High-Voltage Power Systems, EHV AC Transmission Lines & Reactive Compensation
Transmission Line SIL & Ferranti Voltage Rise Formula
Calculate Characteristic Surge Impedance Z_c (Ohms), Surge Impedance Loading SIL (MW), Open-Circuit Receiving-End Ferranti Voltage V_r (kV), and Overvoltage Percentage (%).
Why lightly loaded long 500 kV lines suffer dangerous receiving-end overvoltage—while loading right at SIL achieves flat voltage!
Along a lossless extra-high-voltage (EHV) overhead transmission line with distributed series inductance L (mH/km) and shunt capacitance C (nF/km), electromagnetic waves see Characteristic Surge Impedance Z_c = SQRT((L * 10^-3) / (C * 10^-9)) Ohms (~250–400 Ohms on bundled overhead lines vs ~40–60 Ohms on underground cables)! Delivering active power equal to Surge Impedance Loading SIL = (V_LL_kV^2) / Z_c MW balances distributed inductive reactive consumption I^2 * omega * L against capacitive reactive generation V^2 * omega * C, producing a flat voltage profile along the entire corridor! Conversely, switching off the receiving substation breaker (open-circuit / light night load) unleashes the Ferranti Effect: charging current flowing through series inductance lifts Receiving-End Voltage to V_r = V_s / COS(2 * PI() * f * SQRT(L * C * 10^-12) * Length_km), forcing grid operators to switch in shunt reactors before insulation flashover!
Try this 400 km 500 kV 50 Hz Bundled-Conductor Corridor first—Sending_Voltage V_s = 500.0 kV (B2), Inductance L = 0.90 mH/km (B3), Capacitance C = 12.50 nF/km (B4), Corridor_Length = 400.0 km (B5), and Grid_Frequency f = 50.0 Hz (B6): Surge Impedance Z_c equals 268.33 Ohms (E2), Natural Surge Impedance Loading SIL reaches 931.69 MW (E3), and Open-End Ferranti Receiving Voltage jumps to 547.96 kV (E4)—a +9.59% overvoltage surge above the 5% utility ANSI/IEC ceiling (E5)! In a shared Arte Sheets grid-planning workbook, a `suggester` can test a 600 km corridor extension in Suggest Mode while asking a hint-first AI prompt ("Why does Ferranti voltage rise scale faster than quadratically when corridor length approaches a quarter-wavelength?") before the `owner` sizes shunt MVAR reactors.
| 500 kV EHV Line KPI (L=0.90 mH/km, C=12.5 nF/km, 400 km, 50 Hz) | Formula | Verified Output |
|---|---|---|
| Characteristic Surge Impedance Zc | SQRT((L_mH_km * 1e-3) / (C_nF_km * 1e-9)) | 268.33 Ohms |
| Surge Impedance Loading SIL (MW) | (Sending_kV ^ 2) / Zc_Ohms | 931.69 MW |
| Open-End Ferranti Voltage Vr (kV) | Vs_kV / COS(2*PI()*f*SQRT(L*C*1e-12)*Len) | 547.96 kV (619.80 kV @ 600 km!) |
| Ferranti Overvoltage Rise (%) | ((Vr_kV - Vs_kV) / Vs_kV) * 100 | 9.59% (23.96% @ 600 km!) |
Distributed-Parameter SIL & Exact Hyperbolic-Cosine Lossless Ferranti Formula
Validate Sending_Voltage > 0, Inductance L > 0, Capacitance C > 0, Length > 0, Frequency in [16.7, 400] Hz, and electrical phase angle beta * Length < 1.30 rad (below quarter-wave resonance PI/2)!
=LET(vs_k, B2, lm_h, B3, cn_f, B4, ln_k, B5, fr_q, B6, IF(OR(vs_k <= 0, lm_h <= 0, cn_f <= 0, ln_k <= 0, fr_q < 16, fr_q > 400), "Invalid EHV Line Input", LET(bt_l, 2 * PI() * fr_q * SQRT((lm_h * 0.001) * (cn_f * 0.000000001)) * ln_k, IF(bt_l >= 1.3, "NEAR QUARTER-WAVE RESONANCE", LET(vr_k, vs_k / COS(bt_l), ROUND(((vr_k - vs_k) / vs_k) * 100, 2))))))600 km Uncompensated Corridor Severe Ferranti Surge Check (Set Length = 600.0 km)
Extend Corridor_Length B5 from 400.0 km to 600.0 km without shunt reactor compensation: Z_c (268.33 Ohms) and SIL (931.69 MW) are length-independent, yet Open-Circuit Receiving Voltage E4 surges from 547.96 kV to 619.80 kV (+23.96% in E5)!
=IF(ISTEXT(E5), E5, LET(bt_l, 2 * PI() * B6 * SQRT((B3 * 0.001) * (B4 * 0.000000001)) * B5, ROUND(B2 / COS(bt_l), 2)))Try the example
Use the original values or change them to test your understanding.