Synchronous Machine Armature Windings - 2 | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) PDF Download

Frequency of an A.C. synchronous generator
Commercial a.c. synchronous generators can have many poles and operate at various shaft speeds depending on the prime mover and application. For a two-pole machine, one complete mechanical revolution produces one electrical cycle; hence the output frequency is directly proportional to the mechanical speed (revolutions per second). When the machine has P poles, the number of electrical cycles produced per mechanical revolution equals the number of pole pairs (P/2). Therefore the usual relationship between electrical frequency, number of poles and speed is:

(18)

where:
P is the number of poles; N is the speed in rpm (rev/min); f is the frequency in hertz (cycles per second); ω_m is the mechanical angular speed in rad/s; ω_e is the electrical angular speed in rad/s.

In formula form the commonly used relations are:
f = (P × N) / 120
ω_m = 2πN / 60
ω_e = (P / 2) ω_m

Constructional details of the rotor

The field windings of a synchronous machine are carried on the rotor. For large three-phase synchronous generators there are two principal rotor constructions:

• Cylindrical (non-salient) rotor - used for high-speed machines driven by steam turbines (often called turbo-alternators). Typical synchronous speeds for 50 Hz are 1500 rpm (4-pole) and 3000 rpm (2-pole). Cylindrical rotors have a smooth contour, are compact, resist centrifugal forces at high speeds, and give low windage losses and quiet running.
• Salient-pole rotor - used for low-speed machines, such as hydro generators or low-speed motors. These rotors have projecting pole shoes and require many poles for low speeds (for example, a 50 Hz generator running at 100 rpm needs P = 120 × 50 / 100 = 60 poles).

Figure 14 (next group of images) illustrates two- and four-pole cylindrical rotors and a developed view of field winding for a pair of poles, together with a single salient pole and its field coil. The stator slots (which contain the armature winding) are omitted for clarity. The approximate path of the main field flux (excluding leakage) is shown by dashed lines in the illustration.

Figure 14: Synchronous machines with stator slots and armature windings omitted (a) Two-pole cylindrical rotor, (b) Four-pole cylindrical rotor, (c) Developed view of two-pole cylindrical rotor field structure, (d) Salient pole and field coil

Figure 15: Cylindrical rotor m.m.f. wave and its fundamental of a synchronous machine

When the field winding conductors are considered in the slots and nearly fill the slot area, the resultant magnetomotive-force (m.m.f.) distribution across the air gap is a stepped waveform. Its fundamental component is approximately sinusoidal (dashed sine in Fig. 15). For cylindrical rotors the air-gap is almost uniform, so the stepped m.m.f. produces a flux density space-wave that is closer to sinusoidal because of fringing effects. However, in regions of high m.m.f. the iron may saturate and flatten the flux-density peak.

Excitation systems for synchronous machines

Direct current must be supplied to the rotor field winding to produce the main field flux. Several excitation arrangements are in use; the choice depends on machine size, dynamic requirements of the power system and reliability needs. Common methods are:

• Direct-connected rotating exciter (d.c. generator) mounted on the same shaft as the synchronous machine. The exciter output feeds the field winding through slip-rings and brushes. The exciter field may be varied by a rheostat to adjust the main machine field strength.
• Motor-generator set or separate motor driven exciter where a separate prime mover drives the exciter if shaft-driven excitation is not desired.
• Static (solid-state) excitation where an a.c. supply and rectifiers provide d.c. to the rotor field; often used where a compact, reliable excitation is required.
• Brushless excitation (described in the next section) where rotating rectifiers and a rotating armature eliminate the need for brushes and slip rings.

Typical excitation voltages are about 125 V for units up to around 50 kW, with higher voltages for larger machines. Faster response (important during system disturbances) can be obtained by multi-stage arrangements such as a pilot exciter that drives the main exciter field, providing quicker changes in field current.

Brushless excitation system

A brushless system removes the commutator/collector rings and brushes, improving reliability and reducing maintenance. A typical arrangement consists of:

• a small rotating permanent-magnet pilot exciter with stationary armature,
• a regulator that rectifies/regulates the pilot exciter output,
• a rotating-armature a.c. exciter whose stationary field receives the regulated dc, and
• rotating rectifiers (diode bridge) that convert the ac exciter output to dc to feed the turbo-generator field.

Figure 17: Brushless excitation system

Brushless excitation is widely used in turbo-generators and aircraft generators where reduced maintenance, elimination of brushes in low pressure environments, and high mechanical simplicity are advantageous.

The action of the synchronous machine under load

When a synchronous generator supplies an external load the armature currents alter the air-gap flux distribution produced by the rotor field. The relative phase of the armature current and the induced e.m.f. depends on the load power factor (p.f.):

• For unity power factor (resistive load) current and induced e.m.f. are approximately in phase (neglecting resistance and leakage reactance).
• For lagging p.f. (inductive load) the current lags the induced e.m.f. by up to 90 electrical degrees; the current peak occurs about half a pole pitch behind the e.m.f. peak.
• For leading p.f. (capacitive load) the current leads the induced e.m.f. by up to 90 electrical degrees; the current peak occurs about half a pole pitch ahead of the e.m.f. peak.

