Hi friends in this video we are going to
see the last theorem in our DC circuit
module and that is maximum power transfer
theorem so lets say the statement of
it first it states that the maximum
power from a source can be transferred
when load resistance is made equal to
the internal resistance of the source so
lets see what it is talking about so
first of all its very good if I know
what is the maximum power that I can be
getting from a voltage source so for that
some condition need to be satisfied and
that condition here they are stating as
load resistance should be made equal to
internal resistance or you can say
source resistance so lets see the
condition so what we are going to do
over here we are going to see the
derivation of maximum power transfer
so lets consider a simple practical
voltage source whose open end voltage is
V and a small series internal resistance
that is R so this is a voltage source
so it's a practical voltage source so
for this voltage source I am connecting
a load resistance RL and it is a
variable one that means I can change its
value
so I will write all the notation once
again V is the open end voltage R is
internal resistance of voltage source
and RL is the load resistance so lets
derive condition for maximum power
transfer I want what is the power
transfered from source to load so for
this I should know what is the current
taken by this load so first of all I
will write the expression of I N so IL
will be equal to voltage V divided by
total resistance in the circuit so it is
V upon R plus RL now power transfered
to load resistance I will use a notation
PL simply given by IL square multiplied
by RL so if I substitute the value of IL
I will get PL as V square divided by R plus
RL the whole square multiplied by RL
I will consider this as equation number
one now what I want  this power should
be maximum so I want to maximize this
function but what is the variable over
here by virtue of which I can get this
as a maximum so for this we need to
check the circuit once again in this
circuit voltage V is fixed its internal
resistance R is fixed only variable is
this load resistance so actually I want
to get a power maximum by varying RL so
this will give me a hint of maxima
minima so what I will do here I will
write the expression for getting maximum
power so to get maximum power transfer
I will differentiate equation number 1
with respect to RL because RL is a
variable so I will get D PL by D RL as
differentiation with respect to RL V
square multiplied by RL divided by small
R plus RL the whole square we know the
concept of maximum minima for a maximum
power transfer this derivative should be
equal to zero means this derivative
should be zero so lets differentiate
this V square is a constant I will take
it out and here it is uyv rule of
derivative which say V multiplied by
derivative of u minus u derivative of V
divided by v square so it is R plus RL
whole raise to four equal to zero I will
take this term to this side it becomes
equal to zero since R plus RL the whole
square derivative of RL with respect to
RL is 1 now derivative of R plus RL the
whole square is 2 into R plus RL into
derivative of R plus RL equal to zero so
what I am getting over here now R plus
RL the whole square minus 2 times RL
into R plus RL and derivative of this
term with respect to RL is 1 derivative
of R is 0 because its a constant and
derivative of RL with respect to RL is 1
so from this expression I can take R
plus RL common so I will get R plus RL
minus 2 times RL equal to 0 again I will
take this to other side becomes equal to
0 and finally I will get R minus RL
equal to 0 which ultimately implies load
resistance should be equal to 
internal resistance of the battery or
you can say voltage source so here I
will write load resistance equal to
internal resistance of a voltage source
so I am getting a condition which we
called as condition for maximum power
transfer
now what will be the value of this
maximum power for that we need to
substitute R equal to RL in equation of
PL so equation of PL is like this V
square divided by R plus RL the whole
square into RL so if I substitute RL
equal to R I can get equation as V
square multiplied by R divided by R plus
R the whole square if I simplify I will
get V square multiplied by R upon 2 R
square so it is V square into R
multiplied by 4 R square 1 R get
cancelled so I will get power transfer
to a load from a source is V square
divided by 4 R so every source has its
capacity to transfer a maximum power and
that is given by its open and voltage
square divided by 4 into its internal
resistance now how we are going to use
this while solving a problem so for that
lets go back to a Thevenin theorem in
Thevenin's theorem we will draw Thevenin
equivalent circuit which is having a
voltage named as V TH and a resistance
in series with V TH that we named as R TH
now these are the terminals A and B
where low resistance can be connected if
I compare this with this circuit I will
come to a point where I can say this V TH
R TH is nothing but a voltage source
so I will consider this part as a
practical voltage source where V TH is
open end voltage and R TH will be
internal resistance of voltage source so
now I can say what should be the
resistance connected between terminals
A and B which I can say load resistance
so that maximum power will be
transferred through it so for that the
condition is  load resistance should
be equal to Thevenin equivalent
resistance which is a condition for
maximum power transfer and what will be
that power is nothing but V TH square
upon 4 R TH this is expression of
maximum power transfer so what we have
done basically maximum power transfer
theorem is just a application of
Thevenin theorem where you have to get
the RL which is nothing but R TH to get
maximum power transfer through it and
the expression of that power will be VT
square divided by 4 R TH so it is just a
extension of Thevenin theorem so
whenever we will solve a problem at that
time we will come to know the procedure
is absolutely same as Thevenin theorem
only added feature is this expression so
in subsequent videos we will solve
numerical based on this thank you
