Archive for the ‘Engineering’ Category

The Electrical Power Grid

December 25, 2012

The  Electrical Power Grid

L V Nagarajan

Mr. Contractor and Mr. Engineer are great friends. They are running a reasonably successful construction business for the last twenty years. They decided to construct houses for their own use, but in a small village far away from maddening crowd. The village they chose, did not even have electricity supply as yet. They decided to start construction of their houses adjacent to each other. They had a small diesel generator for construction purposes. After completing the construction they promptly applied for electricity connection. Due to lack of infrastructure, the same was delayed for more than two years. However they wanted to live in their new houses, hence they commissioned two  diesel generators, one for each house and connected the same to their individual main switch boards. They both moved to their new houses

Mr. Contractor started the generator and switched on his lights, fans and heaters etc and as he switched them on, one by one, he observed the whine of the rotating generator getting louder and shriller. But there were other changes happening in the machine which he could not observe. As he put on one appliance after the other, the extra load slowed down the rotating speed of the machine. The speed governor sensed this speed drop and picked up the speed to normal by automatically increasing the fuel input. Just as the speed of the machine is an indicator of generation-load balance, speed of the machine also decides the frequency of AC-power output. The frequency should be maintained close to 50 Hz, as in all AC systems. As the load on the generator kept varying with the use of different appliances, the speed/frequency also followed the changes to maintain the generation-load balance.

There was a time once, when Mr. Contractor’s gen-set developed a problem and got shut down. He suffered a loss of power. He immediately called his neighbor Mr. Engineer, to look into the problem. Mr. Engineer’s gen-set was up and running. He suggested to connect his neighbor’s load also on to his machine, at least till the faulty machine is repaired. He installed a pair of cables to jumper the outputs of both the machines, but took care to put it through a circuit breaker. He requested Mr. Contractor to switch on only essential loads to keep the load on the single machine within its limits. Mr. Engineer checked the faulty machine and found it was a minor fault. He repaired the same and started the machine. He took the machine in service. But he forgot to remove the jumper cable. Then he decided to leave the same on, to increase the reliability of power supply to both of them. Thus was born a GRID, a Power Grid.

When both gen-sets operated as a GRID, they enjoyed the increased reliability and stability of power supply, but they also experienced a few problems:

i) Any load changes in one system affected both the gen-sets.

ii) The response of the two speed-governors being slightly different, there were oscillations in actual power sharing between the two gen-sets.

iii) The power flow through the jumper cable (normally called Tie-line in GRID terminology) was varying widely, some time very close to its full capacity.

Mr. Engineer being very smart, introduced some changes in the governor systems as below:

a) He made the speed governors less sensitive by introducing a speed-load droop response in their systems.

b) He introduced a secondary frequency-control equipment in both the gen-sets, which responds based on ‘Speed/Frequency plus Tie-line flow’, as seen from respective systems. This secondary control helped individual systems in keeping generation/load balance on their own systems. (This secondary control is called Tie-line Bias Control in GRID terminology).

Under normal conditions, the GRID was operating quite well. they derived following advantages:

– Any minor restrictions on the individual gen-sets could be easily managed

– Any gen-set could be released for routine maintenance easily, with only minor load restrictions

– Voltage and frequency were better regulated, even during load surges and drops

There were also a few new problems:

– lack of proper accounting of Tie-line power flows either way, which could enable proper sharing of energy/fuel expenses.

– Mr. Contactor’s gen-set was inherently more prone to frequent failures. This created problems for Mr. Engineer also. Can he be compensated in any way?

– Once, Mr. Contractor’s gen-set-1 started dropping load gradually due to some mechanical problem. But Mr. Contractor ignored to take action and did not switch of his non-essential loads. This lack of discipline resulted in tripping of both the gen-sets giving a total shut down to the GRID.

– Of course there were other times when Gen-set-1 one failed instantly with the same result. Mr. Contractor could not do anything to prevent this and Mr. Engineer had to accept it.