For simplicity many treatments assume:

• stator resistance and leakage reactance are negligible;
• magnetic circuit is linear (no saturation), so flux is proportional to resultant ampere-turns.

Under these idealisations the induced e.m.f. equals the terminal voltage and the phase angle between current and e.m.f. is determined solely by the external load power factor.

Armature reaction

The term armature reaction denotes the effect of the armature (stator) currents on the main field flux. Consider the stretched-out (developed) view of a synchronous machine showing fixed stator armature windings and a rotating field (rotor) carrying dc field windings. When the induced e.m.f. in a given stator coil is at maximum, the coil may carry a current depending on the load p.f. The ampere-turns of this coil distort the main flux: they strengthen the flux on one side of a pole and weaken it on the other, causing a displacement and distortion of the main flux distribution.

For a resistive (unit p.f.) generator load the armature mmf distorts the field so that the resultant main flux axis is shifted slightly against the direction of rotation. The induced e.m.f. E produced by the distorted flux is displaced by an angle δ from the position E₀ the e.m.f. would have if no armature current flowed. This shift is often referred to as cross-magnetization.

Figure 19: Stretched out synchronous motor

When the machine operates as a motor at unity p.f. the armature current direction reverses and the armature mmf distorts the main flux in the opposite sense. The e.m.f. E is then displaced ahead of E₀ by an angle δ (see adjacent phasor diagrams).

Purely inductive or capacitive load cases

For a purely inductive load (current lags E by 90°) the stator ampere-turns act directly to oppose the rotor field ampere-turns, reducing total flux and induced e.m.f. No torque results from the cross-magnetizing component in this case and hence no mechanical power is produced (E and I are in quadrature so VI = 0 for idealised lossless machine).

Figure 20: Stretched out synchronous generator (inductive load)

For a purely capacitive load (current leads E by 90°) the stator ampere-turns assist the rotor field ampere-turns and the total flux and induced e.m.f. increase.

Figure 21: Stretched out synchronous generator (capacitive load)

For intermediate power factors both direct-magnetizing (which increases or decreases net flux depending on sign) and cross-magnetizing (which distorts and shifts the flux axis) components exist simultaneously. The overall effect influences the terminal voltage, the required excitation current and, for motors, the operating power factor.

Behaviour of a loaded synchronous generator

Summarising the basic working:

• A synchronous machine driven as a generator produces e.m.f.s in its armature windings at frequency f = n p (where n is mechanical revolutions per second and p is pole pairs).
• When armature currents flow, their ampere-turns modify the main field flux according to the load p.f., producing mechanical torque and requiring mechanical input power to balance the electrical output.
• In a balanced three-phase machine the fundamental component of the armature reaction is a rotating steady m.m.f. so that the total electromagnetic torque is unidirectional and, for balanced loads, the three-phase torque is practically constant (individual phase torques pulse but sum to nearly constant value).

Figure 22: Synchronous generator supplying a lagging p.f. load

For cylindrical (non-salient) rotor machines with nearly uniform air-gap the armature reaction can be divided conveniently into two orthogonal components: the cross-magnetizing component and the direct-magnetizing component. The armature reaction m.m.f. F_a is resolved into F_aq (quadrature, cross) and F_ad (direct), where F_ad either demagnetizes (lagging currents) or magnetizes (leading currents) the main field, and F_aq shifts the resultant flux axis.

Figure 23: Sinusoidal distribution of the components of armature reaction in a synchronous generator

Figure 24: Elementary synchronous motor action - Attraction of unlike poles keeps the rotor locked to the rotating stator field

Concept of synchronous reactance

To model the effect of armature reaction in circuit form the concept of synchronous reactance is introduced. The resultant flux linkage of an armature phase is split into two components: the flux due to the rotor (field) current alone and that due to the armature current alone. This superposition is approximately valid when the magnetic circuit permeability is constant (i.e. low saturation).

The simplifying assumptions commonly used are:

1. The permeability of all magnetic parts is constant (no significant saturation); the field and armature fluxes can be superposed.
2. The air gap is uniform, so armature flux distribution is independent of rotor angular position (cylindrical rotor approximation).
3. The distribution of the field flux in the air gap is sinusoidal.
4. The armature winding is uniformly distributed and carries balanced sinusoidal currents (harmonics are neglected).

