Experiment 6: Transmission Line Pulse Response

Transcription

1 Eperimen 6: Transmission Line Pulse Response Lossless Disribued Neworks When he ime required for a pulse signal o raverse a circui is on he order of he rise or fall ime of he pulse, i is no longer possible o approimae he circui wih lumped elemens of ordinary circui heory. Insead, i is necessary o rea he circui componens as having spaially disribued resisance, capaciance, and inducance. Pracically, his is required in siuaions where he eciaion frequency is high, he circui elemen has a long physical dimension, and/or shor pulse rise imes are used. In his eperimen, we will eamine he properies of pulse propagaion in long inerconnecs, or ransmission lines. Wave Propagaion I(,) L d I(+d,) V(,) C d V(+d,) +d Figure 1 Infiniesimal secion of a lossless ransmission line Our analysis of a disribued nework begins by considering he infiniesimal secion of Figure 1. In his analysis we assume for simpliciy ha he nework has no series resisance or shun conducance. Wih his assumpion, we mainain he essenial feaures of pulse propagaion on ransmission lines, while avoiding he complicaions of he conversion of recangular pulses ino gaussian or complemenary error funcion pulse shapes due o he line losses. The circui parameers in disribued nework analysis are aken on a per uni dimension basis. Tha is, L in Figure 1 has unis of Henries/meer and C has unis of Farads/meer. On his basis, Kirchhoff s curren law gives Or, using he definiion of a derivaive, (1) (2) Khirchhoff s volage law gives or (3) (4) 1

2 Taking he parial derivaive of equaion (2) wih respec o and he parial derivaive of equaion (4) wih respec o (and he converse), we obain (5) Equaions (5) and (6) are referred o as wave equaions. Propagaion Velociy I is sraighforward o verify by direc subsiuion in equaions (5) and (6) ha general soluions of he equaions are (6) (7) (8) where (9) The funcions wih he + superscrip describe waves raveling in he + direcion while hose wih he superscrip represen waves raveling in he direcion. The velociy of propagaion is v. If we eamine he parameers of a coaial cable, for eample, he capaciance/meer is (10) where ε is he permiiviy of he dielecric, a is he radius of he inner wire, and b is he disance from he cener of he inner wire o he inner edge of he ouer sheah. The inducance/meer is essenially he eernal inducance which is (11) where µ is he magneic permeabiliy of he maerial beween he inner and ouer conducors. In he produc LC, geomeric facors cancel and we have where µ r is he relaive magneic permeabiliy of he insulaing medium beween he conducors, ε r is is relaive dielecric permiiviy, c is he velociy of elecromagneic propagaion in free space, 3 (10) 8 meers/sec., and v is he velociy of he propagaing wave in he cable. These relaions hold rue for all wo conducor uniform cross-secion ransmission lines. (12) 2

3 Characerisic Impedance We consider he relaion beween a curren wave, I + ( /v), and a volage wave V + ( - /v), boh raveling in he + direcion. For hese waves (13) So, from equaion (2) we have (14) which implies ha waves of curren and volage raveling in he + direcion have he same funcional dependence on he variable /v, and differ only by a scale facor, Cv. We can wrie And since he unis of Cv are Ohms -1, hen furher, (15) (16) In he same way, The quaniy, R 0, is he characerisic impedance of he ransmission line. (17) (18) So, he general soluions of he wave equaions for he ransmission line are (19) (20) Noice from he above equaions ha he characerisic impedance is no an impedance in he usual sense ha i is measurable by an ohmmeer, bu is simply a characerisic parameer for a disribued nework. Tha is, R 0 is equal o V(,) divided by I(,) only in cerain special condiions. 3

4 The characerisic impedance of a coaial cable is found by insering equaions (10) and (11 ) ino equaion (18) wih he resul (21) We will be using a RG 58/U coaial cable in he eperimen. This cable has µ r = 1, ε r = 2.3, and R 0 = 50 Ohms. Reflecion Coefficiens Infinie Line R s I(0,) V s () V(0,) = 0 R 0, v Figure 2 Semi-infinie ransmission line The paricular soluions o he wave equaions (19) and (20) are deermined by he boundary condiions on he disribued nework. Consider he semi-infiniely long ransmission line shown in Figure 2. Because i eends o infiniy, here can be no waves raveling in he direcion. From equaions (19) and (20) (22) (23) and V(,)=I(,)R 0 (24) for all values of 0 and. This is one of he special cases where he characerisic impedance, R 0, is acually a resisance. From Kirchhoff s volage law and equaion (24) (25) which is jus a volage divider equaion. So, a he sending end he semi-infinie ransmission line appears as a resisor whose value is he characerisic impedance. 4

