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TELECOMMUNICATIONS

TELECOMMUNICATIONS. Dr. Hugh Blanton ENTC 4307/ENTC 5307. Transmission Lines. Transmission lines. Transmission line parameters, equations Wave propagations Lossless line, standing wave and reflection coefficient Input impedence Special cases of lossless line Power flow Smith chart

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TELECOMMUNICATIONS

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  1. TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

  2. Transmission Lines

  3. Transmission lines • Transmission line parameters, equations • Wave propagations • Lossless line, standing wave and reflection coefficient • Input impedence • Special cases of lossless line • Power flow • Smith chart • Impedence matching • Transients on transmission lines Dr. Blanton - ENTC 4307 - Transmission Lines 3

  4. Transmission line parameters, equations B A VBB’(t) Vg(t) VAA’(t) L A’ B’ VAA’(t) = Vg(t) = V0cos(t), Low frequency circuits: VBB’(t) = VAA’(t) Approximate result VBB’(t) = VAA’(t-td) = VAA’(t-L/c) = V0cos((t-L/c)), Dr. Blanton - ENTC 4307 - Transmission Lines 4

  5. B A VBB’(t) Vg(t) VAA’(t) L A’ B’ • Transmission line parameters, equations Recall: =c, and  = 2 VBB’(t) = VAA’(t-td) = VAA’(t-L/c) = V0cos((t-L/c)) = V0cos(t- 2L/), If >>L, VBB’(t)  V0cos(t) = VAA’(t), If <= L, VBB’(t) VAA’(t), the circuit theory has to be replaced. Dr. Blanton - ENTC 4307 - Transmission Lines 5

  6. B A VBB’(t) Vg(t) VAA’(t) L A’ B’ • Transmission line parameters • time delay VBB’(t) = VAA’(t-td) = VAA’(t-L/vp), • Reflection: the voltage has to be treated as wave, some bounce back • power loss: due to reflection and some other loss mechanism, • Dispersion: in material, Vp could be different for different wavelength Dr. Blanton - ENTC 4307 - Transmission Lines 6

  7. B E • Types of transmission lines • Transverse electromagnetic (TEM) transmission lines B E a) Coaxial line b) Two-wire line c) Parallel-plate line d) Strip line e) Microstrip line Dr. Blanton - ENTC 4307 - Transmission Lines 7

  8. Types of transmission lines • Higher-order transmission lines a) Optical fiber b) Rectangular waveguide c) Coplanar waveguide Dr. Blanton - ENTC 4307 - Transmission Lines 8

  9. Lumped-element Model • Represent transmission lines as parallel-wire configuration A B Vg(t) VBB’(t) VAA’(t) B’ A’ z z z R’z L’z L’z R’z L’z R’z Vg(t) G’z C’z C’z C’z G’z G’z Dr. Blanton - ENTC 4307 - Transmission Lines 9

  10. Transmission line equations • Represent transmission lines as parallel-wire configuration i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z V(z,t) = R’z i(z,t) + L’z  i(z,t)/ t + V(z+ z,t), (1) i(z,t) = G’z V(z+ z,t) + C’z V(z+ z,t)/t + i(z+z,t), (2) Dr. Blanton - ENTC 4307 - Transmission Lines 10

  11. i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z jt jt V(z,t) = Re( V(z) e i (z,t) = Re( i (z) e ), ), • Transmission line equations V(z,t) = R’z i(z,t) + L’z  i(z,t)/ t + V(z+ z,t), (1) -V(z+ z,t) + V(z,t) = R’z i(z,t) + L’z  i(z,t)/ t - V(z,t)/z = R’ i(z,t) + L’  i(z,t)/ t, (3) Rewrite V(z,t) and i(z,t) as phasors, for sinusoidal V(z,t) and i(z,t): Dr. Blanton - ENTC 4307 - Transmission Lines 11

  12. i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z jt e j = Re(i ), - dV(z)/dz = R’ i(z) + jL’ i(z), (4) • Transmission line equations Recall: jt di(t)/dt= Re(d i e )/dt - V(z,t)/z = R’ i(z,t) + L’  i(z,t)/ t, (3) Dr. Blanton - ENTC 4307 - Transmission Lines 12

