Chapter 5: Coaxial Components and Rectangular Waveguide Components
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1 Chpter 5: Coil Components nd Rectngulr Wveguide Components The informtion in this wor hs een otined from sources elieved to e relile. The uthor does not gurntee the ccurc or completeness of n informtion presented herein, nd shll not e responsile for n errors, omissions or dmges s result of the use of this informtion. Septemer Fin Kung Wi Lee References [] D.M. Por, Microwve engineering, nd edition, 998 John-Wile & Sons. 3 rd edition, 005 is lso ville from John-Wile & Sons). [] R.E. Collin, Foundtion for microwve engineering, nd edition, 99, McGrw-Hill. [3] C.A. Blnis, Advnced engineering electromgnetics, 989, John- Wile & Sons. Septemer Fin Kung Wi Lee
2 5. Coil Components Septemer Fin Kung Wi Lee 3 Introduction The microstrip nd stripline structures re importnt for guiding electromgnetic wves on printed circuit ord PCB). For sstem-to-sstem or ord-to-ord, cle is used for guiding electromgnetic wves. The most common cle tpe for this purpose is the coil cle, which consists of two circulr conductors, one is hollow nd the other is usull solid, shring similr center is hence the nme coil). Although generll used for trnsporting high-frequenc electricl signl, the coil cle cn lso e used for low-frequenc signl virtue of it eing two-conductor interconnection. One conductor would serve s the signl nd the other for the return current. The coil cle cn support TEM, TE nd TM modes of propgting electromgnetic wves. Septemer Fin Kung Wi Lee 4
3 Pictures of Coil Cles Rigid co cle Outer conductor Semi-rigid co cle Fleile Co Cle Fleile co cle Inner conductor Semi-rigid Co Cle Dielectric usull uniform) filling the gp etween conductors Septemer Fin Kung Wi Lee 5 Fleile Coil Cles Outer cet Conducting rid Center conductor Dielectric Single rid Outer cet Outer rid Inner cet Dielectric Doule rid Center conductor Inner rid Velocit of Propgtion: Solid Dielectric % of the speed of light in vcuum Fomed Dielectric 7-85% of the speed of light Septemer Fin Kung Wi Lee 6 3
4 Semi-Rigid Coil Cles Solid outer conductor Center conductor Tpicl dielectric tpes re PTFE Poltetrflouroethlene), fom nd ir. Solid Dielectric Dielectric Solid outer conductor Center conductor Air Articulted Dielectric sheet Septemer Fin Kung Wi Lee 7 Coil Cle Prmeters ) The dominnt propgtion mode for electromgnetic wves in coil cle is TEM mode. In this mode the RLCG prmeters under low-loss condition) re given s follows: ε ε ' ε '' D d ε E H ) R D d πσδ c s L µ D d π ln / ) πε' C ln D/ d) Septemer Fin Kung Wi Lee 8 πωε '' G ln D / d).).).).c).d) δs ωσµ c 4
5 Coil Cle Prmeters ) From these sic RLCG prmeters under TEM mode, other prmeters of interest such s chrcteristic impednce, ttenution fctor, velocit of propgtion, mimum power hndling, cut-off frequenc when non-tem modes strt to propgte) cn e derived. Other prmeters which re influenced the mechnicl spects of the coil cle re fleiilit of the cle, operting temperture rnge, connector tpe, cle dimeter, cle noise or shielding effectiveness etc. Of these, the most importnce is the chrcteristic impednce Z c. Under lossless pproimtion, with R G 0, the chrcteristic impednce is given : L µ Z c ln D/ d).3) C π ε ' Tpicl Z c vlues re 50, 75 nd 93. Of these Z c 50 is the most common. Septemer Fin Kung Wi Lee 9 Wh Z c 50 Ohms? ) Most coil cles hve Z c 50Ω under low loss condition, with the 75Ω eing used in television sstems. The originl motivtion ehind these choice is tht n ir-filled coil cle hs minimum ttenution for Z c 75Ω, while mimum power hndling occurs for cle with Z c 30Ω. A cle with Z c 50Ω thus represents compromise etween minimum ttenution nd mimum power cpcit. Ber in mind this is onl true for ir-filled coil cle, ut the trdition previls for coil cle with other tpe of dielectric. In the old ds coil cle with Z c 93Ω is lso mnufctured, these re minl used for sending digitl signl, etween computers. The cpcitnce per unit length C is minimied for this impednce vlue. Septemer Fin Kung Wi Lee 0 5
