Series Resonance in RLC Circuit - YouTube

Hey friends, Welcome to the YouTube channel ALL ABOUT ELECTRONICS.

So, in this video, we will see what is Resonance in the series RLC circuit.

Or simply, Series Resonance in the RLC circuit.

So, first of all, intuitively we will understand what do we mean by this resonance in the series

RLC circuit and we will see the related parameters to this series RLC circuit.

And then after mathematically we will derive all the expressions related to this series

RLC circuit.

So, first of all, let us understand what is resonance in the electrical circuit.

So, in electrical circuit resonance is the phenomenon at which response of the circuit

is the maximum for a given particular frequency.

So, let's say we have one circuit which contains resistor, inductor and capacitor.

Now, let's assume that to this circuit we have applied some AC voltage source of a particular

frequency.

And also assume that this voltage source is tunable.

Now, for the given frequency at the output, we will get some response.

So, now as we tune this voltage source, at one particular frequency it will happen that

the response of the circuit will be maximum.

So, this phenomenon is known as the resonance in the electrical circuit.

And that particular frequency at which we are getting the maximum output is known as

the resonant frequency.

Now, this resonance phenomenon we are seeing in the electrical circuit because of the capacitor

and the inductor.

As, both the elements have the ability to store the energy.

So, now let's understand this resonance phenomenon by taking simple LC circuit.

So, let's say we have one LC circuit and in this LC circuit, we have applied some voltage

source.

And at time t=0, we are removing this voltage source and replacing it with a short circuit.

So, as we know the inductor and capacitor have the ability to store the energy.

So, even if we remove this voltage source, the inductor will continue to supply current

to this capacitor and through that current capacitor will get charged.

And later on, this capacitor will act as a voltage source and it will again supply energy

back to the inductor.

So, in this way, the energy will get transferred between the inductor and capacitor.

And the rate of transfer of this energy depends upon the value of this L and C.

And because of this transfer of energy, we will see the oscillation in our circuit.

So, now if the circuit is ideal and don does not have any resistance involved, then this

oscillation will continue for ever.

But all the circuits, have some resistance involved in the circuit.

So, because of that resistance, the energy will get dissipated across that resistor and

eventually this oscillation will die out.

So, to maintain this oscillation, we need to supply some external source with same frequency

to this LC circuit.

So that the oscillation will continue for ever.

So, now we know that this capacitor and inductor have a tendency to produce the oscillation

in the circuit while this resistor has the tendency to reduce this oscillation.

So, now let's see the resonance condition in the series RLC circuit.

So, now let's say in this circuit we have applied AC voltage source which is having

a value of V0*Sin(wt).

And because of this voltage source, we are getting some current through this circuit.

So, now as we change the frequency of this voltage source, then at a particular one frequency

we will get maximum current that is flowing through this circuit.

And that is known as the resonant condition.

So, if we see graphically this phenomenon, it will look like this.

At lower frequency, there will not be any current that is flowing through the circuit.

And also at a higher frequency, if you see, there will not be any current that is flowing

through the circuit.

So, as we move towards the resonant frequency, you will see that the current that is flowing

through the circuit will suddenly increase.

And resonant frequency, you will find the maximum current that is flowing through the

circuit.

And again as we move away from this resonant frequency, the current will fall down.

And eventually, at higher frequencies, this current will go to zero.

Now, we know that XL can be given as wL and Xc can be given as 1/(wc)

So, at w=0, if you see, this Xc term will be infinity.

And XL will be zero.

So, in this circuit capacitor will act ass open circuit.

And hence you will not find any current.

Similarly, at w is equal to infinity if you see, this XL term will be infinity.

So again at higher frequencies, this inductor will act as an open circuit.

And again you will not find any current that is flowing through the circuit.

So, that is the reason, at lower and higher frequencies, the current that is flowing through

the circuit is minimum.

While at the resonant frequency, XL=XC and hence the impedance of the circuit will be

purely resistive, where you will find the maximum current that is flowing through the

circuit.

we will see more about it when we derive all the expressions related to this RLC circuit.

So, now this RLC circuit basically provides some kind of selectivity for the given particular

frequency.

And this principle is used in radio communication for selecting the particular channel.

Let's say you have FM receiver and you want to tune your FM receiver at one particular

frequency or one particular channel.

So, now let's say we want to change our channel.

