Transmission Line Parameter Calculator

GMR (geometric mean radius) is a conductor's own effective self-distance, already accounting for internal flux linkage within a stranded conductor — it's smaller than the conductor's actual outside radius and comes from published conductor tables (like ACSR data), not calculated from geometry alone. GMD (geometric mean distance) is the effective spacing between the different phase conductors. For inductance, the ratio GMD/GMR matters; for capacitance, the ratio GMD/r matters, using actual outside radius r — not GMR — since capacitance depends only on the external field, not internal flux linkage.

L = (μ₀/2π)·ln(GMD/GMR) per phase, per unit length, where μ₀=4π×10⁻⁷ H/m. For a flat (non-symmetric) spacing, GMD = ∛(D_AB·D_BC·D_CA) — the geometric mean of the three phase-to-phase spacings — which is exactly why real lines are transposed periodically along their length: it makes each phase's average inductance equal despite the physically asymmetric tower geometry, keeping the three-phase system balanced.

C_n = 2πε₀/ln(GMD/r) per phase-to-neutral, per unit length, where ε₀=8.854×10⁻¹² F/m and r is the conductor's actual outside radius. Wider phase spacing (larger GMD) reduces both inductance and capacitance, but in opposite senses for line loading: lower capacitance is generally desirable (less charging current), while lower inductance is also generally desirable (less voltage drop and higher power transfer capability) — so tower geometry is usually driven more by clearance, insulation and right-of-way constraints than by trying to optimize these electrical parameters directly.

Z₀ = √(L/C) is the line's natural impedance — the impedance a wave 'sees' traveling along the line, independent of line length. Surge impedance loading, SIL = V²/Z₀ (V in line-to-line volts, giving SIL in watts), is the power the line delivers to a purely resistive load equal to its own characteristic impedance — a natural reference loading level where the line's own reactive power production (from its capacitance) and consumption (from its inductance) exactly balance, needing no external reactive support. Typical overhead line Z₀ runs roughly 240–400 Ω; underground cable Z₀ is much lower, roughly 1/10th, because cables have much higher capacitance per unit length from their tightly spaced conductors and dielectric insulation.

Lines under roughly 80 km are conventionally treated as electrically short (shunt capacitance effects negligible, simple series R+jX model adequate); medium lines (roughly 80–250 km) need the shunt capacitance included, typically lumped at the line's midpoint or split between both ends (the nominal-π model); long lines (over roughly 250 km) need the full distributed-parameter (hyperbolic) line equations, since voltage and current genuinely vary continuously along the line's length in a way lumped models can't capture accurately. This classification is exactly why SIL and Z₀ — length-independent parameters — are useful reference points regardless of which detailed model a specific study ultimately uses.

For a medium-length line, the distributed series impedance and shunt admittance are approximated as a lumped series impedance Z=R+jX with half the total shunt admittance Y/2 connected at each end (a π-shaped circuit). This gives the two-port ABCD parameters: A=D=1+ZY/2, B=Z, C=Y(1+ZY/4), relating sending-end voltage and current to receiving-end voltage and current: V_s=A·V_r+B·I_r, I_s=C·V_r+D·I_r. Given a receiving-end load (power and power factor), this lets you solve directly for what voltage and current the sending end actually needs to supply.

Voltage regulation compares the receiving-end voltage at no load to its value at full load, as a percentage: VR=(|V_r,no-load|−|V_r,full-load|)/|V_r,full-load|×100, where V_r,no-load=V_s/A (the receiving-end voltage that would result if the same sending-end voltage were maintained but the load, and its current, dropped to zero). Lower voltage regulation means the line holds its voltage better under load — a shorter, lower-impedance line, or a more capacitive (leading) power factor, both improve it. Transmission efficiency is simply P_r/P_s, the fraction of sending-end power that actually reaches the receiving end after resistive losses (P_s−P_r) along the way.

A 230 kV, 50 Hz, 150 km line uses a conductor with GMR = 9.14 mm and outside radius = 11.28 mm, in equilateral spacing at D = 6 m, with AC resistance 0.214 Ω/km. It delivers 100 MW at 0.90 lagging power factor. Find the line parameters and the sending-end conditions.

