Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground lane on one side of a PCB and lands on he oher: Microsri is an examle of a ransmission line, hough echnically i is only an aroximae model for microsri, as we will see laer in his course. Why TLs? Imagine wo ICs are conneced ogeher as shown: When he olage a A changes sae, does ha new olage aear a B insananeously? No, of course no. 2013 Keih W. Whies
Whies, EE 481 Lecure 2 Page 2 of 13 If hese wo oins are searaed by a large elecrical disance, here will be a roagaion delay as he change in sae (elecrical signal) raels o B. No an insananeous effec. In microwae circuis, een disances as small as a few inches may be far and he roagaion delay for a olage signal o aear a anoher IC may be significan. This roagaion of olage signals is modeled as a ransmission line (TL). We will see ha olage and curren can roagae along a TL as waes! Fanasic. The ransmission line model can be used o sole many, many yes of high frequency roblems, eiher exacly or aroximaely: Coaxial cable. Two-wire. Microsri, sriline, colanar waeguide, ec. All rue TLs share one common characerisic: he E and H fields are all erendicular o he direcion of roagaion, which is he long axis of he geomery. These are called TEM fields for ranserse elecric and magneic fields. An excellen examle of a TL is a coaxial cable. On a TL, he olage and curren ary along he srucure in ime and
Whies, EE 481 Lecure 2 Page 3 of 13 saially in he direcion, as indicaed in he figure below. There are no insananeous effecs. E E H Fig. 1 A common circui symbol for a TL is he wo-wire (arallel) symbol o indicae any ransmission line. For examle, he equialen circui for he coaxial srucure shown aboe is: Analysis of Transmission Lines As menioned, on a TL he olage and curren ary along he srucure in ime () and in disance (), as indicaed in he figure aboe. There are no insananeous effecs.
Whies, EE 481 Lecure 2 Page 4 of 13 i, +, - i, How do we sole for (,) and i(,)? We firs need o deelo he goerning equaions for he olage and curren, and hen sole hese equaions. Noice in Fig. 1 aboe ha here is conducion curren in he cener conducor and ouer shield of he coaxial cable, and a dislacemen curren beween hese wo conducors where he elecric field E is arying wih ime. Each of hese currens has an associaed imedance: Conducion curren imedance effecs: o Resisance, R, due o losses in he conducors, o Inducance, L, due o he curren in he conducors and he magneic flux linking he curren ah. Dislacemen curren imedance effecs: o Conducance, G, due o losses in he dielecric beween he conducors, o Caaciance, C, due o he ime arying elecric field beween he wo conducors. To deelo he goerning equaions for, will consider only a small secion V and I,, we of he TL. This is so
Whies, EE 481 Lecure 2 Page 5 of 13 small ha he elecrical effecs are occurring insananeously and we can simly use circui heory o draw he relaionshis beween he conducion and dislacemen currens. This equialen circui is shown below: i, +, - R i, L C G +, - Fig. 2 The ariables R, L, C, and G are disribued (or er-uni lengh, PUL) arameers wih unis of /m, H/m, F/m, and S/m, reseciely. We will someimes ignore losses in his course. A finie lengh of TL can be consruced by cascading many, many of hese subsecions along he oal lengh of he TL. In he case of a lossless TL where R = G = 0 his cascade aears as: L L L L C C C C This is a general model: i alies o any TL regardless of is cross secional shae roided he acual elecromagneic field is TEM.
