Synchronous Generator Phasor Diagram Phasor Diagram π0 π πΉ π½ π π0 = π + π π π + πππ π πππ π π π π π Definitions: (Outer) Power Factor Angle π½ οΌε€οΌεηε ζ°θ§ Inner Power Factor Angle π ε εηε ζ°θ§ Power Angle εοΌηοΌθ§ πΉ Generator Phasor Diagram – Lagging Power Factor Load π0 π πΉ π½ π πππ π π π π π π0 = π + π π π + πππ π Over excited (πΈ0 > π) θΏε± Generator Phasor Diagram – Unity Power Factor Load π0 π π π½=π πΉ πππ π π π π π π0 = π + π π π + πππ π Slightly over excited (πΈ0 > π) Generator Phasor Diagram – Leading Power Factor Load π0 π π½ πππ π π πΉ π π π π π0 = π + π π π + πππ π Maybe under excited (πΈ0 < π) ε―θ½ζ¬ ε± Generator Power Flow (or πππ ) w πππ = ππ΅π π΅π sin πΌ πππ = πππ ππ = 3πΈ0 πΌ cos π π π = ∠π0 − ∠π π = πΌ − 2 π = π0 − π π π − πππ π Output Power (1) π0 = π + π π π + πππ π ≈ π + πππ π, since π π << ππ πππ’π‘ = 3ππΌ cos π π0 πππ π π½ π πΉ π ππ πΌ cos π = πΈ0 sin πΏ π½ π π cos π = πΈ0 cos π πππ ≈ πππ’π‘ πΌ cos π πΈ0 sin πΏ = ππ 3ππΈ0 sin πΏ = 3ππΌ cos π = ππ Output Power (2) π0 = π + π π π + πππ π ≈ π + πππ π, since π π << ππ π π π½ πππ π π0 πΉ π πΌ −πππ π π0 − πππ π ≈ π ππ + ππ = ππππ‘ ζ¬ ε±(πΈ0 < π), capacitive π πΌ = ∠π0 − ∠ −πππ π = π + 2 Induced ElectromagneticTorque πππ 3ππΈ0 sin πΏ = = πππ ππ ππ πππ 3ππΈ0 sin πΏ = ππ ππ πππ = ππ΅πππ‘ π΅π sin πΏ Measurement of SG Model Parameters Measurement of Model Parameters 1. The relationship between field current If and E0 2. The synchronous reactance XS 3. The armature resistance Ra Open Circuit Characteristics (OCC) πΌ=0 πΈ0 ΰ·‘π π ΰ·‘π π·π π πΈ0 = 8 2ππ π0 πΌπ ππππ π2 Saturation for large field current Short Circuit Characteristics (SCC) π0 = π π π + πππ π π0 (or IL ) π π π πππ π ππ π≈ π 2 π= π=0 πΌ= π πΌ =π+ ≈π 2 ππππ‘ ππ π0 π π + πππ πΈ0 π΅π ≈ π΅π Bnet ο» 0 π π2 + ππ 2 π=0 No saturation Measurement of Synchronous Reactance πΌπ π = πΈ0 πΈ0 ≈ 2 2 ππ π π + ππ πΈ0 ⇒ ππ ≈ πΌπ π (1) Therefore, an approximate method for determining the synchronous reactance Xs at a given field current is: 1. 2. 3. Get the internal voltage E0 from the OCC at that field current. Get the short-circuit current flow ISC at that field current from the SCC. Find Xs by applying (1). Unsaturated Synchronous Reactance Saturation for large field current πΈ0 πΌπ π X S ,u No saturation Follow equation ππ ≈ πΈ0 πΌπ π The unsaturated synchronous reactance Xs,u can be found simply by applying ππ ≈ πΈ0 /πΌπ π at any field current in the linear portion (on the airgap line) of the OCC curve. Short-Circuit Ratio Short Circuit Ratio: The ratio of the field current required for the rated voltage at open circuit to the field current required for the rated armature current at short circuit. ππΆπ = πΌππ |rated voltage at OC πΌππΌ |rated current at SC πΈ0 ππππ‘ππ πΌπππ‘ππ πΌππ πΌππΌ ππΆπ = πΌππ πΌππΌ Measurement of Armature Resistance The armature resistance Ra can be approximately measured by applying a DC voltage to the windings while the machine is in stationary and measuring the resulting current flow. Using DC voltage means that the reactance of the windings will be zero during the measurement process. This technique is not perfectly accurate, since the AC resistance will be slightly larger than the DC resistance (as a results of the skin effect at higher frequencies).
