ECE Microwave Engineering

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1 ECE Mirowave Engineering Aapte from notes by Prof. Jeffery T. Williams Fall 18 Prof. Davi R. Jakson Dept. of ECE Notes 7 Waveguiing Strutures Part : Attenuation ε, µσ, 1

2 Attenuation on Waveguiing Strutures For most pratial waveguies an transmission lines the loss assoiate with ieletri loss an onutor loss is relatively small. To aount for these losses we will make this approximation: kz β jα Phase onstant for lossless wave guie Attenuation onstant α α + α Attenuation onstant ue to onutor loss (ignore ieletri loss) Attenuation onstant ue to ieletri loss (ignore onutor loss)

3 Attenuation ue to Dieletri Loss: α Lossy ieletri omplex permittivity k omplex wavenumber k ω µε ω µε (1 jtan δ ) Note: k k jk { } k Re k ω µε ε σ ε j ω ε jε ε jε ε ε 1 j ε ε (1 j tan δ) εε (1 j tan δ) r σ j ω Note : In most books, ε r is enote as ε. r (e.g., ε r.1 for Teflon). 3

Attenuation ue to Dieletri Loss (ont.) An exat general expression for ieletri attenuation is foun from: k β jα k k z k ω µε This is an exat formula for attenuation ue to ieletri loss.

4 Attenuation ue to Dieletri Loss (ont.) An exat general expression for ieletri attenuation is foun from: k β jα k k z k ω µε This is an exat formula for attenuation ue to ieletri loss. It works for both waveguies an TEM transmission lines (k ). Remember: The value k is always real, regarless of whether the waveguie filling material is lossy or not. Note: The raial sign enotes the prinipal square root: ( ) π < arg z < π 4

5 Approximate Dieletri Attenuation First we approximate k: k (1 jtan δ ) ω µε ω µε (1 j tan δ ) Assume a small ieletri loss in meium: tanδ << 1 Use: 1 z 1 z/ for z << 1 ( ) ( 1 j tan / ) k ω µε δ k ω µε k k ( δ ) tan / 5

6 Approximate Dieletri Attenuation (ont.) For the wavenumber of the guie wave: k β jα k k z ω µε (1 jtan δ ) k ω µε k jω µε tanδ Assume a small ieletri loss: ( ) ω µε tan δ << ( ω µε k ) Use: a z a 1 z 1 z 1 ( z/ a) a 1 a a a for z << a 6

7 Approximate Dieletri Attenuation (ont.) k ω µε k jω µε tanδ z ω µε k β β jα ω µε tanδ j ( ω µε k ) β We assume here that we are above utoff. β β ω µε k β k k α ω µε tanδ β Reall : k ω µε α k tanδ β (works for both WGs an TLs) 7

8 Approximate Dieletri Attenuation (ont.) TEM moe We an simply put k in the previous formulas. Or, we an start with the following: k β jα k k jk z β k α k (exat equations) α k tan δ Reall : ( tan ) / k k δ (approximate equation) 8

9 Summary of Dieletri Attenuation TEM moe k β jα k z Waveguie moe (TM z or TE z ) k β jα k k z β α α k k k tan δ β Re k k α β α k k jk ω µε Im k k k k k tanδ β 9

10 Attenuation ue to Conutor Loss Assuming a small amount of onutor loss: We an assume fiels of the lossy guie are approximately the same as those for lossless guie, exept with a small amount of attenuation. We an use a perturbation metho to etermine α. Notes: Dieletri loss oes not hange the shape of the fiels at all in a waveguie or transmission line, sine the bounary onitions remain the same (PEC). Conutor loss oes isturb the fiels slightly. 1

11 Surfae Resistane This is a very important onept for alulating loss at a metal surfae. ε, µ x ε, µσ, Note: The tangential fiels etermine the power going into the metal. z Plane wave in a goo onutor Note: In this figure, z is the iretion normal to the metal surfae, not the axis of the waveguie. Also, the eletri fiel is assume to be in the x iretion for simpliity. 11

12 Surfae Resistane (ont.) Assume σ ωε 1 (for the metal) The we have (for the metal): 1/ 1/ σ σ µσ 1 j ωµσ k ω µ ε j ω µ j ω (1 j) ω ω ω Hene k k jk ωµσ j ( 1 ) Therefore k k ωµσ 1

