Reactive Power Q -can only be defined to Pure sinusoidal Waveforms.

It is the added Taxed /Required Cost of using Real /Active Powe- P.

Reactive power Q-is required to establish the EM/ES Fields and Storage Exchange Between Forms of Energy or simply transferring Power over Transmission Feeders!

It is defined by its relationship to Active Power and Power Factor

Q/P is the equivalent to Tan(Theta) where Theta is Power Factor Angle.

If the Voltage V and Current I are heavily distorted above 10 % , he definition of Reactive POWER- Q is Flawed and requires additional modification to account for Harmonics in V and I.

This definition is not yet fully developed mathematically but can be accounted for by Harmonic Distortion Factor affecting Net Power Factor And Energy Utilization.

Reactive Power can be generated by Switching Devices as well as Any Form of Chopping Waveforms or inherent nonlinearlity or Process-Chaotic Behaviour!!

New definitions of Q are emerging with urgent need for Metering and Accurate Bllings!

AMS-

I would like to to supplement the answer given by Nasir Alfaraj,

Real Power (usually called as Power) is the actual Power supplied to a circuit or an equipment which includes the Power supplied to do the actual work and also the real power loss ( I*I*R) which appears in the form of Heat.

The Reactive Power is the Power supplied to establish the Magnetic Field in the case of an Inductance or Inductive Circuit like Transformer( where the Magnetizing component of the no load current establishes the magnetic Flux in the core), A.C Motors etc or the power supplied to establish the Electric Field in the case of Capacitance or Capacitive Circuits. It is not Real Power and is different from Real Power.

The average Power over one full cycle of A.C. is Zero in the case of energy storing elements like Pure Inductance and Pure Capacitance.

The apparent Power in the case of A.C .is the one that is actually supplied by the A.C. Generators and includes both active and Reactive Power supplied.

Unlike in D.C. where we have only one type of Power called Real Power, which is the one that does the actual work, in A.C. We have three types of Power defined. They are Real Power, Reactive Power and apparent Power.

In D.C. Power means only Real Power and

P=E*I where E is the Voltage across the element or Voltage supplied to the Circuit and I is the current passing through the element or the circuit.; V is in Volts, I is in Amperes and P is in watts (or in Kilo Watts)

In A.C. all the quantities are Complex quantities or Phasors (loosely referred to as Vectors and there is a difference between Phasors and Vectors ! )

Real Power P= E*I*cos(Phi) where the angle (Phi) is the Power Factor Angle ie. The Angle between the Voltage and Current Phasors.

The Reactive Power Q= E*I*Sin(Phi) and is in VAR(Reactive Volt Amperes} or KVAR.

The apparent Power S=E *( I Conjugate ) or S=(E conjugate) * I .

S is in VA or in KVA.

Assume E=(a+ j b) and I= (c-j d) for inductive circuit.,with E as Reference Phasor..

We have are two conventions followed in calculating S. They are

(1) S=E * (conjugate of I ) in which case We get S=(ac+ bd) + j( bc+ ad)

S= ( P+ j Q.) for inductive circuit

S =( P - j Q) for Capacitive circuit in this convention because, with the same complex expression for voltage as reference, I= (c+j b) for capacitive circuit..

Both P and Q are Positive Quantities.

(2) In the second convention on similar lines we will have

S=( conjugate of E) * I and we get

S=(ac+ bd) – j ( bc+ ad)

S =( P – j Q) for inductive circuit and

S=(P+ J Q ) for Capacitive Circuit.

Both P and Q are Positive Quantities.

The sign convention for Q in S is very important and is generally ignored by many.

Unless we know the convention followed for S we will not be able to properly understand the expressions given like in Power Circles drawn, by different authors.

In D.C. P=E * I only because just like in the case of A.C. in a Pure Resistance Circuit,the Current will be in Phase with Voltage phasor. and cos(phi)=1 as f=0

for D.C.

..

Dr.P.S.

I would like to ask one doubt.

