences Theefe the psitive and negative sequence capacitances becme πε πε 2 2 C C2 Faads pe mete ps pm GM2 GMR2 F the ze sequence tem 3 3 aai bbi cci abi acibci p 0 ps 2 pm ab c abacbc 3 Expanding yields p ( aai bbi cci )( abi acibci ) 2 ( )( ) a b ab ac bc ( aai bbi cci )( abi aci bci ) 2 ( )( ) a b ab ac bc ( aai bbi cci )( abi acibci )( bai cai cbi ) ( )( )( ) a b ab ac bc ba ca cb GM0 p 0 3 GMR0 whee ( )( )( ) GM 9 0 aai bbi cci abi aci bci bai cai cbi ( )( )( ) GMR 9 0 a b c ab ac bc ba ca cb The ze sequence capacitance then becmes ε ε C 0 Faads pe mete ps 2 pm 3 GM0 GMR Page 2 f 30

13 _03_EE394J_2_Sping2_Tansmissin_Linesdc which is ne-thid that f the entie a-b-c bundle by because it epesents the aveage cntibutin f nly ne phase Bundled Phase Cnducts If each phase cnsists f a symmetic bundle f N identical individual cnducts an equivalent adius can be cmputed by assuming that the ttal line chage n the phase divides equally amng the N individual cnducts The equivalent adius is N eq [ NA ]N whee is the adius f the individual cnducts and A is the bundle adius f the symmetic set f cnducts Thee cmmn examples ae shwn belw in Figue 8 uble Bundle Each Cnduct Has Radius A eq 2A Tiple Bundle Each Cnduct Has Radius A eq 3 2 3A Quaduple Bundle Each Cnduct Has Radius A eq 4 3 4A Figue 8 Equivalent Radius f Thee Cmmn Types f Bundled Phase Cnducts Page 3 f 30

14 _03_EE394J_2_Sping2_Tansmissin_Linesdc 3 Inductance The magnetic field intensity pduced by a lng staight cuent caying cnduct is given by Ampee's Cicuital Law t be I Hφ Ampees pe mete whee the diectin f H is given by the ight-hand ule Magnetic flux density is elated t magnetic field intensity by pemeability μ as fllws: B μh Webes pe squae mete and the amunt f magnetic flux passing thugh a suface is Φ B ds Webes 7 whee the pemeability f fee space is 4π ( 0 ) μ Tw Paallel Wies in Space Nw cnside a tw-wie cicuit that caies cuent I as shwn in Figue 9 Tw cuent-caying wies with adii I I < > Figue 9 A Cicuit Fmed by Tw Lng Paallel Cnducts The amunt f flux linking the cicuit (ie passes between the tw wies) is fund t be μi μ I I dx μ dx Φ x x π Henys pe mete length Fm the definitin f inductance NΦ L I Page 4 f 30

15 _03_EE394J_2_Sping2_Tansmissin_Linesdc whee in this case N and whee N >> the inductance f the tw-wie pai becmes μ L Henys pe mete length π A und wie als has an intenal inductance which is sepaate fm the extenal inductance shwn abve The intenal inductance is shwn in electmagnetics texts t be μint L int Henys pe mete length 8π F mst cuent-caying cnducts μ int μ s that L int 005µH/m Theefe the ttal inductance f the tw-wie cicuit is the extenal inductance plus twice the intenal inductance f each wie (ie cuent tavels dwn and back) s that L tt μ π μ 2 8π μ π 4 μ π e 4 μ π e 4 It is custmay t define an effective adius eff e and t wite the ttal inductance in tems f it as L tt μ Henys pe mete length π eff Wie Paallel t Eath s Suface F a single wie f adius lcated at height h abve the eath the effect f the eath can be descibed by an image cnduct as it was f capacitance calculatins F a pefectly cnducting eath the image cnduct is lcated h metes belw the suface as shwn in Figue 0 Page 5 f 30

16 _03_EE394J_2_Sping2_Tansmissin_Linesdc Cnduct f adius caying cuent I h Suface f Eath h Nte the image flux exists nly abve the Eath Image cnduct at an equal distance belw the Eath Figue 0 Cuent-Caying Cnduct Abve Eath The ttal flux linking the cicuit is that which passes between the cnduct and the suface f the eath Summing the cntibutin f the cnduct and its image yields F 2h h 2h μ I dx Φ x h dx μ I h x ( 2h ) μ I ( h ) 2 >> a gd appximatin is μi 2h Φ Webes pe mete length h s that the extenal inductance pe mete length f the cicuit becmes Lext μ 2h Henys pe mete length The ttal inductance is then the extenal inductance plus the intenal inductance f ne wie L tt μ 2h μ μ 2h 2h π π π 4 4 μ e using the effective adius definitin fm befe μ 2h Ltt Henys pe mete length eff Page 6 f 30

17 _03_EE394J_2_Sping2_Tansmissin_Linesdc Bundled Cnducts The bundled cnduct equivalent adii pesented ealie apply f inductance as well as f capacitance The questin nw is what is the intenal inductance f a bundle? F N bundled cnducts the net intenal inductance f a phase pe mete must decease as because the N intenal inductances ae in paallel Cnsideing a bundle ve the Eath then L tt μ 2h μ μ 2h μ 2h μ e 4 eq 8πN eq 4N eq N eq 2h e 4 N Facting in the expessin f the equivalent bundle adius eq yields N eqe 4 N N N N [ NA ] N e 4N Ne 4 A [ N A ]N eff Thus eff emains e 4 n matte hw many cnducts ae in the bundle The Thee-Phase Case F situatins with multiples wies abve the Eath a matix appach is needed Cnside the capacitance example given in Figue 6 except this time cmpute the extenal inductances athe than capacitances As fa as the vltage (with espect t gund) f ne f the a-b-c phases is cncened the imptant flux is that which passes between the cnduct and the Eath's suface F example the flux "linking" phase a will be pduced by six cuents: phase a cuent and its image phase b cuent and its image and phase c cuent and its image and s n Figue is useful in visualizing the cntibutin f flux linking phase a that is caused by the cuent in phase b (and its image) Page 7 f 30

18 _03_EE394J_2_Sping2_Tansmissin_Linesdc b a ab bg g ai bg abi bi Figue Flux Linking Phase a ue t Cuent in Phase b and Phase b Image Page 8 f 30

