RMS Voltage Video Lecture | Physics for SSS 3

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going ahead with part 5 so let us start
with adam s voltage so add em s it
stands for root mean square so it means
root mean squared voltage so this three
terms itself defines RMS voltage and it
defines RMS current that means it is
square root of something of what of the
mean of the mean of what of the mean of
squared of each terms so you will
understand it better now so when we talk
of RMS voltage we call it as the
effective voltage so what do I mean by
effective voltage effective voltage
means that it is the effective value of
the alternating current because the
value of the alternating current is
changing with time so we consider that
the RMS value which we get that is the
effective value of the voltage like I
mean this RMS value doesn't talk about
the value at a particular instant of
time it talks about the entire current
as a whole if I say that the RMS voltage
is 20 volts that means it is talking
about the entire alternating current
flowing through a circuit and that is
considered as the effective voltage of
the entire ordinate incurring
irrespective of which instead
irrespective of which time irrespective
of whether that value is the minimum
value or the maximum value of the
alternating running it just tells that
the overall effective value of the
alternating current or the alternating
voltage is this much so load mean
squared voltage is the effective voltage
of the alternating current it is the
square root of the mean of the squared
function of instantaneous values so you
know right what is instantaneous voltage
instantaneous voltage is nothing
but the value of voltage at each instant
of time so let us suppose so as the name
suggests it is root mean square that
means it is square root of the mean of
the squared function of instantaneous
values now as I mentioned in the
previous slide that what is
instantaneous value for example let me
consider this as my alternating voltage
so when I say instantaneous voltage that
means each of these values at each
instant of time they represent
instantaneous voltage values right so
when if I ask you to calculate the RMS
voltage for this alternating voltage so
how do we do we say that let us suppose
the instantaneous voltages at every
instant is v1 v2 v3 v4 these are the
four instantaneous values of voltages at
for instants of time so what would be
the RMS value the RMS value would be
equal to the square root of me of the
squared function of instantaneous values
so what would be the squared function of
instantaneous values it would be B 1
squared V 2 squared V 3 square V 4
squared and mean of these squared
functions so this would be V 1 square
plus V 2 square plus V 3 square plus V 4
square divided by the number of values
which you have considered that is 4 so
this would be your V RMS that is RMS
voltage right so this is how we will
calculate RMS voltage so now you
understand the significance of the name
now it supplies the same power to the
load as an equivalent DC circuit this is
what gave rise to this RMS value of
voltage right we saw in the previous
slide that the power consumption related
to a purely resistive circuit was
earlier found to be half I M square R
that is it was earlier found in terms of
the peak
to a parent but people wanted that thing
should be consistent between AC and DC
so that is where they introduced this
RMS voltage and they said that now power
will be equal to I RMS square R so now
power would be I RMS square into R so it
is consistent with dizzy that means the
power which this value of current will
supply to the load that power will be
equivalent to a DC Circuit right so if
we represent it in in terms of draft so
this meant that the power consumption
here and this is inside it let us
suppose if this is the peak value like
for this AC voltage the maximum value is
the peak value and let us suppose this
value represents the RMS value so the
Animus value has to be something less
than the peak value right because peak
value is the maximum one so let us say
this is the RMS value so it says that
this RMS value will correspond to the
same amount of power consumption that we
will have in case of a DC Circuit so if
we have a DC current somewhat like this
and a DC voltage so in that is this V
RMS value of an alternating current or
an alternating voltage will correspond
to the same value in case of a DC
voltage so this will be equivalent and
that is why we call that RMS voltage is
nothing but the effective voltage of an
alternating voltage this is the
effective value of the alternating
voltage so you can consider it as
equivalent to DC because a DC voltage
and RMS voltage will supply the same
power the power consumption will turn
out to be the same in both the cases so
now we will talk about how do we
evaluate RMS value of voltages or how do
we calculate RMS value
of voltages right so the first case we
will consider is instantaneous discrete
values that means the values and each
distinguishable from the other the
second scenario we will consider for
instantaneous conceived continuous
values so we will take these examples
which will make things clearer so let us
first consider the discrete values let
us suppose we have a voltage whose
values at every instant of time is
mentioned as say V 1 V 2 V 3
so and so till the end so these are each
discrete values of voltages at each
specific instance of time so I have to
calculate the root mean square value of
the voltage so what would be the root
mean square voltage it would be nothing
but the square root of the square
functions me so their square
corresponding squared functions would be
nothing but V 1 square V 2 square V 3
square V n square so this would be V 1
square plus V 2 square plus V 3 square
plus till VN square divided by this is
the mean of these squared functions so
divided by n so this will give you the
RMS value of voltage in this case where
you have instantaneous values at as each
discrete values now instead of discrete
instantaneous values if you have
continuous values that means each
instant of time cannot be this is not
distinguishable from the other instance
right so in that case how would you
evaluate your RMS value of voltage so
for this we can take the example of your
sign voltage right the alternating
current voltage which we have considered
let us suppose you have your voltage
varying as VM sine Omega T that means
the voltage it is changing at every
moment so you do not know the values of
instantaneous voltage at every instance
of time you do not know that you just
know that the voltage value is
like a continuous sine function so how
would you calculate the RMS value of
voltage in this case just that the
summation will now get replaced with an
integration right the way you have
summation for discrete values similarly
you have your integration for continuous
values so in this case we will calculate
the RMS value of voltage as so the RMS
would be equal to root over integration
of V square that is squared function as
I mentioned so this would be B squared
and this V Square will be integrated
over the entire time interval because I
am calculating the average right so this
average for to calculate the average it
has to be integrated over the entire
time period from 0 to the time period
capital T divided by the time period
right so this is how you will calculate
the average so this comes out to be root
over 1 by T integration of you can
replace V with V M square sine square
Omega sine square Omega T DT from 0 to T
so here also we'll apply our
trigonometry so we can write it as VM
square divided by T integration 0 to T
sine square Omega T DT so this entire
thing Deut over so this will be equal to
root over VM square divided by T now
what can you write sine square theta we
know that 1 now we know that 1 minus cos
2 theta is equal to 2 sine square theta
so using this identity we can write sine
square Omega T as 1 minus cos 2 Omega T
the
I did point two so now this we can write
it as root over V M square divided by
two T into T minus integration of cos 2
Omega T would be sine 2 Omega T divided
by 2 Omega and your limits varying from
0 to capital T that is your time period
for one complete cycle so this value
will come out to be VM square divided by
2 T into T minus sine 2 Omega T divided
by 2 Omega now we also know that this
Omega that is your angular frequency is
equal to 2 pi nu this is how it is
related to your linear frequency and
your linear frequency is nothing but
reciprocal of your time period right so
we will use this here so instead of
Omega we can put it as 2 pi by T so this
will give you VM square divided by 2 T I
am sorry I just miss the square root so
this would be VM square divided by 2 T
into T plus sine 4 pi divided by 2 Omega
so this sine 4 PI comes out to be 0 so
that term comes out to be root over V M
square divided by 2 T into T so TT gets
cancelled so you get V M divided by root
2 so what do we find we found that your
V RMS that is the RMS value of voltage
comes on to be 1 by root 2 times the
peak value of voltage so this is how we
can relate the peak value of voltage
with the RMS value of voltage throughout
our alternating current because in
alternating current the instantaneous
voltage will always be represented as a
sinusoidal function so this is an
important conclusion that V RMS is equal
to peak value of voltage divided by root
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