So, we were discussing about the graphs and
tree of a network so this graph we have discussed
directed graph it is a full graph.
Then tree is a is a reduced form of this graph
in which all the nodes will be present, all
the nodes will be connected but there should
not be do not take any elements in the to
form a tree so that it forms a loop, so this
is a valid tree because I omitted the element
4.
If I connect element 4 then only it will be
a closed path generating and so on.
So this is fine now what is done is this these
are called tweaks and the red ones are called
the links cleare.
So, what are the links here?
Element 4 element 5 element 4 element 5 and
element 6 these are missing.
So, what we will be doing is we will be showing
those links by red lines this, this and this,
these are links.
So, this is the blue one is the tree blue
one is the tree and red one is dotted red
is co-tree 
tree means tweaks.
Those elements will be called tweaks associated
with t co-tree elements are links.
How many links are there let let me also show
you show it here that is in this link this
is 4 this is 5 this is 6.
Now the 
what we plan to do is this after I have drawn
the links we will be writing down the KVL
equation of the loops which will be formed
when you when you put 1 links at a time and
then write down the KVL equation there.
So, this method of using co-tree 3 concepts
to analyze the circuit will be basically the
loop analysis method.
Loop analysis method, loop analysis you in
terms of suppose I did not know about graph
theory I know what does it mean.
I will take several different different loops
write down the KVL equations and try to solve
the circuit.
In fact Maze analysis is a form of loop analysis
only.
Ok so 4 5 6 are the links so there are 3 links
which I will call L4 L5 and L6.
Now suppose I say that I want to write down
if L4 was in place here then what will be
the KVL equation in this loop.
KVL equation in this loop so write KVL, KVL
equations for A matrix we do not KCl equation
at the nodes for in this case I will write
KVL equations thinking I am putting one link
at a time.
For example 4 2 and 1 will be a closed loop
forget about these 3 business come here in
the actual circuit.
This will be 1 loop, now in this loop the
some of the here I am not showing it is true
for all the branches.
This is the direction of the current I am
not writing it is ie4 we have already seen
that all the numbers writing is sufficient
to understand this.
So, when I say ve 4 it is with this voltage
across this element you know these things
I will not write henceforth it is understood.
Now in in this loop which is formed by this
link L 4 and these 2 edges this one in this
circuit it will be KVL equation will be e
4 this you follow me carefully e 4 give a
positive sign when you are putting along the
arrow that is the like we will be 4 so e 4
plus then -e2 then -e1 if you traverse this
loop.
So the positive sign to e4 whatever direction
is there e4 then e4 if you have assigned positive
voltage then e2 is in opposition, so it will
be -e2 -v1 and this must be equal to 0.
KVL equation formed by link 4 and these 2
trees.
Similarly I will write down KVL equation where
L5 link 5 will be involved that is this if
I put link 5 this is the loop this is the
loop and what will be the KVL equation in
this loop start from this first go along the
link here that is e5 what about t minus this
must be also plus because it is dc this direction
you have a sign plus.
So, plus plus then also plus plus e3 is equal
to 0 KVL equation is to be satisfied and finally
in loop in third loop which I have called
L6 that is this one we what is that loop now
here you be careful what I mean this loop
which will be formed by this link alone will
be like this start from a L6 then these things
this that is the; in the loop that you are
writing KVL there should be only one link
present, why?
Because while writing down this equation KVL
equation I took only one link L4 and other
tools are the tweaks.
So, one link to 2 tweaks formed loop one L5
in this 1 link and 2 tweaks 2 and 3 this is
the second loop and finally the loop which
will be formed will have only 1 link so this
6 I will take then I cannot form this loop
like this okay we will be satisfied but I
will always follow this rule I will take one
link at a time to form a loop.
So, I go by this then this which is not a
link come down here not a link this is also
not a link this is tweak.
And you are correct loop formation these 3
loops will be independent groups as you can
see so KVL equation in L6 I will write, how
I am writing it this one going in this direction
it will be e6 that I will assign positive
sign.
Then e3 will come do not take e5, e3 so – e3
- e2 because you opposite to that direction
of 6.
Then from this to this –e1 and this must
be equal to 0.
So, this is the KVL equation in 3 well-defined
loops.
How this looks have been formed I have got
the ideas of trees I know the links then I
will take one link at a time form a loop like
this that is the whole idea.
If necessary I will repeat once again but
let me proceed.
Now once I get this these three equations
can be written in a matrix form nicely, what
is that?
