Okay, we are starting on electrical machines. We are going to look at both motors and generators

in this particular course. But it definitely requires some amount of recalling of what

was already done, so let me give you roughly what is the course plan initially and then

I will go over whatever is the recalling that needs to be done.

So if you look at any of the electrical machine normally they are going to be made up of ferromagnetic

materials, because if you look at electromechanical energy conversion it is always through a magnetic

via media. So I am going to have a magnetic via media and I am going to have on one side

let us say electrical energy and I am going to have on the other side mechanical energy.

So it is converting from electrical energy to mechanical energy through magnetic via

media. So when this is being done if electrical energy is converted into mechanical energy

it is going to be motoring operation on one side whereas if I am going to do from mechanical

energy to electrical energy this is known as generator operation.

So I can have either motoring operation or generating operation, the same machine can

work as both generator as well as motor. So all I need to do is I have to do the controlling

such a way or if I give mechanical energy it will convert that into electrical energy

and it can work as a generator or vice versa. So I can in general say the construction wise

and functionality wise both are almost similar. So the same machine can work as generator

as well as motor. But because we are using ferromagnetic material it is very essential

to know about the behavior of ferromagnetic materials; so, which we call as basically

magnetic circuit.

So we call this whatever is the behaviour we try to analyze it by the help of magnetic

circuit concepts. So we will be starting off with this first then we are going to look

at, so if I actually start the flow of the course, the first and foremost thing we will

be doing is review of single phase and three phase circuit. You must have studied this

already you must have seen what is complex notation, what is phasor diagram, and how

the power is measured and so on. So we will be revisiting these things to some extent

first, the second topic is going to be the magnetic circuits, so we will be looking at

magnetic circuits, as the second topic where we will be talking about the behaviour of

ferromagnetic material around that if I wind a wire how it is going to function that is

what we are going to look at.

The third topic after this we are going to look at, basically transformers, so in transformers

we will be actually looking at both single phase and three phase transformers. We will

of course start with constructional details how the transformer is constructed then after

doing that we would be looking at what is the principle of operation and then we will

be actually looking at the phasor diagram of the transformer; how the voltages and currents

are displaced from each other, then we will be looking at equivalent circuit of the transformer.

Once we have the equivalent circuit derived we should be able to really talk about the

performance characteristics of the transformers like voltage regulation and efficiency, these

two we will be talking about and after that we will be looking at actually open circuit

and short circuit test or testing of transformers from which actually we derive the equivalent

circuit parameters. Then after looking at these things we will be actually looking at

the auto transformer, and instrument transformers, that is instruments transformer are generally

meant for instrumentation that is measurements, so we may use potential transformer and current

transformer.

After looking at all these single phase transformer principles then we will be going over to three

phase transformers, in three phase transformers we will be looking at star star, start delta,

delta star and delta delta connection and any other special connections.

I forgot to tell you the number of hours roughly I will be dwelling on each of these topics,

review of the single phase and three phase circuits might take about maximum of 1 to

2 hours, so I am giving myself maximum 2 hours inclusive of introduction. Magnetic circuit

again should take anywhere between 2 to 2.5 hours, nothing more than that, transformers

is a vast topic, so inclusive of three phase transformer I may be requiring in about 8.5

to 10 hours, the entire thing.

Then the next topic that I am going to have which is actually the third topic, fourth

topic I suppose, fourth topic, the fourth topic actually is going to be the electromechanical

energy conversion principle, until now what we had seen in the topic was only static machines,

they are not having any moving parts but if mechanical to electrical energy conversion

or electrical to mechanical energy conversion has to take place through a magnetic via media

we have to have definitely looking at energy conversion principles themselves. So this

is just giving an introduction to how electromechanical energy conversion takes place. This will take

anywhere between about 2 to 2.5 hours.

After this we will be looking at DC machines, because this is one of the simpler machines

in terms of operation, I would not say constructional principle, definitely in operation. So we

will be looking at first of all basic construction after that we look at the classification of

DC machine, working principle, then after that we will be looking at something called

armature reaction, what happens when the armature is also creating some flux along with the

field so this is known as armature reaction.

