Resonance Video Lecture | Physics for JEE Main & Advanced

today we are going to talk about
resonance in AC circuits this is
something you actually use in your real
life because when you are tuning in to a
radio station you're actually using
resonance probably you know this that a
particular radio station transmits
electromagnetic waves of a special
frequency so each radio station has its
own frequency now while you turn the
knob of the radio set what you are doing
is you are wearing the capacitance of a
circuit over here and this in turn
varies the frequency of the LCR circuit
of the radio now when the frequency of
the incoming waves from the broadcasting
station match the natural frequency of
the radio circuit which is the frequency
of this LCR circuit then resonance
happens now how do you detect that that
is the point at which the reception has
maximum clarity and the current in the
circuit is maximum at that instant now
let's look at this a little in detail
if you recall the impedance of an LCR
circuit is given by Z equals square root
of XL minus XC whole square plus R
square in case you are not familiar with
this
I suggest you first go through the
previous lessons which is connecting an
AC across an a resistor and inductor a
capacitor an LCR circuit and then come
back to this lesson now coming back to
this the inductive reactance XL you know
is given by Omega times L and the
capacitive reactance
XC is 1 over Omega C now you can see
that since both of them depend on the
frequency Omega so the total impedance Z
will also change with the frequency C if
I draw a graph for each of them now this
is again something which has been
discussed earlier so in case you're not
familiar just go through it this would
be the graph for XL since the inductive
reactance is directly proportional to
frequency this would be the graph for XC
since the value becomes very low at high
frequencies this is not 0 it tends to 0
this is XC and this is actually the x
axis is actually an asymptote so it
doesn't really touch the axis and the
graph of the frequency would look like
this because the frequency doesn't
change with sorry the resistance does
not change with frequency the total
impedance Zed if I draw a graph it would
look like this this is the value for
this is the graph for Z and you can see
that Z has a minimum value at a
particular frequency which I am going to
call as f naught
if we also recall the phasor diagram
the let's say this is the current in the
circuit this is the voltage across the
resistance this is the voltage across
the inductor which leads the current by
90 and this is the voltage across the
capacitor which lags the current by 90
next let's look at the impedance the
impedance said is the value of Z will
depend on XL minus XC whole Square + R
square you can see that at low
frequencies the circuit is predominantly
capacitive and at high frequencies the
circuit is predominantly inductive now
there will be some value of a frequency
in between when the capacitive reactance
and the inductive reactance cancel each
other out this frequency is the resonant
frequency at resonance only the
resistance R contributes because XL
becomes equal to XC so what you are left
with Z is equal to simply R because this
is root R square which equals R and at
this instance when the impedance of the
circuit is minimum the current which is
given by V over Z will be equal to V
over R so the current will be maximum C
because obviously if the impedance of
the circuit is minimum the current has
to be maximum
now if you are asked what is resonant
frequency of an LCR circuit that is the
frequency at which the impedance Z of an
LCR circuit is minimum the resonant
frequency is also known as the natural
frequency of the circuit if the
frequency of the AC source that is
driving the system matches this natural
frequency of the LCR circuit resonance
occurs for instance in case of a radio
set this is a radio set now the incoming
frequency is say let's say it has a
particular value F and the natural
frequency of the LCR circuit of the
radio Omega is given by 1 over under
root LC this is because at resonance XM
is equal to XC which means Omega L is
equal to 1 over Omega C and Omega square
is 1 over LC and this implies that Omega
equals 1 by under root LC
when Omega is equal to 1 over under root
LC then since Omega is 2 pi F this can
be written as 1 over 2 pi root LC this
frequency is the natural frequency of
the LCR circuit and we are going to
denote this by F naught so when the
frequency of the LCR circuit F naught
matches the frequency of the incoming
waves which is F that which means when F
naught becomes equal to F in that case
resonance is said to occur and the
current in the circuit becomes maximum
so you might have noticed that the
resonant frequency F naught depends on
the inductor and the capacitor but it
does not depend on the resistance of the
LCR circuit it is independent of the
resistance
but the resistance of the LCR circuit
effects another very important factor
which is the quality factor and we need
to take a look at this in detail because
this tells us how selective is the
circuit what is the sharpness of
resonance so let's look at this a little
more in detail let's say we have a
circuit whose inductance is 2 Henry
capacitance is 0.5 micro farad and hence
the resonant frequency Omega is
thousands of thousands per second now
for different values of resistance we'll
plot a graph between the current the
current amplitude and the angular
