Transmission Line Fundamentals
Basic Transmission Line Model
A transmission line is distributed, not lumped.
Per unit length parameters:
- R → resistance (Ω/m)
- L → inductance (H/m)
- C → capacitance (F/m)
- G → conductance (S/m)
Important Equations
(a) Propagation Constant
γ = α + jβ = √[(R + jωL)(G + jωC)]
For lossless line (very common in GATE):
- R = 0, G = 0
- α = 0
- β = ω√(LC)
(b) Characteristic Impedance
Z₀ = √[(R + jωL) / (G + jωC)]
For lossless line:
Z₀ = √(L / C)
Voltage & Current on the Line
General solutions:
V(z) = V⁺e⁻ᵞᶻ + V⁻eᵞᶻ
I(z) = (V⁺ / Z₀)e⁻ᵞᶻ − (V⁻ / Z₀)eᵞᶻ
Reflection Coefficient
At load:
ΓL = (ZL − Z₀) / (ZL + Z₀)
Special cases (memorize):
- Open circuit → Γ = +1
- Short circuit → Γ = −1
- Matched load → Γ = 0
Standing Wave Ratio (SWR)
SWR = (1 + |Γ|) / (1 − |Γ|)
- Perfect match → SWR = 1
- Total reflection → SWR → ∞
Input Impedance of Transmission Line
For a line of length l:
Zin = Z₀ [(ZL + jZ₀tan(βl)) / (Z₀ + jZLtan(βl))]
🔹 Quarter-wave line (l = nλ/4)
Zin = Z₀² / ZL
In the above case for n even:
Zin = ZL
Used for impedance matching
🔹 Half-wave line (l = nλ/2)
Zin = ZL
Quick Table
| Line Length | n | Input Impedance |
|---|---|---|
| λ/4 | Odd | Z₀² / ZL |
| 3λ/4 | Odd | Z₀² / ZL |
| λ/2 | Even | ZL |
| λ | Even | ZL |
Memory Trick
- Odd × λ/4 → Invert
- Even × λ/4 → Repeat
- Any × λ/2 → Repeat