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ALL ABOUT ELECTRONICS.
So, in this video, we will learn phasors and
the phasor diagram.
So, in the very simple way, if I say, this
phasor is nothing but the simple way of representing
a sinusoidal signal.
So, now when we have more than one such sinusoidal
signal, which is having the same frequency
but the different phase and different amplitude
then, we can use this phasor diagram to represent
the phase difference between the sinusoidal
signal.
So, now we know that any sinusoidal signal,
in general form can be expressed by this expression,
that is vm sin(wt + φ)
Where Vm is peak amplitude of this sinusoidal
signal, w represents angular frequency and
φ represents the initial phase of this sinusoidal
signal.
So, now let's represents this sinusoidal signal
using this phasor.
So, now this phone is nothing but vector which
rotates around its origin at the constant
speed of w rad /sec in the anti-clockwise
direction.
And here the length of this vector represents
the maximum amplitude of this sinusoidal signal.
And it's angular velocity represents the angular
frequency of this sinusoidal signal.
So, now if we take the projection of this
rotating vector on the Y-axis then we will
get the instantaneous value of this sinusoidal
signal.
So, now as you can see here, as the vector
rotates in the anti-clock wise direction at
speed of w rad/sec and if we take the projection
of this vector on Y- axis then we will get
the instantaneous value of this sinusoidal
signal.
And if we plot all the values which are possible
for this phasor then we can reproduce this
sinusoidal signal.
So, now let's see how we can represent this
phasor on phasor diagram.
so, now on this phasor diagram different AC
sinusoidal signals which are having the same
frequency but different amplitude and different
phase are represented.
So, while representing these phasors on this
phasor diagram, we only represent them by
their amplitude and initial phase.
So, let's say we have two sinusoidal signals.
The first is, let's say 5 Sin (wt +30)
And the second signal is v2= 10 sin (wt +80)
And we want to represent these two signals
in terms of the phasor on phasor diagram.
So, the first signal is 5 Sin (wt +30). So,
first, we will draw the vector which is having
the amplitude of 5. And it will make an angle
of 30 degrees with the horizontal axis or
reference axis.
Now, if you see the second signal that is
10 Sin (wt +80)
So, here peak amplitude of the signal is 10.
So, we will draw vector which is having the
magnitude of 10 and it will make an angle
of 80 degree with the horizontal axis.
Now, sometimes, this phasor is also represented
by the RMS value of the sinusoidal signal.
The reason is that the most of the measurement
which is being carried out by the Ammeter
and Voltmeter are in terms of the RMS values.
So, sometimes you will find that these phasors
are represented by the RMS values instead
of the peak value.
But in this video, we will consider the peak
value of this sinusoidal signals.
So, now let's see one more example based on
this phasor diagram representation.
So, let's say now we have voltage that is
equal to
10 sin (wt - 30) and the second voltage or
second signal is 5 sin (wt -120)
And we want to represent those signals on
the phasor diagram.
So, the first signal is 10 sin (wt - 30).
So, we will draw a vector which is having
an amplitude of 10 and it will make an angle
of -30 degree with the horizontal axis.
Now, second we will draw the second signal
which is having an amplitude of 5 and it will
make an angle of -120 degree with this horizontal
axis.
So, we can draw a vector of 5 V amplitude
which is having a phase shift of -120 degree.
So, this will be the representation of V2
and this will be the representation of V1.
Now, on this phasor diagram, this anti-clockwise
direction is known as the leading direction
and this clock wise direction is known as
the lagging direction.
So, as you can see here this V1 is leading
the V2 by 120 - 30 that is by 90 degree.
So, here this V1 signal is leading the V2
by 90 degree.
Similarly, in the first case if you see this
V2 signal is leading the V1 signal by 80 - 30
that is by 50 degree.
So, now so far we have seen that how to represent
phasor on phasor diagram.
Now, let's see how we can represent this phasor
in mathematical form.
So, the first form is the polar form, where
we can represent this phasor in terms of r
and θ.
So, suppose we have phasor which is having
amplitude of Vm and it makes an angle of Φ
with the horizontal axis, then in polar form,
we can represent it as Vm <Φ.
So, the next form is the rectangular form.
So, in this form, we can represent any phasor
by the complex number.
that is A + jB. So, the amplitude of the phasor
will be sqrt (A^2 + B^2)
and the phase of this phasor can be given
by the expression tan ^(-1) [B/A]
So, this will be the rectangular form of representation
of the phasor.
Now, the third form of representation is the
exponential form, where we can represent the
phasor in terms of
Vm e^(jΦ)
So, these are the basic three representations
by which we can represent this phasor.
So, now as we have seen how to represent any
phasor in phasor diagram as well as by mathematical
expression, now let's see phasor relationship
between the voltage and current for the basic
circuit elements like resistor, capacitor
and the inductor.
