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good morning welcome back to this lecture
series on a pulse width modulation for power
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electronic converters so we have been you
know first in this course we looked at various
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kinds of power electronic converters such
as dc to dc converters and dc to ac converters
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etcetera
now we are looking at this question of pulse
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width modulation which is beginning to look
at this issue of pulse width modulation in
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fact in todays lecture we just going to see
what is the purpose of pulse width modulation
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so lets what we are going to see in this ah
in the coming couple of lectures so why do
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we need pulse width modulation that is basic
question that we are going to address here
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now if you look at this what you need is you
have a voltage source inverter and that voltage
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source inverter has a dc bus voltage it has
a dc bus voltage and the dc bus voltage is
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being large fixed you know it is not vary
it is not vary remain it is fixed and it could
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possibly be unregulated you mean you can have
a dc source this inverter may be feed from
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a dc source and that could be a dc source
such as an active pwm rectifier or in some
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kind of controlled rectifier in which case
you know that could be well regulated on the
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other hand it could be simply a diode bridge
rectifier with some l filter i mean with the
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c filter or an lc filter in which case the
dc bus voltage could also be unregulated
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so what we have is lets lets ah you know ignore
this ah regulation or unregulation for the
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time being now lets just say it is fixed at
a particular value so we want to realize whatever
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fundamental voltage we need on the ac side
we need certain fundamental voltage lets call
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it v one so given certain dc bus voltage which
we can call it as vdc we want to realize this
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v one and who should do this it is the pwm
converter should do this now and the the the
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inverter as can produce a range of fundamental
voltage for this given vdc
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now you want a specific voltage lets call
it sixty percent or seventy percent or whatever
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so now if the first goal of this pulse width
modulation is to ensure that the fundamental
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voltage has the desired amplitude that it
should have the desired amplitude is one of
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the first goals that it you need to control
the ac side fundamental voltage now so there
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are other cases you know you may also have
to control the frequency of the fundamental
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voltage and there are also both the phase
of the fundamental voltage etcetera so controlling
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the ac side fundamental voltage given a fixed
dc bus is one of the ah goals of this ah you
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know pulse width modulation beyond that what
happens is what you produce on the ac side
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is not a sinusoidal waveform we are trying
to synthesize some sinusoid you know of some
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fifty hertz or whatever modulating frequency
but it is essentially a nonsinusoidal waveform
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we are trying to produce ac using dc so it
is you know what you are playing basically
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is not a sinusoidal waveform we apply various
pulses you you may apply positive pulses during
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the positive of cycle and negative pulses
during the negative of cycle and so on and
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it is not a sinusoidal waveform it is it has
some nonsinusoidal i mean it has certain harmonics
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it is nonsinusoidal so when you say nonsinusoidal
it has certain harmonic voltages
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so these harmonic voltages are going to result
in certain undesirable ah you know going to
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have certain undesirable effects this harmonic
voltages will cause certain harmonic currents
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to flow and these harmonic currents for example
can go about increase the losses this harmonic
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currents will now flow on top of the fundamental
current through all the line side components
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so it could be a line side inductor are could
be an induction motor whatever it could be
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so these are going to increase the losses
and sometimes the core losses in those reactive
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elements and ah the motor can also increase
now
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so what happens as a as an effect of you know
the harmonics is that the harm the the losses
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are increased now and then you in in case
of motor drive you may also have a pulsating
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torque what you normally have in a motor in
an induction motor fed from a sinusoidal voltage
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sources the torque is steady you get a steady
torque because it is a result for interaction
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of two different ah fluxes are one flux and
a current or whatever and both are sinusoidal
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and you know if you look at them through phasors
we will deal with all this at a later in in
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in a greater detail but just to help you grab
this you know you have two fields one pertaining
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to the stator and another pertaining to the
rotor which both revolve at the same frequency
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and they have a fixed phase angle because
of the interaction between the two you produce
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a steady torque
but what could happen in a motor drive is
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that you have not only a sinusoidal voltage
but you also have harmonic voltages getting
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applied them because of these harmonic voltages
you can have a harmonic fluxes and harmonic
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currents the result of interaction between
a fundamental current and harmonic flux are
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between ah steady flux and harmonic currents
you you get pulsating torque as we will see
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later now so these are ah some examples of
the harmful effects of harmonic voltages so
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pulse width modulation what it tries to do
is it first tries to give the desired fundamental
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voltage v one now this voltage v one is achievable
through a number of means
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so now you can you can use this and you select
a particular method of producing this v one
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which results in certain amount of ah reduction
in the harmful effects of ah the harmonic
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voltages now so the pulse width modulation
has its second goal this mitigation of the
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harmful effects of the harmonics sometimes
you may say i do not want specific harmonics
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you can say i dont want fifth harmonic i dont
want seventh harmonic etcetera that what is
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known as harmonic elimination sometimes you
might want to reduce certain harmonic currents
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in certain range and set that sometimes you
might want to reduce the pulsating torque
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so your goal could be different but overall
you know what can be said is you know it aims
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at pulse width modulation aims at reducing
the harmonics and they are harmful side effects
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now so this is what i would you call as the
basic ah purpose of pulse width modulation
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now and we go further to see how we control
fundamental voltage and how to calculate harmonics
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and how to calculate you know the harmonic
currents and things like that now so moment
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i said you know there is a pwm waveform what