19 _03_EE394J_2_Sping2_Tansmissin_Linesdc The linkage flux is Φ a (due t I b and I b image) μi b μ I μ I π bg b abi b abi ab bg 2 ab Cnsideing all phases and applying supepsitin yields the ttal flux μ μ Ia aai Ib abi Ic aci Φ a a ab ac μ Nte that aai cespnds t 2h in Figue 0 Pefming the same analysis f all thee phases and ecgnizing that N Φ LI whee N in this pblem then the inductance matix is develped using Φ Φ Φ a b c μ aai a bai ba cai ca abi ab bbi b cbi cb aci ac bci bc cci c I I I a b c Φ abc LabcIabc A cmpaisn t the capacitance matix deivatin shws that the same matix f natual lgaithms is used in bth cases and that Labc μ Pabc μ ε C abc μεc abc This implies that the pduct f the L and C matices is a diagnal matix with μ ε n the diagnal pviding that the eath is assumed t be a pefect cnduct and that the intenal inductances f the wies ae igned If the cicuit has gund wies then the dimensin f L inceases accdingly Recgnizing that the flux linking the gund wies is ze (because thei vltages ae ze) then L can be Kn ' educed t yield an equivalent 3 x 3 matix L abc T include the intenal inductance f the wies eplace actual cnduct adius with eff Cmputing 02 Inductances fm Matices Once the 3 x 3 L ' abc matix is fund 02 inductances can be detemined by aveaging the ' diagnal tems and aveaging the ff-diagnal tems f L abc t pduce Page 9 f 30

20 _03_EE394J_2_Sping2_Tansmissin_Linesdc LS LM LM avg L abc LM LS LS LM LM LS s that LS 2LM 0 0 avg L 02 0 LS LM LS LM The Appximate Fmulas f 02 Inductancess Because f the similaity t the capacitance pblem the same ules f eliminating gund wies f tanspsitin and f bundling cnducts apply Likewise appximate fmulas f the psitive negative and ze sequence inductances can be develped and these fmulas ae and μ GM2 L L2 GMR2 μ GM0 L0 3 GMR0 It is imptant t nte that the GM and GMR tems f inductance diffe fm thse f capacitance in tw ways: GMR calculatins f inductance calculatins shuld be made with e 4 eff 2 GM distances f inductance calculatins shuld include the equivalent cmplex depth f mdeling finite cnductivity eath (explained in the next sectin) This effect is igned in capacitance calculatins because the suface f the Eath is nminally at ze ptential Mdeling Impefect Eath The effect f the Eath's nn-infinite cnductivity shuld be included when cmputing inductances especially ze sequence inductances (Nte - psitive and negative sequences ae elatively immune t Eath cnductivity) Because the Eath is nt a pefect cnduct the image cuent des nt actually flw n the suface f the Eath but athe thugh a csssectin The highe the cnductivity the nawe the css-sectin Page 20 f 30

21 _03_EE394J_2_Sping2_Tansmissin_Linesdc It is easnable t assume that the etun cuent is ne skin depth δ belw the suface f the ρ Eath whee δ metes Typically esistivity ρ is assumed t be 00Ω-m F μ f 00Ω-m and 60Hz δ 459m Usually δ is s lage that the actual height f the cnducts makes n diffeence in the calculatins s that the distances fm cnducts t the images is assumed t be δ 4 Electic Field at Suface f Ovehead Cnducts Igning all the chages the electic field at a cnduct s suface can be appximated by E q ε whee is the adius F vehead cnducts this is a easnable appximatin because the neighbing line chages ae elatively fa away It is always imptant t keep the peak electic field at a cnduct s suface belw 30kV/cm t avid excessive cn lsses Ging beynd the abve appximatin the Makt-Mengele methd pvides a detailed pcedue f calculating the maximum peak subcnduct suface electic field intensity f thee-phase lines with identical phase bundles Each bundle has N symmetic subcnducts f adius The bundle adius is A The pcedue is Teat each phase bundle as a single cnduct with equivalent adius N / N [ ] eq NA 2 Find the C(N x N) matix including gund wies using aveage cnduct heights abve gund Kn educe C(N x N) t C(3 x 3) Select the phase bundle that will have the geatest peak line chage value ( q lpeak ) duing a 60Hz cycle by successively placing maximum line-t-gund vltage Vmax n ne phase and Vmax/2 n each f the the tw phases Usually the phase with the lagest diagnal tem in C(3 by 3) will have the geatest q lpeak 3 Assuming equal chage divisin n the phase bundle identified in Step 2 igne equivalent line chage displacement and calculate the aveage peak subcnduct suface electic field intensity using E q lpeak avg peak N ε 4 Take int accunt equivalent line chage displacement and calculate the maximum peak subcnduct suface electic field intensity using Page 2 f 30

22 _03_EE394J_2_Sping2_Tansmissin_Linesdc E E ( N A max peak avg peak ) 5 Resistance and Cnductance The esistance f cnducts is fequency dependent because f the esistive skin effect Usually hweve this phenmenn is small f Hz Cnduct esistances ae eadily btained fm tables in the ppe units f Ohms pe mete length and these values added t the equivalent-eath esistances fm the pevius sectin t yield the R used in the tansmissin line mdel Cnductance G is vey small f vehead tansmissin lines and can be igned 6 Undegund Cables Undegund cables ae tansmissin lines and the mdel peviusly pesented applies Capacitance C tends t be much lage than f vehead lines and cnductance G shuld nt be igned F single-phase and thee-phase cables the capacitances and inductances pe phase pe mete length ae and ε C ε Faads pe mete length b a μ b L Henys pe mete length a whee b and a ae the ute and inne adii f the caxial cylindes In pwe cables a b is typically e (ie 2783) s that the vltage ating is maximized f a given diamete F mst dielectics elative pemittivity ε F thee-phase situatins it is cmmn t assume that the psitive negative and ze sequence inductances and capacitances equal the abve expessins If the cnductivity f the dielectic is knwn cnductance G can be calculated using σ G C Mhs pe mete length ε Page 22 f 30

23 _03_EE394J_2_Sping2_Tansmissin_Linesdc Assumptins SUMMARY OF POSITIVE/NEGATIVE SEQUENCE CALCULATIONS Balanced fa fm gund gund wies igned Valid f identical single cnducts pe phase f identical symmetic phase bundles with N cnducts pe phase and bundle adius A Cmputatin f psitive/negative sequence capacitance whee C / ε GM / GMRC / faads pe mete 3 / ab ac bc GM metes whee ae ab ac bc and whee distances between phase cnducts if the line has ne cnduct pe phase distances between phase bundle centes if the line has symmetic phase bundles GMR C / is the actual cnduct adius (in metes) if the line has ne cnduct pe phase GMR N C / N N A if the line has symmetic phase bundles Cmputatin f psitive/negative sequence inductance μ GM L / / GMRL / henys pe mete whee GM / is the same as f capacitance and f the single cnduct case GMR L / is the cnduct gm (in metes) which takes / 4 int accunt bth standing and the e adjustment f intenal inductance If gm is / 4 nt given then assume e and gm Page 23 f 30