They together discussed these problems and decided on the following solutions:

1. Tie line power flow will be metered either way and the net flow of energy from either system to the other will be billed to the receiver at an agreed rate.

2. The above is fine for inadvertent exchanges of power, happening due to system dynamics. For scheduled assistance from one system to the other, the charges could be agreed at a higher rate.

3. For unscheduled short-term emergency assistance of power, the charges could be agreed at a still higher rate.

4. Proper discipline should be followed during abnormal situations, to keep the loads to be within the limits of  ‘own generation plus agreed assistance’.

5. They also agreed to commission automatic load shedding (on under frequency and reverse power) and system separation (i.e. tripping the tie-line) in cases where such load-generation balance could not be achieved by self-discipline.

Both Mr. Engineer and Mr. Contractor (somewhat reluctantly) agreed and implemented these ideas. Surely, there were frequent bickering between the two, but the GRID basically worked well. Before the GRID could be affected by this bickering, fortunately they received the electrical connection from the power utility. Yes, now they are connected to a common grid managed by the utility. Mr. Contractor’s gen-set was disposed off. Mr. Engineer’s gen-set was retained as a stand-by supply till their connections were fully stabilized.

Though the above GRID was a small one, the principles of grid operations are the same even when two bigger systems are connected as a super GRID. To ensure that these principles are always upheld, usually a separate GRID operation department, (or a Load Dispatch Centre) is established, common and neutral to all the systems interconnected. It monitors the grid operation on a 24×7 basis and instructs the component systems to adhere to the principles of grid operation.  Failure by one or many of these connected systems to adhere to these principles will lead to major system black-outs, as occurred in the Northern Grid of India on 31 July 2012, which affected 500 million people. But out of these large population, even 0.01% would not have understood what exactly led to this shut down. With this note, I sincerely hope, at least 0.02% may now understand how any lack of discipline as described above could lead to failure of GRID operation and eventually lead to a black-out. (which means at least 50,000 people should read this blog !). I have kept this write-up intentionally simple to help even non engineers to understand.


25 Dec 2012


Sine of an angle – by Hindu Maths

December 3, 2012

Sine of an angle – by Hindu Maths

L V Nagarajan

Aryabhata (AD 476-550) was the first in the line of great mathematician-astronomers known to us from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya and the Arya-siddhanta. The following stanza in Āryabhaṭiya gives a series of 24 numbers and calls them as Ardha-Jya differences.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

Makhi Bhakhi Phakhi Dhakhi Nnakhi Nyakhi

Ngakhi Hasjha Skaki Kishga Sghaki Kighva

Ghlaki Kigra Hakya Dhaki Kicha Sga

Jhasa Ngava Kla Ptha Pha Cha kala-ardha-jyas

When decoded into numbers it reads thus:

225, 224, 222, 219, 215, 210; 

                               205, 199, 191, 183, 174, 164

154, 143, 131, 119, 106, 93;  

                               79, 65, 51, 37, 22, 7 : Ardha Jya Differences.

You may imagine a bow. The string tying the two ends of the bow is called Jya (or rope) in Sanskrit. (This sanskrit word Jya, for rope, is the root for Geometry, Geology, Geography, as we know them now). Jya-ardha is half of this length.

Please see the figures below. In Figure-1, ABC is an arc of a circle. AC is its Jya. AM is ardha-Jya or half-Jya. It is seen clearly that half-Jya, AM  is nothing but Sine of angle AOB, multiplied by radius OA.

Considering the Figure-2 above, all the vertical steps from bottom to top add up to respective Jya-ardhas or half-jyas of the increasing angles. In the above diagram, the angles are in steps of 15 degrees. Hence, Step 1 = R Sin 15; step1+ Step2 = R Sin 30; Step 1 +2 +3 = R Sin 45 and so on up to 90 degrees. These steps are called half-Jya differences. The above sanskrit verse gives the step-sizes or half-Jya differences for 24 steps of 3.75 degrees each to add up to 90 degrees. Thus it gives a table RSines for 0 to 90 degrees in steps of 3.75 degrees, with R= R sin90 = Sum of all steps 1 to 24 = 3438. These values are found to be highly accurate with the present day values of Sines, as shown in the table and chart given at the end.