these are some examples of waveforms that
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we will see in dc to ac converters so the
first is basically a square wave now you can
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regard this as your pole voltage v r o that
is we look at the pole i mean the midpoint
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of the load terminal of a leg and measured
it with respect to the dc bus neutral or the
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midpoint of the dc bus this is the kind of
waveform and the if the inverter squee you
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know switched in a square wave fashion this
is the kind of voltage waveform you will get
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this is your pole voltage waveform now
so this is a periodic waveform so similarly
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you can modulate in several ways some some
examples which we have already seen now you
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can modulate in all these ways all these are
periodic signals now so many of such simple
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pwm waveforms are periodic signals so what
we can do is we can use fourier series we
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want to know calculate how much harmonic it
has i mean how much fundamental component
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it has how much whether it has a specific
harmonic or not and what is the amplitude
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of that harmonic and so on so what we can
we can use fourier series to do that now
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so what is fourier series you may have any
periodic signal you may have a periodic signal
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and this periodic signal may have any shape
doesnt matter but it can be expressed as a
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sum of several sinusoids it can be expressed
as a sum of several sinusoids so let us define
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the sinusoids a little more closely so these
sinusoids may have certain frequencies firstly
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it could be zero frequency or there can be
dc now next is it can have what is called
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as the fundamental frequency that is the same
periodicity as the actual waveform and it
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can also have harmonic frequencies which are
integral multiples of the fundamental frequencies
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now so periodic signal can be decomposed into
a dc component a fundamental component and
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a set of harmonic components as indicated
here now so here what i have given is some
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f of omega t it has been you know the the
variable instead of time i have taken angle
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at the fundamental frequency here omega is
equal to two pi upon t where t is the time
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period of the waveform in question so i we
have expressed this as a function of the angel
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at the fundamental frequency or what we will
call as the fundamental angle so now this
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f is f of omega t has could have a dc component
which is given by the a naught by two and
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it could have several ah um follow frequency
components one of the frequency component
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could be omega itself
so if you look at the frequencies zero is
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one that is dc omega is another one two omega
is three omega four omega and so on ah this
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can have all these various frequencies now
so fourier series helps us expand any given
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f of omega t into such a series so it helps
us calculate the coefficients of various terms
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i mean once we are able to calculate the coefficients
of various terms we know how much harmonic
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is there for example in this term if we can
calculate a naught we know what is the dc
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value which is a naught by two if we can calculate
a one and b one we know what is the sinusoidal
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component it will be a one squared plus b
one squared and the root would be the amplitude
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of the fundamental component here if you want
the nth harmonic component we must try and
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get this a n b n here
so lets move on to see a few more periodic
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signals now so this periodic signal now these
are all periodic because they repeat over
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this with this periodicity t so now they are
defined over f f of f of t plus capital t
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is equal to f of t and t is this period now
one example that has been given as a wave
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form ramps up and then falls back and then
it is zero and once again the same thing begins
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and this is your time period now this is one
example of a ah periodic signal now another
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example is given here current trices stays
flat falls back and then it is zero for certain
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amount of time and this is your time period
t and the same thing repeats now so these
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are also certain other examples of periodic
signals
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now in this case you know there is a difference
between the previous one and here the previous
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ones were all by enlarge ac waveforms the
ones these were all by enlarge ac waveforms
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and they did not contain any dc component
at all they have a zero average value if you
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look at this it as zero average that is if
you look at the area under this and the area
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enclosed by this they are equal and they have
just have an opposite sign so if you look
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at the total area within over a cycle it is
zero it has no average value but these are
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certain examples of signals which have some
average value so if you look at the average
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here this average is in the average it over
the fifty percent the average is somewhere
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like this if i average it over the entire
cycle then it is something like this
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so in this case actually two point five is
the average value of this ah current that
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you have now so if you look at the other case
the average value here if you look at you
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know over this it is also possible for you
to calculate certain average like this its
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possible for you to calculate certain average
like this so these are examples of some periodic
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signals which have non zero dc values they
have some average values and you know these
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are actually taken from some dc dc convertors
these are some current waveforms in certain
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dc dc converters there they always have certain
dc component flowing whereas in dc to ac converters
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you know on the when you are looking at the
ac side there are i mean there is no dc component
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here they are all ac waveforms these waveforms
would typically have you know fundamental
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and other harmonic frequencies but not the
dc
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so that is what i said here so these are examples
where they have no average value you integrate
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this whole waveform we can consider the first
waveform integrated over the end with respect
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to d omega t start from zero to two pi and
you find that the value is zero you integrate
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this waveform you will also find that it is
zero you can take the third example also you
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do that integration you will find that it
is zero that is because these are waveforms
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on the ac side of dc to ac converter now these
are all essentially ah ac waveforms we ideally
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we wish that they were sinusoidal waveforms
so like you know what we want to see here
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is some sinusoidal waveform like this but
then it has several other non sinusoidal components
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which we cannot help you know because we are
producing ac using dc we are applying basically
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positive dc pulses and negative dc pulses
to get something like this so what we are