Then we will be looking at characteristics of the DC machine, so if I am talking about

motor I have to talk about speed torque characteristics if I am talking about the generator I should

talk about voltage versus current characteristic, so this we will be looking at then, then when

we are talking about motor we will be actually concentrating on starting, how we start a

DC motor, then we will be looking at breaking very briefly and we will also be looking at

speed control, how to control the speed of a DC motor.

These are the different things we will be looking at, of course I have left out testing

of DC machine we should actually look at the testing of DC machines as well. Depending

upon of course time, availability of time, then after completing DC machines we will

be actually taking up induction machines, which is one of the most important AC machines,

if you look at the industries 80 percent of the electrical energy actually in factories

or industries is consumed by induction motors, three phase induction motors.

So we will be first looking at three phase induction motor in three phase induction motor

we will be looking at construction but even before that we should look at revolving magnetic

field theory. How we are going to have when three phase voltages are applied to three

phase time displaced, space displaced winding how a revolving magnetic field is created,

this is how we will start the three phase induction motor, then in the three phase induction

motor itself we will talk about construction, two types of construction, that is squirrel-cage

and wound rotor, then we will talk about principle of operation.

After looking at the principle of operation then we will be looking at phasor diagram

and equivalent circuit, after looking at the equivalent circuit we will be able to derive

the torque expression, torque slip or torque speed characteristics, then we will be clearly

looking at how the characteristics are dependent upon these expressions. So we will be talking

about the speed torque characteristics mainly.

Then after that, we will be looking at starting what are all the different starting methods,

and then we can look at speed control and very little bit about breaking. So these are

the topics that will be covered under induction motor then if time permits we will also look

at single phase induction motor.

Let’s see how it goes, so single phase induction motor we will be looking at basically double

field revolving theory and then we will be looking at again equivalent circuit characteristics,

etc. So these are going to be different topics in single phase induction motor. I would not

like to call this as a separate topic because this will be pretty small.

So for three phase induction motor I might take anywhere between again in 8 to 10 hours

minimum, this might take about 1 to 2 hours, because we are just going to touch upon this,

nothing more than that, then the next topic that we would be looking at will be synchronous

machines. This is also a three phase machine but we will be concentrating on this as a

generator.

So we call a three phase synchronous machine working as a generator as an alternator so

we will be looking at alternator and its working, basically in this particular topic, the synchronous

machines generally we will start with classification then we will look at construction after looking

at construction here we will be concentrating more on the winding arrangement, so we will

talk about something called winding factor and how we influences the voltage generation

in a synchronous machine.

After that we will looking at power angle relationship and equivalent circuit and because

it is working as a generator it is very important to look at VI characteristics of the generator

whether it will be able to maintain a constant voltage in case of disturbance how do we handle

it, these things will be talked about. After this, we will be talking about power

factor control which is one of the most important aspects of the synchronous machine. Then we

will be talking about the same machine functioning as a motor, very briefly that is synchronous

motor under which we will be talking about V and inverted V characteristics. One more

thing I have forgotten we also need to look at the generator as how it can be synchronized,

how the synchronization can be done with the three phase grid. So this is an important

thing we need to do when we want to feed power from the synchronous generator into the three

phase grid, so we will be talking about the synchronization as well.

In my opinion this should not take as long as what it has taken for induction machine,

so it may be anywhere between 6 to 7 hours. So I think on the whole if we actually compute

the total number of hours we will be spending about 6 plus 2, 8, 8 plus 8, 16 and we are

having a electromechanical energy conversion principle plus transformers maybe about 10

12.5 hours, so 16 plus 12, 28, have we left to DC machine, I have not given any time here

that has to be again about 8 to 10 hours, so about 38 hours of 36 hours gone here, then

introduction and review everything together about 2 plus 2, 4 hours, so I think it comes

to about 40 to 42 hours. So that is the way the entire lecture is going to be planned,

okay?

We have some reference books for this electrical machines course, so let me write them out.