6 Wh Z c 50 Ohms? ).5.4 Normlied Vlue Ω stndrd Power hndling cpilit is highest t 30 Ω Attenution is lowest t 77 Ω Chrcteristic Impednce Ω) Septemer Fin Kung Wi Lee Cle Specifictions ) A series of stndrd tpes of coil cle were specified for militr uses in the United Sttes, in the form "RG-#" or "RG-#/U". These re dted c to World Wr II nd were listed in MIL-HDBK-6 96). These designtions re now osolete. The current US militr stndrd is Militr Specifictions MIL-C-7. MIL-C-7 numers, such s "M7/75- RG4," re given for militr cles nd mnufcturer's ctlog numers for civilin pplictions. However, the RG-series designtions were so common for genertions tht the re still used tod lthough the hndoo is withdrwn. Septemer Fin Kung Wi Lee 6
7 Cle Specifictions ) Cle tpe Z o Ω) Dielectric Overll Dimeter inch) Attenution db/00ft@ 3GH) Mimum power W@3GH) Cpcitnce pf/ft) RG-8A 5 Polethlene RG-58C 50 Polethlene RG-74A 50 Polethlene RG-96A 50 Teflon RG-79B 75 Teflon RG Teflon 0.50 S) RG Teflon 0.4 S) RG Teflon S) Septemer Fin Kung Wi Lee 3 Upper Usle Frequenc Since coil cle supports TEM, TE nd TM electromgnetic wve propgtion modes, the TE nd TM modes will come into eistent for sufficientl high operting frequenc. The Upper Usle Frequenc UUF) for coil cle refers to the frequenc where the first non-tem mode comes into eistent. For coil structure, the non-tem mode with the lowest cut-off frequenc f c ) is the TE mode. The UUF cn e estimted []: 4 c d f D d c c c π ε r.4) UUF where c speed of light in vcuum.4) Septemer Fin Kung Wi Lee 4 ε r d D 7
8 Emple For instnce for RG-4 coil cle Ref []), with inner conductor rdius 0.89mm nd outer conductor rdius.95mm, the estimted UUF is: d.78mm D D d mm c d f c 6.78 GH c As sfet precution we usull include some mrgin, s 5%, thus the upper usle frequenc is rted t: f UUF f c GH Septemer Fin Kung Wi Lee 5 Connectors nd Adpters ) The ends of coil cle re fied to connectors. Such connectors re clindricl in shpe, thus the connectors lso ehiit upper usle frequenc limit. Some common emples of connectors/dpters for coil cle re shown elow. Vrious SMA-to-SMA M-to-M nd F-to-F) 3.5 mm/sma connectors connectors PCB to coil dpter BNC to N tpe dpter BNC to SMA dpter Vrious 3.5 mm/sma to N tpe coil dpter Septemer Fin Kung Wi Lee 6 8
9 Connectors nd Adpters ) A comprison of RF/microwve connectors usle frequenc rnge: BNC N connector), for DC to 400 MH. N connector, for DC to 8 GH. SMA Su-miniture version A) connector inner dimeter, D 4.6mm), for DC to 8 GH. 3.5 mm connectors, for DC to round 30 GH..9 mm connectors, for DC to round 40 GH..4 mm connectors, for DC to round 50 GH..8 mm connectors, for DC to round 65 GH. Septemer Fin Kung Wi Lee 7 Attenutors Equivlent to series resistor Equivlent to prllel resistor T-section Conductor Loss mteril for instnce cron sed) π-section A commercil coil ttenutor Septemer Fin Kung Wi Lee 8 9
10 Termintions A termintion is component which sor ll incident wve. This cn e pproimted hving n ttenutor in series with short circuit. For instnce in the emple elow, the reflected wve power will e 40 db or 0000 times) smller thn the incident wve power. 0-dB ttenutor Shortcircuit plte Tpered lod A rnge of of coil termintions Shortcircuit plte Septemer Fin Kung Wi Lee Rectngulr Wveguide Septemer Fin Kung Wi Lee 0 0