The genius of Aryabhata defined length-MB (refer to Figure-1) as Utkrama-Jya, reverse-Sine or Versine. Aryabhata proposed accumulation of the above Jya-differences in the reverse order to get the successive Utkrama-Jyas, as it is obvious from the above step-diagram where steps are symmetrical about 45 degrees. Hence Koti-Jya or (Cos x) was defined by Aryabhata as (1 – Utkrama-Jya).

Aryabhata has actually devised an algorithm to develop this Sine table. The second section of Āryabhaṭiya, titled Ganitapāda, contains the following stanza indicating a method for the computation of the sine table.

rasi lipthashtamo bhaga: prathamam jya-arda muchyathe

thath dwibhakta labdhon mishritham thath dwitheeyakam

aadyenaivam kramaath pindaan bhaktwa labdhon samyutha:

khandaka: syu: chaturvimsa jya-ardha pinda: kramadami.

There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse (by Katz) wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.

“When the second half-chord partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.”

With my highly limited knowledge of Sanskrit, I can guess a few parts of the above Sutra.

Rasi – 12th part of a full circle (30 degrees).

Ashtomo Bhaga – 8th part (of Rasi), 3.75 degrees

prathamam – First; Jya-ardha – Sine; Muchyate – obtained

Dwibhakta – Double or  Add to itself;

Labdhon – Profit, Dividend, Quotient;

Mishritam – together;  Dwiteeyakam -The second;

aadyenaivam – In a similar way

kramath – Successive; Pindan – difference

chaturvimsa – twenty four; Jya-ardha Pinda – Sine differences;

Kramadami; recursively

Now let me try to give a context-based English translation of the above Sutra:

Arc of an eighth of a rasi gives you the first Jya;

That doubled and divided – together gives the second;

Same way successive Jya-differences, together with quotient,

recurrently give all the twenty four Jya-differences.

There are 12 Rasis in Earth’s trajectory around the Sun. Hence each Rasi is 30 degrees. A quadrant arc of a circle subtends an angle of 90 degrees at the centre, or 3 rasis of 30 degrees each. 8th part of rasi will be 3.75 degrees. There will be twenty four such 3.75 degree arc-sectors in a quadrant, totaling to 90 degrees. An arc (or a bow) of 2 x 3.75 degrees will have a Jya (or a rope). Half of this, is Jya-ardha and corresponds to Sine of 3.75 degrees; which will be same as this arc of 3.75 degrees, as per the Sutra. Yes, in modern mathematics, it only means Sin(x) = x, when the angle is as small as 3.75 degrees. Evidently, a basic circle of certain radius must have been considered for this purpose and hence Jya really means Rsin(x). For some reason (to be explained later) the first Jya is taken as 225 :– 4 x 24 x 225 gives, the total circumference of the circle considered as 21,600, with corresponding radius of 3438.  The value of Jya of the first angle having been initiated as 225, the Jyas of all the twenty four angles can be found following the above Sutra. Interestingly the Sutra above gives different rules for the first two Jyas and prescribes the recursive rule only from Jya3. (Jya is used synonymously with Jya-ardha, which actually represents Sine).

Jya1 = Jya-Diff 1 = 225; Jya2 = (225+225) – (225/225) = 449; Jya-Diff 2 = 224

Jya-Diff 3 (as per the Sutra) =  Jya-Diff 2 – Jya2/225 = 224 – (449/225) = 222.0044; Jya3 = 671.0044

Jya-Diff 4 =  Jya-Diff 3 – Jya3/225 = 222.0044 – (671.0044/225) = 219.0222; Jya4 = 819.0266

The whole table of 24 Jya-diff’s, as developed by using the above sutra is given in a table below. Even this table gives good values for RSines, though not as accurate as the earlier table as could be seen from the plot.

