Transmission Line Fundamentals

Transcription

1 ransmission Line Fundamentals b Chris Angove DISCLAIMER he Author makes no representation or warranties with respet to the aura or ompleteness of the ontents of this paper and speifiall dislaims an implied warranties of merhantabilit or fitness for an partiular purpose and shall in no event be liable for an loss of profit or an other ommerial damage, inluding but not limited to speial, inidental, onsequential or other damages. An opinions epressed are those personal opinions of the Author onl and the do not neessaril impl that an detailed and aredited engineering tests have been performed to arrive at them. he reader is advised to hek the information given, formulas and derivations against other soures before using them beause the Author annot guarantee them to be free from errors. ransmission Lines Page 1 of 1 5 Ma 11

2 Aronm Meaning µm mirometer db deibel DC diret urrent RF radio frequen E transverse eletri EM transverse eletri magneti M transverse magneti VSWR voltage standing wave ratio ransmission Lines Page of 1 5 Ma 11

3 Contents 1 ransmission lines What is an Eletrial ransmission Line? 4 1. istor he Elements of a Pratial Uniform EM ransmission Line he otal Voltage Wave he otal Current Wave Charateristi Impedane Voltage Refletion Coeffiient, Voltage Standing Wave Ratio and Return Loss he Variation of Impedane Along a Pratial (Loss) Mismathed ransmission Line he Loss-Free ransmission Line he Loss-Free ransmission Line Equation Phase Constant and Phase Veloit 11 Waveguides 13.1 he Retangular Waveguide [] 13 3 Referenes 1 Figures Figure 1-1 he fundamental element setion of a uniform EM transmission line... 5 Figure 1- he fundamental transmission line onfiguration... 8 Figure -1 he retangular o-ordinate sstem used with retangular waveguides ransmission Lines Page 3 of 1 5 Ma 11

4 1 RANSMISSION LINES 1.1 What is an Eletrial ransmission Line? An eletrial transmission line is a devie for transferring eletrial energ from one loation to another. his must usuall be ahieved with the highest possible effiien. ransmission lines have been developed for operation at pratiall all frequenies from DC to optial. here are three ommon tpes of transmission line used in radio frequen (RF) and mirowave engineering: ransverse eletri-magneti (EM) transmission lines. Condutor waveguides. Dieletri waveguides. EM transmission lines have separate ondutors for the forward and return eletri urrent paths and inlude open wire lines, twisted wire lines, oaial ables and stripline. he ommon transmission line known as mirostrip departs slightl from pure EM as the field assoiated with the line is shared between the substrate material and the air above, so this line is sometimes alled a quasi-em transmission line. A EM transmission line will operate at frequenies from DC upwards. In pratie however, there is an upper frequen limit determined b when the wavelength beomes suffiientl short that non- EM modes start to our. hese are known as waveguide modes and omprise either transverse eletri (E) or transverse magneti (M) modes. Condutor waveguides take the form of preision rigid pipes omprising insulator ores bound b eletrial ondutors. he allow propagation of eletrial energ in the form of non-em wave modes supported b eletrial urrents irulating on the inside walls of the guide and the resulting magneti and eletri fields. he modes are again speifi orders of E or M modes. Dieletri waveguides omprise ores of low loss dieletri material surrounded b a shell of another similar dieletri material but with a slightl higher dieletri onstant. ransmission ours b a series of refletions at the dieletri boundaries. he most ommon eample of the dieletri waveguide is the fiber opti able, tpiall used for the propagation of infrared wavelengths around 1.5 μm 1. istor he theor of transmission lines developed from work performed b James Clerk Mawell, Lord Kelvin and Oliver eaviside. In 1855 Lord Kelvin performed the first distributed analsis of a transmission line [1]. e modelled the tpe of pulsed urrent then used in long distane telegraph ables and orretl predited the poor performane of the trans- Atlanti submarine able whih was laid in In 1885 eaviside published the first papers whih analsed the propagation of telegraph-like signals, arriving at the telegrapher s equations. hese will be desribed in Setion 1.3 Earl telegraph lines were ver rude, eah one omprising a single iron ondutor arried b overhead telegraph poles and forming an eletrial iruit using an earth return path. he hoie of an eletrial ondutor with suh a relativel high resistivit would seem odd toda, but in the mid-nineteenth entur ver little researh had been done is this area and iron was plentiful and relativel heap. After the invention of the telephone at the end of the nineteenth entur, attempts to transmit reasonable qualit audio frequenies (known as telephon) over telegraph lines were not suessful. wo separate wires were found to be muh more effetive, the seond providing the urrent return path instead of a route ransmission Lines Page 4 of 1 5 Ma 11

