Electronics Guide

Stator and Armature Winding

The stator core is a stack of thin, insulated electrical-steel laminations, which limit eddy-current loss. Slots on its inner surface hold a three-phase winding distributed over many slots, with the phases displaced 120 electrical degrees around the air gap. Large machines use preformed coils or bars with mica-based insulation. Generator phases are normally connected in wye, and the neutral is often grounded through an impedance that limits ground-fault current.

Round and Salient-Pole Rotors

The field winding receives direct current through slip rings and brushes or from a brushless exciter on the shaft, and it creates alternate north and south poles. The form of the rotor follows its speed. Steam and gas turbines are high-speed machines, so their generators have two or four poles and a round, or cylindrical, rotor: a solid alloy-steel forging with axial slots for the field winding, which wedges hold along the body and retaining rings hold at the ends. Centrifugal stress limits the diameter of a rotor turning at 3,600 or 3,000 revolutions per minute, so turbo-generators are long and slender, and their nearly uniform air gap gives them almost the same magnetic properties along every axis.

Hydraulic turbines and large engines turn slowly, so their generators need many poles. Their rotors have salient poles: laminated pole bodies that project from a rim, each with a concentrated field coil, on a rotor that is wide in diameter and short along its axis. The air gap is short over the pole faces and long between them, so flux passes more easily along the pole axes. That property is saliency.

Damper Windings and Cooling

Salient-pole rotors usually carry a damper, or amortisseur, winding: bars in the pole faces joined by end rings, like the cage of an induction motor. In balanced steady state it carries no current, because the stator field does not move relative to it. When the rotor swings, or when unbalanced stator currents create a backward-rotating field, currents induced in the damper oppose the change. In a round rotor, eddy currents in the solid body and the slot wedges do much of the same work.

Smaller machines are air-cooled. Large turbo-generators are commonly filled with hydrogen, whose low density and high thermal conductivity cut windage loss and improve heat transfer compared with air, and the largest units also pass water through hollow conductors in their stator bars.

How Three Phases Produce a Rotating Field

Measure the electrical angle θ around the air gap from the axis of phase a, and keep only the fundamental, sinusoidal part of each winding's MMF. With balanced currents, phase a produces F_a = F_m cos ωt cos θ, where F_m is proportional to the peak current and to the winding's effective turns. Phases b and c give the same expression with both ωt and θ reduced by 120° and 240°, respectively. Writing each product of cosines as half the sum of two cosines gives each phase a forward term, ½F_m cos(θ − ωt), and a backward term; the three backward terms are 120° apart and sum to zero. The total is

F(θ, t) = (3/2) F_m cos(θ − ωt)

The wave has a constant peak 1.5 times that of one phase, located at θ = ωt. Interchanging any two supply leads reverses the phase sequence and the direction of rotation. Unbalanced currents add a backward-rotating wave, the negative-sequence field treated in Fault Analysis and Symmetrical Components.

Synchronous Speed

A machine with P poles, always an even number, has P/2 pole pairs. The field advances one pole pair per cycle of the supply, and electrical and mechanical angles are related by θ_e = (P/2)θ_m. The synchronous speed in revolutions per minute (rpm) is

n_s = 120 f / P

where f is the frequency in hertz and P is the number of poles, not pole pairs; 120 is 60 seconds per minute times 2 poles per pair. In radians per second, ω_sm = 4πf/P. Conversely, a generator with P poles driven at n rpm produces f = Pn/120 hertz.

Two-pole machines serve most steam and gas turbines, and four-pole machines the half-speed turbines of many nuclear plants. Hydro-generators, whose speed suits the turbine and the head at the site, have many more poles.

Why Torque Requires Synchronous Speed

The DC field turns with the rotor, so steady torque arises only when the rotor and stator fields turn together at a constant angle. At any other speed the angle between them changes continuously and the torque averages zero, which is why a synchronous motor needs a separate means of starting.

The EMF Equation

Drive the rotor at synchronous speed with the field excited and the armature open, and suppose the rotor produces a sinusoidally distributed flux Φ per pole. The flux linking a concentrated, full-pitch winding of N turns varies as NΦ cos ωt, and Faraday's law gives an induced voltage of ωNΦ sin ωt, with an RMS value of ωNΦ/√2. A real phase winding is spread over several slots, and its coils usually span less than a pole pitch, so its coil voltages are slightly out of phase and their sum is reduced by a winding factor k_w. The RMS voltage per phase is

E_f = √2 π f N k_w Φ ≈ 4.44 f N k_w Φ

where N is the number of series turns per phase and Φ is in webers. E_f is called the excitation, internal, or open-circuit voltage. Below saturation, Φ is proportional to the field current I_f, so E_f is proportional to both speed and field current.