So, first, we will see that the phasor relationship
between the voltage and current for the resistor.
Now, let's say we have applied this sinusoidal
input to this resistor.
So, this voltage signal can be represented
in the phasor form as Vm <Φ
Where Vm is the peak amplitude and the Φ
is an intial phase.
Or simply we can say that this is nothing
but V
Now, according to the Ohm's Law, the current
i(t) that is flowing through this resistor
will be Vm sin (wt +Φ)/R
And if we write this equation in terms of
the phasor form then we can write it as
I = V/R
Where I and V both are in phasor form.
That means, V is nothing but Vm<Φ
So, now as you can see here, both voltage
and current are in phase.
So, if we represent this voltage and current
on phasor diagram then it will look like this.
So, as you can see here, both voltage and
current are in phase.
Now, here the ratio of this voltage over current
is also known as the impedance.
So, for the purely resistive circuit, the
impedance is equal to the resistance.
Similarly, now let's see the phasor relationship
between the voltage and current in the case
of an inductor.
So,, now let's here assume that the current
that is flowing through the inductor is given
by this expression.
That is Im sin(wt +Φ)
Or in terms of the phasor relationship, we
can write it as
Im<Φ or simply I
Now, we know that the inductor voltage can
be given by the expression L*di/dt or we can
write it as
wL*Im cos (wt +Φ)
or we can write it as wL*Im*sin(wt +Φ+90)
So, this will be the expression of the voltage
across the inductor.
So, as you can see here this voltage leads
the current by 90 degree.
So, in terms of the phasor relationship if
we see then the phasor diagram will look like
this.
So, here is the current that is flowing through
this inductor and voltage will lead the current
by 90 degree.
Now, suppose if we represent this signal in
terms of the phasor then we can write it as
wL*e^(jπ/2)*Im<Φ
And we know that e^(jπ/2) is nothing but
j.
So, we can write it as (jwL)I
Where I is nothing but Im<Φ
So, this will be the expression for the voltage
across the inductor in terms of the phasor
form.
So, now we know that the ratio of voltage
over current is nothing but impedance.
So, here we have seen that this voltage can
be given by the expression (jwL)I and here
both V and I are the voltage and across this
inductor in terms of the phasor form.
So, we will get impedance Z as jwL
So, we can say that for the purely inductive
circuit, the impedance or simply reactance
of this inductor will be jwL
So, now similarly let's see the phasor relationship
for the case of the capacitor.
So, here let's once again assume that capacitor
voltage that is applied to this capacitor
is vm*sin(wt +Φ)
And we know that the capacitor current ic(t)
can be given by the expression C*dV/dt
That is C*w*Vm cos(wt +Φ)
That is nothing but wC*Vm* sin (wt +Φ +90)
So, this will be the expression for the current
that is flowing through this capacitor.
And as you can see here the capacitor current
leads the voltage signal by 90 degree.
So, in terms of the phasor representation,
if you see, it will look like this.
So, if this phasor represents the voltage
then the current will lead the voltage by
90 degree.
Now, suppose if we represent this signal in
terms of the phasor form then we can write
it as I= wC*e^(jπ/2)*Vm<Φ
Or we can say that jwC*V
Where V is nothing but Vm<Φ
So, this expression is nothing but the current
that is flowing through this capacitor in
the phasor form.
And as we have seen earlier, the ratio of
Voltage and current is nothing but impedance.
And here as you can see it is nothing but
1/(jwC)
So, we can say that for the purely capacitive
circuit, the impedance or simply reactance
of the circuit will be 1/(jwC)
So, in a summary, we can say that for the
resistive circuit the current and voltage
will be in phase.
while, in case of the purely inductive circuit,
the voltage will lead the current by 90 degree.
And in case of the purely capacitive circuit,
the current will lead the voltage by 90 degree.
So, now suppose if we want to remember this
voltage and current relationship across this
resistor, capacitor, and inductor then we
can remember this just by using this simple
keyword.
That is CIVIL.
Now, if you see this word, if you see the
first three letters, that C, I and V.
So, we can say that in the capacitor the current
leads the voltage by 90 degree.
And if you see the last three letters, that
is V, I and L, then we can say that in inductor
the voltage leads the current by 90 degree.
So, in this way we can remember, in the capacitor,
the current leads the voltage and in case
of an inductor, the voltage leads the current
by 90 degree.
So, now in the next video, we will see that
how to draw the phasor diagram for the RL,
RC, as well as the RLC circuit and we, will
also solve some problems related to this phasors.
So, I hope in this video you understood about
the phasor and the phasor diagram and how
to draw the phasor diagram for the basic circuit
elements like resistor, capacitor, and inductor
