PET4I101 ELECTROMAGNETICS ENGINEERING (4 th Sem ECE- ETC) Module-I (10 Hours)

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1 PT4I LCTROMGNTICS NGINRING (4 th Sem C- TC) Module-I ( Hous). Cte Clindicl nd Spheicl Coodinte Sstems; Scl nd Vecto Fields; Line Sufce nd Volume Integls.. Coulomb s Lw; The lectic Field Intensit; lectic Flu Densit nd lectic Flu; Guss s Lw; Divegence of lectic Flu Densit: Point Fom of Guss s Lw; The Divegence Theoem; The Potentil Gdient; neg Densit; Poisson s nd Lplce s qutions. 3. mpee s Mgnetic Cicuitl Lw nd its pplictions; Cul of H; Stokes Theoem; Divegence of B; neg Stoed in the Mgnetic Field. Module-II (8 Hous). The Continuit qution; Fd s Lw of lectomgnetic Induction; Conduction Cuent: Point Fom of Ohm s Lw Convection Cuent; The Displcement Cuent;. Mwell s qutions in Diffeentil Fom; Mwell s qutions in Integl Fom; Mwell s qutions fo Sinusoidl Vition of Fields with Time; Bound Conditions; The Retded Potentil; The Ponting Vecto; Ponting Vecto fo Fields Ving Sinusoid ll with Time Module-III (8 Hous). Solution of the One-Dimensionl Wve qution; Solution of Wve qution fo Sinusoid ll Time-Ving Fields; Polition of Unifom Plne Wves; Fields on the Sufce of Pefect Conducto; Reflection of Unifom Plne Wve Incident Nomll on Pefect Conducto nd t the Intefce of Two Dielectic Regions; The Stnding Wve Rtio; Oblique Incidence of Plne Wve t the Bound between Two Regions; Oblique Incidence of Plne Wve on Flt Pefect Conducto nd t the Bound between Two Pefect Dielectic Regions; Module-IV (8 Hous). Tpes of Two-Conducto Tnsmission Lines; Cicuit Model of Unifom Two-

2 Conducto Tnsmission Line; The Unifom Idel Tnsmission Line; Wve Reflection t Discontinuit in n Idel Tnsmission Line; Mtching of Tnsmission Lines with Lod. dditionl Module (Teminl mintion-intenl) (8 Hous). Fomultion of Field qutions; Wve Tpes; the Pllel-Plte Wveguide; the Rectngul Wveguide.. Rdition Popeties of Cuent lement; Rdition Popeties of Hlf-Wve Dipole; Ygi Ud ntenn; the Pbolic Reflecto ntenn. 3. The Vecto Mgnetic Potentil; neg stoed in cpcito Gphicl field mpping; Continuit of Cuent in Cpcito; Citicl ngle of Incidence nd Totl Reflection; Bewste ngle. I3I LCTROMGNTIC FILD THORY (3 d Sem I& ngg.) MODUL I (3 Hous). Vectos nd Fields: Cte Coodinte Sstem Clindicl nd Spheicl coodinte sstem Vecto lgeb Scl nd Vecto Fields gdient divegence cul opetions The Lplcin Divegence Theoem Stoke s Theoem Useful vecto identities nd thei deivtions. (selected potions fom. to.5 of TB-).: lectic nd Mgnetic fields: Field due to line/sheet/volume chge Biot _Svt LwGuss s Lw fo lectic Field nd Mgnetic Field Fields of electic nd mgnetic dipoles pplictions of electosttics nd mgnetosttics Fd s Lw mpee s Cicuitl Lw. (potions 3.4.to to 8.8 nd 9. of TB-) 3. Mwell s qutions: Divegence nd Diffeentil Fom Line Integl Sufce Integl nd Integl fom Fds Lw mpee s Cicuitl Lw Guss s Lw fo lectic Field nd Mgnetic Field. (potions 4. to 4.3 of TB-) MODUL II (3 Hous) (Potions 5. to 5.3 of TB-) 4. Wve Popgtion in Fee Spce: The electomgnetic wve eqution nd its solution Unifom Plne Wves Diection coes Concept on TM mode Ponting Vecto nd Powe densit 5. Wve Popgtion in Mteil Medi: Conductos nd Dielectics Mgnetic Mteils Wve

3 qution nd Solution Unifom Plne Wves in Dielectics nd Conductos Polition Bound Conditions Reflection nd Tnsmission of Unifom Plne Wves t the bound of two medi fo noml nd oblique incidence Bewste s ngle. MODUL III ( Hous) 6. Tnsmission Line nlsis: Tnsmission lines Cicuit epesenttion of pllel plne tnsmission line Pllel plne tnsmission lines with loss nd H bout long pllel clindicl conductos of bit s section Tnsmission line theo UHF lines s cicuit elements (potions 7. to 7.6 of TB-) 7. Wve Guide Pinciples: Rectngul guides TM wves in ectngul guides T wves in ectngul guides Impossibilit of TM wve in wve guides wve impednce nd chcteistic impednces ttenution fcto nd Q of wve guides Dielectic Slb Guide (potions 8. to of TB-) PT6J NTNNS & WV PROPGTION (6 th Sem C- TC) MODUL- I lectomgnetic dition nd ntenn fundmentls- Review of electomgnetic theo: Vecto potentil Solution of wve eqution etded cse Hetiin dipole. ntenn chcteistics: Rdition ptten Bem solid ngle Diectivit Gin Input impednce Polition Bndwidth Recipocit quivlence of Rdition pttens quivlence of Impednces ffective petue Vecto effective length ntenn tempetue. MODUL-II Wie ntenns- Shot dipole Rdition esistnce nd Diectivit Hlf wve Dipole Monopole Smll loop ntenns. ntenn s: Line nd Ptten Multipliction Two-element Unifom Polnomil epesenttion with nonunifom cittion-binomil MODUL- III petue ntenns- Mgnetic Cuent nd its fields Uniqueness theoem Field equivlence pinciple Dulit pinciple Method of Imges Ptten popeties Slot ntenn Hon ntenn Pmidl Hon ntenn Reflecto ntenn-flt eflecto Cone Reflecto Common cuved eflecto shpes Lens ntenn

4 MODUL- IV Specil ntenns-long wie V nd Rhombic ntenn Ygi-Ud ntenn Tunstile ntenn Helicl ntenn- il mode heli Noml mode heli Biconicl ntenn Log peiodic Dipole Spil ntenn Mitip Ptch ntenns. ntenn Mesuements- Rdition Ptten mesuement Gin nd Diectivit Mesuements nechoic Chmbe mesuement. DDITIONL MODUL (TRMINL XMINTION-INTRNL) Rdio wve popgtion- Clcultion of Get Cicle Distnce between n two points on eth Gound Wve Popgtion Fee-spce Popgtion Gound Reflection Sufce wves Diffction Wve popgtion in comple nvionments Topospheic Popgtion Topospheic Sctte. Ionospheic popgtion: Stuctue of ionosphee Sk wves skip distnce Vitul height Citicl fequenc MUF lecticl popeties of ionosphee ffects of eth s mgnetic fields Fd ottion Whistles. PL3I LCTROMGNTIC THORY (3 d Sem ) Module I (8 hous) Univesit Potion (8%): Co-odinte sstems & Tnsfomtion: Cte co-odintes cicul clindicl coodintes spheicl co-odintes. Vecto Clculus: Diffeentil length e & volume Line sufce nd volume Integls Del opeto Gdient of scl Divegence of vecto & divegence theoem cul of vecto & Stoke s theoem lplcin of scl (Tet Book : Chpte- Chpte-) College/Institute Potion (%): Field: Scl Field nd Vecto Field. O elted dvnced topics s decided b the concened fcult teching the subect. Module II ( hous) Univesit Potion (8%): lectosttic Fields: Coulomb s Lw lectic Field Intensit lectic Fields due to point line sufce nd volume chge lectic Flu Densit Guss s Lw Mwell s qution pplictions of Guss s Lw lectic Potentil Reltionship between nd V Mwell s qution n lectic Dipole & Flu Lines neg Densit in lectosttic Fields. lectosttic Bound Vlue

5 Poblems: Possion s & Lplce s qutions Uniqueness theoem Genel pocedues fo solving possion s o Lplce s qution. (Tetbook-: Chpte to 5.5) College/Institute Potion (%): Ntue of cuent nd cuent densit the eqution of continuit. O elted dvnced topics s decided b the concened fcult teching the subect. Module III (8 hous) Univesit Potion (8%): Mgntosttic Fields: Mgnetic Field Intensit Biot-Svt s Lw mpee s cicuit lw- Mwell qution pplictions of mpee s lw Mgnetic Flu Densit-Mwell s equtions. Mwell s eqution fo sttic fields Mgnetic Scl nd Vecto potentils. (Tetbook-: Chpte- 6. to 6.8) College/Institute Potion (%): ( hous) neg in Mgnetic Field O elted dvnced topics s decided b the concened fcult teching the subect. Module IV (7 hous) Univesit Potion (8%): lectomgnetic Fields nd Wve Popgtion: Fd s Lw Tnsfome & Motionl lectomgnetic Foces Displcement Cuent Mwell s qution in Finl foms Time Ving Potentils Time-Hmonic Field. lectomgnetic Wve Popgtion: Wve Popgtion in loss Dielectics Plne Wves in loss less Dielectics Powe & pointing vecto. (Tetbook-: Chpte-8. to 8.7 Ch.9. to 9.3 & 9.6) College/Institute Potion (%): Genel Wve qution Plne wve in dielectic medium fee spce conducting medium good conducto nd good dielectic Polition of wve. O elted dvnced topics s decided b the concened fcult teching the subect.

