# Learn power factor correction formula by using capacitor bank

Electricity is mostly generated, transmitted, and distributed by means of alternating currents, which are almost exclusively used as their medium. In electricity generation, the question of the power factor correction formula always comes. This leads immediately to the question of the power factor, which should be taken into consideration. It is important to note that most loads (ex. induction motors, arc lamps) are inductive in nature, which means their power factor is disproportionally low.

It is highly undesirable to have a poor power factor as this causes an increase in current, causing an increase in real power losses in the power system, from the generator in the power station down to the devices used for consumption. Having PF as close to unity as is technically possible is one of the most important factors to take into account when ensuring that the most favorable conditions are created for the supply system. As an electrical engineer, I will present in this article a number of different approaches to improve power factor and to solve the low PF issue by utilizing power factor correction methods.

## Power Factor:

The PF of a circuit is defined as the cosine of the angle between voltage and current in the circuit. It is common to find an AC circuit with a phase difference Phi between the voltage and current. The PF of a circuit is described by the term cos Phi. The PF is referred to as lagging if the circuit is inductive loads because the total current lags behind the source voltage.

On the other hand, in a capacitive circuit, the current often leads the source voltage, and the power factor leads. If we were able to simulate an inductive circuit where a lagging current I is drawn off of the supply voltage V; the lag angle would be φ. An illustration of the circuit’s phasor diagram can be found in Figure 1. There are two components of the circuit, total current I which can be separated into two perpendicular factors:

Figure 1: phasor diagram of the circuit

a) I cos φ in phase with V

(b) I sin φ 90o out of phase with V

## True Power, Reactive Power, and apparent power

There are three components to the equation I cos φ, which is active and wattfull and I sin φ, which is reactive and wattless. In order to measure the PF, we use the reactive component. When reactive components are small, phase angles φ are small, and therefore PFs cos φ will be high. Thus, in a circuit with a low reactive current (i.e., I sin φ), the PF will be high, and in a circuit with high reactive current, the PF will be low.

## Power Triangle

It is also possible to make an analysis of PF by considering the a.c. power draw of a circuit. The power triangle OAB shown in Fig is formed when the voltage of each side of the current triangle OA is multiplied by the volts V.

Figure 2: Power Triangle

The power right triangle can be summarized as follows:

AC circuits have two components of apparent power such as active and reactive power, which are diagonally opposed to each other.

OB2=OA2+AB2

(Apparent power) 2= (active power) 2+ (Reactive power) 2

(KVA)2=(KW)2+(KVAR)2

Power Factor, cosΦ=OA/OB=Active power/Apparent power=KW/KVA

Power Factor=Resistance/Impedance

In ac circuits, the correct power factor has a significant impact as it determines how much power is consumed.

For single phase system, we know that

P=VL*IL*CosΦ

For three phase system, we know that

P=1.73* VL*IL*CosΦ

Based on the above calculation, it can be seen that the load current (amps) inversely proportional to power factor for fixed power and voltage. Load current (amps) increases with a lower power factor. The following disadvantages result from a power factor less than unity:

1. Large kVA rating of the equipment.
2. Greater conductor size
3. Large copper losses.
4. Poor voltage regulation
5. The reduced handling capacity of the system

### Causes of Low PF

In terms of economics, a low PF is a problem in power system. On a supply system that has a whole load, the PF is usually lower than 0·8. PF is low due to the following reasons:

(i) Almost all of the motors in the ac industry are induction motors (1-phase as well as 3-phase), which are characterized by the low lagging PF. On light load, these motors have an extremely low PF (0.2 to 0.3), which increases to 0.8 or 0.9 on full load.

(ii) There is a low lagging power factor for arc lamps, electric discharge lamps, etc which are used in industrial gas furnaces, commercial as well as residential appliances.

(iii) The power system is under varying loads which are higher in the mornings and evenings and lower at other times. As the supply voltage increases during a low load period, the magnetization current increases. Consequently, the power factor is reduced.

For the above reasons, we should always develop PF of power system by using power factor correction methods.

### Power Factor correction capacitor formula

The low PF is mainly due to the fact that most of the power loads are inductive and, therefore, take lagging currents. In order to improve power factor, some devices taking leading working power should be connected in parallel with the load. One such device can be a capacitor. The capacitor draws a leading current and partly or completely neutralizes the lagging reactive component of the current. This raises the power factor of the load.

### Power factor correction diagram

Figure 4: Power factor correction diagram

Figure 5 phasor diagram

The figure shows how the PF of a single-phase load taking a lagging current I increases when a capacitor is added as shown in the figure. There is a parallel connection between the capacitor C and the load. The current (amps) IC of the capacitor has a 90 degree lead on the supply voltage. In this case, I1 represents the phase sum of I and IC, and φ2 indicates the lag angle as shown in Fig (iii). Therefore, cos φ2 is greater than cos φ1, because φ2 is less than φ1. This increases the load’s PF. Here are a few things to note:

Only the reactive component lags behind the active component in PF correction, so the active component remains unchanged before and after the capacitor is placed.

I cos φ1=I1 cos φ2

As a result of the improvement in PF, a reduction of the lagging reactive component occurs and is identical to the difference between the lagging reactive component of load (I sin φ1) and capacitor current (IC)

I1sin φ2=I sin φ1-Ic

From the above formula,

I cos φ1=I1 cos φ2

We can multiply by voltage V and we get

VI cos φ1=VI1 cos φ2

Due to an improvement in PF, real power (kW) is unchanged.

