Methods to determine synchronous reactance — OCC, SCC, ZPF, slip test

Reference notes

Use Next → to walk through the classical methods used to characterize a synchronous machine's reactance: OCC + SCC together give Z_s; ZPF / Potier separates leakage from armature reaction; the slip test gives X_d and X_q for salient-pole machines.

Why X_s matters

The synchronous-machine per-phase equivalent circuit is:

R_a is small and easily measured. X_s is the workhorse parameter — used in voltage-regulation, capability-curve, fault-analysis, and stability calculations. Two flavors:

• Unsaturated Z_s: the slope of the air-gap line vs SCC. Used for fault-current calculations (iron unsaturated during a fault transient).
• Saturated Z_s: from the actual OCC curve vs SCC. Used for steady-state voltage-regulation calculations near rated voltage.

Open-circuit characteristic (OCC)

Spin the machine at rated speed; leave terminals OPEN; slowly raise field current I_f; record terminal voltage E_f. Shape: linear at low I_f (iron's reluctance dominated by air gap), saturating knee as iron approaches B_max, asymptoting to a maximum.

The air-gap line is the tangent through the origin matching the linear region. It represents what E_f would be if the iron NEVER saturated.

Short-circuit characteristic (SCC)

Spin the machine at rated speed; short the armature terminals through ammeters; slowly raise I_f; record I_a. Shape: linear. Reason: armature reaction during short circuit is heavily demagnetizing, so net flux stays well below saturation regardless of I_f. SCC is effectively the air-gap line viewed from the armature side.

EMF method — synchronous impedance Z_s

Read E from the air-gap line at any I_f. Read I from SCC at the same I_f. Ratio = unsaturated synchronous impedance. Typical 1.0–1.5 pu on machine ratings.

For SATURATED Z_s near rated voltage:

ZPF / Potier method — separates X_L from X_ar

The EMF method lumps X_s = X_L (leakage) + X_ar (armature reaction). For voltage-regulation accuracy near rated voltage, we want them separated. Test:

1. Load the machine at rated I_a, zero power factor lagging (use a synchronous motor or shunt reactor as load).
2. Measure V_t vs I_f and plot the ZPF curve.
3. The Potier triangle construction on the OCC reveals:
   • Horizontal side → armature-reaction MMF in I_f units (the additional I_f needed to overcome demagnetization).
   • Vertical side → I_a · X_L (Potier reactance ≈ leakage reactance).

Typical X_L = 0.10–0.20 pu; X_ar = Z_s − X_L is the rest.

Slip test for X_d and X_q (salient-pole machines)

Salient-pole machines have direct-axis reactance X_d (rotor pole aligned with stator MMF) and quadrature-axis reactance X_q (rotor q-axis aligned). X_d > X_q because reluctance through the iron pole is much lower than through the air gap.

Test procedure:

1. Apply reduced three-phase voltage at rated frequency to the armature.
2. Field winding shorted; rotor disconnected from any prime mover.
3. Let the rotor "slip" slowly through the stator's rotating field (mechanically or with very low driving torque).
4. Armature current pulsates: I_max when q-axis aligns with stator field, I_min when d-axis aligns.

Typical X_d ≈ 1.0–1.5 pu, X_q ≈ 0.5–0.7 pu for hydro alternators. Round-rotor (turbo) machines have X_d = X_q = X_s. With distinct X_d and X_q, the two-reaction model gives a power-vs-load-angle (δ) equation with both sin δ and sin 2δ terms — the second term is the reluctance-power contribution.

Fault current shortcut in per-unit

Once Z_s is known on machine-base, the bolted symmetrical fault current at machine terminals (driven by V_th ≈ 1.0 pu and an inductive Z_s) is simply:

The fault current angle is set by the φ of Z_s (essentially 90° lagging since Z_s ≫ R_a). Multiplied by I_base on the machine, this gives breaker-sizing amperes.

Keyboard shortcuts

• → next  ·  ← previous  ·  R replay  ·  M mute  ·  F fullscreen