Is the magnetizing current supplied to establish magnetic field in an Inductor or Transformer correspond to Reactive Power ? If so, I would like to know how it is so.

Thanks everyone.

I will give a similarity for reactive power

A man for lifting a weight , needs some power , it is stored in it, This is the reactive power

if you take in the case of say Buck Converter, the current flowing through the Inductor will also flow through the load (DC Component). Now can you differentiate Reactive component and Active component in this case ???

A Resistor is a Passive Element and does not oppose the input current or input voltage.It accepts the input voltage and causes voltage drop across it. The Power consumed by such an element is the Real Power.Since the current phasor of Resistance will be in phase with the voltage across it,the inphase component of the current ( in phase with voltage phasor I * cos(Phi) contributes to the real Power. This component of the current is also called Power Component or Wattful Component of the Current. The Power consume by Resistance is P=VI cos (phi) in Watts.Resistance does not react to Excitation.(in the sense opposing the excitation)

Inductance and Capacitance are known as Reactive elements.Inductance opposes the input current due to self and mutually induced emf. Capacitance opposes the input Voltage due to the charging of the capacitor building up opposing voltage and hence opposes the current through it.Inductance and Capacitance React immediately to the Excitation,opposing the Excitation.Hence they are called Reactive elements.

Hence we can treat the Power consumed by such Reactive elements(Inductance and Capacitance) as Reactive Power. The Reactive Power consumed by Reactance elements is Q=VI Sin(phi).

In the Impedance Triangle, the Reactance of Inductance or Capacitance will be in quadrature(+ 90 degrees for pure Inductance and - 90 degrees for pure capacitance) with Resistance Phasor. Also I*sin(phi) is called Wattless component of the Current as it does not contribute to Real Power which is measured in Watts.The Reactive Power Q= VI sin(phi).

Dr.P.S.

Previous answers are valid in conditions of linear RLC circuits, and sinusoidal source voltage. In nonlinear circuits (just add simple switch) and harmonics, traditional definition of reactive power, just not hold. Look for new power theories in time domain, such as Generalized Instantaneous Non-active Power Theory.

Reactive power is the rate at which the energy supplied 'circulates' in the system, while active power is the rate at which energy is lost/consumed.

The statement of Prof.Janardhanan Ramakrishnan is also one way of looking at the problem.As seen from the waveforms for v,i,and vi, we can see that the Average Value of Power is zero for Reactive elements and non zero for Power consumed or Lost by Resistive Loads.

P.S.

Reactive power is a circulating power in the electrical system . Electrical element

i. e ideal inductor and ideal capacitor store the energy in the magnetic field and electric field respectively from the supply system and back to the supply system.this power is not consumed by the circuit element.

In a power system reactive power is required to establish, store and maintain energy in the electromagnetic fields within a power system. As such both reactive power and energy oscillate within the system as mentioned above and is necessary to allow active power to be transmitted through the power system as mentioned above.

Reactive power and energy are very important for generation, transmission and distribution without which we get the classical voltage collapse where the electric field and hence power system becomes unstable. It is for this very reason that distributed generation has grown as it adds or strengthens the electric field at the point of supply allowing a more robust power system and lower local losses.

The classical definition is well known but the purpose and function not so well understood and sometimes not well taught.

Synchronous motors / generators and capacitors are typical sources of lagging (capacitive IEEE) reactive power (energy stored in electric field) and are primarily responsible for maintaining the electric field or voltage. Add too much inductive load or leading reactive power (energy stored in the magnetic field) and voltage collapse will result. Classical tap changing transformer a typical example.

Two good simple example are PF correction where by placing a capacitor at the load or terminals of an indcution motor allows its magnetic circuit to source and exchange energy with electric field of the capacitor (electric field - magnetic field - electric field .....) rather than draw such reactive power / energy through the transmisson system and incur transmission losses. This is no different to placing a decoupling cap on a PC board for electronic circuits and IC's.

See Kunder's book for more.

