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Qwestrum Engineering360 · Electrical & Electronics · Power Systems

Transmission Line Parameters

Line inductance and capacitance depend logarithmically on the spacing-to-radius ratio (using GMD and GMR); these set the surge impedance and hence the surge-impedance loading of a transmission line.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Bundled conductors reduce L and increase C
  • GMD and GMR for composite conductor spacing
  • Short line: lumped R, X; long line: distributed ABCD parameters

Topic details

Introduction

Inductance per phase depends on ln(GMD/GMR): larger spacing or smaller effective radius increases inductance. GMR replaces the physical radius to account for internal flux linkage (GMR = 0.7788r for a solid round conductor). Bundling conductors raises the effective GMR, lowering inductance.

Scope in B.Tech and GATE syllabus

Capacitance depends on ln(GMD/r) using the actual radius r, not GMR, because charge resides on the surface. Bundling raises capacitance.

Key relations & formulas

InductanceL=(μ02π)ln(Dr)HmInductance L = (\frac{\mu_{0}}{2\pi}) ln(\frac{D}{r}) \frac{H}{m}
(single phase, GMR concept)

Formulas (Indian textbook notation)

  • CapacitanceC=2πεln(Dr)FmCapacitance C = 2\pi \frac{\varepsilon}{ln}(\frac{D}{r}) \frac{F}{m}

Formulas (Indian textbook notation)

  • CharacteristicimpedanceZc=LC;SIL=V2ZcCharacteristic impedance Z_{c} = \sqrt{\frac{L}{C}}; SIL = \frac{V^{2}}{Z_{c}}

Notation and sign conventions

Relation 1 —
InductanceL=Inductance L =
InductanceL=(μ02π)ln(Dr)HmInductance L = (\frac{\mu_{0}}{2\pi}) ln(\frac{D}{r}) \frac{H}{m}
(single phase, GMR concept)
Write this relation with symbols exactly as in Electrical Power Systems — CL Wadhwa before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
CapacitanceC=2πεlnCapacitance C = 2\pi \frac{\varepsilon}{ln}

Formulas (Indian textbook notation)

  • CapacitanceC=2πεln(Dr)FmCapacitance C = 2\pi \frac{\varepsilon}{ln}(\frac{D}{r}) \frac{F}{m}
Write this relation with symbols exactly as in Electrical Power Systems — CL Wadhwa before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
CharacteristicimpedanceZc=√Characteristic impedance Z_{c} = √

Formulas (Indian textbook notation)

  • CharacteristicimpedanceZc=LC;SIL=V2ZcCharacteristic impedance Z_{c} = \sqrt{\frac{L}{C}}; SIL = \frac{V^{2}}{Z_{c}}
Write this relation with symbols exactly as in Electrical Power Systems — CL Wadhwa before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

GMD (geometric mean distance) for an unsymmetrically spaced three-phase line is the cube root of the product of the three phase-to-phase distances: GMD = ∛(D_ab D_bc D_ca). Transposition makes each phase see the same average inductance.

Governing relations in practice

Surge impedance Z_c = √(L/C) is around 400 Ω for overhead lines and much lower for cables. Surge-impedance loading SIL = V²/Z_c is the natural loading at which the line neither absorbs nor supplies reactive power.

Design and analysis considerations

Model choice matters: short lines (<80 km) ignore capacitance; medium lines use a nominal-π; long lines need distributed ABCD parameters with hyperbolic functions.

Assumptions and validity limits

State assumptions explicitly before using any relation for transmission line parameters — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Power Systems viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Power Systems papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to transmission line parameters.
4. Use equation 1:
InductanceL=Inductance L =
.
5. Use equation 2:
CapacitanceC=2πεlnCapacitance C = 2\pi \frac{\varepsilon}{ln}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Transmission Line Parameters appears in state utilities and industrial substations. In Indian electrical curricula this topic is tested because it connects theory to generation, transmission, and faults.
GATE and semester exams often combine transmission line parameters with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use transmission line parameters?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using physical radius r for inductance (should use GMR) and GMR for capacitance (should use r)
• Forgetting to take the cube root when computing GMD for three phases
• Applying the short-line model to a long line (ignoring charging current)
• Mixing per-metre and per-km values without converting

Quick revision checklist

Before attempting transmission line parameters problems, confirm you can:
1. Bundled conductors reduce L and increase C
2. GMD and GMR for composite conductor spacing
3. Short line: lumped R, X; long line: distributed ABCD parameters
Revise the solved examples in Electrical Power Systems — CL Wadhwa and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Inductance of a single-phase line

Problem

A single-phase line has two conductors of radius 1 cm spaced 1 m apart. Find the loop inductance per km (use GMR = 0.7788r).

Solution

GMR = 0.7788 × 0.01 = 7.788×10⁻³ m; D = 1 m.
L per conductor = (μ₀/2π) ln(D/GMR) = 2×10⁻⁷ × ln(1/7.788×10⁻³).
ln(128.4) = 4.855; L = 2×10⁻⁷ × 4.855 = 9.71×10⁻⁷ H/m per conductor.
Loop (two conductors) = 2 × 9.71×10⁻⁷ = 1.94×10⁻⁶ H/m = 1.94 mH/km.

Conceptual check — Transmission Line Parameters

Problem

In a Power Systems semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of transmission line parameters." What should a complete answer include?

Exams & GATE

CL Wadhwa — calculate Z and Y for given conductor geometry.

📖 Standard books (India)

  • Electrical Power Systems — CL Wadhwa

    Read: Syllabus unit

    Generation, transmission, and fault basics

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