I would like to start with simplest of books, basically for electrical machines, although

some of the topics I mentioned may not be covered very well in this book, nevertheless

for a beginner it is very very easy to read that book so I would rather recommend that

book, this book is written by P.C. Sen and this is called Principles of Electrical Machines

and Power Electronics, of course we will not be going into power electronics portion, no

doubt but this covers electrical machines from a fundamental principle, it starts with

magnetic circuits and so on there are appendices given on a three phase circuits and single

phase circuits . So I think it almost gives you a very good introduction to entire machines

course.

This is from John Wiley and second edition is available as far as I remember, if you

ask me the year I don’t remember, the second book I would rather recommend very strongly

is that there is one book on DC machines, let me probably erase this, let me write this

same order as I have done, so M.G. Say and O. Taylor, there is a book on DC machine by

these two people, it is a vast book that is why I would not give this as a principle reference

book but it is written extremely well and this is from probably ELBS, it used to come

from ELBS, English language books society, low price edition, I don’t know whether

Pearson has taken it already, I am not sure.

Similarly, the counter part in the AC side is again by M. G. Say, this is actually a

Performance and Design of Alternating Current Machines, this is again a very vast book but

the explanations are extremely good and this is also from ELBS, this is a low price edition,

then I would say one more book which is really worthy of reading that is by Fitzgerald and

Kingsley, of course there is a third author, Umans, I have left him out. So this is actually

Electric Machinery. I think seventh edition, or something has come now as far as I remember

and this is from McGraw Hill, there are definitely other books that are available as well from

Pearson, for example there is Electrical Technology by Edward Hughes, this is a very good book.



There is one more which is actually Electric Machinery by Chapman Stephen Chapman, these

are from Pearson, and one more Indian author book, probably Electric Machinery by or by

Electrical Machine by Nagarathan Kothari. I think this is from Tata Macgraw Hills, it

used to be from Tata Macgraw Hills so this is now Macgraw Hill India. So these are some

of the reference books that we might like to refer to whenever we have some confusion

in any of the topics.

So let me start with actually the recalling of single phase AC circuits. So I am starting

with recalling single phase AC circuits, all of you guys know that generally when we talk

about single phase AC circuit we are going to have sinusoidal excitation basically. So

we are going to give basically a sinusoidal excitation, what we mean by excitation is

a voltage, we are going to give a sinusoidal voltage.

So whenever we have a sinusoidal voltage rather than using that as a sign quantity, let us

say this is t, if I am writing this with respect to time I have to add and subtract sinusoidal

quantities, I have to multiply, I have to differentiate, integrate, all of them needs

to be done on this sinusoidal quantity because I am sure fundamental elements in an electric

circuit you guys know so V equal to Ri, this is corresponding to resistance I can write

L di by dt equal to V, this is with reference to inductance C dv by dt equal to current,

this is with reference to capacitance.

So you can very well see that I have V and i without being differentiated or integrated

here and ultimately if I want power I should say V times i, right? Instantaneous power.

So I will have to multiply some of the sinusoidal quantities, I have to differentiate some of

the sinusoidal quantities, I have to integrate if I want actually voltage equal to 1 by C

integral idt, right? So I have to integrate these quantities.

The major advantage of using a sinusoidal is that whether you differentiate, whether

you integrate, whether you multiply, whether you divide, you are going to get ultimately

sinusoidal quantity itself, and in nature most of these wave forms that are available

are sinusoidal in nature, so that is the reason why and it does not have any abrupt rise or

fall and in nature nothing grows abruptly, nothing really you know, may be death is the

only thing which occurs abruptly but other than that everything is not abrupt, that is

the reason why we are using sinusoidal quantities in all our electricity and magnets.

Now, if I have to actually refer to this quantity in the form of a vector or rotating vector,

actually rotating vector is generally known as phasor. So if I want to refer to this as

a phasor then what I can do is I can try to look at the projection along this axis and

try to plot it, for example I am looking at time T equals to 0, at that point I am having

0 value, if I look at corresponding to say 30 degree point, this is 30 degree, I am going

to have half the value of the peak, so I can actually project this as though a vector is

rotating continuously with respect to time, so when I had time t equals to 0 if I actually

specify this vector somewhat like this.

