varying with time, but its phasor amplitude (e.g., A) is constant.

Transcription

1 ntruct Tranmi le have very teretg prpertie, quite ifferent many way t the nrmally aciate with tanar circuit. Thee ifference will be apparent thi cncie verview f the imprtant prpertie an equat ue the uial teay-tate analyi f a gle tranmi le. ( ( tranmi ignal urce le la The tranmi le thi icu i t be cniere tw parallel cnuctr f length, nt t far apart, with unifrm material arun the cnuctr. A tranmi le i al referre t a jut a le r cable. The equat thi icu can be ue fr tw-lea le, caxial cable, me trace n prte circuit bar, an many ther pair f cnuctr. The paive la impeance,, at the utput f the le, can be cmplex (i.e., cnta bth a real an an imagary term: R jx The urce impeance,, can al be cmplex. The equat thi ummary are fr uial teay-tate cnit. Thu, the put urce, hwn a with a urce impeance f, i phar frm. n the real time ma thi urce vltage i j( ωt θ jωt v ( t Ac ( ωt θ ReAe Re e where i the phar repreentat f the urce vltage. Thi urce vltage i nt equal t the vltage at the put f the le at unle. The phar i equal t the amplitue A when θ i er. t i imprtant t recall frm baic circuit that phar are nt a funct f time. Thi implifie the analyi but require the ue f the cmplex peratr j 1. All f the vltage an current variable given thi icu are phar, clug an (, which i a phar that i a funct f. har have bth an amplitue an a phae angle. Travelg an Stang Wave The vltage r current alng a tranmi le riente parallel t the axi can be ecribe by the um f (1 a wave travelg α Ae t c ωt β θ, which i the irect uch a ( referre t a frwar r pitive-travelg wave, an ( a wave α travelg the irect uch a Be t c ( ωt β θ, which i referre t a backwar r negative-travelg wave. The figure that fllw i a plt f a pure travelg wave: a wave that i nly travelg ne irect, thi cae the irect with n attenuat r l (i.e., α. What cmplicate the plttg f thee travelg wave i that they are bth a funct f time, t, an pit,. Nte that the imum amplitue,, an imum amplitue, 1, f the wave (it phar amplitue i the ame. Obviuly, the trength f the ignal i varyg with time, but it phar amplitue (e.g., A i cntant. v(, t t T 4 T 3T 4 pure travelg wave When bth frwar an backwar travelg wave are preent n a le, a tang wave can be pruce. The reultant wave tay ne lcat even a the time creae ce the lcat f the imum an er appear t be fixe. Nte that the ttal imum phar amplitue,, i er fr a tang wave when repreente the time ma fr thi particular ituat. v(, t 1 tang wave Fr ther cnit, hwever, the tw travelg wave nt appear t generate either a pure travelg r tang wave but a k f cmbat f the tw. An example f thi ituat fllw. n thi cae, the imum phar amplitue,, i nt er. v(, t.5 tang an travelg wave t i very cnvenient practice t meaure the rm value f a vltage. The rm value f a ignal, which mathematically vlve the time tegrat f the ignal, i nt a funct f time. The rm value f a ignal a a funct f i cniere a tang wave even thugh it i nt a funct f time. Thi tang wave pattern i evient the fllwg plt. Kenneth. Kaier, er 1/9/7

