Electrical Machines - I

Prof. Tapas Kumar Bhattacharya

Lecture - 03
Ideal Transformer, dot Convention and Phasor Diagram

Welcome to lecture 3. So, in my last class, I was talking about the magnetizing current
necessary to create a flux.

(Refer Slide Time: 00:31)

The flux is decided by the supply voltage and I will be writing many a times this. So, that
you also become accustomed with this one. So, this is the thing.

𝑉1
𝜑𝑚𝑎𝑥 =
√2𝜋𝑓𝑁1

So, 𝜑𝑚𝑎𝑥 gets fixed and then from this I will be able to calculate 𝐵𝑚𝑎𝑥 and from 𝐵𝑚𝑎𝑥 then
I will say, where is your B-H curve? What is the core material? B-H curve is like this
suppose then corresponding to 𝐵𝑚𝑎𝑥 , you read 𝐻𝑚𝑎𝑥 . This axis can be for a given value of
number of turns and magnetic length 𝑙, this can be also treated as 𝑖. 𝐻𝑚𝑎𝑥 or 𝐼𝑚𝑎𝑥 . Is not?
𝑁
I can always do that because come as a constant.
𝑙
So, corresponding to this 𝐵𝑚𝑎𝑥 , you get 𝐻𝑚𝑎𝑥 , hence 𝐼𝑚𝑎𝑥 hence the current drawn from
this supply and the another important thing I told if you have better and better magnetic
material another material, then the current needed will be only this much not this much
have another better material current will be needed this much. So, essentially this 𝜇𝑟 is
increasing ok. Now, I will be telling you about the ideal transformer ok. Therefore, if the
magnetic material is very highly permeable, if 𝜇𝑟 tends to infinity very large that is this
curve will become almost like a vertical line, then to create a flux current needed will be
vanishingly small to establish the flux ok.

(Refer Slide Time: 02:50)

Therefore, we can say that, if this material 𝜇𝑟 tends to infinity what will be the ammeter
reading? You think a bit. Ammeter reading will be vanishingly small ok some finite current
is necessary, but that current you can make it as small as possible provided you do not
have any restriction on using better and better magnetic material.

Important thing is flux is finite like this, 𝜇𝑟 tending to infinity means that this ammeter
reading 𝑖𝑚 magnetizing current will tend to 0 that is all, it only means that. No point in
telling why 0 current how it can create flux ok, it takes a current no matter how good your
magnetic material is, it will definitely take a current, but that current is very small and will
see that, it will be very small compared to the rated current of a transformer.
So, like that therefore, in a ideal situation; very ideal situation will say that as 𝜇𝑟 attains to
infinity the current drawn that is 𝐼 and this is 𝐵. This magnetizing current drawn can be
made as small as possible. So, we are now in a position to talk about ideal transformer ok.

(Refer Slide Time: 04:37)

Now, for the first time I am writing this ideal transformer and if the concept of ideal
transformer is clear, you can deal with any situation you can deal with a practical
transformer very nicely, because the concepts are so interesting.

So, ideal transformer let us say that the first two conditions remains same winding or coil
resistance are 0, 2 no leakage flux, that is all the flux is confined to the core or all the flux
or all the flux is mutual flux of course, this mutual flux what it is I will tell write now. And
number 3 is magnetic core material is extremely good 𝜇𝑟 → ∞ I put a this mark also here
means vanishingly small current extremely good.

So, that magnetizing current necessary to create flux is practically 0. These are the 3
properties. Now, it is I will now draw the core of the transformer from my previous
diagram, which I have already drawn that is I come here and copy it and then please bear
with me till that time, because and is the best; this is the thing. So, this was the thing. Now
you see in this it is not a transformer a single coil. So, far I was discussing.

Now what I will do is this; I will draw another coil suppose here is another coil. So, on the
magnetic circuit, now 2 coils are connected and this two terminals of this second coil I
have not connected anything. Therefore, whatever I discussed in my previous lectures a
single coil excited with a voltage 𝑉1 from frequency 𝑓 remains intact I mean, because this
coil whether it is present or not no one bothers, because nothing have connected no current
in this coil.

Therefore, you and suppose this has got a 𝑁1 number of turns 𝑁2 is that clear? Therefore,
you have created these then flux is created mind you although you require vanishingly
small current, but 𝜑𝑚𝑎𝑥 is finite and its strength is this one. That is what? But current
needed magnetizing current, it will draw magnetizing current and this current is practically
0. If I assume the magnetic material is of very high quality having very large permeability
and all the flux here is confined to the core.

So, what is mutual flux that is what I have to tell you mutual flux is the flux which is
common to both 𝑁1 and 𝑁2 turns. So, mutual flux will be the flux, which is confined to the
core ok. Therefore, same flux will be linking both the primary coil, this I will call now
primary and in this coil no source is connected, this flux is also changing with respect to
time therefore, with respect to I mean applying Faraday’s law I will then also conclude
that these too will become a seat of emf.

