Phasor Diagram/ Phasors (including RMS Voltage and RMS current) Video Lecture | Physics for JEE Main & Advanced

okay everything we've done in
electricity so far has been leading to
this point we are now going to discuss
AC circuits so this was the big battle
between Edison and Tesla Edison favored
the DC circuit where it's all very
simple you make electrons go this
direction and the current goes this
direction and nothing's weird and Tesla
said no let's have it go this way in the
net line this one that wind and this
wind and that way sloshing back and
forth and as you can see Tesla one out
for household distribution and there are
a lot of reasons for that one of them is
the idea of transformers one of them
we'll talk about in a little bit but
transformers work require constantly
changing flux so if you've got it
constantly changing flux a consistently
changing flux right there then you can
make a transformer this is the simplest
one right you just put two coils near
each other and you have them share a
flux like I've got a soft magnetic
material in here and then any any flux
change in one is the same as the flux
change and the other so you can make
voltages dramatically change and
converting from one voltage to another
is a really practical thing so this is
one of the reasons that AC went out and
the other one is for a later time for us
to discuss
but with AC we take this definition we
say that the voltage as a function of
time is the maximum voltage I don't care
what it is but it's going to be
multiplied by a sine function and it's
of course going to be the sine of Omega
times T that's going to be in radians
per second so be careful we're going to
be taking the sine of something in
radians and that gives us a graph of let
me get you the v-max this is voltage and
this is time and it's going to start out
here and go like that so if the voltage
is doing that then this level right here
would be the max if the voltage is doing
that and you've just got a simple
resistor
like I don't know a hot plate or a
toaster or something then the current
will be doing in fact exactly the same
thing the current will be changing as a
function of time this is how hard you're
pushing and this is the way the system
responds so it's kind of like a
stress-strain relationship again and
you're going to have well certainly you
won't have the same value but will
expect the zeros to line up and we'll
expect it to have the same structural
performance here this is current as a
function of time and this is I max right
here and we've still got all of our
favorite relationships like V is I R but
all of these become a function of time
you assume that the resistance doesn't
change with time but if the voltage is
going up and down then the current is
similarly going up and down so inside of
my hot plate I actually have current
going this direction and then that
direction in this direction in that
direction clockwise and counter
clockwise and in the United States it
changes 60 times every second it's going
forward and back 60 times in every
single second and in Europe and India
shout out to India I think they've got
50 times every second that it switches
direction and so if if ooh look at this
look at this what if we say what which
we try to find out what v-max is no
let's not try to find out with the axes
let's just say that current as a
function of time is going to be well I
guess it'll be V as a function of time
divided by R so it's v-max over R times
the same sine function sine of Omega
times T and now it's time to introduce a
really neat concept these things are
called phasers and a phaser looks like
this first you get yourself a coordinate
system and I'm just going to define X
here and Y here and a phasor is a vector
a phasor simply put is just a vector and
the vector is continuously spinning
around the origin that's the definition
of a phasor and this phasor I'm going to
say is v-max long that's how long it is
and I want to say that the projection
projection of phaser on y-axis is
instantaneous value it helps to look at
simulations of these at this point
because I'm not going to be able to
rotate that vector around there but if
you know that that sucker is rotating
and I guess we need to label some of
these angles this angle is going to be
Omega times T because well that's what's
happening here we initially have a y
component a Y projection of zero and it
goes to a Y projection of V Max and it
takes the time the time that it takes to
do one quarter of a revolution gets us
up to right here because that's a
quarter wave right
and then as it goes around it goes back
to zero again yeah that's right here
this maybe I can draw you some lines
that correspond to various points when
yeah yeah I can do that I can show you
the phasor in its orientation so here
the phasor is directly up here the
phasor is directly to the left here the
phasor is directly down here the phasers
directly to the right now it doesn't
matter whether it's left or right at
this point but but whether it's up or
down tells us whether we have a positive
or a negative projection of the vector
so we can put little tick marks all
along here and sort of acknowledge that
it's these points that we're getting
maximum or zero values for our voltage
but remember I said that the current is
doing the same thing but of course it's
it's V Max divided by R so I don't know
what R is but I can draw you a similar
phasor maybe right next to that other
phasor it actually should be right on
top of it and this phasor is length
right here is V Max over R and V Max
over R defines IMAX so in a similar way
at each of these tick mark times
I can say that we have the same
orientation for the IMAX vector what
that means is when there is a maximum
current there is a maximum voltage and
vice versa because those guys oh you got
language for this
these guys are said to be in phase
they're said to be in phase because the
phasers line up with each other so that
means if one change is the other changes
and they have the same time dependence
uh-huh uh-huh now let's think about the
definition of well I don't like the
changing all the time I would like a
value to describe what it actually is
and I'm going to use a value called
