Phasor diagram Series LCR circuit (Part - 16) - Alternating current, Physics, Class 12 Video Lecture

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ahead with path 16 now when I look at
the phasor diagram for series NCR
circuit so this would be interesting
this will tell us that how the concept
of phasor diagrams turned out to be very
useful
I mean how what role actually phasor
diagram please I mean what was the need
of phasor diagram when we already had
graphs to represent the voltage current
relationships so let us try to phasor
diagram for series LCR circuit so the
points which we already know about the
LCR circuit is resistor capacitor and
inductor they are all in series the
applied voltage V is given as VM sine
Omega T the current flowing through the
circuit I is given as I M sine Omega T
plus Phi right so now here since we have
all the circuit elements so what are the
voltages involved here there is a
voltage across the inductor there is a
voltage across the resistor there is a
voltage across the capacitor and there
is a source voltage right so if I
represent it like this it is suppose
this is my n see our circuit this is my
resistor this is my inductor and this is
my capacitor so this is my source
voltage this is the voltage across the
resistor this is the voltage across the
inductor and this is the voltage across
the capacitor right so now let me try to
plot the phasor so first of all now
before that I would say that what does
this V s this VL is nothing but I M into
X and that is I I write similarly this
is nothing but I M into I this VC is
nothing but I M in the XC right and the
corresponding value for this V is
nothing but V M so these are the peak
values for each of them right the peak
value for the voltage across the
inductor would be equal to I into R that
is I into the inductive reactance
similarly for resistor for capacitor and
the peak value for the source voltages
VM that is the voltage amplitude right
so now let us try to draw them phasor
diagram so let us suppose that my source
voltage is represented by this phaser
so if this is my source voltage V that
means what would be where good I
represent my voltage of req across
resistor the voltage across resistor
would also be represented along this
direction because in case of resistor
the voltage and current are in phase so
if this is my current my voltage would
also be along this direction right
because now remember the resistor
circuit in case of resistor V and I are
in phase so they are in phase means V
and I will overlap with each other right
now what would happen in case of V L
that is inductor we know that in case of
inductor what happens with the voltage
in case of inductor the current lags
behind by PI by 2 that means I behind V
so that means current should be behind
voltage right so if current is here
voltage should be ahead of current by PI
by 2 so PI by 2 ahead of current so this
should be the voltage or inductor now
what where should the voltage across
faceted be represented in that is the
voltage should be behind current because
we studied all these things right we
studied that in case of an inductor the
phase difference is PI by 2 and current
lags behind voltage by PI by 2 so that
is what we represent in here
similarly for capacitive circuit voltage
lags behind current by PI by 2 so this
is my current so the voltage should be
somewhere here so that is this should be
PI by 2 so now if you look at this
phasor diagram you see that V L and VC
ad II opposite in direction their linear
and exactly opposite in direction now
the length of the phasers I mean C you
can see that some phaser is small some
phaser is same that is because the
length of the phaser represents these
values that is the peak value right I
told you that the length of phaser
represents the peak value so here V R
represents I M R similarly V L
represents I M into X L similarly VC
represents I M into X C so the length of
phasers length of phasers we present
these values that is why you can see
that VC is comparatively larger than V L
that's because the value of I and X C is
comparatively larger than I M X L
that's because here X L would represent
Omega L and here it would be divided by
Omega C right so for the same amount of
frequency so if the frequency for the
same frequency this X this VC comes out
to be larger when compared to your X
VC comes out to be larger when compared
to your V n right so in this case we see
that V L and VC are exactly opposite to
the each other and they are along the
same line right now
you know that is the part of the phasor
diagram now if you look at the circuit
diagram what do you see you see that the
voltage across the resistance plus the
voltage across inductor plus the voltage
across capacitor is equal to V that is
the source voltage right that is what we
can see from the circuit now with the
help of this we will try to simplify the
phasor diagram so how do we do that
now instead of displaying VL and VC
