Three-phase fault analysis — symmetrical and unbalanced

Reference notes

Use Next → on the narrator above to step through fault analysis: Thevenin reduction, symmetrical 3-φ faults, component sequence impedances, single line-to-ground faults (the most common type), other unbalanced faults, and practical fault-impedance + breaker-rating + motor-contribution considerations.

The Thevenin reduction

Every fault calculation starts by reducing the network behind the fault to a Thevenin equivalent at the fault bus: a pre-fault voltage source V_th in series with a Thevenin impedance Z_th. V_th is typically taken as 1.0 pu (the pre-fault nominal voltage). Z_th is computed by combining generator subtransient reactances, transformer leakage impedances, and line impedances in the appropriate parallel-series fashion, all expressed in per-unit on a common base.

Symmetrical three-phase fault

All three phases short to each other through a low impedance. By symmetry only the positive-sequence network is active:

Worked example on a 100 MVA, 13.8 kV system with Z_+ = 0.10 pu, bolted fault:

• I_pu = 1.0 / 0.10 = 10 pu
• I_base = 100·10⁶ / (√3 · 13 800) ≈ 4184 A → I_fault ≈ 42 kA
• S_SC = 100 / 0.10 = 1000 MVA

Component sequence impedances

Each piece of equipment has three impedances:

Zero-sequence behaviour is dominated by the grounding scheme:

• Solidly grounded Y_n: low Z₀, often comparable to Z₊. SLG can equal or exceed 3-φ fault.
• Resistance-grounded: Z₀ dominated by the grounding resistor, limiting SLG current to tens or hundreds of amps.
• Isolated / ungrounded: Z₀ = ∞; SLG produces only capacitive charging current (a few amps), but the unfaulted phases see V_LL instead of V_LN, stressing insulation.
• Delta winding: presents open circuit to zero-sequence trying to cross the transformer.

Single line-to-ground (SLG) fault

The most common fault type (70–80%). Boundary conditions on a phase-a SLG fault: I_b = I_c = 0 and V_a = Z_f · I_a. Substituting into the Fortescue transform:

The three sequence networks connect in series. The 3·Z_f appears because all three sequence currents flow through the same physical fault impedance.

Other unbalanced faults

• Line-to-line (LL): two phases bolt to each other, no ground. Z₀ is out. Sequence networks: Z₊ and Z₋ in series. I₊ = V_th/(Z₊ + Z₋ + Z_f); I_fault,b = √3 · I₊.
• Double line-to-ground (DLG): two phases bolt to each other AND to ground. All three sequence networks active. Z₊ in series with the parallel combination Z₋ ∥ (Z₀ + 3·Z_f).

Practical considerations

Fault impedance Z_f

Bolted metallic fault: Z_f ≈ 0. Arcing fault: typically a few ohms to tens of ohms (the arc-voltage drop divided by arc current). Tree contact, contact resistance, soil resistance: highly variable. High-impedance ground faults often sit below overcurrent settings and are detected only by sensitive ground-fault relays.

Breaker interrupting rating

The breaker must clear the maximum first-cycle fault current at its location. Standard ratings: 22, 42, 65, 100, 200 kA RMS symmetrical. The asymmetrical (peak with DC offset) rating is roughly 2.6× the symmetrical RMS rating immediately after fault initiation.

Motor contribution

Every running motor over ~50 hp briefly contributes to fault current for the first ~6 cycles via stored kinetic energy and air-gap flux. The contribution magnitude is approximately 4–6× the motor's rated current (governed by X_d″ or locked-rotor reactance), decaying as the motor decelerates. IEEE Std 141 (Red Book) and IEC 60909 give standardised procedures for including motor contribution in industrial fault studies.

Keyboard shortcuts

• → next step  ·  ← previous step
• R replay narration  ·  M mute / unmute  ·  F fullscreen
• On steps 3–6, click the fault-type buttons (3-φ / SLG / LL / DLG) on the canvas to switch the sequence-network interconnection.