Distribution and Pitch Factors

The winding factor is k_w = k_dk_p. The distribution factor accounts for the coils of a phase lying in q adjacent slots per pole, each displaced by the slot angle γ in electrical degrees:

k_d = sin(qγ/2) / [q sin(γ/2)]

The pitch factor accounts for coils that span a fraction ρ of a pole pitch:

k_p = sin(ρ × 90°)

For the hth space harmonic, replace γ with hγ and 90° with h × 90°. A pitch of 5/6 keeps a pitch factor of 0.966 for the fundamental but cuts the fifth and seventh harmonics to 0.259 of their full-pitch values. Third-harmonic voltages are in phase in all three phases, so they cancel in the line-to-line voltages of a wye-connected winding.

A Numerical Example

A four-pole, 60 Hz machine has 48 stator slots and a double-layer winding with coils spanning 10 slots. A pole pitch is 12 slots, so ρ = 5/6; q = 48/(4 × 3) = 4 slots per pole per phase; and γ = 180° × 4/48 = 15° electrical.

1. k_d = sin 30° / (4 sin 7.5°) = 0.9577 and k_p = sin 75° = 0.9659, so k_w = 0.9250.
2. With 4 turns per coil and all 16 coils of each phase in series, N = 64 turns.
3. For a flux of 0.0175 Wb per pole, E_f = 4.443 × 60 × 64 × 0.9250 × 0.0175 = 276 V per phase, or 478 V line-to-line, at 1,800 rpm.

Armature Reaction

Balanced armature current produces its own MMF, rotating at synchronous speed with the rotor, so the air-gap flux, and the voltage E_r it induces, differ from those of the field alone. This armature reaction induces a voltage proportional to the armature current and lagging it by 90°, as an inductive reactance does, so it is represented by a reactance X_a: E_r = E_f − jX_aI_a in the generator convention defined below. Leakage flux that links the armature without crossing the air gap adds a leakage reactance X_l, and the two form the synchronous reactance:

X_s = X_l + X_a

Saturation affects mainly X_a, which is why the synchronous reactance at rated voltage is smaller than its unsaturated value. In a round rotor the direct- and quadrature-axis reactances are nearly equal, and X_s is the direct-axis synchronous reactance X_d that fault studies use for sustained current. Schneider Electric's Cahier Technique no. 158 gives typical values of 1.50 to 2.30 per unit for turbo-generators and 0.70 to 1.20 per unit for salient-pole generators, as tabulated in Per-Unit System and One-Line Diagrams.

Generator and Motor Conventions

In the generator convention, the armature current I_a flows out of the machine, and the per-phase circuit is E_f in series with the armature resistance R_a and jX_s:

E_f = V_t + (R_a + jX_s) I_a

where V_t is the terminal voltage per phase. In the motor convention, I_a flows into the machine, and

V_t = E_f + (R_a + jX_s) I_a

The direction of real power, not the circuit, distinguishes generating from motoring. The model assumes balanced sinusoidal steady state, a linear magnetic circuit, and a round rotor.

Phasor Diagrams

Take V_t as the reference, neglect R_a, and let a generator deliver I_a at a power factor angle φ, positive when the current lags V_t. The drop jX_sI_a leads I_a by 90°, and adding it to V_t places E_f ahead of V_t by the power angle δ, also called the load angle or torque angle. Resolving along and across V_t gives

E_f cos δ = V_t + X_sI_a sin φ

E_f sin δ = X_sI_a cos φ

• Lagging power factor (φ > 0): E_f cos δ exceeds V_t. The machine is overexcited and delivers reactive power.
• Unity power factor (φ = 0): E_f = √[V_t² + (X_sI_a)²].
• Leading power factor (φ < 0): E_f cos δ is less than V_t. The machine is underexcited and absorbs reactive power.

In the motor convention E_f = V_t − jX_sI_a, so the E_f of a motor lags V_t by δ. An overexcited motor draws leading current and supplies reactive power to the network, as an overexcited generator does.

The relations also give the voltage regulation: the rise in terminal voltage when full load is removed at constant field current, as a fraction of rated voltage. A machine with X_s = 1.60 per unit delivering rated current at 0.85 power factor lagging has E_f = 2.29 per unit, a linear-model regulation of 129 percent. Saturation keeps the actual rise far smaller.

