Polyphase Synchronous Machines - 1

Table of Contents
1. Introduction
2. Constructional Features
3. Generated emf in Armature
4. Field Winding (Rotor)
5. Flux and MMF Phasors
6. Cylindrical-rotor Synchronous Motor
7. Open-Circuit and Short-Circuit Characteristics
8. Zero Power-Factor Characteristic and Potier Triangle
9. Voltage Regulation of an Alternator
10. Practical Notes and Summary Points
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Introduction

Rotating machines that rotate at a speed fixed by the supply frequency and the number of poles are called synchronous machines.

A three-phase synchronous machine is a doubly-excited AC machine: its field winding (rotor) is energised from a DC source (for example DC exciter, static excitation) and its armature winding (stator) is connected to the AC supply or load.

Under steady-state conditions the operating speed of a synchronous machine depends on the frequency of the armature current and the number of field poles.

Synchronous speed

The synchronous speed of a machine relates electrical frequency and number of poles.

Formula (rotational speed):

n_s (rps) = 2 f / P

Alternative (rpm): n_s (rpm) = 120 f / P

Where P = number of field poles, f = supply frequency (Hz), and n_s = synchronous speed.

Remember:

• In a synchronous machine the three-phase armature winding is on the stator and the field winding is on the rotor.

Constructional Features

Synchronous machines are designed so that the stator carries the armature winding and the rotor carries the field winding. There are two common mechanical arrangements for AC rotating machines: (a) rotating field and stationary armature, (b) rotating armature and stationary field. Modern large alternators typically use rotating field and stationary armature for practical advantages.

• With armature on the stator and field on the rotor only two slip rings are required to transfer DC excitation to the rotor; this is simpler and more reliable than transferring large AC power on the rotor.
• Stationary armature windings can be insulated to higher voltages more conveniently than rotating windings.
• Stationary armature windings allow more efficient cooling arrangements.
• Field winding on the rotor is low-power (DC) and requires less copper and insulation; this reduces rotor weight and inertia and permits higher permissible rotor speeds for a given centrifugal stress.
• Placing high-voltage three-phase armature windings in the stator slots makes the machine bracing and mechanical support simpler and more rigid against electromagnetic forces.

Rotor types

Synchronous machines are classified by rotor geometry:

• Salient-pole type - poles are projecting ("salient"). Field winding on each pole is generally a concentrated winding. The air-gap is non-uniform (smaller under pole faces, larger between poles).
• Cylindrical (round) rotor type - rotor surface is nearly cylindrical; field winding is distributed in rotor slots. The air-gap is approximately uniform (neglecting slot openings). This type is used for high-speed machines (steam-turbine driven alternators).

Synchronous generators are usually three-phase because of advantages in generation, transmission and utilisation of power. For three-phase generation at least three coils displaced by 120 electrical degrees in space are required.

Generated emf in Armature

In a rotating machine the relative motion between an armature coil and the flux-density wave causes time variation of flux linkage and an induced emf in the coil.

• Let ø be the total flux per pole and k_w be the winding distribution factor (since armature winding is distributed); the induced emf expression contains k_w.
• The generated phase emf in a synchronous machine is proportional to flux per pole and speed, and is reduced by winding factors.

Field Winding (Rotor)

The field winding of a synchronous machine is always energised with direct current under steady-state operation.

Field current relation:

I_f = V_f / r_f

Where V_f is the DC field voltage and r_f is the field winding resistance.

Flux and MMF Phasors

We use phasor diagrams in space and time to represent the relationships between field mmf, armature mmf, fluxes and induced emfs. The following discussion is primarily for cylindrical-rotor machines where the air-gap is essentially uniform and phasor addition of mmfs is straightforward.

• The no-load terminal voltage of an alternator is adjusted to rated value by varying the field current.
• The emf generated by a flux (field) lags that flux by 90° in time: the induced voltage is 90° behind the flux wave which creates it. Thus E_f lags ø_f by 90°.
• Field mmf per pole F_f = I_f N_f (neglecting saturation the flux ø_f is in phase with field mmf F_f).

Case 1 - No-load / Excitation

The emf produced by the field flux alone is called the excitation voltage.

When Alternator is Connected to Balanced 3-Phase Load

• Balanced three-phase generated emfs result in balanced armature currents.
• At unity power factor the armature current I_a and excitation voltage are maximum at the same instant; armature mmf produced is perpendicular to the field flux and therefore is cross-magnetising.
• The mmf set up by armature current is called armature-reaction mmf F_a.
• The phasor sum F_f + F_a gives the resultant air-gap mmf F_r.
• For generator operation the prime-mover torque must oppose the electromagnetic torque produced.

Case 2 - Unity power-factor load

Case 3 - Zero power-factor (lagging) load

• When load power factor is zero and lagging, the armature mmf produced opposes the field mmf (demagnetising effect).
• E_f lags ø_f by 90°, I_a lags E_f by 90°, therefore F_a opposes F_f.
• Under this condition the armature mmf is entirely demagnetising in nature for a generator.

Case 4 - Zero power-factor (leading) load

• For zero power-factor leading the armature mmf is entirely magnetising in nature (it aids the field mmf).

