Synchronous Generator Easy Note, Cheat Sheet of Electrical Engineering

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Synchronous Generator Reference

June 16, 2026

Contents

• 1 Introduction to Synchronous Generators
• 2 EMF Equation & Voltage Generation
  • 2.1 Induced EMF
  • 2.2 Simplified EMF Equation
  • 2.3 EMF versus Field Current (Open Circuit Characteristic)
• 3 Armature Reaction
  • 3.1 Definition
  • 3.2 Load Conditions and Their Effects
• 4 Synchronous Impedance
  • 4.1 Definition
  • 4.2 Determining Xs Using Open Circuit (OC) and Short Circuit (SC) Tests
  • 4.3 Determining Armature Resistance
• 5 Power & Torque in Synchronous Generators
  • 5.1 Power Output Equation
  • 5.2 Maximum Power
  • 5.3 Power as a Function of Load Angle

2 EMF Equation & Voltage Generation

2.1 Induced EMF

The core of voltage generation in synchronous machines lies in Faraday’s law of electromagnetic induction. The induced electromotive force (EMF) per phase is calculated as:

Ea = 2πNeΦf

with:

• Ea: RMS induced EMF per phase.
• Ne: Number of turns in each phase winding.
• Φ: Magnetic flux per pole (Weber).
• f : Electrical frequency (Hz).

2.2 Simplified EMF Equation

Rearranging and considering mechanical angular velocity ωm, the EMF can be expressed as:

Ea = kΦωm

where the machine constant k is defined as:

k = N 2 eP

This form offers insight into how mechanical speed and flux directly influ- ence the generated voltage.

2.3 EMF versus Field Current (Open Circuit Characteristic)

The open circuit characteristic (OCC) graphically represents the relation- ship between the induced EMF Ea and the field current If :

Ea ∝ If (withinlinearregion)

However, under load conditions, the terminal voltage Vp differs from Ea due to several factors:

• Armature reaction : Distortion of the main flux due to armature current.
• Self-inductance : Reactance of armature coils.
• Resistance : Ohmic voltage drops in armature windings.
• Salient pole effects : Uneven air-gap magnetic fields.

3.2 Load Conditions and Their Effects

Power Factor Armature Reaction Effect

MMF

Interaction

Terminal Voltage Behavior Lagging Demagnetizing Opposes field MMF

Voltage drops significantly; Ea > Vp Leading Magnetizing Aids field MMF Voltage may rise; Ea < Vp Unity Cross- magnetizing

Distorts flux axis

Flux magnitude nearly constant

4 Synchronous Impedance

4.1 Definition

Synchronous impedance Zs is a complex impedance representing the op- position to current flow within the synchronous machine, combining the ar- mature resistance Ra and synchronous reactance Xs:

Zs =

√ R^2 a + X s^2

Here:

• Ra: Armature resistance per phase.
• Xs: Synchronous reactance per phase.

4.2 Determining Xs Using Open Circuit (OC) and Short Cir-

cuit (SC) Tests

The synchronous reactance can be approximated from the ratio of open circuit voltage to short circuit current at the same field current:

Test Type Description Open Circuit Test Machine runs at synchronous speed; terminals open (Ia = 0); vary field current If ; measure terminal voltage Voc = Ea. Short Circuit Test Machine runs at synchronous speed; terminals shorted; vary field current If ; measure armature current Isc.

The synchronous reactance is then approximated by:

Xs ≈ E Iaa = V Iϕ,OCSC

Usually, Ra is small compared to Xs and often neglected.

4.3 Determining Armature Resistance

Armature resistance Ra is measured by a DC test:

Pmax =^3 V XpEsa

However, for stable operation, synchronous generators typically operate within a load angle of about 15 ◦^ to 20 ◦.

5.3 Power as a Function of Load Angle

The power output varies sinusoidally with load angle:

Pout =^3 V XpEsa sin δ

This relationship highlights the critical role of load angle in controlling power delivery and machine stability.