Active Power, Reactive Power and Power Factor

1.0   Introduction

Many practicing electrical engineers, some even in the utility, do not have a clear  understanding of the concepts of Active and Reactive Powers and the Lagging and Leading Power Factors in electrical supply lines. Many do have an implicit knowledge of them, adequate under any normal circumstances. In this note an attempt is made to derive these concepts from basic principles of Ohm’s law and I2R power. This will also lead to a better understanding of quality issues of electrical power as supplied to customers.

2.0   Basics

Power in an electrical circuit is commonly understood as the product I2R of resistance and current-squared. By Ohms law, it is also expressed as VI or V2/R where I, V and R are the usual representations for Current, Voltage and Resistance. The above expressions remain largely true as long as we consider direct current (DC) circuits. When you consider alternating currents, the input voltage is alternating between a positive and a negative voltage, as a sine wave, (normally) at a frequency of 50 or 60 cycles per second. In this dynamic situation, two other major elements of the circuitry gain importance, namely, the Inductance (L) and the Capacitance (C). They are together called as Reactance (X) and they, along with the Resistance (R), affect the flow of current in a circuit profoundly. When a voltage is applied to a circuit with reactance (X), it takes some time for the current to get established to a steady state condition, due to induced voltage across the inductance and due to charging up of the capacitance. Even in the case of AC voltage input, the resulting alternating current reaches a steady state condition, but due to the effects of induced voltage and capacitance charging, there is a displacement between the current and voltage waveforms. This displacement is known as phase angle between the AC-voltage and the current. Coming back to our discussion on electrical power, V*I is still the power, but in this case, it is an alternating power. Initially let us consider an AC circuit only with a resistive load. As before I2R is the power consumed in the circuit. As the current is alternating the power also will be ‘alternating’. So the average power in the circuit will be R multiplied by the average of I2 over a cycle of the alternating current. This average of I2 over a cycle is knows as the Mean Square value. The square root of this current is known as the Root Mean Square value or IRMS . Same way, we can define a VRMS for the voltage wave form. Without going into rigours of mathematics, Power in a AC circuit with a resistive load, can be expressed as:

Power, P = IRMS2.R =   VRMS2/R =  VRMS * IRMS .

For a pure sinusoidal waveform, RMS value = Peak Value/ √2

3.0   Complex Power

Now let us consider an AC circuit with Resistance (R) and Reactance (X). To represent resistance and reactance together, we have a term known as Impedance (Z). As discussed earlier, power can be expressed as I2Z or V2/Z. To enable AC circuit analysis, all these parameters are expressed as vectors or complex numbers, as below:

Voltage V= V e jo= V + j0  —- (Reference)     

Current C = I e – jØ   =  Ia – jIr

Impedance Z = Z e jØ = R + jX

Total Power = V* C*  = V * I e jØ = P + jQ

[where Ø = arctan(X/R)]

The Total Power as mentioned in the above expression is normally known as Apparent Power, S, expressed in units of Volt-Ampere (VA). In Z, if reactance X is zero, then Ir will be zero, hence Ir is known as reactive current. Same way if R is zero, Ia will be zero, hence it is known as resistive current, or more commonly known as active current.

Now we have, from above,

S = V*(I cos Ø + j I sin Ø) = P + jQ = V*Ia + jV*Ir

This angle Ø is immediately recognised as the phase displacement between voltage and current waveforms introduced by the presence of reactance X in the circuit. At the instant, when ‘V’ achieves its peak value of the sine wave from, ‘I’ will lag behind and will have a value of only I cosØ. Active power, P, is the actual Active Power in the circuit, whereas Q is the imaginary power generated by the induced emf in the inductance (and the charging emf in the capacitance), as a reaction to the (sinusoidally) varying applied voltage. Hence Q is termed as Reactive Power, expressed in units of Volta-Ampere-Reactive (VAR). 

Now we are ready to write the full expressions for Power in the AC circuit with resistance and reactance as,

The Magnitude of Apparent Power |S| =  VRMS . IRMS       (VA)    

Active Power       P  =  VRMS . IRMS cos Ø,     (Watt)

Reactive Power    Q = VRMS . IRMS sin Ø        (VAR)

 The term ‘cos Ø’ is known as the Power Factor.

4.0   Effects of Frequency and Distortion

Another important factor is that the value of reactance X is frequency dependant.  The inductive reactance XL increases directly as frequency whereas capacitive reactance XC decreases inversely as frequency. The modern power systems have consumers whose loads include many more dynamic elements in addition to L and C in the form of rectifiers, non linear loads and switched mode power supplies for electronics circuitry, etc. These loads tend to distort the current and voltage wave forms away from a pure sinusoid. To analyse such circuits, the current and voltage forms are considered to have several harmonics components superimposed over the basic sine wave of 50 or 60Hz. The power calculations get further complicated if these harmonics are considerably high. Even in DC circuits the so-called ripples create similar ill-effects on power calculations.