PF correction by placing a capacitor at the load or terminals of an indcution motor is valid only in conditions of linear RLC circuits, and sinusoidal source voltage. In nonlinear circuits (e.g. switching power supply unit for your PC or frequency converter for induction motor) it's not true. In the presence of harmonics we need to use other Reactive Power Theory.

It matters very little whether the system is linear, non-linear, lumped or distributed as when transmitting power from source to load energy may either be stored within the EM fields of the transmission system or transmitted from source to load via the transmission system.

That component which is stored is referred to as reactive energy and that component which is transmitted referred to as active energy. The harmonic content and which particular harmonic component or combination thereof which constitute these will not change this.

The new theories mentioned above do not attempt to redefine this but rather provide a more precise and consistent mathematical description to account for which harmonic components and which combination of harmonic components constitute or contribute towards this.

In the case of voltage sourced variable speed drive feeding an induction motor the reactive energy is drawn from the DC link capacitor and not the mains supply so it is no different to PF correction in the classical sense and makes no difference whether PWM is used IGBTS or thyristors etc. PF correction was used as an example of energy storage.

Similarly in a PC the output of the power supply is fed to and "smoothed" by electrolytic capacitors which store electrical energy for use by the microprocessor and other motherboard components as are the various decoupling capacitors deployed at the various IC's on the motherboard.

Thus active power will remain the power required to transfer energy through the system and reactive power the power required to store electrical energy within the system.

To illustrate it: Assume you have a damp less-mass spring system. Lets say that the mass was stretched at t=0. As time passes, the stored potential energy is converted into kinetic energy and potential. When the mass comes to the rest position, the energy stored in the system will be all kinetic. The "transfer" of energy from potential to kinetic and then back to potential,...etc is an energy stored in the system. Its flow rate from one form to another is the reactive power. Thus reactive power represents the flow of energy within the system.

In EM, this may be more subtle, although it is basically the same.

Reactive power (energy) is part of the electromagnetic energy in a power system that is not converted into another form of energy (heat, mechanical work ...), it remains all the time in the system in the form of electromagnetic energy (stored in the electric and magnetic fields in the power system).

Active power (energy) is part of the electromagnetic energy that is transmitted through the power system, and at some point in the system converts into another form of energy (heat, mechanical work ...)

Apparent power (energy) is the energy in power system that could be converted into another form of energy (heat, mechanical work ....), but it does not convert into another form of energy completely.

Dear Marinko, Reactive power is the part of Apparent Power so how Apparent power be converted into another form of energy. Reactive power is amount of power stored int passive elements.

Ahsok power cannot be stored only energy. Apparent power is used to define the true power loss incurred in the transmission system due to both active and reactive components of power (current) when. Any machine (motor, transformer etc) must be designed to consider both components. This is the basis for basic power factor correction.

Real power: Power flows from source to load

Reactive power : Power flows from load to source.

At the beginning, I apologize for my English. Perhaps this is reason what a definition of apparent power that I gave is not clear enough.

On the topic:

The physical there is instantaneous power p(t) only, in an electrical network. This instantaneous power is speed of electric energy change (derivation of energy per time) or using voltage and current, it is a product of instantaneous voltage (u(t)) and current (i(t)).

All other powers (P, Q, and S) in AC networks are mathematical concepts. Active (P), apparent (S) and reactive (Q) power are mathematically defined from the current and voltage in the following way.

First we define the active power (P): it is average value of instantaneous power and it is equal to half of the product of voltage and current magnitudes and cosine of the angle (phase shift) between voltage and current. This is the DC component of instantaneous power.

After that we define the apparent power (S). The apparent power in AC networks is equal to magnitude of the AC component of the instantaneous power. This is also half of the product of voltage and current magnitudes.

At the end of the defined reactive power (Q). Reactive power is the square root of the difference of squares of apparent power and active power.

For the case when no reactive power in the system, active power is equal to the apparent power. From this it follows that the apparent power is the maximum amount of active power when would not be reactive power in the system. The state without reactive power can be achieved by applying compensation.

