2/5/13. y H. Assume propagation in the positive z-direction: β β x

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1 /5/3 Retangular Waveguides Mawell s Equatins: = t jω assumed E = jωµ H E E = jωµ H E E = jωµ H E E = jωµ H H = jωε E H H = jωε E H H = jωε E H H = jωε E

2 /5/3 Assume prpagatin in the psitive -diretin: e jβ = jβ E + jβe = jωµ H E jβe = jωµ H E E = jωµ H H + jβ H = jωε E H jβ H = jωε E H H = jωε E 3 E ωµ E + jβe = jωµ H E = H + j β β E ωµ E jβe = jωµ H E = H + j β β E E = jωµ H H + jβ H = jωε E H jβ H = jωε E H H = jωε E 4

3 /5/3 E + jβe = jωµ H E jβe = jωµ H E E = jωµ H H ωε H + jβh = jωεe H = E + j β β H ωε H jβh = jωεe H = E + j β β H H = jωε E 5 E H ω µ ε β H = jωε jβ ( ) E H ω µ ε β H = jωε jβ ( ) E H ω µ ε β E = jβ + jωµ E H ω µ ε β E = jβ jωµ Let = ωµε β Knwn as the utff wavenumber 6 3

4 /5/3 E H H = j ωε β E H H = j ωε + β E H E = j β + ωµ E H E = j β + ωµ = ωµε β 7 Useful dempsitin: E = E e(, ) + e â e jβ H = H h(, ) + h â e jβ Transverse (Vetr) Mde Funtin Lngitudinal Cmpnent 8 4

5 /5/3 Transverse Eletrmagneti Wave (TEM wave): E =, H = E H H = j ωε β E H H = j ωε + β E H E = j β + ωµ E H E = j β + ωµ Pssible nl if = ωµε β = β =± ω µ ε 9 Als reall E + E = = ω µε E E + E + E + E = + E β E + E = E + E + = E = E + E = 5

6 /5/3 E + E = Laplae s Equatin (Eletrstatis) Sine E =, the E-field has E, E mpnents, H =, the H-field has H, H mpnents: define t = + then t E = similarl t H = Fr TEM waves, the transverse fields satisf Laplae s equatin just as in the eletrstati ase. TEM Waves (Transmissin Line Waves) Reall: E = Φ and i E = ρ hene: i E = i Φ with = Φ = = V, Φ( = ) = = A + B Φ = d Integrate twie: Φ Φ( ) = B = Φ( d) = V Ad = V A = V d Φ = V d, E = Φ = V d â, V H = η d â 6

7 /5/3 Transverse Eletri (TE) Waves: E =, H Transverse Magneti (TM) Waves: H =, E 3 Parallel Plate Waveguide w d ( w = ) d σ = ε, µ Can supprt TEM waves (alread disussed) as well as higher mdes (TM and TE waves). 4 7

8 /5/3 TM Waves: H =, E Assume = (fields d nt var in -diretin) E + E + E = E + E = = ωµε β e jβ Assumed E = Asin + Bs 5 E = ωε H H j β E = ωε H j β H + H = E = β E j ωµ H + E = E H E = j β + ωµ ωε E H = j β E E = j 6 8

9 /5/3 E = Asin + Bs ωε E H = j β E E = j ωε H = j As Bsin β E = j As Bsin 7 E = Asin + Bs Bundar nditins: E =, =, d, all B, d = nπ, n =,,, 3, Mdes 8 9

10 /5/3 = β, = ω µ ε β = β nπ = ω µ ε d nte: n= β = E = TEM 9 β = Fr < β is real, wave is prpagating as jβ β is pure imaginar, wave is evanesent r e e β = e α still lssless! Define the utff frequen as ω = β π f = = = β = = ω µ ε ω n =, f = = µε π µε d µε n, n, parallel plate guide

11 /5/3 Pnting Vetr: S = ( E H * )i â H *â = E â + E â i â = E H *â + E H *â i â * = E H Average pwer arried in -diretin: P = Re Crss Setin w S da d = Re E H * dd = Average pwer arried in -diretin: ωε H = j As d w βωε * P = Re s AA d w βωε = Re A d β E = j As s d nπ = d

