# Power Factor

## Contents

### Introduction

Power factor is defined as the cosine of the power angle $\cos \theta \,$, the difference in phase between voltage and current, or simplified as the ration of the real power (P) and the apparent power (S). People will often refer to power factor as leading or lagging. This is because the power angle can only range between -90° and +90°, and the cosine of an angle in the fourth quadrant (between 0 and -90°) is always positive. Therefore the power factor is also always positive and the only way to distinguish whether the power angle is negative or positive from the power factor is to denote it leading or lagging.

• Lagging power factor: when the current lags the voltage, this means that the current waveform comes delayed after the voltage waveform (and the power angle is positive). In the context of the power generation it means that the generator injects reactive power (over-excited regime), with respect to the Q-maximum defined by the capability curve (i.e. S = P + jQ).
• Leading power factor: when the current leads the voltage, this means that the current waveform comes before the voltage waveform (and the power angle is negative). The generator absorbs reactive power (under-excited regime), with respect to the Q-minimum defined by the capability curve (i.e. S = P - jQ).
• Unity power factor: refers to the case when the current and voltage are in the same phase.

The physical significance of power factor is in the load impedance. Inductive loads (e.g. coils, motors, etc) have lagging power factors, capacitative loads (e.g. capacitors) have leading power factors and resistive loads (e.g. heaters) have close to unity power factors.

A power factor of one or "unity power factor" is the goal of any electric utility company since if the power factor is less than one, they have to supply more current to the user for a given amount of power use. In doing so, they incur more line losses. They also must have larger capacity equipment in place than would be otherwise necessary. As a result, an industrial facility will be charged a penalty if its power factor is much different from 1.

Industrial facilities tend to have a "lagging power factor", where the current lags the voltage (like an inductor). This is primarily the result of having a lot of electric induction motors - the windings of motors act as inductors as seen by the power supply. Capacitors have the opposite effect and can compensate for the inductive motor windings. Some industrial sites will have large banks of capacitors strictly for the purpose of correcting the power factor back toward one to save on utility company charges.

### Synchronous Machine PF Contribution

Information about the operation of synchronous machines is often determined by analysis of the armature circuit phasor diagram. Taking the armature circuit equation:

$\overrightarrow{E} = \overrightarrow{V} + \overrightarrow{I_A}R_A + j\overrightarrow{I_A}X_A$

Where;

$\overrightarrow{E} = E \angle \delta$

E is the open-circtuit induced voltage, also called excitation, and delta is the load angle.

$\overrightarrow{V} = V \angle 0^{\circ}$

$\overrightarrow{I_A} = I_A \angle \phi$

The phasor diagram is constructed by taking the terminal voltage (V) as the phase reference. Once the terminal voltage is drawn, additional phasors for current, resistive voltage drop, reactance voltage drop and induced voltage can be added. The shape of the phasor diagram is dependent on the phase of the current relative to the terminal voltage. Examples of leading, lagging and unity power factor are shown below.

#### Capability Curve

Alternator capability curve - Green area is normal operating range of a typical synchronous machine, yellow is abnormal but not damaging and operating in red regional will cause damage or misoperation.

The ability of any generator to absorb the kVAR is termed as reverse kVAR limit. This ability is defined as reactive capability curve. Figure above shows typical generator reactive capability curve. X-axis is the kVAR produced or absorbed (positive to the right). Y-axis indicates the kW (positive going up). kVAR and kW are shown as per unit quantities based on the rating of the alternator (not necessarily the generator set, which may have a lower rating.

The normal operating range of a generator set is between 0 and 100 percent of the kW rating of the alternator (positive) and between 0.8 and 1.0 power factor (green area on curve). The black lines on the curves show the operating range of a specific alternator when operating outside of normal range. Notice that as power factor drops, the machine must be de-rated to prevent overheating. On the left quadrant, you can see that near-normal output (yellow area) can be achieved with some leading power factor load, in this case, down to about 0.97 power factor, leading. At that point, the ability to absorb additional kVAR quickly drops to near zero (red area), indicating that the AVR is “turning off” and any level of reverse kVAR greater than the level shown will cause the machine to lose control of voltage.

A good rule of thumb for generators is that it can absorb about 20% of its rated kVAR output in reverse kVAR without losing control of voltage. However, since this characteristic is not universal, it is advisable for a system designer to specify the reverse kVAR limit used in his design, or the magnitude of the reverse kVAR load that is expected.

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