If I look at the vertical projection or projection along the vertical axis the magnitude is 0

which corresponds to the sinusoidal quantity also at time t equals to 0, when I look at

30 degrees I can just take it a long, may be it would have rotated exactly by 30 degrees

here and this is actually 30 degree and if I call this point as A this point is going

to correspond to A.

So if I try to look at the vertical projection it will exactly have this value. So similarly

I should be able to draw basically this corresponds to 90 degree, this corresponds to 180 degrees,

and this corresponds to 270 degrees, and this corresponds to 360 degrees again. So if I

say that this is actually oscillating at a frequency of f, in one second I am going to

see f such oscillations. So if I am traversing from 0 degree to 360 degree completely, or

if I go from small that is 0 to capital T here, then I would have traversed this by

2 pi radiance or 360 degrees.

So if I am saying that f times it is going to oscillate in 1 second I will have 2 pi

times f as the radiance that is traversed within 1 second. So this is the angle that

is traversed in 1 second. So we call this as angular velocity, which is omega. So we

are going to say this as omega equal to 2 pi f. So then we are plotting with respect

of angle, we can write this as omega t rather than t because I can say 2 pi f is the angular

velocity so 2 pi f time t will be the angle that is traversed, so every time when we draw

actually the wave form of voltage or current we will always write the angle write the x

axis as omega t rather than t, this is what we are going to do.

Now, if I am having 1 inductance probably let us say L di by dt equal to VL, voltage

across the inductance, lets say I am going to have i equal to some Im sin omega t, right?

So if I do that I am going to have di by dt equal to Im omega cos omega t, right? So this

is multiplied by L, I have to do this also multiply by L and this is going to be my VL,

and because of which cosine wave always leads the sin wave by 90 degrees.

So I am going to have this voltage leading the current by 90 degree, it is going to lead

by 90 degrees, so if I draw the two quantities here I should probably show the voltage like

this and I have to show the current as though it is 90 degrees lagging behind like this,

this is how it will be. So if I try to plot both of these in the form of phasor at any

instant if I say this is what is my current, then I am going to have the voltage leading

by 90 degrees. So this is what we call as the phasor diagram when we draw two of the

quantities which are alternating in nature, we say at any instant of time how they are

displaced from each other.

So we call this as the phasor quantity. So if I have a circuit for example that I am

going to have an alternating voltage then may be a resistance may be an inductance,

another resistance may be a capacitance and another resistance, it is not going to be

very easy for me to draw for each of them, let us say I call this as iL, I call this

as iR1, and I call this as iR2, let me call this as may be i through C and let me call

this is iR3 and when I add these that is these two currents I will be able to get what is

iR3 and when I add the voltage drops across these two elements and this element I will

be able to get what is the voltage source, but it’s not going to be easy for me to

draw all these phases and add them.

So people wanted to find out what is the easier way out, because you are mentioning voltages

and currents, maybe another voltage something is here, I may mention the voltage like this.

Why can’t I mention this in the form of complex numbers, that is the thought that

came into people’s mind. So that is the reason why if I say this is some VC or something

it can have a horizontal component, it can have a vertical component, the horizontal

component we call as the real part of the complex number and the vertical component

we call as the imaginary part of the complex number, imaginary part of the complex number.

So we are essentially now diversifying from mentioning a sinusoidal quantity as a phasor

to indicating that as a complex number. So the complex notation mainly gives us the major

advantages if I say for example iL is going to be some a plus jb and let us say I am going

to have iC as some C plus jd. Please note I am not using i here, normally in complex

number we use a plus ib and c plus id but i we use normally for current, so I am using

j.



So a plus jb and c plus jd, so I can say iL plus iC will give me what is iR3, it will

be very easy for me to add it. I can say a plus C plus j times b plus d, very easy to

add, so whenever we have a complex notation, the algebraic operation become much simpler

that is the reason why we have chosen to have the complex number notation, right?

