Sharpness 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. 


The Sharpness of Resonance Definition:

The depletion of an oscillating wave with respect to time is called the sharpness of the resonance. 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 characterize the centre frequency and bandwidth of resonator and the underdamped resonator. 

It is represented mathematically as:

Q= Estored/Elost 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.