Field and Wave Electromagnetic

Transcription

1 Field and Wave Eletromagneti Chapter Waveguides and Cavit Resonators Introdution () * Waveguide - TEM waves are not the onl mode o guided waves - The three tpes o transmission lines (parallel-plate, two-wire, and oaial) are not the onl possile wave-guiding struture. - Attenuation onstant α or loss line C L α ( R + G ) R L C Rs π μ R = ( ) = w w σ Attenuation o TEM waves tends to inrease monotoniall with requen prohiitivel high in the mirowave range. Eletromagneti Theor

2 Introdution () - TEM waves : E = H = - TM waves : transverse magneti waves ( H =, E ) - TE waves : transverse eletri(te) waves - single ondutor wave guide : retangular and lindrial wave guide. - dieletri-sla waveguide : surae waves Eletromagneti Theor 3 General Wave Behaviors along Uniorm Guiding Strutures () * General wave ehaviors along uniorm g uiding strutures - straight guiding strutures with a uniorm ross setion. - Assume that the waves propagate in the + diretion with a propagation onstant γ = α + jβ - For harmoni time dependene on and t or all ield omponents : e e = e = e e γ j ω t ( j ω t γ ) α j( ω t β ) Eletromagneti Theor 4

3 General Wave Behaviors along Uniorm Guiding Strutures () - For a osine reerene Et ee e ( jωt γ) (,, ; ) =R [ (, ) ] where E (, ): two-dimensional vetor phasor - jw, γ, in using a phasor representation t - In the harge-ree dieletri region inside, Helmholt's equations should e satisied E+ k E = where H + k H =, k = ( ) E ( ) E = + = ω με - In Cartesian oordinates, retangular wave guide + E = E+ γ E Eletromagneti Theor 5 General Wave Behaviors along Uniorm Guiding Strutures (3) E+ ( γ + k ) E = H + ( γ + k ) H = ) E+ ( γ + k ) E = ( E + E + E ) + ( γ + k )( E + E + E ) = i.e E + ( γ + k ) E =, E + ( γ + k ) E = E + ( γ + k ) E = The solution o aove equations depends on the ross-setional geometr and the oundar onditions ) instead o or waveguides with a irular ross setion rφ Eletromagneti Theor 6

4 General Wave Behaviors along Uniorm Guiding Strutures (4) - Interrelationships among the si omponents in Cartesian oordinates E = jωμh H = jωε E E 4 + γe = jωμh + γh = jωεe E γ E E 3 6 = jωμh 5 γ H = jωε E E = jωμh = jωε E ) E E jωμh E E = jωμh = j H ωμ E E E E jωμh E γ γ where and is surpressed e Eletromagneti Theor 7 General Wave Behaviors along Uniorm Guiding Strutures (5) - Transverse omponent an e epressed in terms o longitudinal omponents. E) Comining and 5 : E + γe = jωμh 5 : γh = j E eliminating E rom,5 ' : ωε E jωε + γ jωε E = ω με H 5' : γ γ = γ ωε E ( k + γ ) H = jωε γ H j E E H = ( j + ) h = k h ωε γ where + γ Eletromagneti Theor 8

5 General Wave Behaviors along Uniorm Guiding Strutures (6) - i.e H H E E = + h ( jωε γ ) E = + + h ( jωε γ ) E = + h ( jωμ γ ) E E ( j ) h k h = ωμ + γ where = + γ note First, solve E+ h E = and H + h H = then ind H, H, E, E or longitudinal omponents using aove equation Eletromagneti Theor 9 TEM Waves () * TEM wave - E =, H = or TEM waves E = E = H = H = h = unless - TEM waves eist onl where or γ TEM TEM + k = γ = jk = jω με : propagation onstant o a uniorm plane wave on a lossless transmission line. - Phase veloit μ ptem ( ) ω = = k με Eletromagneti Theor

6 TEM Waves () E jωμ γ ΤΕΜ - Wave impedane ZTEM = = = rom,4 H γ jωε μ = = η ε ΤΕΜ note Z Z TEM TEM is the same as the intrinsi impedane o the dieletri medium E μ = = H ε H = E Z TEM Eletromagneti Theor TEM Waves (3) * Single-ondutor waveguides annot support TEM waves. Wh?. B lines alwas lose upon themselves. For TEM waves to eist, B and H lines would orm losed loops in a transverse plane. 3. B the Ampere's iruital law. Hdl i = I + I I I d d transverse plane : ondutor urrent : displaement urrent 4. Without an inner ondutor I = Eletromagneti Theor

