- Phasor Diagram Definition: A phasor diagram for a synchronous motor shows the relationships between various electrical quantities like voltage and current.
- Reference Phasor: Vt is the reference phasor, with armature current and excitation voltage being plotted in relation to it.
- Opposite Phases: The armature current is in phase opposition to the excitation emf in a synchronous motor.
- Power Factor Operations: Different power factor operations (lagging, unity, leading) affect the expressions for the excitation emf, using components of terminal voltage and armature current.
- Advantages of Phasor Diagrams: They provide physical insight and help derive mathematical expressions for the synchronous motor’s operations.
The phasor diagram for a synchronous motor shows the steady-state relationship between per-phase voltage, current and internal emf. This article uses these symbols:
Ef is the internal excitation, or counter-emf
Vt is the terminal voltage
Ia is armature current, defined into the motor
Θ is the power-factor angle between terminal voltage and armature current
ᴪ is the angle between internal emf and armature current
δ is the torque angle between terminal voltage and internal emf
ra is armature resistance per phase.
Take Vt as the reference phasor when drawing the phasor diagram for synchronous motor. Then apply two conventions:
(1) Define armature current as entering the motor. This differs from the outward-current reference often used for a generator and is unrelated to an asynchronous motor classification.
(2) In normal motoring operation, internal excitation emf lags terminal voltage by the positive torque angle δ.
For a cylindrical-rotor model, the per-phase phasor equation is terminal voltage equals internal emf plus armature current times synchronous impedance.
The first diagram uses the outward current reference associated with generator equations, so motor current appears with the opposite reference direction.
The second diagram instead defines current into the motor. This is a change of reference, not an omitted mathematical sign. Use one current direction consistently throughout the phasor equation and diagram.
(a) Lagging-power-factor motoring, normally associated with underexcitation at a fixed load.
(b) Unity-power-factor motoring at the corresponding field current.
(c) Leading-power-factor motoring, normally associated with overexcitation at a fixed load.
The diagrams below use the same motor-current and angle conventions for all three cases.
(a) Motoring at lagging power factor: Resolve the terminal-voltage phasor along and perpendicular to armature current Ia. Its component along the current is VtcosΘ.
With the diagram’s chosen axes, the armature-resistance voltage drop contributes –Iara, so the component used is (VtcosΘ – Iara). The perpendicular voltage component is (Vtsinθ – IaXs) under this sign convention. Triangle BOD then gives the magnitude of the internal emf:
(b) Motoring at unity power factor: Armature current Ia is in phase with terminal voltage, so theta is zero and ᴪ = δ. Triangle BOD in the second diagram gives
(c) Motoring at leading power factor: Resolve terminal voltage relative to Ia again. The component along current is VtcosΘ. The resistance contribution is (–Iara), giving (VtcosΘ – Iara) along armature current. The perpendicular component becomes (Vtsinθ + IaXs) for the leading-current diagram. Triangle BOD gives
Advantages of Drawing Phasor Diagrams for Synchronous Motor
(1) A phasor diagram makes voltage drops, torque angle and power factor visible for synchronous motors.
(2) It provides a direct geometric check on the per-phase voltage equation and the derived internal-emf magnitude.