All of the above applies only to steady state for monoharmonic voltage and currents. For voltages and currents that contains higher harmonics this can be specified for each harmonic separately. In transient states is defined only instantaneous power. For example, when we turn off electric circuit all the energy stored in magnetic field in the moment of turn off will be dissipated in form of heat in a switch(circuit breaker) and all the energy stored in electric field will be stay stored in capacitive elements of the circuit.

P.S. Direction of power flow does not show in full if it is active or reactive power. At some point, power flows from the source to the consumer and if the consumer has an active and reactive elements the power is contained active and reactive power components. For the consumer that consumes active power only, power flows in one direction all the time. For consumer that consumes reactive power, power flows in one direction in one part of the time and in opposite direction in other part of the time.

Your English is fine and well understood as is your explanation. I think we all realise the issues you raise about instantaneous power and P,Q, S but that was not the point of the original question being asked.

The question asked was that given a mathematical concept like Q how best would / could one define it and give a physical meaning for it.

Even when using instantaneous power theory in the presence of harmonics or non-sinusoidal waveforms if no average power is present how does this differ from the mathematical concept of Q.

If pure sinusoidal waveforms produce the same average power as highly non-linear waveforms with many harmonics how does one tell the difference and which explanation is more relevant.

They are equivalent but not the same and the "truth" lies in how one tells the story.

Your PS is saying the same as what I have said.

All of us know what is concept of reactive power but I asked this question to find a comprehensive definition which can be understood by undergraduate students. very thanks for your valuable responses. can anyone tell us an example to describe this concept? (for example: It can be compared to the foam in a glass of beer : it is not the real stuff, but there is no way to avoid it and the glass must be oversized unless you’ll have overflow.)

You can find some analogies with other physical systems here:

http://www.leonardo-energy.org/tutorial/explaining-reactive-power-4-further-analogies

http://www.coned.com/reactivepower/understanding_power_factor.pdf

What is Reactive Power?Peter W. Sauer.

I hope that this will be of help.

Good online references explaining Real,Reactive and Apparent Power,

P.S.

Dear Marinko, very thanks for these valuable references. in my idea they are very useful for students.

I think that the above analogies, are not so good. They not reveal the physical phenomenons. The reactive power is the energy, which the reactive elements (LC) can store in their magnetic and electric fields. This power circulating between source and load with double frequency and zero mean. Although the reactive power reduces flow capacity of supply lines, she is necessary for the operation of the electrical equipment (for example - electrical motors). My point is that the reactive power is not unnecessary part of the apparent power.

Dear Iman:

In my opinion reactive power is part of total energy supplied to a system or machine which do not do work. It is stored in form of magnetic field. Contrary, active power is part of energy supplied to a machine that do work. In this regarding, the power factor (PF) is a measure of efficiency in the use of energy supplied to a system o machine. This index represent what portion of the energy supplied to machine do work, FP

http://wiki.answers.com/Q/What_is_reactive_power

Reactive power is simply this: when a coil or capacitor is connected to an AC power supply, the coil or capacitor stores electrical energy during one-fourth of an AC cycle. But then during the next quarter-cycle, the coil or capacitor dumps all the stored energy back into the distant AC power supply. Ideal coils and capacitors consume no electrical energy, yet they create a significant electric current. This is very different from a resistor which genuinely consumes electrical energy, and where the electrical energy flows continously in one direction; moving from source to load.

Why is reactive power so confusing? Well, the math is daunting if not entirely obscure. And the concept of "imaginary power" puts many people off. But this is not the only problem. Unfortunately most of us are taught in grade school that an electric current is a flow of energy, and that energy flows back and forth in AC power lines. This is completely wrong. In fact the energy flows constantly forward, going from source to load. It's only the charges of the metal wires which flow back and forth.