2 vrm ( λ rm, rm, ariable an Unit variable name an S unit * cmplex cnjugate negative-travelg wave icatr phae angle f a cmplex number pitive-travelg wave icatr.5 A B ne-way attenuat B when ( C G G ( m c capacitance per unit length (F/m itance t la (m cnuctance per unit length (S/m cnuctance f la when la i purely real phar current the upper cnuctr f the le at (A imagary part cient t the la j cmplex peratr 1 uctance per unit length (Η/m imum imum n teger clug er (, ± 1, ± c pen-circuite la ( average pwer (W Q reactive pwer (AR R reitance per unit length (Ω/m Re real part ref reflecte frm the la R reitance f la when la i purely real rm rt mean quare S cmplex pwer (A vltage tang wave rati (SWR c hrt-circuite la ( tan -1 vere tangent r arctangent v velcity (m/ ( phar vltage acr the put f the le al referre t a the eng-en vltage ( ( phar vltage acr the utput f the le al referre t a the receivg-en vltage ( ( phar vltage acr the le at ( cient r frwar cmpnent f travelg vltage phar ( reflecte r backwar cmpnent f travelg current phar ( lcat alng the le (m put impeance (Ω put impeance with the la (Ω α β γ Τ λ φ ω lcat f firt put impeance imum frm la (m la impeance (Ω characteritic impeance (Ω mimatche matche urce impeance (Ω attenuat cntant (neper/m phae cntant (ra/m prpagat cntant α jβ (1/m tranmi cefficient wavelength (m (vltage reflect cefficient phae angle f reflect cefficient (ra raian frequency (ra/ umpe Mel arameter R ( ( G ( C ( An electrically mall egment f a le f length ( λ 1 << i hwn. The parameter R,, G, an C are per unit length. The reitance f bth cnuctr i mele by R. The reitance f the ielectric between the tw cnuctr i mele by the cnuctance, G, which i nt equal t 1/R. (A G ecreae, the ielectric le ecreae. The capacitance between the cnuctr i mele by C, an the uctance f the path generate by the tw cnuctr i mele by. A le i cniere lle when bth R an G are er (r negligible. A tranmi le i nt a gle lumpe circuit a hwn but the um f many f thee lumpe circuit. (The number f thee egment fr a le f length wul be /. The reultant repreentat fr the le a apprache er i referre t a a itribute circuit. Characteritic mpeance The characteritic impeance f a le,, i prbably the mt imprtant parameter f a le. Thi i the value ften prte n cmmn caxial cable (e.g., 5 Ω r 75 Ω. The characteritic impeance i nt the ttal reitance r impeance f the cable, althugh it i ften referre t a the le impeance, an it i nt a funct f the length f the cable. R jω ( RG ω C jω ( G RC ( ω G jωc G C Generally, i a cmplex value, but at higher frequencie it i apprximately C. Thi real expre i the mt cmmnly ue equat fr the characteritic impeance. The fllwg expre are ueful ce they nt vlve the quare rt f a cmplex number r funct. Kenneth. Kaier, er 1/9/7

3 R G if R >> ω, R ( ωc ( 1 j if R >> ω, ω ( G ( 1 j if R << ω, C if R << ω, Reflect Cefficient The reflect cefficient i a meaure f the clene f t. t i generally al a cmplex quantity unle bth the la i purely reitive an the le i lle. When the la i equal t the le impeance, the le i referre t a matche an the reflect cefficient i er. A wave n a le impgg n a matche la will nt reflect ff the la. 1 if 1 if if φ 1 if le lle & la purely reactive 1 1 At the la the reflect cefficient i the rati f the reflecte vltage cmpnent,, t the cient vltage cmpnent,. (A reflect cefficient can al be efe fr the current. Althugh nt ue thi ummary, the tranmi cefficient i efe a 1. rpagat Cntant The prpagat cntant etere hw quickly a wave attenuate r ecay ver the length f the le. t al etere hw the phae f the wave change ver the length f the le. The attenuat r l i etere by α an the phae change by β. γ α jβ ( R jω( G jωc ( RG ω C jω ( RC G c 1 R jω tanh 1 c ( ω G j C The fllwg apprximat are ueful ce they nt vlve the quare rt f a cmplex number r funct. RG if R >> ω, ωrc ( 1 j if R >> ω, γ ωg ( 1 j if R << ω, jω C if R << ω, le e Baic f the le f a le are mall, which i ften the cae if the frequency i nt t mall r t high, then the fllwg relathip are ue. Nte that the velcity f a wave n the le, v, an the characteritic impeance,, are nt a funct f the frequency. α, β π λ ω C, v 1 C ω β, C, 1 nput mpeance (le e The impeance lkg t a tranmi le i nt necearily equal t r. The impeance lkg twar the la alng the le i periic with a peri f λ/. When the la i either a hrt r an pen circuit, then the put impeance i either uctive r capacitive with n real cmpnent ce there are n reitive urce alng the lle le. When the la i reitive r cmplex, then the put impeance can be nearly anythg. t value i a funct f the frequency, le impeance, la impeance, phae cntant, an length. Fr the ften eirable matche cnit,, the put impeance i alway. When λ/4, the tranmi le i referre t a a quarter-wave tranfrmer. ( j tan ( β 1 e j tan β 1 e ( ( j β j β ( β ( β j tan if j tan if if ( ( if 8 j j λ if λ 4 if λ if λ When the le i electrically hrt an the la i purely reitive, the le impeance i reitive an uctive r reitive an capacitive epenent n the ie f the la: ω ( { ω ( } R j 1 RC if β << 1, >> R 1 G j C 1 G C if β << 1, << R nteretgly, the imum amplitue f the impeance lkg twar the la i at the vltage imum alng the le. The imum amplitue f thi impeance i at the vltage imum alng the le. Furthermre, thee imum an imum value are entirely real at thee lcat. n rer t bta thee extreme value, the le mut be ufficiently lng that thee value are reache.,,, tan ( β tan ( β ( β ( β j c c c j 1 j tan j tan The characteritic impeance can al be btae by meaurg the put impeance f the le when the la i a hrt circuit an an pen circuit. c c ltage (le e The vltage acr the le i ecribe by a ifferential equat referre t a the tranmi le equat: ( β ( When thi ifferential equat i lve, it i etere that the Kenneth. Kaier, er 1/9/7