So once we have the complex number notation let us say I have V equal to a plus jb and

let us say I have i equal to c plus jd, okay? I may be able to mention this as some V angle

theta 1, and I may be able to mention this as V angle, I angle, sorry let me just erase

this, I angle theta 2, right?

All of you know basically when we look at the power we generally say VI theta 1 minus

theta 2 angle, this is what happens to be normally the power, right? This is what we

normally write as the power. What I mean is VI cos of theta 1 minus theta 2 will be real

power. This is what you must have learned in circuits, AC circuits. Similarly VI sin

of theta1 minus theta 2 will be the reactive power.

So if want actually how to calculate this power in the complex form let me call this

as P, let me call this as Q and ultimately what I want is P plus jQ, this is what I want.

So if I have to write this, I have to write this as VI cos of theta 1 cos theta 2 plus

sin theta 1 sin theta 2, right?

And similarly Q will be VI sin theta 1 cos theta 2 minus cos theta 1 sin theta 2, so

finally when I get the power which is actually we call that as apparent power, so we want

to get the apparent power in the format of P plus jQ. So if I actually try to do it as

V angle theta 1 multiplied by I angle theta 2, it’s not going to work properly, so I

have to take the conjugate of this particular I.

If you try to calculate backwards you will see that you are getting this expression for

P and this expression for Q, so you are going to have basically in this particular case,

if I write this as VI conjugate which means I am going to have V cos theta 1 plus j sin

theta 1, Euler’s formula I am using directly plus multiplied by I, I

and I should say I conjugate it so it will be cos theta 2 minus j sin theta 2, right?

So when you multiply you are going to get VI commonly cos theta 1 multiplied by cos

theta 2 and j sin theta 1 multiplied by minus j sin theta 2, j multiplied by j square which

is minus 1, so I will get plus sin theta 1 sin theta 2, so this becomes my real part

which is corresponding to the real power and I am going to have the imaginary part, which

is going to have j , so j sin theta 1 cos theta 2 and I am going to have minus of sin

theta 2 cos theta 1.

So you can see very clearly that these two are matching each other, if I look at P and

Q expressions that I have got, they are exactly matching with this. So normally we write the

apparent power expression S as either VI conjugate, IV conjugate and so on. So we have one of

them conjugate; that is what is important. So complex notation gives us a very easy way

for calculating power, calculating summation of currents or multiplication of currents,

multiplication of voltages and so on.



All those things are done quite easily when we have complex notation, right? So in general

single phase quantities when we look at single phase AC quantity or single phase sinusoidal

AC quantities, they are represented mainly in two forms, one is in complex form and the

complex form indicated is also indirectly translating into phasor diagram.

So we will be repeatedly using in these form getting our power or phasor diagram equivalence

circuit, all of them in transformers, AC machines and so on. So this is really important to

know. Now let us recall some of the principles corresponding to three phase circuits. So

if you look at many of the electrical machines that are available in the market, AC machines,

they are all three phase in nature.

This was initially actually the three phase configuration was initially introduced by

Tesla, it was invented by Tesla and he had made the induction motor sometime in 1890’s

or something, so induction motor design had not changed until day, it is almost remaining

the same ever since 1890. So if you actually look at the three phase the major advantages,

if I look at the advantages of three phase because I have in three phase I am going to

have Va, Vb, Vc and in all the three circuits I am going to have ia, ib and ic and the overall

power is going to be Va ia plus Vb ib plus Vc ic, this is going to be the power.

So if one of them is low, the other one may be high, so all of them balance with each

other and ultimately you get a constant power, or constant torque, ultimately. So if I say

constant power, power divided by omega is going to be torque in any rotating machine,

so the torque also becomes a constant quantity roughly, because you have the power and torque

to be constant it is very suitable for high capacity applications, what is mean is mega

watt level application or even 100 of kilowatt level applications, you do not want the torque

to change continuously, if the torque changes continuously you are going to have vibrations

in the shaft, which can not be tolerated especially at high capacity applications.