Now let us check this recursive calculation

Difference, D(n) = Sin (n+1)x – Sin nx

= Sin nx Cos x + Cos nx Sin x – Sin nx

Similarly, D(n -1) = Sin nx – Sin (n -1)x

= Sin nx – Sin nx Cos x + Cos nx Sin x

D(n) – D(n-1) = 2 Sin nx Cos x – 2 Sin nx = -2 Sin nx (1 – cos x)

We know,  Jya(n) = J(n) = R Sin nx

Hence Jya Difference, DJ(n) = R * D(n)

Similarly, DJ(n -1) = R * D(n-1)

Hence, DJ(n) – DJ(n-1) = R * [D(n) – D(n-1)] = – 2 J(n) (1- cos x)

i.e., DJ(n) – DJ(n-1) = -J(n)/K,

K being a constant and equals [1/2(1-cosx)]

With x = 3.75 degrees, It works out that K = 233.5374.

If we start the recursive process with J(1) = K = 233.5374, we will get a very accurate table of Jyas, but with R = K x 48/Pi = 3568. However with values of Pi and (Cos x) as available in ancient Hindu period, values of 225 and 3438 for K and R were good enough, as can be seen from the plot below. Or was there a reason to choose these values? The whole circle is 360 degrees. Each degree can be divided into 60 minutes. Now the whole circle is 360×60 = 21600 minutes. Hence perhaps, the radius of the circle was taken as 21600/2Pi = 3438. An arc of 3.75 degrees will be 225 units long.

The ancient Hindus however knew the exact values of Jya for angles of 30, 45, 60 and 90 degrees and Aryabhata could have very well used them to apply corrections to the above table as required. Hence the value of R may not be very critical. Aryabhatta’s table of Sines, given earlier, is the corrected and improved version of the table as developed by his own formula, and hence is much closer to actual values, especially at 30, 45, 60, 90 degrees.

The whole table looks as below:

As per the recursive fromula

Aryabhata’s  Final Table


Angle   x

Quotient   Jx/225

 Ardha Jya Diff RSine(n)  Jya-n  Ardha Jya Diff RSine(n)  Jya-n

R=3438   RSine(n)


0.00     0   0 0


3.75 1.0000 225 225 225 225 224.86
2 7.50 1.9956 224 449 224 449



11.25 2.9822 222.004 671.004 222 671 670.72


15.00 3.9557 219.022 890.027 219 890 889.82
5 18.75 4.9115 215.067 1105.093 215 1105



22.50 5.8455 210.155 1315.248 210 1315 1315.67
7 26.25 6.7536 204.309 1519.558 205 1520



30.00 7.6316 197.556 1717.114 199 1719 1719.00
9 33.75 8.4757 189.924 1907.038 191 1910


10 37.50 9.2822 181.449 2088.486 183 2093



41.25 10.0473 172.166 2260.653 174 2267 2266.83


45.00 10.7679 162.119 2422.772 164 2431 2431.03
13 48.75 11.4405 151.351 2574.123 154 2585



52.50 12.0624 139.911 2714.033 143 2728 2727.55
15 56.25 12.6306 127.848 2841.882 131 2859


16 60.00 13.1427 115.218 2957.099 119 2978



63.75 13.5963 102.075 3059.174 106 3084 3083.45


67.50 13.9896 88.479 3147.653 93 3177


19 71.25 14.3206 74.489 3222.142 79 3256



75.00 14.5880 60.168 3282.310 65 3321


21 78.75 14.7906 45.580 3327.891 51 3372



82.50 14.9275 30.790 3358.681 37 3409


23 86.25


15.862 3374.543 22 3431


24 90.00


0.864 3375.407 7 3438


The above is a complete table of Sines as per ancient Hindu mathematicians. The last column gives the value of Rsine, (i.e. Jya) as calculated using the current accurate values. The closeness of the values can be observed in the following chart.