That is in matrix laws this equation will
be written like this there will be all these
6 voltages of the elements will parties may
participate in this equations.
So here it will be 1 2 3 4 5 and 6 here I
will write v1 v2 v3 element voltages v4 v5
and v6.
And here it is loop for 3 loops are there
L4 does not L6 does not mean I am considering
6 loops I have just identified name the loops
in terms of the link number that is all.
So, it is 1 loop L4 formed by link 4 another
link 4 5 6, so I will now put so this is outside
this matrix it is for correctly writing down
correctly and quickly writing down the KVL
equations.
So for example in loop for it is e4 is present.
So, +1 -e2 -e1 -e2 -e1 other entries are 0
and this will be equal to 0 that is all in
loop in this loop number 5 it will be 2 3
and 5 involved so 2 3 and 5 all other entries
are 0.
And in loop number 6 this is loop number 6
mine do in loop number six it is a 6 is present
so +1 -e3 -e3 -1 and that – e2 which is
also -1 and -e1 hopefully I have written correctly
this is the thing.
And this all this row has to be 0.
So the KVL equations in these three fundamental
loops they are called once you have chosen
a tree then the links you know and involving
links one link only write down the KVL equation
in the original network.
Because we have identified the links so I
will call this is the fundamental loop.
This I will L6 only one link should be present
then I will say this is loop 5 and similarly
loop 6 and so on.
So, this is the equation okay - 1 - 1 0 1
0 0 0 1 1 1 like that here also the entries
are only +1 and -1, now this equation this
matrix is called a matrix B and which we call
it as a cut set matrix and this is the element
column voltages representing voltages across
each element and this is equal to 0.
We got similar things in incidence matrix
a into ie is equal to 0 castle and b into
vb equal to 0 where from we got it from KVL,
KVL in which loops I wrote the loops which
is formed by at least only one only one one
and only one link and links have been identified
after drawing the tree of these graph that
is the thing so this is b matrix understood.
Now after you get these KVL equations we will
try to see how the link currents will be related
with element currents.
The next question I asked because it is loop
analysis I will be doing essentially by drawing
a tree etc at the end we will show that but
in loop analysis what you need you have to
write down the KVL equations.
And you try to solve for the loop currents
is not and if loop currents are known then
all the branch currents can be calculated
may be some of the loop currents difference
of the loop currents and so on.
So, next thing is how to express element currents
in terms of loop currents that will be the
study.
So, the question is so these diagrams I will
be once again reading so I will copy this
diagram.
It is like this 
so I copied go to next page and paste it.
(Refer Slide Time: 17:13)
So, this diagram is very much needed, now
only thing it was not copied is a we see other
things so this is the tree or 3 etcetera.
Now what I am trying to do is this expressing
this step is most important and you must understand
this expressing element currents element currents
in terms of loop currents but the point now
where is the loop currents loop currencies
one loop I as I told you will be like this
involving this link that is this link if you
put only a loop will be formed so this is
one loop current.
This is iL4 you know this will be the loop
current what is the next loop current so in
this diagram if I show it will be like this
let me use different color so this is iL4
in this loop iL4 where will be iL5 only this
link will be this one so this will be this
iL5 this will be this now what is this loop
current formed by this link L6 it will be
like this I will use a different color so
iL6 will go this way this way this way come
back here iL6 this you must understand it
very clearly.
So loops has been so here also if I show it
is iL5 I have assumed it to be this iL5 and
iL6 is this one, this is iL6 then you can
easily see how to write down the element currents
in terms of loop currents.
So, how do I write it down element currents
ie1 which one is one branch ie1 is a branch
1 what will be there are 2 loop currents flowing
here okay, first let me okay i1, i1 in terms
of i1 direction is this way you know.
So, there are two loop currents involved in
deciding this magnitude this arrow let me
put very clearly.
This arrow will be dictated by the arrow of
the links so this is how I have put it this
arrow of iL4 was dictated by the arrow of
this branch ab understood.
Similarly this arrow was dictated by this
link so this is how it was put.
Therefore ie1 as you can see will be -iL4
-iL6 that is all ie1 will be -il 4 -iL6, ie2
where is e 2, e 2 is here ie2 in ie2 if you
see what will be the loop currents involved
its direction is like this.
What are the loop currents that are present
in this branch 2 or H 2 or element 2 it is
iL5 this is the loop current.