6 MODUL-I lement Coodinte Sstems We know tht the clindicl coodintes e given b. Bsicll the plne is tnsfomed into pol coodintes while the is emins the sme in both the clindicl nd the ectngul sstem of coodintes. We solve couple of numeicl to eplin the bsics of the thee othogonl coodinte sstems nmel ectngul clindicl nd spheicl... Relte the diffeentils in clindicl coodintes to the ectngul coodintes. tn.. Compute. in clindicl coodintes Soln: Substitution of the esults in.. in the bove epession gives us This is equl to... Theefoe

7 ..3. Tnsfom the eqution into pol coodintes. Soln: We know tn Hence. nd. tn Thus Thus.. Theefoe

8 Simill dding the two tems we get..4 Deive suitble epession fo in clindicl coodintes. Soln: We hve tn Now.

9 . Now 3 3. Substitution of this in the bove gives us...5 Fo spheicl wve the displcement is function of nd t whee is the mgnitude of the distnce fom fied point. Obtin genel eqution fo the spheicl wve. Soln: We need to show the following We undestnd tht

10 Now Let us go fo the second ode diffeentil which is obtined b diffeentiting the bove function once moe with espect to.thus 3 Simill we get fo the component 3 nd fo the component 3 Theefoe

11 Plne m cies sufce chge densit of nc/m. Wht is the electic field t the oigin? Soln: The unit noml to the plne of m is. s this is bove the oigin the unit noml fo the electic field t the oigin becomes. Thus the electic field stength becomes s V/m.7. Two point chges (Q=QQ=Q) nd n infinite gound plne e shown in the following figue. Find the tio of the foces epeienced b both Q nd Q. Soln: The gound plne cts like mio fo the two chges nd chges of opposite polit with sme mgnitude e induced on both sides of the plne. Hence the digm looks s the following Q Q d Q Q d Q Q

12 Q Q d d d Q Q d Foce epeienced b Q would be due to thee chges; Q Q t d. The net foce is F 4 Q 4 4d Q. 4 d Q. 4 d Q Q QQ Q d 3d d. Q Q.Q Q Q d d Q Q t d Q Q t 3d nd.8. The electic field stength t distnce point P due to point chge +q locted on the oigin is V/m. If the point chge is now enclosed b pefectl conducting metl sheet sphee whose cente is t the oigin compute the electic field stength outside the sphee. Soln: With conducting metl sheet suounding the point chge +q equl nd opposite chges e induced in inne nd oute sufces of the sphee. The net chge enclosed b Gus sufce of dius is gin +q. Thus D. ds q Q Q q D 4 q 4 Thus the electic field is sme with nd without the conducting sphee.

13 .9. stight cuent cing conducto nd two conducting loops nd B e shown in the following digm. Wht e the diections of the induced cuents in nd B if the cuent in the stight conducto is deceg? B Soln: The diection of the cuent s shown poduces mgnetic field in the nticlockwise diection tht encicles the stight wie. It becomes nticlockwise in B whees hs clockwise encicled mgnetic flu. If the cuent is deceg in the stight conducto the induced cuents in the loops would be in such diection so s to oppose this chnge. Hence B would get cuent in the clockwise diection nd would hve its induced cuent in the nticlockwise diection.. In spheicl coodintes egion is egion is b nd egion 3 is b. Regions nd 3 e fee spce while 5. Given B.ˆ find H in ech egion. Soln: We know tht the noml components e elted s H H 3H 3 Hence B. H H /m 4. 4 H.4 5 nd the coesponding mgnetic field in egion 3 is. H 3H 3 H H 3 /m.. Obtin the unit vecto long the diection of popgtion of wve the t 3 4 5t. displcement of which is given b Soln: The unit vecto long the diection of popgtion of the wve is coesponding to the plne 3 4 constnt Hence it becomes

14 ˆ ˆ ˆ ˆ 9 4 ˆ ˆ.. The displcement ssocited with thee-dimensionl wve is given b 3 t k k t. Find the diection of wve popgtion. Soln: The genelied wve eqution is t Re epk. t Fo the given poblem the two components of the vecto k e k k k nd the two components of the position vecto e The wve popgtes in the plne mking n ngle of 3 with the positive is nd 6 with the positive is..3 Find the foce pe unit e on the sufce of conducto with sufce chge densit in the pesence of n electic field n is the unit outwd noml to the sufce. Soln: The electic field t given point due to the sufce chge densit is n Foce pe unit e due to nothe sheet chge is F. q. n n.4 Inside ight cicul clinde. The eteio is fee spce. If B.5φˆ (T) inside the clinde detemine B ust outside. Soln: We know tht the tngentil component of H is continuous s the bound. Thus B B nd hence B φˆ.5 mt.5. In fee spce B B e m t ˆ. Find the coesponding electic field.

15 Soln: We know tht B B H t t t If we epnd the bove epession we get ˆ ˆ ˆ ˆ ˆ ˆ B B B B B B B B B In the bove epession we undestnd tht the mgnetic field hs n component onl nd futhe. Theefoe t m e B B nd ˆ ˆ t m t m e B e B s Substitution of the bove in the electic field epession we get ˆ ˆ ˆ t m t m t m e B e B e B This epession shows tht the electic field nd the mgnetic field H e in spce qudtue but in time phse. Both of the fields popgte with the sme velocit in the positive diection.

16 ..6 Find the electic field in the egion between the two cones s shown in Fig... V V V 6 Fig.. Figue fo..6 Soln: The potentil is obseved to be function of the imuthl ngle onl nd is constnt with espect to nd. Hence Lplce s eqution becomes d dv d d d dv d d dv K d dv K csc d V K ln csc cot The tem C csc cot. tn

17 C K K K V K d dv ln cot. ln ln ln cot ln csc csc. csc tn ln cot K K K V C K V V The potentil between the two cones equidistnt fom the oigin is theefoe ln cot... K d K K d K d V l Theefoe the solution to the potentil becomes C K C K V tn ln cot csc ln Futhe we obseve tht V V t nd V t. Theefoe C K V tn ln nd tn ln tn ln K C C K Fom these two we obtin

18 tn ln tn ln tn ln tn ln V K K K V The potentil is theefoe tn ln tn ln tn ln tn ln V V The electic field is evluted s.88 tn8 ln tn ln tn ln tn ln. sec cot ln tn V V V K K K V C K V V The genel electic field is evluted s. sec cot ln tn K K K C K V V Hence the totl electic flu bounded b the conicl sufce is Q KR d d K d D R S.. s The totl chge enclosed b the sufce. Theefoe the cpcitnce pe unit length is given s

19 Q K C V K lncot lncot The integl of csc is ln csc cot In spheicl coodintes unde the ssumption tht the potentil t given point is function of the dil distnce fom the souce then we hve V V V K V K K V C The two most impotnt theoems often used to evlute dited fields due to vious conducto configutions e Guss s divegence theoem nd Stoke s theoem. The fome s the nme suggests is elted to the divegence of vecto nd the ltte is elted to the cul of vecto. The divegence theoem sttes tht D. ds dv. Ddv S V This gives us. D V V Souce point ' ' ' R Obsevtion point Oigin ' ' Fig.. Pth diffeence between souce point nd n obsevtion point The potentil V due to continuous chge distibution contined in the volume V is epessed s

20 ' ' 4 dv R V V (B) whee ' ' ' ' d d d dv. The distnce R is computed s follows: ' ' ' ' ' ' ' ' ' ' ' ' R (c) We cn simplif the distnce b witing '... ' 3 ' ' ' ' ' m m m P R (D) We note tht the tem ' lso epesents the poection of ' onto nd theefoe cn be epessed s '.. Fig..3 n electic dipole tht mkes use of the pinciple outlined in Fig.. (n infinitesiml electic dipole) Two ppoimtions e usull cied out fo the f field of the ntenns. These e mde to the mplitude nd phse of the dited fields. The ppoimtions e: d q q P

21 R ' fo the phse tem nd R fo the mplitude tem The chge dipole hence m be epessed s d d q The f field potentil s obtined fom (D) m be epessed s 4V ' '. dv V q qd V d d ' d' d' d' This is due to the fct tht d nd d d d d d d This follows fom the fct tht the delt function ssumes vlue of unit onl t nd is eo t ll othe vlues of. Futhe the e bounded b dic delt function is equl to one. d This esult due to n electic dipole consisting of two fied sttic chges gives us the potentil t fw point. We m obseve tht the negtive of the gdient of the potentil gives us the electic field evluted s V V qd 3 4 V qd 3 4 V nd These fields e sttic nd the e constnts both in mgnitude s well s in diection t cetin distnce fom the souce. The lso v invesel s the cube of the distnce which mens tht t ell f distnces fom the souce thei mgnitude becomes too insignificnt to c eneg o powe. These cnnot ct s dited fields s equied in powe tnsfe b wieless methods. We must hve field tht vies t most s the invese of the squed distnce fom the souce. Moeove the field needs to be time ving so tht we get dited fields. lso mgnetic field is equied to c eneg

22 fom one point to nothe in wieless mnne. We tke up the genetion of the electomgnetic fields fist to undestnd this wieless tnsfe of eneg b electic nd ssocited mgnetic fields geneted due to time ving souces. Stoke s lw s elted to the cul of vecto sttes tht H. ds H. ds S The stndd wve equtions e V J nd V t t Unde the ssumption of usoidl sted stte the wve equtions become J V The mgnetic vecto potentil t given obsevtion point f w fom cuent souce simill cn be evluted with the id of Fig. () ecept tht the chge s souce should be eplced b cuent souce tht hs cuent densit of J /m estblished nd theefoe we cn simill wite J ' dv' 4 R V s we m obseve the distnce of the obsevtion point fom the souce R is given b (B). Tnsmission Lines The tnsit time effect is n impotnt issue in the chcteition of high fequenc behvio of electic cicuits. t low fequencies the wvelength is long nd the time peiod of given signl is lso long. This mens tht if souce in the fom of peiodic wvefom is pesent t one pticul point of the cicuit t given point in time it would be vilble on some diffeent point lmost t the sme time.

