Resonant Frequency Explained: Key Concepts & Applications

The phenomena of resonance is common among systems that have a tendency to oscillate at a particular frequency. This frequency is called the natural frequency of oscillation of the system. If such a system is driven by an energy source, whose frequency is equal to the natural frequency of the system, the amplitude of oscillation becomes large and resonance is said to have occurred.

For example, a child on a swing has a natural frequency for swinging back and forth. If another child pushes the swing at regular intervals and the frequency of the pushes is almost the same as the frequency of swinging, the amplitude of the swing becomes very large. We say that resonance has occurred.

Hermann Helmholtz

Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physicist and physician who made significant contributions in a number of domains. The Helmholtz Association, Germany's largest association of research institutions, is named after him.

He is most known in physics for his theories on energy conservation, electrodynamics, acoustics, and a mechanical underpinning of thermodynamics.

Helmholtz released Sensations of Tone in 1863, reflecting his continued interest in sensory physics. The musicologists of the twentieth century were affected by this book. To determine the various frequencies or pitches of the pure sine wave components of complex sounds containing numerous tones, Helmholtz devised the Helmholtz resonator.

What is Resonance?

Objects, mechanical systems, and charged particles all have a frequency when they vibrate at. This frequency is referred to as the resonance frequency or natural frequency. When a light or sound wave impacts an item that is already vibrating at a specific frequency, and that frequency happens to match the resonant frequency of the thing being struck, you get resonance. When the matching vibrations of another item enhance the amplitude of an object's oscillations, this is known as resonance.

What is Resonant Frequency?

In physics, the resonant frequency is defined as a physical process in which an external applied force or any vibrating system imposes the force on another system such that it vibrates with greater amplitude at a particular frequency of operation. Such frequency is known as the resonant frequency.

The resonant frequency formula varies depending on the type of wave and the system we are using. For example

1. The resonant frequency of a single continuous wave can be determined by the following formula:
   $v=\lambda f$

$f=\dfrac{v}{\lambda}$

Where,

v - The velocity of the wave

𝜆 - The wavelength of the wave used

2. The resonant frequency of a spring system can be determined by using the given formula:

$f_0=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$

Where.

k - The spring constant

m - The mass attached to the spring

3. Electrical resonant frequency can be determine by following equation:

$\omega=\dfrac{1}{LC}$

Expression for electrical resonant frequency in case of a series resonance circuit

A circuit in which inductance L, capacitance C, and resistance R are connected in series and the circuit admits maximum current corresponding to a given frequency of a.c., is called resonance circuit. The circuit diagram of the series resonance is shown below,

An RLC circuit's impedance (Z) is given by,

$Z=\sqrt{R^2+\left(L\omega-\dfrac{1}{c\omega}\right)^2}$…………(1)

At very low frequencies the inductive reactance $X_L=L\omega$ is negligible, but the capacitive reactance $X_C=\dfrac{1}{C\omega}$ is very high. As the frequency of alternating e.m.f applied to the circuit is increased, XL goes on increasing and XC goes on decreasing. For a particular value of ⍵=⍵r (say) then XL=XC. So we can write,

$\omega_r L=\dfrac{1}{C\omega_r}$

Taking the wr to one side of the equation and the rest of the terms to the other side,

$\omega_r^2=\dfrac{1}{LC}$

So, $\omega_r=\dfrac{1}{\sqrt{LC}}$

As we know that $\omega_r=2\pi\nu_r$, so putting this value into the above equation,

$2\pi\nu_r=\dfrac{1}{\sqrt{LC}}$

Taking the 2𝜋 to other side of the equation,

$\nu_r=\dfrac{1}{2\pi\sqrt{LC}}$

As XL=XC at this particular frequency $\nu_r$, therefore, we can write the equation (1) as,

$Z=\sqrt{R^2}$

After solving we can get,

Z = R = minimum

That means in a series RLC circuit the impedance is minimum and hence current will be maximum. This frequency is called resonance frequency. The variation of circuit current with the changing frequency of applied voltage is shown in the figure below,

It is clear that for frequencies greater than or less than wr the value of current is less than the maximum value (I0). Further at ⍵ = ⍵r , the current is maximum. The value of maximum current is inversely proportional to R. For lower R values, I0 is large and vice-versa. As a series resonance circuit admits maximum current through it, it is called an acceptor circuit.