average the problem is the app what's
the average of this graph well let's see
it spends half its time up and half its
time down the average of that graph over
one full rotation nothing and the
average of that over a next rotation
nothing so the average isn't going to
help us at all but what if we square it
do you know what the square of the sign
function looks like square the sine
functions like so that's time and that's
voltage for instance but it's voltage
squared and so voltage squared is uh
let's look at that it would be v-max
squared times sine square of Omega times
T so these squared still depends on time
but what if I take the average of that
now take mean and if I take the mean of
that sucker right there I'm going to say
the squared averaged no I can't average
in then square because then it just gets
zero but if I take the square and then
average it then I'm looking for this
value right here what do you think that
is you see this value up at the top this
top value is what that top value is
simply
Vmax square right that's what it means
the squared max is v-max squared so
that's fine but these squared averaged
is just half that big look at it isn't
it half that big of course it's half
that big it's symmetric around the
halfway point so this is one half now
careful I'm going to write 1/2 v-max
squared ok all right now here's my next
plan my next plan is to take the square
root of this business right here so I'm
going to say V squared averaged and then
screwed it is well the screwed of that
business right there so this is 1 over
screw 2 times V Max oh that's a lovely
conclusion Wow
turns out that every excuse me huh every
relationship where you've got a sign has
the same functionality right here if you
square it and then average it and then
screwed it you will get 1 over root 2 of
the maximum value of that function
that's really lovely so I can say in a
general sense this will work for current
and it will work for well frankly
anything that is changing sinusoidal e
or cosinusoidal e I take any variable
and I square it and I average the square
and I screwed it and this is 1 over root
2 times the maximum value of that
variable and this thing right here this
squaring and then meaning and screwing
is called the RMS value stands for root
mean squared because I'm taking the root
and I'm taking the mean and I'm squaring
it not in that order though I'm square
mean rooting it but it's called root
mean square so you have to get used to
that and this is just whatever the
maximum value of that sucker is divided
by root 2 so let's let's say a couple
things and then I'll make some some
shout outs to you there's um Wow well
there's power right
and you know that power in a resistor
like my hot plate power is I score times
R and if power is I squared times R then
it's simply going to be I max square
times R oh and then there's this sine
square of Omega T but that you see this
works instantaneously this is the power
at any instant but the voltage and the
current are changing sinusoidally so at
these instants right here the power
delivered by the circuit to my resistor
is actually zero and here this is huge
power and this is also huge power
doesn't matter that the currents going
the other direction in a resistor it's
going to heat up either way huge power
huge power no power no power no power no
power this is another advantage of
decent of AC / DC actually even though
it seems like it just makes it more
complicated it makes it really lovely so
this is true instantaneously but I can
tell you another equation like power
average equals current RMS square times
R and so that means wait a second the
average power wait if I take I RMS this
thing right here and I square it I'm
going to get I square wait a second
that's IMAX square times R divided by 2
into resting alright and this is true at
any time this doesn't have any time
dependence because I'm looking for
average the beautiful thing is we've had
all this complication and we get back
all of our standard equations power is I
V as long as I use RMS RMS I'm still
good I can find the average power and I
get power is I squared times R if I use
ms current and then I will get average
power and we can also say that power is
v square over R if you use RMS voltage
then you'll get average power yeah
everything we knew already oh and also V
is IR it's true that this is true
instantaneously but also the RMS voltage
is the RMS current times the resistance
of our resistor yay we got everything
back now the only oh shoot
this right here average power this right
here I squared max times R is maximum
power so the maximum power divided by 2
is the average power because power
depends on the square of the current
function or depending on how you look at
it I guess it depends on the square of
the voltage function so you've sometimes
got zero power and you sometimes got
maximum power but your average power is
right smack in the middle of the average
and the maximum and that makes sense I
think so let's look at just one example
of this beautiful equation here the RMS
voltage in fact us says that V RMS is uh
what do they say they say the RMS
voltage is 120 volts from my electrical
supplier and I know that the RMS voltage
is the maximum voltage divided by the
scroot of 2 so I can find that the
maximum voltage is Oh what do you get
170 volts or so whoa so they've been
telling me it's 120 volts but really
it's 170 no it's 170 some of the time
here's voltage as a function of time and
it's making this sine graph the maximum
value is 170 but the RMS is 1 over root
2 of that so the RMS is like right here
that's the RMS voltage and of course
it's not the same thing as the average
because the average voltage is 0 but
it's a way to indicate to us the amount
of voltage that is there on a root mean
squared kind of level I
any better way to say it and hello
everybody in India they tell me that
your RMS voltage is what do they say I
think they say that it's 220 volts and
that's your maximum voltage divided by
the screw of two so the v-max in India
is about 311 volts
Wow okay so I think in Europe they'd
find it to be 320 or 330 volts that's
just the difference between the maximum
value here and the RMS value that I've
shown in green goodbye