So three phase is really really good for high capacity applications, and one more thing

is that I am sure you guys know that three phase can be connected either in star or delta,

so if I have Va, Vb and Vc, all the three phases voltages they can be applied to either

star connected circuit or delta connected circuit. So throughout if I look at the transmission

lines that are going I can just have three transmission lines, that’s it.

And I will be able to disseminate power which is three times single phase power where as

if I am disseminating the whole thing in the form of single phase power I am essentially

going to require two conductor for disseminating just P whereas here I am disseminating 3 P

for which I require only 3 conductor, so the cost what is involved in copper conductors

that comes down drastically when you go from single phase to three phase, of course in

most of the cases we will be having a neutral conductor also, if you look at most of the

transmission lines we will be having a neutral conductor.

So even then four conductors are able to transmit 3 P whereas here two conductors, two conductors

are transmitting only P, so obviously we are saving on the cost of copper. So three phase

is very very commonly used for generation transmission and as well as distribution except

for domestic application, only for domestic application we are going to use single phase,

right? So it’s very important to learn about three phases quantities, how we measure power,

and how we are going to actually justify measuring the power and reactive power, so how we are

going to measure these things that is what we are going to see next.

So we are going to look at three phase power measurement if it is balanced load, I am not

going to take the case of unbalanced load condition because most of the electrical machines

normally are balanced so there is no need to really talk about t unbalanced. There are

different power measurement methods, it can be with single watt meter but this can work

only when neutral is available, most of the times neutral may not be available so we may

not be using it. Similarly, three watt meter method which will

also be used only if the neutral is available, otherwise it’s not going to be easy if it

is delta connected we can still think of using this probably, three watt meter but two watt

meter method is a very very common method which is used for using, used for measuring

power in a three phases balanced as well as unbalanced system. In both the case it will

work quite well.

So what is this method all about that is what we are going to see now, so let me draw the

circuit diagram corresponding to the 2 watt meter method. Let us say I have a three phase

supply here, so let me call this as VA, this as VB and this as VC and this is the neutral,

okay? Now I am going to have probably a load here, the load can be delta connected or star

connected, so I am showing the load just like a black box.

So I have only 2 watt meters with me so I am going to connect one of the watt meter

current coil here another watt meter current coil here and I am going to connect the voltage

coil or the pressure coil like this and the pressure coil is connected between the phase

the current coils are connected and the other phase which is not having any current coil,

then this is connected directly to the load.

I am sure you guys have seen already the watt meters so there will be the current coil which

will be actually connected between M and L normally this stands for load and M stands

for the mains and similarly I am going to have here also M and L, let me call this as

M1 and L1, okay. So this is the current coil 1 and I am going to have the pressure coil

connected between the common point, which is common between the current coil and pressure

coil and this is going to be the V.

So similarly here also I am going to have common 1 and V1, so this is the connection

that we are going to have in the three phase two watt meter method. Let me first of all

try to justify that this will give me three phase power. So normally if you look at the

three phase power you are going to have V1 i1 or Va ia plus Vb ib plus Vc ic. This is

the instantaneous three phase power, right? But if I try to look at what is the actual

value of power normally we write, we write this as P equal to root 3 VL IL cos phi, right?

Where phi, cos phi is the power factor of the load.

So load power factor I am calling as cos phi in this particular case, this is going to

be the power factor, okay? So let us try to see what is it that I am going to get in watt

meter one and watt meter two respectively. So in watt meter one I have here voltage of

A phase and here voltage of B phase so I am going to get VAB as the voltage across the

pressure coil of this particular watt meter and a current through this is going to be

iA.

So I have to write this as iA, please note, I am writing RMS quantity so all of them are

RMS quantities so this is VAB multiplied by iA, right? And between VAB and iA whatever

is the angle that is present I should say cos of the angle between VAB and iA. This

is going to be the reading of W1. Similarly W2 is going to have the reading to be this

is voltage of C and this is B so I am going to have VCB and the current that is flowing

here is iC so this is going to be iC cos of VCB and iC, right? This is going to be W2.