4 ttal vltage acr the cnuctr i given by the um f a j pitive-travelg wave, e β, an a negative-travelg wave, j e β. Althugh the vltage wavefrm i patially periic, repeatg every wavelength, λ, it amplitue repeat every λ/. jβ jβ jβ e e e 1 ( ( [ ] jβ j β ( j β e 1 e where e 1 cβ ( φ ( ( λ ( c β ( if β ( if if 1 1 jβ ( ( ( ( e 1 1 jβ ( ( ( ( e The phae angle at λ/4 multiple alng the le relative t the phae angle at the la i 1 ( ( ( λ 4 tan ( 9 ( ( ( λ 18 1 ( ( ( 3λ 4 tan ( 9 1 where tan ( X R ( ( X R X R The imum amplitue f the vltage ccur at the la when > an i purely reitive: ( ( ( 1 ( 1 ( π φ ( β ccur at n ccur at if > & real The imum amplitue f the vltage ccur at the la when < an i purely reitive: ( ( ( ( 1 1 { ( π φ ( β } ccur at n 1 ccur at if < & real When the la an le impeance are equal, the imum an imum amplitue are equal: matche Current (le e The current alng the le i al ecribe by the tranmi le equat: ( β ( The lut t thi ifferential equat fr the current i imilar t the vltage expre. t i nt, hwever, merely unle the le i matche. Nte the ign equal t ( frnt f the negative-travelg term. The amplitue f the current, a with the vltage, ha a patial peri f λ/. jβ jβ jβ e e e ( e jβ 1 e jβ ( [ 1 ( ] ( 1 c ( β φ ( ( λ ( ( β ( if ( cβ ( if if 1 λ 4 9 tan X R ( ( ( ( ( ( ( λ 18 1 ( ( ( 3λ 4 9 tan ( X R ( 1 ( 1, The current amplitue imum i lcate at the vltage amplitue imum, an the current imum at the vltage imum. The relathip between the current, vltage, an put impeance i hwn the given figure fr a mimatche le (the variable are cale by ifferent factr. ( λ ( ( wer (le e The egree f mimatch between the la an le etere the pwer that i actually elivere t the la. nteretgly, the time-average pwer alng a lle le i nt a funct f. Thi i reanable ce there are n le le. Hwever, the reactive pwer, Q, i a funct f. The ign f thi imagary pwer i an icat f whether the tre energy i uctive ( r capacitive ( fr the given lcat,. The pwer abrbe by the la i given a,. 1 ( S ( jq Kenneth. Kaier, er 1/9/7

