Synchronous equivalent circuit
E_f = V_t + I_a(R_a + jX_s) — one source, one resistance, one reactance, and the phasor diagram that tells you everything.
Step 1 — Per-phase equivalent: E_f source behind R_a + jX_s
Reference notes
Use Next → on the narrator above to walk through six conditions of the synchronous machine: no-load through unity PF, lagging, and leading loads.
Per-phase equivalent circuit (cylindrical rotor)
For a balanced cylindrical-rotor synchronous machine, the per-phase equivalent reduces to one EMF source Ef in series with the armature resistance Ra and the synchronous reactance Xs:
This is the working equation for almost every alternator calculation you'll do.
What Xs actually is
Xs is called the synchronous reactance and lumps together two physical effects:
- Armature leakage reactance xl — leakage flux that doesn't link the rotor.
- Armature reaction reactance Xar — the equivalent reactance representing the Fa-vs-Ff interaction we built up in the previous lesson.
Xs is typically a much larger number than a transformer's leakage reactance — armature reaction is the dominant term. Ra is usually small compared to Xs and is often neglected in problems.
The power (load) angle δ
δ is the angle by which Ef leads Vt in the phasor diagram. It's called the load angle because it grows as load grows: more shaft input → more current → bigger drops → bigger angle. δ is also the key variable in the power-angle equation (next lesson).
- No-load: Ia = 0, no drops, δ = 0, |Ef| = |Vt|.
- Lagging load: |Ef| ends up larger than |Vt| (the operator must boost field current to maintain Vt).
- Leading load: |Ef| can become smaller than |Vt| — terminal voltage rises above the source EMF.
- Stability limit: δ = 90° is the maximum stable operating angle. Beyond that the rotor slips out of step (power-angle lesson covers this).
Why Xs is bigger than you might expect
In a transformer, leakage reactance per-unit is usually 5–10 %. In a synchronous machine, Xs per-unit is typically 0.8–2.0 — that is, 80–200 %! The reason is purely armature reaction: the stator MMF Fa directly modifies the rotor's view of the gap, and on the equivalent-circuit level this looks like a very large reactance in series with the source.
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