23 v B C D F G H G F D C B T t t We obseve tht the points mked -G equie finite mount of time t to ech the destintion end lso known s the lod. This is known s the etce time. The time peiod of the signl is T. We m see tht these two times e lmost compble. B the time point eches the destintion the souce signl chnges to point mked lod end. s the voltge goes to diffeent vlues between the souce nd the lod end which e some distnce pt we get phse diffeence between these two points. simil sitution holds good fo ll othe points. This is due to the fct tht the time to ech the lod nd the time peiod of the souce e compble. If we incese the fequenc of the souce we m nticipte tht the phse chnge would be moe. To descibe it mthemticll we define the electicl length of the cicuit s L whee L is the phsicl length between the souce nd the lod nd is the wvelength of the signl (voltge o cuent) impessed t the souce end. The qusi-sttic egime is defined fo L which is tpicl of low fequenc lumped cicuit pmetes. s we m see the wvelength of the signl is much longe thn the phsicl distnce between the souce nd the lod. Let us define the tnsit time s t L v. Hence in this cse L L t L v. T f t L T This simpl mens tht the signl impessed t the souce end eches the lod in ve smll mount of time. In othe wods the tnsit time is such tht duing this time the signl t the souce end does not chnge much s it hs longe time peiod s comped to the tnsit time.

24 Howeve s the fequenc of the souce inceses the time peiod gets coespondingl shote nd b the time one pticul point of the souce signl eches the lod end the souce signl chnges to diffeent vlue The signl mplitudes e t diffeent vlues between the souce nd the lod end nd this gives ise to the esonnce egion t chcteied b L ~. The tnsit time becomes n ppecible fction of the time T peiod which mens tht the souce nd the lod ends epeience diffeent mplitudes of the signl which gives ise to phse diffeence between them nd it becomes function of the phsicl distnce o the electicl length of the cicuit. Due to the phse diffeence between two diffeent points of the cicuit (obseved t the sme time of couse) the effect is distibution of fequenc sensitive elements like inductnce nd cpcitnce ove the length of the cicuit. s pcticl cicuits ehibit losses thee e esistnces nd conductnces s well distibuted thoughout the length of the cicuit. This gives ise to the so clled distibuted cicuit nlsis. V R L I I G CV These equtions cn be witten s V I R LI nd R LV Fo the lumped cicuit nlsis to hold good. Doing so we get V dv lim R LI nd simill d I di lim G CV d The voltge nd the cuents on tnsmission line e govened b two coupled fist ode diffeentil equtions s given bove. To solve those let us diffeentite both w..t. so tht we hve d V d d I d R L I nd G C V d d d d Substituting the epessions fo the fist ode diffeentils in these d V R LG CV d d I R LG CI d Let us wite R LG C So tht now the two equtions become d V V d nd d I I d

25 Both the equtions e govened b the sme second ode diffeentil equtions. s we might see the constnt is constnt t given fequenc. The solutions to the voltge nd cuent long the line e: V V ep V ep I I ep I ep The V V I I e bit constnts tht e to be evluted b ppopite bound conditions. These constnts e in genel comple nd thei phses epesent tempol phses with espect to some efeence time. s we m see in these epessions the time hmonic function is implicit. The totl solution is theefoe V V ep V ep ep t I ep I ep ep t I Fom these two epessions we see tht the epesent stnding wve on the line; one ep with n mplitude of V nd the othe goes long the positive -is given b long the negtive -is given b ep hving mplitude of V. simil condition holds fo the cuent flowing in the line. Wht we see is tht insted of tlking bout voltge nd cuent given point on the line now we hve two wves in two opposite diections with two diffeent pek mplitudes. The most impotnt obsevtion is tht both voltge nd cuent e functions of the given point in spce; is. This is unlike the low fequenc chcteition of voltge nd cuent whee both e functions of time onl. The e tempol functions. Howeve fo the tnsmission lines the voltge nd cuents e lso sptil functions. Theefoe we m guess tht the impednce is lso point specific; it is function of spce. Let us wite The constnt is known s the ttenution constnt of the line; s it ttenutes the wve s it popgtes long the line. The unit of is nepe/m. This mens tht if the voltge tvels length of m fom the souce it decs to vlue of e of its mplitude t the souce. The ttenution in db is theefoe loge db. To evlute the fou constnts we poceed s V ep V ep R LI ep I ep ep quting espective tems on both sides we hve fo the coefficient of V R LI nd simill fo the coefficient of ep V R LI V R L R L R L I R LG C G C nd V R L R L I The quntit G C R L hs the dimensions of impednce denoted s G C

26 Z R G L C This Z is known s the chcteistic impednce. The pmetes nd Z e known s the second pmetes of the tnsmission line. MODUL-II. GNRTION OF LCTROMGNTIC WVS n ntenn is stuctue usull mde fom good conducting mteil tht hs been designed to hve shpe nd sie such tht it will dite M powe in n efficient mnne. Time-ving cuents dite M wves. Thus n ntenn is stuctue on which time-ving cuents cn be ecited with eltivel lge mplitude when the ntenn is connected to suitble souce usull b mens of tnsmission line o wveguide. In ode to dite efficientl the minimum sie of the ntenn must be compble to the wvelength. geneic stuctue fo the genetion of the M wves is shown in Fig... Tnsmitting ntenn High Powe mplifie Fom modulto Tnsmission line/ Wveguide Spheicl wve fonts High fequenc feed voltge/cuent Fig.. Geneic Block Digm of genetion of M wves though low fequenc to high fequenc convesion t fequencies bove 4MH communiction is essentill limited to line-of-sight pths. tpicl LOS link is tht used fo TV bodcsting. nothe emple is the LOS micowve link used in the telephone sevice. In ode fo n ntenn to dite into smll ngul egion nd theeb povide highe concenttion of powe t the eceiving site it must be phsicll lge in tems of wvelength. In the micowve bnd whee the wvelength is in the nge of 3 to 3 cm lge eflecto ntenns with gins s lge s 4 to 5dB e quite common. With lge vilble gin the tnsmitte powe cn be educed ccodingl. It is not unusul to use tnsmitte powes of few wtts o even s

27 low s few milliwtts in the micowve bnd. Thee is lso less tmospheic noise t the highe fequencies so smlle signl levels cn be used. The wveguide is stuctue tht guides electomgnetic wves fom souce to destintion. The tnspot electomgnetic eneg ove long distnces with minimum mount of signl loss. The wveguides bsicll e of two tpes: metl bsed nd dielectic bsed. The fequenc nge of opetion of the metllic wveguides nge fom few tens of kh to few tens of GH. Beond these fequencies these wveguides hve ecessive losses nd become inefficient in the tnspottion of eneg. Dielectic wveguides on the othe hnd beond millimete wvelength opetion e used to tnspot electomgnetic eneg howeve in the fom of opticl signls o light wves. We hve eithe slb o clindicl wveguides. The slb wveguides e used in thin film pplictions nd integted opticl devices. The opticl fibes e clindicl wveguides tht hve been widel used in both communiction nd instumenttion pplictions due to numbe of dvntges. The inteested ede cn efe to stndd tets on opticl fibe bsed sstems fo n in depth knowledge bout the woking nd pplictions of the opticl fibes.. Vecto Potentil ppoch in foming field epession The vecto potentil set up b the infinitesiml cuent element t distnce fom the oigin of the coodintes sstem unde considetion is e P Idl (.) 4 We m obseve fom (.) tht epesents the wve numbe given s. Fig.. Reltionship between the unit vectos nd Fom Fig.. in spheicl coodintes Substitution of (.) in (.) we hve (.)

