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Transmission line modeling — short, medium-π, long with ABCD

Three lumped models by length, the hyperbolic ABCD form, surge impedance Z_c, SIL = V_LL²/Z_c, and the Ferranti effect on long lightly-loaded lines.

Junior ~12 min

Step 1 — Four distributed parameters: R, L, G, C per km

Animation speed 0.55×

model — Z_c — SIL —

Reference notes

Use Next → on the narrator to walk through the three transmission-line lumped models (short / nominal-π / long with hyperbolic ABCD), the ABCD-parameter algebra that lets you cascade two-ports, and the surge-impedance / SIL / Ferranti concepts that govern line operation.

The four distributed parameters

Every transmission line is described by four per-unit-length parameters:

R — series resistance per km (Ω/km). Source of I²R losses.
L — series inductance per km (H/km). Source of voltage drop V = jωL·I.
G — shunt conductance per km (S/km). Insulation leakage; usually negligible.
C — shunt capacitance per km (F/km). Source of charging current and Ferranti rise.

Compact form: series impedance z = R + jωL (Ω/km), shunt admittance y = G + jωC (S/km).

Model choice by line length

Short line (< 80 km / 50 mi)

Shunt admittance is small and neglected. Only series Z = z·ℓ matters.

V_s = V_r + I_r · Z I_s = I_r

ABCD: A = D = 1, B = Z, C = 0

Medium line (80–250 km)

Use nominal-π: full series Z in middle, total shunt Y = y·ℓ split into Y/2 at each end.

A = D = 1 + (Y·Z)/2 B = Z C = Y · (1 + Y·Z/4)

Nominal-T is the alternative — Z/2 at each end, full Y in middle. Both have similar accuracy.

Long line (> 250 km)

Solve the telegrapher equations to get exact ABCD via hyperbolic functions of the propagation constant γ:

γ = √(z · y) Z_c = √(z / y)

A = D = cosh(γ·ℓ) B = Z_c · sinh(γ·ℓ) C = sinh(γ·ℓ) / Z_c

For a lossless approximation (drop R, G): γ = jω√(LC) is purely imaginary, and Z_c = √(L/C) is purely real. Useful for surge-impedance / SIL calculations.

ABCD parameters

ABCD are the standard two-port parameters for transmission elements. The matrix equation:

[V_s; I_s] = [A B; C D] · [V_r; I_r]

Two key constraints for passive transmission lines:

Reciprocity: A·D − B·C = 1.
Symmetry: A = D (if the line looks the same from either end).

Cascaded networks combine by matrix multiplication: [A_total] = [A_1] · [A_2]. That's the whole reason ABCD is the standard tool — chaining transformers, lines, and shunt compensators is just sequential 2×2 multiplication.

Surge impedance, SIL, and the Ferranti effect

The characteristic (surge) impedance is the value that, if used as the load impedance, produces a flat voltage profile along the line.

Z_c = √(L / C) (lossless approximation)

Typical Z_c values:

Overhead 3-φ transmission line: 350–450 Ω.
Underground HV cable: 30–80 Ω (much lower because C/km is much higher).

The surge-impedance loading is the real-power load at which |V_s| = |V_r|:

SIL = V_LL² / Z_c

For a 230 kV line with Z_c = 380 Ω: SIL = 230²/0.380 ≈ 139 MW.

Loading below SIL → line is capacitively dominant → voltage RISES along the line → Ferranti effect. Worst at no load on a long line.
Loading above SIL → line is inductively dominant → voltage drops along the line.

Operator response:

Light load on long lines → connect shunt reactors at the receiving end to absorb excess MVAR and pull voltage down.
Heavy load → connect shunt capacitors to provide reactive support and prop voltage up.

Voltage regulation

%Reg = (|V_r,NL| − |V_r,FL|) / |V_r,FL| × 100 %

where V_r,NL is the no-load and V_r,FL is the full-load receiving-end voltage with sending-end voltage held constant. On a short line with lagging PF, Reg is dominated by I·X·sin θ. On long lines, the calculation needs the full ABCD.

Take-away. Pick model by length: short (<80 km, just Z), nominal-π (80–250 km, with Y/2 at each end), long (>250 km, hyperbolic ABCD). Z_c = √(L/C) is the surge impedance, ~400 Ω for overhead lines, much lower for cables. SIL = V_LL²/Z_c is the flat-profile loading. Below SIL → Ferranti rise; above SIL → voltage drop. ABCD cascades by matrix multiplication, with AD − BC = 1 for any reciprocal passive line.

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