7 TEM Waves (4) 5. For TEM wave, E = no longitudinal displaement urrent ) I d D = ids = in the diretion s t E = 6. Thereore there an e no losed loops o magneti ield lines in an transverse plane 7. Assuming peret ondutors, a oaial transmission line having an inner ondutor an support TEM waves 8. When the ondutors have losses, no longer TEM waves Eletromagneti Theor 3 TM / TE Waves () For TM waves H = For TE waves E = solving solving E + h E = or E H H E E =. jωε E h jωε E = h γ E = h γ E = h with proper or H oundar onditions oundar onditions : : : 3 : 4 H + h H = H H E E. γ = h γ = h = = jωμ h jωμ h with proper : 5 : 6 : 7 : 8 Eletromagneti Theor 4

8 TM / TE Waves () γ ( E ˆ ˆ T ) TM = E + E = TE h E, E are given, H, H an e determined rom the wave impedane or the TM mode rom 3, and 4, E E γ Z TM = = = H H jωε jωμ ) ZTM γ γ or TM is not equal to jω με, whih is γ H = ( ˆ E) Z TM TEM γ ( H ) = H ˆ + H ˆ = H h similar wa, E, E an e T TE T otained rom, H E = ZTE ( ˆ H ) = Z ( H ˆ ) TE H rom 67, and 58, E E jωμ Z TE = = = H H γ γ ) ZTE ZTM = jωε Eletromagneti Theor 5 TM / TE Waves (3) solution o E + h E = or a given oundar ondition are possile onl or disrete values o h ininit o h's ut solutions are not possile or all values o h eigenvalues or harateristi values h = γ + k γ = h k = h ω με H + h H = eigen values Eletromagneti Theor 6

9 TM / TE Waves (4) or γ =, ω με = h h = : uto requen π με ) The value o or a partiular mode in a waveguide depends on the eigenvalue o this mode H + h H = eigen values γ = h ( ) Eletromagneti Theor 7 TM / TE Waves (5) γ = h ( ) (a) ( ) > or > in this range, > and is imaginar h γ = jβ = jk ( ) k = jk ( ) ωμε propagation mode with a phase onstant β h γ β = k ( ) (rad/m) Eletromagneti Theor 8

10 TM / TE Waves (6) - Guided wavelength π λ π λg = = > λ where λ = = = β k με ( ) let uto wavelength, λ = then ( ( = ) ) g λ λ = ( ) = g u + = λ λ λ g u λ λ λ λ u Eletromagneti Theor 9 TM / TE Waves (7) - Phase veloit u g u = ω λ p u u β = = λ > ( ) ). Phase veloit o guided wave is aster than that o unounded medium.. Phase veloit depends on requen so that single ondutor waveguides are dispersive transmission sstems Eletromagneti Theor

11 TM / TE Waves (8) - Group veloit λ = = = < ug u ( ) u u dβ/ dω λg uu = u g p ) dk d π με dβ = = dω dω d( π ) ( ( ) [ ( ) ] d = ( ) d u = u ( ) Eletromagneti Theor - Z TM jk γ = = jωε ( ) jωε TM / TE Waves (9) μ = ( ) = η ( ) ε ; purel resistive and less than the intrinsi impedane o the dieletri medium - Z TE jωμ = = γ jk jωμ ( ) μ η = = ε ( ) ( ) ; purel resistive larger than the intrinsi impedane o the dieletri medium Eletromagneti Theor

12 TM / TE Waves () () ( ) < or < γ = α = h ( ) e e γ α = to e evanesent waveguide : high-pass ilter : real numer wave diminishes rapidl with and is said h ( ) γ h ZTM = = = j ( ), < jωε jωε ωε purel reative no power low assoiated with evanesent mode Z TE jωμ = = γ j h ωμ ( ) : purel reative. no power low. Eletromagneti Theor 3 TM / TE Waves () - ω β diagram ω ω β = ( ) u ω Eletromagneti Theor 4