When I was researching for this blog, I came across the works of late Sri T S Kuppanna Sastri, an expert in Sanskrit and Ancient astronomy. I was naturally feeling proud, since I have met him about 30 years back. He is an uncle of my wife and he is the father of my friend Dr. T K Balasubramanian, a retired scientist of BARC. Sri Kuppanna Sastri was a professor of Sanskrit and Astronomy in several colleges. His renowned major works are two books namely Pañcasiddhāntikā of Varāhamihira and Vedāṅga jyotiṣa of Lagadha.   I dedicate this blog to the memory of late Sri T S Kuppanna Sastri.



2. History of Ancient Indian Mathematics, C N Srinivasiengar, The World Press Private Ltd. Calcutta (1967)

LVN/26 Nov 2012

“Reactive Power – A Strange Concept”

July 10, 2010

L V Nagarajan 

It was very interesting to read a paper by R.Fetea and A. Petroianu (of University of Cape Town), on the above subject. (refer:

Reactive Power as a concept is really strange, but is very necessary for power system management. So we have to live with it. But we need some kind of reconciliation with many valid points raised by the authors of the above paper. Let me try the same in the following paragraphs. 

Let us start from their first two equations:

v = Vmax Cos(ωt)

 i = Imax Cos(ωt – θ)

Actual Instantaneous power = vi = Vmax Cos(ωt) Imax Cos(ωt – θ)

Let us now take the expression for current:

 I = Imax Cos(ωt – θ)

    = {Imax cos θ cos ωt + Imax sin θ sin ωt}

    = {Imax cos θ cos ωt – Imax sin θ cos (ωt – π/2)}  

We may now recognize the two terms in the above expression as:  in-phase and 90-deg-lagging components of current, I.

We say,

Active component:      Ia = Imax cos θ cos ωt

Reactive component: Ir = Imax sin θ cos (ωt – π/2)


Root Mean Square value of v, V= (Vmax/√2)

Root Mean Square value of  i,  I = (Imax /√2)


(instantaneous) Active power, p = v Ia

 i.e.,    p =  Vmax cos(ωt) Imax cos θ cos ωt

                 = (Vmax/√2) (Imax/√2) cos θ (2cos2 ωt)

                 = VI cos θ (2cos2 ωt)

                 = P (1 + cos 2ωt),        where P = VI cos θ

This is a positive sinusoidal function with an average value of P, the active power, transferred through the circuit.

(instantaneous) Reactive Power, q = v Ir

i.e.,  q = Vmax cos(ωt) Imax sin θ sin ωt

              = (Vmax/√2) (Imax/√2) sin θ (2 sin ωt cos ωt)

              = VI sin θ (sin 2ωt)                                        

              = Q (sin 2ωt), where Q = VI sin θ

This is a sinusoidal function with an average value of zero. But still we say a reactive power of value Q is transferred through the circuit!! Why at all?

Assume a power source and a load connected as below

If the load R is purely resistive,

            V —————————- R

there will be only active power flow through the circuit, as voltage and current will be in phase.

Suppose we add an inductive load XL. Now the current will lag the voltage by an angle, and hence there will be some reactive power flow also.  

            V —————————- R+ XL

Though average of this reactive power will be zero, the source will still have to ‘supply’ this reactive power flow also.

Suppose we now add a capacitative load Xc, to exactly compensate this inductance.

            V —————————- R + XL +  Xc

Then the source need not ‘supply’ any reactive power, as the same is ‘compensated’ by the capacitance. (i.e, the lag by inductance is nullified by the lead created by capacitance). However there will be reactive power flow between XL and Xc. Sometimes it is said, Xc ‘produces’ lagging reactive power to ‘supply’ XL

It is exactly in this way, we handle the reactive power Q, even though its average is always zero.

Actual instantaneous Power

                          = Active power p + Reactive power q

                          = P (1 + cos 2ωt) + Q (sin 2ωt)

                          = P + P (cos 2ωt) + Q (sin 2ωt)

The average values of both second and third terms above are zeros over a voltage cycle. Hence P become actual average power transmitted over the circuit.