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trying to do is now the width of these pulses
you can see controls the fundamental voltage
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which is one of the purpose that we were talking
about earlier and now if you look at here
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this could also have some ac waveform but
in such you know there is some fundamental
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component such cases this is one example where
the harmonics could be a little lower then
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let us say fall the pulses were of equal bits
you need to do a complete fourier analysis
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to you know expand them as fourier series
ah to get the exact magnitudes of how much
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etcetera here
so once again going back to fourier series
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these are examples that we have no average
value now so the average is taken off there
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is there is no dc component now so then let
us look at the fundamental component now so
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we want to understand how much what is the
amplitude of the fundamental voltage so if
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you look at this waveform the waveform certainly
has a fundamental component like this so the
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fundamental component is a sinusoid of the
same freak periodicity as the original parent
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waveform right and we are also able to judge
the phase this should be the phase of this
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ah sinusoidal waveform what we do not know
is this amplitude let me call this as v one
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we do not know what is that amplitude
so how can you calculate that so what we need
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to do is going by fourier series this waveform
has several frequencies you know like zero
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omega two omega etcetera now we know that
for sure that it has no zero so it could have
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all these frequencies omega two omega three
omega etcetera it could have um it could have
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all these frequencies so we still do not know
now we are interested in finding out the amplitude
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of the fundamental frequency so how do we
do that so what we need to do is if you multiply
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this waveform by a sine let us say we multiply
this waveform by a sine waveform let me just
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choose a different ah color here maybe this
let me multiply this by a sinusoid sinusoid
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of the same frequency as the fundamental component
why do i do that if i do that why do i choose
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the same frequency as the fundamental component
now this waveform in question has several
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components let me write the mass v one sine
omega t for example plus some other you know
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let let me say certain v n or you know some
v ok let us say v three sin three omega t
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plus so on it has all these various components
now if i multiply this by sin omega t if i
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multiply this by sin omega t what happens
you first have this product let me once again
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change the color to something a little brighter
so now lets say this sin omega t and sin omega
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t you have this product gives you sin squared
omega t it gives you a sin squared omega t
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term
so what you will have is basically this will
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become v one sin squared omega t and the second
term will become v three sin three omega t
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sin omega t and so on so you will get some
waveforms if you look at this if you look
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at this waveform and in a if you find out
its dc value its average of this term over
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cycle it will have some dc value
so this average value is a measure of what
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is your v one if you consider this waveform
for example sin three omega t into sin omega
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t you look at the product waveform and you
integrated over a cycle its average value
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will be zero so all the other components also
you know their averages all the product terms
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that you will have here their averages over
cycle will be zero except for this term so
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how to get this what you need to do is you
given wave form you multiplied by sin of the
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same frequency and the sin can be unit amplitude
now what is the phase of the sin in this case
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the sin is defined as sin omega t now what
we are trying to do in this case is we are
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trying to do v of omega t multiplied by sin
omega t d omega t and we we are integrating
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this from zero to two pi and this is two upon
two pi this is what we do to get our amplitude
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of the fundamental component this is what
you need to do to get this now we are multiplying
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it by sin omega t and because you know this
is what is called as one way to put this is
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this is an example of what is called as an
odd function so if you extend the wave form
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to the left side whatever is its value at
some theta at this minus theta it will have
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00:18:17,890 --> 00:18:22,990
the negative of it so in these kind of cases
the wave form has only the sin components
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00:18:22,990 --> 00:18:27,220
in does not have the cosine components so
for example if you go through the previous
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00:18:27,220 --> 00:18:30,620
ones
so it will have only this b and coefficients
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00:18:30,620 --> 00:18:37,350
listed in it it will not have this a and coefficients
listed here now so this is one so you know
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00:18:37,350 --> 00:18:44,799
that you have to multiply this by a sin function
here now so lets look at it more let us say
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00:18:44,799 --> 00:18:51,940
the wave form is considered differently you
know there are you know the same voltage wave
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00:18:51,940 --> 00:18:59,929
form lets take it like that this is the omega
t the same plus v and the same minus v
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00:18:59,929 --> 00:19:05,379
now what we have is this is ninety degrees
this is two hundred and seventy and one eighty
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00:19:05,379 --> 00:19:11,370
comes in between now this is the natural wave
form now so in this wave form i am trying
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00:19:11,370 --> 00:19:19,690
to see what is the fundamental component so
what should i do now the same way but what
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00:19:19,690 --> 00:19:25,850
i have to see is it is fundamental i can see
that it is fundamentally is something like
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00:19:25,850 --> 00:19:32,690
that is fundamental has a has its phase like
this its fundamental is something like that
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00:19:32,690 --> 00:19:38,590
so i need to multiply this by a unit sine
i need to multiply this by a unit sine as
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00:19:38,590 --> 00:19:44,679
shown here
so what i do here is i have v of omega t as
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00:19:44,679 --> 00:19:51,110
defined here i multiply this by cos omega
t and i integrate this with respect to omega
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00:19:51,110 --> 00:20:00,289
t from zero to two pi and two upon two pi
i do this to get the fundamental amplitude
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00:20:00,289 --> 00:20:06,499
so here this is the case of what is called
as an even function this is a case example
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00:20:06,499 --> 00:20:11,340
of what is called as an even function if you
look at the negative values go along the negative
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00:20:11,340 --> 00:20:16,419
axis i extend this to the negative horizontal
axis also you will see that this waveform
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00:20:16,419 --> 00:20:24,110
is symmetric about the ah vertical axis so
vf minus omega t will be basically equal to
217
00:20:24,110 --> 00:20:28,249
vf omega t so that is the kind of symmetry
and that is what is called as an even function
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00:20:28,249 --> 00:20:33,720
in such cases the b terms do not exist but
the a terms a coefficients in the fourier
