The nature of reactive power

Reactive power is a crucial aspect of alternating current (AC) electrical systems and is intimately linked with the behavior of inductive and capacitive elements within those systems.

1. Inductive Nature: Reactive power arises due to inductive elements in AC circuits, such as motors, transformers, and solenoids. When an alternating voltage is applied to an inductive load, such as a motor, it creates a magnetic field around the inductor. As the voltage changes, the magnetic field also changes, inducing a voltage in the opposite direction to the applied voltage, according to Faraday's law of electromagnetic induction. This induced voltage opposes the change in current, causing the current to lag behind the voltage in inductive circuits.

2. Capacitive Nature: Capacitive elements in AC circuits, such as capacitors, also contribute to reactive power. When an alternating voltage is applied to a capacitor, it charges and discharges according to the voltage waveform. The current leads the voltage in capacitive circuits because the capacitor initially draws current to charge up and then releases stored energy when the voltage decreases. This leading current creates a phase shift between voltage and current in capacitive circuits.

3. Energy Exchange: Reactive power represents the exchange of energy between the source and the reactive components (inductive or capacitive) in the circuit. Energy is alternately stored and released in the magnetic or electric fields associated with these components without performing useful work in the load.

4. Voltage and Current Relationship: In AC circuits, both real power (which performs useful work) and reactive power (which facilitates energy exchange) contribute to the total apparent power. The relationship between real power, reactive power, and apparent power is described by the power triangle or by the equation ${\text{Apparent Power}}^2={\text{Real Power}}^2+{\text{Reactive Power}}^2$.

5. Importance: Reactive power is essential for maintaining voltage stability and ensuring efficient power transmission and distribution. However, excessive reactive power can lead to voltage fluctuations, power losses, and decreased system efficiency. Therefore, managing reactive power is crucial for optimizing the performance of AC electrical systems.

Understanding the nature of reactive power is vital for designing, operating, and maintaining AC power systems effectively. Proper management of reactive power helps to improve power quality, reduce energy losses, and enhance the reliability of electrical networks.

Definition of reactive power

Reactive power is a component of electrical power that oscillates between the source and the load in an alternating current (AC) circuit without being consumed by the load to perform useful work. Unlike real power, which is responsible for doing useful work such as driving motors or lighting bulbs, reactive power facilitates the transfer of energy between the source and the load without actually being converted into useful work.

Reactive power arises primarily due to the presence of inductive or capacitive elements in the circuit. Inductive loads, such as electric motors and transformers, consume reactive power by storing energy in magnetic fields during one part of the AC cycle and returning it to the system during another part. Capacitive loads, on the other hand, supply reactive power by storing energy in electric fields.

Reactive power is measured in volt-amperes reactive (VAR) and is essential for maintaining the voltage levels and stability of AC power systems. While reactive power does not contribute to the real work performed by the load, it is necessary for maintaining the electromagnetic fields in motors and transformers and ensuring the efficient transmission of electrical energy.

Utilities and industries often need to manage reactive power to maintain the stability and efficiency of the power system. Power factor correction techniques, such as installing capacitors or inductors, are used to minimize reactive power and improve the overall efficiency of the system.

For most electrical loads like motors, the current I is lagging behind the voltage V by an angle φ.

If currents and voltages are perfectly sinusoidal signals, a vector diagram can be used for representation.

In this vector diagram, the current vector can be split into two components: one in phase with the voltage vector (component I_a), one in quadrature (lagging by 90 degrees) with the voltage vector (component I_r). See Fig. L1.

I_a is called the active component of the current.

I_r is called the reactive component of the current.

Fig. L1 : Current vector diagram

The previous diagram drawn up for currents also applies to powers, by multiplying each current by the common voltage V. See Fig. L2.

We thus define:

• Apparent power: S = V x I (kVA) 
• Active power: P = V x Ia (kW)
• Reactive power: Q = V x Ir (kvar)

Fig. L2 : Power vector diagram

In this diagram, we can see that:

• Power Factor: P/S = cos φ

This formula is applicable for sinusoidal voltage and current. This is why the Power Factor is then designated as "Displacement Power Factor".

• Q/S = sinφ
• Q/P = tanφ

A simple formula is obtained, linking apparent, active and reactive power:

• S² = P² + Q²

A power factor close to unity means that the apparent power S is minimal. This means that the electrical equipment rating is minimal for the transmission of a given active power P to the load. The reactive power is then small compared with the active power.

A low value of power factor indicates the opposite condition.

Useful formulae (for balanced and near-balanced loads on 4-wire systems):

• Active power P (in kW)

  -  Single phase (1 phase and neutral): P = V.I.cos φ
  -  Single phase (phase to phase): P = U.I.cos φ
  -  Three phase (3 wires or 3 wires + neutral): P = √3.U.I.cos φ

• Reactive power Q (in kvar)

  -  Single phase (1 phase and neutral): P = V.I.sin φ
  -  Single phase (phase to phase): Q = U.I.sin φ
  -  Three phase (3 wires or 3 wires + neutral): P = √3.U.I.sin φ

• Apparent power S (in kVA)

  -  Single phase (1 phase and neutral): S = V.I
  -  Single phase (phase to phase): S = U.I
  -  Three phase (3 wires or 3 wires + neutral): P = √3.U.I

where:
V= Voltage between phase and neutral
U = Voltage between phases
I = Line current
φ = Phase angle between vectors V and I.

An example of power calculations (see Fig. L3)

Type of circuit		Apparent power S (kVA)	Active power P (kW)	Reactive power Q (kvar)
Single-phase (phase and neutral)		S = VI	P = VI cos φ	Q = VI sin φ
Single-phase (phase to phase)		S = UI	P = UI cos φ	Q = UI sin φ
Example	5 kW of load	10 kVA	5 kW	8.7 kvar
	cos φ = 0.5
Three phase 3-wires or 3-wires + neutral		S =  UI	P =  UI cos φ	Q =  UI sin φ
Example	Motor Pn = 51 kW	65 kVA	56 kW	33 kvar
	cos φ= 0.86
	ρ= 0.91 (motor efficiency)

Fig. L3 : Example in the calculation of active and reactive power

The calculations for the three-phase example above are as follows:

Pn = delivered shaft power = 51 kW
P = active power consumed

S = apparent power

So that, on referring to diagram Figure L3 or using a pocket calculator, the value of tan φ corresponding to a cos φ of 0.86 is found to be 0.59

Q = P tan φ = 56 x 0.59 = 33 kvar (see Figure L15).

Alternatively:

Fig. L2b : Calculation power diagram