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 -
\[\nu = \lambda f\]
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 on Sharpness of Resonance
1. What is meant by the sharpness of resonance in a series LCR circuit?
Sharpness of resonance describes how quickly the current in a series LCR circuit falls from its maximum resonant value as the frequency of the AC source is changed from the resonant frequency. A sharp resonance means the current peak is very narrow and high, indicating the circuit is highly selective to a specific frequency. This occurs when the damping in the circuit is low.
2. How is the sharpness of resonance measured?
The sharpness of resonance is quantitatively measured by a dimensionless parameter called the Quality Factor or Q-factor. A higher Q-factor signifies a sharper, more defined resonance peak, while a lower Q-factor indicates a flatter, broader resonance. The formula for the Q-factor in a series LCR circuit is given by Q = ω₀L / R, where ω₀ is the resonant angular frequency, L is the inductance, and R is the resistance.
3. What is the key difference between a sharp resonance and a flat resonance?
The primary difference lies in their frequency selectivity and the effect of damping. Here’s a comparison:
- Sharp Resonance: Occurs in circuits with low resistance (low damping). The current amplitude is very high at the resonant frequency but drops off steeply for other frequencies. It has a high Q-factor and a narrow bandwidth.
- Flat Resonance: Occurs in circuits with high resistance (high damping). The current amplitude at resonance is lower, and the peak is broad, meaning the circuit responds to a wider range of frequencies around the resonance point. It has a low Q-factor and a wide bandwidth.
4. How does damping in an LCR circuit affect the sharpness of its resonance?
Damping, which is primarily caused by the resistance (R) in the circuit, is inversely related to the sharpness of resonance. When resistance is low, damping is minimal, and less energy is dissipated per cycle. This allows the current amplitude to build up to a high, narrow peak, resulting in a sharp resonance. Conversely, when resistance is high, damping is significant, more energy is lost, and the current peak becomes broad and less pronounced, leading to a flat resonance.
5. Why is a high sharpness of resonance crucial for tuning a radio or a TV set?
High sharpness of resonance is crucial for tuning because it allows the device to be highly selective. A radio receiver's tuning circuit (an LCR circuit) is designed to have a very high Q-factor. This creates a sharp resonance, enabling the circuit to respond strongly to the frequency of the desired station while effectively rejecting the signals from all other stations at nearby frequencies. This ensures a clear reception without interference.
6. What is the relationship between the bandwidth and the sharpness of resonance?
Bandwidth and sharpness of resonance are inversely proportional. Bandwidth is the range of frequencies over which the power in the circuit is at least half of the maximum power. A circuit with a sharp resonance has a very narrow bandwidth, meaning it is sensitive to a small range of frequencies. A circuit with a flat resonance has a wide bandwidth, making it responsive to a larger range of frequencies.
7. What are some real-world examples where the principle of resonance is applied?
Resonance is a fundamental phenomenon with many applications, including:
- Musical Instruments: The body of a guitar or violin resonates at the frequencies produced by the strings, amplifying the sound.
- Microwave Ovens: They use microwaves at a frequency that matches the natural resonant frequency of water molecules, causing them to vibrate rapidly and generate heat.
- Swings: Pushing a swing at its natural frequency (in time with its motion) is an example of mechanical resonance, which increases the swing's amplitude efficiently.
- Bridges: While resonance is useful, it can be destructive. Engineers must design bridges to avoid resonating with external forces like wind or marching soldiers, which could cause structural failure, as seen in the Tacoma Narrows Bridge collapse.