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00:20:33,720 --> 00:20:39,269
series v exist so you go about doing it now
so lets take it more generally now let us
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00:20:39,269 --> 00:20:58,679
say my waveform is something like that
this starts at some angle phi and it goes
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00:20:58,679 --> 00:21:08,799
on till some angle one eighty plus phi r pi
plus phi and this is three sixty plus phi
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00:21:08,799 --> 00:21:12,980
in such cases if we do not know what this
phi is are you know if r we are unable to
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00:21:12,980 --> 00:21:18,230
see ah i mean are in any any any kind of a
general waveform what we need to be doing
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00:21:18,230 --> 00:21:25,419
is we need to multiply this waveform by sin
component by sin omega t and do the integration
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00:21:25,419 --> 00:21:30,720
over the cycle and also similarly we must
multiply the waveform by cos omega t also
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00:21:30,720 --> 00:21:36,600
and get this so this multiplication by sin
and averaging over a cycle gives one component
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00:21:36,600 --> 00:21:40,980
of that and by doing it you know the other
component is available what are called as
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00:21:40,980 --> 00:21:45,620
a one and b one both these components have
to be evaluated and then from a one square
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00:21:45,620 --> 00:21:49,100
plus b one square under root will give you
this amplitude that is your actual way of
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00:21:49,100 --> 00:21:52,489
doing it now
so what i can see is we can kind of generalize
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00:21:52,489 --> 00:21:58,739
now in the in the earlier case you multiply
it by a unit sine as shown here in the second
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00:21:58,739 --> 00:22:02,850
case you are multiplying it by a unit sine
as shown here now here what you should be
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00:22:02,850 --> 00:22:09,280
doing you must actually be multiplying it
by a unit sine you must actually be multiplying
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00:22:09,280 --> 00:22:16,250
it by a unit sine like this if you can multiply
it by unit sine such as shown here you do
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00:22:16,250 --> 00:22:22,639
your integration over a cycle or take the
average of the product then we will get the
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00:22:22,639 --> 00:22:27,240
amplitude i mean the the average value is
proportional to the amplitude of the fundamental
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00:22:27,240 --> 00:22:31,549
voltage so what is being then if the phase
is known that is what we are trying to do
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00:22:31,549 --> 00:22:35,629
is you know we are trying to multiply by a
sine wave what is the frequency of the sine
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00:22:35,629 --> 00:22:41,119
wave it is the fundamental frequency that
is the same periodicity as the parent ah waveform
240
00:22:41,119 --> 00:22:45,419
in question right
now what should be the amplitude of the sin
241
00:22:45,419 --> 00:22:50,900
it is unit amplitude now what should be the
phase the sin actually should have the same
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00:22:50,900 --> 00:22:58,239
phase as the fundamental component of the
original waveform in this case we are able
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00:22:58,239 --> 00:23:02,630
to judge that in certain cases we may not
be able to judge that if we are able to judge
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00:23:02,630 --> 00:23:07,899
that or if we know that then we can multiply
it by a sin who which is in phase with the
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00:23:07,899 --> 00:23:12,610
fundamental component of the original waveform
and we can do that integration of the product
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00:23:12,610 --> 00:23:17,100
and get the amplitude out of that when we
are not aware of the exact phase what we should
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00:23:17,100 --> 00:23:21,179
be doing is we must multiplied this by sin
omega t and also by cos omega t you know get
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00:23:21,179 --> 00:23:25,220
those both those a one and b one terms and
from there get the overall amplitude from
249
00:23:25,220 --> 00:23:30,130
doing it here now
so this is how you determine the fundamental
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00:23:30,130 --> 00:23:35,509
compute of the fundamental component now let
us say how do you determine the amplitude
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00:23:35,509 --> 00:23:43,009
of second harmonic so here we multiplied as
we saw a little earlier we multiplied this
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00:23:43,009 --> 00:23:48,100
whole thing what is within the bracket represents
the waveform we multiplied by sin omega t
253
00:23:48,100 --> 00:23:53,350
because we were interested to see whether
there is any omega t you know sin omega t
254
00:23:53,350 --> 00:23:57,900
term exist we wanted to find out the amplitude
of the fundamental frequency now
255
00:23:57,900 --> 00:24:01,659
now if you see that you know there is a second
harmonic could be present here what we must
256
00:24:01,659 --> 00:24:07,029
be doing is we should be multiplying this
by a second harmonic we should multiply this
257
00:24:07,029 --> 00:24:11,779
by a second harmonic sine wave so let me say
that i am multiplying it by a second harmonic
258
00:24:11,779 --> 00:24:27,289
sine wave like this um right well you know
this is a second harmonic sine wave now i
259
00:24:27,289 --> 00:24:32,149
want to get the product of these two what
would be the product of these two like now
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00:24:32,149 --> 00:24:40,269
let me consider some point let me consider
certain instant here let me call this instant
261
00:24:40,269 --> 00:24:46,190
as theta and let me consider another instant
which is one eighty degrees away so what do
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00:24:46,190 --> 00:24:51,889
i find these two instance i find that the
even harmonic has the same value at these
263
00:24:51,889 --> 00:24:56,899
two instance i consider theta and i consider
pi plus theta r one eighty plus theta at both
264
00:24:56,899 --> 00:25:02,679
this insta[nce]- you know instances second
harmonic has the same value because this pi
265
00:25:02,679 --> 00:25:06,580
is measured at the fundamental angle one eighty
degrees for the fundamental is one complete
266
00:25:06,580 --> 00:25:10,820
cycle for ah i mean what is one half cycle
for the fundamental is one full cycle for
267
00:25:10,820 --> 00:25:17,350
a second harmonic so it has the same value
now how about the waveform in question that
268
00:25:17,350 --> 00:25:22,590
is the square wave here that we have considered
let us let me take this so the square wave
269
00:25:22,590 --> 00:25:29,539
has this value and what is the value here
so here it is plus v here it is minus v so
270
00:25:29,539 --> 00:25:35,630
the value at one eighty plus theta is negative
of whatever it is at theta if you take this
271
00:25:35,630 --> 00:25:39,919
product this product is something at theta
and what is the product at one eighty plus
272
00:25:39,919 --> 00:25:47,690
theta it also has the same value but it has
a negative sign when you add these two these
273
00:25:47,690 --> 00:25:53,370
two add up to zero so in such a wave this
its not only at this instant you take any
274
00:25:53,370 --> 00:25:59,279
other instant so you take here and you take
the corresponding instant there you will always
275
00:25:59,279 --> 00:26:02,970
find this is some other theta dash in one
eighty plus theta dash in even we looking
276
00:26:02,970 --> 00:26:07,730
at some arbitrary angles
so you will find that at both these instants
277
00:26:07,730 --> 00:26:11,740
the even harmonic i mean the second harmonic
or any even harmonic for that matter would
278
00:26:11,740 --> 00:26:17,269
have the same value but now the waveform in
question is such that you know its value is
279
00:26:17,269 --> 00:26:21,070
the negative of the other at one eighty plus
theta is the negative of whatever it is at
280
00:26:21,070 --> 00:26:27,509
theta so such a waveform as no even harmonic
you consider the same thing here also you
281
00:26:27,509 --> 00:26:34,730
consider this angle and this is at one eighty
plus theta so this waveform also has no even
282
00:26:34,730 --> 00:26:39,320
harmonic you consider this waveform you consider
this angle and you consider this about one
283
00:26:39,320 --> 00:26:45,070
eighty degrees later the waveform here whatever
value it has at v of theta at one eighty plus
284
00:26:45,070 --> 00:26:48,779
theta it has a different value
so whatever the waveform is at one eighty
285
00:26:48,779 --> 00:26:53,929
plus theta it is negative of its value at
theta this is what we observe in all these
286
00:26:53,929 --> 00:27:00,760
now and whenever this condition is satisfied
there are no even harmonics this is what we
287
00:27:00,760 --> 00:27:06,269
call as half wave symmetry i should caution
new that you know symmetries are defined in
288
00:27:06,269 --> 00:27:09,580
slightly different terms by various authors
and all that so we are trying to define it
289
00:27:09,580 --> 00:27:15,279
for the purpose of course here when we say
half wave symmetry what we mean is that the
290
00:27:15,279 --> 00:27:20,249
wave form in question four satisfies this
property if we consider the wave form at theta
291
00:27:20,249 --> 00:27:24,710
and at one eighty plus theta where this angle
is a fundamental angle angle at the fundamental