5 through the ground. It was soon disovered that using opper wires, a muh better eletrial ondutor than iron, redued the series loss affeting the signal. eaviside also showed that the addition of series indutors, regularl spaed ever mile or so, ompensated for the apaitane of the line, inreased its effetiveness and allowed a finer gauge of wire to be used between the indutors than previousl. 1.3 he Elements of a Pratial Uniform EM ransmission Line Even toda the priniple of the transmission line has not hanged signifiantl from eaviside s time. It omprises elements of serial apaitane, indutane, resistane and ondutane distributed as evenl as possible along the line. he resistane and ondutane require minimising as the ause unwanted attenuation of the signal. he apaitane and indutane require areful ontrol to balane eah other out as muh as possible. A pratial EM transmission line will ontain elements of series resistane R, series indutane L, parallel ondutane G and parallel apaitane C distributed along the line. Consider a small element of suh a transmission line of short elementar length, as shown in Figure 1-1. he lower ase represents the distane measured along the line whih is also the diretion of propagation. Figure 1-1 represents the setion from an unbalaned transmission line, suh as a oaial able whih is the most ommon form of transmission line. In unbalaned transmission lines the return path omprises a metalli sreen surrounding the inner oaial ondutor used for the forward path. In this ase the return path will have low resistane and indutane, whih is assumed negligible and therefore not shown in Figure 1-1. Figure 1-1 he fundamental element setion of a uniform EM transmission line Man suh elementar setions ma be asaded to form a setion of a pratial, uniform transmission line. he units of the elementar parameters are defined per unit length as follows: ' R ' Ohms per metre ( / m ); ' L' enries per metre ( / m); ' G ' Siemens per metre ( S / m ); ' C ' Farads per metre ( F / m); he impedane of the series setion represented b Z, with an upper ase Z, not to be onfused with the length whih is a lower ase is: Z R jl (1.1) ransmission Lines Page 5 of 1 5 Ma 11

6 and the admittane of the setion is represented b Y, where Y G jc (1.) In (1.1) and (1.) the term is the angular frequen of the applied sinusoidal waveform in radians per seond ( rad / s ). is related to the frequen in hert ( ) of the waveform b: f (1.3) B Kirhoff s laws using the definitions of voltages and urrent shown in Figure 1-1: V V I Z (1.4) 1 1 I I V Y (1.5) 1 If the voltage differene therefore aross Similarl 1 1 is V V V V I Z (1.6) V I1Z V V lim IZ I YV Differentiating (1.7) with respet to : V I Z Z ( YV ) YZV V YZV (1.7) (1.8) (1.9) his differential equation epresses the voltage variation along the line in terms of position. Similarl, the following differential equation ma be derived in terms of the urrent. I YZI 1.4 he otal Voltage Wave (1.1) A pratial linear transmission line will simpl omprise a asade of networks of the tpe shown in Figure 1-1. It will have some loss beause finite values for resistane R and ondutane G. Eah of these parameters will dissipate heat and therefore waste some of the power intended for propagation along the transmission line. Eah will also hange with frequen, the resistane and ondutane tending to inrease with inreasing frequen beause of the skin effet. Eessive heat dissipation is not usuall a problem at low power levels but ma beome so at high power levels if, for eample, the transmission line was used to feed an antenna from a high power transmitter. Indutane also hanges with frequen, tending to inrease with inreasing frequen. Using a transmission line at high powers also inreases the risk of breakdown and/or overheating. Breakdown an our ver quikl, often being initiated b a ver narrow pulse of high instantaneous peak power. Overheating is normall the diret result of ransmission Lines Page 6 of 1 5 Ma 11