11 Introduction As we hve seen erlier wveguides refer to n structure tht cn guide electromgnetic EM) wves long its il direction, which include trnsmission line. Here we consider wveguide s specificll refers to long metllic structures with onl piece of conductor etween the source nd lod end. These metllic structures re usull hollow, so tht EM fields re confined within the hollow nd e guided long the il direction. Appling Mwell s Equtions with the proper oundr conditions see Appendi) shows tht propgting EM wves cn e supported the wveguide. Due to the sence of center conductor, onl TE nd TM mode eist. Septemer Fin Kung Wi Lee Rectngulr Wveguide ) Emples of rectngulr wveguides sections Bend E E E ~ ~ E E E $ t Septemer Fin Kung Wi Lee
12 EM Wves Propgting Modes in Wveguide E E E E E E TE 0 TE TE Cutoff frequenc for TE mn or TM mn mode The corresponding cutoff wvelength for TE mn or TM mn mode λ m ) n ) f c c, mn c, mn m / ) n / ) Thus we see tht for m, n0, the TE 0 hs the lowest cut-off frequenc, nd is the dominnt mode. Septemer Fin Kung Wi Lee 3 Dominnt Mode for Rectngulr Wveguide β ω c E TE 0 mode For TE 0 mode, m, n0, thus the cut-off wvelength is: λ c, 0 m / ) n / ) Recommended operting frequenc rnge for rectngulr wveguide Septemer Fin Kung Wi Lee 4
13 Wveguide Wll Currents The current on the wll of the rectngulr wveguide t certin instnce in time for TE 0 mode. λ g / λ g Guide wvelength We cn crete nrrow slot prllel to the center is of the wveguide without disturing the current flow, hence the internl EM fields. Septemer Fin Kung Wi Lee 5 Slotted-Line Proe A short metl proe cn e inserted into the slot with miniml disturnce to the EM fields of the TE 0 mode. This proe cn e used to mesure the reltive strength of the electric field in the wveguide cvit. Usull the microwve EM fields will e modulted low-frequenc signl, nd the diode/cpcitor pir cts s envelope detector to demodulte this low-frequenc signl. Vrile short-circuit tuner Rectifier µa SWR meter Metl proe to detect electric field intensit) Rectngulr wveguide with slot Septemer Fin Kung Wi Lee 6 3
14 Bends nd Twists E E-end The orienttion of electric E) field is ltered. H H-end The orienttion of mgnetic H) field is ltered. Septemer Fin Kung Wi Lee 7 Mtched Termintions The grdul trnsition from the rectngulr wveguide to the loss mteril ensures miniml reflection of the electromgnetic wve. Short circuit Rectngulr wveguide Loss mteril Rectngulr wveguide Usull the hetsin will e ttched to the eterior for het dissiption Wveguide Hetsin Septemer Fin Kung Wi Lee 8 4
15 5 006 Fin Kung Wi Lee Attenutors An emple of wveguide ttenutor, here the loss mteril is shped so s to provide grdul chnge in the wveguide internl geometr, resulting in smll reflection of incident wve. Tpered loss mteril Septemer Fin Kung Wi Lee V V V V V V V V V V V V Wveguide Tees Septemer E-plne Tee 3 H-plne Tee 3 E 3 3 Often used s power dividers
16 Circultor Emple of rectngulr wveguide circultor see Chpter 4): Permnent mgnets B field Ferrite post Septemer Fin Kung Wi Lee 3 Wveguide to Co-il Adpter Coil cle A wveguide to coil dpter Wveguide Short circuit Metllic Ridges Front view of the dpter Note: for modern design the ridges cn e replced with tper structure. Septemer Fin Kung Wi Lee 3 6
17 Appendi.0 Solution of Electromgnetic Fields for Rectngulr Wveguide Septemer Fin Kung Wi Lee 33 Introduction Rectngulr wveguides re one of the erliest wveguide structures used to trnsport microwve energies. Becuse of the lc of center conductor, the electromgnetic field supported wveguide cn onl e TM or TE modes. For rectngulr wveguide, the dominnt mode is TE, which hs the lowest cut-off frequenc. Septemer Fin Kung Wi Lee 34 7