28 e P Idl (.3) 4 the vecto mgnetic potentil s defined bove gives coesponding mgnetic field in the egion outside of the cuent souce s H (.4) which gives the (.5) The vlidit of (.5) follows fom the fct tht cuent flowing though conducto long the positive is poduces mgnetic field in the fom of loops contined in the plne nd hs component onl. We theefoe hve fom element mgnetosttics I H (.6) Fom (. 3) we note tht e Idl nd 4 (.7) e Idl 4 (.7b) e Futhe Idl 4 (.8) e Thus Idl 4 (.9) e nd Idl 4 (.9b) Use of (.9) gives (.5) s e e Idl Idl Idl e (.) If cuent flows though the positive- diection we note tht (.) Unde the ssumption tht we wite (.) s (.) Idl Thus Idl If then Idl (.3) 4 4 This gives us

29 (.4) The esult given in (.4) gives the f field distibution fo most of the pcticl ntenns. The element mgnetic field due to n element cuent Idl flowing in the positive -diection t distnce fom the oigin is given s Idl R dh (.5) 4R s we note fom bove fo cuent in the positive -diection this becomes Idl R Id R dh (.6) 4R 4R Substitution of (.) in (.6) gives us Id Id Idl Id dh (.7) s fo the cuent flow such s this d dl (.7) lso gives the noted Biot-Svt lw in the spheicl coodinte. Fo cuent element Idl the phso mgnetic vecto potentil fom (.3) is Idl e (.8) 4 Diffeentiting (.8) w..t. we obtin Idl Idl Idl e e e (.9) Hence the mgnetic field becomes Idl H e (.) 4 We note fom (.) tht the mgnetic field hs component due to cuent flow in the -diection. The ssocited electic field is given s H dt (.) H H H (.) Fom (.) we undestnd tht the electic field hs n nd component. The component is obtined fom the fist tem in the RHS b tking its cul nd subsequent integtion of it with espect to time. This becomes then

30 H Idl 4 t v v t v v t 3 v t v t t t Idl v v v 3 4 v v (.3) The component of the electic field is obtined s the time integl of (.3). Thus Idl H dt (.4) 4 This is t t t Idl v v v (.5) 3 4 v v Let t t' (.6) v Then (.5) is epessed s Idl t' t' H (.7) 4 v H Idl t' t' Simill. dt dt. (.8) 4 v This becomes Idl t' t' Idl t' t'. dt (.9) 4 v 4 v 3 v

31 Fig..3 Schemtic illusttion of fomtion of wves The electic field lines between two conductos show ptten simil to tht of the voltge souce connected s them. Fo emple if the voltge is usoidl then the lines of foce would ehibit simil usoidl ptten in the spce between the two conductos nd this is illustted in Fig..3. The epetition of the positive nd negtive hlf ccles of the electic field tke plce t te equl to the fequenc of the pplied souce. The eltive stength of the field is indicted b the closeness o the fness of the lines in Fig.. nd the diection is shown b the ows. n upwd going ow indictes the positive hlf ccle nd negtive going ow indictes the negtive hlf ccle of the souce. The bunching togethe of the lines t cetin points show the occuence of positive nd negtive peks. s the electic field vies in time it lso genetes n ssocited mgnetic field s indicted b Mwell s lw. We theefoe hve both electic nd mgnetic fields inside the spce between the two conductos t the fled end of the conductos the wves see chnge in impednce s esult of which some of them e eflected nd tvel towd the souce. The intefeence of the fowd tveling wve nd the eflected wve would poduce stnding wve ptten which is not desible fom n ntenn point of view. This is simpl due to the fctt tht we would like the ntenn o the sstem of conductos to dite wves o mke possible onl fowd flow of eneg. Fom ll the bove discussions we undestnd tht fo cuent flow in the positive - diection the mgnetic field hs component whees the electic field hs components both long the nd the diection. This implies tht the electic field is contined in the plne which is pependicul to the mgnetic field oiented completel long the diection. One field gives ise to the othe nd hence the eist simultneousl. The H field s given in (.) hs two components; i.e. the fist tem being popotionl to invese of the distnce fom the oigin nd the second tem being popotionl to the sque of the invese distnce. Fo smll close to the cuent cing conducto the second tem pedomintes nd this is clled induction field. This field is lso the mgnetic field tht would be poduced b cuent of I t due to the ppliction of Biot-Svt lw ecept fo the fcto of t '. This field is function of t ' is eplined b the fct tht finite time fo popgtion is equied on the pt of the field. This cuses n eneg tht is stoed in the field duing one qute of ccle nd etuned to the cicuit duing the net. The fist tem is known s dition o distnt field nd domintes fo lge; i.e. t gete distnces fom the cuent element. This field is not pesent fo sted cuents. This esults fom the finite time of popgtion which is of no concen in the sted-field cse. This tem is esponsible fo flow of eneg w fom the souce. The two fields hve equl mplitudes t tht vlue of which mkes (.3) v v v which implies tht (.3) f 6

32 The mplitudes of the dition fields of n electic cuent element Idl e fom the second tem of (.5) Idl Idl 6Idl (.3) 4 v The fist tem of (.7) gives us the mplitude of the mgnetic field s Idl Idl nd H (.33) 4v These two eltions in(.3) nd (.33) show tht the mplitudes of the electic field nd the mgnetic field e in time phse but spce qudtue. The e elted b (.34) H which is constnt known s the intinsic impednce of the medium. Fo emple the intinsic impednce coesponding to fee spce is found to be Ω. Knowing one field will help in computing the othe. Fom the bove it is undestood tht mgnetic field is poduced onl when the cuent is chnging nd these two fields eist simultneousl. The two fields e in spce qudtue. sttic chge poduces sted electic field which does not chnge with espect to time nd hence cnnot genete n induced mgnetic field. Simill dc cuent flowing in nd ound conducto poduces sted mgnetic field tht emins constnt in mgnitude nd diection t given distnce fom the conducto. This kind of sted mgnetic field cnnot induce n electic field. We theefoe see tht sted electic nd mgnetic fields e independent of ech othe nd do not contibute to poducing wve of n kind.. Poof of the outwd eneg flow due to the dition tem The instntneous powe flow pe unit e t the point P is given b the Ponting vecto t tht point. The dil Ponting vecto is epessed s t't' t' t' t' I dl c c P H 6 t' t't' t' c c c t' t' t' I dl c c P (.35) 6 t' 3 c Hence the vege dil powe flow is given b the integl of the instntneous vlue ove one complete ccle. Doing so we undestnd fom bsic theo tht the fist second thid nd the fifth tem will ield eo. Hence we obtin I dl I dl P v... T (.36) 3 3 T 6 c 3 c

33 whee T is the time peiod of oscilltion of the cuent element ndt. The element e in spheicl coodintes is d the totl powe dited becomes powe = sufce P d v Idl 4c I dl 6c 3 I dl d 6c I dl c 3 d (.37) We note fom (.37) tht the dited powe is invesel popotionl to the sque of opeting fequenc nd cuent. If the cie fequenc inceses the powe dited into spce lso inceses s its sque. n incese in cuent gives ise to stonge mgnetic nd electic fields s is evident fom the epessions fo H nd. Hence the dition field is stonge fo highe cuents. In the bove epession the pek vlue of cuent hs been used. The effective o the RMS cuent is fo usoidl distibution I I ms (.38) Substitution of (.38) in (.37) ields I msdl Powe = 6c dl 8 I ms dl dl 8 I ms (.39) Fom (.39) it ppes tht the tem hs the ppence of esistnce s powe is I R. This esistnce is defined s the dition esistnce nd fo cuent element dl 8 R d (.4) In (.4) the dipole cuent is unifom. Howeve with no end loding the cuent must be eo t the ends nd if the dipole is shot the cuent tpes lmost linel fom mimum t the cente to eo t the ends with n vege vlue of of the mimum. We lso note tht fo lengths smll comped to wvelength the dition esistnce is dl ve smll. Fo emple if. then R d.8. The vlue of the dition esistnce is n indicto of the powe dited b the ntenn into spce. Thus this tpe of ntenn is not good dito of M powe into spce. pcticl ntenn such s n element dipole is cente-fed ntenn. Its length is ve shot s comped to wvelength t the cie fequenc. Fo the sme cuent I t the teminls the pcticl dipole of length l dites onl one-qute s much powe s the cuent element of the sme length which hs the cuent I thoughout its length. The dition esistnce of pcticl shot dipole is one-qute tht of the cuent element of the sme length. Tht is

34 l l R d shot dipole (.4).3 Rdition Resistnce of Loop ntenn The field dited b smll mgnetic dipole is the dul of tht dited b smll electic cuent dipole shot cuent filment. The souce point (the element cuent length) is ssumed to be in the plne hving coodinte of ' ' nd the obsevtion point is ssumed to be t. The contibution of the cuent filment of stength Id to the totl vecto potentil is found b the use of the eqution Idl d (.4) 4 ' Fig..4 loop in the plne cing cuent I tht poduces field t (.4) tkes the fom of I d ep Rd d' (.43) 4R This is due to the eson tht the element cuent length is in the plne. Howeve we m note immeditel fom Fig.. tht the contibution to the field due to the component will be cnceled b dimeticll opposite cuent element. Theefoe (.43) is simplified to I d ep Rd' (.44) 4R Howeve ' nd ' (.45)

35 Substitution of (.44) into (.43) gives us R I ' ' d ' e d (.46) 4R / whee R ' ' Method-I (.47) I Thus ep R 4 ' R ' d' (.48) We e pimil inteested in the f field so tht. It is futhe ssumed tht so tht the loop m be teted s point souce. In the spheicl coodinte sstem R ' ' ' ' ' ' '... ' s nd ' nd ' (.46) is ppoimted s R ' ' ' (.47) / s Hence substitution of (.49) into (.48) gives us I ' e e ' ' d 4 I e 4 ' e ' ' d' (.49) (.48) (.49) We futhe mke n ssumption tht. Hence the eponentil tem m be epnded in seies nd the highe odes in the epnsion m be neglected. We obtin I e ' ' d' 4 (.5) I e ' ' d' 4

