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Synchronous Generator β Zero Power Factor Characteristics and Potier Triangle
The zero power factor characteristics (ZPFC) is the graph plotted between the armature terminal voltage per phase and the field current, when the machine being operating with constant rated armature current at synchronous speed and zero lagging power factor. The ZPFC is also known as Potier Characteristics after its originator.
In order to maintain the power factor very low, the alternator is loaded by means of reactors or by an under-excited synchronous motor. The shape of the ZPFC is very much like that of the open-circuit characteristic (O.C.C.) displaced downwards and to the right.
Phasor Diagram
Figure-1 shows the phasor diagram of the alternator corresponding to zero power factor lagging load.
Here, the terminal voltage per phase (V) is taken as the reference phasor. At zero power factor lagging, the armature current (πΌπ) lags behind the voltage V by 90°. The voltage drop (πΌππ π) in the armature resistance is drawn parallel to the current (πΌπ) and the drop (πΌππππΏ) in the leakage reactance perpendicular to (πΌπ).
Hence, the generated voltage per phase is then,
$$\mathrm{πΈ_{π} = π + πΌ_{π}π _{π} + πΌ_{π}π_{πL} … (1)}$$
If
$πΉ_{ππ}$ = Armature reaction MMF (in phase with $πΌ_{π}$)
$πΉ_{π}$ = Field MMF
$πΉ_{π}$ = Resultant MMF in the air gap
The three MMF phasors πΉπ , πΉπ and πΉππ are in phase and their magnitudes are related by the following equation −
$$\mathrm{πΉ_{π} = πΉ_{π} + πΉ_{ππ} … (2)}$$
If the armature resistance (π π) is neglected, the resulting phasor diagram will be as shown in Figure-2. From Figure-2, it can be seen that the terminal voltage per phase (V), the reactance voltage drop (πΌππππΏ) and the generated voltage (πΈπ) are all in phase. Hence, the terminal voltage (V) is practically equal to the arithmetical difference between πΈπ and πΌππππΏi.e.,
$$\mathrm{π = πΈ_{π} − πΌ_{π}π_{ππΏ} … (3)}$$
The arithmetical expressions given in eqns. (2) and (3) form the basis for the Potier triangle.
Also, Eqn. (2) can be transformed into its equivalent field current form by dividing both sides by the effective number of turns per pole (ππ) on the rotor.Thus,
$$\mathrm{\frac{πΉ_{π}}{π_{π}}=\frac{πΉ_{r}}{π_{π}}+\frac{πΉ_{ππ}}{π_{π}}}$$
$$\mathrm{\Rightarrow\:πΌ_{π} = πΌ_{π} + πΌ_{ππ} … (4)}$$
Potier Triangle
The O.C.C. and ZPFC are shown in Figure-3.
Consider a point b on the ZPFC corresponding to the rated terminal voltage (V) and a field current of (ππ = πΌπ).
If under this operating condition, the armature reaction MMF (πΉππ) has a value expressed in equivalent field current of (πΏπ = πΌππ), then the equivalent field current of the resultant MMF (πΉπ) would be (ππΏ = πΌπ).
This resultant field current (ππΏ = πΌπ) would result in a generated voltage (πΏπ = πΈπ) from the O.C.C. Since, for zero power factor lagging operation, the generated voltage is given by,
$$\mathrm{π¬_{π} = π½ + π°_{π}πΏ_{ππΏ} … (5)}$$
Here, the vertical distance ac must be equal to the leakage reactance voltage drop (πΌππππΏ) where πΌπ is the rated armature current. Therefore, the armature leakage reactance is,
$$\mathrm{π_{ππΏ} =\frac{Voltage\:ππ\:per \:phase}{Rated\:armature\:current}… (6)}$$
Now, the triangle formed by the vertices a, b and c is known as the Potier Triangle.
In the synchronous machine, the effect of the field leakage flux in combination with the armature leakage flux gives rise to an equivalent leakage reactance ππ, known as Potier Reactance. It is given by,
$$\mathrm{π_{π} =\frac{Voltage\:drop\:per\:phase (voltage\:ππ)}{ZPF\:rated\:armature\:current\:per\: phase (πΌ_{π})}… (7)}$$
For a cylindrical rotor synchronous machine, the Potier reactance (ππ) is approximately equal to the armature leakage reactance (πππΏ) while in a salient-pole machine, the ππ may be as large as 3 times πππΏ.
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