Synchronous Generator Fundamentals
The machine behind every grid, boiled down to its essence: an internal EMF E behind a synchronous reactance Xs, tied to an infinite bus at V = 1.0 pu. Two knobs run it — the governor pushes watts through the load angle δ, the exciter pushes vars by stretching |E| — and the phasor diagram shows exactly why the two jobs don't mix. Drive it toward pull-out and watch the coupling let go.
Field current sets |E|. More field → more EMF → the machine "pushes" vars into the bus.
The turbine can't push watts the magnetic coupling won't carry: pull-out sits at Pmax = E·V / Xs.
Model the round rotor as a voltage source E∠δ behind the synchronous reactance Xs, feeding an infinite bus V∠0° that pins terminal voltage and frequency. Kirchhoff around that one loop is the whole machine:
The two-knob mental model. Alone, a generator's governor sets frequency and its exciter sets voltage. On an infinite bus both are already fixed — so the knobs move what's left. The governor feeds in mechanical power; the rotor advances until (E·V/Xs)·sin δ absorbs it, so Pm picks the angle δ and hence the watts. The exciter stretches |E|; since E·sin δ = P·Xs/V is pinned by the governor, extra excitation goes almost entirely into E·cos δ — that is, into Q. Watch it on the phasor panel: at fixed P, the tip of E slides along the dashed horizontal constant-P locus. Excitation moves vars; the governor moves watts. Cross-coupling is second-order.
Why over-excited = lagging = "exporting vars". With E·cos δ > V the machine's EMF more than covers the bus voltage, and the surplus drives magnetizing current out into the grid: Q > 0, and in generator convention the armature current lags V — the machine looks like the grid's source of vars, holding voltage up. Under-excited (E·cos δ < V) the bus back-fills the machine's own magnetization: Q < 0, current leads, and the operating point drifts toward larger δ, thinner stability margin, and — in a real machine — stator end-region heating and the under-excitation limiter.
Pull-out. P(δ) peaks at δ = 90°: Pmax = E·V/Xs. Ask the shaft for more than that and no angle can transmit it — the rotor accelerates, slips poles, and the unit must trip. Excitation is the rescue knob: raising |E| lifts the whole sine curve.