opposite to each other we can substitute
it like this let us suppose this is I
this is V R this is VC now since VC is
greater than VL as far as the magnitude
is concerned therefore VC will dominate
so we can say that this length is
nothing but VC minus VL because we are
reversing VL from this direction to this
direction so instead of showing VL and
VC in two opposite directions along the
same line I can display it as VC minus
VL along one direction so now this is V
R and this is VC minus VL that is 90
degree so the resultant can be
calculated from this triangle and this
would be the resultant so the line
joining these two would be the resulting
right so here we can say that this is a
right angle triangle so we get a right
angle triangle we obtain a right
triangle with the help of phasers whose
hypotenuse is V that is the source
voltage so the hypotenuse of this right
triangle is the source voltage so now
from Pythagoras theorem so using
Pythagoras theorem we can write that V R
square plus VC minus V L whole square
is equal to V M squared because here the
length of the resultant represents the
peak value of the source voltage so peak
value of the source voltages V M right
so we can write V M square is equal to
what is V ad V R is nothing but I M into
R so I M R whole square plus what is VC
VC is I M into X C minus I M into X L
whole square so this comes out to be I M
square R square plus I M square X C
minus X L whole square so this comes out
to be I M square M 2 R square plus XE
minus X and whole square so this is V M
square so this can be written as V M is
equal to I M and root over R square plus
XC minus X L whole square so if you look
at this equation and compare it with the
basic resistor equation V is equal to I
R so in this circuit you can see that if
this is the voltage this is the current
then this term acts as resistance so
this entire term acts as resistance so
from here we obtain that the impedance
of the circuit is equal to root over R
square plus XC minus X n whole square so
now you understood the importance of
phasor diagram also with the help of
phasor diagram how easily we could
obtain the relationship between the
voltages across all the different
circuit elements because instead of
phasor diagram if you would have plotted
it in terms of graph it would have been
really it would have looked really
complicated and confusing because you
will have some 3/4 waves moving and then
it would have been difficult to find out
which one is in phase and which one is
out of phase so here you understood so I
will quickly review it once what we did
here is we first
to do a vector or the first do of things
are for currently then we do the phasers
for voltage across resistor voltage
across inductor and voltage across
capacitor the voltage across a resistor
is in phase with the current so it
overlapped with current voltage across
inductor is ahead of current so it is
ahead of current by 90 degrees voltage
across capacitor is behind current by 90
degree so it is plotted behind current
by 90 degree so this is what we obtained
initially then we saw that since VL and
BC are opposite in direction and they
are along the same line so what we did
we took both of them along one direction
so we took VL from this direction to the
opposite direction so made it minus L so
this vector now became VC minus VL and
this vector was already VI and they were
at 90 degrees to each other so we joined
the other ends of the vectors and
obtained a right angle triangle such
that the hypotenuse became the source
voltage that is V like so using
Pythagoras theorem we calculated the
value of the result source voltage that
is VM and we found that VM came out to
be I M into a value which acted as the
resistance for this LCR circuit and that
value is known as the impedance of the
circuit and is given as root over R
square plus XC minus XL whole square
right so now let us look at so that
means we can say that we can represent
the phasor diagram for an LCR circuit
somewhat like this let us suppose if
this is my BF this is just a different
representation of the same thing I am
just showing it so that you do not get
confused if I say that current is in
this direction it is not necessary that
I always take current along this
direction so if current is along this
direction V R will be along the same
direction as current VL will be ahead of
current by 90
Vesey will be behind current by 90
degree and the resultant of all this
would be this one that is how do we get
this one now CC VL and BC we can take it
in one direction and then we can say
that this is the resultant of these two
or this is another way of representation
that we have taken the current along
this direction this is V R this is VL
and this is VC so we have taken VL + VC
along this direction so the resultant V
is along this direction right so I hope
that the concept is clear to you now ok
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