24 _03_EE394J_2_Sping2_Tansmissin_Linesdc f bundled cnducts N N GMRL / N gm A if the line has symmetic phase bundles Cmputatin f psitive/negative sequence esistance R is the 60Hz esistance f ne cnduct if the line has ne cnduct pe phase If the line has symmetic phase bundles then divide the ne-cnduct esistance by N Sme cmmnly-used symmetic phase bundle cnfiguatins A A A N 2 N 3 N 4 Assumptins ZERO SEQUENCE CALCULATIONS Gund wies ae igned The a-b-c phases ae teated as ne bundle If individual phase cnducts ae bundled they ae teated as single cnducts using the bundle adius methd F capacitance the Eath is teated as a pefect cnduct F inductance and esistance the Eath is assumed t have unifm esistivity ρ Cnduct sag is taken int cnsideatin and a gd assumptin f ding this is t use an aveage cnduct height equal t (/3 the cnduct height abve gund at the twe plus 2/3 the cnduct height abve gund at the maximum sag pint) The ze sequence excitatin mde is shwn belw alng with an illustatin f the elatinship between bundle C and L and ze sequence C and L Since the bundle cuent is actually 3I the ze sequence esistance and inductance ae thee times that f the bundle and the ze sequence capacitance is ne-thid that f the bundle Page 24 f 30

25 _03_EE394J_2_Sping2_Tansmissin_Linesdc I I 3I I 3I I V I C bundle V I L bundle 3I I I 3I I 3I I L V I C C C V I L L 3I Cmputatin f ze sequence capacitance C 0 ε faads pe mete 3 GMC 0 GMR C0 whee GM C0 is the aveage height (with sag facted in) f the a-b-c bundle abve pefect Eath GM C0 is cmputed using GM C0 9 i i aa bb cc i ab i ac i bc i metes whee is the distance fm a t a-image i is the distance fm a t b-image and s i aa fth The Eath is assumed t be a pefect cnduct s that the images ae the same distance belw the Eath as ae the cnducts abve the Eath Als GMR C0 9 GMRC / ab ac bc ab metes whee GMR C / ab ac and bc wee descibed peviusly Page 25 f 30

26 _03_EE394J_2_Sping2_Tansmissin_Linesdc Cmputatin f ze sequence inductance μ δ L0 3 Henys pe mete GMR L0 ρ whee skin depth δ metes μ f The gemetic mean bundle adius is cmputed using GMR L0 9 GMRL / ab ac bc metes whee GMR L/ ab ac and bc wee shwn peviusly Cmputatin f ze sequence esistance Thee ae tw cmpnents f ze sequence line esistance Fist the equivalent cnduct esistance is the 60Hz esistance f ne cnduct if the line has ne cnduct pe phase If the line has symmetic phase bundles with N cnducts pe bundle then divide the ne-cnduct esistance by N Secnd the effect f esistive Eath is included by adding the fllwing tem t the cnduct esistance: f hms pe mete (see Begen) whee the multiplie f thee is needed t take int accunt the fact that all thee ze sequence cuents flw thugh the Eath As a geneal ule C / usually wks ut t be abut 2 picf pe mete L wks ut t be abut mich pe mete (including intenal inductance) / 0 C is usually abut 6 picf pe mete L 0 is usually abut 2 mich pe mete if the line has gund wies and typical Eath esistivity abut 3 mich pe mete f lines withut gund wies p Eath esistivity The velcity f ppagatin is appximately the speed f light (3 x 0 8 m/s) f psitive LC and negative sequences and abut 08 times that f ze sequence Page 26 f 30

27 _03_EE394J_2_Sping2_Tansmissin_Linesdc Electic Field at Suface f Ovehead Cnducts Igning all the chages the electic field at a cnduct s suface can be appximated by E q ε whee is the adius F vehead cnducts this is a easnable appximatin because the neighbing line chages ae elatively fa away It is always imptant t keep the peak electic field at a cnduct s suface belw 30kV/cm t avid excessive cna lsses Ging beynd the abve appximatin the Makt-Mengele methd pvides a detailed pcedue f calculating the maximum peak subcnduct suface electic field intensity f thee-phase lines with identical phase bundles Each bundle has N symmetic subcnducts f adius The bundle adius is A The pcedue is 5 Teat each phase bundle as a single cnduct with equivalent adius N / N [ ] eq NA 6 Find the C(N x N) matix including gund wies using aveage cnduct heights abve gund Kn educe C(N x N) t C(3 x 3) Select the phase bundle that will have the geatest peak line chage value ( q lpeak ) duing a 60Hz cycle by successively placing maximum line-t-gund vltage Vmax n ne phase and Vmax/2 n each f the the tw phases Usually the phase with the lagest diagnal tem in C(3 by 3) will have the geatest q lpeak 7 Assuming equal chage divisin n the phase bundle identified in Step 2 igne equivalent line chage displacement and calculate the aveage peak subcnduct suface electic field intensity using E q lpeak avg peak N ε 8 Take int accunt equivalent line chage displacement and calculate the maximum peak subcnduct suface electic field intensity using Emax peak Eavg peak ( N ) A Page 27 f 30

28 _03_EE394J_2_Sping2_Tansmissin_Linesdc 345kV uble-cicuit Tansmissin Line Scale: cm 2 m 57 m 78 m 85 m 76 m 76 m 44 m 229 m at twe and sags dwn 0 m at midspan t 29 m Twe Base uble cnduct phase bundles bundle adius 229 cm cnduct adius 4 cm cnduct esistance Ω/km Single-cnduct gund wies cnduct adius 056 cm cnduct esistance 287 Ω/km Page 28 f 30

29 _03_EE394J_2_Sping2_Tansmissin_Linesdc 500kV Single-Cicuit Tansmissin Line Scale: cm 2 m 39 m 5 m 5 m 33 m 30 m 0 m 0 m Cnducts sag dwn 0 m at mid-span Eath esistivity ρ 00 Ω-m Twe Base Tiple cnduct phase bundles bundle adius 20 cm cnduct adius 5 cm cnduct esistance 005 Ω/km Single-cnduct gund wies cnduct adius 06 cm cnduct esistance 30 Ω/km Page 29 f 30

30 _03_EE394J_2_Sping2_Tansmissin_Linesdc ue Wed Feb 22 Use the left-hand cicuit f the 345kV line gemety given n the pevius page etemine the L C R line paametes pe unit length f psitive/negative and ze sequence Then f a 00km lng segment f the cicuit detemine the P s Q s I s VR and δr f switch pen and switch clsed cases The geneat vltage phase angle is ze Q L absbed P jq I R jωl P 2 jq 2 I 2 200kVms jωc/2 Q C pduced jωc/2 Q C2 pduced V R / δ R 400Ω One cicuit f the 345kV line gemety 00km lng Page 30 f 30

