Chapter 5 Waveguides and Resonators

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1 5-1 Chpter 5 Wveguides nd Resontors Dr. Sturt Long

2 5- Wht is wveguide (or trnsmission line)? Structure tht trnsmits electromgnetic wves in such wy tht the wve intensity is limited to finite cross-sectionl re In this chpter we will focus on three types of wveguides: Prllel-Plte Wveguides. Rectngulr Wveguides. Coil Lines.

3 5-3

4 5-4 Prllel Plte Wveguide

5 5-5 Prllel Plte Wveguide Assume both pltes to be perfect conductors. Assume wveguide to be very lrge in ŷ direction (w>>λ). Field vectors hve no y-dependence 0. y Neglect ny fringing fields t y0 nd yw. Propgtion long the ˆ + direction.

6 5-6 Prllel Plte Wveguide From Mwell Equtions ( E, H, H ) TE (Trnsverse Electric) E 0 y or ( H, E, E ) TM (Trnsverse Mgnetic) H 0 y (Sect. 5.1)

7 5-7 TE Wves in Prllel Plte Wveguides (,, ) E H H B.C. t 0 nd E 0 y y Remember: 0 y The wve eqution for the TE cse derived from Mwell's Equtions is + + ω με Ey 0

8 5-8 TE Wves in Prllel Plte Wveguides For ppg. in +ˆ direction with E E sin( k ) e y 0 jk (stisfies B.C t 0) (5.5) + ω με k k k (5.6) To stisfy B.C. ( E 0) t y k mπ ( m is ny integer ecept 0) (5.7)

9 5-9 TE Wves in Prllel Plte Wveguides The Electric field cn be epressed in nother form by substituting in for k E with k mπ E sin e y 0 jk 1 1 mπ mλ ω με ω με 1 where λ π k

10 5-10 TE Wves in Prllel Plte Wveguides The guided wve propgtes with the phse velocity v p ω 1 1 k με mλ 1 Note: k is imginry for k mπ < k or ω με < or λ > m If k becomes imginry, the wve will ttenute eponentilly, nd the velocity for the guided wve will lso become undefined.

11 5-11 TE Wves in Prllel Plte Wveguides ω c π m με f c ωc m ( cutoff frequency of TE mode) m π με frequency t which becomes imginry k eponentil ttenution when f < fc λ c m ( of TE mode) cutoff wvelength wvelength t which becomes imginry m k

12 5-1 Physicl Interprettion TE mode E mπ E sin e y 0 jk E y je0 e jk jk e jk jk (5.8) wve trv. in +ˆ nd +ˆ dir. coef f je 0 wve trv. in - ˆ nd +ˆ dir. coeff - je 0

13 5-13 Physicl Interprettion of Guided Wves k mπ 3π π k π 3.5π For this cse k so modes m 1,,3 will ppg. but for m 4, k is imginry. k k (fig 5.3) k k mπ 1 (5.9)

14 5-14 TM Wves in Prllel Plte Wveguides - ( H y,e,e) cn find Hy H0cos k e jk E ~ H y E ~ sin k For B.C. t ( E 0) 0, k mπ

15 5-15 TM Wves in Prllel Plte Wveguides For ppg. in +ˆ direction H H cos y 0 ( ) k e jk (stisfies B.C t 0) with + ω με k k k To stisfy B.C. ( E 0 ) t k mπ ( m is ny integer including 0 )

16 5-16 TM Wves in Prllel Plte Wveguides The field solutions cn be epressed in nother form by substituting in for wi th k k H mπ H cos e y 0 jk 1 mπ mλ ω με ω με 1 1 where λ π k

17 5-17 TM Wves in Prllel Plte Wveguides since E ~ H y k mπ E H0sin e jωε jk E k mπ cos e ωε H0 jk

18 5-18 TM Wves in Prllel Plte Wveguides Remember tht for TM wves we cn now hve m 0. For TM 0 m 0. k 0 k ω με k For this cse E nd H re both perpendiculr to the direction of propgtion ( ˆ ). We refer to this cse s TEM mode (Trnsverse Electromgnetic)

19 5-19 TM 0 mode The field solutions for the TM or TEM mode re given by 0 H E y 0 0 E 0 He ηh e jk jk (5.13) (5.13b) or equivlently E Ee 0 jk H y E0 e jk η

20 5-0 Physicl Interprettion TEM mode J s n J s n (on bottom plte) J s E nˆ H ˆ yˆ e ˆ 0 η E e η 0 jk 0 jk (5.13c) ρ nˆid ˆiεE εe e s 0 jk (5.11d) (on top plte n - ˆ ) - J s s ˆ E0 e η ρ εe e 0 jk jk

