hi friends in this video we are going to
see the most important parameter of AC
voltage waveform and that is RMS or
effective value lets see the
definition of it the effective or RMS
value of an alternating quantity is
given by that steady current which when
flowing through a given circuit for a
given time produces the same amount of
heat as produced by the alternating
current which when flowing through the
same circuit for the same time meaning
suppose I perform my experiment where I
connect a DC supply to a lamp and I will
measure a power by passing a DC current
for some amount of time now instead of a
DC I will replace AC and I will check
the same effect for same amount of time
the moment I will get a same effect
that value of AC I noted down which is
nothing but RMS value so in short I can
say RMS value of AC quantity is nothing
but its DC equivalent meaning if I
replace the RMS value of AC with the
same amount of DC I will get same effect
so lets move to the next point the
next point is I will elaborate the same
concept with example so what I have over
here I have a AC supply given by the
socket and I will measure the effect
suppose it is giving a 240 volt AC 240
volt AC is doing the same amount of work
as if 240 volt DC so this particular
example will clear the idea of RMS so
what is the advantage of RMS value
everywhere AC is denoted by its RMS
value because RMS value of AC is nothing
but a useful component of AC voltage or
AC current which is responsible
for production of actual power or
responsible for doing a work done so
everywhere whenever we see a voltage
that is always a rms value just take an
example whenever we have a domestic
household supply that is single-phase
230 volt AC 50 Hertz so this 230 volt is
nothing but RMS value of AC supply now
effective value can be calculated by two
ways one is a graphical and second is a
analytical way so let us say a graphical
method so in graphical method I have
considered only a half cycle right so
its RMS is nothing but a root mean
square so what we have to do we have to
take a mean of square of every
instantaneous values and then we have to
take the square root of it so what I
have done only half wave I have
considered only half cycle I have consider
so same effect can be true for a full
cycle so in half cycle I have considered
this 12 instance so this 12 instance
says giving you to an instantaneous
values of voltages v1 v2 v3 like that so
as per the definition it is a root of
mean of square of instantaneous values
so that will give you VRMS equal to
root of V1 square plus V2 square plus
V3 square like that till with 12 square
divided by 12 because I have considered
12 values in general if I have n number
of values I will get RMS voltage at V
RMS equal to root of V1 square plus V2
square like that till VN square divided
by n for a current if I replace V by I 
 I will get I RMS as root of I1 square
plus I2 square plus I2 square up to In
 square divided by N
so in a graphical way what we do a
waveform be splitted into number of
instances and for every instant we get
the instantaneous value square it
likewise we take addition of all the
squaring of all the instantaneous values
divided by number of instances we are
considered and then ultimately we take a
root we will get a rms value by a
graphical method suppose I want to find
out rms value by another technique which
is analytical method see how we are
going to do that so the current is given
by I equal to IM sine theta a standard
AC current waveform then let's square it
so I get I square equal to IM square
sine square theta so in now this figure
I have shown the Im sine theta like
this and squaring of it will be like
this so what we are doing we are
considering the area under this curve
which is nothing but a integral 0 to pi I
Square D theta and length of this is
nothing but pi because we are just
considering half cycle so if I solve
further what I will get average value of
square of the current over half cycle is
given by area of curve over half cycle
divided by length of curve for half
cycle so it gives me integral 0 to pi I
Square D theta divided by length is PI
so that is ultimately 1 over PI I can
take it out 0 to PI I Square D theta
then 1 upon PI 0 to PI is a limits for
integral I we know its IM  sine theta
and squaring is IM  square sine square
theta D theta we know sine square theta
is given by 1 minus cos 2 theta divided
by 2 so I have replaced that constant I
will take out of integration so I will
get this expression IM  square divided
by PI integral exist from 0 to PI
I 1 minus cos 2 theta divided by 2 into
D theta integral of 1 is theta if I take
a 1 by 2 common so ultimately I will get
1 minus cos 2 theta inside an integral
integral of 1 is Theta integral of cos 2
theta is sine 2 theta divided by 2 if I
apply the limits ultimately I will get IM
 square divided by 2 but that is
nothing but a average value of square
of the current what we want root mean
square so RMS value of current is
nothing but the root of this value so I
RMS equal to root of
IM square divided by 2 if I further
simplify I will get IRMS as IM
divided by root 2 so in a voltage case I
will get VRMS as VM by root 2 and we
know 1 by root 2 as 0.707 so finally I
can say IRMS is 0.707
IM VRMS equal to 0.707
 VM RMS is very important
parameter for AC because all the meters
voltage are volt meter for a voltage
measurement emitter for current
measurement are designed in RMS values
and whenever I see any AC voltage or any
AC current I am talking about RMS
voltage or RMS current only thank you