The Open-Circuit Characteristic

With the armature open, the field current is raised in steps and the terminal voltage is recorded. The open-circuit characteristic, or open-circuit saturation curve, starts along a straight line, the air-gap line, on which nearly all the field MMF drives flux across the air gap. It bends below that line as the iron saturates.

The Short-Circuit Characteristic

With the terminals shorted through ammeters, the field current is raised until the armature current reaches about its rated value. The machine's own impedance is almost purely reactive, so the current lags E_f by nearly 90° and its armature reaction directly opposes the field. The net air-gap flux stays small, the iron stays unsaturated, and the short-circuit characteristic is almost a straight line.

Synchronous Reactance and Short-Circuit Ratio

At a given field current, the unsaturated synchronous reactance X_d(unsat) is the air-gap-line voltage divided by the short-circuit current. The short-circuit ratio (SCR) is the field current for rated voltage on the open-circuit characteristic divided by the field current for rated current on the short-circuit characteristic. Because the short-circuit characteristic is linear, the SCR equals the per-unit short-circuit current at the field current that gives rated open-circuit voltage, so

X_d(sat) ≈ 1 / SCR

in per unit, where the saturated synchronous reactance X_d(sat) suits calculations near rated voltage. Consider illustrative results for a 100 MVA, 13.8 kV, 60 Hz round-rotor generator, with a rated current of 4,184 A and a base impedance of 13.8²/100 = 1.9044 Ω. The air-gap line reaches rated voltage at a field current of 400 A and the open-circuit characteristic at 460 A, and the short-circuit characteristic reaches rated current at 736 A.

1. Unsaturated reactance. At 736 A the air-gap line gives 736/400 = 1.84 per unit of voltage while the short circuit carries 1.00 per unit of current, so X_d(unsat) = 1.84 per unit, or 3.50 Ω per phase.
2. Short-circuit ratio. SCR = 460/736 = 0.625.
3. Saturated reactance. X_d(sat) ≈ 1/0.625 = 1.60 per unit, or 3.05 Ω per phase.

Both values fall in the Cahier Technique range for turbo-generators, and the examples below use X_s = 1.60 per unit for this machine. A high SCR, meaning a low synchronous reactance, raises the power a machine can carry before its stability limit and reduces its voltage change with load, at the cost of a longer air gap, more field ampere-turns, and a larger machine.

Other Tests and the Standards

A zero-power-factor test, with the generator delivering rated current to an inductive load, separates leakage reactance from armature reaction and yields the Potier reactance. A slip test gives the quadrature-axis reactance of a salient-pole machine: with the field open and a reduced balanced voltage applied, the rotor is driven slightly off synchronous speed, and the ratio of voltage to current swings between X_d and X_q. A sudden short circuit gives the transient and subtransient quantities described later.

IEEE 115-2019, the IEEE Guide for Test Procedures for Synchronous Machines Including Acceptance and Performance Testing and Parameter Determination for Dynamic Analysis, was published in March 2020, and IEEE has authorized its revision. Standstill frequency response testing, which identifies dynamic-model parameters without rotating the machine, was first standardized in IEEE 115A-1987, a supplement to the 1983 edition of IEEE 115. The IEC counterpart, IEC 60034-4-1:2018, describes methods for three-phase machines rated 1 kVA and larger whose field winding is supplied through slip rings and brushes. It notes that brushless machines need special effort for some tests and that its tests apply only in part to permanent-magnet machines, which must be protected against irreversible demagnetization.

Real Power

Let δ be the angle by which E_f leads V_t. Multiplying the second phasor relation by V_t/X_s gives the real power per phase, V_tI_a cos φ = V_tE_f sin δ / X_s. For three phases, with per-phase RMS voltages,

P = 3 V_t E_f sin δ / X_s

In per unit the factor 3 disappears. This is the transfer relation P = V₁V₂ sin δ / X of Power System Analysis, with the internal voltage as bus 1 and the terminals as bus 2, and the sign convention is the same: a positive δ, with E_f leading, means that the machine is generating, and a negative δ that it is motoring. Through an external reactance X_e to a network voltage V, the same form holds with V in place of V_t, X_s + X_e in place of X_s, and δ taken as the angle by which E_f leads V.

The torque is T = P/ω_sm. The angle δ is electrical; the rotor is displaced by δ divided by the number of pole pairs, so a 72-pole hydro-generator at a power angle of 38° runs only about 1.06 mechanical degrees ahead of its no-load position.