21 5-1 Microstrip Lines microstrip line dielectric substrte ground plne

22 5- Microstrip Lines The time-verge Poynting power density cn be found by: 1 S Re E H S 1 - jk E0 + jk Re Ee 0 e ( ˆ ˆ) y η S ˆ E 0 η

23 5-3 Microstrip Lines ε ε 0 qusi TEM-fields k ω με ε > ε 0

24 5-4 Rectngulr Wveguide y b

25 5-5 Rectngulr Wveguide MAX EQN HELMHOLTZ EQN (wve eqution) E+ ωμεe 0 H+ ωμεh 0 6 EQNS like H H H ωμεh y 0

26 5-6 Rectngulr Wveguide For seprble solution ssume 1 X 1 Y 1 Z H ( y,, ) X( ) Y( y) Z( ) + + X Y y Z ω με ech term is constnt -k - k - k ωμε y 1 X X -k X 0 + k X

27 5-7 Rectngulr Wveguide X c sin k + c cos k 1 Y c sin k y+ c cos k y 3 y 4 y Z ce + ce jk + jk 5 6 H X Y Z Using M. eqns. cn solve for trnsverse fields (E, E in terms of longtitudinl one s (E,H ) y, H, H ) y

28 5-8 Rectngulr Wveguide jk E jωμ H jk E jωμ H E E + k k k k ; y c c y c y c jk H jωε E jk H jωε E H + H k k k k ; y c c y c y c where c + y ω με k k k k

29 5-9 Rectngulr Wveguide Specil cses 1) when E 0 ; H 0 Trnsverse Electric mode ( TE) ) when H 0 ; E 0 Trnsverse Mgnetic mode ( TM) If E H 0 ll comp. 0 Trnsverse ElectricMgnetic mode ( TEM) cn not eist

30 Rectngulr Wveguide TE-Cse (E 0) B.C. tngentil E 0 t 0, nd y 0, b H Side wlls E y ~ 0 (t 0, ) mπ E y ~ kc 1cosk kc sink c1 0 nd k H Top nd bottom wlls E ~ 0 ( t y 0, b) y E ~ kc cosky kc y 3 y y 4 nπ sin kyy c3 0 nd ky b y b 5-30 Let ccc H H XYZ H H 0 mπ nπy cos cos e b jk

31 5-31 Rectngulr Wveguide TE-Cse (E 0) y b Propgtion constnt k k - k c where k c mπ nπ + b substituing in for k c k mπ nπ ωμε- - b (5.19) Note: propgtion ( k rel) for k > k c eponentil ttenution ( k imginry) for k < k c

32 5-3 Rectngulr Wveguide TE-Cse (E 0) y b f c ω 1 c mπ π + π π με b 1 n of m ( cutoff frequency TE ode) mn frequency t which k becomes imginry (5.1) eponentil ttenution when f < fc λ g π π ( of TE mode) guide wvelength k k - kc wvelength t which becomes imginry mn k (5.0) defines doubly infinite set of modes TE mn

33 5-33 Rectngulr Wveguide TM-Cse (H 0) y b mπ E E0sin sin nπy b e jk (5.18) Sme k, k, f, λ s c c g TE For TM m 0 ; nd n 0 lowest order (smllest f ) is TM c 11 The lowest order of ll is TE clled DOMINANT MODE for rectngulr wveguides ( no TEM possible) (there eist frequency rnge were only it cn propgte) 10

34 5-34 Rectngulr Wveguide TE 10 mode (m1, n0) y b H H 0 π cos e jk jωμ H jωμ H jωμ π E 0 E sin y k ; y H 0 kc kc c e jk jk H jk π jk H H sin H 0 k y jk H 0 e ; y kc c kc

35 5-35 Rectngulr Wveguide TE 10 mode (m1, n0) y b k π k 1 λc ; fc ; kc με π λ g π k λ λ 1 Note: these re the eqns. found in slide 5-3 nd 5-33 with m1 nd n0

36 Physicl Interprettion TE 10 mode (m1, n0) y b 5-36

37 k ω ω versus k grph 5-37 TM 11 ω c11 TE 11 TE 10 k ω με ω c10 v p θ ω tnθ (function of frequency ) k where c k k k nd v g ω k

38 5-38 Design of Prcticl Rectngulr Wveguide (Emple 5.4) y f c 1 1 mπ nπ + π με b b f f f c c c TE 10 TE 01 TE με 1 1 με b 1 με

39 5-39 Design of Prcticl Rectngulr Wveguide (Cont. e.g 5.4) y b for b> for b< TE 01 b TE 10 TM 0 b TE10 TM0 TE 01 for b TE 10 TE TM 01 0 TE TM f c f c TE 10