292
00:27:24,710 --> 00:27:29,359
frequency then whatever is its value at one
eighty plus theta will be the negative of
293
00:27:29,359 --> 00:27:34,100
its value at theta that is so one example
of this is the square wave and all the other
294
00:27:34,100 --> 00:27:38,779
wave pwm wave forms there are a couple of
other pwm wave forms we saw earlier also satisfy
295
00:27:38,779 --> 00:27:42,909
this particular property
so this half wave symmetry basically means
296
00:27:42,909 --> 00:27:49,499
no even harmonics so if you multiply this
one eighty plus theta here at sin to omega
297
00:27:49,499 --> 00:27:54,080
t and sin omega d which is what we just saw
right now you know if you do that integration
298
00:27:54,080 --> 00:27:57,879
over a half a cycle or a complete cycle you
will see that the product i mean you multiply
299
00:27:57,879 --> 00:28:06,789
this waveform by sin two omega t or sin four
omega t or sin six omega t or anything like
300
00:28:06,789 --> 00:28:14,909
that you will always integrate it over a cycle
you will find that the product has zero average
301
00:28:14,909 --> 00:28:19,979
so it has no even harmonics so significance
of half wave symmetry whenever waveform satisfy
302
00:28:19,979 --> 00:28:24,820
this condition one eighty plus theta is equal
to minus v of theta you have no even harmonics
303
00:28:24,820 --> 00:28:29,440
present there
now let us look at certain examples of half
304
00:28:29,440 --> 00:28:34,389
wave symmetry i mean examples of wave forms
that exhibit half wave symmetry sine wave
305
00:28:34,389 --> 00:28:39,159
of course has half wave symmetry because sin
of one eighty plus theta let us say if you
306
00:28:39,159 --> 00:28:45,749
want to take sin of ninety degrees is basically
minus sin ten degrees so it has that so all
307
00:28:45,749 --> 00:28:51,059
the sinusoidal kind of waveforms whatever
phase shifts you consider they all have this
308
00:28:51,059 --> 00:28:55,580
cosine whatever you might call now certain
other examples that you can think of or i
309
00:28:55,580 --> 00:29:02,909
am just giving some arbitrary examples lets
say a trapezoidal wave lets say a trapezoidal
310
00:29:02,909 --> 00:29:08,129
wave this waveform also has half wave symmetry
i mean the slopes are equal here there are
311
00:29:08,129 --> 00:29:12,809
certain drawing inaccuracies but the slopes
are actually equal it is sum plus v and it
312
00:29:12,809 --> 00:29:20,129
is sum minus v here
so if you see here whatever is the value at
313
00:29:20,129 --> 00:29:25,519
theta the wave form has the same thing at
one eighty theta this is three hundred and
314
00:29:25,519 --> 00:29:31,139
sixty degrees are what corresponds to t in
terms of time and this is one eighty degrees
315
00:29:31,139 --> 00:29:37,159
or what corresponds to t by two so this is
one eighty degrees this is one eighty degrees
316
00:29:37,159 --> 00:29:43,090
so whatever it is at theta if you take one
eighty plus theta the wave form has this property
317
00:29:43,090 --> 00:29:48,080
that at one eighty plus theta its value is
the negative of whatever it was at so this
318
00:29:48,080 --> 00:29:53,539
is one example of half wave symmetry like
this you can go about forming several such
319
00:29:53,539 --> 00:30:00,760
examples that you can ah go on giving now
let me say shall i draw let us consider a
320
00:30:00,760 --> 00:30:07,789
triangular wave this is a complete triangular
wave this is the time t now this is time t
321
00:30:07,789 --> 00:30:13,270
by two here also if i see is and i consider
another instant is one eighty degrees away
322
00:30:13,270 --> 00:30:18,009
whatever is the value here and the value here
they are equal in magnitude but opposite in
323
00:30:18,009 --> 00:30:22,750
sin they just have you know one is negative
of the other the same way if i go another
324
00:30:22,750 --> 00:30:27,289
one eighty degrees later i will have the same
thing the value will be the same just the
325
00:30:27,289 --> 00:30:31,609
sin goes on changing
so this is what we have a half wave symmetry
326
00:30:31,609 --> 00:30:37,119
you can go about constructing certain other
examples also which anyway we will revisit
327
00:30:37,119 --> 00:30:41,899
see and let me just give you one example of
half wave symmetry you know let me say there
328
00:30:41,899 --> 00:30:51,320
is a wave form of signal rises like this and
then it is flat now it falls down ok now let
329
00:30:51,320 --> 00:31:03,909
us say from here it takes the shape
now this is t this is t by two the signal
330
00:31:03,909 --> 00:31:10,919
rises from zero to v in some you know some
duration over some duration and then it is
331
00:31:10,919 --> 00:31:17,909
flat at v till t by two then it comes down
to zero and and zero it goes to minus v with
332
00:31:17,909 --> 00:31:22,019
some rate which is same as here except for
the sin the slope here and here are equal
333
00:31:22,019 --> 00:31:26,009
but just for the sin now
so it goes down like this and it is here now
334
00:31:26,009 --> 00:31:31,619
is this wave form half wave symmetric is the
half wave symmetric lets try and apply the
335
00:31:31,619 --> 00:31:37,649
condition now so let us say let me just change
the ink color for some clarity let me consider
336
00:31:37,649 --> 00:31:45,999
certain angle theta let me consider one eighty
plus theta so whatever is the value here it
337
00:31:45,999 --> 00:31:54,539
is the same here but for a opposite sin let
me take consider certain other instant let
338
00:31:54,539 --> 00:32:00,149
me consider this instant if i go one eighty
degrees later i have this so what you you
339
00:32:00,149 --> 00:32:05,879
see the values here and there they are the
same except for the sin that is this value
340
00:32:05,879 --> 00:32:11,820
and this value are the same except for the
sin so this waveform also has half wave symmetry
341
00:32:11,820 --> 00:32:16,450
there is something not so symmetric it doesnt
appeal to the i there is something not so
342
00:32:16,450 --> 00:32:20,299
symmetric we there are some additional symmetries
which will come to a little later but this
343
00:32:20,299 --> 00:32:25,239
waveform nevertheless has half wave symmetric
because it satisfies the property that v of
344
00:32:25,239 --> 00:32:30,059
hundred and eighty plus theta is equal to
minus v of theta and this waveform you can
345
00:32:30,059 --> 00:32:36,129
be very sure that it has no even harmonic
you can try doing the multiplication check
346
00:32:36,129 --> 00:32:42,289
it around yourself now
maybe if you want you can try that you multiply
347
00:32:42,289 --> 00:32:51,399
this by a second harmonic ah sine so it would
be like this this is half cycle this is one
348
00:32:51,399 --> 00:32:57,879
cycle of a second harmonic and so this is
like this now so if you take this green instant
349
00:32:57,879 --> 00:33:02,769
whatever is the value here the same value
the second harmonic has here but if you look
350
00:33:02,769 --> 00:33:09,139
at the original waveform they are opposite
in sin so if you add the product at this instant
351
00:33:09,139 --> 00:33:12,970
to the product at this instant that is add
the product at theta to the product at one
352
00:33:12,970 --> 00:33:19,409
eighty plus theta the sum is zero this is
valid for any theta so on the whole this reduces
353
00:33:19,409 --> 00:33:23,999
to zero so this is an example of a wave form
which has half wave symmetry you know it is
354
00:33:23,999 --> 00:33:27,940
could be little deceptive you may the first
side many students generally tend to say well
355
00:33:27,940 --> 00:33:33,010
it does not have a half wave symmetry but
it has and you know that is the property and
356
00:33:33,010 --> 00:33:37,400
that is how we define half wave symmetry here
and it has the meaning of having no even harmonic
357
00:33:37,400 --> 00:33:43,970
now let us go further now now within half
is this wave form half wave symmetric yes
358
00:33:43,970 --> 00:33:47,979
because you take this angle at theta and you
take the corresponding angle at one eighty
359
00:33:47,979 --> 00:33:53,519
plus theta you know one is the inve ah reverse
of the other now what else does it have it
360
00:33:53,519 --> 00:34:00,909
also has an interesting property that you
know let me consider the middle here let me
361
00:34:00,909 --> 00:34:06,970
consider the middle of one half cycle one
half cycle of its you know fundamental of
362
00:34:06,970 --> 00:34:11,970
its you know let us say its positive cycle
now if i travel some distance from the middle
363
00:34:11,970 --> 00:34:18,099
and travel the same distance from this side
the wave form has the same value is on top
364
00:34:18,099 --> 00:34:22,690
of quarter wave symmetry half wave symmetry
it has this property [reme/remember] remember
365
00:34:22,690 --> 00:34:27,010
this wave form the last wave form we do here
did not have this property whereas if you
366
00:34:27,010 --> 00:34:30,970
look at here you travel to the left or you
travel to the right by the same distance long
367
00:34:30,970 --> 00:34:35,419
as you have travelled the same distance it
will have you know the same value so that
368
00:34:35,419 --> 00:34:40,510
is what we call as quarter symmetry let me
call this as ninety degrees or pi by two so
369
00:34:40,510 --> 00:34:45,680
whatever it is ninety plus theta is the same
at ninety minus theta so whatever it is at
370
00:34:45,680 --> 00:34:50,220
two seventy plus theta the value is same as
two seventy minus theta you you also have
371
00:34:50,220 --> 00:34:54,419
the symmetry observed around two hundred and
seventy degrees from two hundred and seventy
372
00:34:54,419 --> 00:34:58,410
degrees you go some distance to the right
or you go to same distance go the same distance
373
00:34:58,410 --> 00:35:04,140
to the left the wave form will have same values
so you can look at certain other examples
374
00:35:04,140 --> 00:35:11,680
also before we ah go into that now from the
previous examples we had constructed we had
375
00:35:11,680 --> 00:35:18,000
constructed a trapezoidal waveform some of
this you know this was one example we considered
376
00:35:18,000 --> 00:35:24,420
this is a trapezoidal waveform one time period
t so this is t by two so the wave form has
377
00:35:24,420 --> 00:35:29,740
half wave symmetry as we already seen and
now it also has quarter wave symmetry it also