  13. Transmission line equations • Represent transmission lines as parallel-wire configuration i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z V(z,t) = R’z i(z,t) + L’z  i(z,t)/ t + V(z+ z,t), (1) i(z,t) = G’z V(z+ z,t) + C’z V(z+ z,t)/t + i(z+z,t), (2) Dr. Blanton - ENTC 4307 - Transmission Lines 13

  14. i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z jt jt V(z,t) = Re( V(z) e i (z,t) = Re( i (z) e ), ), • Transmission line equations i(z,t) = G’z V(z+ z,t) + C’z V(z+ z,t)/t + i(z+z,t), (2) - i (z+ z,t) + i (z,t) = G’z V(z + z ,t) + C’z  V(z + z,t)/ t -  i(z,t)/z = G’ V(z,t) + C’  V(z,t)/ t, (5) Rewrite V(z,t) and i(z,t) as phasors, for sinusoidal V(z,t) and i(z,t): Dr. Blanton - ENTC 4307 - Transmission Lines 14

  15. i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z jt e j = Re(V ), - d i(z)/dz = G’ V(z) + jC’ V(z), (7) • Transmission line equations Recall: jt dV(t)/dt= Re(d V e )/dt - i(z,t)/z = G’ V(z,t) + C’  V(z,t)/ t, (6) Dr. Blanton - ENTC 4307 - Transmission Lines 15

  16. i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z - d i(z)/dz = G’ V(z) + jC’ V(z), (7) - dV(z)/dz = R’ i(z) + jL’ i(z), (4) - d²V(z)/dz² = R’ di(z)/dz + jL’ di(z)/dz, (8) • Telegrapher’s equation in phasor domain Take d /dz on both sides of eq. (4) Dr. Blanton - ENTC 4307 - Transmission Lines 16

  17. - dV(z)/dz = R’ i(z) + jL’ i(z), (4) - d²V(z)/dz² = R’ di(z)/dz + jL’ di(z)/dz, (8) • Telegrapher’s equation in phasor domain - d i(z)/dz = G’ V(z) + jC’ V(z), (7) substitute (7) to (8) d²V(z)/dz² = (R’ + jL’) (G’+ jC’)V(z), or d²V(z)/dz² - (R’ + jL’) (G’+ jC’)V(z) = 0, (9) d²V(z)/dz² - ²V(z) = 0, (10) ² = (R’ + jL’) (G’+ jC’), (11) Dr. Blanton - ENTC 4307 - Transmission Lines 17

  18. i(z,t) i(z+z,t) L’z R’z V(z,t) V(z+ z,t) G’z C’z - d i(z)/dz = G’ V(z) + jC’ V(z), (7) - dV(z)/dz = R’ i(z) + jL’ i(z), (4) • Telegrapher’s equation in phasor domain Take d /dz on both sides of eq. (7) - d² i(z)/dz² = G’ dV(z)/dz + jC’ dV(z)/dz, (12) Dr. Blanton - ENTC 4307 - Transmission Lines 18

  19. - dV(z)/dz = R’ i(z) + jL’ i(z), (4) • Telegrapher’s equation in phasor domain - d i(z)/dz = G’ V(z) + jC’ V(z), (7) - d² i(z)/dz² = G’ dV(z)/dz + jC’ dV(z)/dz, (12) substitute (4) to (12) d² i(z)/dz² = (R’ + jL’) (G’+ jC’)i(z), or d² i(z)/dz² - (R’ + jL’) (G’+ jC’) i(z) = 0, (9) d² i(z)/dz² - ²i(z) = 0, (13) ² = (R’ + jL’) (G’+ jC’), (11) Dr. Blanton - ENTC 4307 - Transmission Lines 19

  20.  = Re (R’ + jL’) (G’+ jC’) ,  = Im (R’ + jL’) (G’+ jC’) , • Wave equations d²V(z)/dz² - ²V(z) = 0, (10) d² i(z)/dz² - ²i(z) = 0, (13)  =  + j, Dr. Blanton - ENTC 4307 - Transmission Lines 20

  21.  = Re (R’ + jL’) (G’+ jC’) ,  = Im (R’ + jL’) (G’+ jC’) , z -z -z z e e e e • Wave equations d²V(z)/dz² - ²V(z) = 0, (10) d² i(z)/dz² - ²i(z) = 0, (13)  =  + j, Solving the second order differential equation + - V(z) = V0 (14) + V0 + - + i(z) = I0 (15) I0 Dr. Blanton - ENTC 4307 - Transmission Lines 21