5 Terminaed Line R s I(0,) I(L,) V s () V(0,) R 0, v V(L,) R L = 0 =L Figure 3 - Terminaed ransmission line In a finie ransmission line, we have o keep all erms of equaions (19) and (20) o allow for possible reflecions. An eaminaion of Figure 3 shows ha (26) Defining a load reflecion coefficien as he raio of he refleced volage wave o he inciden volage wave gives (27) And equaion (26) becomes (28) which gives (29) for an inciden volage wave on he receiving or load end. Using a similar analysis, he source reflecion coefficien is (30) 5

6 for an inciden wave on he sending or source end. In hese analyses, we have no eamined reflecion coefficiens for curren waves bu hey are a sraighforward eension of he volage wave argumens. Noice in equaions (29) and (30) ha when R L and R S are equal o he characerisic impedance, R 0, he reflecion coefficiens on boh ends of he line are zero. Therefore, here are no refleced waves on he ransmission line and he line is said o be properly erminaed. This is also rue when one end has a zero reflecion coefficien and he line is energized from he oher end hrough a source wih a non-zero reflecion coefficien. Any oher siuaion will give a refleced wave on he line. For eample, a zero impedance erminaion or shor circui will have a reflecion coefficien of 1 and he refleced wave will eacly cancel he inciden wave a he shor circui erminaion. An open circui erminaion has a reflecion coefficien of +1 and so he refleced wave adds o he inciden wave a he open circui erminaion. In his laer case, he refleced curren wave cancels he inciden curren wave a he erminaion so as o give zero curren hrough he open circui. The resuling waveforms a he sending and receiving ends of a ransmission line can be mos easily deermined wih he aid of reflecion diagrams. Reflecion Diagrams and Volage Wave Forms 0 L 0 T V(0,) V(L,) 2T 3T ρ L =0 Figure 4 R L and R S = R 0 Figure 4 shows he reflecion diagram, inpu, and oupu volage waveforms for a line erminaed a boh generaor and load ends wih R 0. Because of volage division, ½ of he inpu signal, +, ravels down he line and reaches he load afer a ime equal o he ime delay of he line, which is T. (31) A he load, he reflecion coefficien is zero and here is no refleced wave. 6

7 0 L 0 T - 2T V(0,) V(L,) 3T ρ L = -1 Figure 5 - R L = 0 (SC) and R S = R 0 In Figure 5, + propagaes down he line and reaches he shor circui in ime T. Wih a reflecion coefficien of 1, - is immediaely refleced, canceling he inciden wave of +/2, leaving 0 Vols a he load. The refleced wave of ravels back o he source and cancels he iniial + a ime 2T. 0 L 0 T 2T V(0,) V(L,) 3T ρ L = +1 Figure 6 - R L = (OC) wih R S = R 0 In Figure 6, + propagaes down he line and reaches he open circui a ime T. Wih a reflecion coefficien of +1, + is immediaely refleced, adding o he inciden + and leaving Vols a he load. The refleced wave of + ravels back o he source and adds o he iniial + a ime 2T. 0 L 0 T - 2T V(0,) V(L,) 3T Figure 7 Capacior as load wih R S = R 0 7

8 The reflecion diagram and wave forms associaed wih an uncharged load capaciance are indicaed in Figure 7. An uncharged capacior has no volage across i and so appears o a pulse as an iniial shor circui. A fully charged capacior has he maimum circui volage across i and appears o a pulse as an open circui. Considering hese effecs, he reflecion coefficien for an uncharged load capacior goes from 1 o +1 wih a ime consan of R 0 C. Jusificaion for he ime consan R 0 C comes from applying Thevenin s heorem o he ransmission line, generaor, and generaor resisance (R 0 ) as seen from he capacior. Capaciive erminaions are common in high speed digial inegraed circuis. 0 L 0 T V(0,) V(L,) 2T 3T - Figure 8 Inducor as load wih R S = R 0 The reflecion diagram and wave forms associaed wih an uncharged load inducor are indicaed in Figure 8. An uncharged inducor has no iniial curren hrough i and so appears o a pulse iniially as an open circui. A fully charged inducor has he maimum curren hrough i and so appears o a pulse as a shor circui. The reflecion coefficien for an uncharged load inducor herefore goes from +1 o 1 wih a ime consan of L/R 0. 0 L 0 T V(0,) V(L,) 2T 3T Figure 9 Diode as load wih R S = R 0 The reflecion diagram and waveforms associaed wih a diode or base-emier juncion of a bipolar ransisor load can be seen in Figure 9. The resisance and he reflecion coefficien for hese nonlinear devices varies wih he volage change ha occurs during he finie rise ime of he pulse. Juncion capaciy also plays a par. For hese devices, he reflecion coefficien will have he form (32) 8

