Transmission Line Distributed Parameters
1. Distributed Model (The Core Idea)
A transmission line is modeled using tiny segments of length dx:
- Resistance: R dx
- Inductance: L dx
- Conductance: G dx
- Capacitance: C dx
Per-unit-length parameters:
- R: resistance per unit length (Ω/m)
- L: inductance per unit length (H/m)
- G: conductance per unit length (S/m)
- C: capacitance per unit length (F/m)
2. Capacitance per Unit Length (C)
How much charge can be stored per meter between conductors.
C = Q / (V · length)
Example:
C = (2π ε) / ln(d/r)
3. Inductance per Unit Length (L)
Magnetic energy stored per meter.
L = Φ / (I · length)
Example:
L = (μ / 2π) ln(d/r)
4. Conductance per Unit Length (G)
Leakage current through dielectric.
G = ω C tanδ
5. Resistance per Unit Length (R)
Ohmic loss of conductor.
R = ρ / A
6. Telegrapher’s Equations
∂V/∂x = -(R + jωL) I
∂I/∂x = -(G + jωC) V
7. Characteristic Impedance (Z₀)
Z₀ = sqrt((R + jωL) / (G + jωC))
Lossless Case:
Z₀ = sqrt(L / C)
8. Propagation Constant
γ = sqrt((R + jωL)(G + jωC))
9. Wave Velocity
v = 1 / sqrt(LC)
10. Intuition
- C: electric field storage
- L: magnetic field storage
- R: conductor losses
- G: dielectric leakage
Summary
All of this comes from:
- Maxwell’s equations → fields
- Kirchhoff’s laws → circuits
- Continuous approximation of physical systems