TOPIC: Electrostatics and Magnetostatics (ENEL475) Q.1

Transcription

1 TOPIC: Eletosttis nd Mgnetosttis (ENEL475 Q. We e inteested in inding the potentil ( t point on the -xis P = (0, 0, o uniom hge density ρ S distibuted on disk o dius = lying in the xy-plne nd ented ound the oigin, ssuming the eeene is hosen s ininity. ( Dw le sketh o this poblem to nlye the geomety. You sketh should lely indite the ield point P, the ield position veto, nd the soue position veto. (b Solve o the potentil ( t point on the -xis P = (0, 0, Note: You will pobbly equie one o the ollowing integls in ode to solve this poblem ( x + dx = ln x + x + x + dx = ( x tn x x + dx = x + x x + dx = ln ( x +

2 Q. n ininitely long oxil ble hs hollow inne onduto o dius, whih hs ldding o dius b whih is mgneti mteil o pemebility µ, s depited in the igue below. The inne onduto ies the totl supply uent I, nd the oute onduto ies the totl etun uent I; both e distibuted uniomly ound thei espetive ylindil sue in the dietions indited in the igue. ( Choose n ppopite oodinte system nd, using mpee s lw, show tht the mgneti ield eveywhee in the egion < ρ < b is given by H = I πρâφ m (b Using the given ield in pt (, lulte the stoed mgneti enegy in the ldding egion o length d o the ble nd o uent I. ( I the eltive pemebility o the ldding is µ = 0, wht is the indutne o length d o this ble?

3 Q.3 Stting om Guss lw in integl om, podue deivtion o the onept o Divegene by showing tht D d S D = lim v 0 S v Q.4 Stting om simple miosopi/tomi bsis, explin how eh o the ollowing mosopi popeties is mniest: ( Condutivity σ (b Pemittivity ε ( Pemebility µ

4 TOPIC: Eletomgneti wves nd pplitions (ENEL476 Q. Conside the loop ontining esisto s shown below. The loop is pled in mgneti lux density desibed by: B=-0 os(00t-/3 mwb/m y 0. m m x Find the EMF (em b Clulte the indued uent in the loop. Indite the dietion o uent low duing the ist qute peiod on the igue bove. Q. gound penetting d system is modeled s uniom plne wve in ee spe impinging on the gound t noml inidene. The inident eleti ield (in ee spe, so popeties e o, o, =0 is given by: E i (x,t=0 os(0 9 t-3.3xy /m Find the wvelength. The gound hs popeties o =4, =, nd =0. S/m. b Clulte the eletion ( nd tnsmission (T oeiients. Find n expession o the eleted eleti ield (E (x,t. d Find n expession o the tnsmitted eleti (E t (x,t nd mgneti ields (H t (x,t.

5 Q.3 distotionless tnsmission line hs R=5 /m, L=0 H/m, C=30 nf/m nd is opeted t 0 MH. Clulte the ollowing quntities: G b the impedne o the line, o the ttenution o the line, d the phse onstnt o the line, e the wvelength on the line, lod o L=30-j40 is tthed to 6 m o the tnsmission line. Using the ppopite eqution, ind in o the setion o tnsmission line teminted by the lod. Q.4 lod o impedne L=30-j60 is tthed to tnsmission line with 75 hteisti impedne (o=75. The equeny o opetion is 5 GH nd the wvelength on the line is 6 m. Use the Smith Cht to solve the ollowing questions. Find the eletion oeiient t the lod (. b Find the stnding wve tio (s. eiy with the ppopite eqution Find the input impedne in o line o length o 5.0 m tthed to the lod. d Find the dmittne o the lod (YL. e Find the distne om the lod to the ist voltge minimum. Find the shotest distne to puely esistive lod. Q.5 lod o impedne L=70+j5 is tthed to tnsmission line with 00 hteisti impedne (o=00. The equeny o opetion is 900 MH nd the wvelength on the line is 67 m. To mth the lod to the line, design seies stub tune with n open temintion on the stub, nd shunt stub tune with shot temintion on the stub.

6 TOPIC: RF/Miowve tive Ciuits (ENEL574 Q. Detemine the S pmetes o two pot netwok onsisting o seies esistne R teminted t its input nd output pots by the hteisti impedne o. Q. Input mthing Netwok Output mthing Netwok 0 = 50 Ω 5 0 Ω 0 = 50 Ω 0 = 50 Ω Γ L = = 50 Ω 5 0 Ω Γ S = Design the input mthing netwok by giving the lengths o 50 Ohms tnsmission lines s untion o the wvelength, λ s shown in the bove igue to podue the soue eletion oeiient Γ S = -70 t GH. b Design the output mthing netwok by giving the lengths o 50 Ohms tnsmission lines s untion o the wvelength, λ s shown in the bove igue to podue the lod eletion oeiient Γ L = t GH. Q.3 n mpliie is diven by modulted signl hving 0 MH bndwidth is onstituted by the sde o thee mpliies, nd 3 nd hving the ollowing hteistis: mpliie : G =3 db, IP3_ =3 dbm, NF = db. mpliie : G =7 db, IP3_=4 dbm, NF =3 db. mpliie 3: G 3=3 db, IP3_3=50 dbm, NF 3=5 db. The pmetes IP3_N e speiied t the output o eh mpliie nd the eeene noise tempetue T o is 90 K. b Detemine the equivlent noise igue NF e o the powe mpliie s well s its equivlent noise tempetue T e. Clulte the thid ode inteeption point IP 3 t the output o the powe mpliie. Dedue the P db o the powe mpliie.