18 Wh No TEM Mode in Wveguide? From Mwell s Equtions, the mgnetic flu lines lws close upon themselves. Thus if TEM wve were to eist in wveguide, the field lines of B nd H would form closed loop in the trnsverse plne. However from the modified Ampere s lw: r r H J t r D C r r r H dl I The line integrl of the mgnetic field round n closed loop in trnsverse plne must equl the sum of the longtitudinl conduction nd displcement currents through the loop. Without n inner conductor nd with TEM mode there is no longtitudinl conduction current nd displcement current inside the wveguide. Consequentl this leds to the conclusion tht there cn e no closed loops of mgnetic field lines in the trnsverse plne. t S r r D ds Septemer Fin Kung Wi Lee 35 Rectngulr Wveguide The perspective view of rectngulr wveguide is referred elow. The following slides shll illustrte the stndrd procedures of otining the electromgnetic EM) fields guided this structure. µ ε 0 Assume to e ver good conductor PEC) nd ver thin Septemer Fin Kung Wi Lee 36 8
19 9 Septemer Fin Kung Wi Lee 37 TE Mode Solution ) To otin the TE mode electromgnetic EM) field pttern, we use the sstemtic procedure developed in Chpter Advnced Trnsmission Line Theor. We strt solving the pttern function for -component of the mgnetic field: Prolem.) is clled Boundr Vlue Prolem BVP) in mthemtics. Once we now the function of h,), the EM fields re given :, 0 β o c c t h h e h e e H h c h c ˆ ˆ ˆ β β β β β r e e E h c h c ˆ ˆ β ωµ β ωµ r nd oundr conditions.).).) Note: Here we onl consider propgtion in positive direction, tretment for negtive propgtion is similr. Septemer Fin Kung Wi Lee 38 TE Mode Solution ) Epnding the prtil differentil eqution PDE) of.) in crtesin coordintes: Using the Seprtion of Vriles Method, we cn decompose h,) into the product of functions nd c to e the sum of constnts: Putting these into.), nd fter some mnipultion we otin ordinr differentil equtions ODEs): ) ) ) Y X h, ) 0, h c 0 0 c Y Y X X c Y X XY X Y c X X Y Y.).4).4) Y Y X X.3).3)
20 TE Mode Solution 3) From elementr clculus, we now tht the generl solution for.4) nd.4) re: X ) Acos ) Bsin ).5) Y ) C ) Dsin ).5) Thus h,) is given : h [ ][ C cos ) Dsin ) ] ) Acos ) Bsin ),.6) Septemer Fin Kung Wi Lee 39 TE Mode Solution 4) A, B, C nd D in.6) re unnown constnts, to e determined ppling the oundr conditions tht the tngentil electric field must vnish on the conductive wlls of the wveguide. From.): E 0, ) E, ) 0 0, ), ) h.7) h 0 r E E E h e h e ˆ ωµ ˆ β ωµ ˆ β ˆ c c 0,0) E, ) 0,0), ) E h.7) h 0 Septemer Fin Kung Wi Lee 40 0
21 TE Mode Solution 5) Using.6) nd ppling the oundr condition.7): h [ Acos ) Bsin ) ][ C sin ) D cos ) ] h,0) h, ) 0 D 0 0, n 0,,,3L Using.6) nd ppling the oundr condition.7): h [ ] [ Asin ) Bcos ) ] C cos ) Dsin ) h ) ) 0, h 0 B 0, 0, m 0,,,3L In the ove equtions, we cn comine the product of A C, let s cll it R. Common sense tells us tht R would e different for ech pir of integer m,n), thus we should denote R : R mn Septemer Fin Kung Wi Lee 4 TE Mode Solution 6) From.3), c nd the propgtion constnt β re given : c ) ) o c o ) ) Since c nd β lso depends on the integer pirs m,n), it is more pproprite to write these s: β cmn ) ).7) ) ), ω µε βmn o o.7) Septemer Fin Kung Wi Lee 4
22 TE Mode Solution 7) With these informtion, nd using.) nd.), we cn write out the complete mthemticl epressions for the EM fields under TE propgtion mode for rectngulr wveguide: E ωµ mn sin c mn ) m ) n β R cos π π ) e mn E ωµ mn cos cmn ) m ) n β R sin π π ) e mn H βmn mn cos cmn ) R ) β sin ) e mn H βmn mn sin cmn H m n mn Rmn cos π β sin π e ) R ) β cos ) e mn ) ).8).8).8c).8d).8e) Septemer Fin Kung Wi Lee 43 Cut-Off Frequenc for TE Mode ) Notice from.7) tht the propgtion constnt β mn is rel when: m ) n ) o > π π When β mn is imginr the EM fields cnnot propgte. Since ωπf, we cn define limit for the frequenc f s follows: m ) n ) o ω µε > π π f > π µε ) ) The lower limit of this frequenc is clled the Cut-off Frequenc f c. fc TE mn π µε ) ).9) Septemer Fin Kung Wi Lee 44
