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Salient-pole two-reaction theory

Blondel's d/q decomposition + reluctance-power term explains hydro alternators and the reluctance motor.

Freshman ~9 min

Step 1 — Cylindrical vs salient-pole rotors: the air gap isn't uniform

Animation speed 0.55×

X_d/X_q — P/P_max — δ_peak —

Reference notes

Use Next → on the narrator above to step through Blondel's two-reaction analysis for salient-pole synchronous machines.

Why a salient-pole rotor needs special treatment

A cylindrical (non-salient) rotor — typical of high-speed turbo-alternators — has a uniform air gap, so the magnetic reluctance is the same in every direction. One reactance, X_s, characterises everything.

A salient-pole rotor — typical of low-speed hydro generators and many synchronous motors — has projecting poles, with a much smaller air gap directly under each pole than between them. Two different air-gap reluctances exist, giving two different reactances:

X_d (direct-axis reactance) — for flux aligned along the pole axis. Small air gap → low reluctance → large inductance → large X_d.
X_q (quadrature-axis reactance) — for flux between poles. Large air gap → high reluctance → small inductance → small X_q.

Typical ratio X_d/X_q is 1.5–2.0 for hydro generators; turbo-alternators are nearly 1 (cylindrical).

Two-reaction (Blondel) decomposition

Resolve the armature current I_a into two components along the rotor's d- and q-axes:

I_d = I_a·sin(δ + θ) (component along d-axis)
I_q = I_a·cos(δ + θ) (component along q-axis)

Each component is then analysed against its own reactance. This decoupling is the heart of Blondel's two-reaction theory (1899) — it makes salient-pole machines tractable.

The salient-pole power-angle equation

Plugging the decomposition into the power transfer expression gives:

P = (E_f·V_t / X_d)·sin δ + (V_t² / 2)·(1/X_q − 1/X_d)·sin 2δ

Two terms. δ is the same load angle we met in the cylindrical-rotor case — the angle by which E_f leads V_t in the phasor diagram. Compare with the cylindrical-rotor limit (X_d = X_q = X_s): the second term vanishes and we recover the familiar P = (E_f·V_t/X_s)·sin δ.

The reluctance-power term

The (V_t² / 2)·(1/X_q − 1/X_d)·sin 2δ term is reluctance power. Three remarkable properties:

Independent of excitation — there's no E_f in it. The reluctance power flows even with zero field current.
Peaks at δ = 45° — sin 2δ has period 180°, so it tops out at half the cylindrical-rotor peak angle.
Shifts the overall peak to below 90° — superimpose the two terms and the total P-δ curve peaks somewhere around 70–80° (depends on excitation and the saliency ratio).

Reluctance motor — the extreme case

What if you remove the field winding entirely (set E_f = 0)? The first term in the power-angle equation vanishes, leaving only:

P = (V_t² / 2)·(1/X_q − 1/X_d)·sin 2δ

This is the principle of the reluctance motor — a synchronous machine with no DC excitation, running entirely on the d/q reluctance difference. It produces a useful torque (limited but nonzero), runs at synchronous speed, and needs no slip rings or brushes. Used in timers, fans, and small precision drives.

Why this matters for hydro generators

Hydro alternators run at low speed (small n_s, large pole count) and use salient-pole rotors. Their P-δ curves are visibly different from turbo-alternators':

Steeper rise from δ = 0 (the reluctance term contributes alongside the main term).
Peak slightly before 90°.
Higher synchronising-power coefficient near the operating point → faster restoration after disturbances.

This is also why hydro generators tend to have higher short-circuit ratios and better voltage regulation than turbo-alternators of the same MVA rating.

Take-away. Two-reaction theory adds a single sin(2δ) term to the power-angle equation, but that term is the only reason reluctance motors exist and the reason hydro alternator dynamics differ from steam-turbine ones. One algebraic refinement, two pieces of new physics.

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