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

Further Reading

1.