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

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) Lecture 4

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 Lecture 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) Lecture 4

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 Lecture 4

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 Lecture 4

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

Expressions will be derived in later chapters

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

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

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) Lecture 4

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 ), Lecture 4

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) Lecture 4

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) Lecture 4

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) Lecture 4

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) Lecture 4