So,

𝑑𝜑
𝑒1 = 𝑁1
𝑑𝑡

𝑑𝜑
𝑒2 = 𝑁2
𝑑𝑡

same flux. Rms value of the induced voltage, see polarity of that voltage I have found it
out without I mean thinking so, much about that negative sign. I have applied physical
reasoning and told that if supply voltage at any time is increasing, then flux is increasing
this terminal has to become plus and minus so far as 𝐸1 is concerned.

Similarly, and

𝐸1 = √2𝜋𝑓𝜑𝑚𝑎𝑥 𝑁1
which of course, happens to be equal to 𝑉1, because KVL is to be satisfied. Similarly, in
the second one; if I say induced voltage in the second coil 𝐸2 in which way it will be
different same flux only 𝑁2 comes in therefore, it will be

𝐸2 = √2𝜋𝑓𝜑𝑚𝑎𝑥 𝑁2

That is all. Is not I am not deriving once again differentiating and trying to write. This is
going to be what else. So, rms value of the induced voltage in the primary, which happens
to be equal to 𝑉1, because there is no resistance, no leakage flux, 𝑉1 = 𝐸1.

On the secondary side this will be this one. Now I should not of course, jump to the
conclusion that, this is plus, this is minus will see now that one. So, at a particular instant,
if this is plus this is minus, what will be the polarity of this induced voltage 𝐸2 here;
whether, this is plus this is minus or this is minus this is plus, that I have to decide ok. The
polarity of the induced voltage will be such that, it will try to oppose the very cause for
which it is due, I am repeating the same statement. Means what this flux about this flux,
what I told the flux is positive and it is increasing ok.

Now, there are only two possibilities, either this is plus this is minus or this is minus this
is plus, either of one of this is true correct, that way if you think. Suppose I say that, I am
not sure, suppose this has happened, when this is plus this is minus this has become plus
this is minus; suppose, let us assume it is like that. Now let us verify whether this
assumption is correct or not it has become a seat of emf.

Now you imagine that, this 𝐸2 I will allow it to act; 𝐸2 as it is open circuited nothing
happens no current etc, only thing it has become a seat of EMF and somebody says this is
plus this is minus. Now I am telling you that, if it 𝐸2 is allowed to act on a circuit, then it
will deliver current at that instant and what will be the direction of this current? If you
imagine that you have connected some loads; some resistance here I will connect it some
resistance here.

Then at that instant current supplied by the coil will be like this is not it will go this is the
source like a battery current will go like this and the direction of the current in the windings
will be like this. Now this is what in accordance with Lenz’s law the answer is no why?
Because the cause of the voltage induced is that 𝜑 was increasing is not? 𝜑 was positive
𝑑𝜑
and increasing was positive that was our assumption.
𝑑𝑡

Now I find at that instant if current flows the flux in the core produced by this current will
be; flux produced by core in this coil will be also in this same direction; are you getting?
This was the flux created by 𝐼𝑚 , it was going up in this direction now also flux. So, the
flux is strengthened, the cause for which it is due is strengthened, but that is not in
accordance with Lenz’s law.

It will try to oppose the very cause for which it is due. What is the very cause? Phi from
top to bottom was increasing. So, the polarity of this induced voltage will be such that, if
this 𝐸2 is allowed to act, it should pass current through this winding in such a direction,
that it will try to oppose this flux that is what has to be. That is therefore, it looks like this
is not correct no. So, this cannot be like this, what is the other possibility? Other
possibilities is this has become plus, this has become minus.

(Refer Slide Time: 18:12)

Let us see, whether this will be consistent mind you this is the continuous thing I have
been only corrected no.

Student: Ok.
What is the other possibility? This is plus this is minus is it consistent? Let us see, if this
emf is allowed to act on its own to an external circuit like 𝑟 you have connected, then at
that instant it will try to send current in this way. Because this is battery it is the source,
current comes current goes current goes current goes like this current goes. So, this coil,
when it delivers current it will create flux in a direction opposite to these 𝜑(𝑡) which has
been created by this one.

So, it is trying to oppose the very cause for which it is due. Therefore, at a given instant of
time, if this terminal is plus this terminal is also plus. Let me repeat this point in a nicer
diagram; it is like this.

(Refer Slide Time: 19:50)

So, let me sketch it that will be faster. So, I sketch because let us spend some time on this;
sorry, I have not selected I will cut it out. See so, let us do it like this, suppose this is your.
Now I had got 2 coils let me repeat this point emphasize this point. This is suppose one
coil having 𝑁1 turns, this is suppose another coil with 𝑁2 turns, you have applied I will not
just simply say that you have applied 𝑉1 rms voltage of frequency 𝑓 with this is plus this
is minus.