Announcements Candidates Visiting Next Monday 11 12:20 Class 4pm Research Talk Opportunity to learn a little about what physicists do

Announcements Candidates Visiting Next Monday 11 12:20 Class 4pm Research Talk Opportunity to learn a little about what physicists do Wed., /11 Thus., /1 Fi., /13 Mn., /16 Tues., /17 Wed., /18 Thus., /19 Fi., / 17.7-9 Magnetic Field F Distibutins Lab 5: Bit-Savat B fields f mving chages (n quiz) 17.1-11 Pemanent Magnets 18.1-3 Mic. View

More information

CHAPTER 24 GAUSS LAW

CHAPTER 24 GAUSS LAW CHAPTR 4 GAUSS LAW LCTRIC FLUX lectic flux is a measue f the numbe f electic filed lines penetating sme suface in a diectin pependicula t that suface. Φ = A = A csθ with θ is the angle between the and

More information

5/20/2011. HITT An electron moves from point i to point f, in the direction of a uniform electric field. During this displacement:

5/20/2011. HITT An electron moves from point i to point f, in the direction of a uniform electric field. During this displacement: 5/0/011 Chapte 5 In the last lectue: CapacitanceII we calculated the capacitance C f a system f tw islated cnducts. We als calculated the capacitance f sme simple gemeties. In this chapte we will cve the

More information

A) N B) 0.0 N C) N D) N E) N

A) N B) 0.0 N C) N D) N E) N Cdinat: H Bahluli Sunday, Nvembe, 015 Page: 1 Q1. Five identical pint chages each with chage =10 nc ae lcated at the cnes f a egula hexagn, as shwn in Figue 1. Find the magnitude f the net electic fce

More information

OBJECTIVE To investigate the parallel connection of R, L, and C. 1 Electricity & Electronics Constructor EEC470

OBJECTIVE To investigate the parallel connection of R, L, and C. 1 Electricity & Electronics Constructor EEC470 Assignment 7 Paallel Resnance OBJECTIVE T investigate the paallel cnnectin f R,, and C. EQUIPMENT REQUIRED Qty Appaatus 1 Electicity & Electnics Cnstuct EEC470 1 Basic Electicity and Electnics Kit EEC471-1

More information

ELECTROMAGNETIC INDUCTION PREVIOUS EAMCET BITS

ELECTROMAGNETIC INDUCTION PREVIOUS EAMCET BITS P P Methd EECTOMAGNETIC INDUCTION PEVIOUS EAMCET BITS [ENGINEEING PAPE]. A cnduct d f length tates with angula speed ω in a unifm magnetic field f inductin B which is pependicula t its mtin. The induced

More information

CHAPTER GAUSS'S LAW

CHAPTER GAUSS'S LAW lutins--ch 14 (Gauss's Law CHAPTE 14 -- GAU' LAW 141 This pblem is ticky An electic field line that flws int, then ut f the cap (see Figue I pduces a negative flux when enteing and an equal psitive flux

More information

Electric Charge. Electric charge is quantized. Electric charge is conserved

Electric Charge. Electric charge is quantized. Electric charge is conserved lectstatics lectic Chage lectic chage is uantized Chage cmes in incements f the elementay chage e = ne, whee n is an intege, and e =.6 x 0-9 C lectic chage is cnseved Chage (electns) can be mved fm ne

More information

Example

Example hapte Exaple.6-3. ---------------------------------------------------------------------------------- 5 A single hllw fibe is placed within a vey lage glass tube. he hllw fibe is 0 c in length and has a

More information

Magnetism. Chapter 21

Magnetism. Chapter 21 1.1 Magnetic Fields Chapte 1 Magnetism The needle f a cmpass is pemanent magnet that has a nth magnetic ple (N) at ne end and a suth magnetic ple (S) at the the. 1.1 Magnetic Fields 1.1 Magnetic Fields

More information

WYSE Academic Challenge Sectional Mathematics 2006 Solution Set

WYSE Academic Challenge Sectional Mathematics 2006 Solution Set WYSE Academic Challenge Sectinal 006 Slutin Set. Cect answe: e. mph is 76 feet pe minute, and 4 mph is 35 feet pe minute. The tip up the hill takes 600/76, 3.4 minutes, and the tip dwn takes 600/35,.70

More information

Electromagnetic Waves

Electromagnetic Waves Chapte 3 lectmagnetic Waves 3.1 Maxwell s quatins and ectmagnetic Waves A. Gauss s Law: # clsed suface aea " da Q enc lectic fields may be geneated by electic chages. lectic field lines stat at psitive

More information

A) 100 K B) 150 K C) 200 K D) 250 K E) 350 K

A) 100 K B) 150 K C) 200 K D) 250 K E) 350 K Phys10 Secnd Maj-09 Ze Vesin Cdinat: k Wednesday, May 05, 010 Page: 1 Q1. A ht bject and a cld bject ae placed in themal cntact and the cmbinatin is islated. They tansfe enegy until they each a final equilibium

More information

Fri. 10/23 (C14) Linear Dielectrics (read rest at your discretion) Mon. (C 17) , E to B; Lorentz Force Law: fields

Fri. 10/23 (C14) Linear Dielectrics (read rest at your discretion) Mon. (C 17) , E to B; Lorentz Force Law: fields Fi. 0/23 (C4) 4.4. Linea ielectics (ead est at yu discetin) Mn. (C 7) 2..-..2, 2.3. t B; 5..-..2 Lentz Fce Law: fields Wed. and fces Thus. (C 7) 5..3 Lentz Fce Law: cuents Fi. (C 7) 5.2 Bit-Savat Law HW6

More information

A) (0.46 î ) N B) (0.17 î ) N

A) (0.46 î ) N B) (0.17 î ) N Phys10 Secnd Maj-14 Ze Vesin Cdinat: xyz Thusday, Apil 3, 015 Page: 1 Q1. Thee chages, 1 = =.0 μc and Q = 4.0 μc, ae fixed in thei places as shwn in Figue 1. Find the net electstatic fce n Q due t 1 and.

More information

Consider the simple circuit of Figure 1 in which a load impedance of r is connected to a voltage source. The no load voltage of r

Consider the simple circuit of Figure 1 in which a load impedance of r is connected to a voltage source. The no load voltage of r 1 Intductin t Pe Unit Calculatins Cnside the simple cicuit f Figue 1 in which a lad impedance f L 60 + j70 Ω 9. 49 Ω is cnnected t a vltage suce. The n lad vltage f the suce is E 1000 0. The intenal esistance

More information

Summary chapter 4. Electric field s can distort charge distributions in atoms and molecules by stretching and rotating:

Summary chapter 4. Electric field s can distort charge distributions in atoms and molecules by stretching and rotating: Summa chapte 4. In chapte 4 dielectics ae discussed. In thse mateials the electns ae nded t the atms mlecules and cannt am fee thugh the mateial: the electns in insulats ae n a tight leash and all the

More information

5.1 Moment of a Force Scalar Formation

5.1 Moment of a Force Scalar Formation Outline ment f a Cuple Equivalent System Resultants f a Fce and Cuple System ment f a fce abut a pint axis a measue f the tendency f the fce t cause a bdy t tate abut the pint axis Case 1 Cnside hizntal

More information

Introduction. Electrostatics

Introduction. Electrostatics UNIVESITY OF TECHNOLOGY, SYDNEY FACULTY OF ENGINEEING 4853 Electmechanical Systems Electstatics Tpics t cve:. Culmb's Law 5. Mateial Ppeties. Electic Field Stength 6. Gauss' Theem 3. Electic Ptential 7.