Case 5 - General lagging power-factor load

Let armature current I_a lag the excitation voltage E_f by an electrical angle ψ. Then the armature reaction mmf F_a lags the field mmf F_f by a space angle of (90° + ψ). The resultant mmf F_r is the phasor sum of F_f and F_a. The current I_a lags E_f by ψ because load power factor measured with respect to E_f is cos ψ lagging.

Cylindrical-rotor Synchronous Motor

• In motor operation the rotating armature mmf phasor F_a is in phase with I_a and the armature-reaction flux ø_a is in phase with F_a.
• The resultant mmf F_r is the phasor sum of F_f and F_a. For motor operation the field poles must be dragged behind the resultant air-gap flux by the load torque.

When I_a lags E_f by 90°, in an alternator the armature mmf is magnetising; in a synchronous motor it is demagnetising.

Cylindrical Rotor Alternator - Resultant Flux

• The air-gap flux actually present is due to the resultant mmf of field and armature currents.
• For cylindrical rotors the phasor addition of field mmf F_f and armature mmf F_a is valid because both mmfs are approximately sinusoidally distributed and there is no relative motion between their spatial patterns at synchronous speed.

Open-Circuit and Short-Circuit Characteristics

These tests are essential for finding machine parameters and predicting performance.

Open-Circuit Characteristic (OCC)

Drive the alternator at rated speed with armature open-circuited and increase field current from zero while noting the terminal voltage. Plot terminal voltage (generated emf) versus field current (or field mmf).

• At small field current the iron requires negligible mmf so nearly all applied mmf appears across the air-gap; the OCC is approximately linear for small excitations.
• As field mmf increases iron parts require growing mmf (saturation) and the OCC curve bends and becomes nonlinear.

Short-Circuit Characteristic (SCC)

Drive the alternator at rated speed and short-circuit the armature terminals through an ammeter. Increase field current and note the short-circuit armature current.

• During the short-circuit test the machine typically operates under unsaturated conditions and SCC is approximately a straight line.

Zero Power-Factor Characteristic and Potier Triangle

The zero-power-factor characteristic (ZPFC) is the plot of armature terminal voltage versus field current when the machine supplies a fixed armature current at essentially zero (purely inductive) power factor. ZPFC combined with OCC is used to find armature leakage reactance X_al and armature reaction mmf F_a.

• Procedure to obtain ZPFC: run machine at rated speed, connect a purely inductive load, increase field current until full-load armature current flows, and record terminal voltage. Vary load in steps and adjust field to maintain full-load current; plot terminal voltage versus field current.

In the zero-power-factor lagging condition the terminal voltage V_t and air-gap voltage E_r are nearly in phase and satisfy the approximate relation

V_t = E_r - I_a X_al

The resultant mmf F_r and field mmf F₁ are related by simple algebraic relations on the potier diagram.

Voltage Regulation of an Alternator

Voltage regulation is the change in terminal voltage when the machine goes from no-load to full-load at constant speed and field excitation. In large machines it is difficult to load them to rated output for direct measurement, so test methods are used to determine machine constants and compute regulation.

Methods for Computing Voltage Regulation

Electromotive Force (emf) Method (Synchronous impedance method)

This is also known as the synchronous impedance method. It can be applied readily to cylindrical-rotor synchronous machines because the resultant air-gap flux is not affected by rotor angular position.

Assumptions:

• Magnetic circuit iron is assumed to have constant permeability (saturation neglected).
• mmfs can be treated as proportional to corresponding fluxes and thus to induced emfs.

Field mmf F_f generates an emf E_f which lags F_f by 90°. Similarly resultant mmf F_r generates air-gap voltage E_r lagging F_r by 90°, and armature-reaction mmf F_a generates E_ar lagging F_a by 90°.

Define a constant K (slope of the air-gap line) and a constant C so that armature-reaction mmf is proportional to I_a. The armature-reaction effect can be represented as an equivalent emf or as an equivalent reactance.

In phasor form the air-gap emf E_r is given by the phasor sum of terminal voltage V_t, armature drop I_a r_a and reactive drop I_a X_al.

By grouping constants we obtain an equivalent reactance due to armature reaction X_ar. The total synchronous reactance

X_s = X_al + X_ar

and the per-phase synchronous impedance

Z_s = r_a + j X_s

X_ar is a fictitious reactance that accounts for the voltage generated by armature-reaction mmf. The term (r_a + j X_s) is called the synchronous impedance.

Remember: for an alternator the current and power leave the machine; for a synchronous motor the current and power enter the machine.

Measurement of Z_s and X_s

• Open-circuit (OCC) and short-circuit (SCC) characteristics are required to determine Z_s and X_s.
• In the open-circuit test the armature current is zero so V_t = E_f (the generated emf equal to terminal voltage).
• In the short-circuit test the entire emf E_f is consumed in circulating the short-circuit current I_sc through synchronous impedance Z_s.

In the short-circuit test the phasor diagram shows E_f across Z_s carrying I_sc.