378
00:35:29,740 --> 00:35:35,490
has quarter wave symmetry lets this is the
middle point this is t by four this is t by
379
00:35:35,490 --> 00:35:43,140
four and you go some distance to the right
goes the equal distance to the left the value
380
00:35:43,140 --> 00:35:47,400
is the same
so lets say you go further you go further
381
00:35:47,400 --> 00:35:53,000
the same it is equal so it is symmetric about
this t by four or ninety degrees similarly
382
00:35:53,000 --> 00:35:57,100
it is also symmetric about if you can just
consider one half cycle and you you take the
383
00:35:57,100 --> 00:36:03,730
central instant at the half cycle what is
that now so what is it that we are saying
384
00:36:03,730 --> 00:36:07,380
we are saying that you know these are symmetric
about ninety degrees and this is symmetric
385
00:36:07,380 --> 00:36:11,040
about two seventy degree what is special about
nineteen what is special about two hundred
386
00:36:11,040 --> 00:36:17,830
and seventy degrees if you look at the wave
form lets say here where is the fundamental
387
00:36:17,830 --> 00:36:25,830
component line excuse me where is the fundamental
component line so the fundamental component
388
00:36:25,830 --> 00:36:31,030
is something like this the fundamental component
is something like that
389
00:36:31,030 --> 00:36:34,910
so what we are saying as ninety degrees here
and what we are saying as two hundred and
390
00:36:34,910 --> 00:36:39,130
seventy degrees here they are nothing but
the instants at which the fundamental component
391
00:36:39,130 --> 00:36:44,480
is at its peak the instant at which the fundamental
components at peak now let us say theta p
392
00:36:44,480 --> 00:36:49,920
is the instant at which the fundamental component
is at peak any any instant at which the fundamental
393
00:36:49,920 --> 00:36:54,910
component is at its peak now so if if this
property is satisfied in addition to half
394
00:36:54,910 --> 00:37:00,340
wave symmetry we call this quarter wave symmetry
let me just construct one other waveform here
395
00:37:00,340 --> 00:37:11,870
let us say the wave form like this this is
t this is t by two and this is t by four t
396
00:37:11,870 --> 00:37:17,470
by four and these values are lets say if it
is v one v two and these are the same values
397
00:37:17,470 --> 00:37:22,480
with negative sign minus v one and minus v
two
398
00:37:22,480 --> 00:37:30,960
so this has half wave symmetry and also has
quarter wave symmetry how do we say that so
399
00:37:30,960 --> 00:37:36,800
you consider this instant theta you consider
one eighty degrees away or t by two away so
400
00:37:36,800 --> 00:37:41,890
what you have it is the function as the same
value but for opposite sign that is half wave
401
00:37:41,890 --> 00:37:50,530
symmetry it has no even harmonic on top of
that you look around this point you go some
402
00:37:50,530 --> 00:37:57,080
distance and you go the same distance on this
side you find the values to be equal you find
403
00:37:57,080 --> 00:38:01,980
the values to be equal so this is quarter
wave symmetry
404
00:38:01,980 --> 00:38:07,510
so someone can ask now well what is the significance
of quarter wave symmetry in the significance
405
00:38:07,510 --> 00:38:13,030
of half wave symmetry is that you know there
are no even harmonics what is the significance
406
00:38:13,030 --> 00:38:18,230
of quarter wave symmetry one answer to this
question is if you have half wave symmetry
407
00:38:18,230 --> 00:38:23,180
as in any of these cases and if you want to
find out the fundamental amplitude what you
408
00:38:23,180 --> 00:38:28,430
normally do is you carry out an integration
from zero to two pi instead you can only do
409
00:38:28,430 --> 00:38:34,690
an integration over zero to pi and you can
take this factor to be two upon pi you can
410
00:38:34,690 --> 00:38:41,800
lets say f of omega t you multiply this by
sin omega t d omega t and you do an integration
411
00:38:41,800 --> 00:38:45,490
to get the fundamental component you dont
have to do that you know if wave form as half
412
00:38:45,490 --> 00:38:49,710
wave symmetric you dont have to consider the
wave form over zero to two pi you can consider
413
00:38:49,710 --> 00:38:54,570
it from zero to pi or any from any theta to
pi plus theta you must consider one half cycle
414
00:38:54,570 --> 00:38:57,940
of the waveform you can calculate an average
over that
415
00:38:57,940 --> 00:39:02,910
so that is one of the advantages now so it
reduces your calculation if you have quarter
416
00:39:02,910 --> 00:39:08,530
wave symmetry what you can further say is
you need to consider only one quarter for
417
00:39:08,530 --> 00:39:14,620
example you can only consider this quarter
and you you you have this v of omega t defined
418
00:39:14,620 --> 00:39:18,560
only over this quarter that is zero to ninety
degrees and you perform an integration that
419
00:39:18,560 --> 00:39:26,150
is v of omega t multiplied by sin of omega
t this is integrated with respect to d omega
420
00:39:26,150 --> 00:39:32,700
t starting from zero to pi by two and then
it is two upon pi by two or four by pi this
421
00:39:32,700 --> 00:39:37,460
will give you the value now
so this spares you the trouble of defining
422
00:39:37,460 --> 00:39:42,870
v of omega t from zero to nineteen or that
what you need to do but if you want to care
423
00:39:42,870 --> 00:39:48,350
you know use do the calculation or entire
cycle you have to define v between zero to
424
00:39:48,350 --> 00:39:53,730
ninety which is some equation you have to
define v between ninety to one eighty it is
425
00:39:53,730 --> 00:39:58,590
another equation this is another straight
line if you have to define v between one eighty
426
00:39:58,590 --> 00:40:02,700
to two seventy it is yet another equation
between two seventy and three sixty it is
427
00:40:02,700 --> 00:40:05,710
yet another equation
if you have half wave symmetry you dont have
428
00:40:05,710 --> 00:40:10,230
to define the wave form over the entire cycle
we can just consider one half cycle do that
429
00:40:10,230 --> 00:40:14,060
if you have quarter wave it is enough if you
define the wave form just from zero to ninety
430
00:40:14,060 --> 00:40:18,470
degrees and you perform this integration over
zero to ninety degree you can come up with
431
00:40:18,470 --> 00:40:21,750
your component
the same thing is possible for any nth harmonic
432
00:40:21,750 --> 00:40:25,410
component also if for nth harmonic component
you are going to multiply this by certain
433
00:40:25,410 --> 00:40:31,280
sin n omega t and you going to do this now
so your calculation burden reduces that is
434
00:40:31,280 --> 00:40:37,910
one of the advantages that you might have
with quarter wave symmetry now then what could
435
00:40:37,910 --> 00:40:41,900
be the other one like you know what are the
significance is what we have been talking
436
00:40:41,900 --> 00:40:50,350
about now what you can say is in a wave form
that lacks quarter wave symmetry you know
437
00:40:50,350 --> 00:40:54,640
or we may be we should first look at some
examples where you have half wave symmetry
438
00:40:54,640 --> 00:40:59,200
but no quarter wave symmetry of course one
of the examples i had already constructed
439
00:40:59,200 --> 00:41:07,200
this is a wave form this is a wave form which
has half wave symmetry but no quarter wave
440
00:41:07,200 --> 00:41:12,820
symmetry once again to make my point clearer
i would say you consider this theta and you
441
00:41:12,820 --> 00:41:17,600
consider this pi plus theta the values you
look at the values so one is the negative
442
00:41:17,600 --> 00:41:24,760
of the other so it has half wave symmetry
but if you look at quarter wave symmetry you
443
00:41:24,760 --> 00:41:28,470
look at ninety degrees you take t by four
i dont know where it is ninety degree is you
444
00:41:28,470 --> 00:41:33,720
know strictly speaking but you see anywhere
from here the wave form is not really symmetric
445
00:41:33,720 --> 00:41:37,870
the waveform is not symmetric about a particular
line here
446
00:41:37,870 --> 00:41:45,500
so this lacks quarter wave symmetry in such
cases what happens is it is a little difficult
447
00:41:45,500 --> 00:41:51,230
for us to judge where the fundamental phases
in a previous example let us say if the same
448
00:41:51,230 --> 00:41:57,930
thing happens to be a trapezoidal wave symmetric
trapezoidal wave we very easily say that the
449
00:41:57,930 --> 00:42:03,611
fundamental component is here we say that
this is the fundamental component the phase
450
00:42:03,611 --> 00:42:07,240
of the fundamental component is clear to us
whereas the phase of the fundamental component
451
00:42:07,240 --> 00:42:12,320
is not very clear to us and in cases where
there is quarter wave symmetry wherever the
452
00:42:12,320 --> 00:42:17,560
fundamental has zero crossing the harmonics
will also have zero crossings the harmonics
453
00:42:17,560 --> 00:42:25,960
will also have you you may have another harmonic
the harmonic will have a zero crossing here
454
00:42:25,960 --> 00:42:33,510
like this sorry m naught yeah whereas where
there is no quarter wave symmetry wherever
455
00:42:33,510 --> 00:42:37,930
the fundamental has a zero crossing the harmonics
need not have their zero crossing that is
456
00:42:37,930 --> 00:42:43,560
another issue it gives you certain information
about the phase of the harmonics so they are
457
00:42:43,560 --> 00:42:47,990
now whenever the fundamental crosses zero
the harmonic also crosses zero maybe you know
458
00:42:47,990 --> 00:42:52,050
from negative to positive or positive negative
but they also cross there is yet another thing
459
00:42:52,050 --> 00:42:56,740
that you have when you have you know when
you when you do not have quarter wave symmetry
460
00:42:56,740 --> 00:43:01,520
so this is one example that i gave to show
that you know waveform could have a half wave
461
00:43:01,520 --> 00:43:07,920
symmetry and thereby you know have no even
harmonics but still not have quarter wave
462
00:43:07,920 --> 00:43:18,340
ah symmetry i would give another one other
example
463
00:43:18,340 --> 00:43:25,860
this is a waveform this is time t this waveform
somehow looks nice but has no half wave symmetry
464
00:43:25,860 --> 00:43:36,030
if you consider this instant if you consider