12 /5/3 d d nπ, n d = d d, n= s w βωε d P = Re A wd ωε = A 4 Re β wd ωε,, β A f f β real n 4 wd ωε =,, β A f f β real n =, f < f ( β imaginar) 3 Z in ~ I in + V in Lssless Netwr If Z in = jx in, n average pwer is transferred int the b, but vltage and urrent are still present in the netwr.. 4

13 /5/3 Wave impedane (TM mdes) Z TM ωε E H = j β E E = j E = H β = ωε β, f f ( β real) ωε = β j, f < f ( β imaginar) ωε Remember, we are assuming a lssless waveguide. 5 An alternate view f waveguide prpagatin: Plane wave dempsitin E E H E π E = Asin e d jβ 6 3

14 /5/3 π E = Asin e d jβ π A β = e d e j π j j j jβ d Plane wave prpagating in the = π d â + βâ diretin. Plane wave prpagating in the = π d â + βâ diretin. The tw plane waves interfere (frm a null) at =, d. 7 = π d â + βâ = π d â + βâ π d θ β π = = sinθ d = β = sθ = ω µ ε v v v p p p ω = ω vp = = sinθ sinθ v ω vp = = sθ sθ p 8 4

15 /5/3 Lss plates Lss fatr: α = Pwer lss per unit length Pwer applied = P P Upper & lwer plate Time-average Surfae Resistane P = R s w = J s d where J s = H t =,d ωε ωε H(, d) = As( nπ ) = A 9 P = R s w = J s d = w A ωε R s Previusl fund wd ωε β A 4 = wd ωε β A n= P, n, P ωε ωε Rs α = = R, α = R = P βd βd ηd l n s s ωµ σ R s = 3 5

16 /5/3 Read abut TE mdes in the tet. 3 Retangular Waveguide b σ = TE Mdes: E H H H =, H ε, µ H = H H + β H + = H H H + + H = σ = a Wh H? = β, = ω µ ε 3 6

17 /5/3 Separatin f variables gives ( ) H (, ) = As + Bsin Cs + Dsin H H = j β H H = j β H E = j ωµ H E = j ωµ 33 ( ) H (, ) = As + Bsin Cs + Dsin ( As Bsin ) Cs Dsin + + H = j β ( As Bsin ) Cs Dsin + + H = j β ( As Bsin ) Cs Dsin + + E = j ωµ ( As Bsin ) Cs Dsin + + E = j ωµ 34 7

18 /5/3 ( ) H (, ) = As + Bsin Cs + Dsin H = j β A + B C + D ( sin s) s sin H j A B C D ( s sin ) sin s = β + + E j A B C D ( s sin ) sin s = ωµ + + E = j ωµ A + B C + D ( sin s) s sin 35 Bundar Cnditins: E =, =, b, all E =, =, a, all E j A B C D ( s sin ) sin s = ωµ + + E = j ωµ A + B C + D ( sin s) s sin E =, =, all E =, =, all E = j ωµ As + Bsin D = D ( ) ( ) E = j ωµ B Cs + Dsin = B 36 8

19 /5/3 s H = AC s H = j β AC sin s H j AC sin = β s E j AC sin = ωµ s E = j ωµ AC sin s 37 s H = H s H = j β H sin s H j H sin = β s E j H sin = ωµ s E = j ωµ H sin s 38 9

20 /5/3 E =, = b, all E =, = a, all ωµ nπ E b = jh ( ) ( b) = = b (, ) s sin ωµ mπ E ( a, ) = jh sin ( a) s( ) = = a As alwas + + = mπ nπ + + β = a b 39 TE mde Eletri Field Magneti Field 4

21 /5/3 TE mde Eletri Field Magneti Field 4 TE mde Eletri Field Magneti Field 4

22 /5/3 β mπ nπ = βmn, =± = ωµε a b If a > b, the first TE mde t prpagate is the ne fr whih m =, n =. This is the dminant mde (the TE mde). Wave Impedane: Z TE mn E E η = = = H H β m, n 43 Lss walls α = P P P = R s u J du P = Re E H * wavegiude rsssetin i â da b H Tp (,b) H Left Side (, ) H Right Side (a, ) H Bttm (,) a 44