As phase angle θ varies from 0 to 90, the value of ‘active’ power P reduces from maximum value of VI to zero and ‘reactive power’, Q increases from zero to VI. However all through, the magnitudes of AC voltage and AC current remain the same and hence the value of VI. For this reason VI is known as apparent power, S.

 S2 = VI2(cos2 θ + sin2 θ)

       = (VI cosθ) 2 +  (VI sinθ) 2  = P2 +  Q2  

S = √ (P^2 + Q^2) = VI, is an important parameter known as VA which is widely used for specifying power ratings of electrical devices such as generators, transformers and even major loads. Now, in phase component of S, S cosθ is same as active power P. The cross phase component of S, S sinθ is called reactive power, Q, which does not appear very strange now.

Now we may use complex algebra to revisit the same concepts as above. 

Phasor Representation

By using Euler’s Formula, any sinusoidal function can be expressed as
V = Vmax cos(ωt- θ) = Real Part of [Vmax ej(ωt- θ)]

                                           = Re[Vmax e-jθ  ej(ωt) ]

[Vmax e-jθ] is known as V’, the phasor representation of V(ωt), represented as V∟-θ

Now, V = Re [V’ ej(ωt) ]

The time-dependency has been effectively factored out, in the phasor representation as it deals with only the static quantities of amplitude and phase angle.

By phasor representation as above, we may write,

V’ = V∟0 and I’ = I∟-θ

Apparent Power S’= V.I* = V.{I e-jθ }* = VI∟θ

Alternately, if V’ = V∟-θ1 and I’=∟-θ2

Apparent Power S’= V.I* = V e-jθ1.{I e-jθ2 }*

                                      = VI∟(θ2- θ1) = VI∟θ,

where θ = (θ2- θ1), the actual phase difference.

 (i.e)    S’ = VI e-jθ   = VI (cos θ + j sin θ)

                  = VI cos θ + j VI sin θ

                   = P + j Q

As we see the real part of S’ is ‘Active Power’, VI cos θ; and imaginary part of S’ is Reactive Power, VI sin θ. In this representation the Reactive Power does not seem as strange as earlier. 

Let us again consider only a resistive load, R. As in this case current will be in phase with voltage, θ = 0. Hence,

 S’ = V’. I’* = V∟0 . I∟0 = VI cos 0 + j VI sin 0

(i.e)      S∟0 = P + j0, where P = VI

Impedance Z’ = V∟0 / I∟0  = V/I cos 0 + j V/I sin 0

                             = R + j0

Now consider only an inductive load of XL   In this case the current will lag voltage by an angle of 90 degrees, i.e., π/2. Now,

S’ = V’ I’* = V∟0. I∟π/2  = VI cos π/2 + j VI sin π/2

(i.e)      S∟π/2 = 0 + jQ, where Q = VI

Impedance Z’ = V∟0 /  I∟-π/2  = V/I cos π/2 + j V/I sin π/2

                              = 0 + j XL

Now consider an combined load of R and XL . In this case there will be a phase difference of, say, θ, lagging. Hence,

S’ = V’ I’* = V∟0. {I∟- θ}*  = VI cos θ + j VI sin θ

(i.e)       S∟θ = P + j Q, where P = VI cos θ and Q = VI sin θ

Impedance = Z’ = V∟0 /  I∟- θ  = V/I cos θ + j V/I sin θ

                                 = R + j XL

               We recognize R = V/I cos θ and XL = V/I sin θ    

When a capacitor is added to the load, we have

                Impedance Z’ = R + j XL – j Xc

Here also we observe the mutually nullifying effect of XL and Xc. Hence we are able to represent combined resistive and ‘reactive’ loads conveniently as a phasor or a complex number, known as impedence, Z.

It appears that Reactive Power concept, though somewhat strange, is very useful. By this concept, complicated trigonometric functions have been reduced to simple(!) complex algebra.

Yet another set of strange things about reactive power is its direction of flow and sign. If you still have appetite for further confusion you may refer to my blog:   and

We may meet again later.

L V Nagarajan

Active Power, Reactive Power and Power Factor

February 19, 2010

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.