Reactive Power

Resolving along V_t gives the reactive power delivered at the terminals:

Q = 3 V_t (E_f cos δ − V_t) / X_s

A generator supplies reactive power when E_f cos δ exceeds V_t, the overexcited condition, and absorbs it otherwise. This is the machine form of the pairing, noted in Power Flow Analysis, of real power with angle and reactive power with voltage magnitude.

Pull-Out Power and the Steady-State Stability Limit

At fixed E_f and V_t, the power-angle curve peaks at the pull-out power, 3V_tE_f/X_s, when δ = 90°. Its slope is the synchronizing power coefficient:

P_s = dP/dδ = 3 V_t E_f cos δ / X_s

Below 90° the slope is positive: if the rotor advances slightly, the electrical power rises and pulls it back, so the machine is stable against small, slow changes. Beyond 90° an advance lowers the electrical power, the rotor gains on the field, and the machine pulls out of step. With constant field current, the steady-state stability limit of a round-rotor machine connected directly to an infinite bus is therefore δ = 90°. Operators keep a margin below it, and because the limiting power rises with E_f, an underexcited machine is less secure. The response to large, sudden disturbances, such as a fault, also depends on rotor inertia and clearing time; power system stability studies analyze it with the swing equation.

Synchronizing

Before its breaker closes, an incoming generator must match the network in phase sequence, frequency, voltage magnitude, and phase angle; closing with a large angle or voltage difference causes a current surge and a torque shock. An automatic synchronizer, or a synchroscope with a synchronism-check relay, governs the closing, as Hydroelectric Power Electronics describes. Grid-tied converters solve the equivalent problem with phase-locked loops, as Grid Synchronization and Control explains.

Controlling Real Power

After an ideal synchronization, E_f equals V_t and the machine carries no current. Opening the turbine's valves or wicket gates raises the mechanical torque; the rotor advances until the electrical power matches the input, and the machine settles at synchronous speed with a larger δ. The governor setpoint thus controls real power. At constant field current, raising the real power also lowers the reactive power, because E_f cos δ falls as δ grows.

Controlling Reactive Power

At constant mechanical power, E_f sin δ is fixed, so the tip of the E_f phasor moves along a line parallel to V_t. Raising the field current lengthens E_f, reduces δ, and raises E_f cos δ, so the machine delivers more reactive power. Lowering it does the reverse until the machine absorbs reactive power and, at δ = 90°, reaches its stability limit.

Worked Example

The generator of the test example, with X_s = 1.60 per unit on its 100 MVA rating, is connected to an infinite bus at its terminals, V_t = 1.0 per unit, and delivers 80 MW and 40 Mvar. The base current is 4,184 A, and the base phase voltage is 13.8/√3 = 7.967 kV.

1. Power. P = 0.80 and Q = 0.40 per unit, so the power factor is 0.80/√(0.80² + 0.40²) = 0.894 lagging.
2. Current. With V_t as reference, I_a = (S/V_t)^* = 0.80 − j0.40 = 0.8944 e^−j26.57° per unit, or 3,742 A.
3. Excitation voltage. E_f = V_t + jX_sI_a = 1.0 + j1.60(0.80 − j0.40) = 1.64 + j1.28 = 2.0804 e^j37.97° per unit. The excitation voltage is 2.0804 × 7.967 = 16.58 kV per phase, and the power angle is 37.97°.
4. Check. P = 2.0804 sin 37.97° / 1.60 = 0.800 per unit, and Q = (2.0804 cos 37.97° − 1.0) / 1.60 = 0.400 per unit.
5. Stiffness. The synchronizing power coefficient is 2.0804 cos 37.97° / 1.60 = 1.025 per unit per radian, or 1.79 MW per electrical degree.

Holding 80 MW and treating E_f as proportional to field current, as the linear model allows, gives the operating points in the table.

A 10 percent increase in field current raises the reactive output by 40 percent and the armature current by 9 percent. Cutting the field current to 70 percent turns the machine from a source of 40 Mvar into a sink of 19 Mvar. The least excitation that can carry 80 MW is E_f = PX_s/V_t = 1.28 per unit. If instead the governor raises the output to 90 MW at the base-case field current, δ grows to 43.80° and Q falls to 0.313 per unit.

V-Curves

Plotting armature current against field current at constant real power gives the V-curves. Each has its minimum, I_a = P/V_t, at unity power factor, where E_f = √[V_t² + (PX_s/V_t)²]. To the right of the minimum the machine is overexcited and delivers reactive power; to the left it is underexcited, until the curve ends at the stability limit, E_f = PX_s/V_t. The table gives points for the example machine, with E_f standing in for field current; beyond the limit no steady operating point exists.

