TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

Transmission Lines

Dr. Blanton - ENTC Transmission Lines 3 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

Dr. Blanton - ENTC Transmission Lines 4 1.Transmission line parameters, equations Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L V AA’ (t) = Vg(t) = V 0 cos(  t), V BB’ (t) = V AA’ (t-t d ) = V AA’ (t-L/c) = V 0 cos(  (t-L/c)), V BB’ (t) = V AA’ (t) Low frequency circuits: Approximate result

Dr. Blanton - ENTC Transmission Lines 5 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) = V 0 cos(  (t-L/c)) = V 0 cos(  t- 2  L/ ), Recall:  =c, and  = 2  If >>L, V BB’ (t)  V 0 cos(  t) = V AA’ (t), If <= L, V BB’ (t)  V AA’ (t), the circuit theory has to be replaced.

Dr. Blanton - ENTC Transmission Lines 6 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 treated 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

Dr. Blanton - ENTC Transmission Lines 7 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

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

Dr. Blanton - ENTC Transmission Lines 9 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

Dr. Blanton - ENTC Transmission Lines 10 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)

Dr. Blanton - ENTC Transmission Lines 11 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 ),

Dr. Blanton - ENTC Transmission Lines 12 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)

Dr. Blanton - ENTC Transmission Lines 13 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)

Dr. Blanton - ENTC Transmission Lines 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) - 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)

Dr. Blanton - ENTC Transmission Lines 15 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)

Dr. Blanton - ENTC Transmission Lines 16 V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) i (z,t) i (z+  z,t) - d i(z)/ dz = G’ V(z) + j  C’ V (z), (7) - d V(z)/ dz = R’ i (z) + j  L’ i (z), (4) Telegrapher’s equation in phasor domain Take d /dz on both sides of eq. (4) - d² V(z)/ dz² = R’ d i (z)/dz + j  L’ d i (z)/dz, (8)

Dr. Blanton - ENTC Transmission Lines 17 - d i(z)/ dz = G’ V(z) + j  C’ V (z), (7) - d V(z)/ dz = R’ i (z) + j  L’ i (z), (4) Telegrapher’s equation in phasor domain substitute (7) to (8) d² V(z)/ dz² = ( R’ + j  L’) (G’+ j  C’)V(z), - d² V(z)/ dz² = R’ d i (z)/dz + j  L’ d i (z)/dz, (8) d² V(z)/ dz² - ( R’ + j  L’) (G’+ j  C’)V(z) = 0, (9) or d² V(z)/ dz² -  ² V(z) = 0, (10)  ² = ( R’ + j  L’) (G’+ j  C’), (11)

Dr. Blanton - ENTC Transmission Lines 18 V(z,t) R’  z L’  z G’  z C’  z V(z+  z,t) i (z,t) i (z+  z,t) - d i(z)/ dz = G’ V(z) + j  C’ V (z), (7) - d V(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’ d V (z)/dz + j  C’ d V (z)/dz, (12)

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

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

Dr. Blanton - ENTC Transmission Lines 21 Wave equations d² i(z)/ dz² -  ² i(z) = 0, (13) d² V(z)/ dz² -  ² V(z) = 0, (10)  =  + j ,  = Re ( R’ + j  L’) (G’+ j  C’),  = Im ( R’ + j  L’) (G’+ j  C’), V(z) = V 0 (14) Solving the second order differential equation + e -z-z + - V0V0 e zz i(z) = I 0 (15) + e -z-z + - I0I0 e zz

Dr. Blanton - ENTC Transmission Lines 22 Wave equations V(z) = V 0 (14) + e -z-z + - V0V0 e zz i(z) = I 0 (15) + e -z-z + - I0I0 e zz where: + V0V0 - V0V0 andare determined by boundary conditions. + I0I0 - I0I0 andare related to - V0V0 + V0V0 andby characteristic impedance Z 0.