23 /5/3 H Tp = H (,b)â + H (,b)â H Bttm = H (,)â + H (,)â H Left Side = H (, )â + H (, )â H Right Side = H (a, )â + H (a, )â nˆ Tp = aˆ nˆ = aˆ nˆ = aˆ Left Side Right Side nˆ Bttm = aˆ ˆn H = J s 45 nˆ Tp = aˆ nˆ = aˆ nˆ = aˆ Left Side Right Side nˆ Bttm = aˆ ˆn H = J s J Tp = â H (,b)â + H (,b)â J Bttm = â ( H (,)â + H (,)â ) J Left Side = â H (, )â + H (, )â J Right Side = â H (a, )â + H (a, )â 46 3

24 /5/3 J Tp = H (,b)â + H (,b)â J Bttm = H (,)â H (,)â J Left Side = H (, )â + H (, )â J Right Side = H (a, )â H (a, )â J Tp = H s( )s( b)â + j β H sin ( )s( b)â J Bttm = H s( )s( )â j β H sin ( )s( )â J Left Side = H s( )s( )â + j β H s ( )sin( )â J Right Side = H s( a)s( )â j β H s ( a )sin( )â 47 J Tp = H s b J Bttm = H J Left Side = H J Right Side = H s a s â + j β sin ( )â s( )â j β sin ( )â s( )â + j β sin ( )â s( )â j β sin ( )â 48 4

25 /5/3 J Tp = J Bttm = H J Left Side = J Right Side = H s s + β sin ( ) β sin ( ) + P = a a b b R J s Tp d + J Bttm d + J Left d + J Right d P = H Rs m,n s ( ) + β a b sin ( ) d + s ( ) d 49 P = a a b b R J s Tp d + J Bttm d + J Left d + J Right d P = H Rs m,n s ( ) + β a b sin ( ) d + s ( ) d Fr the dminant TE mde m =, n = : π π π π π = m =, = m + n = a a a b a P = H, P = H, R s R s a + β + b a + β a π + b 5 5

26 /5/3 P = Re E H * P = Re P = wavegiude rsssetin a H m,n b = = i â da E H * * E H i â d d a b βωµ sin ( )s = = Fr the dminant TE mde m =, n = : d d P 3 βωµ ab H ab π H = = 4,, βωµ 5 α, = P P = α, = P P = π a 3 R a 3 bβωµ s a + β π + b β = β, = ω µ ε π a a 3 bβη R s a 3 + π b 5 6

27 /5/3 TM mdes (see tet): ( ) E (, ) = As + Bsin Cs + Dsin E (, ) = E (,) = A= C sin E (, ) = BDsin sin = E sin mπ nπ E( a, ) = E(, b) = =, =, m, n a b 53 E (, ) = E sin sin ωε E H = j ωε E H = j β E E = j β E E = j = ωµε β 54 7

28 /5/3 ( ) ωε sin ωε H j E j E ( ) = sin sin s = ωε sin ωε H = j E sin = j E s sin β s β E = j E sin = j E s sin β sin β E j E j E = sin sin s = 55 Partiall filled waveguide b ε, µ t σ = ε, µ a σ = Cnsider the TE m mdes. = 56 8

29 /5/3 b Pieewise hmgeneus: In regin : In regin : ε, µ t σ = ε, µ a σ = H t + =, H t a + =, = ωµε β = β same β! = ωµε β = β 57 As befre: H = As( ) + Bsin( ), t H = Cs( ) + Dsin( ), t a H H = j β H H = j β H E = j ωµ H = ωµ E j = = 58 9

30 /5/3 H = As( ) + Bsin( ) β H = j ( Asin( ) + s), B t ωµ E = ( sin + s) j A B H = Cs( ) + Dsin( ) β H = j ( Csin( ) + s) D, ωµ E = j ( Csin( ) + s) D t a 59 Bundar nditins: E =, = B E =, = a Csin( a) + Ds( a) = D = Ctan( a) H = H, = t As( t) Cs( ) Dsin( t) = E = E, sin ( sin s) = t A + + = t C t D t As( t) Cs( t) Ctan( a)sin( t) = Asin( t) + ( Csin( ) + tans) = t C a t 6 3

31 /5/3 s( t ) s( t ) tan( a )sin( t) A = sin sin + tans t t a t C ( + ) s( t) sin( t) tan( a)s( t) sin( t) s( t) tan( a)sin( t) = tan( t) tan( a) tan( t) = + tan tan a t tan( t) tan + = a t 6 Eigenvalue Equatin fr TE m mdes: tan( t) tan + = a t = ωµε β = β = ωµε β = β Eigenvalue Equatin Charateristi Equatin Dispersin Relatin Fr anther urse 6 3