Step 1 — Inductance. L = (μ₀/2π)·ln(GMD/GMR) = (4π×10⁻⁷/2π)·ln(6/0.00914) = 1.2974 mH/km.
Step 2 — Capacitance. C = 2πε₀/ln(GMD/r) = 2π(8.854×10⁻¹²)/ln(6/0.01128) = 8.86 nF/km.
Step 3 — Total series impedance. R = 0.214 × 150 = 32.10 Ω. X_L = 2π(50)(1.2974×10⁻³)×150 = 61.14 Ω. So Z = 32.10 + j61.14 Ω.
Step 4 — Total shunt susceptance. B = 2π(50)(8.86×10⁻⁷)×150,000 = 417.68 μS.
Step 5 — Characteristic impedance and SIL. Z₀ = √(L/C) = √(1.2974×10⁻³/8.86×10⁻⁹) = 382.6 Ω. SIL = V²/Z₀ = 230,000²/382.6 = 138.3 MW. At 150 km (under 250 km), this is a medium line.
Step 6 — Receiving-end current. With P_r = 100 MW at 0.90 lagging PF and V_r = 230 kV: I_r = P_r/(√3·V_r·PF) = 100×10⁶/(√3×230,000×0.90) = 278.9 A, lagging V_r by cos⁻¹(0.90) = 25.84°.
Step 7 — ABCD and sending-end voltage. A = 1+ZY/2 = 0.9872 + j0.0067. Applying V_s = A·V_r+B·I_r and converting the resulting phase voltage back to line-to-line gives V_s = 254.79 kV.
Step 8 — Voltage regulation. VR = (|V_s/A|−|V_r|)/|V_r|×100 = 12.21%.

Enter these same values into the calculator above (they are its defaults) to see every one of these results reproduced exactly.

Inductance depends on the magnetic flux linkage both outside and inside the conductor, and GMR is specifically defined to give the correct total (internal plus external) inductance using the same simple formula as if all the flux were external — but capacitance depends only on the electric field outside the conductor's actual physical surface, so it correctly uses the real outside radius rather than the GMR.

In a real (non-equilateral) tower configuration, each of the three phases has a different physical spacing to the other two, which would give each phase a slightly different inductance and capacitance — an inherently unbalanced three-phase system even under perfectly balanced load. Transposition (physically rotating which conductor occupies which position at intervals along the line) makes each phase spend an equal length of the route in each position, equalizing the average inductance and capacitance across all three phases and restoring balance.

A line loaded well below its SIL is a net reactive power source (its capacitance dominates), while a line loaded well above SIL is a net reactive power sink (its inductance dominates) — operating a line near its own SIL minimizes the reactive power exchange with the rest of the system, which is part of why SIL is used as a practical reference point for how much real power a given line can carry efficiently, alongside its actual thermal current limit.

No — this uses the standard flat-earth GMD/GMR approximation, which is accurate to within a few percent for typical transmission towers with ground clearance above about 10 m. A more rigorous treatment uses Carson's equations, which explicitly model the earth as an imperfect conducting return path and become more important at lower conductor heights or when precise zero-sequence (ground-fault) parameters are needed.

A lagging (inductive) load draws current that lags voltage, which interacts with the line's own series inductive reactance to produce a larger voltage drop along the line — the sending end has to supply noticeably more voltage than the receiving end needs. A leading (capacitive) load's current leads voltage, which can partially or fully offset that same reactive voltage drop, sometimes even making receiving-end voltage exceed sending-end voltage (negative regulation) — exactly why utilities use capacitor banks and other reactive compensation to improve voltage support on heavily loaded lines.

The nominal-π model is accurate for the vast majority of practical overhead line lengths (roughly up to 250 km) and is dramatically simpler to compute than the full distributed-parameter hyperbolic line equations, which require evaluating complex hyperbolic sine and cosine functions of the propagation constant. For a genuinely long line, the nominal-π model's accuracy degrades since it approximates continuously distributed capacitance as two lumped points rather than truly distributed — the line classification result tells you when this matters.

Resistance depends on the conductor's material, cross-sectional area and temperature — not on tower geometry — so unlike inductance and capacitance, it can't be derived from spacing and GMR alone. This calculator asks for AC resistance directly from published conductor tables (which already account for skin effect at power frequency), rather than trying to calculate it from first principles.