Dr. Blanton - ENTC Transmission Lines 23 recall: - d V(z)/ dz = R’ i (z) + j  L’ i (z), (4) V(z) = V 0 (14) V0V0 e zz i(z) = I 0 (15) + e -z-z + - I0I0 e zz e -z-z e -z-z V0V0 + e zz V0V0 - - = ( R’ + j  L’) i (z), (16) i (z) = ( R’ + j  L’)  e -z-z (V0(V0 + e zz V0V0 - - ) I0I0 + =  V0V0 + I0I0 - = -- V0V0 - (17) (18) Characteristic impedance Z 0

Dr. Blanton - ENTC Transmission Lines 24 Characteristic impedance Z 0 I0I0 + = ( R’ + j  L’)  V0V0 + I0I0 - = -- V0V0 - (17) (18) = ( R’ + j  L’)  + Z 0  Define characteristic impedance Z 0 I0I0 + V0V0 =  = ( R’ + j  L’) (G’+ j  C’) ( R’ + j  L’) (G’+j  C’) recall:

Dr. Blanton - ENTC Transmission Lines 25 Summary:  = ( R’ + j  L’) (G’+ j  C’) + Z 0  I0I0 + V0V0 = ( R’ + j  L’) (G’+j  C’) V(z) = V 0 (14) V0V0 e zz i(z) = I 0 (15) + e -z-z + - I0I0 e zz e -z-z (19) (20)

Dr. Blanton - ENTC Transmission Lines 26 Example, an air line : solution: R’ = 0 , G’ = 0 / , Z 0 = 50 ,  = 20 rad/m, f = 700 MHz L’ = ? and C’ = ? Z0Z0 = ( R’ + j  L’) (G’+j  C’) = j  L’ j  C’ = 50   = ( R’ + j  L’) (G’+ j  C’) = jj L’C’  =  + j ,  =  L’C’ = 20 rad/m

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

Dr. Blanton - ENTC Transmission Lines 28 lossless transmission line :  = 0  =  L’C’ lossless line Z0Z0 = ( R’ + j  L’) (G’+j  C’) = j  L’ j  C’ Z0Z0 = L’ C’  = 2  / = 2  /  =  L’C’ 1 Vp =  /  = L’C’ 1

Dr. Blanton - ENTC Transmission Lines 29 For TEM transmission line : Vp = L’C’ 1 L’C’ =  =  1  =  L’C’ =   = rrrr c Z0Z0 = L’ C’ summary : V(z) = V V0V0 e jzjz i(z) = I 0 + e -j  z + - I0I0 e jzjz e = rrrr c Vp = L’C’ 1  =  =  

Dr. Blanton - ENTC Transmission Lines 30 Voltage reflection coefficient : Vg(t) VLVL A z = 0 B l ZLZL z = - l Z0Z0 ViVi i(z) = V(z) = V V0V0 e jzjz - e -j  z e + V0V0 Z0Z0 e jzjz - V0V0 Z0Z0 V L = + = - V0V0 + V0V0 V(z) z = V0V0 Z0Z0 - V0V0 Z0Z0 i(z) z = 0 i L = = ZLZL = VLVL iLiL = + - V0V0 + V0V0 - + V0V0 Z0Z0 - V0V0 Z0Z0 + V0V0 - V0V0 = ZLZL Z0Z0 - ZLZL Z0Z0 +

Dr. Blanton - ENTC Transmission Lines 31 Voltage reflection coefficient : - V0V0 + V0V0 = ZLZL Z0Z0 - ZLZL Z0Z0 +    Current reflection coefficient : - i0i0 + i0i0 = -  i  - V0V0 + V0V0 = -  Notes : 1.|  |  1, how to prove it? 2.If Z L = Z 0,  = 0. Impedance match, no reflection from the load Z L.

Dr. Blanton - ENTC Transmission Lines 32 An example : A’ z = 0 A Z 0 = 100  R L = 50  C L = 10pF f = 100MHz Z L = R L + j/  C L = 50 –j159 = ZLZL Z0Z0 - ZLZL Z0Z0 + 