So, what we used to do, we used to change the knob on the receiver.

So, eventually, what we are doing, we are changing the capacitor value in the circuit.

So, if we change the value of this capacitor, then this curve will get shifted to some another

frequency.

Let's say Wr2, and in this way, we are able to change the channel or frequency of the

given receiver.

So, this principle is used mainly in radio communication for selecting a particular channel.

So, now for the good selectivity, the resonant curve should be as narrower as possible.

So, that you will not find any interference from the another channel.

And to achieve this, the value of resistance in the circuit should be as low as possible.

So, here as you can see, we have three different resonant curves for the different values of

the resistor.

As, the value of resistance increases, the curve is getting wider and wider.

And as the value of resistance decreases, the curve is also getting narrower.

So, for the good selectivity of a channel, the value of resistance should be as low as

possible.

in the circuit.

Also, at the lower resistance, you will see that peak current that we are getting is also

maximum.

So, as the value of resistance reduces, the current that you get at resonant frequency

will also increase.

So, to define the sharpness of the curve or the selectivity of the circuit, two parameters

are widely used for the resonant circuit.

which are Quality factor and Bandwidth.

So, let's see them one by one.

So, now, first of all, let's talk about the quality factor.

So, this quality factor is defined as 2*pi* (Maximum energy that is stored in the

element in the circuit divided by the energy that is dissipated across the resistor.

So, as we already know, as the value of resistance in the circuit increases, the circuit will

have less tendency to oscillate.

And because of that if you see the resonant curve, it will also get wider.

So, basically, this quality factor defines the sharpness of this resonant curve.

Or simply to defines the selectivity of the series RLC circuit.

So, we will derive the expression for this quality factor in terms of the resistor, capacitor

and inductor at the later part.

Similarly, now let's see about the bandwidth.

So, now this bandwidth is defined as the difference between the -3dB frequencies in the given

circuit.

Or it also, defined as the difference between the half power frequencies.

Now, let's, first of all, understand what do we mean by this -3 dB frequencies or half

power frequencies.

So, this half power frequency is the nothing but the frequency at which the power of the

circuit is getting reduced to the half.

Or if you see the current, in terms of the current the value of current is getting reduced

to the 1/√2 of the maximum value.

So, this f1 is known as the lower cut off frequency and this f2 is known as the upper

cut off frequency.

So, the difference between this f2 and f1 is known as the bandwidth.

Or if we talk about in terms of the angular frequency, the difference between w2 and w1

defines the bandwidth of the circuit.

So, for the good selectivity, the bandwidth of the circuit should be as narrower as possible.

So, we will also derive this bandwidth in terms of the resistor, capacitor and inductor

of the given circuit.

So, so far we have intuitively seen that what do we mean by the resonance in the electrical

circuit and what are the different parameters related to this series resonant circuit.

So, now let's derive mathematical expressions for this series resonant circuit.

So, let's once again assume that we have applied voltage

V= V0*sin(wt) in our series RLC circuit.

And because of that current, I is flowing in the circuit.

So, now this current I can be defined as V/Z, where Z is nothing but R+ XL + XC.

That is the series combination of R, XL and XC.

Where XL is nothing but jwl and XC is nothing but (-j/wc)

So, we can write it as R+ j[wl- (1/wc)] So, that is how we can define the impedance

of the circuit.

So, we know that at the resonant frequency, we are getting the maximum amount of current.

So, to get the maximum amount of current, the value of impedance should be minimum.

And this minimum value of impedance is possible when this term is zero.

That means, when wl - 1/wc is zero, then we will get minimum value of impedance.

And the value of impedance will be nothing but R.

So, this is known as the resonant condition.

That means, when XL = XC or wl = 1/wc, then you will get minimum value of impedance in

the circuit and hence the maximum amount of current will flow through the circuit.

And that maximum current can be defined as V/R.

Now, we can write this wL=1/wc as w^2 = 1/LC

Or we can say that w= 1/√LC So, this frequency is known as the resonant

frequency of the circuit.

So, if you write in terms of the frequency, then we can write it as

f= 1/(2π√LC), so this frequency is known as the resonant frequency for the given circuit.

So, now let's derive the expression for the bandwidth in terms of the R, Land C.

So, as we have seen so far, this f1 and f2 are the lower and upper cut off frequencies

respectively.

And those frequencies, the value of current will be nothing but 1/√2 of the maximum

value.