Whies, EE 481 Lecure 2 Page 6 of 13 Howeer, he PUL-arameer alues change deending on he secific geomery (wheher i is a microsri, sriline, wo-wire, coax, or oher geomery) and he consrucion maerials. Transmission Line Equaions To deelo he goerning equaion for,, aly KVL in Fig. 2 aboe (ignoring losses) i,, L, (2.1a),(1) Similarly, for he curren i, aly KCL a he node, i, C i, (2.1b),(2) Then: 1. Diide (1) by :,, i, L (3) In he limi as 0, he erm on he LHS in (3) is he forward difference definiion of deriaie. Hence,, i, L (2.2a),(4) 2. Diide (2) by :
Whies, EE 481 Lecure 2 Page 7 of 13,, 0, i i C (5) Again, in he limi as 0 he erm on he LHS is he forward difference definiion of deriaie. Hence, i,, C (2.2b),(6) Equaions (4) and (6) are a air of couled firs order arial differenial equaions (PDEs) for, and i,. These wo equaions are called he elegraher equaions or he ransmission line equaions. Reca: We exec ha and i are no consan along microwae circui inerconnecs. Raher, (4) and (6) dicae how and i ary along he TL a all imes. TL Wae Equaions We will now combine (4) and (6) in a secial way o form wo equaions, each a funcion of or i only. To do his, ake (4): of (4) and of (6): 2, 2 i, L 2 (7)
Whies, EE 481 Lecure 2 Page 8 of 13 2 i, 2, (6) : C 2 Subsiuing (8) ino (7) gies: 2, 2, LC 2 2 Similarly, 2 i, 2 i, LC 2 2 (8) (9) (10) Equaions (9) and (10) are he goerning equaions for he and deendence of and i. These are ery secial equaions. In fac, hey are wae equaions for and i! We will define he (hase) elociy of hese waeforms as 1 [m/s] (2.16) LC so ha (9) becomes 2, 1 2, (11) 2 2 2 Volage Wae Equaion Soluions There are wo general soluions o (11): 1., (12)
Whies, EE 481 Lecure 2 Page 9 of 13 is any wice-differeniable funcion ha conains,, and in he form of he argumen shown. I can be erified ha (12) is a soluion o (11) by subsiuing (12) ino (11) and showing ha he LHS equals he RHS. Equaion (12) reresens a wae raeling in he + direcion wih seed 1/ LC m/s. To see his, consider he examle below wih = 1 m/s: A = 1 s, focus on he eak locaed a = 1.5 m. Then, 1.5 s 1 0.5 1 The argumen s says consan for arying and. Therefore, a = 2 s, for examle, hen 2 s 0.5 1 Therefore, 2.5 m So he eak has now moed o osiion = 2.5 m a = 2 s.
Whies, EE 481 Lecure 2 Page 10 of 13 Likewise, eery oin on his funcion moes he same disance (1 m) in his ime (1 s). This is called wae moion. The seed of his moemen is 1 m m 1 1 s s 2., (13) This is he second general soluion o (11). This funcion reresens a wae moing in he - direcion wih seed. The comlee soluion o he wae equaion (11) is he sum of (12) and (13), (14) and can be any suiably differeniable funcions, bu wih argumens as shown. Curren Wae Equaion Soluions A similar analysis can be erformed for curren waes on he TL. The goerning equaion for i, is 2 i, 1 2 i, (15) 2 2 2
Whies, EE 481 Lecure 2 Page 11 of 13 The comlee general soluion o his curren wae equaion can be deermined in a manner similar o he olage as i, i i (16) + wae - wae Furhermore, he funcion i can be relaed o he funcion and i can be relaed o. For examle, subsiuing i and differeniaing hen inegraing gies 1 i C ino (6), or i C (17) Bu, 1 C C C LC L We will define L Z0 [] (2.13),(18) C as he characerisic imedance of he ransmission line. (Noe ha in some exs, Z 0 is denoed as R c, he characerisic resisance of he TL). Wih (18), (17) can be wrien as
Whies, EE 481 Lecure 2 Page 12 of 13 Similarly, i can be shown ha i i 0 (19) Z 0 (20) The minus sign resuls since he curren is in he - direcion. Finally, subsiuing (19) and (20) ino (16) gies 1 1 i, (21) Z Z This equaion as well as (14) Z 0 0, (22) are he general wae soluions for and i on a ransmission line. Power Flow These olage and curren waes ransor ower along he TL. The ower flow carried by he forward wae, is 2,,, i, (23) Z0 which is osiie indicaing ower flows in he + direcion. Similarly, he ower flow of he reerse wae is
Whies, EE 481 Lecure 2 Page 13 of 13 2,,, i, (24) Z which is negaie indicaing ower flows in he - direcion. 0