Imagine that we connect a battery to a light bulb. Electric charges already present inside the wires will begin to flow in the circle, and then electrical energy moves almost instantly to the light bulb. The charge flow is circular like a belt, but the energy flow is one-way. Now imagine that we suddenly reverse the connections to the battery. The voltage and current will reverse... but the energy still flows in the same direction as before. It still goes from battery to bulb. If we keep reversing the battery connections over and over, we'd have an AC system. So, in an AC system, only the voltage and current are "alternating," while the electrical energy flows one-way, going from source to load. Where AC resistive loads are concerned, electrical energy does not "alternate." To understand energy flow in AC systems, it's critically important that we understand the difference between charge flow (current, amperes) and energy flow (power, watts.)

What is imaginary power? Simple: it's the unused power which flows backwards and forwards in the power lines, going back and forth between the load's coil or capacitor and the distant AC generator. If your appliance was a pure capacitor or inductor, then it would consume no electrical energy at all, but instead all the flowing energy would take the form of "sloshing energy," and we'd call it "imaginary power." Of course it's not actually imaginary. Instead it's reflected by the load.

If power is the rate of change of energy can you please explain how power goes back and forth but not energy?

Also how does resonance then take place?

There is a difference between average and instantaneous power/energy as already mentioned elsewhere above.

In the wiki example the flow of energy is toward the bulb from the battery but in opposite directions in the wires between the battery and bulb or through the bulb itself depending where the connections are reversed.

Reactive Power Q -can only be defined to Pure sinusoidal Waveforms.

It is the added Taxed /Required Cost of using Real /Active Powe- P.

Reactive power Q-is required to establish the EM/ES Fields and Storage Exchange Between Forms of Energy or simply transferring Power over Transmission Feeders!

It is defined by its relationship to Active Power and Power Factor

Q/P is the equivalent to Tan(Theta) where Theta is Power Factor Angle.

If the Voltage V and Current I are heavily distorted above 10 % , he definition of Reactive POWER- Q is Flawed and requires additional modification to account for Harmonics in V and I.

This definition is not yet fully developed mathematically but can be accounted for by Harmonic Distortion Factor affecting Net Power Factor And Energy Utilization.

Reactive Power can be generated by Switching Devices as well as Any Form of Chopping Waveforms or inherent nonlinearlity or Process-Chaotic Behaviour!!

New definitions of Q are emerging with urgent need for Metering and Accurate Bllings!

AMS-

Agreed, whilst the basic definition remains, the same cannot be said for the mathematical formulation as has been said earlier.

As such in the last sentence it would be more preferable to say "New mathematical formulations to more accurately quantify Q .....".

GD

Active power for each period of the fundamental frequency of the voltage source is transferred to production settings in one of two pulsations in action. Then it turns into mechanical and thermal energy (light) or chemical energy and does some useful work.

Starting a constant reactive power inductive type is the average energy of the magnetic field that is formed and disappears in the inductor. At each period, the fundamental frequency reactive power transferred to the electrical circuit four pulsations of electric power with alternating forward and reverse directions of action. For a quarter period, it strengthens the magnetic field in the inductor, but not further converted into mechanical and thermal energy (light) or chemical, and during the next quarter period back to the voltage source.

Similarly, the initial constant capacitive reactive energy is the average energy of the electric field that forms and disappears in the tank. At each period, the fundamental frequency reactive power transferred to the electrical circuit four pulsations of electric power with alternating forward and reverse directions of action. For a quarter period, it increases the electric field in the tank, but not further converted into mechanical and thermal energy (light) or chemical, and during the next quarter period back to the voltage source.

The opposite signs of periodic functions show that every point in the transmission line is a partial mutual compensation of inductive and capacitive reactive power flow. If they are equal, then there is full mutual compensation of inductive and capacitive flow (current response), and the resulting jet flow in the lines at all times equal to zero.

Reactive power is measured around the world to assess the damage it brings mains. Such physical damage accumulate in time.

Note that in practice the analysis, design and operation of elec-tion networks and systems of periodic functions of electrical parameters are not used. They are used solely for the theoretical study of physical processes.

Why Reactive power is so useful in the Grid that so expensive FACTs devices are used  to compensate it?