13 Surfae Resistane (ont.) k k ωµσ Denote 1 δ p skin epth epth of penetration k Note : 1 e.37 (For z δ, fiels are own to 37% of their values at the surfae.) Then we have δ p k k ωµσ 1 δ At 3 GHz, the skin epth for opper is about 1. mirons. 13

14 Surfae Resistane (ont.) δ ωµσ Example: opper (pure) µ µ π [H/m] 7 σ [S/m] Note: A value of [S/m] is often assume for pratial opper. Frequeny δ 1 [Hz] 6.6 [m] 1 [Hz].1 [m] 1 [Hz] 6.6 [mm] 1 [khz].1 [mm] 1 [khz].66 [mm] 1 [khz] 1 [mm] 1 [MHz] 66 [µm] 1 [MHz].1 [µm] 1 [MHz] 6.6 [µm] 1 [GHz].1 [µm] 1 [GHz].66 [µm] 1 [GHz].1 [µm] 14

15 Surfae Resistane (ont.) x S z Fiels evaluate on this plane P time-average power issipate / m on S P 1 1 E H zˆ EH * * Re( ) Re( x y ) z 15

16 Surfae Resistane (ont.) Insie onutor: where E x ηh y E H x y Note: z, air µ µ µ ωµ η j ε σ σ ε j j σ ω ω E H 1+ j ωµ (1 + j) σ (1 + j) Rs Z x s ωµ σ y plane wave in metal ( ˆ ) E Z n H t s t Surfae resistane (Ω) R s ωµ σ Surfae impeane (Ω) Z Note: To be more general: ˆn outwar normal s ( 1 ) + j R s 16

17 Surfae Resistane (ont.) Summary for a Goo Conutor ( ˆ ) E Z n H (fiels at the surfae) t s t Z R s s (1 + j) R ωµ 1 σ σδ s δ ωµσ 17

18 Surfae Resistane (ont.) eff J s Conutor ˆn ( ˆ ) E Z n H t s t E t H t tangential eletri fiel at surfae tangential magneti fiel at surfae nˆ outwar unit normal to onutor surfae Effetive surfae urrent ( ˆ ) Hene we have eff Js n Ht eff E Z J t s s For the effetive surfae urrent ensity we imagine the atual volume urrent ensity to be ollapse into a planar surfae urrent. The surfae impeane gives us the ratio of the tangential eletri fiel at the surfae to the effetive surfae urrent flowing on the objet. 18

19 Surfae Resistane (ont.) R s ωµ 1 σ σδ Example: opper (pure) µ µ π [H/m] 7 σ [S/m] Note: A value of [S/m] is often assume for pratial opper. Frequeny R s 1 [Hz] [Ω] 1 [Hz] [Ω] 1 [Hz] [Ω] 1 [khz] [Ω] 1 [khz] [Ω] 1 [khz] [Ω] 1 [MHz] [Ω] 1 [MHz] [Ω] 1 [MHz].61 [Ω]6.6 1 [GHz].85 [Ω] 1 [GHz].61 [Ω] 1 [GHz].85 [Ω] 19

20 Surfae Resistane (ont.) Returning to the power alulation, we have: P 1 1 (( ) ) 1 Re( EH ) Re ZH H R H * * x y z s y y s y In general, P 1 R H s t eff J nˆ H For a goo onutor, s t Hene P 1 R J eff s s This gives us the power issipate per square meter of onutor surfae, if we know the effetive surfae urrent ensity flowing on the surfae. PEC limit: J eff s PEC J eff PEC s Perturbation metho : Assume that Js Js

21 Perturbation Metho for α Power flow along the guie: ( ) Pz Pe α z Power z is alulate from the lossless ase. Power loss (issipate) per unit length: P l ( ) P z z ( Pabs Pflow ( z) ) P z Pe P z αz l ( ) α α ( ) Pl( z) Pl() α Pz ( ) P Note: α α for onutor loss 1

22 Perturbation Metho: Waveguie Moe α P () l P There is a single onuting bounary. S C 1 ˆ * P Re E H z S S Rs Pl() Js C Surfae resistane of metal onutors: z R s z ωµ σ For these alulations, we neglet loss when we etermine the fiels an urrents. J nˆ H s On PEC onutor

23 Perturbation Metho: TEM Moe α P () l P There are two onuting bounaries. S Re 1 1 * P E H z S S Z I 1 ˆ z z 1 Pl() Rs Js C + C Surfae resistane of metal onutors: R s ωµ σ For these alulations, we neglet loss when we etermine the fiels an urrents. ( Z Z lossless ) J nˆ H s On PEC onutor 3