  22. + - - + - + and are related to V0 V0 I0 V0 V0 I0 and by characteristic impedance Z0. -z -z z z e e e e • Wave equations + - V(z) = V0 (14) + V0 + - + i(z) = I0 (15) I0 where: and are determined by boundary conditions. Dr. Blanton - ENTC 4307 - Transmission Lines 22

  23. + - V(z) = V0 (14) + V0 + - + i(z) = I0 (15) I0 + - - V0 V0  + - - dV(z)/dz = R’ i(z) + jL’ i(z), (4) i(z) = ) - (V0 V0 = (R’ + jL’) i(z), (16) (R’ + jL’) -z -z z z -z z z -z e e e e e e e e  - - + + - I0 I0 = = V0 V0 (R’ + jL’) (R’ + jL’) • Characteristic impedance Z0 recall: (17) (18) Dr. Blanton - ENTC 4307 - Transmission Lines 23

  24. (R’ + jL’) V0  = (R’ + jL’) (G’+ jC’) +  I0 (R’ + jL’) (G’+j C’)  - + - - + I0 I0 = = V0 V0 (R’ + jL’) (R’ + jL’) • Characteristic impedance Z0 (17) (18) Define characteristic impedance Z0 recall: + Z0  = = Dr. Blanton - ENTC 4307 - Transmission Lines 24

  25. + - V(z) = V0 (14) + V0 + - + i(z) = I0 (15) I0 + V0  = (R’ + jL’) (G’+ jC’) Z0  + I0 (R’ + jL’) (G’+j C’) = z -z -z z e e e e • Summary: (19) (20) Dr. Blanton - ENTC 4307 - Transmission Lines 25

  26. jL’ = Z0 j C’  = (R’ + jL’) (G’+ jC’) j = L’C’ L’C’ (R’ + jL’) (G’+j C’) = • Example, an air line : R’ = 0 , G’ = 0 /, Z0 = 50,  = 20 rad/m, f = 700 MHz L’ = ? and C’ = ? solution: = 50  =  + j,  =  = 20 rad/m Dr. Blanton - ENTC 4307 - Transmission Lines 26

  27. = (R’ + j L’ ) (G’+ jC’) j  = = L’C’ L’C’ • lossless transmission line :  =  + j, = (R’ + jL’) (G’+ jC’) If R’<< j L’ and G’ << jC’,   = 0 lossless line  Dr. Blanton - ENTC 4307 - Transmission Lines 27

  28. jL’ = Z0 j C’  = L’C’ L’C’ L’C’ (R’ + jL’) Z0 1 (G’+j C’)  = 2/ = =  1 L’ = = Vp = / C’ • lossless transmission line : lossless line  = 0   = 2/ Dr. Blanton - ENTC 4307 - Transmission Lines 28

  29. + - V(z) = V0 + V0 c 1 c 1 1 + = = = = = - + i(z) = I0 I0 Z0     = = = = L’C’ L’C’ rr  rr L’C’  L’C’  L’ -jz jz -jz jz e e e e = C’ • For TEM transmission line : L’C’ =  Vp  • summary : Vp  Dr. Blanton - ENTC 4307 - Transmission Lines 29

  30. B A Z0 VL Vg(t) Vi ZL l z = - l z = 0 + + + - - - V0 V0 V0 V0 V0 V0 + - V(z) = V0 + V0 + - + + - - VL iL - i(z) = V0 V0 V0 V0 V0 V0 Z0 Z0 Z0 Z0 Z0 Z0 i(z) V(z) z = 0 z = 0 - ZL Z0 -jz jz jz -jz e e e e = + ZL Z0 • Voltage reflection coefficient : VL = = + - iL = = + ZL = = - Dr. Blanton - ENTC 4307 - Transmission Lines 30

  31. + + + i0 V0 V0 i  = - = -  - - - i0 V0 V0 • Voltage reflection coefficient : - ZL Z0   = + ZL Z0 • Current reflection coefficient : • Notes : • || 1, how to prove it? • If ZL = Z0, = 0. Impedance match, no reflection from the load ZL. Dr. Blanton - ENTC 4307 - Transmission Lines 31

  32. An example : A RL = 50 f = 100MHz Z0 = 100 A’ CL = 10pF z = 0 ZL = RL + j/CL = 50 –j159 - ZL Z0  = + ZL Z0 Dr. Blanton - ENTC 4307 - Transmission Lines 32

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