40 5-40 Design of Prcticl Rectngulr Wveguide ( Emple 5.3) y Trnsmitting Power in wveguide 1 S Re E H [ TE mode ] 10 b S 0 E k ˆ π sin ωμ Time-verge Poynting vector P b 00 S i ˆ ddy P E 0 bk 4ωμ Totl trnsmitted power

41 5-41 Design of Prcticl Rectngulr Wveguide (Cont. e.g 5.3) y b Note: Do not wnt b > for bndwith Do not wnt b smll for power hndling P 0 E bk 4ωμ Usul cse b

42 5-4 Design of Prcticl Rectngulr Wveguide y b X-Bnd Wveguide (8-1 GH) f c TE GH f c TE GH f c TE GH 0.9 b 0.4

43 5-43 Rectngulr Wveguide TE 10 mode y b H H 0 π cos e jk E jωμ H π y 0 π sin e jk jk π H H0sin π e jk

44 5-44 Rectngulr Wveguide TE 10 mode y b or E E y 0 π sin e jk where E jωμ H π 0 0 k π H E0sin ωμ e jk (5.3) H π E0 π cos jωμ e jk

45 5-45 Emple (Emple 5.5 & 5.6) Design Rectngulr wveguide with 10 GH ( λ 3cm) t Mid-Bnd nd b BW < f < 8 8 let cm b b 1.15 cm

46 5-46 Emple (Cont. e.g 5.5 & 5.6) M power hndling P E b 0 4Z TE where Z TE ωμ k [ Ω] E BD ir 10 6 V m tke E m 10 5 V m (sfety fctor of 10) Z TE 10 7 (π 10 )(4π 10 ) Ω k ( π ) [ ]

47 5-47 Emple (Cont. e.g 5.5 & 5.6) note: k ( ) ( ) π 1 λ π λ 3 k cm m P m 5 ( 10 ) (0.05)(0.0115) 4(505.8) [ KW]

48 5-48 Coil Trnsmission Lines ε b

49 5-49 Cylindricl Coordintes ρ ẑ φˆ ˆρ y φ ( fig. 5.16) ˆ ρ φ ˆ ˆ

50 5-50 Coil Trnsmission Lines ε b Mwell Eqns. E V0 e ρ jk ˆ ρ ( 5.48) H V0 e ηρ jk ˆ φ ( 5.48b) k ω με ( 5.49) η μ ε ( 5.49b)

51 The Del Opertor in Cylindricl Coordintes ρ ẑ φˆ ˆρ 5-51 y φ ρˆ φ ˆ ˆ ( fig. 5.16) ia ( ρ ) ( ρ ) 1 Aρ 1 Aφ A + + ρ ρ ρ φ (5.44) ( ρ A ) 1 A A φ Aρ A ˆ 1 φ A ρ A ˆ ρ + - φ + ˆ ρ φ ρ ρ ρ φ (5.45)

52 5-5 Currents on Coil Lines H ε J s ˆn b J s ˆ V nˆ H ˆ ρ φ e ρ ηρ 0 jk 0 ρ ˆ V η e jk ( 5.50) I π J s 0 ρ φ πv η 0 e jk ( 5.50b)

53 5-53 Short-Circuited Co for < 0 both Incident Fields nd Reflected Fields eist 0 (Fig. 5.18) E V ˆ ρ e ρ V + e ρ 0 jk 1 + jk ( 5.5) H ˆ φ V e ρη V e ρη 0 jk 1 + jk ( 5.5b) B.C t 0 tngentil E 0 E 0 V V ρ 1 0

54 5-54 Short-Circuited Co ( 5.53) ( jk jk E ˆ ρ ) Similrly V + jv e e ˆ ρ k ρ sin ρ 0 (Fig. 5.18) ( 5.53b) H ˆ V φ ηρ 0 c osk To mesure stnding wve pttern cut longitudinl slot in outer conductor for moveble proble. Note: H totlly in ˆ φ direction so J s totlly in ˆ direction

55 5-55 Emple (Emple 5.8) ε b Clculte the trnsmitted power long coil line. V0 E e ρ V0 H e ηρ jk jk ˆ ρ ˆ φ TEM mode in coil line 1 1 V V ˆ V0 Re R ˆ ˆ e 0 jk 0 jk S E H e ρ e φ ρ ηρ ηρ P π V0 b ln η

56 5-56 Trnsmission Lines ε or more conductors, (TEM possible ) (long with TE nd TM) b Coil two-wire (shielded)

57 5-57 Trnsmission Lines or more conductors, (TEM possible ) (long with TE nd TM) Prllel Plte microstrip stripline Conductor Dielectric

58 5-58 Wveguides Rectngulr Ellipticl Ridged Circulr Opticl Fiber Dielectric Slb

Waveguide Guide: A and V. Ross L. Spencer

Waveguide Guide: A and V. Ross L. Spencer Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it