7 d Detemine the ie to thid ode intemodultion poduts tio (C/IMD 3 t the output the powe mpliie when it is diven with two-tone signl sped by 0 MH nd hving totl input powe o -8 dbm Q.4 The stteing nd noise pmetes o Gs FET tnsisto t GH e: S = , S = 3. 40, S = nd S = F min =.5 db, Γ opt = , n = 0.95 Study the stbility o the devie nd dw the input nd output stbility iles in the Smith Cht, b Cn the devie be onsideed uniltel? Dw the opeting powe gin ile o G P = 0 db. d Detemine the soue nd lod eletion oeiients equied to design n mpliie to hve n opeting powe gin o 0 db. Explin you hoies o Γ S nd Γ L.

8 TOPIC: RF/Miowve Pssive Ciuits (ENEL575 Q. n mpliie is tthed to n ntenn though 9 m long setion o oxil ble. ssume the ox line hs no loss, nd its pmetes e C=96 pf/m nd L=40 nh/m. The ntenn hs dition impedne o 50 Ω (you n ssume the ntenn is you geneto, nd its dition impedne the geneto intenl impedne, howeve the mpliie is not mthed to the system. Mesuements showed tht the mp hs n input impedne o (5+j0 Ω. The system opetes t 0 GH. You hve been sked to impove the system peomne. Hee e you tsks:. ssume tht the ntenn genetes voltge g. Compute the voltge nd the uent t the input teminl o the tnsmission line (so om the ntenn side, emembe, you ntenn ts s geneto hee, ind out the mplitudes o owd nd bkwd tvelling wves.. Compute vilble powe. Wht peentge o vilble powe t the geneto is deliveed to the mp? 3. Now mth the system using lumped elements. Wht peentge o vilble powe is deliveed to the mp? Remembe to lulte the tul vlues o neessy indutnes o pitnes.

9 Q. Conside etngul wveguide with the ollowing dimensions: =.0mm, b=0.0mm. The wveguide wlls e mde o peet onduto, wveguide is illed with i (lossless. Fequeny is 0 GH. How muh the TE 0 mode will be ttenuted t tht equeny ove distne o m (expessed in db b. The wveguide is loded with n impedne o (54.6+j54.6Ω. Mth the system using double stub seil tune. The tunes e shot-iuited, nd septed by 5/4λ (lmbd. You e llowed to dd o setions o λ/8 setion o wveguide between the tune nd the impedne (only i you ind it neessy. Dw the stutue, py ttention to the esibility o you stutue (tht is, ould you tully build it? Show ll dimensions (in mm. In ptiul, show how the seil onnetions o the wveguide nd the tunes look like.

10 Q.3 Design thee-pot esistive divide o n equl powe split nd 75 ohm system impedne (lets ll this system. Deive S mtix o this system. Now teminte pot 3 o the 3-pot system with 50 ohm esisto eetively tuning the initil 3 pot stutue into pot stutue with mtix S B. Compute S B in this new system.

11 Field o Study Exmintion, Feb 4 07 Subjet e: Rdio Fequeny Systems nd pplied Eletomgnetis This question ppe hs pges (not inluding this ove pge. This question ppe hs 4 questions. nswe minimum o one question nd t most thee questions om this subjet e. Use septe booklet (i.e., blue booklet o the nswes to questions in this subjet e.

12 . This question hs 4 pts (-(d. wieless eeive is onstituted by low noise mpliie (LN with gin G = 35 db nd noise igue NF = 3 db; nd P db = 3 dbm, bnd pss ilte hving db insetion loss (IL, nd pssive mixe hving onvesion loss (CL o 4 db s shown in the lineup below. The eeive pmetes e: noise eeene tempetue: T 0 = 90 K; signl bndwidth: B = MH; output thid-ode inteept point: IP 3 = 7 dbm. Boltmnn onstnt k = J/K. G = 35 db NF = 3 db IL = db CL = 4 db S i, N i S o, N o Low noise mpliie Bndpss ilte Mixe ( Clulte the ovell noise igue (NF, the gin nd the equivlent noise tempetue (T e o the eeive. (b Clulte the noise powe t the output o the eeive. ( Clulte the dynmi nge (DR o the eeive. (d Find the ie to the thid-ode intemodultion poduts tio (C/IMD3 t the output the eeive when it is diven with two-tone signl hving totl powe o 0 dbm. /

