Sharpness of Resonance

Whenever a system is acted upon with a frequency equal or close to the natural frequency of the system, it has a tendency to vibrate with an increase in amplitude, which is known as resonance. The resonance frequency, also known as the resonant frequency, is the frequency at which the amplitude is the greatest. The resonance’s sharpness is determined by the Q factor. The term ‘resonance’ comes from the field of acoustics and is most usually linked with sympathetic resonance in musical instruments, which occurs whenever one string begins to vibrate and produce sound after another is touched. Studying resonance can help you appreciate the sharpness of resonance. We’ll learn about resonance sharpness and the Q factor in this essay.

Resonance

Resonance is defined as a system’s tendency to vibrate with an increase in amplitude when it is subjected to frequency excitation. The resonance frequency, or resonant frequency, is the greatest frequency at which the amplitude is comparatively maximal. The sharpness of the resonance is determined by the Q factor.

Sharpness of Resonance

The sharpness of the resonance is the time-dependent depletion of an oscillating wave. The most essential factor in determining it is the Q factor. Resonance sharpness is mostly governed by two factors:

Amplitude

The height of a wave going in a straight line is referred to as amplitude. The amplitude has an inverse connection with the sharpness of the resonance. As the amplitude grows, the sharpness of the resonance reduces. As the amplitude lowers, the sharpness of the resonance increases.

Damping

Damping is a phenomenon where the amplitude of a wave reduces with time. Damping has a direct relationship with the sharpness of the resonance. The sharpness of the resonance increases whenever the damping is increased, and vice versa.

What is the Q-Factor?

The Q factor is an acronym for the quality factor. It’s used to figure out the resonator’s centre frequency, bandwidth, and underdamped resonator.

Mathematically, it is written as:

Q=EstoredElost per cycle

Resonance in LCR Series Circuit

The phenomenon of resonance can be observed in a series LCR circuit. The circuit is in resonance at its resonance frequency fr , that occurs when XL=Xc. The resonant frequency is calculated using the following formula:

fr=0.5πLc

Now, f Equals fr at maximum current, based on the circumstances.

The following requirements apply to the RLC circuits in the provided sequence:

f < fr — Purely capacitive

f > fr — Purely inductive

f = fr — Purely resistive

Circuit in Resonance

The following properties are noticed whenever the circuit is in resonance:

The circuit’s impedance is equal to R if Z = R, and it is at its lowest value at resonance.

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

Quality Factor in Sharpness of Resonance

The quality factor (Q) is a measure of the sharpness of the resonance in a series RLC circuit. It’s written as:

Q=(rLR)

Power Factor

In an AC circuit, the power factor is described as the ratio of actual to perceived power dissipation, given as cos∅=R/Z.

The power factor of an AC circuit can range from 0 to 1.

It is 0 if the circuit is totally inductive, and 1 if the circuit is entirely resistive.

The Sharpness of Resonance Diagram

The sharpness of the resonance diagram is given below:

The Sharpness of Resonance Formula

The resonance frequency is independent of resistance R, while resonance sharpness is inversely proportional to R. The sharpness of resonance is ω02ω. Although ω0LR is referred to as the circuit’s quality factor Q.

Things to Remember

Resonance describes the tendency of a system to vibrate as the amount of the stimulation of frequencies increases.

The sharpness of the resonance is measured using the Q factor, which quantifies how quickly energy decays in an oscillating system.

As damping increases or decreases, the sharpness of resonance increases or decreases, and as the amplitude expands, the sharpness of resonance reduces.

The Q factor, also referred as the quality factor, is a dimensionless parameter which describes an underdamped resonator’s bandwidth and centre frequency.

fr=0.5Lc is the formula for computing the resonant frequency in an LCR series circuit.

The quality factor (Q) in a series RLC circuit is computed as Q=(rL/R).

The power factor in an AC circuit is the ratio of actual to perceived power dissipation.

Conclusion

Resonance is defined as a system’s tendency to vibrate with an increase in amplitude when it is subjected to frequency excitation. The resonance frequency, or resonant frequency, is the greatest frequency at which the amplitude is comparatively maximal. The sharpness of the resonance is the time-dependent depletion of an oscillating wave. The most essential factor in determining it is the Q factor. The quality factor (Q) is a measure of the sharpness of the resonance in a series RLC circuit. It’s written as:

Q=(ω0L/R)