9 where (33) 9

10 Eperimen Equipmen Lis 1 Prined Circui Board wih SN74LS241 Ocal Buffer/Driver 1 Prined Circui Board Fiure 1 Coil of RG58/U Coaial Cable wih 50 Ω Characerisic Impedance, ε r = Shor circui erminaion 1 50Ω Terminaion 1 93 Ω Terminaion 1 2 nf Terminaion µh Terminaion 1 Blue Terminaion 1 Blue I Terminaion Procedure Ep. No. 9 Prined Circui Board Ω 50 Ω 1 BNC 4 BNC VDC 22 BNC 18 BNC Inpu 10 kω Figure 9 Ocal Buffer/Driver Circui for Coaial Cable Transmission Lines Se he HP funcion generaor o a 20 khz square wave wih an oupu ha varies from 0 o a posiive value of approimaely 4V and urn i off. Se a DC power supply o a posiive value of 5V and urn i off. Now make he connecions o he ocal buffer/driver indicaed in Figure 9 wih he HP funcion generaor applied o he inpu and urn he dc power supply and he funcion generaor on. Observe and copy he 5, 50, and 10 kω oupu signals from he buffer/driver. A shor (1 meer) 50 Ω coaial cable may be used. For proper operaion hese signals should be square waves wih ampliude of approimaely 0.4 V and period of 50 µs. (a) Connec a long (coiled) 50 Ω coaial cable o he 50 Ω oupu of he buffer/driver wih provisions for measuring he inpu and oupu waveforms of he coaial cable on he oscilloscope. Make some marking on he coil you use in his projec so ha you can idenify i and use he same coil for projec 7. A shor (1 meer) 50 Ω coaial cable may be used o measure inpus o he long cable. Trigger he oscilloscope on he rising edge of he inpu waveform a a ime scale of around 50 ns per division and eperimen wih 10X, 1X probes, a shor (1 meer) coaial cable, and a ee connecion o see which gives he bes oupu waveforms. 10

11 Observe and copy he magniudes and ime relaionships beween he inpu and oupu waveforms for long (coiled) cable erminaions of open circui, shor circui, 50 Ω, 93 Ω, 2 nf, 4.7 µh, blue, and blue I. (b) Connec he long (coiled) 50 Ω coaial cable o he 5 Ω oupu of he buffer/driver. Observe and copy he magniudes and ime relaionships beween he inpu and oupu waveforms for long (coiled) cable erminaions of open circui, shor circui, 50 Ω, and 93 Ω. (c) Connec he long (coiled) 50 Ω coaial cable o he 10 kω oupu of he buffer/driver. Observe and copy he magniudes and ime relaionships beween he inpu and oupu waveforms for an open circui long (coiled) cable erminaion. Repor (a) (i) Eplain how you are able o deermine he characerisic impedance of he long (coiled) coaial cable ransmission line from your measuremens. Calculae he impedance of his ransmission line using your measuremens for each of he source impedance s in pars (a), (b), and (c) (ii) Eplain how you are able o deermine he lengh of he long (coiled) coaial cable ransmission line from your measuremens. Wha did you obain for is value? (iii) Calculae he capaciance and inducance per uni lengh of he long (coiled) coaial cable ransmission line. (iv) Presen and compare he heoreically epeced and eperimenal magniudes and ime relaionships beween he inpu and oupu waveforms for oupu erminaions of open circui, shor circui, 50 Ω, 93 Ω, 2 nf, and 4.7 µh wih a 50 Ω source resisance. (v) From your eperimenal daa derive equivalen circuis wih componen values for blue and blue I erminaions. (b) Presen and compare he heoreically epeced and eperimenal magniudes and ime relaionships beween he inpu and oupu waveforms for long (coiled) coaial cable erminaions of open circui, shor circui, 50 Ω, and 93 Ω wih a 5 Ω driver oupu resisance. (c) Presen and compare he heoreically epeced and eperimenal magniudes and ime relaionships beween he inpu and oupu waveforms for an open circui long (coiled) coaial cable erminaion wih a 10 kω driver oupu resisance. Show your calculaion in deail. (d) Consider he juncion beween properly erminaed 50 Ω and 93 Ω ransmission lines. Design a resisive nework ha will eliminae reflecions from he juncion back ino eiher line. (Noe: There will be some loss of power a he maching nework.) References and Suggesed Reading 11

12 1. Fawwaz T. Ulaby, Fundamenals of Applied Elecromagneics, (Pearson Prenice Hall, Upper Saddle River, New Jersey, 2004) 2. C. W. Davidson, Transmission Lines for Communicaions, (Wiley, New York, 1978) 3. R. E. Maick, Transmission Lines for Digial and Communicaion Neworks, (McGraw- Hill, New York, 1969). 4. G. Mezger and J. P. Vabre, Transmission Lines wih Pulse Eciaion, (Academic Press, New York, 1969) 12