36 We note tht the integtion of the fist tem in (.5) gives us eo nd tht of the second tem. Hence we obtin the component of the vecto potentil s I e (.5) 4R The quntit I is the poduct of the e of the cicul loop of dius nd the cuent flowing though is nd is known s the dipole moment of the smll cicul loop. Method-II The sme epession could hve been deived b nothe w in the following mnne. s befoe unde the ssumption of nd ug the seies epnsion of the eponentil tem we obtin I e ' ' ' d' 4R (.5) The integtion of the fist two tems in the bove epession gives us eo. Ug ' ' ' nd ug in the bove epession fo the second tem we obtin ' ' ' d ' (.53) I I Hence e e (.54) 4R 4R We lso wite R ' Hving obtined the vecto potentil we cn obtin the mgnetic field nd the coesponding electic field. I M H e e (.55) 4 4 The coesponding electic field is given s M H e (.56) 4 The dited powe is 4 M 3 P H Re H dd dd 4 M * ' (.57) 6 The dition esistnce of the loop m be found b equting I R 4 M I 4 = I I R to P. So we get (.58) Hence R (.59) Let us conside loop with dius of cm opeting t fequenc of MH nd substitution of these vlues into the bove epession gives us dition esistnce of

37 R Ω. So smll loop ntenn is ve poo dito. If N tuns of wie e used the dition esistnce is incesed b fcto of N. Smll loop ntenns e often used s eceiving ntenns fo potble dios. lthough the e ve inefficient the do give n cceptble pefomnce becuse of the lge vilble signl level. t low fequencies tmospheic noise is often the limiting fcto so moe efficient ntenn does not necessil give bette eception. Of couse smll loop ntenn would not be used fo tnsmitting puposes unless ve shot distnces wee involved nd the poo gin could be toleted. The gin of smll loop ntenn is ve low becuse the ohmic esistnce of the wie is genell much gete thn the dition esistnce...: Two diected Hetin dipoles e in phse nd distnce of d pt. The electic field intensit is given b Idl d e In deiving this epession use hs been mde of The coesponding mgnetic field stength is given s Idl d H e Thus the powe dited becomes Re * Idl Idl P vg H.. The dited powe is obtined b tking the integl of the bove ove n element sufce. Idl Idl 3 P d.. d d. Idl d d 8 Mking use of the integl n n n d d we note tht n n 3 d 3 3 d P d Idl. d. Idl.. Idl We mke use of in the bove to get P d Idl. 3 4 Idl

38 .4 DIRCTIVITY The bilit of n ntenn o n of ntenns to concentte the dited powe in given diection o convesel to bsob effectivel incident powe fom tht diection is specified viousl in tems of its gin powe gin diective gin o diectivit. The mgnetic field is obtined fom the mgnetic vecto potentil b tking its cul. In spheicl coodintes it is epessed s (.6) Unde the ssumption of the wve popgting long the dil diection fom the cuent souce we note tht the electic nd the mgnetic fields would lie in the plne. Theefoe (.6) educes to (.6) This is due to the fct tht the dil component does not eist o. s the mgnetic vecto potentil hs components constined to lie in the plne we cn lso epess (.6) s the following. Let us define vecto potentil T (.6) Ug the fct tht nd (.63) We m obseve tht T (.64) T (.65) Let us conside ve geneic epessions fo the two components of the mgnetic field vecto s the following: ep (.66)

39 nd simill ep (.66b) Theefoe the fist tem of (.6) becomes ep ep ep (.67) Simill ep ep ep (.67b) The dited field components coespond to the tems contining onl. Hence we etin the tem ep fom (.67) nd ep fom (.67b). The totl contibution to the mgnetic vecto potentil is B ep ep (.68) We hve mde use of (.67) in deiving (.68). We theefoe obtin the mgnetic field s T c f c B H (.69) whee we hve mde use of (.65). Futhe c (.7) Hence the mgnetic field cn lso be epessed s T T c H (.7) The electic field ssocited with (.7) is evluted s

40 c c H.. (.7) To evlute (.7) we note tht simil epession ppes in (.6). Hee the dil component is noneistent. Hence H (.73) We m note tht ep ep ep (.74) simil epession m be obtined fo. Combining these two togethe we hve n epession fo the dited electic field s T c c ep ep (.75) The geneic epessions fo the dited electic field nd the mgnetic field e given b (.75) nd (.7) espectivel when the wve popgtes long dil diection fom the souce. The powe dited pe unit e in n diection is given b the Ponting vecto P. Fo the distnt o dition field fo which the electic field nd the mgnetic field e othogonl in plne pependicul to the dius vecto nd fo which H v the powe flow pe unit e is given b H H P v v (.76) The dition intensit in given diection is defined s the powe pe unit solid ngle in tht diection. This tkes the fom of

41 P wtts/unit solid ngle (.77) v The dition intensit is independent of the dil distnce. The totl powe dited is W d wtts (.78) Since thee e 4 setdins in the totl solid ngle the vege powe dited pe W t unit solid ngle is. 4 The diective gin o the diectivit is defined s the tio of the dition intensit in tht diection to the vege dited powe. Tht is 4 4 G (.79) W v t Wt d 4 Fo cuent element Idl the distnt field in the diection of mimum dition is fom (.3) dl 6 I (.8) The cuent equied to dite wtt is theefoe fom (.39) I (.8) 8dl Substitution of (.8) in (.8) gives us coesponding field stength in the diection of mimum dition s 6 dl 6.. V/m (.8) 8dl 8 The dition intensit is 6 3 (.83) 8 8 so tht the diectivit o mimum gin of the cuent element is g d m 4. 5 (.84) The bove epession sttes tht fo cuent element the computed diectivit is.5. Net let us evlute the diectivit of hlf wve dipole b consideing its dited field epessions. The Ponting vecto powe densit fo the hlf-wve dipole is mimum t nd hence the effective o the RMS vlue of the electic field is 6I ms ms (.85) The dited powe is ms 6I ms P (.86) 3I ms The powe input to the dipole is R I I (.87) d ms 73 ms

42 Fom (.87) we obseve tht the dition esistnce of hlf wve dipole is 73Ω. Hence the diective gin of the hlf wve dipole is 3I ms 4. G.64 73I ms 73 (.88) We cn lso evlute the diectivit of hlf wve dipole in the following mnne. Fo hlf-wve dipole the mimum field stength is 6I m V/m (.89) The effective o the RMS vlue of the electic field is theefoe 6I ms ms (.9) The cuent equied to dite wtt is 73 mps. This cn be veified fom (.87). The coesponding field stength in the diection of mimum dition is obtined b substituting this vlue in (.9) nd this is theefoe (.9) The dition intensit is (.9) Hence its diectivit is 6 6 G m (.93) We net elte the electic field stength nd the effective dited powe. s the powe dited is popotionl to the sque of the RMS vlue of the cuent W Thus I ms (.94) 36.5 Substitution of the bove vlue of the RMS cuent in the field stength epession gives us 6 W W V/m (.95) 36.5 One mile = metes = Km The bove RMS field stength fo distnce of mile is (.96) W W mv/m Net we set up the vecto potentil due to tveling wve cuent distibution in the I I e m diection given b

43 MODUL-III 3. LINR NTNN line ntenn is ssumed to be mde up of lge numbe of ve shot conductos connected in seies. conducto hving length of is shown in Fig.. The two ends of the conducto e t opposite voltges. The RF eneg fom the tnsmitte is fed t its cente becuse the dipole ntenn is smmeticl ntenn in which the two ends e t equl potentils eltive to the mid-point. t the open-ends the cuent is eo nd the voltge is mimum. The dition ptten epesents the field stength in vious diections of n open ended hlf-wve ntenn. Fo the opened out hlf wve conducto the mgnetic field will be mimum long line etending fom the its cente nd electic fields will be pependicul to it. The hlf-wve dipole (Fig to be inseted) deives its nme fom the fct tht its length is hlf wvelength l. It consists of thin wie fed o ecited t its midpoint b voltge souce connected to the ntenn though tnsmission line. The field due to the dipole m be consideed to be consisting of chin of Hetin dipoles. The mgnetic vecto potentil t point P due to diffeentil length dl d of the dipole cing phso cuent I is d I I s ' e (3.) 4 ' usoidl cuent distibution hs been ssumed hee fo two esons. Fist it is due to the tnsmission line model of the dipole. Second the cuent must be eo t the ends of the dipole. Thus one m conside tingul distibution of the cuent lso. Howeve it is likel to ield less ccute esult. We wite ' (3.) o '

44 P I R Fig.3. n electic dipole The diffeence between nd ' is quite significnt. Hence the bove mgnetic vecto potentil is witten s I d e (3.3) 4 The totl mgnetic vecto potentil becomes I e e d (3.4) We undestnd tht e b b b e bd c (3.5) b Substitution of (3.5) in the totl mgnetic vecto potentil given s in (3.4) gives I e e b 4 (3.6) 4 4 Futhe ce o nd then 4 I e e e 4 (3.7) I e In this section we will deive genel epession fo the distnt electic field of dipole ntenn of n hlf-length H. The totl vecto potentil t point P is