5.0   3-Phase Power

So far we have confined our discussions to single phase AC circuits. Now let us move on to 3-Ph AC.

From now on, V and I mean only RMS values unless otherwise specified.

Trivially we may write for 3-ph AC,

P = 3 (V I) cosØ

However we should specify that both V and I are per-phase values. In a normal situation voltage between phases (known as line voltage) is more important than voltage of each phase, (Phase Voltages). In a 3-phase system,

V = V(line) = √3 * V(Phase), and hence,

P = √3 (V I) cosØ, and Q = √3 (V I) sinØ

6.0   Lagging and Leading Phase Angle

Ø is already recognised as the angular displacement between the voltage and current sinusoids of the circuit. This displacement is the result of the presence of inductance and/or capacitance in the circuit. The induced voltage across the inductance makes the current to lag behind voltage by a phase angle Ø, whereas the delay in charging up the capacitance  makes the current to lead the voltage by a  phase angle Ø. Accordingly the phase angle Ø will be (+) positive  or (-) negative. The active power P remains positive in either case, whereas the reactive power Q changes sign as per the inductance or capacitance in the circuit. It may be observed that the lagging reactive power Q is rendered as positive in the earlier expressions for complex power. Lagging Q is considered as consumption of lagging reactive power. The leading reactive power is negative and is sometimes considered as generation of lagging reactive power.

7.0   Active and Reactive Power.  

Active power is the real power resulting in actual work done. Reactive power is a necessary nuisance. The inductive load requires a higher current for the same amount of power and thus the power source also needs to supply this increased current. As this increased current does not result in any actual work done, it is termed as reactive current, Ir. The current I in the circuit is resolved into two components: one component Ia is in phase with Voltage and another component, Ir, with a phase angle of 90 degrees lag with the Voltage. This lagging reactive power requires to be compensated by the source, by ‘generating’ this reactive power. This is done dynamically by the following process: with active power remaining same (say), if reactive load increases, it results, (a) in a demand for higher current, (b) which drops the voltage all along, (c) voltage regulator at the generator end senses this, (d) generator terminal voltage gets picked up automatically or manually (for essentially the same output power), (e) phase angle between voltage and current increases, resulting in higher generation of reactive power as required by the system. But generators in the system have capacity limitations on reactive power generation and total volt-ampere generation. These may ultimately result in lower voltages all through the system, when the system reactive power requirement exceeds the total reactive capacity of generators in the system. Generation of reactive power is comparatively cost free. But to generate the same at the generator end and then to transmit it to the load end where it is required, is costing the power utility in terms of higher transmission losses. Hence the reactive power compensation is more effectively done at the load end, by using shunt capacitor banks. We know that capacitors act as leading-reactive loads. But in this context, we use them as lagging-reactive source. In general, in a utility power system, – just like we balance the active power requirement by active power generation by using frequency as our index -, reactive power requirement is balanced by reactive power generation by using system voltage as the index. In this process, in addition to generators, the shunt capacitors also contribute as lagging reactive sources. For voltage/reactive control of power systems, utilities also use a device known as Synchronous Condensers, loosely described as AC generators-without-a-prime-mover, which can generate only reactive power, both leading and lagging.

8.0   Direction of flow of Active and Reactive Power

Even though the AC current flows alternatively in both the directions, the direction of AC current is always rendered positive in the direction of power flow. In power balance calculations at any node in the power system, by a convention adopted by most of the utilities, the outgoing power from the node is taken as positive and incoming power as negative. For a detailed discussion on directions of active and reactive power flow please refer to the link below

Direction of flow of Active and Reactive Power 

The link also includes a drawing showing the quadrant principle of power factor.

9.0   Power Factor Monitoring

The power factor has already been defined earlier as cosine of phase angle between voltage and current wave forms in an AC electrical circuit. This is an important parameter that affects the quality of power supply and also the performance of the power system. Hence power factor requires to be monitored at all the important nodes in a power system and also at all bulk power supply points. But what is a power factor? It is just a measure of reactive power requirement as demanded by the various types of connected loads. In a three phase AC power utility system, the power factor is rather an ambiguous measurement for the following reasons – the phase angle between current and voltage wave forms is very likely to differ significantly among the three phases – both current and voltage waveforms may not remain strictly sinusoidal due to the presence of harmonics thus affecting the phase angle and power factor. In a way to solve some of these ambiguities in the power factor as defined (called some times as displacement power factor), another term, true power factor is defined as the ratio of total active power to total apparent power. Utility penalties and other decisions to improve performance of the power system are based on this true power factor.

