EE243 Advanced Electromagnetic Theory Lec # 13: Waveguides and sources
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1 Applied M Fall 6, Neureuher Lecure #3 er /8/6 43 Advanced lecromagneic Theor Lec # 3: Waveguides and sources Source Free Region: ecor Poenials A and F Single direcion componen of A and F Give TM and T Are Adequae Perfecl lecricall Conducing Guides Source ciaion via Reciproci Reading: Jacson Ch , 8. arringon 3., 3. Collin 4. Coprigh 6 Regens of Universi of California
2 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Overview A ver general T/TM separaion ino wo scalar problems is firs developed as a woring ool. This T/TM separaion wih respec o he (propagaion) direcion is applied o perfecl elecricall conducing waveguides. The waveguide modes, phase and group velociies are hen derived. General represenaions for fields in waveguides are hen developed. Reciproci is hen used o deermine he eciaion of a given waveguide mode. Coprigh 6 Regens of Universi of California
3 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Overview: T/TM Separaion In a homogeneous source free region, i is adequae in general o independenl solve for TWO SCALAR FUNCTIONS insead of 6 componens simulaneousl (3 componens of and 3 componens of B). These scalar funcions are he componens of he vecor poenials A and F in a common direcion. Under he Loren Gauge hese scalar funcions saisf he wave equaion and heir boundar condiions generall differ. The scalar componen of A gives fields ha are ransverse magneic and he scalar componen of F gives he fields ha are ransverse magneic (boh wih respec o he common direcion). Coprigh 6 Regens of Universi of California 3
4 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Magneic/lecric Duali lecric Sources arringon 3. Magneic Sources A 4π iωε iωµ A Je i J d 3 F iωε F 4π iωµ Me M i d 3 Dual equaions for problems in which onl an elecric source J or onl a magneic source M are presen. Coprigh 6 Regens of Universi of California 4
5 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Simulaneous lecric and Magneic Sources F iωε A iωµ ( A J ) ( F M ) A 4π i Je d 3 F 4π i Me d 3 Superimpose conribuions for he wo source pes. Coprigh 6 Regens of Universi of California 5
6 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Source Free Region arringon 3. Represenaion is sill valid F ( A) iωε in a source free regions (A and F are viewed as being ( ) produced b sources A F iωµ ouside of he region). Wih he Loren gauge A and F saisf he wave equaion and he fields are give b A F A F F iωµ A iωε A iωf iωµ ( A) ( F ) Coprigh 6 Regens of Universi of California 6
7 Coprigh 6 Regens of Universi of California 7 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Onl lecric Poenial in Direcion This conribuion is ransverse magneic (TM) o he direcion ˆ i i i A ωε ωε ωε
8 Coprigh 6 Regens of Universi of California 8 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Onl Magneic Poenial in Direcion This conribuion is ransverse elecric (T) o he direcion ωµ ωµ ωµ ˆ i i i F
9 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Toal Soluion The addiion of he TM and T soluions is sufficien o epress an arbirar field in a source free region. Furher, he choice of he direcion abou which o mae he T/TM can be an direcion. In emp waveguides and on dielecric slabs he T/TM separaion is ofen he direcion of propagaion. An ecepion is a pariall filled waveguide where he separaion direcions is perpendicular o he dielecric filling. The scalar funcion describing he T and TM soluions picall have differen boundar condiions Perfec elecric conducor Coprigh 6 Regens of Universi of California B n S S 9
10 Coprigh 6 Regens of Universi of California Applied M Fall 6, Neureuher Lecure #3 er /8/6 M for Uniform Guided Waves Guided waves have a uniform longiudinal phase variaion e i Facor he differenial operaors ino ransverse operaors and longiudinal algebraic mulipliers. φ φ φ φ φ φ φ φ φ ) ( ˆ ˆ ˆ ˆ ˆ ˆ i i i i Jacson 8.
11 Applied M Fall 6, Neureuher Lecure #3 er /8/6 M for Uniform Guided Waves (Con.) [ ( )] B µεω Jacson 8. ˆ ( ) [ ] ωˆ B µεω ( ) [ ] B µεωˆ µεω and B saisf wave equaion wih ransverse operaor and Brea up and B ino longiudinal and ransverse Transverse fields can be found from and B. Coprigh 6 Regens of Universi of California
12 Applied M Fall 6, Neureuher Lecure #3 er /8/6 TM Degenerac Occurs when boh and B are ero TM Soluion B maes Curl TM maes Div TM TM is a soluion o an elecrosaic problem in D. Consequences Propagaes wih of plane wave in free space B TM and TM are relaed as in plane wave in free space Can onl eis wih more han one conducor Coprigh 6 Regens of Universi of California
13 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Boundar Condiions n n B B n S S For a perfec elecric conducor here are no fields inside and an is coninuous For a perfec elecric conducor B is ero inside and B normal is coninuous Coprigh 6 Regens of Universi of California 3
14 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Z Z γ B TM T ± ˆ Z ωε i ± γ i ± γ µω µεω Waveguide Simplificaions B µ ε µ ε Firs find and B Then use gradien o find he associaed ransverse B or Use impedance o find he associaed ransverse and B Poeniall here are 5 erms each for T and TM bu ofen here are onl 3 erms each for he T and TM Coprigh 6 Regens of Universi of California 4
15 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Generic Wave Properies The soluion for he scalar funcion or B inroduces eigenvalues Phase propagaion in he direcion onl occurs once µεω eceeds he eigenvalue. Cuoff frequenc Phase veloci Group veloci Modes ha are cuooff are said o be evanescen due o heir eponenial deca Coprigh 6 Regens of Universi of California 5
16 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Recangular Waveguide ample (TM) b S i ± Z TM γ γ ± ˆ Z ωε a µ ε Coprigh 6 Regens of Universi of California γ mn mn m π a iπ γ a mπ nπ sin( )sin( ) a b cos iπ sin γ b mn Z mn Z TM TM iπ iπ γ n b γ mn mn () sin() () cos() a sin b () cos() cos() sin() 6
17 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Recangular Waveguide Orhonormal Modes The T and TM modes are orhogonal o each oher in an combinaion The T and TM modes are orhogonal amongs hem selves The can be normalied as has been don in Jacson 8. Coprigh 6 Regens of Universi of California 7
18 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Represenaion of an Arbirar Field The mos general field in a source free region is a summaion of T and TM modes If he source is o he lef hen onl fields propagaing o he righ need o be included FYI: ach mode b iself can be wrien as a sum of 4 planewaves epand he sinusoidal behavior in and as eponenials Compue he -vecor wave direcions Coprigh 6 Regens of Universi of California 8
19 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Fields Generaed b a Localied Source n n Source J To find ampliude of a given mode propagaing o he righ General epansion for modes o righ Assume he given mode of arbirar ampliude is propagaing o lef from ouside he boundar Appl reciproci o he volume for hese wo ses of fields Coprigh 6 Regens of Universi of California 9
20 Applied M Fall 6, Neureuher Lecure #3 er /8/6 Fields Generaed b a Localied Source (Con.) n n Source J Appl reciproci o he volume for hese wo ses of fields Inegral over wall is ero Inegral over lef cu-plane is ero as all modes going same direcion Inegral over righ cu plane is proporional o ougoing mode ampliude imes incoming mode ampliude Inergral over he source measures he componen of he source wih he, eigenfuncion variaion as well as phase coherence wih and is proporional o incoming mode ampliude Raio cancels incoming ampliude and gives ougoing ampliude. Coprigh 6 Regens of Universi of California
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