Application of resonant frequency

The practical application of series resonance circuits is in radio and TV receiver sets. A radio/TV antenna picks up signals from a variety of broadcasting channels. To receive a certain radio station or television channel, we adjust the capacitance of a capacitor in the receiver's tuning circuit until the circuit's resonance frequency is identical to the chosen station's frequency. Therefore, resonance occurs. The amplitude of current with the frequency of the signal from the desired station becomes maximum and it is received in our set.

Solved Examples

1. What is the resonance frequency of a circuit with a 30 mH inductance and a 5 F capacitance?

Sol: To solve this problem we have to use the resonance frequency equation and by putting the values of capacitance and inductance we can find the answer. So let’s start the solution by writing the resonance frequency expression as,

$\nu_r=\dfrac{1}{2\pi\sqrt{LC}}$

From the question the value of inductance is $L=30 ~mH=30\times10^{-3}~H=3\times10^{-2}~H$ and capacitance is C = 5 F. Now putting these value to the resonance frequency equation as,

$\nu_r=\dfrac{1}{2\pi\sqrt{3\times10^{-2}\times 5}}$

After putting the value of 𝜋 = 3.14 and doing the calculations we can write,

$\nu_r=\dfrac{1}{2\times3.14\times\sqrt{3\times10^{-2}\times 5}}$

After solving the terms of the square root we write the expression as,

$\nu_r=\dfrac{1}{6.28\times0.387}=\dfrac{1}{2.432}=0.411~Hz$

Hence, the resonance frequency of a circuit with 30 mH inductance and 5F capacitance is 0.411 Hz.

2. Determine the value of capacitance of a circuit if the inductance of a circuit is 20 mH and its resonant frequency is 0.51 Hz.

Sol:

In order to solve the above mentioned problem we have to use the resonant frequency formula and by putting the values of given quantities in the expression we can find the correct answer. So the expression of resonance frequency is given as,

$\nu_r=\dfrac{1}{2\pi\sqrt{LC}}$

We can also write the expression as,

$\nu_r^2=\dfrac{1}{4\pi^2LC}$

After rearranging we can write,

$C=\dfrac{1}{4\pi^2L\nu_r^2}$

According to the question, the value of resonance frequency is 𝜈r = 0.51 Hz and inductance is, $L=20 ~mH=20\times10^{-3}~H=2\times10^{-2}~H$.

Now putting these value in the above expression,

$C=\dfrac{1}{4(3.14)^2(0.51)^2(2\times10^{-2})}$

After solving the denominator terms we can write,

$C=\dfrac{1}{4\times9.85\times0.26\times(2\times10^{-2})}$

On further solving we can write,

$C=\dfrac{1}{20.48\times10^{-2}}=4.88~F$

Hence, the value of capacitance of a circuit is 4.88 F.

Interesting facts

• A guitar string that generates a distinctive noise when touched is an example of the helpful impact of the resonant frequency. The amount of noise produced is determined by the string's thickness or slackness.

• Any likelihood of resonance in earthquake engineering, on the other hand, is damaging to the building structure.

• Orbital resonance is responsible for the Kirkwood gap, as well as other relationships between orbits.

Conclusion

We conclude that resonance is basically the tendency of a system to vibrate with increasing amplitudes at specific frequencies of excitation. These are the resonant frequencies of the system (or resonance frequencies). The resonator can have any number of harmonics and a fundamental frequency. We also learned that depending on the type of system and wave under consideration, the method of determining the resonant frequency also varied.