FAQ on Electrical Power

November 6, 2008


What is Power?

Power is the capacity to do any work. For the same work, a less powerful man will take more time and a more powerful man will take less time. Power is the rate at which work is done.


What is work?

A body remains at rest unless it is moved by a force. The distance the body moves is a measure of work done. The amount of force required to move the body and to keep it moving depends on forces that resist movement, like friction. More friction means more work need to be done to move the body by the same distance. This may require higher force. Hence force multiplied by the distance gives a full measure of the work done. How quickly the body can be moved by a distance depends on how much force is exerted and for how long. If the force is just enough to overcome the friction, it will barely move. If the force is increased it will move faster. With the higher force the work is done faster. The capacity to exert this higher force over a period of time is known as power. Power is measured as the rate of work done.


What is electrical power?

Electrical power is rarely used directly as electrical power. It is always converted to: mechanical power in pumps and drills, heat power in furnaces and heaters and light power in lamps etc. Efficiency of conversion varies as per the devices. Raw electrical work is done by charges moving in an electrical field created by voltage (Electro Motive Force). The medium, like an electrical conductor, resists the movement of charges. This property of electrical conductivity is known as resistance. The amounts of charges that move overcoming this resistance represent the raw electrical work done. More voltage, more charges and more work. Hence voltage multiplied by amount of charges moving gives us total electrical work done. The electrical power is the rate of this work. Rate of movement of charges is known as electrical current. Now it is easy to see that raw electrical power is just voltage multiplied by current.


Ohm’s Law

In general, when Voltage(V) increases the Current(I) increases. For a particular conducting material, (eg) aluminum, I, the current is proportional to applied voltage, V.

                      i.e. V = kI.

This constant k is known as resistance (R), measured in ohms in the name of the scientist who found this law,

           Ohm’s law, V= I R.

Now we can express electrical power, P as,

            P =     VI     =      I2 R    =      V2/R      measured in Watts.


What is Direct Current and Alternate Current Electricity?

All naturally available forms of electricity are direct current.

(eg-1) Static electricity obtained by charging up certain materials. When an ebony rod is wiped vigorously by a silk cloth, the ebony rod gets charged up to a voltage. When touched to earth it discharges by passing a current to earth.

(eg-2) When a lightning strikes, a highly charged lower cloud arcs through to a high point on the earth, discharging a heavy current with a flash of arc.

Other man-made devices, like thermo-couples and batteries also provide an easy source of direct current. About Alternate Current Electricity we will see later.


What is a Generator?

For heavier applications we needed a more powerful source of electricity. Movement of an electrical conductor in a magnetic field was found to produce an electrical current in the conductor. This phenomena lead to the discovery of a device known as a dynamo. When a conducting coil of a dynamo was rotating between the poles of magnets, it was initially found to produce a current in alternating directions as they pass through north and south poles of the magnet. But it was converted to direct current by a device called split ring. This, in contact with metal brushes, eventually reversed the circuit for every half rotation of the coil mounted on the shaft, thus producing the unidirectional ‘direct current’. These dynamos were the earliest electricity generators, used in place of batteries, for heavier applications such as arc lamps and heaters.


What is a motor?

In a generator, ‘movement’ in a ‘magnetic field’ produced ‘electricity’. In a motor electricity’ in a ‘magnetic field’ produced ‘movement’. When a current was injected into a coil mounted between the poles of a magnet, the coil along with the shaft was found to rotate. The device, known as a DC motor, found major applications in traction and drives in factories, textile mills and transportation. More and more powerful DC generators were manufactured and DC grids were formed to supply different customers. Thomas Alva Edison was the great engineer who pioneered such grids, starting with New York City, way back in 1880. (Reference quoted).


Then, why we did not continue with DC?

When we started, we talked about ‘power’ and ‘work’. No matter how powerful you are, you exert less force for a smaller work and more force for a heavier work. Same is the case with electric power. We need lower voltage for smaller devices like bulbs and heaters, whereas we need higher voltage for electric traction and arc furnaces etc. Further for transmitting power from a generating station to a consumer substation, it would be preferable to use a much higher voltage. This will result in lower currents and higher efficiency of transmission. The DC electricity once generated at a specific voltage is not suitable for ‘transforming’ into another usable voltage.