In addition there are problems in online monitoring of the power factor. Power factor varies in the range of 0 to 1. The value as such does not say whether it is lagging or leading.  Some utilities use a range of ‘-1 to 0 to +1’ for power factor to go from lagging to leading PF ! In this representation the middle range of, say, -0.5 to +0.5, is a non acceptable range. The ends of this range, -1 and +1, are essentially same representing unity PF with no phase lag or lead. Such a representation for power factor as a mesurment appears ridiculous. (even the limits for LOLO, LO, HI and HIHI conditions cannot be defined for this parameter).

Some energy meter manufacturers use a range of 0 to 100 to 200 for pf; 0 to 100 representing ‘lagging pf 0 to 1’ and 100 to 200 representing ‘leading pf 1 to 0’. Many utility engineers are not comfortable with this usage. The author of this note has solved this problem in an Indian utility by defining two pfs, namely ‘Leading pf’ varying from 0 to 1 and ‘Lagging pf’ varying from 0 to 1. Both were derived as calculated points from the actual measurement of pf.

Furthermore, pf is a not an easily measurable parameter and it is a highly fluctuating parameter.  For all the above reasons, the author of this note feels pf may not serve well as a parameter to monitor and we may think of other ways to achieve monitoring of reactive power requirement in a system. May be tan Ø, instead of cos Ø, will serve this function better. Tan Ø varies from -(infinity)  to 0 to +(infinity) , as Ø varies from -90 to 0 to +90. It gives the ratio of reactive power to active power and hence may be termed as ‘Reactive Factor’. Utility penalties and other decisions to improve performance of the power system can be based on this reactive factor. This reactive factor can be easily monitored. This is only a suggestion for further consideration by power system operators and experts.

10.0   Conclusion

An attempt has been made in the above note to resolve some of the ambiguities as felt by many practicing utility and industrial electrical engineers in understanding the concepts of Reactive Power and Power Factor. The effect of high reactive requirement on the utility system and the need to penalise low pf consumers are also explained. I will be glad to receive suggestions and comments.



38 Responses to “Active Power, Reactive Power and Power Factor”

  1. Mustafa Alam Says:

    That’s good information.

  2. “Reactive Power – A Strange Concept” « OPEN MIND Says:

    […]   and […]

  3. Nagarajan Says:

    I found this blog to be very popular. Lot of people have seen this and hopefully benefitted. I have added one more blog on this subject to explain the ‘strangeness’ of teh concept of Reactive Power.

  4. Nagarajan Says:

    You also see the following blog.

  5. balamurugan Says:

    now i am working in siemens as testing engineer..i have one small dought wihin that..can u help me..please.. u give ur contact number is 8098950526

  6. vdailyk Says:

    А вы слышали про интересные материалы

  7. subir Says:

    what is d meaning of power factor? Why we need dis ?

    • Nagarajan Says:

      Normally in an electrical circuit, Voltage mutiplied by Current gives you a measure of electrical power. There are two major types of electrical circuits known as Direct Curent (or DC) and Alternating Current (or AC). In DC, Power is always given by Voltage multiplied by Current. But in electrical utilities AC systems are in use. In AC systems Voltage and Current are varying in a sine wave fashion, and hence, they are also influenced by inductances and capacitances in the circuit. This causes the current and voltage wave forms to go out of phase with each other. This directly results in power trasferred through the circuit, to be reduced by a factor, (equal to cosine of the phase difference). This factor is called Power Factor, and is always needed to calculate power through AC cicuits. Kindly read my other blogs on the same subject availble in this site, which may enlighten you more on this subject.

  8. maruza Says:

    Ussually capacitor is used to improve power factor, but must used correct size, there is article about power factor improvement by capacitor, including Excel spreadsheet with formula to calculate capacitor size.

    Definition for: efficiency, power factor, watt, voltampere; are described here:

  9. nichm Says:

    thanks for the info on kw and kva

  10. Nagarajan Says:

    You may also see my latest blog on Power Grid. I am sure you will like it.

  11. Nagarajan Says:

    During this week this blog has topped 100 views. You may also see my latest blog on Power Grid. I am sure you will like it.

  12. SHEHR YAR UMER Says:

    where is the dierivation of active power???

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  14. lloyd irvin rape Says:

    When I originally commented I clicked the “Notify me when new comments are added” checkbox and now each time a comment is added I get
    several e-mails with the same comment. Is there
    any way you can remove people from that service?


    • Nagarajan Says:

      Thanks for your interest in my blog. However I am sorry you are disturbed by repeated comments. I think it is a ‘blip’ in WordPress software. You may ask for help from wordpress

  15. kannan ps Says:

    Hai sir can you explain how can we calculate short circuit rating for a panel board

  16. Don Howard Says:

    If a utility installs capacitor banks in substations and charges a penalty to poor (below 85%) PF customers to pay for the capacitor banks, should a customer who has installed its own capacitor that results in Leading reactive demand also be charged a penalty (the same charge that the lagging poor power factor customers would be charged) to help with poor power factors? would the capacitor banks do anything to help with the Leading reactive demand created by this customers installed capapcitors?

