Pref (0)
standing wave solution
V ( z ) = V o+ e − j β z + V o− e j β z I (z) = I e
+ o − jβ z
+I e
− o
jβ z
V o+ − jβ z V o− j β z e e = − Zo Zo
2-6
路
• Reflection coefficient (applications in measurement, radar, and remote sensing)
2-8
路
(derivation) 1 T 1 T v(z, t )i(z,t )dt = ∫ Re[V (z)e jwt ]Re[ I ( z)e jwt ]dt T ∫0 T 0 1 T1 1 = ∫ [V (z)e jwt + V * ( z)e− jwt ] [ I ( z)e jwt + I * ( z)e− jwt ]dt T 0 2 2 1 T = ∫ [V (z) I ( z)e j 2 wt + V * ( z) I * ( z)e− j 2 wt + V ( z) I * ( z) + V * ( z) I ( z)]dt 4T 0 T 1 1 = [V ( z) I * ( z) + V * ( z) I ( z)]∫ dt = 2Re[V ( z) I * ( z)]T 0 4T 4T 1 = Re[V ( z) I * ( z)] 2 1 1 1 = Re{[Vo+ e− jβz + Vo−e jβz ] [Vo+ e− jβz − Vo−e jβz ]*} = Re{[Vo+ e− jβz + Vo−e jβz ][Vo+*e jβz − Vo−*e− jβz ]} Zo 2 2Z o Pin (z) =
V − ( − l ) V o− e − jβ l Γ (−l ) ≡ + = + jβ l = Γ L e − j 2 β l = e − jβ l Γ L e − jβ l V (−l ) Vo e V o− ∵ Γ (0) = + ≡ Γ L Vo
• Input impedance (applications in circuit design) Z + jZ o tan β l V (−l ) = Zo L Z in ( − l ) ≡ I (−l ) Z o + jZ L tan β l
+ − o
wave
G + jwC
I o+
I o−
( derivation ) γ dV ( z ) = −γVo+ e − γz + γVo− e γz = − ( R + jwL ) I ( z ) → I ( z ) = (Vo+ e − γz + Vo− e γz ) dz R + jwL Vo+ − γz Vo− γz R + jwL =I ( z ) + I ( z ) = e + e ⇒ Zo = = Zo Zo γ 2-3
2-5
路
2.3 The terminated lossless transmission line
Zo
Source match
I(-l) Zo, Γs=0 + V(-l) Γ(-l) Z in ( − l ), Pin ( − l ) -l ΓL ZL
0 z Pinc (0) Pin (0), or Ptrans (0)
ΓL
e jβ l (1 + Γ L e j
ΓL
e − j 2β l )
e − j 2βl Γ L − 2 β l = π → V m in = V o+ (1 − Γ L )
Γ L − 2 β l = 0 → V m ax = V o+ (1 + Γ L ),
• Time-average input power flow
→ Z in (0) = Z in
( Z L + Z o ) e jβ l + ( Z L − Z o ) e − jβ l Z L ( e jβ l + e − jβ l ) + Z o ( e jβ l − e − jβ l ) e jβ l + Γ L e − jβ l = Z o jβ l = Zo = Zo e − Γ L e − jβ l ( Z L + Z o ) e jβ l − ( Z L − Z o ) e − jβ l Z L ( e jβ l − e − jβ l ) + Z o ( e jβ l + e − jβ l ) Z L cos β l + jZ o sin β l Z + jZ o tan β l = Zo L jZ L sin β l + Z o cos β l Z o + jZ L tan β l
e
0m
− αz
P + ( z ) = P + (0) e −2 αz
1m
z
2-4
路
2. phase constant phase velocity
β=
2π w = vp λ w β
wavelength group velocity
vp =
2π β d β −1 vg = ( ) dw λ=
3. characteristic impedance (wave impedance)
( derivation ) Z in
l=0
V o+ e jβ l + V o− e − jβ l V o+ e jβ l (1 + Γ L e − 2 jβ l ) 1 + Γ L e − 2 jβ l V (−l ) = = Z o + jβ l = Z o + jβ l = Zo I (−l ) V o e − V o− e − jβ l V o e (1 − Γ L e − 2 jβ l ) 1 − Γ L e − 2 jβ l Z − Zo 1+ ΓL V (0) ≡ ZL = Zo ⇒ ΓL = L I (0) 1− ΓL ZL + Zo
+ −
R + jwL G + jwC
路
time-domain solution v ( z,t ) = Vo+ e − αz cos( wt − βz + ∠ Vo+ ) + Vo− e αz cos( wt + βz + ∠ Vo− ) i ( z,t ) = I o+ e − αz cos( wt − βz + ∠ I o+ ) + I o− e αz cos( wt + βz + ∠ I o− )
Chapter 2 Transmission Line Theory
2.1 The lumped-element circuit model for a transmission line transmission line or telegrapher equation, traveling wave solution 2.3 The terminated lossless transmission line Zin, Γ, VSWR time-average power flow 2.4 The Smith chart Zin-plot conformal mapped on Γ-plot 2.5 The quarter-wave transformer frequency response, TDR 2.6 Generator and load mismatches impedance match, conjugate match 2.7 Lossy transmission lines low loss line, distortionless line perturbation method
= Zo
2-7
路
• Voltage standing wave ratio VSWR ≡ Vmax = 1 + ΓL
Vmin 1 − ΓL
V o+
( d eriva tio n ) V ( − l ) = V o+ e j β l + V o− e − j β l = V o+ e j β l (1 + Γ L e − j 2 β l ) = V o+ e j V ( − l ) = V o+ 1 + Γ L e j
incident
• Traveling wave solution
V ( z ) = V + ( z ) + V − ( z ) = Vo+ e − γz + Vo− e γz
I+(z) I(z)
I¯(z)
reflected wave
+ + + V V V+(z) V(z) V¯(z) I ( z ) = I + ( z ) + I − ( z ) = I o+ e − γz + I o− e γz = o e − γz − o e γz Zo Zo - + − V R + jwL Vo z ⇒ Z ≡ = = − o : characteristic impedance
Discussion: 1. attenuation constant
V + (1m) = V + (0m) e −α α = ln V + (0m) V (1m)