13 . This question hs pts (-(l. t suiient distne om n ntenn, the ields dited by the ntenn my be epesented using uniom plne wve. ssume tht the ields e tveling in soue-ee egion o ee spe (ɛ = ɛ 0, µ = µ 0, σ = 0. The eleti ield is given by: E(x, t = 00 os ( t βx y /m Find the: ( equeny ( (b wvelength (λ ( phse onstnt (β (d mgneti ield (H(x, t The ntenn system is being used o thu-wll inspetion (i.e. wves tvel though wll nd detet objets on the othe side. The wll hs ɛ = 3, σ = 0. S/m nd µ =. Given tht the wve is nomlly inident on the wll, ind the: (e ttenution oeiient (α ( phse onstnt (β (g intinsi impedne (η (h eletion oeiient (Γ (i tnsmission oeiient (T (j veloity t whih the wve tvels in the wll (k Fo the inident ield given bove, ind n expession o the eleti (E (x, t nd mgneti (H (x, t ields eleted om the wll (l Fo the inident ield given bove, ind n expession o the eleti (E t (x, t nd mgneti (H t (x, t ields ssoited with the wve tnsmitted into the wll /

14 3. This question hs pts (-(b. pito is mde up o onenti sphees with the inne sphee hving dius nd hge +Q oulombs. The oute sphee hs dius b nd hge Q oulombs. NOTE tht, b nd Q e onstnts. The gp between the pitos is divided into 3 equl spes with pemittivities o ɛ, ɛ nd 3ɛ. ( Using ield nlysis, deive n expession o the pitne C o this pito. (b I the pemittivity is ɛ = ɛ 0 ɛ with ɛ = nd = 5 mm nd b = 9 mm, wht is the vlue o the pitne o the onenti sphees. 3/

15 4. Design qute wve tnsome using etngul wveguide tehnology. The tnsome is to opete t 0 GH nd be positioned between setion o wveguide with dimensions =.86 mm b=0.6 mm (i illed, nd lod with 800 Ω impedne (wve impedne. Design the tnsome using hnge in bod dimension. Ignoe eets t the ones/juntion between the tnsome setion nd egul wveguide, in you design, but desibe them qulittively. Dw the stutue, show ll dimensions o the tnsome. 4/

16 Mxwell s Equtions integl om point om I ~E d~l = ~B d~s Fdy s Lw ~ ~E = ~B I ~D ~H d~l = + d ~S mpee s Lw ~ ~H = ~J + D I ~D d~s = = Guss Lw o Eleti Fields ~ ~D = I Eleti Fields nd Potentil ~B d~s =0 Guss Lw o Mgneti Fields ~ ~B =0 Coulomb s Lw ~F = 4 0 ~R b Wok = ~E d~l Potentil ield o distibuted hges (~ = (~ ~ ~ Potentil o Dipole (~ = ~p ~ ~ ~ ~ 0 3 Mgneti Fields, Foes, nd Toque ~F = Biot-Svt Lw d~l ~ ~ 0 4 ~ ~ 0 3 ~K ~ ~ 0 ~H(~ = 4 ~ ~ ~J ~ ~ 0 4 ~ ~ Mgneti Foe on Cuents I ~F = d~l ~B (line uent ~K ~B 0 (sue uent density 0 ~F = ~J ~B 0 (volume uent density 0 Eleti Field ~E(~ = (~ 0 ~ ~ ~ ~ Potentil Dieene = ~E d~l Eleti Field om Potentil ~E = ~ Fields om Sheet o Chge ~E(~ = b Fields om Line Chge ~E(~ = b Fields om Line Cuent ~H(~ = b Fields om Sheet o Cuent Density ~H(~ = ~ K b Loent Foe Eqution ~F = ( ~E + ~v ~B Toque ~T = ~S ~B = ~m ~B 5/

17 Cuent nd Condutos Continuity o Cuent I integl om ~J d~s = point om ~ ~J = Mteil Reltions nd Boundy Conditions Cuent Density ~J = ~E 0 = F m Dieleti Mteil Reltions ~D = 0 ~E + ~P = ~E =(+ 0 = 0 Eleti Boundy Conditions ³ noml D ~ ~D b = ³ tngentil E ~ ~E b =0 ngles t Intee tn = stn µ ~D = ~D os + sin s µ ~E = ~E sin + os 0 =4 0 7 H m Mgneti Mteil Reltions ³ ~B = 0 H ~ + ~M = ~H =(+ 0 = 0 Mgneti Boundy Conditions ³ noml B ~ ~B b =0 ³ tngentil b H ~ ~H = ~K Polition Chge Reltions bound sue hge density = ~P b bound volume hge density = ~ ~P Ciuit Pmetes, Powe nd Enegy = = Ψ = = Resistne = Cpitne I ~D d~s ~E d~l Indutne ~B d~s = I ~H d~l ~E d~l ~J d~s = = Powe Loss (Joule/Ohmi Loss ³ = J ~ ~E Enegy Stoed in Eleti Field = ³ E ~ ~D = Enegy Stoed in Mgneti Field = ³ B ~ ~H 6/

