Sharpness of Resonance

Meaning of Resonance

To define the Sharpness of Resonance, it’s necessary to understand what Resonance is? The term ‘Resonance’ derived from the field of acoustics, especially the sympathetic Resonance detected in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck. Let’s know the definition of Resonance in brief to have a better understanding on the ‘sharpness of Resonance’. 

Resonance Definition:

As the amplitude increases with the excitation of frequencies, a system's tendency to vibrate also increases. This is defined as Resonance. The maximum frequency following which the amplitude is also maximum is called resonant frequency. To define the Sharpness of Resonance, the Q factor is used. When the matching vibrations of an object make another object’s oscillations increase its amplitude, this procedure is called Resonance.

In Physics, Resonance can also be defined as a phenomenon in which an external force or an external vibrating system makes another vibrating system (lying near the external vibrating system), vibrate at a higher level of frequency and amplitude at a specified frequency of operation than the system's natural set of frequency (natural harmonic frequency).

The formula used to calculate the resonant frequency of a single continuous-wave can be defined as -

= f

There are many types of Resonance that exist in this Physical world, some of them are as follows:

Mechanical Resonance: One great example of mechanical Resonance is swings.

Acoustic Resonance: The classic example of acoustic Resonance, can be the breaking of glass with nothing but sound at the precise resonant frequency of the glass.

Electrical Resonance: The various types of Resonance that happen in electrical circuits are all examples of electrical Resonance.

 

The Sharpness of Resonance Definition:

The depletion of an oscillating wave with respect to time is called the sharpness of the Resonance. It relates to the energy decay that happens in an oscillating system. It is mainly defined by the Q factor. 

The Sharpness of Resonance is dependent on mainly two factors. These are:

  • Amplitude

  • Damping

Amplitude:

It is defined as the height of a wave which is moving in a uniform motion. The amplitude varies inversely with the sharpness of the Resonance. Higher the amplitude, less is the sharpness of the Resonance. And lesser the amplitude, higher is the sharpness of the Resonance. 

Damping:

It is defined as the effect in which the amplitude of the wave is reduced with time. It can be both artificial or natural. Damping is directly related to the sharpness of the Resonance. An increase in damping leads to an increase in Resonance's sharpness, and the converse also holds true.

 

Definition of Q Factor:

After knowing Resonance and Sharpness of Resonance, let us help you understand what Q factor is and its use. 

Q factor stands for the quality factor. It does not have any dimensions. It is used to characterise the centre frequency and bandwidth of the resonator and the underdamped resonator. 

It is represented mathematically as:

Q= Restore/Lost per cycle

For an RF resonant circuit, the Q factor is given with:

Q= F0/F3dB

 

Resonance in Series LCR Circuit:

The phenomenon of Resonance can be observed in an LCR Circuit arranged in series. The circuit is in Resonance during its Resonance frequency fr which happens when XL=Xc

The resonant frequency is given by the formula:

fr= 1/2π√LC

Now, according to the conditions:

When current is maximum: f=fr

The following conditions are applicable in the given series of RLC circuit:

  • f<fr: Purely capacitive

  • f>fr: Purely inductive

  • f=fr: Purely resistive

During Resonance:

The following aspects are observed when the circuit is in Resonance:

  • Z=R, this implies that during Resonance, the impedance of the circuit is equal to R and is at its minimum value. 

  • The RMS (root mean square) value in the circuit is at its maximum, and the Resonance is equal to Vrms/R. 

  • The current and applied voltage are in phase.

  • The power dissipation taking place in the circuit is in their maximum value.

 

Quality Factor: Sharpness of Resonance

The quality factor (Q) is a measure of the Sharpness of Resonance in the series RLC circuit. It is given by:

Q= (⍵rL/R)

Power Factor:

In an AC circuit, the power factor is defined as the ratio of true power dissipation to the apparent power dissipation, represented using:

cos Φ= R/Z

The power factor for the AC circuit lies between the range of 0 and 1. 

Purely inductive circuit= 0

Purely resistive circuit= 1

FAQs (Frequently Asked Questions)

1. What is Wattless Current?

The phase difference between the current and voltage is 90° when there is only inductance or capacitance in the electric circuit. In such circuits, the average power dissipation that takes place is zero.

2. What is the Bandwidth of Resonance Circuits?

The bandwidth of resonance circuits is defined as the frequencies at which the power passed through the circuit has dropped to half the value which was passed at resonance. The formula gives it:

B= ⍵2-⍵1= R/L

3. What is the Quality Factor in a Circuit? Give an Example.

The quality factor is a parameter for the measure of resonance. The formula gives it:

Q= ⍵0L/R

Where, ⍵0= 1/√LC, which is the resonant frequency. 

What is the quality factor of the LCR circuit, where L-C-R are 4mH, 40pF, 100Ω?

Q=1/R*√L/C =100. 

Q is a dimensionless parameter. 

4. What are some real-life examples of Resonance?

Resonance can be observed in our daily lives, some examples of Resonance can be as follows.

Swing: A swing, when pushed, moves forward and backwards and with a series of pushes, the motion can be built. When pushing the swing, Note that the push should be given at the same interval which is the natural rotation of the swing, which will result in the increase in dimension of that swing.

Once the reaches its natural oscillation frequency, here because of the effect of Resonance, a gentle push will be enough to maintain its amplitude, which previously required more push. On the contrary, the swing might start to vibrate hardly, if the push is given irregularly and out of sync motion will never lead to Resonance, and the swing can not reach any higher in amplitude.

Musical Instruments and Sound: Music instruments have special arrangements to produce certain sound Resonance. The musical instrument is set into oscillation or vibration at the natural frequency of vibration of the instrument, whenever they are hit or struck. All specific instruments have a  unique standing wave pattern. These unique natural frequencies are also known as harmonics of that specific musical instrument. If two interconnected objects oscillate at a specified frequency, then the first object can be forced to vibrate at a higher level of frequency than its natural harmonic frequency; this phenomenon is known as Resonance in sound or musical instruments.

Other real-life examples of Resonance can be found in Pendulums and Bridges.

5. What are the advantages and disadvantages of Resonance and Sharpness of Resonance?

We knowingly or unknowingly stumble upon numerous examples of Resonance in everyday life. We had discussed some examples of Resonance on FAQs written above, let us now discuss how they provide any advantage or disadvantage to us. A swing can be considered a great example of mechanical Resonance in everyday life. If a series of regular pushes are given to the swing at the right moments, then its movement can be built up tremendously. But in case, pushes are given irregularly, the swing will hardly vibrate.

A major disadvantage of Resonance can be considered its potential to break apart a bridge. A group of marching soldiers are asked to break their marching steps, as their synchronised march has the potential to break the bridge apart if the synchronised footsteps resonate with the natural frequency of the bridge.

6. What is the quality factor of the LCR Circuit, where L-C-R are 4mH, 40pF, 100Ω?

The formula is Q = 1rLC

Applying values Q = 1100440

Ans = 1/10010

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