23 Cut-Off Frequenc for TE Mode ) The TE mode electromgnetic field is usull leled s TE mn since the mthemticl function of the field components depend on the integer pir m,n). The pir m,n) cnnot e oth eros, otherwise from.8) to.8e), E, E, H, H, H re ll ero, no fields t ll! This is trivil solution, lthough vlid one. The smllest comintion of m,n) re m,n),0) or 0,). Since > the lterl dimensions of the rectngle), we see tht m,n),0) produces smller f c, thus lower cutoff frequenc. Therefore the TE propgtion mode TE 0 is the dominnt mode for TE wves. It s corresponding cut-off frequenc is given : f c π TE0 π µε ) µε Onl ecittion frequenc greter thn TE µε will cuse EM wves to propgte within the rectngulr wveguide. Septemer Fin Kung Wi Lee 45 f c 0 Emple Consider rectngulr wveguide, 45.0mm, 35.0mm, filled with ir ε ε o, µ µ o ). ε o µ 4π 0 7 o f c TE µ oεo Septemer Fin Kung Wi Lee 46 3
24 Phse Velocit in Wveguide Since phse velocit v p depends on propgtion constnt β mn, it too depends on the integer pir m,n) hence the propert of the TE mode fields. v p ω ω > ω βmn o ).0) o ) Speed of light in dielectric of µ,ε) We thus oserve tht the phse velocit of TE mode hs the peculir propert of trveling fster thn the speed of light! Septemer Fin Kung Wi Lee 47 Group Velocit in Wveguide The velocit of energ propgtion, or the speed tht informtion trvel in wveguide is given the Group Velocit v g. Thus from: Since v p > c, β mn ω v g v βmn µεω v g ω β o µεω ) ) mn µεω β c ) µε ) v p c ) c c ω βmn g < v p The group velocit is thus less thn speed of light in vcuum, mintining the ssertion of Reltivit Theor tht no mss/energ cn trvel fster thn speed of light. Septemer Fin Kung Wi Lee 48 4
25 TM Mode Solution ) The procedure for otining the EM field solution for TM propgtion is similr to the TE procedure. We strt solving the pttern function for the -component of the electric field: t c e 0, β c o e nd oundr conditions.) As in solving TE mode prolem, the Seprtion of Vriles Method is used in solving.), nd integer pir m,n) needs to e introduced in the TM mode solution. The mthemticl epressions for the EM field components thus depends on the integer pir m,n), nd is denoted TM mn field. The derivtion detils will e omitted here due to spce constrint. You cn refer to reference [] for the procedure. Septemer Fin Kung Wi Lee 49 TM Mode Solution ) The complete epressions for the TM mn field components re shown elow: E βmn mn sin cmn ) R ) β cos ) e mn E βmn mn cos c mn ) R ) β sin ) e mn H ωε mn cos cmn ) R ) β sin ) e mn H βmn mn sin c mn E m n mn Rmn sin π β sin π e ) R ) β cos ) e mn ) ) Septemer Fin Kung Wi Lee 50.).).c).d).e) 5
26 TM Mode Solution 3) Where c mn ) ).3) ) ), ω µε βmn o o.3) Septemer Fin Kung Wi Lee 5 Cut-Off Frequenc for TM Mode Since the propgtion constnt β mn is similr for oth TE mn nd TM mn mode, cut-off frequenc lso eists for TM mn : fc TM mn π µε ) ).4) Oserve tht from.) to.e) the EM field components ecome 0 if either m or n is 0. Thus TM 00, TM 0 nd TM 0 do not eist. The lowest order mode is TM. f ) ) c π π > π ) f TM c π µε π µε TE0 It is for this reson tht we consider TE 0 to e the dominnt mode of rectngulr wveguide. Septemer Fin Kung Wi Lee 5 6
27 Appendi.0 Plots of Electromgnetic Fields for Rectngulr Wveguide Septemer Fin Kung Wi Lee 53 TE 0 Mode H E Instntneous E field mgnitude Cross section visulition of the EM fields Instntneous H field mgnitude TE 0 Septemer Fin Kung Wi Lee 54 7
28 TE 0 H E Instntneous E field mgnitude Cross section visulition of the EM fields Instntneous H field mgnitude TE 0 Septemer Fin Kung Wi Lee 55 TE H E Instntneous E field mgnitude TE Instntneous H field mgnitude Septemer Fin Kung Wi Lee 56 8
29 TM H E Instntneous E field mgnitude TM Instntneous H field mgnitude Septemer Fin Kung Wi Lee 57 9
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