24 Wheeler Inremental Inutane Rule The Wheeler inremental inutane rule gives an alternative metho for alulating the onutor attenuation on a transmission line (TEM moe): It is useful when you have a formula for Z. α R Z s Zη The formula is applie for eah onutor an the onutor attenuation from eah of the two onutors is then ae. In this formula, (for a given onutor) is the istane by whih the onuting bounary is reee away from the fiel region. E E The top plate of a PPW line is shown being reee. H. Wheeler, Formulas for the skin-effet, Pro. IRE, vol. 3, pp ,

25 Calulation of R for TEM Moe From α we an alulate R (the resistane per unit length of the transmission line): α P () l P i( zt, ) + v( zt, ) R z L z G z C z ( + zt, ) i z + ( + zt, ) v z 1 P Z I Pl ( ) 1 RI - Assume onutor loss only. - z α R Z R Z α 5

26 Summary of Attenuation Formulas Transmission Line (TEM Moe) Metho #1 ( )( ) α Re R+ jωl G+ jωc Note: Set R for α. Set G for α. Metho # α α + α α k α P () l P P The two methos are relate, sine we have: 1 Z I R Z α R R P J J k ω µε s1 s l() s + s C1 z C z 6

27 Summary of Attenuation Formulas (ont.) Waveguie (TM z or TE z Moe) α α + α α Im α P () l P k k C 1 Re ˆ * P E H z S S S ( µε, ) z k ω µε Rs Pl() J s C z 7

28 Comparison of Attenuation Frequeny RG59 Coax Approximate attenuation in B per meter WR975 WG WR159 WG Waveguies are getting smaller WR9 WG WR4 WG WR19 WG WR1 WG 1 [MHz].1 NA NA NA NA NA NA 1 [MHz].3 NA NA NA NA NA NA 1 [MHz].11 NA NA NA NA NA NA 1 [GHz].4.4 NA NA NA NA NA 5 [GHz] 1. OM.4 NA NA NA NA 1 [GHz] 1.5 OM OM.11 NA NA NA [GHz].3 OM OM OM.37 NA NA 5 [GHz] OM OM OM OM OM 1. NA 1 [GHz] OM OM OM OM OM OM 3. OM overmoe NA below utoff Typial single-moe fiber opti able:.3 B/km Typial multimoe fiber opti able: 3 B/km 8

29 Comparison of Waveguie with Wireless System (Two Antennas) Waveguie: ( ) ( ) P z P e αz (attenuating wave) Antenna: A r ( ) P r (spreaing spherial wave) For small istanes, the waveguie elivers more power (no spreaing). For large istanes, the antenna (wireless) system will eliver more power. 9

30 Wireless System (Two Antennas) Here we examine a wireless system in more etail. Two antennas (transmit an reeive): P P t r t er 4π r GA P t power transmitte P r power reeive Mathe reeive antenna: Aer G λ r 4π (from antenna theory) A er effetive area of reeive antenna Hene, we have: P r P t GG λ π t r 4πr 4 3

31 Wireless System (Two Antennas) (ont.) Pr P t GG λ 4π r t r Friis transmission formula Total B of attenuation: Hene, we have: B 1log Pr 1 Pt λ B 1log log + log 4π ( GG ) ( r) 1 t r 1 1 The B attenuation inreases slowly with istane 31

32 B Attenuation: Comparison of Waveguiing system with Wireless System Waveguiing system: B 8.686( αz) λ 4π Wireless system: B 1log ( GG ) log + log ( r) 1 t r 1 1 Examples: (a) Two Half - Wavelength Dipole Antennas: G G π (b) Dish antenna +Dipole antenna: G, ( / ) t Aet A 1.64 et π D Gr λ transmit reeive t r D iameter of ish (hoose 34 meters) 3

33 B Attenuation: Comparison of Waveguiing System with Wireless System 1 GHz RG59 Single Moe Two Dipoles 34m Dish+Dipole Distane Coax Fiber Wireless Wireless 1 m m m km km km km , km , km ,, km ,, km ,, km

ECE Microwave Engineering

ECE Microwave Engineering ECE 5317-6351 Mirowve Engineering Apte from notes Prof. Jeffer T. Willims Fll 18 Prof. Dvi R. Jkson Dept. of ECE Notes 1 Wveguiing Strutures Prt 5: Coil Cle 1 TEM Solution Proess A) Solve Lple s eqution