13 Parallel-Plate Waveguide () - Parallel plate waveguide an support TM and TE waves as well as TEM waves * TM waves etween parallel plates. Assuming ε and μ. Ininite in etent in the - diretion 3. TM waves ( H = ) 4. ( j t ) e ω γ Eletromagneti Theor 5 Parallel-Plate Waveguide () E E e (no variation along γ (, ) = ( ) - diretion) d E ( ) + he( ) = d where B.C. E ( ) = at = and = nπ nπ E( ) = Ansin( ) rom h= where A depends on the strength o eitation o n h = γ + k the partiular TM wave Eletromagneti Theor 6

14 Parallel-Plate Waveguide (3) jωε E jωε nπ H ( ) = = A os( ) h h n jωε E H ( ) = = h γ E E ( ) = = h γ E γ nπ E ( ) = = A os( ) h h n nπ γ = ( ) ω με Cuto requen that makes γ = = n με Eletromagneti Theor 7 Parallel-Plate Waveguide (4) ) = or TM mode with n= με = or TM mode with n= με ) TM mode is the TEM mode with E = = - Dominant mode o the waveguide = the mode having the lowest uto requen - For parallel plate waveguides, the dominant mode is the TEM mode Eletromagneti Theor 8

15 E. -3() ) Field line : the diretion o the ield in spae i.e. dl = d ˆ + d ˆ + d ˆ = ke = k( E ˆ + E ˆ + E ˆ ) d d d = = = k E E E ield line Eletromagneti Theor 9 E. -3() d E (, ; t = ) in the plane = d E (, ; t = ) For TM mode at t=, ωε π H (, ;) = A os( )sinβ π At = and = - There are surae urrents eause o a disontinuit in the tangential magneti ield. - There are surae harges eause o the presene o a normal eletri ield Eletromagneti Theor 3

16 E. -4 () (a) A propagating TM wave = the superposition o two plane waves ouning ak and orth oliquel etween the two onduting plates proo> A E A e e e j A ( / ) ( / ) [ e j β π e j β+ π = ] j π jβ jπ/ jπ/ (, ) = sin( ) = ( ) e jβ Eletromagneti Theor 3 E. -4 () Term : A plane wave propagating oliquel in the + and π diretions with phase onstants β and Term : A plane wave propagating oliquel in the + and + diretions with the same phase onstants H = H ˆ E = E ˆ sinθ E ˆ osθ E = E ˆ sinθ E ˆ osθ i i i i i r r i r i = E ˆ sinθ + E ˆ osθ i i i i β = ˆβ osθ + ˆβ sinθ β = ˆβ osθ + ˆβ sinθ i i i r i i Eletromagneti Theor 3

17 E. -4 (3) E (, ) = E os θ ( e e ) e i i π βsin θi = β, βosθi = jβ osθ jβ osθ jβ sinθ i i i π π β = β ( ) = ω με ( ) π λ osθi = = solution eists onl or λ β λ u at =, = = uto requen λ με then θ = i waves oune ak and orth in the - diretion and no propagation in the - diretion TM mode propagates onl when λ < λ = or >. λ λ u osθi = = sinθi = = = ( ) λ λ u g p Eletromagneti Theor 33 TE Waves etween Parallel Plates () * TE waves E =, = d H ( ) + hh ( ) = d We note that H = H e (, ) ( ) γ jωμ B.C. E = = h dh ( ) i.e = at = and = d nπ H( ) = Bnos( ) Eletromagneti Theor 34

18 TE Waves etween Parallel Plates () γ H ( ) = = ( = ) Z h γ γ nπ H ( ) = = B sin( ) h h n jωμ jωμ nπ E ( ) = = B sin( ) h h n jωμ E ( ) = = ( = ) Z h nπ γ = h k = ( ) ω με the same as that or TM waves The uto requen is the same For n=, H = and E = Eletromagneti Theor 35 TE Waves etween Parallel Plates (3) i.e, TE mode doesn't eist ) TM = TEM ) TM or TM ) TM or TM does not eist mπ mπ E (, ) = E sin( )sin( ) a does eist or the retangular waveguide mπ mπ H (, ) = H os( )os( ) a Eletromagneti Theor 36