More information

Solution: (a) C 4 1 AI IC 4. (b) IBC 4

Solution: (a) C 4 1 AI IC 4. (b) IBC 4 C A C C R A C R C R C sin 9 sin. A cuent f is maintaine in a single cicula lp f cicumfeence C. A magnetic fiel f is iecte paallel t the plane f the lp. (a) Calculate the magnetic mment f the lp. (b) What

More information

Outline. Steady Heat Transfer with Conduction and Convection. Review Steady, 1-D, Review Heat Generation. Review Heat Generation II

Outline. Steady Heat Transfer with Conduction and Convection. Review Steady, 1-D, Review Heat Generation. Review Heat Generation II Steady Heat ansfe ebuay, 7 Steady Heat ansfe wit Cnductin and Cnvectin ay Caett Mecanical Engineeing 375 Heat ansfe ebuay, 7 Outline eview last lectue Equivalent cicuit analyses eview basic cncept pplicatin

More information

Hotelling s Rule. Therefore arbitrage forces P(t) = P o e rt.

Hotelling s Rule. Therefore arbitrage forces P(t) = P o e rt. Htelling s Rule In what fllws I will use the tem pice t dente unit pfit. hat is, the nminal mney pice minus the aveage cst f pductin. We begin with cmpetitin. Suppse that a fim wns a small pa, a, f the

More information

Microelectronics Circuit Analysis and Design. ac Equivalent Circuit for Common Emitter. Common Emitter with Time-Varying Input

Microelectronics Circuit Analysis and Design. ac Equivalent Circuit for Common Emitter. Common Emitter with Time-Varying Input Micelectnics Cicuit Analysis and Design Dnald A. Neamen Chapte 6 Basic BJT Amplifies In this chapte, we will: Undestand the pinciple f a linea amplifie. Discuss and cmpae the thee basic tansist amplifie

More information

ME 3600 Control Systems Frequency Domain Analysis

ME 3600 Control Systems Frequency Domain Analysis ME 3600 Cntl Systems Fequency Dmain Analysis The fequency espnse f a system is defined as the steady-state espnse f the system t a sinusidal (hamnic) input. F linea systems, the esulting utput is itself

More information

March 15. Induction and Inductance Chapter 31

March 15. Induction and Inductance Chapter 31 Mach 15 Inductin and Inductance Chapte 31 > Fces due t B fields Lentz fce τ On a mving chage F B On a cuent F il B Cuent caying cil feels a tque = µ B Review > Cuents geneate B field Bit-Savat law = qv

More information

Example 11: The man shown in Figure (a) pulls on the cord with a force of 70

Example 11: The man shown in Figure (a) pulls on the cord with a force of 70 Chapte Tw ce System 35.4 α α 100 Rx cs 0.354 R 69.3 35.4 β β 100 Ry cs 0.354 R 111 Example 11: The man shwn in igue (a) pulls n the cd with a fce f 70 lb. Repesent this fce actin n the suppt A as Catesian

More information

Phy 213: General Physics III

Phy 213: General Physics III Phy 1: Geneal Physics III Chapte : Gauss Law Lectue Ntes E Electic Flux 1. Cnside a electic field passing thugh a flat egin in space w/ aea=a. The aea vect ( A ) with a magnitude f A and is diected nmal

More information

AIR FORCE RESEARCH LABORATORY

AIR FORCE RESEARCH LABORATORY AIR FORC RSARCH LABORATORY The xtinctin Theem as an xample f Reseach Vistas in Mathematical Optics Mach Richad A. Albanese Infmatin Opeatins and Applied Mathematics Human ffectiveness Diectate Bks City-Base

More information

Physics 111. Exam #1. January 26, 2018

Physics 111. Exam #1. January 26, 2018 Physics xam # Januay 6, 08 ame Please ead and fllw these instuctins caefully: Read all pblems caefully befe attempting t slve them. Yu wk must be legible, and the ganizatin clea. Yu must shw all wk, including

More information

Work, Energy, and Power. AP Physics C

Work, Energy, and Power. AP Physics C k, Eneg, and Pwe AP Phsics C Thee ae man diffeent TYPES f Eneg. Eneg is expessed in JOULES (J) 4.19 J = 1 calie Eneg can be expessed me specificall b using the tem ORK() k = The Scala Dt Pduct between

More information

The Gradient and Applications This unit is based on Sections 9.5 and 9.6, Chapter 9. All assigned readings and exercises are from the textbook

The Gradient and Applications This unit is based on Sections 9.5 and 9.6, Chapter 9. All assigned readings and exercises are from the textbook The Gadient and Applicatins This unit is based n Sectins 9.5 and 9.6 Chapte 9. All assigned eadings and eecises ae fm the tetbk Objectives: Make cetain that u can define and use in cntet the tems cncepts

More information

Chapter 15. ELECTRIC POTENTIALS and ENERGY CONSIDERATIONS

Chapter 15. ELECTRIC POTENTIALS and ENERGY CONSIDERATIONS Ch. 15--Elect. Pt. and Enegy Cns. Chapte 15 ELECTRIC POTENTIALS and ENERGY CONSIDERATIONS A.) Enegy Cnsideatins and the Abslute Electical Ptential: 1.) Cnside the fllwing scenai: A single, fixed, pint

More information

Application of Net Radiation Transfer Method for Optimization and Calculation of Reduction Heat Transfer, Using Spherical Radiation Shields

Application of Net Radiation Transfer Method for Optimization and Calculation of Reduction Heat Transfer, Using Spherical Radiation Shields Wld Applied Sciences Junal (4: 457-46, 00 ISSN 88-495 IDOSI Publicatins, 00 Applicatin f Net Radiatin Tansfe Methd f Optimizatin and Calculatin f Reductin Heat Tansfe, Using Spheical Radiatin Shields Seyflah

More information

Section 4.2 Radians, Arc Length, and Area of a Sector

Section 4.2 Radians, Arc Length, and Area of a Sector Sectin 4.2 Radian, Ac Length, and Aea f a Sect An angle i fmed by tw ay that have a cmmn endpint (vetex). One ay i the initial ide and the the i the teminal ide. We typically will daw angle in the cdinate