7 eessive mean power being dissipated in the R and G elements. Either an ause permanent damage to the transmission line. he series element an be represented as an impedane Z, given b: Z R jl (1.11) and the parallel element an be represented b an admittane Y given b: Y G jc (1.1) (1.9) is a differential equation whih has an eponential solution of the tpe V Ae Be (1.13) where A and B are onstants and is the omple quantit given b ZY R jl G jc (1.14) Furthermore is known as the propagation onstant and is given b j (1.15) where is the attenuation onstant epressed in Nepers per metre ( Np / m ) and is the phase onstant epressed in radians per metre ( rad / m ). A Neper is a logarithmi wa of epressing a power ratio, with a similar definition to the deibel, but using a natural logarithm instead of a ommon logarithm. he bases of logarithms ma be hanged using the following equation: ln b log log 1 (1.16) 1 hus (1.16) ma be used to onvert Nepers to deibels. b e (1.13) is the epression for the total voltage on the transmission line, the sum of the forward ( V F ) and reverse ( V R ) waves. herefore VF VR Ae (1.17) Be (1.18) 1.5 he otal Current Wave V V V (1.19) F R Differentiating the total voltage from (1.13) with respet to gives: But V Ae V IZ Be (1.) (1.1) herefore, using (1.14), (1.) and (1.1), the total urrent waveform is given b: Ae Be I (1.) Z Z Y Y ransmission Lines Page 7 of 1 5 Ma 11

8 1.6 Charateristi Impedane Another wa of writing the total urrent waveform in terms of where Z as: VF and V R IZ V F VR (1.3) is known as the harateristi impedane of the transmission line and is defined Z Z R jl (1.4) Y G jc As Z is an impedane it has dimensions of Ohms ( ). For some transmission lines operating at modest radio frequenies, R and G are negligible ompared to the magnitudes of jl and j C, so Z R jl L G jc C is: (1.5) he harateristi impedane is a basi propert of the line determined b its phsial onstrution and it desribes the fundamental eletrial mathing apabilit of the transmission line. For a transmission line of infinite length, the impedane measured at one end will be Z. If the transmission line is of finite length and terminated with a resistive load Z L idential to Z, the impedane seen at the other end will also be Z. he ondition for maimum power transfer is when the soure impedane equivalent to both Z and Z L. herefore, for maimum power transfer Z Z Z (1.6) S L Z S is also real and Figure 1- he fundamental transmission line onfiguration his is shown shematiall in Figure 1-, in this ase represented b a oaial transmission line, though it would appl to an tpe of transmission line. he soure is shown b a hevenin equivalent iruit of soure impedane Z. he reiproal of harateristi impedane is harateristi admittane Y, where: Y 1 (1.7) Z Although Z is in more ommon usage, Y is often useful when dealing with mathing problems where the design requires swithing between impedanes and admittanes. 1.7 Voltage Refletion Coeffiient, Voltage Standing Wave Ratio and Return Loss Most pratial sstems whih inlude transmission lines will inevitabl have some degree of mismath. A mismath ours in a transmission line when either ZS or Z L differs from S ransmission Lines Page 8 of 1 5 Ma 11