What about AC?

One man, Nikola Tesla of Hungary, wondered why at all the naturally produced alternating current in a dynamo should be converted to DC?  He invented AC motors which could run on alternating current. In the process he faced a major problem of stalling of the motor. Hence it was not easy to promote the use of AC systems, though improved designs of AC generators were developed.

The famous technocrat, George Westinghouse of USA was not interested initially so much in electricity. He found no future in electricity business, unless one finds a way of transporting electricity through a large distance. But, he happened to see a piece of report about a device developed in Europe to transform AC voltages from one value to another. This gave him a spark of an idea to harness AC power for generating, transporting and utilizing at different user levels. This idea changed his entire career.  He bought up this device known as transformer, and made suitable design changes. In March 1886 Westinghouse Electric Company commissioned the first AC grid, in Massachusetts, USA. This led to several AC grids to be commissioned in USA.



What is 3-Phase and Single Phase AC?

The war against Edison’s DC grid was still on, not won. AC cannot win unless reliable AC motors were developed to power the drives of the factories and transports.

Mr. Tesla was delighted at the success of AC grid and promised Westinghouse he will give him the AC motors he needed. They entered into a pact to develop the AC motors. Mr. Tesla dealt a master stroke when he reversed the roles of stator and rotor of the motor. The magnets in the rotor were made to follow the alternating magnetic field generated by the current coils on the stator. Immediately he found the solution for stalling. What is required is only to find a way to produce a rotating magnetic field. The induced magnetism in the rotor will automatically try to follow the rotating magnetic field in the stator and in the process will try to rotate as fast as the rotating magnetic field. One sure way Tesla found, for creating the rotating magnetic field, was to think of a poly-phase AC windings which are out of phase with each other in a way to complement one another, (very much like pistons in the cylinders of the car). This finally led to AC induction motors and 3-phase AC generators. This made a large electrical network possible through devices called transformers. High voltage was used for long distance transmission of electricity and gradually lowering the voltages to lower levels for smaller networks and then to domestic level of 230V or 110V.  It is like carrying Rs 100,000, (a) as 100 nos. of Rs.1000 notes while traveling, and changing them to (b) 1000 No. of Rs 100 notes at the time of spending. Finally we have the 3Ph AC grids as existing now everywhere. For a normal domestic use one phase of this 3-Phase AC is used. All such single phase loads are distributed equitably among all the three phases to obtain a balanced 3-phase load.

(Reference quoted).


In the AC system when the current is alternating either way, how at all the power gets transferred to the device?

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. Coming back to our discussion on electrical power, V*I is still the power, but in this case, it is an alternating power, or is it? Let us consider an AC circuit with a resistive load such a heater. As before I2R is the power consumed in the circuit. Though the current is alternating, it will still heat up the coil either way. The power also will continue to be transferred from AC power source to the heater, irrespective of alternating current. 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 current 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 still be expressed as:


           P =   VI    =    I2 R   =    V2/R      measured in Watts.


But however herein, V and I are RMS values of Voltage and Current.

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

In an AC electrical system, when we refer to magnitudes of voltage and current, we normally refer only to there RMS values.


What is energy?

Work done over a period of time consumes energy of the source. A man spends his energy when he executes a manual work. When he is exhausted (of his energy) he is not able to work any more. Similarly when we use electricity we consume electrical energy and convert them to useful work in devices like heaters, motors etc. Electrical energy is measured as:

            Power x Time or         Voltage x current x time

Voltage is measured in Volts and Kilo Volts

Current is measured in Amps and milliamps

Power is measured in Volt-Amperes or Watts or Kilowatts

Energy is measured in Watt-hours or Units (Kilo Watt-Hours)



THE GRID, by Phillip F. Schewe, Joseph Henry Press, Washington, D.C. (2007)