So, now we can say that at frequency w1, let's say the current that is flowing in the circuit

is I1.

And it can be defined as Imax/√2 Now, this current I1 can be defined as

V/√(R^2+〖(ω1L-1/ω1C)〗^2 ) Now, squaring up at both sides we will get,

I1^2= V^2/(R^2+(ω1L-1/ω1C)^2 )= Imax^2/2 Which is nothing but V^2/(2R^2)

So, now if we compare this two terms then we can

write it as, R^2=(ω1L-1/ω1C)^2 Or we can say that (ω1L-1/ω1C)= ±R

Similarly, we can get another expression for the frequency w2 as (ω2L-1/ω2C)= ±R

Which is the expression at frequency w2.

Now, this w1 is less than wr. that is this frequency is less than the resonant frequency.

And this frequency w2 is greater than wr.

That means this w2 is greater than the resonant frequency.

So, for the frequencies which are lesser than the resonant frequency, the circuit will have

capacitive reactance.

Because this XC is given as 1/wc.

And this term will be dominant over this w1L.

So, the value of w1L - 1/w1C will be negative.

Because this term is greater than this w1L.

Similarly, the frequencies which are greater than the resonant frequencies, at higher frequencies

if you see.

this inductive term will be dominant.

So, this w2L term will be greater than 1/w2c term.

And because of that if you see, this w2L - 1/w2c term will be positive.

So, in this way, we got two different equations for the upper and lower cutoff frequencies.

So, let's simplify these two equations.

So, if we simplify this equation then we will get,

〖ω1〗^2 LC-1=-Rω1 C And if we simplify it then we will get,

〖ω1〗^2+R/L ω1-1/LC=0 And if we simplify it then we will get two

roots of this equation, that is ω1= (-R)/2L±√(〖(R/2L)^2+1/LC)〗 Now, the frequencies can not be negative so

we will not consider the negative root.

So, we will get ω1= (-R)/2L+√(〖(R/2L)^2+1/LC)〗 And similarly, we can derive the expression

for w2 as, ω2= (R)/2L+√(〖(R/2L)^2+1/LC)〗

Now, the bandwidth can be defined as

the difference between this upper and the lower cutoff frequencies.

So, if we subtract these two equations then we will get,

R/2L -(-R/2L)= R/L So, we can define the bandwidth as nothing

but R/L. So, now let's multiply this w1 and w2.

So, if we multiply these two equations, then we can get

w1w2 = 〖(R/2L)^2+1/LC)-(R/2L)^2.

In short, we will get w1w2= 1/LC that is nothing but wr^2

That is square of the resonant frequency.

So, in this way, we will get wr^2= w1w2 So, this is the relation between the resonant

frequency and the upper and the lower cutoff frequencies.

So, now let's derive the expression for this quality factor in terms of the circuit components.

So, as we have discussed earlier, this quality factor is defined as

2*Pi * maximum amount of energy that is stored in the element over the energy that is dissipated

across the resistor.

And this quality factor is defined as Q. So, this Q is nothing but energy that is stored

across either inductor or capacitor divide by the energy that is dissipated across the

resistor.

So, we can say that I^2 XL/(I^2 R) Or simply, we can write it as XL/R

Now, at resonant frequency, this XL = XC So, we can also write it as XC/R.

Now, we know that XL is nothing but wL/R where w is the resonant frequency.

So, we can also write it as 1/wcR So, this equation of quality factor in terms

of the circuit components.

So, now we know that at resonant frequency, this wr can be defined as 1/√LC)

So, we can write this equation as 1/R √(L/C) So, this is the equation of the quality factor

in terms of the circuit components.

Now, if you see here, this R/L is nothing but the bandwidth of the circuit.

So, we can also write this quality factor Q as

w/Bandwidth Or wr/bandwidth

Where wr is the resonant frequency and this R/Lis nothing but bandwidth of the circuit.

So, in this way, we can write this quality factor as

1/R √(L/C) And also in terms of the bandwidth, we can

write it as wr/bandwidth

So, in summary, if you see, these are the equations that we got for the series resonant

circuit.

So, I hope in this video, you underestood what do we mean by the resonance in the series

RLC circuit.

And what are the different parameters which are related to this sereis RLC circuit like

quality factor and the bandwidth.

So, now in the next video we will see the resoance for the parallel RLC circuit.