Hi friends in this video we are going to
apply a technique called as mesh
analysis to solve a circuit having more
than two loops that means we can have a
problem of three loops and based on the
mesh currents we are going to calculate
all the branch currents in a circuit so
lets take a numerical
so two batteries are given off 7
volt and 6 volt this value of
resistance is 1 ohm this is 2 3 ohm 2
ohm once again and this is one ohm the
objective is to find out all loop
currents and hence all branch currents
that mean current flowing through this 1
ohm 2 ohm 3 ohm 2 ohm and 1 ohm using
mesh analysis so here I am having 3
loops 1 2 & 3 so quickly I will mark
loop currents I 1 I 2 I 3 based on the
directions of I 1 I 2 I 3 I will mark all
voltage that got developed across
resistances so because of this I 1 the
drops are like this plus minus for 1 ohm
and this plus minus for 2 Ohm
for this I 2 the drops are plus minus
plus minus and plus minus like this for
this third loop because of I 3 in the
direction of I 3 I am having drops like
this and already batteries polarities
are mentioned no need to alter that so
lets apply KVL to loop number 1
so loop one I will start from this point
tracing  these two resistances this
battery and this battery and we'll come
to the same point so in this direction I
have plus minus that is minus I 1 minus
plus that is plus I 2 plus minus minus 2
I 1 minus plus plus 2 I 3 and plus minus
its a voltage drop that is minus 6 and
last element minus Plus this is plus 7 equal
to 0 after simplifying this I will get
minus I 1 minus 2 I 1 will become minus
3 I 1 plus I 2 will remain as it is plus
2 I 3 will remain as it is minus 6 plus
7 will become plus 1 equal to 0 I will
take this constant to other side of
equation so finally I will get equation
minus 3 I 1 plus I 2 plus 2 I 3 equal to
minus 1 this is equation number 1
similar way I can apply KVL to loop
number 2 lets go back to the original
circuit so in this circuit I am applying
cable to loop number 2 I am starting
from this point and tracing this part I
will come to that point once again so in
this direction of current I have this
drop this drop and this drop along with
these drops I am having two voltage
Rises also that means minus plus this
voltage rise and minus plus this voltage
is so keeping this in mind lets apply
KVL to loop 
number 2 so I will have minus 2 I 2
minus 3 I 2 plus 3 I 3 minus I 2 plus I 1
equal to 0 since I am not having any
voltage source in a loop so that term
won't be there after simplifying this I
will get I 1 as it is minus 2 minus 3
and minus 1 I 2 will become minus 6 I 2
plus 3 I 3 will remain as it is equal to
0
equation number 2 in similar manner I
can apply KVL to loop number 3 so lets
go back to the original circuit once
again now I am considering this loop and
starting from this point and will come
back to the same point after tracing
this path so number of voltage drops and
Rises are there so let's write that
minus 3 I 3 plus 3 I 2 minus I 3 plus 6
minus 2 I 3 plus 2 I 1 equal to 0 let's
simplify this equation so I will take
first I 1 terms so 2 I 1 then I 2 terms
3 I 2 and I 3 terms are minus 3 minus 1
and minus 2 so it will be minus 6 I3 and
constant is 6 I can take to other side
it will become minus 6 so this is
equation number 3 after solving
equations 1 2 & 3 I will get values of
loop currents as I 1 equal to 3 ampere I
2 equal to 2 ampere and finally I 3
equal to 3 ampere now having these
values of mesh currents I can get all
the branch currents so lets calculate
the values of all branch currents so I
will start with this resistance first so
current flowing through 1 ohm resistance
I am talking about this 1 ohm resistance
so here I will draw a small pictorial
view of this resistance and here two
currents are flowing I 1 in downward
direction and I 2 upward direction now I
got the values of I 1 I 2 and I 3 as 3
ampere 2 ampere and 3 ampere
respectively
so I 1 ohm will become I 1 minus I 2
because different directions of currents
and I 1 is more than I 2 so I can say 3
minus 2 which is 1 ampere and since I am
having I 1 more than I 2 so the
resultant current will have the downward
direction same as the I 1 now I will
calculate the current flowing through
this 2 ohm resistance only I 2 is flowing
through this 2 ohm resistance so
through into 2 ohm resistance I will
have same as I 2 which is equal to 2
ampere and I am getting a positive
answer
so whatever direction of current I have
mentioned are correct so for this 2
ampere it is again a downward because if
I am considering this loop current this
loop current is nothing but a branch
current for this 2 ampere and flowing
downward direction let's calculate
current flow include 3 ohm so for 3 ohm
the resistance is given like this this 3
ohm resistance and Here I am having two
currents one current is flowing like
this which is I 2 and another current is
flowing like this which is I 3 so
current flowing through the 3 ohm
resistance will be subtraction of these
two currents since I 3 is more than I 2
I can say is I 3 minus I 2 which is 3
minus 2 which is 1 ampere and the
resultant current will have the
direction same as I 3 because I 3 is
more than I 2 so the direction will be
like this
let's calculate current flowing through
this 1 ohm resistance so since there is
a conflict between this 1 ohm and 1 ohm
so better I will mention the node A and
B and now I have to get a current
flowing through this 1 ohm resistance
which is connected between points A and
B so I will have the notation like this
it is a 1 ohm resistance connected
between points A and B only I 3 is
flowing through that particular
resistance so this loop current is
nothing but a branch current for this 1
ohm so the answer will be very simple is
equal to I 3 is I 3 is given as 3 ampere
and the current direction will be
downwards similarly let's calculate
current flowing through this 2 ohm
resistance now again there is a conflict
between this 2 ohm and this 2 ohm
so better I will give notation as C and
D so I need to find out current flowing
through the 2 ohm resistance which is
connected between Point C and D so this
2 ohm resistance is subjected to two
currents one current is I 1 flowing
downward and another current is I 3 which
is flowing upward so obviously current
flowing through 2 ohm resistance is I 1
minus I 3 now coincidently I am having I
1 and I 3 both are of the same value and
that is 3 ampere so 3 minus 3 the answer
is 0 ampere so here we have seen how
Mesh analysis   technique can be used to
calculate all loop currents and by
knowing all loop currents how we can get
branch currents in subsequent videos we
will see more numericals based on mesh
analysis thank you
