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# Synchronous Generator – 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 (πΌ_{π}) lags behind the voltage V by 90°. The voltage drop (πΌ_{π}π
_{π}) in the armature resistance is drawn parallel to the current (πΌ_{π}) and the drop (πΌ_{π}π_{ππΏ}) in the leakage reactance perpendicular to (πΌ_{π}).

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 πΉ_{π} , πΉ_{π} and πΉ_{ππ} 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 (πΌ_{π}π_{ππΏ}) and the generated voltage (πΈ_{π}) are all in phase. Hence, the terminal voltage (V) is practically equal to the arithmetical difference between πΈ_{π} and πΌ_{π}π_{ππΏ}i.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 (π_{π}) 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 = πΌ_{π}).

If under this operating condition, the armature reaction MMF (πΉ_{ππ}) has a value expressed in equivalent field current of (πΏπ = πΌ_{ππ}), then the equivalent field current of the resultant MMF (πΉ_{π}) would be (ππΏ = πΌ_{π}).

This resultant field current (ππΏ = πΌ_{π}) would result in a generated voltage (πΏπ =
πΈ_{π}) 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 (πΌ_{π}π_{ππΏ}) where πΌ_{π} 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 π_{π}, 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 (π_{π}) is approximately equal to the armature leakage reactance (π_{ππΏ}) while in a *salient-pole machine*, the π_{π} may be as large as 3 times π_{ππΏ}.

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