The minima fall at E_f = 1.00, 1.28, and 1.62 per unit. At zero real power the machine is a synchronous condenser drawing |E_f − V_t|/X_s.

Armature Current Limit

Stator winding temperature limits the armature current. At rated voltage, constant current means constant apparent power, so the limit is a circle about the origin with a radius of rated MVA, 1.0 per unit.

Field Current Limit

Rotor winding temperature limits the field current, and with it E_f, to the constant-excitation circle through the rated operating point, where the machine delivers rated MVA at rated power factor. If the example machine is rated at 0.85 power factor lagging, its rated point has E_f = 2.290 per unit at δ = 36.43°, so the field limit is a circle centered at Q = −0.625 per unit with a radius of 1.431 per unit. At zero real power it allows 0.806 per unit of reactive output, less than the armature could carry. Above rated real power the armature limit governs: at P = 0.95 per unit it allows 0.312 per unit of reactive power, against 0.446 per unit for the field. The limits meet at the rated point because the field winding is designed for the excitation that point requires.

Underexcited Limits

The theoretical steady-state stability limit, δ = 90°, is the line Q = −V_t²/X_s, or −0.625 per unit for the example machine, when the machine connects directly to an infinite bus. With an external reactance X_e, it becomes a circle centered at Q = (V_t²/2)(1/X_e − 1/X_s) with a radius of (V_t²/2)(1/X_e + 1/X_s), whose lower arc rises with real power and shrinks the stable underexcited region at high output. Behind X_e = 0.20 per unit, the example machine's limit rises from Q = −0.625 per unit at zero output to −0.477 per unit at P = 0.90 per unit. Both forms assume constant field current; a continuously acting voltage regulator can extend the practical limit.

The second limit is thermal. When a generator is underexcited, the low field current leaves the rotor retaining rings less saturated, and the end-winding leakage fluxes of stator and rotor combine to raise the axial flux entering the end packets of the stator core, where eddy currents heat the laminations. This core-end heating can severely restrict underexcited operation, particularly in round-rotor machines, and an underexcitation limiter in the excitation system keeps the machine inside both limits.

Prime Mover Limit

The turbine's maximum output adds a vertical line, and some turbines also have a minimum stable output. Hydrogen-cooled generators have one curve for each hydrogen pressure.

Starting

At standstill the stator field sweeps past the poles too quickly for the rotor to follow, so the motor must be brought near synchronous speed by other means.

• Induction starting. A line-started motor accelerates as an induction motor on its damper winding, with the field winding short-circuited through a discharge resistor to limit the high voltage that the slipping stator field would induce in its many turns. Near synchronous speed the controller applies direct current to the field, and the rotor pulls into step; the largest load torque at which the motor can pull its connected inertia into step is the pull-in torque. Reduced-voltage starting limits inrush current where the supply is weak.
• Variable-frequency starting. A converter raises the stator frequency from near zero along with the rotor, keeping the machine synchronized. Pumped-storage units in pumping mode start this way, as do gas-turbine generators, which run as motors to accelerate their turbines, and motors that run permanently from a drive.
• Auxiliary motor. A small pony motor accelerates the unloaded machine, a method found mainly on older installations.

Power Factor Correction and Synchronous Condensers

An overexcited synchronous motor supplies reactive power while it drives its load. Suppose a plant draws 1,500 kW at 0.80 power factor lagging, or 1,125 kvar, and a new synchronous motor drawing 300 kW is to raise the plant's power factor to 0.95 lagging.

1. The plant will draw 1,800 kW, so at 0.95 power factor its reactive demand can be at most 1,800 × tan(cos⁻¹ 0.95) = 591.6 kvar.
2. The motor must supply the difference, 1,125 − 591.6 = 533.4 kvar.
3. The motor therefore runs at √(300² + 533.4²) = 612 kVA, at a power factor of 0.49 leading.

At unity power factor the same motor would leave the plant at 0.85 lagging. The motor must be rated, and its field winding sized, for the leading duty.

A synchronous condenser is a synchronous motor with no mechanical load, operating on the zero-power V-curve. Unlike a capacitor bank, whose output falls with the square of the voltage, a condenser under voltage control raises its output when the voltage sags, and it adds inertia and short-circuit current. Condensers are again being installed where the retirement of synchronous generation has weakened the network, sometimes beside converter-based compensators, as Static VAR Compensators describes.