frequency Omega which means as we vary
Omega we want to plot the value of the
current amplitude when the resistance is
2000 ohms which means it's a very large
value we get a graph which looks like
this now you can see that at very low
frequencies and at very high frequencies
the impedance Z is going to be very
large so the current is very small the
current is almost zero whereas at some
value in between which is the resonant
frequency and which in this case is
thousand radians per second the value of
the current has a maximum value now if
we take a lower value of resistance then
what do we find if we draw the graph
once again at very low frequencies and
at very high frequencies the value of
the current will be low but the value of
the current is maximum at a resonant
frequency see if you recall I is equal
to V over R next if we take a still
lower value of resistance then the value
of the current at resonance is going to
increase so we have a higher value of
the current at resonance
so now if you draw a graph once again at
very low frequencies and at very high
frequencies the value of the current is
very low and at resonance the value of
the current is maximum so this is when
let's say the resistance is 200 ohms and
this is when the resistance is 500 ohms
and this is the case when the resistance
is 2000 ohms
now if you recall at the beginning we
talked about tuning a radio set where
you are trying to match the frequency of
the incoming waves with the frequency of
the LCR circuit of the radio now
this response curve determines how
easily you can tune the radio set see if
you have a high value of resistance then
the curve falls off slowly and the
current doesn't change quickly as you
move away from the resonant frequency
this means it's going to be difficult to
tune in to a particular incoming
frequency because over the wide range of
frequency the amplitude of current is
pretty much the same it doesn't really
drop off sharply if you are not able to
detect resonance that means you will not
be able to catch a particular frequency
so the circuit has low selectivity but
in case the resonance is sharp then what
happens in this case the selectivity is
good it is easy to tune in to this
particular frequency because as you move
away from it the current Falls sharply
see we have two advantages at resonant
frequency the current is very large and
as you move away it falls off rapidly
which means at resonance you get a large
current which means at resonance the
station will be very clear and you as
you move away slightly then the clarity
becomes much lesser so it's easy to tune
into that radio station now if you want
to give a numerical value to the
sharpness of resonance then that is
called the quality factor cue
the quality factor Q is defined as Omega
naught by twice Delta Omega C Omega
naught is the resonance fleet is the
resonant frequency so this value is
Omega naught and two Delta Omega is
equal to Omega 1 minus Omega 2 and now
how have we chosen these points Omega 1
and Omega 2 Omega 1 and Omega 2 are on
either side of the resonant frequency
and these are points at which the power
in the circuit drops down to 1/2 at the
points Omega 1 and Omega 2 the average
power in the circuit is 1/2 its maximum
value
so these frequencies Omega 1 and Omega 2
are also known as half power points the
points at which the power falls down to
half in the circuit shown over here Q
for a circuit with the smallest
resistance which is 200 ohms is more
than Q for a for the circuit which has
the largest resistance so which means
that Q for the 200 ohms circuit is more
than Q for 500 and which is more than Q
for 2,000 on Whiston s'
if you look at this expression over here
you will see that Omega naught does not
depend on resistance but these two
frequencies which is Omega 1 and Omega 2
are going to depend on the resistance of
the circuit so the quality factor is
affected by the resistance in the
circuit now if you work this out and if
you use some algebra you finally get the
quality factor Q as Omega naught L over
R now a circuit is more selective with a
high value of Q which occurs when the
resistance is low or the resonant
frequency Omega naught times inductance
is made large
now the main points that we've discussed
today is the resonant frequency F naught
is equal to 1 by 2 pi under root LC at
resonance the total impedance Z is equal
to the resistance of the circuit at
resonance the current becomes equal to V
over R because the value of Z is our at
resonance the quality factor Q is
defined as Omega naught L by r just
notice this is a dimensionless quantity
higher the value of Q more selective the
circuit which means a high value of Q
gives us a sharp resonance and it
responds to a narrow range of
frequencies there is a low Q value
responds to a wide range of frequencies
and the resonance is not sharp here are
a couple of questions to test your
understanding try these questions the
answers are discussed a short while
later
now for the answers the phase of the
voltage is zero degrees relative to the
current resonant frequency depends upon
l and c the quality factor is this is
true because this is the same as Omega
naught l by r since at resonance Omega
naught L is equal to 1 by Omega naught C
impedance is minimum at resonance and it
is equal to our current is maximum and
it is equal to V over our resonance
occurs only in AC circuits are not in
mechanical systems no this is false that
can this can also occur in mechanical
systems