18 Cylindil Coodintes = os = sin = = p + =tn = Field Component Tnsomtions os sin 0 = sin os = os sin 0 sin os Dieentil Elements Dieentil d~l = d~s = one o = Elements b + b + b ( b ( b ( b Note: igues on this pge e epodued om Eletomgneti Fields nd Wves, nd Edition, Isknde, om Wvelnd Pess. 7/

19 Spheil Coodintes =sin os = sin sin = os = p + + =os p + + =tn Field Component Tnsomtions sin os sin sin os = os os os sin sin sin os 0 = sin os os os sin sin sin os sin os os sin 0 Dieentil Elements Dieentil d~l= d~s=one o = Elements b + b + sin b sin b (sin b (b sin Note: Þgues on this pge e epodued omeletomgnetifieldsndwves,ndedition, Isknde, om Wvelnd Pess. 8/

20 eto Opetos Ctesin Cylindil Gdient ~ = b + b + b Divegene ~ ~F = Cul ~ ~F = Lplin = b + b + b + + ( + ( + b b b b b b µ Gdient ~ = Divegene ~ ~F = Cul ~ ~F = Lplin = µ Spheil b + b + sin b + (sin + sin sin b b sin b sin sin + µ sin + sin sin 9/

21 o = 4 x 0-7 H/m / o = 8 x 0 5 m/h o = 0 o=8.85x0 - F/m x =sin os x =os os x =-sin y =sin sin y =os sin y =os =os = -sin =0 x =os x = -sin x =0 y =sin y = os y =0 = 0 0 = x = os y = sin Ctesin Cylindil Spheil dxx+dyy+d ddd d + d + sin d (dydx ( d d ( sin d d (dxdy (d d ( sin d d (dxdy ( d d ( d d dx dy d d d d sin d d d = x y x = sin os = os - x y y = sin sin = tn - y x = os y x y x sin y y x E E x E E E E E E E sin E sin sin E E y x ( sin (sin sin ( y x y x y x sin sin sin 0/

22 Mgnetosttis mteils Idl H R M mh L 4R M J M W M H Bdv =+ m N T=m x B L I N M F Idl xb I Loent oe Continuity F=q(E+vxB J v t Eletosttis Mteils vdv R l /( S E( = R 4o R P eoe ( ' dv' ( = v C 4o ' P pv P E dl C n ps E =+ e W E E Ddv v C=Q/ Boundy onditions D n -D n =s (D -D =s E t =E t x (E -E =0 B n -B n =0 (B -B =0 x (H -H =J s x (H -H =J s Mxwell s equtions E dl B ds B ds 0 t E d s s o s v dv s v H dl J ds D ds s t s B 0 D v H J D E B t t B em ds ( vxb dl s t L 0 J s is sue uent (lso denoted s K Note: o stti ields, time deivtives e eo. In phso om, time deivtives beome jtems. Time-vying ields: UPW E s Es 0 H s Hs 0 j oo T / E / H o o v p E(, t Eoe os( t x Eo H(, t e os( t y / / 4 tn o 45 P * vg ( Re( E s ( H s ( P(,t=E(,t x H(,t T ost os i ost osi T osi ost osi osi ost osi ost T osi osi ost eto wve equtions o timehmoni ields in lossy medium. With lossless medium o ee spe, =0. Uniom plne wve in lossless medium. Fo ee spe, = nd =. One exmple o E nd H ields in lossy medium. Pmetes desibing UPW in lossy medium. good onduto: (>> Poynting veto Tnsmission nd eletion oeiients: noml inidene Tnsmission nd eletion oeiients: oblique inidene i sint sin i /

23 Wves nd T/R ontinued s sin B tn B sin L /( w R Tnsmission lines: os o o ve e P Wveguides: g ' u u v p p ' u u v g g u' v v g p Distotionless tnsmission lines: R/L=G/C RG LC G R R o Lossless line: ' d (0.5 ' 0 b b R s TE ( ( (( ' n m b n m b b b b R s TE ( ( ' 3 3 n m b n m b b R s TM j L j R C j G C j G L j R o o s e e I ( s e e ( 0 0 L L l min mx SWR l j e l (0 ( SWR o in _ min SWR o o in _ mx tn( tn( l j l j L o o L o in /