Lecture Outline. Introduction Transmission Line Equations Transmission Line Wave Equations 8/10/2018. EE 4347 Applied Electromagnetics.

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1 8/10/018 Course Insrucor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) E Mail: rcrumpf@uep.edu EE 4347 Applied Elecromagneics Topic 4a Transmission Line Equaions Transmission These Line noes Equaions may conain copyrighed maerial obained under fair use rules. Disribuion of hese maerials is sricly prohibied Slide 1 Lecure Ouline Inroducion Transmission Line Equaions Transmission Line Wave Equaions Transmission Line Equaions Slide 1

2 8/10/018 Inroducion Transmission Line Equaions Slide 3 Map of Waveguides (LI Media) Single Ended Differenial Transmission Lines Conains wo or more conducors. No low frequency cuoff. Though of more as a circui clemen Homogeneous Has TEM mode. Has TE and TM modes. coaxial sripline buried parallel plae Inhomogeneous Suppors only quasi (TEM, TE, & TM) modes. microsrip coplanar coplanar srips Waveguides Confines and ranspors waves. Suppors higher order modes. Meal Shell Pipes Homogeneous Inhomogeneous Pipes Enclosed by meal. Does no suppor TEM mode. Has a low frequency cuoff. Suppors TE and TM modes recangular circular dual ridge Suppors TE and TM modes only if one axis is uniform. Oherwise suppors quasi TM and quasi TE modes. Has one or less conducors. Usually wha is implied by he label waveguide. Dielecric Pipes Composed of a core and a cladding. Symmeric waveguides have no low frequency cuoff. Channel Waveguides Confinemen along wo axes. TE & TM modes only suppored in circularly symmeric guides. opical Fiber phoonic crysal Slab Waveguides rib Confinemen only along one axis. Suppors TE and TM modes. Inerfaces can suppor surface waves. shielded pair Transmission Line Equaions sloline no uniform axis (no TE or TM) uniform axis (has TE and TM) dielecric Slab large area parallel plae inerface Slide 4

3 8/10/018 Transmission Line Parameers RLGC We can hink of ransmission lines as being composed of millions of iny lile circui elemens ha are disribued along he lengh of he line. In fac, hese circui elemen are no discree, bu coninuous along he lengh of he ransmission line. Transmission Line Equaions Slide 5 RLGC Circui Model I is no echnically correc o represen a ransmission line wih discree circui elemens like his. However, if he sie of he circui is very small compared o he wavelengh of he signal on he ransmission line, i becomes an accurae and effecive way o model he ransmission line. Transmission Line Equaions Slide 6 3

4 8/10/018 L Type Equivalen Circui Model Disribued Circui Parameers R (/m) Resisance per uni lengh. Arises due o resisiviy in he conducors. L (H/m) Inducance per uni lengh. Arises due o sored magneic energy around he line. G (1/m) Conducance per uni lengh. Arises due o conduciviy in he dielecric separaing he conducors. 1 G R C (F/m) Capaciance per uni lengh. Arises due o sored elecric energy beween he conducors. There are many possible circui models for ransmission lines, bu mos produce he same equaions afer analysis. R L G C Transmission Line Equaions Slide 7 L Type Equivalen Circui Model Disribued Circui Parameers R (/m) Resisance per uni lengh. Arises due o resisiviy in he conducors. L (H/m) Inducance per uni lengh. Arises due o sored magneic energy around he line. G (1/m) Conducance per uni lengh. Arises due o conduciviy in he dielecric separaing he conducors. 1 G R C (F/m) Capaciance per uni lengh. Arises due o sored elecric energy beween he conducors. There are many possible circui models for ransmission lines, bu mos produce he same equaions afer analysis. R L G C Transmission Line Equaions Slide 8 4

5 8/10/018 L Type Equivalen Circui Model Disribued Circui Parameers R (/m) Resisance per uni lengh. Arises due o resisiviy in he conducors. L (H/m) Inducance per uni lengh. Arises due o sored magneic energy around he line. G (1/m) Conducance per uni lengh. Arises due o conduciviy in he dielecric separaing he conducors. 1 G R C (F/m) Capaciance per uni lengh. Arises due o sored elecric energy beween he conducors. There are many possible circui models for ransmission lines, bu mos produce he same equaions afer analysis. R L G C Transmission Line Equaions Slide 9 L Type Equivalen Circui Model Disribued Circui Parameers R (/m) Resisance per uni lengh. Arises due o resisiviy in he conducors. L (H/m) Inducance per uni lengh. Arises due o sored magneic energy around he line. G (1/m) Conducance per uni lengh. Arises due o conduciviy in he dielecric separaing he conducors. 1 G R C (F/m) Capaciance per uni lengh. Arises due o sored elecric energy beween he conducors. There are many possible circui models for ransmission lines, bu mos produce he same equaions afer analysis. R L G C Transmission Line Equaions Slide 10 5