Under these assumptions the effect of armature reaction can be represented by a phasor voltage E_r that is in quadrature with the armature current I_a and proportional to it. Writing the actual induced e.m.f. per phase as E, and the no-load induced e.m.f. (due to field only) as E_o, we have the phasor relation:

E = E_o + E_r         (19)

E_r behaves like the voltage across an inductive reactance caused by the armature current. Define the fictitious reactance of the armature reaction as

x_a = E_r / I_a

The armature leakage reactance is x_l. The sum

x_s = x_a + x_l

is the synchronous reactance. The per-phase synchronous impedance is then

Z_s = r_a + j x_s

where r_a is the armature resistance. For most machines x_s ≫ r_a, so the synchronous impedance magnitude is dominated by x_s. The synchronous reactance model leads to the usual per-phase equivalent circuit for a synchronous machine from which voltage regulation, short-circuit behaviour and power angle relations can be derived.

Approximation of the saturated synchronous reactance

In practice the magnetic circuit is saturated under normal operation, so the linear assumptions are only approximate. A useful approximate synchronous reactance can be obtained from open-circuit and short-circuit tests carried out on the machine. The saturated synchronous reactance is normally determined from rated open-circuit voltage and the short-circuit current produced by the same field current.

Graphically, using the open-circuit characteristic (O.C.C., magnetization curve) and the short-circuit characteristic (S.C.C.), the synchronous impedance Z_s (per phase) under test conditions is approximately the ratio of the open-circuit per-phase voltage to the short-circuit per-phase current obtained for the same field current. This gives a practical estimate of Z_s that accounts approximately for the machine saturation in service.

(20)

Because synchronous reactance varies with saturation, the value obtained from tests is an approximation adequate for many engineering uses.

Open-circuit and short-circuit tests

Open-circuit and short-circuit tests provide the data necessary to obtain the magnetisation (O.C.C.) and short-circuit characteristics. These curves show the effect of saturation and are employed to estimate the saturated synchronous reactance.

Figure 26: Synchronous generator (a) Open circuit (b) Short circuit

When a constant voltage source has constant impedance, the impedance equals the open-circuit terminal voltage divided by the short-circuit current. However, for a synchronous machine the impedance depends on flux saturation; thus both O.C. and S.C. characteristics are required.

Open-circuit characteristic (O.C.C.)

With the machine driven at rated speed and the armature terminals open, line-to-line voltage is measured for various values of field current. The O.C.C. (voltage versus field current) shows the magnetisation curve and the effect of iron saturation. The ideal linear (air-gap) line is the unsaturated reference; per-unit representation is often used so that 1.0 p.u. field current corresponds to the field current that would produce rated voltage if there were no saturation.

Figure 27: (a) Open circuit characteristic and (b) Short-circuit characteristic

Short-circuit characteristic (S.C.C.)

For the short-circuit test, the three armature terminals are shorted (usually through current transformers and ammeters) and the machine is driven at rated speed. Armature currents are measured for various field currents, typically up to and somewhat beyond the rated armature current. The S.C.C. (armature current versus field current) is approximately linear up to rated currents for conventional synchronous machines because the iron remains largely unsaturated under short-circuit flux distributions.

Figure 28: Connections for short-circuit test

Unsaturated synchronous impedance

Plotting the O.C.C. and S.C.C. together (often in per-unit) allows determination of the unsaturated synchronous reactance. For a given field current the open-circuit (air-gap) line gives the corresponding voltage and the short-circuit curve gives the armature current. The unsaturated synchronous reactance per phase (for a star-connected armature) is therefore given approximately by the ratio of the per-phase air-gap voltage to the per-phase short-circuit current produced by the same field current.

Figure 29: Open-circuit and short-circuit characteristic

In symbols (per phase):

(23)

When the O.C.C., air-gap line and S.C.C. are plotted in per-unit, the per-unit unsaturated synchronous reactance equals the per-unit value on the air-gap line corresponding to the field current that produces rated short-circuit current (1.0 p.u.). This per-unit approach makes comparisons between different machines straightforward.

Open remarks and practical considerations

Key practical points:

• Because x_s is generally much larger than r_a, the synchronous machine voltage regulation and short-circuit behaviour are largely reactive phenomena.
• Field excitation must be adjusted to maintain terminal voltage under varying load and power factor conditions; the O.C.C. and an estimate of x_s help choose the correct excitation current.
• Brushless excitation adds reliability and reduces maintenance - useful in turbine generators and aircraft applications.
• Salient-pole machines deviate from the cylindrical assumptions (non-uniform air-gap and non-sinusoidal flux distributions); more detailed models are required for accurate prediction of their behaviour, especially under transient conditions.

Summary
The frequency of a synchronous machine is determined by speed and number of poles. Rotor construction (cylindrical or salient) is chosen according to speed and mechanical stresses. Excitation systems supply the dc field current; brushless systems avoid brushes and slip-rings. When loaded, the stator currents produce armature reaction that distorts and/or changes the magnitude of the main flux; its effect is modelled by the synchronous reactance and synchronous impedance. Open-circuit and short-circuit tests provide practical data for estimating synchronous reactance, taking account of saturation.