So let us try to take a look at exactly how this looks when we are having the phasor diagram

completely drawn. So let us say this is VA, this is going to be VB and this is going to

be VC, these are the three phase voltages, they are phase shifted by each by 120 degree

each when I want VAB, I can take VB on the other side. So this is actually minus VB.

So when I try to look at what is VAB I have to just take the resultant of VA and VB added

together, so I will have this as VAB.

Similarly, I should be able to draw VAC which should be mid, in between these two, I can’t

go beyond this, so this is will be VCB, okay? Now let us say I am going to have a lagging

load, I am just arbitrary taking this as some at a point factor angle phi this is going

to be IA, and if I assume it as a balance load I am going to have IB again at an angle

of phi and IC somewhere here which is also at an angle of phi, right?

Now, I have to look at what is the angle between VAB and IA if I want to get the power of W1,

so I am going to have W1 equal to VAB IA, this is 30 degrees, right? So this is going

to be cos of 30 degrees plus phi. Similarly, if I try to write what is W2, W2 is going

to be VCB IC but this angle is again 30 degrees, so I am going to have cos of 30 degrees minus

phi, right? So I am going to have when I calculate W1 plus W2 it will be VAB or VCB I can call

them as line voltage, VL directly.

Similarly, I can say IA or IC, if it is a balance load condition and I can call that

as IL, I can say this is cos of 30 plus phi plus cos of 30 minus phi, right? So this is

going to be VL IL cos, cos is actually going to be cos of 30 and cos of phi. Cos of 30,

we are going to have two times, right? 2 cos 30 cos phi, so this is going to be root 3

VL IL cos of theta, because cos 30 is root 3 by 2.

So we wrote the power expression earlier as root 3 VL IL cos phi, we are having essentially

the same thing, root 3 VL IL cos phi coming out as W1 as W2. So this is essentially the

three phase power expression. So we able to measure the three phase power just by using

two watt meters here and this is one of the most reliable methods as far as the three

phase system is concerned.



But we wrote V1 equal to VL IL cos of 30 plus phi and W2 we wrote as VL IL cos of 30 minus

phi, if the power factor angle happens to be greater than 60 degrees, very clearly I

am going to have this to be cos of 90 or greater than cos of 90. So if the power factor angle

happens to be 60, I will have one of the watt meter reading to be 0 and the other watt meter

reading will have a value which is positive because 30 minus whatever is the angle cos

of theta equal to cos of minus theta so I will have a positive value.

Whereas if I am going to have the power factor angle to be greater than 60, W1 will be negative

and W2 is going to be positive, that is why you will see that in some of the three phase

experiments when you done of the watt meters might be kicking back, so in which case was

either the voltage or the current you will be able to get a positive reading but that

you have to treat as a negative value and subtract it, you can not, in that case you

have to say W1 magnitude whatever you get the some value X, whereas W2 is going to be

a positive value, let us say Y, then you are going to say Y minus X is the net power in

that particular case, right?

Similarly, if I get W1 equal to W2 by chance, right? That means I am going to have essentially

W1 plus W2 as the total power, no doubt but I can also say from these expressions because

I am having W1 to be VL IL cos 30 plus phi and W2 to be VL IL cos 30 minus phi. W1 minus

W2 I can write this as VL IL cos of 30 plus phi minus cos of 30 minus phi and I can write

this as VL IL sin of 30 and sin of phi, right?

Sin 30 is half, so I am going to get basically VL IL sin phi, so if I try to write what is

sin phi divided by cos phi which is actually tan phi, right? That will be W1 minus W2 divided

by W1 plus W2, I will get sin phi by root 3 cos phi, so I have to multiply this by root

3. This will give me tan phi. So if W1 equal to W2 then tan phi is going to be 0.

From here I can say tan phi is 0, which means it is going to be unity power factor condition.

When W1 equal to W2 it will be unity power factor condition. So we would be able to deduce

the power factor of the load if I have these two reading with me, W1 and W2, these two

readings with me. So in general two watt meter method is a very very advantageous method

from various viewpoints you will be able to determine what is the power factor of the

load, you will be able to get the accurate power whether it is balanced load condition

or unbalanced load condition normally with the help of just two watt meters alone.