More information

Exercises for Differential Amplifiers. ECE 102, Fall 2012, F. Najmabadi

Exercises for Differential Amplifiers. ECE 102, Fall 2012, F. Najmabadi Execises f iffeential mplifies ECE 0, Fall 0, F. Najmabai Execise : Cmpute,, an G if m, 00 Ω, O, an ientical Q &Q with µ n C x 8 m, t, λ 0. F G 0 an B F G. epeat the execise f λ 0. -. This execise shws

More information

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1) EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq

More information

Chapter 4 Motion in Two and Three Dimensions

Chapter 4 Motion in Two and Three Dimensions Chapte 4 Mtin in Tw and Thee Dimensins In this chapte we will cntinue t stud the mtin f bjects withut the estictin we put in chapte t me aln a staiht line. Instead we will cnside mtin in a plane (tw dimensinal

More information

Electric Fields and Electric Forces

Electric Fields and Electric Forces Cpyight, iley 006 (Cutnell & Jhnsn 9. Ptential Enegy Chapte 9 mgh mgh GPE GPE Electic Fields and Electic Fces 9. Ptential Enegy 9. Ptential Enegy 9. The Electic Ptential Diffeence 9. The Electic Ptential

More information

Subjects discussed: Aircraft Engine Noise : Principles; Regulations

Subjects discussed: Aircraft Engine Noise : Principles; Regulations 16.50 Lectue 36 Subjects discussed: Aicaft Engine Nise : Pinciples; Regulatins Nise geneatin in the neighbhds f busy aipts has been a seius pblem since the advent f the jet-pweed tanspt, in the late 1950's.

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes.

More information

Lecture #2 : Impedance matching for narrowband block

Lecture #2 : Impedance matching for narrowband block Lectue # : Ipedance atching f nawband blck ichad Chi-Hsi Li Telephne : 817-788-848 (UA) Cellula phne: 13917441363 (C) Eail : chihsili@yah.c.cn 1. Ipedance atching indiffeent f bandwidth ne pat atching

More information

Surface and Interface Science Physics 627; Chemistry 542. Lecture 10 March 1, 2013

Surface and Interface Science Physics 627; Chemistry 542. Lecture 10 March 1, 2013 Suface and Inteface Science Physics 67; Chemisty 54 Lectue 0 Mach, 03 Int t Electnic Ppeties: Wk Functin,Theminic Electn Emissin, Field Emissin Refeences: ) Wduff & Delcha, Pp. 40-4; 46-484 ) Zangwill

More information

Phys102 Second Major-182 Zero Version Monday, March 25, 2019 Page: 1

Phys102 Second Major-182 Zero Version Monday, March 25, 2019 Page: 1 Monday, Mach 5, 019 Page: 1 Q1. Figue 1 shows thee pais of identical conducting sphees that ae to be touched togethe and then sepaated. The initial chages on them befoe the touch ae indicated. Rank the

More information

Journal of Theoretics

Journal of Theoretics Junal f Theetics Junal Hme Page The Classical Pblem f a Bdy Falling in a Tube Thugh the Cente f the Eath in the Dynamic They f Gavity Iannis Iaklis Haanas Yk Univesity Depatment f Physics and Astnmy A

More information

( ) ( ) ( ) ( ) ( z) ( )

( ) ( ) ( ) ( ) ( z) ( ) EE433-08 Planer Micrwave Circuit Design Ntes Returning t the incremental sectin, we will nw slve fr V and I using circuit laws. We will assume time-harmnic excitatin. v( z,t ) = v(z)cs( ωt ) jωt { s }

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

Lecture 2 Date:

Lecture 2 Date: Lectue 2 Date: 5.1.217 Definition of Some TL Paametes Examples of Tansmission Lines Tansmission Lines (contd.) Fo a lossless tansmission line the second ode diffeential equation fo phasos ae: LC 2 d I

More information

Analytical Solution to Diffusion-Advection Equation in Spherical Coordinate Based on the Fundamental Bloch NMR Flow Equations

Analytical Solution to Diffusion-Advection Equation in Spherical Coordinate Based on the Fundamental Bloch NMR Flow Equations Intenatinal Junal f heetical and athematical Phsics 5, 5(5: 4-44 OI:.593/j.ijtmp.555.7 Analtical Slutin t iffusin-advectin Equatin in Spheical Cdinate Based n the Fundamental Blch N Flw Equatins anladi

More information

PHYS 1444 Lecture #5

PHYS 1444 Lecture #5 Shot eview Chapte 24 PHYS 1444 Lectue #5 Tuesday June 19, 212 D. Andew Bandt Capacitos and Capacitance 1 Coulom s Law The Fomula QQ Q Q F 1 2 1 2 Fomula 2 2 F k A vecto quantity. Newtons Diection of electic

More information

ELECTRIC & MAGNETIC FIELDS I (STATIC FIELDS) ELC 205A

ELECTRIC & MAGNETIC FIELDS I (STATIC FIELDS) ELC 205A LCTRIC & MAGNTIC FILDS I (STATIC FILDS) LC 05A D. Hanna A. Kils Assciate Pfess lectnics & Cmmnicatins ngineeing Depatment Faclty f ngineeing Cai Univesity Fall 0 f Static lecticity lectic & Magnetic Fields

More information

MEM202 Engineering Mechanics Statics Course Web site:

MEM202 Engineering Mechanics Statics Course Web site: 0 Engineeing Mechanics - Statics 0 Engineeing Mechanics Statics Cuse Web site: www.pages.dexel.edu/~cac54 COUSE DESCIPTION This cuse cves intemediate static mechanics, an extensin f the fundamental cncepts

More information

Sensors and Actuators Introduction to sensors

Sensors and Actuators Introduction to sensors Senss and Actuats Intductin t senss Sande Stuij (s.stuij@tue.nl) Depatment f Electical Engineeing Electnic Systems AMPLIFIES (Chapte 5.) Infmatin pcessing system nncntact sens cntact sens abslute sens

More information

n Power transmission, X rays, lightning protection n Solid-state Electronics: resistors, capacitors, FET n Computer peripherals: touch pads, LCD, CRT

n Power transmission, X rays, lightning protection n Solid-state Electronics: resistors, capacitors, FET n Computer peripherals: touch pads, LCD, CRT .. Cu-Pl, INE 45- Electmagnetics I Electstatic fields anda Cu-Pl, Ph.. INE 45 ch 4 ECE UPM Maagüe, P me applicatins n Pwe tansmissin, X as, lightning ptectin n lid-state Electnics: esists, capacits, FET

More information

Phys 332 Electricity & Magnetism Day 3. Note: I should have recommended reading section 1.5 (delta function) as well. rˆ rˆ