Note: If there were no saturation Z_s would be constant. In practice Z_s varies and decreases as saturation increases on the OCC.

For voltage regulation calculations one value of Z_s must be chosen; the lowest value (from largest possible short-circuit current) is commonly used which gives a conservative estimate for regulation by emf method.

Armature resistance and effective resistance

• The dc resistance r_dc of one phase is measured by voltmeter-ammeter method.
• If the armature winding is star-connected and neutral is not available, use the appropriate formula for conversion.

The effective armature resistance per phase in AC operation is larger than the measured DC resistance due to skin effect and heating; approximate relation commonly used is

r_a = (1.2 to 1.3) r_dc

Phasor diagrams differ for lagging, leading and unity power-factor loads. Representative diagrams are:

The percentage voltage regulation is calculated from phasor relations and usually expressed as:

NOTE: Because the emf method uses the unsaturated synchronous impedance it tends to give a higher (pessimistic) value of voltage regulation than the actual value since under load the machine flux paths are more saturated and effective reactance is lower.

The Magnetomotive-Force (mmf) Method

This method replaces emfs by equivalent mmfs (assumes uniform air-gap and neglects saturation). The voltage equation of an alternator can be converted into an mmf equation by dividing by a suitable constant.

• The field mmf F_f induces an emf E_f lagging it by 90°; resultant mmf F_r induces an emf E′ lagging F_r by 90°.
• The armature-reaction mmf is in phase with I_a; the armature-reaction reactance drop I_a X_ar can be transformed into an equivalent mmf F_a.

Note: The combined mmf (F_al^- + F_a) is in phase with armature current I_a.

Let α denote the angle by which I_a lags E′ and also the angle between the normal line of F_r1 and (F_a + F_al).

To obtain voltage regulation by the mmf method:

1. Plot OCC and SCC.
2. Find the required point on the OCC corresponding to the resultant air-gap mmf F_r1 obtained from SCC.
3. Find (F_a + F_al) from SCC.
4. Calculate the required phasor relationships and obtain field mmf F_f.

Corresponding to F_f, obtain E_f from OCC and thus compute the voltage regulation of the alternator.

Zero Power-factor Method (Potier method)

• This method handles emfs as voltages and mmfs as field ampere-turns.
• The armature reaction mmf F_a and armature leakage reactance X_al are determined from the Potier triangle constructed using OCC and ZPFC.

Remember: ZPFC method requires OCC and ZPFC and gives accurate results for voltage regulation.

New ASA (American Standards Association) Method

This method is a modification of the mmf method and gives satisfactory results for both cylindrical and salient-pole machines. It requires OCC and ZPFC and only two points on the ZPFC are sufficient (points labelled A and F′ on a standard construction).

• Point A is obtained by loading the over-excited alternator with an under-excited synchronous motor until full-load armature current at rated voltage flows.
• Point F′ is the field excitation required to circulate full-load armature current when the alternator is short-circuited.
• Armature leakage reactance X_al is determined from the Potier reactance drop BC on the potier triangle.

Using the potier triangle and OCC allows determination of the air-gap voltage E_r, the saturation effect and the total field mmf F_f, leading to E_f and the voltage regulation.

Saturation Synchronous Reactance Method

In the emf and mmf methods saturation was neglected. In practice the magnetic circuit is partially saturated under load. The effect of saturation can be included by introducing a saturation factor k which modifies the magnetising (armature-reaction) reactance only.

• The synchronous reactance X_S consists of leakage reactance X_al (practically unaffected by saturation) and armature-reaction reactance X_ar (affected by saturation): X_S = X_al + X_ar.
• Apply saturation factor k to X_ar to obtain the saturated synchronous reactance X_ss. The leakage reactance X_al remains unchanged.

The unsaturated synchronous reactance X_sag can be calculated for any field current from OCC and SCC data and then X_ar = X_sag - X_al.

Procedure to use the saturated synchronous reactance method:

• Compute air-gap voltage E_r for the operating condition.
• Mark E_r on the OCC and find corresponding unsaturated air-gap voltage on the air-gap line.
• Obtain saturation factor K as ratio of saturated to unsaturated values (BD/BA in the standard diagram).
• Calculate X_sag (unsaturated synchronous reactance).
• Find saturated synchronous reactance X_ss = X_al + k X_ar.
• Draw phasor diagram replacing X_s by X_ss, compute E_f and obtain voltage regulation.

Practical Notes and Summary Points

• The generated emf in a phase lags the flux that produces it by 90°.
• Armature reaction may be cross-magnetising, demagnetising or magnetising depending on load power factor.
• OCC, SCC and ZPFC are the principal characteristic curves used for machine parameter extraction.
• Synchronous reactance X_s = X_al + X_ar; synchronous impedance Z_s = r_a + j X_s.
• Emf (synchronous-impedance) method tends to overestimate (pessimistic) voltage regulation because it ignores saturation under load.
• More accurate regulation calculations use mmf/Potier/ZPFC-based methods or include a saturation correction (saturation synchronous reactance method).
• Effective armature resistance in AC can be approximated by r_a ≈ (1.2 to 1.3) r_dc to account for skin effect and heating.