9 Z. When a transmission line is not perfetl mathed the mismath ma be measured b the voltage refletion oeffiient, V. his is a quantit represented in magnitude and phase b the omple ratio of the refleted voltage wave ( V R ) divided b the forward voltage wave ( V R ), or: V Be R V Ce (1.8) V Ae F where C is also a onstant. A similar epression ma be obtained for the urrent refletion oeffiient, but this is less frequentl used. It is normall understood that the simple epression refletion oeffiient refers in fat to the voltage refletion oeffiient. he more perfetl the transmission line is mathed, the smaller the refleted wave will be. herefore V will approah ero magnitude for a perfetl mathed line. For the unmathed transmission line, the presene of in the eponent indiates that the refletion oeffiient varies with the position along the line at whih it is measured. If the line is loss free ( in (1.15)) the eponent beomes purel imaginar and the magnitude of the refletion oeffiient V is unhanged at an position along the line. owever, its phase will var along the line. An unmathed transmission line would set up a standing wave: one in whih V F and V R are finite. he standing wave is the resultant wave formed from the forward and reverse waves, VF and V R respetivel. he standing wave variation with position on the line would be desribed b (1.13). A ommon salar measure of the degree of mismath of a transmission line is the voltage standing wave ratio (VSWR), whose smbol is the lower ase s. his is defined as the ratio of the voltage maima to voltage minima, so VF V R 1V 1 R V F V s V V 1V V 1 F R F R V (1.9) VSWR is often epressed as a ratio in olon notation relative to unit. For eample if s 1.15, then VSWR 1.15 :1. (1.9) relates the VSWR to the magnitude of the voltage refletion oeffiient. he VSWR is a linear salar measure of math qualit. It does not have an units and is alwas greater than or equal to unit whih represents a perfet math. Alternativel the refletion oeffiient magnitude ma be epressed in terms of the VSWR b reorganising (1.9) as follows: s 1 V (1.3) s 1 A ommonl used logarithmi salar measurement of mismath is known as return loss (RL) and is defined as: RL db (1.31) log1 V Return loss is measured in deibels (db) as a positive quantit in the same wa as the loss of an attenuator is a positive quantit. he larger the return loss the better the math. ransmission Lines Page 9 of 1 5 Ma 11

10 1.8 he Variation of Impedane Along a Pratial (Loss) Mismathed ransmission Line For a pratial (loss) transmission line, mismathed at the distant end, the voltage refletion oeffiient will var at points along the line in both magnitude and phase. If the length of the line is l, then substitution into (1.8) for and l will ield the following equation for the refletion oeffiient measured at the input to the line ( ) in terms of the refletion oeffiient at the load onneted to the end of the line ( ). e l (1.3) Dividing (1.19) b (1.3) gives the general epression relating the impedane Z onneted to the end of the line with the forward and refleted waves V F and V R respetivel. VR 1 V Z VF V R VF 1 Z V I Z VF V R R 1 1 V F (1.33) Notie that all terms in this equation are atual omple quantities. In terms of (1.33) beomes: Substituting for Z Z (1.34) Z Z into (1.3) gives l Z Z Z l Z e (1.35) where l is the refletion oeffiient at the input of a line of length l terminated in an impedane Z. A similar equation to ma be written but this time relating the input impedane of line Z IN to the assoiated refletion oeffiient : Z IN (1.36) Z 1 1 Substituting for from (1.3) and epanding Z Z l 1 e l Z 1 IN e Z Z Z Z Z Z e l l Z 1 e Z Z l Z Z Z Z e 1 e Z Z 1 e 1 e Z Z Z Z tanhl tanhl 1 e l Z Z l l 1 e Z Z l he resulting equation is often known as the (loss) transmission line equation: l (1.37) ransmission Lines Page 1 of 1 5 Ma 11

11 Z Z Z tanh l IN Z Z Z tanh l (1.38) his is used to alulate the input impedane of a loss transmission line of harateristi impedane Z, length l, propagation oeffiient whilst it is terminated with an impedane Z. 1.9 he Loss-Free ransmission Line he Loss-Free ransmission Line Equation Often in transmission line problems it is adequate to assume the line to be loss free as this adds several ver welome simplifiations in the algebra suh as dealing with trigonometri funtions instead of hperboli funtions. For a loss free transmission line the attenuation onstant will be ero, so (1.15) beomes j j (1.39) Substituting for j in the loss-free ase and using the Euler identities j j e e os j j e e j sin (1.4) the hperboli tangent simplifies as follows l j l 1e 1e tanh l l j l 1e 1e jl j l e e j sin l jl j l e e os l j tan l (1.41) Performing this substitution ields the transmission line equation for the loss-free ase Z Z jz tan l IN Z Z jz tan l (1.4) his ma be used to alulate the impedane looking into a loss-free transmission line of harateristi impedane Z, length l and terminated at the distant end with an impedane of Z Phase Constant and Phase Veloit In this ase the phase onstant is given b rad / m (1.43) where is the wavelength within the transmission line. his equation is used to determine the spatial phase variation ( l ) along a transmission line. For the loss free ase R and G therefore, b equating (1.14) with (1.39): ransmission Lines Page 11 of 1 5 Ma 11