So, iL5 it will be plus because this way it
goes then also iL4 and iL6 in the opposite
direction so - iL4 - iL6 this will be element
currents too in, so this this is the this
thing then ie3 in terms of loop current ie3
is this way and here if these are the loop
currents then ie3 in terms of that loop current
will be IL5 + no, iL5 this is iL5 it is a
loop current iL5, ie5 I am writing which is
this way so iL5 and then - iL6 this is 6 IL
6 - iL6 this will be that is all only these
2 loop currents are present in this; to decide
the branch current here.
Ie4 means this branch current, in this branch
current iL6 is not involved only iL4 is involved
neither 6 not the other branch currents 5
is involved only iL4 is iL 4 that is all then
got the point this current is only IL 4 then
I if I will be equal to that is the current
in the element 5 all the iL5 is involved no
other loop currents are going this way.
So, this will be equal to iL5 and finally
ie6 will be equal to iL6.
So, once again repeating very quickly so in
to find out the element current in one this
current i1 what I do is I see what are the
loop currents that are crossing this present
in this branch as well.
I find 2 loop currents are present iL4 and
iL4 but they are in opposite direction of
ie1 so, -iL4 iL6, ie2 is this current whose
direction already assumed this way which are
the loop currents that will decide ie2 all
the 3 are there vertically.
So, this is iL5 is flowing like this now it
will complete circular.
So, here all the three loop currents will
be there ie2 of which iL5 will be in the same
direction as ie2 therefore it will be iL5
-iL4 -iL6.
Similarly ie3 you know this branch current
here only 2 loops are crossing ie 3 that is
iL5 this is the arrow iL5.
And -iL6 that is all and ie4 i4 i5 i6 there
happens to be such that only the respective
loop currents are only one loop currents are
involved so this is the thing.
Now this can be written in the form of a matrix
what is that matrix?
This matrix will be because I want to express
the element currents in terms of loop currents.
So, it will be if you write down it in terms
of a matrix it will be a given ie1 ie2 ie3
4 5 and 6 these element currents.
It will be equal to a matrix 
and this will be decided by the loop currents
all the 3 loop currents are there they are
iLl4 iL5 and iL6 once again telling you iL4
iL5 iL3 that iL1 also exists I just named
them depending upon the number of the number
that has been associated with that particular
links only 3 equations mind you.
And this thing here I will write iL4 also
iL5 iL6 this is for my own benefit quickly
and correctly I will write ie1 is -iL4 so
-1 here - iL6 -1 here iL5 no contribution
ie2 iL5 is +1 and 4 and 6 -1 -1 -1 -1 then
ie3 L5 and L6, L5 + 1, L6 is -1 and this is
0, ie 4 is only iL4, ie5 only iL5 and finally
ie6 is only this one.
So let me put it in the proper perspective.
So, this is the matrix what will be the size
of the matrix?
The size of the matrix will be the number
of elements which are present that is number
of rows is number of elements so e times how
many loops fundamental loops will be present
it is equal to the number of links which are
present and number of links.
And we have seen that number of links is equal
to total number of elements minus the number
of tweaks this is the thing.
So, this is it will be like that and these
of course will be into 6.
Now I have already defined the B matrix you
see here this was the B matrix okay is there
any space for copying these b matrix let me
copy that in the previous page.
So, that this is the nature of the b matrix
copy it go to next page and paste it.
So, let me put the B matrix here okay to get
the idea let me superimpose out the statement.
Now you see this was the B matrix got earlier
now look at the elements here the first row
was -1 -1 0 1 0 0 the first column is -1 -1
0 1 0 0.
Second row was 0 1 1 0 1 0 0 1 1 1 0 1 0 third
row -1 -1 -1 and then 0 0 1.
So, so this I then can write in terms of matrix
it is no new matrix.
The element currents ie is nothing but b transpose
into the loop currents what is ie?
ie is this column vector comprising of all
the element current information what is iL?
IL is this loop current once again a column
vector and what is B transpose, b transpose
is that earlier B matrix which I wrote here
which here we got from the KVL equations.
Therefore in case of loop analysis we have
got these 2 fundamental equation that I am
rewriting here one is B into the element voltages
is equal to 0 that I got in the last page
that is the B into Ve is equal to 0 and this
is 1 and the second one is B transpose into
the loop currents is equal to the element
voltages.
So, these two are the equations which we will
be using if we want to solve the network by
by loop analysis method.
Therefore please go through this 3, last 3
lecture notes carefully hopefully you will
understand and then I will tell you in the
next class there is another way of solving
the network which is based on the cut-set
method, that I will discuss in the next class.
thank you.