Phys 332 Electricity & Magnetism Day 3. Note: I should have recommended reading section 1.5 (delta function) as well. rˆ rˆ Phs 33 lecticit & Magnetism Da 3 Mn. 9/9 Wed. 9/ Thus 9/ Fi. 9/3 (C.-.5,.8). &.5;..-.. Gauss & Div, T Numeical Quadatue (C.-.5,.8)..3 Using Gauss (C.-.5,.8)..3-.. Using Gauss HW quipment Bing in ppt s

More information

TUTORIAL 9. Static magnetic field

TUTORIAL 9. Static magnetic field TUTOIAL 9 Static magnetic field Vecto magnetic potential Null Identity % & %$ A # Fist postulation # " B such that: Vecto magnetic potential Vecto Poisson s equation The solution is: " Substitute it into

More information

Steady State Analysis of Squirrel-Cage Induction Machine with Skin-Effect

Steady State Analysis of Squirrel-Cage Induction Machine with Skin-Effect Steady State Analysis f Squiel-Cage Inductin Machine with Skin-Effect D.-Ing. O. I. Ok Depatment f Electical Engineeing Univesity f Nigeia, Nsukka Enugu State, Nigeia. Email: gnnayak@htmail.cm ABSTACT

More information

PHYS 1444 Section 501 Lecture #7

PHYS 1444 Section 501 Lecture #7 PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos

More information

REPORT ITU-R SA Protection of the space VLBI telemetry link

REPORT ITU-R SA Protection of the space VLBI telemetry link Rep. ITU-R SA.65 REPORT ITU-R SA.65 Ptectin f the space VLBI telemety link CONTENTS Page Intductin... Space VLBI system.... Space VLBI telemety signal, nise and intefeence..... Signal... 3.. Nise and intefeence...

More information

Review: Electrostatics and Magnetostatics

Review: Electrostatics and Magnetostatics Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion

More information

15 B1 1. Figure 1. At what speed would the car have to travel for resonant oscillations to occur? Comment on your answer.

15 B1 1. Figure 1. At what speed would the car have to travel for resonant oscillations to occur? Comment on your answer. Kiangsu-Chekiang College (Shatin) F:EasteHolidaysAssignmentAns.doc Easte Holidays Assignment Answe Fom 6B Subject: Physics. (a) State the conditions fo a body to undego simple hamonic motion. ( mak) (a)

More information

Strees Analysis in Elastic Half Space Due To a Thermoelastic Strain

Strees Analysis in Elastic Half Space Due To a Thermoelastic Strain IOSR Junal f Mathematics (IOSRJM) ISSN: 78-578 Vlume, Issue (July-Aug 0), PP 46-54 Stees Analysis in Elastic Half Space Due T a Themelastic Stain Aya Ahmad Depatment f Mathematics NIT Patna Biha India

More information

CHAPTER 25 ELECTRIC POTENTIAL

CHAPTER 25 ELECTRIC POTENTIAL CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When

More information

MAGNETIC FIELDS & UNIFORM PLANE WAVES

MAGNETIC FIELDS & UNIFORM PLANE WAVES MAGNETIC FIELDS & UNIFORM PLANE WAVES Nme Sectin Multiple Chice 1. (8 Pts). (8 Pts) 3. (8 Pts) 4. (8 Pts) 5. (8 Pts) Ntes: 1. In the multiple chice questins, ech questin my hve me thn ne cect nswe; cicle

More information

Journal of Solid Mechanics and Materials Engineering

Journal of Solid Mechanics and Materials Engineering Junal f Slid Mechanics and Mateials Engineeing Vl. 4, N. 8, 21 Themal Stess and Heat Tansfe Cefficient f Ceamics Stalk Having Ptubeance Dipping int Mlten Metal* Na-ki NOD**, Henda**, Wenbin LI**, Yasushi

More information

CHAPTER 17. Solutions for Exercises. Using the expressions given in the Exercise statement for the currents, we have

CHAPTER 17. Solutions for Exercises. Using the expressions given in the Exercise statement for the currents, we have CHATER 7 Slutin f Execie E7. F Equatin 7.5, we have B gap Ki ( t ) c( θ) + Ki ( t ) c( θ 0 ) + Ki ( t ) c( θ 40 a b c ) Uing the expein given in the Execie tateent f the cuent, we have B gap K c( ωt )c(

More information

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D.

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D. ELETROSTATIS::BHSE 9-4 MQ. A moving electic chage poduces A. electic field only. B. magnetic field only.. both electic field and magnetic field. D. neithe of these two fields.. both electic field and magnetic

More information

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces

More information

Current, Resistance and

Current, Resistance and Cuent, Resistance and Electomotive Foce Chapte 25 Octobe 2, 2012 Octobe 2, 2012 Physics 208 1 Leaning Goals The meaning of electic cuent, and how chages move in a conducto. What is meant by esistivity

More information

Faraday s Law (continued)

Faraday s Law (continued) Faaday s Law (continued) What causes cuent to flow in wie? Answe: an field in the wie. A changing magnetic flux not only causes an MF aound a loop but an induced electic field. Can wite Faaday s Law: ε

More information

Objectives: After finishing this unit you should be able to:

Objectives: After finishing this unit you should be able to: lectic Field 7 Objectives: Afte finishing this unit you should be able to: Define the electic field and explain what detemines its magnitude and diection. Wite and apply fomulas fo the electic field intensity

More information

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b . REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but

More information

Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter).

Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter). Three charges, all with a charge f 0 are situated as shwn (each grid line is separated by meter). ) What is the net wrk needed t assemble this charge distributin? a) +0.5 J b) +0.8 J c) 0 J d) -0.8 J e)

More information

University of Illinois at Chicago Department of Physics. Electricity & Magnetism Qualifying Examination

University of Illinois at Chicago Department of Physics. Electricity & Magnetism Qualifying Examination E&M poblems Univesity of Illinois at Chicago Depatment of Physics Electicity & Magnetism Qualifying Examination Januay 3, 6 9. am : pm Full cedit can be achieved fom completely coect answes to 4 questions.