12 j j LC LC (1.44) herefore, equating (1.43) with (1.44) and using the relationship relating temporal and angular frequen: ields where v P f (1.45) LC f LC 1 LC f v p f 1 LC (1.46) is the phase veloit of propagation within the transmission line. If the transmission line is air spaed then v P is idential to the veloit of eletromagneti radiation in free spae ( v p ). If it inludes a solid dieletri of relative permittivit (dieletri onstant) r, then v P is related to b v p (1.47) r If the dieletri is partiall air-spaed, then it ma be assigned an effetive dieletri onstant k where eff v p (1.48) k he value of k would be found b measurement to be a value between unit (air onl dieletri) and (solid dieletri). r he assumption in these definitions is that the phase veloit is onstant and not a funtion of frequen. ransmission Lines Page 1 of 1 5 Ma 11

13 WAVEGUIDES A waveguide is a transmission line onstruted to guide eletromagneti energ from one loation to another. here are two forms of pratial waveguide: dieletri-ondutor and dieletri-dieletri. he dieletri-ondutor waveguide omprises a ondutor surrounding a dieletri through whih the radio frequen (RF) energ is direted. A dieletri-dieletri waveguide is of similar onstrution with one of the dieletris surrounding the other, and the two dieletris have differing refrative indies, relative permittivities or dieletri onstants..1 he Retangular Waveguide [] he retangular waveguide is a form of ver low loss transmission line used aross the range of mirowave frequenies from approimatel tpiall from about 1 G to over G. It omprises a rigid, preision retangular hollow pipe onstruted from a good eletrial ondutor with the inner hollow formed from a dieletri material. he dieletri must ehibit a suitabl low loss at the maimum frequen to be propagated through the waveguide. piall this might be dr air at normal atmospheri pressure or another gas suh as dr nitrogen at a slightl elevated pressure ompared to atmospheri. he walls are often onstruted from opper, as this is a good eletrial ondutor, relativel heap and hemiall stable. Sometimes the internal walls ma be plated with silver. Silver is a better eletrial ondutor that opper and, provided that the plating thikness is suffiient, the transmission loss will be slightl less that that with an equivalent opper waveguide. Although retangular waveguide is epensive and bulk ompared to the alternative transmission lines suh as mirostrip or oaial ables, it is used widel for high power transmitter feeds, millimetre wave omponents and preision measuring equipment. A differene between a waveguide and a EM transmission line of the tpe desribed in Setion 1.3 is that the waveguide omprises a single ondutor so there is no possible path to form an eletrial iruit requiring separate go and return urrent paths. It annot therefore support EM waves. Unlike EM waves whih propagate down to DC, a retangular waveguide will onl support frequenies above a threshold and then onl in the form or transverse eletri (E) waves or transverse magneti (M) waves. he geometr of a ross setion of the retangular waveguide is shown in Figure -1. he longer and the shorter internal transverse dimensions are a and b respetivel ( a b ). For the most ommon tpe of retangular waveguide a b. he and aes are aligned with the longer and shorter dimensions respetivel. he diretion of propagation through the waveguide, is parallel to the ais and mutuall perpendiular to both the and aes. he eletrial properties of the material within the guide are desribed b its permittivit and permeabilit, where and In (.1) and (.) is the absolute permeabilit r is the relative permeabilit is the absolute permittivit r (.1) r (.) ransmission Lines Page 13 of 1 5 Ma 11