a particular instant here and you consider
465
00:43:36,030 --> 00:43:42,950
the next instant here this is not the negative
of the other this has no half wave symmetry
466
00:43:42,950 --> 00:43:46,330
so this is not basically an example of half
wave symmetry but no quarter wave symmetry
467
00:43:46,330 --> 00:43:51,930
this has no half wave symmetry this is an
example of half wave symmetry but no quarter
468
00:43:51,930 --> 00:43:58,250
wave symmetry this is an example of both half
wave symmetry and quarter wave symmetry
469
00:43:58,250 --> 00:44:04,060
so you can certainly construct ah various
examples let me just take a triangular case
470
00:44:04,060 --> 00:44:10,150
let me say the waveform goes like this and
then comes back like this this is time t and
471
00:44:10,150 --> 00:44:17,570
it continues so this is t by two so if i if
you take any instant like here you consider
472
00:44:17,570 --> 00:44:23,710
some instant and a corresponding instant half
period away the values are one is the negative
473
00:44:23,710 --> 00:44:31,880
of the other so it has even i mean ah it has
half wave symmetry and has no even harmonic
474
00:44:31,880 --> 00:44:36,280
but this waveform once again lacks ah quarter
wave symmetry
475
00:44:36,280 --> 00:44:42,530
so these are some examples that you can really
think of and as i mentioned earlier this definition
476
00:44:42,530 --> 00:44:46,550
slightly vary i know some authors tend to
use different definitions now that is why
477
00:44:46,550 --> 00:44:51,190
i reemphasize you know for the purpose of
this course you will define quarter wave symmetry
478
00:44:51,190 --> 00:44:56,260
as here and half wave symmetry as here that
is v of one eighty plus theta as minus v of
479
00:44:56,260 --> 00:45:01,180
theta in the wave form satisfies that property
that is half wave symmetry and if we have
480
00:45:01,180 --> 00:45:05,320
something like v of ninety plus theta is ninety
minus theta or more generally v of theta plus
481
00:45:05,320 --> 00:45:10,550
plus theta is equal to v of theta p minus
theta where theta p is the instant at which
482
00:45:10,550 --> 00:45:15,260
the fundamental component is at its peak maybe
the positive peak or may be the negative peak
483
00:45:15,260 --> 00:45:19,330
theta p is the instant when the fundamental
component is at its peak this is theta p and
484
00:45:19,330 --> 00:45:24,350
this is also a candidate for theta p ok
then we call this as quarter wave symmetry
485
00:45:24,350 --> 00:45:28,520
if we have quarter wave symmetry over on top
of half wave symmetry then what we can say
486
00:45:28,520 --> 00:45:33,410
is the wave form not only has you know you
we can say that it has no even harmonics and
487
00:45:33,410 --> 00:45:38,320
you can say that you know you need to consider
only one quarter of the waveform starting
488
00:45:38,320 --> 00:45:43,211
from one zero crossing to the peak for example
or any of the four quarters like this of the
489
00:45:43,211 --> 00:45:48,050
ah basically the fundamental cycle to do your
pwm calculations you dont have to define the
490
00:45:48,050 --> 00:45:51,820
entire waveform over the entire cycle it is
enough if you define it it mathematically
491
00:45:51,820 --> 00:45:56,190
over one quarter so your calculation burden
simplifies and you can say something about
492
00:45:56,190 --> 00:46:00,050
the face of all the harmonics whenever the
fundamental cross is zero the harmonics also
493
00:46:00,050 --> 00:46:04,240
cross zero so these are a few things that
you can say when you have quarter wave symmetry
494
00:46:04,240 --> 00:46:07,760
to now
so now this is all about you know a kind of
495
00:46:07,760 --> 00:46:12,290
a review of fourier series so that we can
use fourier series effectively to calculate
496
00:46:12,290 --> 00:46:18,900
ah the fundamental voltages and harmonic voltages
of a several different ah you know pwm waveforms
497
00:46:18,900 --> 00:46:23,220
so now the next issue is when the harmonic
voltages are there what are their effects
498
00:46:23,220 --> 00:46:27,710
is what we are trying to see now now what
gets supplied to an induction motor for example
499
00:46:27,710 --> 00:46:34,280
you know like a an induction motor fed from
ah voltage source inverter ah is not only
500
00:46:34,280 --> 00:46:37,900
the sinusoidal fundamental component also
harmonics get fed there
501
00:46:37,900 --> 00:46:42,480
now let us say we consider only the fundamental
component if you see only the fundamental
502
00:46:42,480 --> 00:46:49,080
component the in it is it is a sinusoidal
quantity so the fundamental component sees
503
00:46:49,080 --> 00:46:53,960
the induction motor as its fundamental equivalent
circuit so which as the standard things this
504
00:46:53,960 --> 00:46:59,760
is your ah you know rotor ah stator winding
resistance and this is the stator leakage
505
00:46:59,760 --> 00:47:04,590
inductance and this is the magnetizing inductance
and you have a rotor leakage and then this
506
00:47:04,590 --> 00:47:08,340
rr is once again the rotor resistance and
s is the slip now
507
00:47:08,340 --> 00:47:13,980
so this rr by s can be divided into two parts
as rr and the remaining the rr alone stands
508
00:47:13,980 --> 00:47:18,860
for the rotor resistance and the remaining
part stands for the mechanical power developed
509
00:47:18,860 --> 00:47:23,910
the whatever power is dissipated through this
resistor is equal to the mechanical power
510
00:47:23,910 --> 00:47:29,440
developed by the machine now so this is an
equivalent circuit what do we mean by that
511
00:47:29,440 --> 00:47:36,200
so when applied you know when some fundamental
voltage is applied the current drawn by the
512
00:47:36,200 --> 00:47:42,610
machine is equal to whatever the this fundamental
circuit draws the same amount of current now
513
00:47:42,610 --> 00:47:47,610
that is there is a you know the terminal relationships
are the same now for a certain amount of fundamental
514
00:47:47,610 --> 00:47:52,040
voltage whatever is the current drawn by the
actual machine is the same as what is drawn
515
00:47:52,040 --> 00:47:56,060
by this circuit
so this circuit gives us a measure of what
516
00:47:56,060 --> 00:48:00,490
is the stator current and what is the magnetizing
current and what is the rotor current all
517
00:48:00,490 --> 00:48:05,340
these measures can be obtained from here now
and we can also see how much power gets dissipated
518
00:48:05,340 --> 00:48:09,250
here and from there you can also come up with
how much mechanical power is developed and
519
00:48:09,250 --> 00:48:13,730
etcetera so this is the fundamental equivalent
circuit the fundamental voltage the the inverter
520
00:48:13,730 --> 00:48:17,460
voltage has fundamental as well as harmonic
components the fundamental component c is
521
00:48:17,460 --> 00:48:23,230
the machine as some such equivalent circuit
here where the slip s is given by whatever
522
00:48:23,230 --> 00:48:30,400
is your synchronous speed omega s minus omega
r divided by omega s so omega s is the you
523
00:48:30,400 --> 00:48:35,290
know synchronous frequency that is if you
apply the stator frequency like fifty hertz
524
00:48:35,290 --> 00:48:40,340
or so ah the the revolving magnetic field
revolves at its synchronous ah frequency which
525
00:48:40,340 --> 00:48:46,960
is decided by you know the ah ah the frequency
of the applied voltage and the number of poles
526
00:48:46,960 --> 00:48:52,290
there there is something like that
so this is the synchronous ah speed the machine
527
00:48:52,290 --> 00:48:57,100
will also the synchronous speed if the machine
is not loaded but you know any machine as
528
00:48:57,100 --> 00:49:00,880
you know i even under the so called no load
condition has some amount of loading on that
529
00:49:00,880 --> 00:49:05,140
and when you actually load it what happens
is there is a slip the rotor speed is a little
530
00:49:05,140 --> 00:49:09,470
lower than the synchronous speed so that is
what you get by the term omega s minus omega
531
00:49:09,470 --> 00:49:14,100
r that is normalized with respect to omega
s gives you the slip so slip is zero under
532
00:49:14,100 --> 00:49:19,080
no load condition and the slip has some rated
value of a few percentage something like five
533
00:49:19,080 --> 00:49:23,030
percent or so when it is under rated condition
many typical machines now
534
00:49:23,030 --> 00:49:28,110
so what happens is the slip is really low
and therefore the term rr by s is significantly
535
00:49:28,110 --> 00:49:33,760
high that is what we have right so we move
on when we look at the harmonic voltage lets
536
00:49:33,760 --> 00:49:37,970
say some instead of the fundamental component
some nth harmonic voltage may be the fifth
537
00:49:37,970 --> 00:49:43,330
harmonic or may be the seventh harmonic is
applied now so what happens now in this case
538
00:49:43,330 --> 00:49:49,470
you as you find the slip the synchronous speed
is higher now the fundamental has fifty hertz
539
00:49:49,470 --> 00:49:52,870
the fifth harmonic is two hundred and fifty
hertz the seventh harmonic is three hundred
540
00:49:52,870 --> 00:49:58,470
and fifty hertz and fifth harmonic also revolves
in a direction opposite to that of the fundamental
541
00:49:58,470 --> 00:50:02,240
and so you know the seventh harmonic revolves
in the same direction
542
00:50:02,240 --> 00:50:06,710
now the synchronous speed to start with is
not omega s but it is n times omega s where
543
00:50:06,710 --> 00:50:13,110
n is the harmonic order and the relative speed
between the synchronous speed and the rotor
544
00:50:13,110 --> 00:50:20,570
is now it could be n omega s plus omega r
or minus omega r it is n omega s plus r minus
545
00:50:20,570 --> 00:50:26,170
omega r so if it were the fifth harmonic for
example this omega and the synchronous ah
546
00:50:26,170 --> 00:50:30,110
the the revolving magnetic field that the
fifth harmonic produces are in the opposite
547
00:50:30,110 --> 00:50:35,810
directions so the relative speed is n omega
s five omega s plus omega r if it were the
548
00:50:35,810 --> 00:50:40,030
seventh harmonic then the seventh harmonics
revolving magnetic field and the rotor revolve
549
00:50:40,030 --> 00:50:44,300
in the same direction it will be seven omega
s minus omega r so if this is what you have
550
00:50:44,300 --> 00:50:50,600