I1sin φ2=I sin φ1-Ic

Multiplying by voltage and we get

VI1sin φ2=VI sin φ1-Vic

Net kVAR after p.f. correction = Lagging kVAR before p.f. correction − leading kVAR of equipment

### Power factor correction methods

Typical power factors for large generators range from 0·8 to 0·9. The PF can be lower in some cases due to factors beyond our control, and in these cases, we should take special measures to improve it. The following equipment can be used to achieve this:

1. Static Capacitors
2. Synchronous condenser

#### Static capacitors:

The power factor of the equipment operated at a lagging level can be improved by linking capacitors in parallel. The capacitor in this connection provides a leading current that partly or completely offsets the reactive current that is lagging in the load system. By doing this, the load power factor is boosted. There are two types of capacitors connection that can be used for three-phase loads: delta and star, as shown in the figure. Power factor improvement in factories is often carried out by using static capacitors.

Figure 6: Static capacitor with delta connection

Figure 7: Static capacitor with Y connection

Losses are low for them. In spite of the fact that they do not have rotating parts, they require little maintenance. Due to their lightweight and no need for a foundation, they are easy to install. In ordinary conditions of atmospheric pressure, they will be able to work.

Their service lives can be as short as eight to ten years. In the event that voltage is exceeded by more than the rated value, they are easily damaged. Once a capacitor has been damaged, it is not economically feasible to repair it.

#### Synchronous condenser

Over-excited synchronous motors take a leading current, which is why they behave like capacitors. Synchronous condenser refers to an overexcited synchronous motor that runs without load. A parallel connection of such a machine neutralizes the reactive component of the load in part by taking a leading current. This results in an improved power factor.

Figure 8: Power factor improvement by synchronous condenser method

It’s possible to change the motor’s current draw by varying the field excitation. The power factor can be controlled steplessly in this manner. Short-circuit currents are not a problem for the motor windings due to their thermal stability. A simple fix can be found for the faults.

The motor is experiencing considerable losses as a result of this. There is a high cost associated with the maintenance of the equipment. There is a lot of noise produced by it. Its cost is generally higher than a static capacitor of the same rating, except for sizes over 500 kVA. A piece of auxiliary equipment must be provided for self-starting since synchronous motors have no self-starting torque.

Note. During the course of synchronous motor operation, a massive amount of reactive power is consumed resulting from the operation of a DC field as well as the mechanical load required to move the motor. A synchronous motor is one that takes up the maximum amount of leading power with a maximum excitation as well as zero loads.

An induction motor’s power factor can be improved with phase advancers. Induction motors are characterized by a low power factor because their stator winding produces an exciting current that lags 90 degrees behind the supply voltage. A motor’s power factor may be improved if it can get its exciting ampere turns from another AC source, thus relieving the stator winding of current. Phase advancers, which are simply AC exciters, are responsible for this job. Phase advancers are connected to the rotor circuit of the main motor and mounted on the same shaft.

During the slip frequency, it energizes the rotor circuit by providing ampere-turns. A leading power factor can be achieved by providing more amp turns than is necessary by inducing an overexcited synchronous motor through the induction motor. There are two primary benefits associated with phase advancers. Due to the high slip frequency of the exciting ampere-turns, this reduces the lagging kVAR drawn by the motor. Phase advancers are a convenient alternative to synchronous motors where synchronous motors cannot be used. In addition, phase advancers do not cost-effectively advance phases for motors with horsepower below 200.

In a nutshell, we can develop our PF of a circuit/power system by utilizing the above power factor correction methods.

## Calculations of Power Factor Correction

Assume that the inductive load is taken into account by the power factor φ1 at a lagging current I. If the circuit’s power factor needs to be improved, then the circuit should be connected in parallel with equipment that takes a leading reactive component as well as partially eliminates the lagging reactive component of the load.

On figure (i) we can see a parallel capacitor that has been attached across the load i.e parallel capacitor. A capacitor usually has an input current IC that is 90 degrees ahead of the supply voltage V. Fig (ii) shows a phasor diagram and a phasor analysis of the current IC and the lagging reactive component of the load current that partly cancel out the reactive component. φ2 is the angle of lag of the resultant circuit current, I′. It is evident that φ2 is less than φ1, so the new PF cos is greater than φ1.

Figure 9: a capacitor connected across the load

Figure 10: Phasor diagram

From the above phasor graph, we can see that the lagging reactive components of the load is decreased to I1sin φ2.

I1sin φ2=I sin φ1-Ic

Ic= I sin φ1- I1sin φ2

The required capacitance of the capacitor to develop p.f. from cos φ1 to cos φ2

Xc=V/Ic=1/Wc

Here, Xc= Capacitive reactance

## Example of Calculate Power Factor Correction:

In a real-world example, an alternator was used to supply 500 kW at 0.55 PF load. What happens if the targeted power factor is increased to unity? How much more power can the alternator deliver if the kVA loading is the same? Calculate power factor?

Solution:

We know need to calculate power factor correction as follows:

PF, cos φ=KW/KVA

KVA=KW/ cos φ

KVA=500/0.55

=910KVA

Here, KW at 0.55PF=500KW

Therefore,

KW at 1.0/unity PF=910Killo Watts

So the increased Power is

910KW-500KW

=410KW.

## Summary

The PF is a measure of the efficiency of a power system. Having a basic understanding of power systems will prepare you to understand the concepts of PF that apply directly to your home. It is crucial to be able to determine PF for both your energy use and cost analysis, no matter how your system is configured. So, it is necessary to learn deeply by power factor correction methods.

Applications of capacitors
battery. In electric power distribution, capacitors are used for power factor correction. Such capacitors often come as three capacitors connected as a

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