14 r is the relative permittivit also known as the dieletri onstant In most ases, for a pratial waveguides, the filling will be of a non-magneti ( r 1). Figure -1 he retangular o-ordinate sstem used with retangular waveguides he differential form of Farada s Law, also known as the Mawell - Farada equation, is B E (.3) t t in vetor form, where the bold tpe indiates a vetor quantit in this and all subsequent equations. E is the eletri field vetor and is the magneti field vetor. he magneti field is a time varing quantit so its temporal (time dependent) phase ma be epressed, for eample, in the following eponential form (.4) j t e where is the angular frequen in radians per seond ( rad / s ), t is the time in seonds and is the value of at t. Differentiating (.4) with respet to t gives j e t jt j Similarl, the eletri field is also time varing, so E je e t jt je herefore the Mawell-Farada equation beomes (.5) (.6) B E j (.7) t t Another one of Mawell s equations, also known as the Mawell-Ampere equation, is D E J J t t (.8) where J is the ondution urrent vetor and D is the surfae harge densit. ransmission Lines Page 14 of 1 5 Ma 11

15 For the dieletri in the hollow setion of a retangular waveguide, no ondution urrent an flow but an alternating urrent an, also known as a onvetion urrent. herefore J and (.8) simplifies to D E j E t t (.9) (.7) and (.9) are used to derive the differential equations that desribe transverse omponents of both the eletrial and magneti fields as desribed in the net setion ransverse Eletri and ransverse Magneti Fields B epanding the url epression in (.7) and epressing the field in terms of its retangular vetor omponents, and : a a a E j a a a E E E (.1) where a, a and a are the unit vetors in the, and diretions respetivel. As propagation is in the diretion, the spatial phase dependene is also in the diretion, so a similar simplifiation that was adopted with the temporal part in (.6) ma be applied to the spatial part. An eletri field E ma be represented in the eponential spatial phase format as: E (.11) j E e Differentiating this with respet to gives E j je e je (.1) he vetor url equation in (.1) ma be epanded and substitutions made from (.6) and (.1) for the, and oeffiients of E and, followed b equating oeffiients for eah of the unit vetors to ield the following si equations. E je j (.13) E je j E E j j j E j j E j E (.14) (.15) (.16) (.17) (.18) ransmission Lines Page 15 of 1 5 Ma 11

16 Equations (.13) to (.18) ma be solved for the transverse field omponents of E ( E and E ) and ( and ) to give the following where E E j E k j E k j E k j E k (.19) (.) (.1) (.) k k (.3) k is the utoff wavenumber or phase onstant speifi to the waveguide. k is the phase onstant of a plane (EM) wave propagating in an unbounded medium eletriall desribed b and. is the phase onstant in the diretion of propagation along the waveguide. Equation (.3) ma be epressed in terms of wavelengths b where (.4) g is the utoff wavelength for the retangular waveguide under onsideration. is the EM wavelength for an equivalent unbounded plane wave onsidered as propagating through the medium desribed b and. g is the guide wavelength in the diretion of propagation. he results presented in (.19) through (.) are the differential equations whih define transverse eletri (E) fields and transverse magneti (M) fields ransverse Eletri Waves ransverse eletri (E) waves, b definition, are those for whih E and so the differential terms of E in (.19) through (.) beome ero, giving the following equations E E j k j k (.5) (.6) ransmission Lines Page 16 of 1 5 Ma 11

17 j k j k (.7) (.8) ransverse Magneti Waves ransverse magneti (M) waves, b definition, are those for whih E and so the differential terms of in (.19) through (.) beome ero, giving the following equations E E j E k j E k j E k j E k E and M Modes (.9) (.3) (.31) (.3) o eamine the modes that ma be supported b retangular waveguides, we need to first return to the modified version of the Mawell-Farada equation (.7). aking the url of both sides of (.7) and substituting the modified version of the Mawell- Ampere equation from (.9): E j j j E E (.33) he net step is to appl the following vetor identit for a general vetor A to (.33). giving A A A (.34) E E = E (.35) Sine there is no stored harge, the volumetri harge densit beomes ero ( ) and the Mawell Gauss equation D also beomes ero, so and E (.36) E (.37) Substituting (.37) into (.35) gives the elmholt wave equation in terms of eletri field: E E (.38) A similar elmholt epression in terms of the magneti field an be obtained starting with the Mawell-Ampere equation (.8) applied to the internal medium of the waveguide ransmission Lines Page 17 of 1 5 Ma 11

