Lesson 9 of 10

Synchronous Machines

Almost every conventional power plant — coal, gas, nuclear, hydro — ends in the same machine: a synchronous generator. It is also the machine behind every idea in the Grid Stability track: inertia, rotor angle, governors, excitation, power system stabilizers. This lesson builds the model you need for all of them, starting from the hardware.

Stator, rotor, excitation

Stator (armature): The stationary outer part, carrying a three-phase winding connected to the grid through the generator step-up transformer. This is where the output power flows.
Rotor (field): The rotating inner part, carrying a winding fed with direct current. Spinning, it produces a rotating magnetic field that induces the three-phase voltage in the stator.
Excitation system: Supplies the rotor's DC field current, controlled by the automatic voltage regulator (AVR). More field current means a higher internal voltage.
Turbine and governor: The prime mover delivers mechanical torque to the shaft; the governor adjusts steam, gas or water flow and therefore the mechanical power.

There are two rotor designs. Steam and gas turbines run fast, so their generators use a long, slim round (cylindrical) rotor with two or four poles. Hydro turbines run slowly, so their generators use a short, wide salient-pole rotor with many poles bolted around the rim.

Why it is called synchronous

In steady state the rotor turns at exactly the speed set by the grid frequency and its number of poles — no faster, no slower:

n_s = 120 · fp

Synchronous speed in rpm (f in Hz, p = number of poles)

A two-pole turbogenerator on a 50 Hz grid therefore spins at 3000 rpm, a four-pole one at 1500 rpm, and a 40-pole hydro generator at only 150 rpm. Every synchronous machine on the same grid is electrically locked to the same frequency — which is exactly what the Grid Stability track means by "staying in synchronism."

Mechanical in, electrical out: the torque balance

The turbine pushes the rotor forward with mechanical torque; the grid holds it back with electrical torque, proportional to the power being delivered. In steady state the two are equal and the speed is constant. If they differ — because a fault suddenly removes the electrical load, or a governor opens the steam valves — the rotor accelerates or decelerates. In per unit, that is the swing equation:

2Hω_s · d²δdt² = P_m − P_e

Swing equation (per unit, H = inertia constant in s, δ = rotor angle)

This single equation connects the machine to two later lessons: H is the inertia constant from the inertia lesson, and δ is the rotor angle whose behaviour after a fault decides transient stability.

The phasor model: a voltage behind a reactance

For steady-state studies a round-rotor machine is represented by an internal voltage E — set by the field current — behind its synchronous reactance X_s, connected to a terminal voltage V. With δ the angle between E and V, the power it delivers is:

P = E · VX_s · sin δ | Q = E · V · cos δ − V²X_s

Active and reactive power of a round-rotor machine (resistance neglected)

The first equation is the power-angle curve used throughout the Grid Stability track. The second shows what the field current does: raise E above V and the machine supplies reactive power (over-excited); lower it and the machine absorbs reactive power (under-excited).

Two independent knobs

A generator operator has two controls, and they act on different quantities. The governor sets mechanical power, which shifts δ and therefore active power P. The excitation system sets E, which mainly changes reactive power Q and the terminal voltage. Frequency control works through the first knob; voltage control through the second.

Three reactances for three time scales

When a fault occurs, currents induced in the rotor's field and damper windings briefly oppose the change, so the machine looks much stiffer at first than in steady state. That is modelled with three reactances; typical per-unit values for a round-rotor turbogenerator, on the machine's own rating, are:

Subtransient X″d: About 0.12–0.25 p.u. Governs the first few cycles of a fault, so it sets the peak short-circuit current that breakers must interrupt.
Transient X′d: About 0.15–0.4 p.u. Governs roughly the first second, which is why classical transient-stability models use it.
Synchronous Xd: About 1.0–2.3 p.u. Governs steady state, long after the damper and field transients have decayed.

The same machine therefore delivers a large, rapidly decaying fault current: the next lesson uses X″d as the generator's positive-sequence impedance when calculating it.

The capability chart

A generator cannot deliver any combination of P and Q. Its P–Q capability chart is bounded by the stator current limit (a circle of constant MVA), the rotor heating limit at high over-excitation, the turbine's maximum power, and an under-excitation limit that protects stability and the stator core ends. Operators and grid codes work inside that envelope — which is also why reactive power support has a real cost.

Key takeaways

A synchronous generator has a three-phase stator connected to the grid and a DC-excited rotor driven by a turbine.
It turns at synchronous speed n_s = 120 f / p — 3000 rpm for a two-pole machine at 50 Hz.
The swing equation links mechanical-electrical power imbalance to rotor acceleration through the inertia constant H.
Active power follows P = (E·V/X_s)·sin δ and is set by the governor; reactive power is set mainly by the excitation.
Subtransient, transient and synchronous reactances describe the machine on fault, stability and steady-state time scales.