📘 Transmission Line Parameters – R, L, C Derivation
Transmission line performance depends on resistance, inductance and capacitance. Derivations are very important for GATE.
🔹 1️⃣ Resistance of Transmission Line
Resistance per phase:R = ρ l / A
Where: ρ = resistivity l = length A = cross-sectional area Important: Resistance increases with temperature.🔹 2️⃣ Inductance of Single Phase Line
Magnetic flux linkage: λ = (μ₀ I / 2π) ln(D / r') Inductance:L = (μ₀ / 2π) ln(D / r')
Where: D = spacing between conductors r' = GMR (Geometric Mean Radius) For 3-phase line (equilateral spacing):L = (2 × 10⁻⁷) ln(D / r') H/m
Very important formula.🔹 3️⃣ Capacitance of Transmission Line
Capacitance between two conductors:C = (2π ε₀) / ln(D / r)
For 3-phase line:C = (2π ε₀) / ln(D / r) F/m
Where: ε₀ = Permittivity of free space D = spacing r = conductor radius🔹 4️⃣ Important Concepts
- Increasing spacing reduces inductance
- Increasing conductor radius reduces inductance
- Capacitance increases when spacing reduces
- GMR used instead of physical radius in inductance
🔹 5️⃣ Example Problem
Given: D = 2 m r' = 0.01 m Inductance: L = (2 × 10⁻⁷) ln(2 / 0.01) = (2 × 10⁻⁷) ln(200) ln(200) ≈ 5.3 L ≈ 2 × 10⁻⁷ × 5.3L ≈ 1.06 × 10⁻⁶ H/m
🎯 GATE Important Points
- Remember L formula clearly
- Understand role of GMR
- Capacitance formula similar but inverted
- Spacing affects both L and C
Transmission Line Parameters = Foundation of Power Systems
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