45 4 H I m H H e I H R d 4 Becuse of the ppoimtion R we wite (.4) s H I H e H e m d d 4 H We mke use of the following integl to evlute the bove. e e bd b bb b In the fist tem in the RHS of (3.9) it is noted tht nd b Thus with this the fist tem in the RHS of (3.9) becomes. I m 4 I m 4 H H e I m e d 4 H H e m R H Simill the second tem in the RHS of (3.9) becomes nd b e d (3.8) (3.9) H H H I m H e I m e d H H 4 4 which is equl to I m H H H e 4 Combining the two we obtin I m I me H H H H 4 (3.) We ecll tht when the cuent is entiel in the diection H Diffeentition of (3.) w..t. the dil distnce gives us the epession fo the mgnetic field stength t distnt point s I me H H H (3.) whee onl the invese distnce tem hs been etined. The electic field stength coesponding to the dition field will be 6I me H H (3.) Fom this we note tht the electic field nd the mgnetic field e in time phse. The powe densit is epessed s H H

46 3 H H I H P m (3.3) esonnt ntenn coesponds to esonnt tnsmission line nd the dipole ntenn is esonnt ntenn. Such n ntenn m be viewed s n open-cicuited tnsmission line open-cicuited t the f end of the esonnt length i.e. multiple of qutewvelength so tht the length of the ntenn is multiple of hlf-wvelengths. The dition ptten is line dwn to oin points in spce which hve equl field intensit due to the souce. 3. NONRSONNT NTNN nonesonnt ntenn is like nonesonnt tnsmission line on which thee e no stnding wves. These e suppessed b the use of coect temintion to ensue tht no powe is eflected thus ensuing the pesence of tveling wve onl. In coectl mtched tnsmission line ll the tnsmitted powe is dissipted in the teminting esistnce. When n ntenn is teminted simil to tnsmission line bout two-thids of the input powe is dited the emining one thid is dissipted in the ntenn nd none is eflected to the input. n emple is Rhombic ntenn tht is used fo point-topoint woking in the HF nge spnning fequenc nge of 3-to 3-MH. It is bodbnd ntenn. Fo the teminted ntenns hving onl tveling wve distibution of cuent we deive net the ptten of these net. Fo distnt point P locted t distnce of R fom the oigin we note tht the element vecto potentil is epessed s d e R I d R m 4 R Fo the distnt field clcultion R Hence d e e I d m 4 Thus L m L m L e e I d e e I d. 4 4 vluting the integl we obtin L m L m e e I e e I We ecll tht when the cuent is entiel in the diection H Hence the mgnetic field stength in the pependicul diection is

47 I me L I m H. e.. e 4 4. This becomes fte simplifing I m. e L H e 4 The electic field intensit is epessed s 3I m. e H L L 4 Thus the mgnitude of the electic field is 3I m L It is obseved fom the bove epession tht with tveling wve the ptten is no longe smmeticl bout the degees plne but insted the dition tends to len in the diection of the cuent wve. The ngle between the is of the ntenn nd the diection of mimum dition becomes smlle s the ntenn nd the diection of mimum dition becomes smlle s the ntenn becomes longe. Fo the stnding wve cuent distibution the ptten is lws smmeticl bout the degees plne. 3.3 LOOP ND FRRIT ROD RCIVING NTNNS The loop ntenn is mde up of one o moe tuns of wie on fme which m be ectngul o cicul nd is ve much smlle thn one wvelength s. The ntenn is popul fo two esons: () it is eltivel compct lending itself to use with potble eceives; nd () it is quite diective lending itself to use with diection-finding equipment. loop ntenn mde of sevel tuns of wie ound ectngul fme wee popul fo elie model bodcst eceives with the loop being mounted in the bck of the cbinet. Recentl these hve been eplced b feite-od ntenns. When the loop is ligned fo mimum signl stength the mgnetic flu linkges e BN whee B is the ms mgnetic flu densit in Tesl is the phsicl loop e in sque mete nd N is the numbe of tuns. The induced emf is given b V S NB When the loop is tuned b mens of n etenl cpcito to the eceived fequenc the voltge t the cpcito teminls is mgnified b the qulit fcto Q of the cicuit. Hence the cpcito voltge becomes Vm VSQ QNB Since the loop is usull much smlle thn the eceived wvelength the induced voltge m be quite smll. It m be incesed b inceg n one of the fctos s shown bove. The Q is detemined b the desied selectivit. The e must be kept smll; inceg the numbe of tuns inceses the coil inductnce nd chnges Q nd even chnging the flu densit ffects the Q. Howeve chnging the flu densit b ug L

48 mgnetic coe cn be chieved with miniml chnge of Q ug feite coes. This hs been pefeed now. The loop ntenn is usull used s diection finding device. The feite-od ntenn is mde b winding coil of wie on feite od simil to the one shown in Fig. Feites e mteils tht ehibit the popeties of feomgnetism. The mteils ehibit high eltive pemebilit in the sme mnne s mgnetic mteils do but unlike the feomgnetic metls the lso hve high bulk esistivit. This mens tht t high fequencies edd cuents induced within the mteils e pcticll noneistent nd high-q coils cn be used. high length-to-dimete tio fo the od gives high pemebilit which is desible. The sie of the coil is compomise mong sevel fctos. If the coil is too long comped to the od length the chnge of pemebilit with tempetue will cuse noticeble chnge in the inductnce. If it is too shot the Q will be low. Positioning the coil on the coe is citicl s well ce the effective pemebilit is function of position of the od chnging fom mimum t the cente to minimum t eithe end. The coil is usull plced ne the qute-point llowing dustment in eithe diection to tim the coil inductnce. When moe thn one coil is mounted on the sme od the must be plced t opposite ends to minimie intection between them. The coil of wie on the feite od is bsicll modified loop ntenn so the induced mimum emf ppeing t its teminls is given b Vs BNF whee F is the modifing fcto ccounting fo coil length nging fom unit fo shot coils to bout.7fo one tht etends the full length of the od is the effective eltive pemebilit of the od s mesued fo the ctul coil position nd is the od ssectionl e Since the voltge ppeing t the teminls is of moe impotnce in eceiving ntenn the fcto Ql is often given s figue of meit fo od ntenns. The diectionl eff popeties of the feite-od ntenn e simil to those of the loop ntenn lthough the null m not be quite so ponounced. tem tht hs specil significnce fo the eceiving ntenns is its effective e (sometimes lso clled the effective petue). It is defined in tems of the diective gin of the ntenn though the eltion g d 4 The effective e is thus the tio of powe vilble t the ntenn teminls to the powe pe unit e of the ppopitel polied incident wve. Tht is W R P whee W R is the eceived powe nd P is the powe flow pe sque mete fo the incident wve. When the diectivit is used in the epession of the effective e it is ssumed tht ll of the vilble powe is deliveed to the lod. This is the cse fo pe cent efficient coectl mtched eceiving ntenn with the pope polition

49 chcteistics. Fo n effective field stength pllel to the ntenn the powe pe sque mete in the linel polied eceived wve is P wtts/sq m The powe is bsobed in popel mtched lod connected to the ntenn would be V l oc eff WR 4R 4R d d Fo cuent element l eff dl nd hence dl WR 4Rd Substitution of the vlue of the dition esistnce in the bove gives us W R dl dl. 4Rd 4.8 dl 3 Hence the mimum effective e is WR.5 P TLVISION NTNN The bsic ntenn fo TV tnsmission nd eception tht mke use of the VHF bnd of fequencies (3-3MH) is the hlf-wve dipole ntenn. This is clled esonnt ntenn s it gives out its best pefomnce onl t pticul fequenc eltive to its length. Fo ntenns close to eth veticll polied M wves ield bette SNR. Howeve when the ntenn is sevel wvelengths bove the gound hoiontll polied wves ield bette SNR. The TV signls e tnsmitted b spce wve popgtion nd hence the ntenn height must be s high s possible in ode to incese the line-of-sight distnce. Hoiontl polition is the stndd fo TV bodcsting. Hoiontl polition indictes the plne of the electic field. Such polition is pefeed becuse of the vilbilit of good signl-to-noise tio (SNR) when ntenns e plced quite high bove the sufce of eth. Fo mimum signl pick-up the eceiving ntenn should hve the sme polition s tht of the tnsmitted signl nd this is the eson wh the TV eceiving ntenns e ligned hoiontll. LONG WIR NTNNS In the fequenc bnd fom to 3 MH long wies (sevel wvelengths long) suppoted b suitble towes m be used s efficient ntenns. The best known tpes e the hoiontl V ntenn the hoiontl hombic ntenn the veticl V nd sloping hombic ntenns the veticl inveted V o hlf hombic ntenn nd the gle hoiontl-wie ntenn. Most long wie ntenns cn be opeted s esonnt ntenns in which cse the cuent on the wie will be stnding wve with the usoidl vition. These ntenns usull opete stisfctoil onl t pticul fequenc nd hmonics of this fequenc. The input impednce will be highl fequenc selective so onl now-bnd