𝐸1 has appeared here, it will have polarity like this here also I applied the Lenz’s law and
the flux in the core is 𝜑(𝑡). Then this two terminals also become a seat of emf. The value
of this voltage 𝑉2 is nothing, but 𝐸2 only 𝑉2 is the terminal voltage there is no distinction
between 𝐸2 and 𝑉2 right now.
So,

𝑉2 = 𝐸2 = √2𝜋𝑓𝜑𝑚𝑎𝑥 𝑁2

and also I know

𝑉1 = 𝐸1 = √2𝜋𝑓𝜑𝑚𝑎𝑥 𝑁1

and it is easy to show that

𝑉1 𝐸1 𝑁1
= =
𝑉2 𝐸2 𝑁2

This is one famous formula ok. So, that is the thing, but I was discussing about the polarity
of this induced voltage instantaneous polarity.

If at any instant of time this is plus this terminal has to become plus no way; the other
possibility that at that instant this is plus this is minus will violate Lenz’s law and this ratio
𝑁
of this voltage is this one and from this many things can be told it is merely the ratio of 𝑁1
2

𝑁1
manipulating this ratio 𝑁 you can step up the voltage this is called secondary coil or step
2

down the voltage depending upon, whether 𝑁2 is greater than 𝑁1 or 𝑁1 is greater than 𝑁2
depending on that.

But anyway, this is the thing. Now people use you know some dot marking to communicate
this particular important aspect of a transformer by using known as dot convention. It
merely tells you that any given instant of time, if the polarity of this induced voltage is this
is plus, induced voltage in the other coil instantaneous polarity will be also this is plus and
this is done by dot convention.

Now I must also point out see, I have drawn the core material and drew the coils primary
coils and the secondary coils rather carefully. What do I mean by carefully; that you can
draw suppose you have a magnetic circuit like this as a you draw the coils like these. It is
for better than this, what I mean to say that this coils I have drawn. So, that you know what
is the sense of the winding, you take the coil this way then turn it like that, but from this
sense you cannot figure out, how this coils went somebody draws like this.
Then of course, you cannot point out if this is dot which one is dot this side. No out of
question you cannot do that. Perhaps by doing some experiments you will be able to do,
but as such on pen and paper you cannot predict ok, this is plus this will be plus that you
can only do provided you know the sense of the winding; sense of winding the coil. If that
is meticulously shown, then you can figure out. Before I tell some more interesting thing;
also you note that if these two are dots after some time; you know because it is after all
AC supply, it will be better if we say dot terminals are those terminals which have like
polarities of the induced voltage.

For example if you say this is plus this will be plus, you rub this off for the same
transformer I am talking this point you listen, suppose this dot I remove if somebody says
no this is dot. Then also he is correct are you getting like terminals, if it is minus that will
be minus whenever it will become plus because polarity reverses this will be also plus.

In other words, what I am telling no point in showing so many dotteds it is understood

So, we have we had discussing about ideal transformer and the ideal transformer is that
transformer, who which requires vanishingly small magnetizing current to create a finite
flux of strength

𝑉1
𝜑𝑚𝑎𝑥 =
√2𝜋𝑓𝑁1

current necessary, that is magnetizing current is vanishingly small 0.

Since, 𝜇𝑟 is infinitely large, that is the idea. Apart from the fact that all flux is confined to
the core there is no winding resistance, which allows me to write 𝑉1 = 𝐸1 = √2𝜋𝑓𝜑𝑚𝑎𝑥 𝑁1
and the last thing I will tell in this class, then if I draw the phasor diagram, what should I
draw this is the applied voltage is not?

Let me just draw it give you some idea, then the current drawn; magnetizing current drawn
although vanishingly small it will be lagging 90°. So, magnetizing current will be along
this line, because after all pure inductance it has got only inductance; inductance value is
very large if 𝜇𝑟 is tending to infinity we have shown wrote some expression for inductance,
if 𝜇𝑟 goes to infinity inductance goes to infinity, you can now interpret the things from
different angle inductance point of view very large 𝜇𝑟 , but the current drawn from the
supply will be 90° lagging and this length is pretty small 0, so, this is 𝑉1.

So, primary current is this where is 𝐸1 ? 𝐸1 will be also like this same as these one and
𝑑𝜑
where will be 𝐸2 same, because same 𝑁 𝑑𝑡 . So, 𝐸2 will be also like this, all voltages

induced voltages, applied voltages this is 𝑉1 they will be all in time phase and this is the
direction of 𝜑. Why 𝜑? Because 𝜑 is proportional to 𝐼𝑚 although 𝐼𝑚 is vanishingly small,
but it creates a finite flux. So, flux phasor will be along this line. We will continue with
this in the next lecture.

Thank you.