  17. Nagarajan Says:

    A customer installs capcitor banks for any or all of these three reasons: a) to improve his power factor and escape from penalty by the utiltiy
    b) to reduce his MVA Maximum demand to avoid paying for non-productive reactive energy
    c) to improve supply voltage at his end.
    These improvements are obtained at the cost of some small amount of extra active energy consumed by the capacitor banks. Normally these (small) capacitor banks at the consumer end reduces the reactive power requirement but does not totally nullify it. If it happens, usually consumer suffers high volrtage problems Consumers, on their own interest, switch of some or all of their capacitor banks at low-load/high-voltage periods. There are auto switching capacitor banks to avoid these problems. Occasionally utility also requests customers to switch of their capacitor banks in case of high voltages. This leading PF problem is not that serious to warrant a penalty. Even the penalty on low power factor is levied only on the basis of average power factor over a billing period. Hope this answers your query. Thanks for your comment.

  18. Noble chishiba Says:

    monitoring power using vamp 96 relay. why is it that active power is read with a nagetive sign

  19. blog Says:

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  20. M.Balanurugan Says:

    sir, iam only ITI, B license holder and 15 year practical experiance , but idont no teritical knelage pls clear the this dowt

  21. vivek Says:

    i still couldnt understand 😦

  22. pradeep Says:

    I have seen a lot of ele.bills with avg pf .999 but they have very high units of RKVAh (which idealy should be zero if PF is unity).
    Also elec. bill with lead pf has mentioned to convert KVAh to KWh,so with lead pf ele.bill it is better to keep pf somewhat lag .990 then go for 0.999.

    • Nagarajan Says:

      From what I understand from your comment it appears the average power factor is 0.999 , near unity. But if he is a big consumer, he might be operating with permanently switched-in capacitor banks to compensate for his reactive requirement. But during off peak hours, especially at nights, he may be exporting reactive power to the system. The tri-vector meter will read this as leading reactive power consumed. While average PF may remain close to 1 due to the export and import of reactive power cancelling each other, tri-vector will have arithmetically added them to obtain higher Rkvah and hence higher Kvah will be billed. To avoid this consumer should switch-off capacitor banks at nights and if required switch-in more capacitors in the peak period.

  23. pradeep Says:

    thank you sir.
    I am trying to convince them to turn off the fixed cap and use automatic switching cap.I want to show them that they are actually paying more bill,and it will be better to have .995 PF than 0.999.
    How Lag/Lead PF decided by energy meter?

  24. Nagarajan Says:

    There is a special energy meter, specially for 3-phase consumers known as Tri-Vector meter to record kWH, kVarH, kVAH, MAx Demand in both directions and calculate instantaneous and average power factors etc. The present generation digital energy meters are very sensitive and very accurate and calculate many more factors and record them in their memory for many months. The utility can download these data any time later. Generally you should also have facility to read these data and adjust your consumption pattern accordingly. Please check with the utility.

  25. Ella Lewis Says:

    terrific use of words within the piece, it really did help when i was reading

  26. satish Anjaneyappa Says:

    Good Article

  27. Shyam Singh rawat Says:

    Nagarajan sir thanks a lot very much to describe Power factor.kvarh and all those things.

  28. Sourav Das Says:

    sir,its really a very useful article.But i have a doubt while reading the article.When regulator in the generator side detects an extra reactive power demand from load,then how increase in generation voltage increases the phase angle between voltage and current???

    • Nagarajan Says:

      As written in the write-up, – “if reactive load increases, it results, (a) in a demand for higher current, (b) which drops the voltage all along, (c) voltage regulator at the generator end senses this, (d) generator terminal voltage gets picked up automatically or manually (for essentially the same output power)”. The terminal voltage is picked up by increasing the excitation of the machine. The prime-mover power remains the same. Hence with active power remaining same, when the voltage increases, it essentially means the increase in voltage is vectorially at 90 degrees to current. Hence the lagging phase angle increases, thus increasing reactive power output. Hope this clears your doubt.

  29. Prakash Dhende Says:

    Sir, This is prakash. One of my protection and control release of ACB is showing Active power as negative. Is it possible? If yes what may be the reason for it?

  30. Prakash Dhende Says:

    hi sir. one of my protection and control release of ACB is showing active power as negative. Is It possible? If yes what may be the reason for this?

  31. Nagarajan Says:

    Yes, it is possible. It indicates the direction of power is reversed. If you want to indicate positive for this particular direction of power flow, then you should reverse the polarity of VT or CT connections to your device. Please let me know when it works.

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