5 1 Re ( ( Re ( rm rm ( ( 1 1 m ( ( m ( Q ( β ( φ, ( 1 c, ref,, c, ref, 4 1 c, c, ref, c,, ref c ( 1 The cient pwer i that pwer cntae the wave travelg twar the la, while the reflecte pwer i that pwer cntae the wave reflecte ff the urce the irect f the put f the le. ref, return lb 1lg 1lg c, 1 lg lg 1 reflecte l B, 1lg 1lg 1 c, ( Stang Wave Rati (le e The tang wave rati i ften meaure t etere the relative egree f mimatch between the la an le. t can be efe term f the rati f the vltage magnitue extreme r current magnitue extreme. The tang wave rati i alway greater than r equal t ne. When the le i matche, the tang wave rati i equal t ne, the bet that it can be. Generally, it i eirable t have a lw tang wave rati if if 1 if if > & la purely real if > & la purely real 1 j[ β ± π ] e y e Baic Fr a ly le where α, the wave nt exactly repeat with ce there i attenuat alng the le. The phae velcity i general a funct f frequency, v ω β. When the velcity i a funct f frequency, itrt will ccur fr ignal with mre than ne frequency cmpnent (e.g., a quare wave r expnential pule. Fr the pecial cae where R C G, the velcity i epenent f the frequency an i real. Thi i referre t a a itrtle le, even thugh there are le n the le. Fr a ly le the reflect cefficient can be greater than ne (but n greater than.41. ltage (y e The vltage acr a ly le i ecribe by a ifferential equat that i imilar t the lle cae: ( γ ( The vltage acr the le can be written term f the hyperblic e an ce funct: γ γ γ e e e 1 ( ( [ ] γ ( γ e 1 e ( ch ( γ ( h ( γ ( ch γ ( ( h γ ( When the le i matche t it la, there i n reflect, nly a ampe frwar-travelg wave. Fr the three pecial cae that fllw, the urce impeance i equal t the le impeance: ( α e chγ if, α ( e h γ ( if, α ( e if, 1 1 γ ( ( ( ( e 1 1 γ ( ( ( ( e Current (y e ( γ ( The current alng the le can be written term f the hyperblic e an ce funct: γ γ γ ( e e e [ 1 ( ] e γ 1 e γ ( ( ( ch ( γ h ( γ ( ( ch γ ( hγ ( When the le i matche t it la, there i n reflect, nly a Kenneth. Kaier, er 1/9/7

6 ampe frwar-travelg wave. Fr the three pecial cae that fllw, the urce impeance i equal t the le impeance: α e hγ ( if, α ( e ch γ ( if, α ( e if, nput mpeance (y e tanh ( γ γ 1 tanh ( γ γ 1 tanh ( γ if tanh ( γ if e e if The parameter, α, an β f a ly le can be etere by meaurg the put impeance fr bth a hrt-circuite an an pen-circuite la. ntea f ug an pen-circuite la t etere, an arbitrary can be ue. An fite la impeance i nt that imple t bta at higher frequencie ue t the la h nner capacitance. ( γ ( γ tanh c tanh c c c c c c c 1 1 ln α c 1 c c 1 c c 1 β nπ c c 1 wer (y e n general, the frwar an reflecte pwer cannt be merely ae a with the lle le: 1 S ( ( jq 1 * α ( ( * S e ( α A 1 lg e 8.68α B Stang Wave Rati (y e Generally, there i nt a gle value fr the SWR n a ly le ce the value f the imum an imum amplitue are varyg alng the le. The imum an imum pible value f thee amplitue (i.e., the envelpe are α α( ( e 1 e, pible α α ( ( 1 e e, pible Hwever, if the le i ufficiently lng that there are many patial wave f the vltage/current, then an ajacent imum an imum can be ue t etere a reanable lcal SWR. α ( ( 1 e α ( 1 e ( AB if AB The ne-way l r attenuat alng a ly le B, A B, i ften given per unit length (e.g., 1.6 B/1 m aug a matche la. The ttal l B fr a matche la i then thi l multiplie by the length f the le. When the la i nt matche t the le, the l i greater than thi matchel value: a the tang wave rati creae, the le creae. The ifference between the imum an imum vltage magnitue (an current ecreae when mvg away frm the la. Even when the le i matche t the la, the magnitue f the vltage varie alng the length f the le, beg larget at the urce r put. ( (,pible ly tranmi le many wavelength lng (,pible Althugh the SWR might be mtly cnveniently meaure at the put f the le, the SWR at the la can be ignificantly greater fr a ly le. Auxiliary Relathip tan ( A jb A B B A A B 1 ( 1 A jb A B tan B A ( 1 tan ( 1 A B B A 1 ( A jb A jb A B tan ( B A jωt jωt ( t Re Se Re( A jb e ω ( ω β ( 1 A B c t tan B A ( α jβ jωt α ( ω β ( t, Re Ae e Ae c t α Ae c t X rm ecayg wave travelg the ± irect vali fr uial ignal with er c ffet R R, jω, C 1 jωc lg, 1lg B jx ω π f, λ v f, e c x j x, j 1 x x ( x ( e e x x ( x ( e e x x ( x ( e e x x ( e e ch, h tanh X R C B m 1 ft, 1 mile 5,8 ft, c 3 1 m/ Kenneth. Kaier, er 1/9/7