1. 16.360 Lecture 4 Transmission lines 1.Transmission line parameters, equations 2.Wave propagations 3.Lossless line, standing wave and reflection coefficient 4.Input impedence 5.Special cases of lossless line 6.Power flow 7.Smith chart 8.Impedence matching 9.Transients on transmission lines

2. 1.Transmission line parameters, equations Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L V AA’ (t) = Vg(t) = V0cos(  t), V BB’ (t) = V AA’ (t-t d ) = V AA’ (t-L/c) = V0cos(  (t-L/c)), V BB’ (t) = V AA’ (t) Low frequency circuits: Approximate result V BB’ (t) = V AA’ (t) 16.360 Lecture 4

3. 1.Transmission line parameters, equations Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L V BB’ (t) = V AA’ (t-t d ) = V AA’ (t-L/c) = V0cos(  (t-L/c)) = V0cos(  t- 2  L/ ), Recall:  =c, and  = 2  If >>L, V BB’ (t)  V0cos(  t) = V AA’ (t), If <= L, V BB’ (t)  V AA’ (t), the circuit theory has to be replaced. 16.360 Lecture 4

4. 1.Transmission line parameters, equations Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L  = 2  f  t = 0.06  e. g:  = 1GHz, L = 1cm Time delay  t = L/c = 1cm /3x10 10 cm/s = 30 ps Phase shift V BB’ (t) = V AA’ (t)  = 2  f  t = 0.6   = 10GHz, L = 1cm Time delay  t = L/c = 1cm /3x10 10 cm/s = 30 ps Phase shift V BB’ (t)  V AA’ (t) 16.360 Lecture 4

5. Transmission line parameters Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L time delay V BB’ (t) = V AA’ (t-t d ) = V AA’ (t-L/v p ), Reflection: the voltage has to be treat as wave, some bounce back power loss: due to reflection and some other loss mechanism, Dispersion: in material, V p could be different for different wavelength 16.360 Lecture 4

6. Types of transmission lines Transverse electromagnetic (TEM) transmission lines B E E B a) Coaxial lineb) Two-wire linec) Parallel-plate line d) Strip linee) Microstrip line 16.360 Lecture 4

7. Types of transmission lines Higher-order transmission lines a) Optical fiber b) Rectangular waveguidec) Coplanar waveguide 16.360 Lecture 4

8. Lumped-element Model Represent transmission lines as parallel-wire configuration Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B zz zz zz Vg(t) R’  z L’  z G’  z C’  z R’  z L’  z G’  z R’  z C’  z L’  z G’  z C’  z 16.360 Lecture 4

9. Expressions will be derived in later chapters

10. Definitions of TL dimensions TEM (Transverse Electromagnetic): Electric and magnetic fields are orthogonal to one another, and both are orthogonal to direction of propagation

11. 16.360 Lecture 4 Lumped-element Model Represent transmission lines as parallel-wire configuration Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B zz zz zz Vg(t) R’  z L’  z G’  z C’  z R’  z L’  z G’  z R’  z C’  z L’  z G’  z C’  z

12. Transmission line equations Represent transmission lines as parallel-wire configuration V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) V(z,t) = R’  z i (z,t) + L’  z  i (z,t)/  t + V(z+  z,t), (1) i (z,t) i (z+  z,t) i (z,t) = G’  z V(z+  z,t) + C’  z  V(z+  z,t)/  t + i (z+  z,t), (2) 16.360 Lecture 4

13. Transmission line equations V(z,t) = R’  z i (z,t) + L’  z  i (z,t)/  t + V(z+  z,t), (1) V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) i (z,t) i (z+  z,t) -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) : V(z,t) = Re( V(z) e jtjt ), i (z,t) = Re( i (z) e jtjt ), 16.360 Lecture 4

14. Transmission line equations V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) i (z,t) i (z+  z,t) Recall: di(t)/dt = Re(d i e jtjt )/dt ),= Re(i jtjt e jj -  V(z,t)/  z = R’ i (z,t) + L’  i (z,t)/  t, (3) - d V(z)/ dz = R’ i (z) + j  L’ i (z), (4) 16.360 Lecture 4

15. Transmission line equations Represent transmission lines as parallel-wire configuration V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) V(z,t) = R’  z i (z,t) + L’  z  i (z,t)/  t + V(z+  z,t), (1) i (z,t) i (z+  z,t) i (z,t) = G’  z V(z+  z,t) + C’  z  V(z+  z,t)/  t + i (z+  z,t), (2) 16.360 Lecture 4

16. Transmission line equations V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) i (z,t) i (z+  z,t) - 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) : V(z,t) = Re( V(z) e jtjt ), i (z,t) = Re( i (z) e jtjt ), i (z,t) = G’  z V(z+  z,t) + C’  z  V(z+  z,t)/  t + i (z+  z,t), (2) 16.360 Lecture 4

17. Transmission line equations V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) i (z,t) i (z+  z,t) Recall: dV(t)/dt = Re(d V e jtjt )/dt ),= Re(V jtjt e jj -  i(z,t)/  z = G’ V (z,t) + C’  V (z,t)/  t, (6) - d i(z)/ dz = G’ V (z) + j  C’ V (z), (7) 16.360 Lecture 4