19 Energ-transport Veloit () * Energ-transport veloit - Wave guide high pass ilter - Broadand signal. low requen omponents ma e elow uto. high requen omponents will travel widel dierent veloit - Energ-transport veloit : veloit at whih energ propagates along a waveguide u en ( P) = W av av (m/s) ( P) = P ids av s av : the time average power Eletromagneti Theor 37 Energ-transport Veloit () W av = [( we ) av + ( wm ) av ] ds : the time average stored u en s = u ( ) energ per unit length [ H.W] prove that ε * ( we) av= ReEE ( i ) 4 μ * ( wm) av= Re( HiH ) 4 Eletromagneti Theor 38

20 Energ-transport Veloit (3) ε nπ β nπ ( we) av= An[sin ( ) + os ( )] 4 h * ) ( EE i ) jβ i( jβ) = β ε β ε ( w ) d A [ ] k A 8 h 8h e av = n + = n ( wm) av= ( ) A os ( ) n μ ω ε nπ 4 h μ ε ( w ) d = ( ωε ) A = k A 8h 8h m av n n ( P) = P id ˆ av av = ωεβ A h nπ ωεβ os ( ) d = 4h n A n Eletromagneti Theor 39 Energ-transport Veloit (4) * ) Pav = R e( E H ) = Re E ˆ H + E ˆ H * ( Pav) i = Re( E H ) ωεβ nπ = A os ( ) n h * * ( ) u en ωβ ω β = = ( ) = u ( ) k k k Eletromagneti Theor 4

21 Attenuation in Parallel-plate Waveguides () * Attenuation in parallel-plate waveguide - Losses are ver small - α = α + α d Ohmi losses Dieletri losses For TEM mode ) For a loss transmission line the time-average power loss per unit length V PL ( ) = [ I ( ) R+ V ( ) G ] = ( R+ G Z ) e Z α V P ( ) = R ev [ ( I ) ( )] = Re Z * α Eletromagneti Theor 4 Attenuation in Parallel-plate Waveguides () P ( ) = PL ( ) = α P( ) P α = = + ( ) L ( ) ( R G Z ) P R G σ μ σ αd = R = = η ( Z R or low loss ondutor) ε ω G = σ where independent o requen R = η ω Eletromagneti Theor 4

22 Attenuation in Parallel-plate Waveguides (3) R π ε α = = R σ μ ) R = η = ω ω ε R = ω π μ σ For TM mode to ind dieletri losses, α σ - εd = ε + ( ) jω d at > Eletromagneti Theor 43 Attenuation in Parallel-plate Waveguides (4) jσ nπ γ = j[ ω με( ) ( ) ] ωε / nπ nπ = j ωμε ( ) { jωμσωμε [ ( ) ] } / nπ jωμσ nπ j ωμε ( ) { [ ωμε ( ) ] } nπ Assumption that ωμσ ω με ( ) Eletromagneti Theor 44

23 Attenuation in Parallel-plate Waveguides (5) For uto requen nπ = π nπ ω ωμε ( ) = ω με ( ) ω = ω με ( ) σ μ γ = αd + jβ = + jω με ( ) ε ( ) με Eletromagneti Theor 45 Attenuation in Parallel-plate Waveguides (6) We otain and ση αd = ( ) β = ω με ( ) dereases when requen inreases To ind α PL ( ) α = P ( ) * P ( ) = w ( E)( ) H d wωεβ A nπ A = ( ) os ( ) d = w ( ) π π n n n ωεβ n Eletromagneti Theor 46

24 Attenuation in Parallel-plate Waveguides (7) P ( ) = w( J ) L s Rs ωεan ωεa = w( ) Rs where Js = H( = ) = nπ nπ P ( ) ωε R L s α = = Rs = P ( ) β η ( ) R s = π μ σ n πμ α = η σ ( )[ ( ) ] Eletromagneti Theor 47 Attenuation in Parallel-plate Waveguides (8) TE modes α : the same as TM α d P w E H d wωμβ B nπ B = ( ) sin ( ) d = w ( ) P ( ) = w( J ) L s Rs * : ( ) = ( )( ) n n ωμβ nπ nπ ( ) n s = wh = R = wbr PL ( ) Rs nπ Rs α = = ( ) = P ( ) ωμβ η ( ) dereases monotoniall as requen inreases Eletromagneti Theor 48

25 Attenuation in Parallel-plate Waveguides (9) Eletromagneti Theor 49 Homework H.W -, -4, -5, -8, -9, -, -4 Eletromagneti Theor 5