Examples of Transmission Lines: controlled geometry lines
Circuit component size compared to
operating wavelength, λ
TL Analysis is needed if size is not too small compared to wavelength
RLCG Physically based Model (linear) - Distributed parameters
Use KVL and KCL to develop voltage-current relationships
Analysis in both Time and Frequency Domains
Second order differential equation for voltage as function of position and time
V(z)=V+e−γz+V−e+γz I(z)= (1/Zo)⋅{V+e−γz−V−e+γz} Zo=√{Rz+jωLz}/{Gz+jωCz} γ=α+jβ=√{Rz+jωLz}⋅{Gz+jωCz}
Using Laplace Transform
v(z,t)=|V+|e−αzcos(ωt−βz+φ+v)+|V−|e+αzcos(ωt+βz+φ−v)
i(z,t)=|V+/Zo|e−αzcos(ωt−βz+φ+i)−|V−/Zo|e+αzcos(ωt+βz+φ−i)
Only for Lossless Line Rz=Gz=0
v(z,t)=v+(t−[√Lz⋅Cz]⋅z)+v−(t+[√Lz⋅Cz]⋅z))
i(z,t)={1/√Lz⋅Cz}{v+(t−[√Lz⋅Cz]⋅z)−v−(t+[√Lz⋅Cz]⋅z)}
Two traveling waves: ''+z'' wave and ''−z'' wave both have same Zo and γ
Zo: Characteristic impedance, Ohms -- γ=α+jβ: Complex propagation
constant, m−1
α: Attenuation
coefficient, Nepers/m -- β: Phase constant, rad/m
Phase velocity
cph=ω/β, m/s -- Wavelength
λ=cph/f=2π/β, m
V+ and V− are determined by boundary
conditions at TL source and load ends
Define Reflection coefficient
Γ(z)= (V−e+γz)/(V+e−γz)
=''−z wave'' /''+z wave''
Γ(z)=Γ(ℓ)⋅e+2γ(z−ℓ) -- Γ(l)=ΓL=[ZL−Zo]/[ZL+Zo]
Γ(z)=[Z(z)−Zo]/[Z(z)+Zo] --
Z(z)=Zo[1+Γ(z)]/[1−Γ(z)]
Reflections are caused by impedance discontinuities "mismatch -- not equal to Zo"
ZL=Zo,
ΓL=0→Γ(z)=0→Z(z)= Zo:
V−=0, only "+z wave", pure traveling wave, |V (z)|=
|V + |e −αz
Magnitude constant except for decay due to attenuation (loss)
ZL=0+jXL,
|ΓL|=1→|Γ(z)|=1→Z(z)=
0+jX(z): |V+|= |V−|, pure standing
waves, |V(z)|=2|V+|cos(βz+ φ)
Wave does not travel, Magnitude has peaks (=2|V+|) and valleys of zeroes (nodes)
ZL≠Zo,
ΓL≠0→Γ(z)≠0→Z(z)≠
Zo: combination of "+z'' and reflected "-z'' waves
Mix of traveling {|V+|-|V−|}, and standing
{2|V−|} waves
Standing Wave Ratio, SWR = Vmax/Vmin=[1+|Γ|]/[1+|Γ|] -- |Γ|=0→SWR=1, |Γ|=1→SWR=∞
Reflections (standing waves) reduce power delivered to load - also degrade signal quality
Mismatch causes dispersion to communication signals-Reflections limit bandwidth of digital communications
For optimum power & signal delivery, impedance matching at both source and load ends is necessary.