18 whih is an insulator so J and the modified version (.9) is used. A further substitution from (.7) produes the result (.39) (.39) ma be redued to its retangular omponents sine, for the left hand side a a a (.4) and a a a (.41) o appl the equations for the E waves (.5) to (.8), the elmholt equation must be etrated from (.4) and (.41): he spatial phase dependene of Sine k omponent of the in the diretion ma be desribed b (.4) j,, h (, ) e (.43) h is not atuall a funtion of, it ma be treated as a onstant when with respet to, so differentiating Substituting (.44) into (.4) and also using (.44) k k (.3) gives the result k, h (.45) he partial differential equation (.45) ma be solved b the method known as separation of the variables, letting then, h X Y (.46) h X Y h Y X Substituting (.46), (.47) and (.48) into (.45) gives the following result (.47) (.48) 1 d X 1 d Y k (.49) X d Y d he priniple of separation of the variables requires eah of the terms in (.49) to be equal to a onstant thus generating two more differential equations d X k X d (.5) ransmission Lines Page 18 of 1 5 Ma 11

19 d Y k Y d (.51) where k and k are onstants suh that k k k (.5) he solutions of (.5) and (.51) are and X A osk B sin k (.53) Y C os k Dsin k (.54) where A, B, C and D are onstants, so h (, ) XY ( Aos k B sin k )( C os k Dsin k ) (.55) Boundar Conditions for E Modes he waveguide walls are ver good ondutors so will not support an tangential omponents of eletri field. Using the retangular oordinate sstem given in Figure -1, this must our at the waveguide walls as follows: e (, ) at and b (.56) e (, ) at and b (.57) e and e are obtained b differentiating (.55) with respet to both and and using results in (.6) and (.5) respetivel. h k ( Asin k B os k)( C osk D os k ) h k ( Aos k B sin k )( C sin k D os k ) (.58) (.59) B substituting (.58) and (.59) into (.6) and (.5) respetivel, the solutions for and e are j e k ( os sin )( sin os ) A k B k C k D k (.6) k j e k ( sin os )( os os ) A k B k C k D k (.61) k Using the boundar onditions in (.56) and (.57), from (.6), D and Similarl, from (.61), B Using (.43) the solution for n k for n,1,... (.6) b and m k for m,1,... (.63) a is therefore e ransmission Lines Page 19 of 1 5 Ma 11

20 where m n j (,, ) Amn os os e (.64) a b A mn is a onstant originating from the onstants A and C. he solutions for the transverse E and omponents are obtained b differentiating (.64) with respet to or as appropriate and substituting into (.5), (.6), (.7) and (.8) to give the following equations. hese equations will desribe the j n m n j E A os sin mn e (.65) k b a b j m m n j E A sin os mn e (.66) k a a b jm m n j A sin os mn e (.67) k a a b j n m n j A os sin mn e (.68) k b a b Emn mode. he propagation onstant, from (.3) b substituting (.6) and (.63) is m n k k k a b he propagation onstant is real when and the utoff ondition is (.69) k k (.7) k m n a b he general epression for the utoff frequen f mn (.71) for an ombination of m and n is f mn k 1 m n a b (.7) he fundamental mode is that with m 1, n 1, known as E, so 1 f 1 1 (.73) a ransmission Lines Page of 1 5 Ma 11

21 3 REFERENCES 1. Chipman, Robert A.; ransmission Lines; Shaum s Outline Series in Engineering; MGraw ill Book Compan;. Poar, David M.; Mirowave Engineering - hird Edition; John Wile & Sons In; ISBN (5); pp ransmission Lines Page 1 of 1 5 Ma 11