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

On the structure of MHD shock waves in a viscous gas

On the structure of MHD shock waves in a viscous gas On the stuctue f MHD shck waves in a viscus gas On the stuctue f MHD shck waves in a viscus gas R. K. Anand and Haish C. Yadav Depatment f Physics, Univesity f Allahabad, Allahabad-, India e-mail: anand.ajkuma@ediffmail.cm

More information

ev dm e evd 2 m e 1 2 ev2 B) e 2 0 dm e D) m e

ev dm e evd 2 m e 1 2 ev2 B) e 2 0 dm e D) m e . A paallel-plate capacito has sepaation d. The potential diffeence between the plates is V. If an electon with chage e and mass m e is eleased fom est fom the negative plate, its speed when it eaches

More information

16.1 Permanent magnets

16.1 Permanent magnets Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and

More information

THE SPATIAL CROSS-CORRELATION OF. Mobile and Portable Radio Research Group

THE SPATIAL CROSS-CORRELATION OF. Mobile and Portable Radio Research Group FFCTS OF MULTIPATH AGULAR SPRAD O TH SPATIAL CROSS-CORRLATIO OF RCIVD VOLTAG VLOPS Gegy D. Dugin and Thede S. Rappapt Mbile and Ptable Radi Reseach Gup Badley Depatment f lectical and Cmpute ngineeing

More information

which represents a straight line whose slope is C 1.

which represents a straight line whose slope is C 1. hapte, Slutin 5. Ye, thi claim i eanable ince in the abence any heat eatin the ate heat tane thugh a plain wall in teady peatin mut be cntant. But the value thi cntant mut be ze ince ne ide the wall i

More information

Combustion Chamber. (0.1 MPa)

Combustion Chamber. (0.1 MPa) ME 354 Tutial #10 Winte 001 Reacting Mixtues Pblem 1: Detemine the mle actins the pducts cmbustin when ctane, C 8 18, is buned with 00% theetical ai. Als, detemine the dew-pint tempeatue the pducts i the

More information

Lecture 4. Electric Potential

Lecture 4. Electric Potential Lectue 4 Electic Ptentil In this lectue yu will len: Electic Scl Ptentil Lplce s n Pissn s Eutin Ptentil f Sme Simple Chge Distibutins ECE 0 Fll 006 Fhn Rn Cnell Univesity Cnsevtive Ittinl Fiels Ittinl

More information

EKT 345 MICROWAVE ENGINEERING CHAPTER 2: PLANAR TRANSMISSION LINES

EKT 345 MICROWAVE ENGINEERING CHAPTER 2: PLANAR TRANSMISSION LINES EKT 345 MICROWAVE ENGINEERING CHAPTER : PLANAR TRANSMISSION LINES 1 Tansmission Lines A device used to tansfe enegy fom one point to anothe point efficiently Efficiently minimum loss, eflection and close

More information

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system

More information

Q1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.

Q1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2. Phys10 Final-133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed

More information

School of Chemical & Biological Engineering, Konkuk University

School of Chemical & Biological Engineering, Konkuk University Schl f Cheical & Bilgical Engineeing, Knkuk Univesity Lectue 7 Ch. 2 The Fist Law Thecheisty Pf. Y-Sep Min Physical Cheisty I, Sping 2008 Ch. 2-2 The study f the enegy tansfeed as heat duing the cuse f

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

Chapter 19 8/30/2010 ( ) Let s review what we have learned in PHY College Physics I. Electric Potential Energy and the Electric Potential

Chapter 19 8/30/2010 ( ) Let s review what we have learned in PHY College Physics I. Electric Potential Energy and the Electric Potential 8/3/ Chapte 9 Electic Ptential Enegy and the Electic Ptential Gals Chapte 9 T undestand electical ptential enegy. T deine electicalptential. T study euiptential suaces. T study capacits and dielectics.

More information

Module 4: General Formulation of Electric Circuit Theory

Module 4: General Formulation of Electric Circuit Theory Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated

More information

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax .7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical

More information

Experiment I Voltage Variation and Control

Experiment I Voltage Variation and Control ELE303 Electicity Netwoks Expeiment I oltage aiation and ontol Objective To demonstate that the voltage diffeence between the sending end of a tansmission line and the load o eceiving end depends mainly

More information

CHE CHAPTER 11 Spring 2005 GENERAL 2ND ORDER REACTION IN TURBULENT TUBULAR REACTORS

CHE CHAPTER 11 Spring 2005 GENERAL 2ND ORDER REACTION IN TURBULENT TUBULAR REACTORS CHE 52 - CHPTE Sping 2005 GENEL 2ND ODE ECTION IN TUULENT TUUL ECTOS Vassilats & T, IChEJ. (4), 666 (965) Cnside the fllwing stichiety: a + b = P The ass cnsevatin law f species i yields Ci + vci =. Di

More information

AT622 Section 15 Radiative Transfer Revisited: Two-Stream Models

AT622 Section 15 Radiative Transfer Revisited: Two-Stream Models AT6 Sectin 5 Radiative Tansfe Revisited: Tw-Steam Mdels The gal f this sectin is t intduce sme elementay cncepts f adiative tansfe that accunts f scatteing, absptin and emissin and intduce simple ways

More information

On the Micropolar Fluid Flow through Porous Media

On the Micropolar Fluid Flow through Porous Media Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM On the Micpla Fluid Flw thugh Pus Media M.T. KAMEL 3, D. ROACH, M.H. HAMDAN,3 Depatment f Mathematical

More information

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.

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. 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

More information

e = 1.60 x 10 ε 0 = 8.85 x C 2 / Nm 2 V i...) F a = m Power =

e = 1.60 x 10 ε 0 = 8.85 x C 2 / Nm 2 V i...) F a = m Power = Equations: 1 1 Constants: q q v v F = k F = qe e = 1.6 x 1-19 C q 1 q 1 9 E = k = k = = 9 1 Nm / C 4πε 4 πε Φ = E da Φ V Total v v q = E da = = V f V i = W q enclosed ε = E ds U e V = q q V V V V = k E

More information

Chapter 6. Dielectrics and Capacitance

Chapter 6. Dielectrics and Capacitance Chapter 6. Dielectrics and Capacitance Hayt; //009; 6- Dielectrics are insulating materials with n free charges. All charges are bund at mlecules by Culmb frce. An applied electric field displaces charges

More information

Design of Analog Integrated Circuits

Design of Analog Integrated Circuits Design f Analg Integated Cicuits Opeatinal Aplifies Design f Analg Integated Cicuits Fall 01, D. Guxing Wang 1 Outline Mdel f Opeatinal Aplifies Tw Stage CMOS Op Ap Telescpic Op Ap Flded-Cascde Op Ap Refeence

More information

Phys 222 Sp 2009 Exam 1, Wed 18 Feb, 8-9:30pm Closed Book, Calculators allowed Each question is worth one point, answer all questions

Phys 222 Sp 2009 Exam 1, Wed 18 Feb, 8-9:30pm Closed Book, Calculators allowed Each question is worth one point, answer all questions Phys Sp 9 Exam, Wed 8 Feb, 8-9:3pm Closed Book, Calculatos allowed Each question is woth one point, answe all questions Fill in you Last Name, Middle initial, Fist Name You ID is the middle 9 digits on

More information

AP Physics Kinematic Wrap Up

AP Physics Kinematic Wrap Up AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in tw-dimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x

More information