Synchronous Generator Operation - 2 - Electrical Engineering SSC JE (Technical)

Salient Pole Rotor Machine

Synchronous reactance x_s is commonly used to represent the effect of armature reaction and leakage flux on the behaviour of a synchronous machine under load. It is expressed as the sum of a fictitious reactance representing armature reaction and the leakage reactance:

x_s = x_a + x_l

Here x_a is a fictitious reactance representing armature reaction and x_l is the leakage reactance. The above representation is adequate for a cylindrical-rotor (non-salient) synchronous machine because the armature and main field magnetomotive forces act on essentially the same magnetic circuit and saturation effects are neglected.

In a salient-pole synchronous machine, however, the magnetic paths seen by different components of the armature mmf are different. Consequently, the cylindrical-rotor approach (single synchronous reactance) does not give accurate results for salient-pole machines. The more appropriate model for salient-pole machines is Blondel's two-reaction theory, which treats the direct- and quadrature-axis effects of armature reaction separately.

Theory of Salient-pole Machines (Blondel's Two-reaction Theory)

In a salient-pole machine, the armature mmf can be resolved into two orthogonal components:

• Direct-axis component (F_ad) - parallel to the main pole axis (field axis). This component acts through a magnetic circuit similar to that of the main field and therefore produces a relatively large flux for a given current.
• Quadrature-axis component (F_aq) - along the interpolar (magnetic neutral) axis. This component acts through interpolar air gaps and other high-reluctance portions of the magnetic circuit, so its flux effect for the same current is smaller and its flux distribution is different from the direct-axis flux.

Because the two components act through different magnetic circuits, their effects cannot be lumped into a single reactance. Blondel's two-reaction theory assigns separate armature-reaction reactances to these components, denoted x_ad (direct-axis armature reaction reactance) and x_aq (quadrature-axis armature reaction reactance). The leakage reactance x_l may be treated separately or added to each armature-reaction reactance (approximately valid since leakage affects both components similarly). Thus the combined synchronous reactances for the direct and quadrature axes are:

(these correspond to the direct- and quadrature-axis synchronous reactances)

Typically, x_sq < x_sd because the quadrature axis magnetic path has higher reluctance (interpolar spaces) and therefore produces less flux per unit current than the direct axis path.

Direct-axis and Quadrature-axis Currents and Phasors

Let I_a be the phasor armature current. Resolve it into components along the direct and quadrature axes:

I_a = I_ad + j I_aq

Here I_ad is the component along the direct axis and I_aq is the component along the quadrature axis. Note that I_ad and I_aq are phasors in quadrature with one another, but these are different from the active and reactive components of current (often written as I_ar and I_aa), which are referred to the terminal voltage phasor.

The phasor diagram of a salient-pole synchronous generator supplying a lagging power-factor load is illustrated in the figure below.

The angle σ is the torque (power) angle between the induced emf and terminal voltage. The load power factor angle is φ. In the diagram the two reactance voltage drops I_aq × x_sq and I_ad × x_sd are shown; each is in quadrature with the respective current component.

The armature resistance drop I_a × R_a is in phase with the armature current I_a. While one may split the resistance drop into direct- and quadrature-axis parts (I_aq × R_a and I_ad × R_a), for most practical purposes it is added in series with the total current phasor.

The phasor diagram is best constructed geometrically if terminal voltage V, load power-factor angle φ, and the synchronous reactances x_sd and x_sq are known. A standard construction is to draw the current phasor, add the resistance drop from the tip of the voltage phasor in the direction of the current, then add the quadrature and direct reactance drops at right angles to their respective current components. The resulting vector from the origin to the final point represents the induced emf E_t.

Power Relations in a Salient-pole Synchronous Machine

Neglecting armature winding resistance, the real power delivered to the load is

P = V × I_a × cos φ      (expression in terms of terminal quantities)

By using the phasor relationships and Blondel's two-reaction model, this power can be expressed in terms of the torque (power) angle σ and the direct- and quadrature-axis synchronous reactances. Using standard derivation (see phasor geometry shown in the figures), one obtains the well-known expression for the electrical power developed by a salient-pole synchronous generator (ignoring R_a):

P = (V E₀/x_sd) sin σ + (V²/2) (1/x_sq - 1/x_sd) sin 2σ

In this expression:

• E₀ is the internal induced emf due to excitation (field current).
• x_sd and x_sq are the direct- and quadrature-axis synchronous reactances respectively.
• The first term, (V E₀/x_sd) sin σ, is the familiar power term that also appears for cylindrical-rotor machines.
• The second term, (V²/2)(1/x_sq - 1/x_sd) sin 2σ, is characteristic of salient-pole machines and arises because of the difference between direct- and quadrature-axis reactances.