6 8/10/018 Relaion o Elecromagneic Parameers,, Every ransmission line wih a homogeneous fill has: LC G C Transmission Line Equaions Slide 11 Example RLGC Parameers RG 59 Coax CAT5 Twised Pair Microsrip R 36 mω m L 430 nh m G 10 m C 69 pf m Z 75 0 R 176 mω m L 490 nh m G m C 49 pf m Z R 150 mω m L 364 nh m G 3 m C 107 pf m Z 50 0 Surprisingly, almos all ransmission lines have parameers very close o hese same values. Transmission Line Equaions Slide 1 6

7 8/10/018 Transmission Line Equaions Transmission Line Equaions Slide 13 E & H V and I Fundamenally, all circui problems are elecromagneic problems and can be solved as such. All wo conducor ransmission lines eiher suppor a TEM wave or a wave very closely approximaed as TEM. An imporan propery of TEM waves is ha E is uniquely relaed o V and H and uniquely relaed o E. V Ed I H d L This le s us analye ransmission lines in erms of jus V and I. This makes analysis much simpler because hese are scalar quaniies! L Transmission Line Equaions Slide 14 7

8 8/10/018 Transmission Line Equaions The ransmission line equaions do for ransmission lines he same hing as Maxwell s curl equaions do for unguided waves. Maxwell s Equaions H E E H Transmission Line Equaions V I RI L I V GV C Like Maxwell s equaions, he ransmission line equaions are rarely direcly useful. Insead, we will derive all of he useful equaions from hem. Transmission Line Equaions Slide 15 Derivaion of Firs TL Equaion (1 of ) + V, 1 R L 3 I, G C 4 + V, Apply Kirchoff s volage law (KVL) o he ouer loop of he equivalen circui: I, V, I, RL 1 V, 0 3 Transmission Line Equaions Slide

9 8/10/018 Derivaion of Firs TL Equaion ( of ) We rearrange he equaion by bringing all of he volage erms o he lef hand side of he equaion, bringing all of he curren erms o he righ hand side of he equaion, and hen dividing boh sides by. I, V, I, RL V, 0 V, V, I, RI, L In he limi as 0, he expression on he lef hand side becomes a derivaive wih respec o. V, I, RI, L Transmission Line Equaions Slide 17 Derivaion of Second TL Equaion (1 of ) + V, R L I, G C I, + V, Apply Kirchoff s curren law (KCL) o he main node he equivalen circui: V, I, I, GV, C Transmission Line Equaions Slide

10 8/10/018 Derivaion of Second TL Equaion ( of ) We rearrange he equaion by bringing all of he curren erms o he lef hand side of he equaion, bringing all of he volage erms o he righ hand side of he equaion, and hen dividing boh sides by. V, I, I, GV, C 0 I, I, V, GV, C In he limi as 0, he expression on he lef hand side becomes a derivaive wih respec o. I, V, GV, C Transmission Line Equaions Slide 19 Transmission Line Wave Equaions Transmission Line Equaions Slide 0 10

11 8/10/018 Saring Poin Telegrapher Equaions Sar wih he ransmission line equaions derived in he previous secion. V, I, RI, L I, V, GV, C ime domain For ime harmonic (i.e. frequency domain) analysis, Fourier ransform he equaions above. dv R jl I di G jcv frequency domain Noe: The derivaive d/ became an ordinary derivaive because is he only independen variable lef. These las equaions are commonly referred o as he elegrapher equaions. Transmission Line Equaions Slide 1 Wave Equaion in Terms of V() dv R jl I To derive a wave equaion in erms of V(), firs differeniae Eq. (1) wih respec o. dv di R jl Eq. (3) Transmission Line Equaions Slide di Eq. (1) G jcv Eq. () Second, subsiue Eq. () ino he righ hand side of Eq. (3) o eliminae I() from he equaion. dv R jlg jcv Las, rearrange he erms o arrive a he final form of he wave equaion. dv R jlg jcv 0 11

12 8/10/018 Wave Equaion in Terms of I() dv R jl I To derive a wave equaion in erms of jus I(), firs differeniae Eq. () wih respec o. d I dv G jc Eq. (3) Transmission Line Equaions Slide 3 di Eq. (1) G jcv Eq. () Second, subsiue Eq. (1) ino he righ hand side of Eq. (3) o eliminae V() from he equaion. d I G jcr jli Las, rearrange he erms o arrive a he final form of he wave equaion. d I G jcr jli 0 Propagaion Consan, In our wave equaions, here is he common erm G jc R jl. Define he propagaion consan o be j G jc R jl Given his definiion, he ransmission line equaions are wrien as dv V 0 d I I 0 Transmission Line Equaions Slide 4 1

13 8/10/018 Soluion o he Wave Equaions If he wave equaions are handed off o a mahemaician, hey will reurn wih he following soluions. dv V 0 d I I V V e V e 0 0 I I e I e Forward wave Backward wave Boh V() and I() have he same differenial equaion so i makes sense hey have he same soluion. Transmission Line Equaions Slide 5 13

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