50 opetion is possible. Most long-wie ntenns cn lso be opeted s tveling wve o nonesonnt stuctues b teminting the f end of the wie in suitble esistnce hving vlue equl to the chcteistic impednce of the ntenn viewed s tnsmission line. In this mode of opetion the useful fequenc bnd cn be quite lge with m cceptble impednce mtch ove the whole nge of fequencies. Vious tpes of long-wie ntenns e used fo commecil shotwve tnsmission in the fequenc nge fom to 3 MH when popgtion is b mens of inonospheic eflection. Fo these pplictions the optimum ngle of dition is usull fom to 3 eltive to the hoiontl line in the diection of eceiving sttion. Since the long-wie ntenns e locted in the pesence of gound the ltte hs n impotnt effect on the dition ptten nd must be tken into ccount in the design of ntenn configution. In genel the design poblem is one of obtining diective bem t the desied ngle eltive to the gound fo optimum long-distnce communiction vi eflection fom the ionosphee long with cceptble input impednce chcteistics tht will fcilitte mtching the ntenn to its feed line. monopole ntenn consists of one-hlf of dipole ntenn mounted bove the eth o gound plne. It is nomll one-qute-wvelength long ecept with spce estictions o othe fctos dictte shote length. The veticl monopole ntenn is used etensivel fo commecil M bodcsting (55-5KH) in pt becuse it is the shotest efficient ntenn to use t these long wvelengths ( to 6m) nd lso becuse veticl polition suffes less popgtion loss thn hoiontl polition does t these fequencies. The monopole ntenn is lso widel used fo the lnd mobilecommuniction sevice. Qute wvelength ntenns e widel used in mobile communictions with the vehicle itself poviding the equied gound plne. In the 7- MH citien bnd qute-wvelength monopole ntenn is.77m long. Mn CB bnd dio uses find n ntenn of this length undesible. Consequentl ntenns fo the CB dio e often onl to 5m long nd use eithe bse loding o cente loding to tune the ntenn to esonnce. The ovell efficienc will not be s get s fo the fulllength ntenn ce the dition esistnce is educed quite mkedl nd the unvoidble dissiptive losses in the tuning coil gound sceen nd the ntenn itself will consume significnt fction of the input powe. Popgtion below MH is mde possible b the sufce wves fo which the hoiontll polied wves e ttenuted much moe pidl thn veticll polied fields. Fo this eson hoiontll oiented long wie ntenns e nomll not used below MH. In the shotwve bnd fom to 3MH whee popgtion is vi ionospheic eflection long-wie ntenns e effective nd becuse of thei simple stuctues e commonl used. Rhombic nd V ntenns lso find some pplictions t fequencies fom 3 to 6 MH. t fequencies bove bout 3MH slot ntenns cut in metllic sufce such s the skin of n icft o the wll of wveguide often pove to be convenient ditos. The

51 slot m be fed b geneto o tnsmission line connected s it o in the cse of the wveguide b guided wve incident upon the slot. t ve high fequencies (3-3 GH) the sie of the diting elements become ve smll. It is then convenient to use the concept of cuent sheet ditos such s slot ntenns hons nd pboloid ntenns. The dition fom these kind of cuent sheet ditos cn be evluted fom the individul cuent elements ectl s we hve done fo the line ditos povided the cuent distibution is known o cn be estimted. Howeve in mn cses the cuent distibution is neithe known no cn be esil estimted. In tht cse method often used to compute the dited fields due to continuous cuent distibutions is the petue of the ntenn. The discone ntenn the helicl ntenn e used in the VHF-UHF nge. The fome is used to dite n omnidiectionl ptten with veticl polition. It is bodbnd ntenn with usble chcteistics ove fequenc nge of nel :. It is usull designed to be fed diectl fom 5-Ω coil line nd is mounted diectl on the end of tht line. This tpe of ntenn is idel fo bse-sttion opetion fo ubn mobile communiction sstems ce it gives good omnidiectionl ptten is phsicll ve compct nd ugged nd is quite inepensive to constuct. Its diectionl gin long the hoiontl plne is compble to tht of dipole ntenn. The log peiodic ntenn is bsicll n of dipoles fed with ltenting phse lined up long the is of dition. The element lengths nd thei spcing ll confom to tio given s L n X n Ln X n The ngle of divegence is given s Ln tn X n The open-end length L must be lge thn if high efficienc is to be obtined. The impednce of this ntenn is peiodic function of the logithm of the fequenc-hence its nme. The ntenn chcteistics e bodbnd nd it hs the diectionl chcteistics of dipole. This tpe of ntenn is often used fo mobile-bsesttion opetions whee mn chnnels must be hndled ove gle ntenn sstem with good diective chcteistics. 3.5 PRTUR NTNNS The litel mening of n petue is n opening o slot in closed sufce. n petue ntenn theefoe is mde out of closed sufce b mking smll opening o slot tht is mde to c time-ving cuent nd hence to dite. Computtion of the fields is not s stightfowd s fo the simple geomet ntenns tht we coveed peviousl. The electomgnetic fields in souce fee lossless egion e completel specified b the tngentil components of the electic nd the mgnetic fields on the sufce enclog the egion. s the egion is consideed to be souce fee the tngentil

52 fields on the sufce nd the fields inside the egion e poduced b souces etenl to the egion. s we m obseve lte the petue ntenn is nled b mking use of the tngentil components of the electic field nd the mgnetic field. We know tht s the bound of two medi the tngentil components of the electic field nd H. If the fields in plne petue e nd H then J S nˆ H nd M S nˆ whee J is the sufce cuent densit nd S M S is the mgnetic cuent densit. The concept of mgnetic cuent densit m sound bsud in the fist plce s it would equie the flow of mgnetic chges in closed pth nd this is not possible becuse of the fct tht mgnetic chges o monopoles do not eist. Howeve the concept of one would help nling the behvio of stuctues tht dite fom petues. s we hve seen peviousl the wie cuents e suitbl ecited b high fequenc cuents to poduce dited fields. Howeve we cnnot ppl the sme pinciple to n petue simpl becuse of the fct tht it is n opening o cvit (shpe does not mtte now) in n othewise closed sufce nd conventionl electic cuent cnnot flow in such n opened out stuctue. We theefoe need diffeent ppoch to evlute the dited fields due to n petue like stuctue mde in metllic sufce. This ppoch mkes use of Hugen s second wve pinciple following which the two epessions e witten. lge sque sufce of plne wve font cts like ectngul of Hugen s souces ll fed in phse. The dition ptten of the is theefoe obtined b multipling the unit ptten of the element b the fcto. The mgnetic vecto potentil is epessed s 4 J ' ' ' V ' S ep R dv' R We hve peviousl seen fom (.75) tht the electic field cn lso be evluted in tems of the mgnetic vecto potentil. Combined with the gdient of the scl potentil we hve V. H Simill we m define n electic vecto potentil due to the mgnetic cuent flow s the following: ep R F S ' ' ' dv' 4 M R V ' Hee V ' is the volume tht contins the mgnetic souce cuent m wite F Mking use of F F F The electic field in tems of the electic vecto potentil is epessed s M S. In simil lines we

53 F F F F F F F F ep ep nd the coesponding mgnetic field s F F H. The totl electic field due to the mgnetic vecto potentil electic vecto potentil mgnetic scl potentil nd electic scl potentil is epessed s F. nd simill F F H. Now we cn epess the dited field component due to ll the fctos combined togethe s F F Septing the nd the components we hve F nd F The coesponding mgnetic fields e epessed s H H (B) Net we wish to evlute the dited fields due to ectngul petue of sie b plced in the plne nd thee is n electic field long the is nd mgnetic field long the is. Futhe these fields e ssumed to be constnts ove the petue. Let nd H H. The mgnetic sufce cuent though the petue is S n ˆ M Simill the electic cuent densit though the petue is S H H n ˆ H J The electic vecto potentil is

54 ' ' ep. 4 ' ep 4 ' ep ' ' ' 4 ' ' ' d d R R dv R R dv R R V V V S M F () Hence the integl educes to computing the eponentil tem. We find tht little mnipultion of (.49) esults in the following simplified epession fo the distnt point locted R units w fom the oigin. Theefoe ' '... ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' / R Substituting this in () we hve GI d d d d d d R R S S S ' ' ' ' ep ep 4 ' ' ' ' ep. 4 ' ' ep. 4 F whee I epesents the integl ove the petue sufce nd G is the fmili scl spheicl wve function given s G 4 ep The integl I is evluted s

55 ep ' d' ep c Theefoe ' ep ep S ep ' ' d' d' b bs S whee S.The electic vecto potentil hs component long the is onl. In simil mnne we cn evlute the mgnetic vecto potentil lso b witing the following: ep R S ' ' ' dv' 4 J R 4 V ' H V ' ep R. H d' d' 4 R S This tuns out to be ep R. H d' d 4 R ep R dv' R S H GI We know fom element coodinte geomet tht nd F F ' F F Fom ll these bove discussions it is cle tht the mgnetic vecto potentil hs component long the is onl. We now combine these two esults nd wite the ovell dited fields s Theefoe

56 F F bg bg H GI H GI b S S b S S This is due to the fct tht H nd. nd c. nd futhe.this is equl to fte the substitution c c of the necess epessions tht b bg S S b b S S Simill F H F GI GI b GH S S b Gb S S Fom the epessions fo the electic field in the plne we obseve tht the mimum vlue of S is obtined fo. This coesponds to the cse of. The ssocited mgnetic fields e given s (B). In the f field egion the dition intensit is times the time-veged powe densit nd this is epessed s U Substitution of the necess epessions in gives us U (D) b b S S 4