n omega s plus or minus omega r and you normalize
it with respect to n omega s
551
00:50:50,600 --> 00:50:56,040
if you do this this is almost equal to one
because n omega s is much higher than omega
552
00:50:56,040 --> 00:51:01,660
r so you you have something which is almost
close to one for high values of n so what
553
00:51:01,660 --> 00:51:07,890
do you have is rr instead of rr by s now rr
by s is big because s is small therefore rr
554
00:51:07,890 --> 00:51:13,830
by s is big so if if s is point zero five
rr by s is some twenty times rr now it is
555
00:51:13,830 --> 00:51:18,360
only rr rr by itself is only the winding resistance
of the rotor which is a very small number
556
00:51:18,360 --> 00:51:24,010
so what happens is it reduces slip is close
to one the the rotor side resistance is now
557
00:51:24,010 --> 00:51:27,780
very low ok
so you move to the next point so what you
558
00:51:27,780 --> 00:51:32,230
have is you basically had only the stator
winding resistance on the rotor winding resistance
559
00:51:32,230 --> 00:51:36,190
and when you are looking at harmonic frequencies
when you look at harmonic frequencies this
560
00:51:36,190 --> 00:51:42,800
omega l this actually offers n times omega
l where omega could be the fundamental frequency
561
00:51:42,800 --> 00:51:47,980
the harmonic frequencies n times omega n times
omega l is the reactance seen by this and
562
00:51:47,980 --> 00:51:53,270
n times omega l r is the reactance seen by
that so this is the total reactance now if
563
00:51:53,270 --> 00:51:58,380
you see these reactances this n omega ls for
example is much bigger than the rotor resistance
564
00:51:58,380 --> 00:52:06,370
rs and similarly this n omega lr the reactance
pertaining to lr is much larger than rr at
565
00:52:06,370 --> 00:52:10,740
the harmonic frequencies therefore you can
ignore those two for for the point of view
566
00:52:10,740 --> 00:52:15,420
of calculating how much harmonic current is
being drawn you can ignore those two now
567
00:52:15,420 --> 00:52:19,690
if you go further this inductance is much
larger than this inductance so this inductance
568
00:52:19,690 --> 00:52:23,520
is practically like an open circuit of the
parallel combination of this two almost dominated
569
00:52:23,520 --> 00:52:28,050
by this it is almost close to this value of
inductance and that is what we get here this
570
00:52:28,050 --> 00:52:31,950
is what is called as a harmonic equivalent
circuit of an induction motor the harmonic
571
00:52:31,950 --> 00:52:39,940
voltage that is applied essentially sees an
induction motor as an equal as its total leakage
572
00:52:39,940 --> 00:52:43,460
inductance
so if you want to calculate the i n so this
573
00:52:43,460 --> 00:52:52,220
i n is simply equal to your v n divided by
n omega l s plus l r is that right so this
574
00:52:52,220 --> 00:52:56,680
is the reactance pertaining to the leakage
reactance so you can very easily calculate
575
00:52:56,680 --> 00:53:01,450
that now you must remember that this equivalent
circuit is mainly to calculate i n you can
576
00:53:01,450 --> 00:53:05,210
calculate i n considering the resistances
for example you can consider the various other
577
00:53:05,210 --> 00:53:10,990
things but i n wont be significantly different
even if you neglected those but but it you
578
00:53:10,990 --> 00:53:16,830
know and you use this i n it does not mean
that you know there is no loss this is to
579
00:53:16,830 --> 00:53:21,440
calculate once you have calculated this i
n you can use this to calculate your copper
580
00:53:21,440 --> 00:53:24,160
loss
so now the current harmonic what what we see
581
00:53:24,160 --> 00:53:30,230
is if you take the stator winding resistance
like what is indicated here what flows through
582
00:53:30,230 --> 00:53:34,550
the actual stator winding is not only the
fundamental current also several harmonics
583
00:53:34,550 --> 00:53:41,890
flow so when i n flows that it also produces
i n squared r s will also produce you know
584
00:53:41,890 --> 00:53:44,970
nth harmonic current also produces certain
loss
585
00:53:44,970 --> 00:53:49,940
so the harmonic currents also produce the
stator copper loss is given by i squared r
586
00:53:49,940 --> 00:53:53,590
and in fact the resistance corresponding to
harmonics could be a little higher because
587
00:53:53,590 --> 00:53:57,960
of high frequency effects just the skin effect
so the resistance could really be higher too
588
00:53:57,960 --> 00:54:02,500
but we ignore those and ah you know what we
want to do is we want to consider the same
589
00:54:02,500 --> 00:54:07,630
resistance and want to be able to calculate
a value of i n and this is an approximation
590
00:54:07,630 --> 00:54:11,630
that is reasonably valid for calculating i
n now this is simplified analysis you can
591
00:54:11,630 --> 00:54:15,090
you can add more and more details to it but
simplified analysis gives you a first cut
592
00:54:15,090 --> 00:54:18,650
figure which you can relay on and also it
gives you a certain amount of insight what
593
00:54:18,650 --> 00:54:23,330
is that insight we will use this extensively
the harmonic you know well the fundamental
594
00:54:23,330 --> 00:54:28,560
equivalent circuit sees the machine as impedance
and that is a function of slip that is a function
595
00:54:28,560 --> 00:54:33,050
of slip the machine is seen as some impedance
which is a function of slip that is fuzzy
596
00:54:33,050 --> 00:54:35,640
fundamentally
but if you look at the harmonic the harmonic
597
00:54:35,640 --> 00:54:41,220
equivalent the harmonic voltage sees the machine
simply as reactance so if you take a particular
598
00:54:41,220 --> 00:54:44,970
harmonic voltage and a particular harmonic
current they are related as though you know
599
00:54:44,970 --> 00:54:49,430
their relationship is similar to that of the
voltage and current through an inductor now
600
00:54:49,430 --> 00:54:53,990
this is of consequence as we will see later
this will help us in analyzing the harmonic
601
00:54:53,990 --> 00:54:57,940
currents very effectively later on
now instead of a motor drive application if
602
00:54:57,940 --> 00:55:02,060
you take it to be a grid connected converter
you know the line side is basically connected
603
00:55:02,060 --> 00:55:06,060
to the the ac side of the inverter is basically
connected to the mains through line inductor
604
00:55:06,060 --> 00:55:10,820
l in that case what you have is vg is the
mains voltage this is what we saw in you know
605
00:55:10,820 --> 00:55:14,670
one or two lectures back we were discussing
front end converters and we were discussing
606
00:55:14,670 --> 00:55:19,551
statcom kind of applications this is the grid
voltage and this is the fundamental component
607
00:55:19,551 --> 00:55:22,850
of the inverter voltage and this is the ripple
part of the inverter voltage the ripple has
608
00:55:22,850 --> 00:55:27,390
fundamental i mean the the inverter output
voltage the terminal voltage has fundamental
609
00:55:27,390 --> 00:55:32,020
which is that line frequency plus some ripple
added to the the ripple is the sum of all
610
00:55:32,020 --> 00:55:35,290
the harmonics
if you want the fundamental equivalent circuit
611
00:55:35,290 --> 00:55:39,130
that is you want to calculate the fundamental
current drawn through this then you can just
612
00:55:39,130 --> 00:55:44,050
ignore the ripple here this is your equivalent
circuit now and if you are drawing certain
613
00:55:44,050 --> 00:55:49,390
amount of current i you can even draw a phasor
diagram this is vg and this is i that you
614
00:55:49,390 --> 00:55:54,270
want to draw and the oh the drop across the
inductance ah due to i is going to be some
615
00:55:54,270 --> 00:55:59,560
j omega l you subtract that this is going
to be minus j omega l i and this is what is
616
00:55:59,560 --> 00:56:03,320
going to be our v inverter voltage you can
very easily draw when the phasor diagram and
617
00:56:03,320 --> 00:56:06,350
you can relate all the various quantities
here
618
00:56:06,350 --> 00:56:09,890
so this is the fundamental equivalent circuit
and you move over this is the harmonic equivalent
619
00:56:09,890 --> 00:56:15,820
circuit which also we saw so here if the fundamental
components the grid of the grid and the inverter
620
00:56:15,820 --> 00:56:20,940
do not make any effect here and you can calculate
your nth harmonic current simply using your
621
00:56:20,940 --> 00:56:26,290
nth harmonic voltage and the inductance here
so these are the models which will help us
622
00:56:26,290 --> 00:56:30,240
calculate the harmonic currents in various
applications when based on those harmonic
623
00:56:30,240 --> 00:56:34,830
currents we may be able to calculates you
know these the copper losses and they now
624
00:56:34,830 --> 00:56:39,260
we may be able to calculate many other things
so what we have been looking at is these is
625
00:56:39,260 --> 00:56:43,400
a measure for calculating harmonic current
and from there we will be able to ah quantify
626
00:56:43,400 --> 00:56:48,620
some of the undesirable effects due to of
harmonics that is such as the increased copper
627
00:56:48,620 --> 00:56:51,640
loss etcetera now
so as i mentioned the purpose of pulse width
628
00:56:51,640 --> 00:56:56,330
modulation is to control the fundamental voltage
and to mitigate the harmonics and their harmful
629
00:56:56,330 --> 00:57:01,880
side effects so we have reviewed fourier series
which tells us how to calculate the fundamental
630
00:57:01,880 --> 00:57:07,510
and the harmonic components and we have developed
models so that you know given a fundamental
631
00:57:07,510 --> 00:57:11,741
voltage or a fundamental current you know
or harmonic voltage we are able to calculate
632
00:57:11,741 --> 00:57:14,510
the corresponding fundamental current and
harmonic currents
633
00:57:14,510 --> 00:57:20,430
so lets continue ah this in the next class
and get into or ah business of understanding
634
00:57:20,430 --> 00:57:25,240
pwm effectively i thank you very much for
your interest and for your time and your patience
635
00:57:25,240 --> 00:57:31,540
and i hope ah that you will you know have
continued interest in this lecture series
636
00:57:31,540 --> 00:57:33,390
and ah you know see you again in the next
lecture
637
00:57:33,390 --> 00:57:33,490
thank you very much