The presence of the sin 2σ term means that the power-angle characteristic of a salient-pole machine differs from that of a cylindrical rotor. One direct consequence is that even with zero excitation (E₀ = 0) a non-zero power (and torque) may be produced due to the second term; this is called the reluctance torque. The magnitude of this reluctance contribution is, however, small compared with that produced by finite excitation.

Experimental Determination of x_sd and x_sq (Slip Test)

The unsaturated values of the direct- and quadrature-axis synchronous reactances of a three-phase synchronous machine can be obtained experimentally by the slip test. The essential procedure is as follows:

1. Drive the synchronous rotor at a speed close to synchronous speed (by a prime mover such as a DC motor), but ensure the speed is not exactly synchronous. Drive in the intended direction of rotation.
2. Keep the field circuit open (field current zero).
3. Supply a low balanced three-phase voltage to the stator (armature) through a variac and vary the applied voltage.
4. Observe the armature current waveform and note its minimum and maximum values as the rotor mmf sweeps through different magnetic reluctance paths (from interpolar region to pole axis region).
5. The ratio of the applied voltage to the minimum armature current gives an estimate of the direct-axis synchronous reactance x_sd. The ratio of the applied voltage to the maximum armature current gives the quadrature-axis synchronous reactance x_sq.

For more accurate determination, the applied voltage and armature current oscillograms can be recorded and analysed. The slip test yields unsaturated reactance values since the field is open and the machine operates with small flux levels.

Figure 34: Phasor diagram of a generator - Two-reaction theory

Losses and Efficiency

When calculating the efficiency of a synchronous generator under load, all significant losses must be identified and quantified. The principal losses are:

• Rotational losses - friction and windage losses. These are essentially constant because the synchronous machine runs at nearly constant speed. They can be determined from no-load tests.
• Core (iron) losses - hysteresis and eddy-current losses in the magnetic circuit due to alternating flux density. Core loss is measured by comparing the driving power with and without excitation when the generator runs at no load; the difference gives the core loss.
• Copper losses - I²R losses in the armature windings and in the field winding. The armature copper loss per phase is I_a²R_a; multiply by the number of phases for total armature copper loss. The field copper loss is V_fI_f (or I_f²R_f), where V_f and I_f are the field winding voltage and current respectively.
• Load (stray) losses - additional losses caused by leakage flux producing local eddy currents and increased iron loss where the flux distribution is distorted by loaded armature MMF. These stray losses are often accounted for by using an effective armature resistance larger than the dc resistance when computing I²R losses.

Remarks on individual losses:

• Rotational losses are nearly constant because of constant synchronous speed.
• Core loss is frequency- and flux-dependent and is obtained from no-load measurements with and without excitation.
• Copper losses vary with load current; they are computed from winding resistances and measured currents. Use per-phase values carefully and multiply appropriately for total losses in a three-phase machine.
• Stray losses are difficult to separate precisely by simple tests; they are commonly included with copper losses or represented by an effective additional resistance.

Once all losses have been evaluated, efficiency η (in per unit or percent) is given by the ratio of the real output power to the input power from the prime mover:

where η = efficiency,

kVA = apparent power rating of the generator,

PF = power factor of the load,

and the product (kVA × PF) is the real power output (kW) delivered to the load.

The input power from the prime mover is

P_in = P_out + P_losses

Summary

Blondel's two-reaction theory is required for accurate analysis of salient-pole synchronous machines because the armature reaction has distinct direct-axis and quadrature-axis effects. These are modelled by separate synchronous reactances x_sd and x_sq, which lead to an extra sin 2σ term in the power-angle characteristic and permit the existence of reluctance torque. The slip test provides an experimental method to determine unsaturated direct- and quadrature-axis reactances. Accurate accounting of rotational, core, copper and stray losses is required to compute the generator efficiency.