57 Theefoe the mimum dition intensit is obtined b substituting in (D) nd hence we obtin The dited powe psg though the petue e is found out b evluting the Ponting vecto nd integting it ove the petue. Doing this we obtin s H d d U S. s nd H H the powe dited though the petue becomes... b b d H d S S s s H The diectivit educes to m 4 4 b b b D The epession fo diectivit shows tht tio of the diectivit to the petue e is constnt which is equl to 4 nd is independent of the ntenn sie. This depends onl on the wvelength o fequenc of opetion. Less the wvelength highe is the vlue. Howeve if the cuent distibution is not unifom then n equivlent unifom petue e is defined s the mimum effective petue e nd then the epession fo diectivit gets modified s e D m b b b b S S b U

58 whee e is the effective petue e. Fo unifom cuent distibution the effective e is the sme s tht of the ctul e of the petue. 3.6 Rectngul Wveguides ectngul wveguide is used to c eneg fom souce to destintion t micowve nd opticl fequencies. This is hollow metllic pipe with ectngul s section. It is ssumed tht the wlls of the wveguide hve infinite conductivit nd the medium filing up the spce is n idel dielectic with given pemebilit pemittivit nd no conductivit. The wveguide is ssumed to hve ectngul s section of dimension b nd is infinitel long. We m ssume hee tht b nd the bode dimension; is oiented long the is nd the shote dimension b is ligned long the is. The length of the wveguide is ligned long the is. We futhe ssume wve popgtion to tke plce long the positive is. b hollow ectngul wveguide with dimension b We m note tht the wve popgtion inside hollow stuctue such s the one shown in Fig. tkes plce in the fom of modes. It is no longe the unifom plne wves o the TM mode tht is possible in n unbounded homogeneous medium. s we m see the wlls of the wveguide e mde of good conducto while the hollow spce is filled with some kind of dielectic. Hence n wve popgting inside the spce hs to be govened b bound conditions long the two dimensions; nd the es. s we shll see ppliction of the bound conditions t the s section gives ise to modes fo the wves. mode is chcteied b n intege numbe of hlf usoids long the pticul dimension unde considetion. The simplest being ust hlf usoid tpicll known s the dominnt mode. The wves hve to be tnsvese electic (T) tnsvese mgnetic wves o some definite combintion of these two known s hbid modes. s we know the longitudinl electic field is eo while thee eists noneo longitudinl mgnetic field fo the T wves. Let us nle the T wves fist.

59 Tnsvese lectic (T) Wves s the mgnetic field hs noneo component long the becomes H H pnsion of this gives us H H Let H t X Y Z ep t Pefoming the double diffeentition we obtin d X d Y d Z YZ ZX XY XYZ d d d d X d Y d Z X d Y d Z d Dividing both sides b XYZ Now we cn equte ech of these tems to constnt s shown below d X d Y d Z B nd X d Y d Z d The diffeentil eqution becomes d X d Y X d Y d B h h B In the bove eqution we hve mde use of h We hve theefoe d X X d X C C Simill d Y B Y d is the wve eqution Y C3 B C4 B The complete solution of the longitudinl mgnetic field theefoe becomes t C C C B C B ep ep t H 3 4 We know tht the tngentil electic fields e mgnetic field H in the following mnne: nd which e elted to the

60 H h C C C 3B B C4B Bep ep t h nd simill H h C C C 3 B C4 Bep ep t h The wveguide wlls e metllic sufces bounded b to on the bode side nd to b on the nowe side. The tngentil components of the electic fields e eo t the wlls. Theefoe both nd e eo howeve t diffeent sufces. The field is eo t nd b while the is eo t nd Substituting these into the two tngentil electic fields we hve C obtined fo nd C fo. Simill t the ends of the s section we get C C C B Bbep ep t 4 h n B b Simill C C B C Bep ep t 3 4 h m The esultnt solution theefoe becomes n m n C C B ep ep t 4 h b b n m n K ep ep t h b b We hve epessed C C4 K Simill m m n C C ep ep t 4 h b m m n K ep ep t h b The tngentil mgnetic fields e epessed s 3

61 t b n m K m h t b n m K m h H h h H ep ep ep ep ep ep t b n m b n K h H h h H nd finll t b n m K t b n C m C t H ep ep ep ep 4 We m futhe veif tht b n m B h The integes m nd n togethe define the ode of the mode the wve tkes on to popgte down the wlls of the wveguide. The mode is designted s mn T mode. We obseve fom the epessions fo the tnsvese fields tht eithe of the integes m be eo fo these to be noneo s ech of the fou fields contin hlf coe tem given b m o b n. If both of the integes e eo then the longitudinl mgnetic field H becomes non eo howeve ll the tnsvese fields become eo. s we know mgnetic field vition in spce is not possible without coesponding electic field nd vice ves. Thus thee cn be no T mode. We cn hve s the lowest ode modes coesponding to eithe m giving ise to T mode o T mode coesponding to n. Pob: stblish the field eltions fom the fou bsic Mwell s equtions. Pob: stblish the dimensionl of h. Wht is its significnce? Pob: Sketch m nd b n mk ou obsevtions.

62 Tnsvese Mgnetic (TM) Wves Simil to the T wves the TM wves e chcteied b longitudinl electic field; nd the mgnetic field H. Theefoe the eqution govening the TM mode is epessed s pnsion of this gives us Let t X Y Z ep t Pefoming the double diffeentition we obtin d X d Y d Z YZ ZX XY XYZ d d d d X d Y d Z X d Y d Z d Dividing both sides b XYZ Now we cn equte ech of these tems to constnt s shown below d X d Y d Z B nd X d Y d Z d The diffeentil eqution becomes d X d Y X d Y d B h In the bove eqution we hve mde use of h We hve theefoe d X X d X C C Simill d Y B Y d Y C3 B C4 B The complete solution of the longitudinl electic field theefoe becomes t C C C B C B ep ep t () 3 4 The field is tngentil field long the is which is ssumed to be infinite in etent. lso we note tht t nd nd likewise t nd b Substitution of these in () gives us C nd futhe C 4. gin we hve

63 m nd b n B t the othe ends of the ectngul wll. The complete solution to the longitudinl field then is t b n m K t b n C m C t ep ep ep ep 3 In the bove we hve epessed K C C 3 The othe tnsvese fields e obtined s follows: t b n m K m h h ep ep t b n m K b n h h ep ep t b n m K b n h H h H ep ep nd t b n m K m h h H ep ep s befoe we note tht b n m B h The integes m nd n togethe define the ode of the mode the wve tkes on to popgte down the wlls of the wveguide. The mode is designted s mn TM mode. We obseve fom the epessions fo the tnsvese fields tht if eithe of the integes is eo then the fields become eo s ech of the fou fields contin hlf e tem given b m o b n. Thus thee cn be no TM mode. Simill s the lowest ode modes coesponding to eithe m giving ise to TM mode o TM mode coesponding to n e lso not possible. The lowest ode mode theefoe is ttined onl when both the integes e t lest equl to one n m. This is the lowest ode o the fundmentl mode of TM wve known s TM mode.

64 MODUL-IV ecise: numete the fundmentl diffeences between T wve nd TM wve. We obseve fom ll the bove discussions tht fo both the wves fields eist in discete pttens of hlf usoids o cousiods coesponding to onl intege vlues of m nd n. Theefoe m nd n cn tke on onl discete vlues nd not continuum of vlues. This mkes the field ptten fo ech of the sets of m nd n unique nd lso the fundmentl mode of popgtion. We cn lso see tht fo the guided wves othe thn the fundmentl mode ll othe modes if llowed to eist e the highe ode modes. This essentill mens n intege numbe of hlf ccles contined long the ppopite dimension. Fo emple if mode is given s T it simpl mens tht thee is one full ccle of field vition (two hlf ccles of ltenting polit) ech long the is nd the is. t this point we would like to find out which modes cn be suppoted b given wveguide. In ode to nswe tht we note tht b n m Theefoe b n m We note tht the b n m in ode fo to be el. nd fo give mode to eist in the wveguide the vlue of must be el. The fequenc t which chnges fom el to imgin is known s the cut off fequenc given s b n m f b n m c c (D) It follows fom (D) tht the fundmentl mode fo T wve hs the following cut off fequencies coesponding to T s T f c Simill b b T f c nd fo the TM wve the fundmentl mode hs cut off fequenc given s

65 b b TM f c s b theefoe we hve TM f T f T f c c c We theefoe see tht the lowest fequenc tht cn popgte inside ectngul wveguide coesponds to the T mode. No powe tnsfe is possible coesponding to fequenc lowe thn the cutoff fequenc. This is simpl due to the fct tht t fequencies lowe thn the cut off fequenc the popgtion constnt becomes imgin. Thus the tem ep ep does not epesent wve. In this spect then wveguide cn be thought of s high pss filte llowing fequencies bove the cut off fequenc. We cn futhe see tht s the ode of the mode inceses the cutoff fequenc simill inceses. The T is known s the dominnt mode of ectngul wveguide. Simil to the cutoff fequenc the cut off wvelength fo given mode is given s b n m b n m v v f v c c This is due to the fct tht the intinsic velocit of wve in wveguide is given s v. It follows fom the bove discussion tht the cutoff wvelength coesponding to the dominnt T mode is T T c c The longest wvelength tht cn popgte inside ectngul wveguide coesponds to the dominnt mode nd is equl to twice the sie of its bode dimension.

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