Resonance Formula
We know that LCR circuits change their frequency depending upon the supplied voltage. Also, in the LCR series circuit source voltage is always a fixed frequency steady-state sinusoidal supply. In series LCR circuits two or more sinusoidal signals can be combined together with the help of phasors providing that they have the identical frequency supply. In a series LCR circuit, there will be a particular frequency point where the inductive reactance of the inductor will be equal to the capacitive reactance of the capacitor, i.e., \[X_{c}=X_{L}\].
The frequency at which the capacitive inductance will be equal to the inductive reactance is known as the resonant frequency and the corresponding formula is known as the resonant frequency formula. In this article, we will learn resonant frequency formula, resonance formula, resonance frequency equation along with a few exercise problems that will boost our understanding.
From the definition of the resonant frequency, it is understood that it is the frequency of a circuit under the resonant. The LCR circuits are also known as the tuning circuits because they are used to generate oscillations of desired frequencies. The resonant frequency is an important parameter in the series LCR circuit in which the capacitive inductance will be equal to the inductive reactance.
A Resonant electronic circuit is also familiarly known as the LC circuit or tank circuit or tuned circuit. In the resonant electronic circuit, the inductor and capacitor components will be connected parallel to each other. The resonant circuits are mainly used to create the desired frequency or to filter out a certain frequency from a complex circuit.
Resonance Frequency Equation
The resonance frequency formula or the resonant frequency equation is given by the following equation:
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}\]
Where,
L- The inductance of the inductor
C - The capacitance of the capacitor
Let us have a look at how we arrived at the resonance formula. Consider a series LCR circuit that is connected by an AC power supply. We know that the resonant frequency is achieved when the capacitance and inductance are equal. Then the impedance of the circuit is given by:
\[\Rightarrow Z = R + j\left ( \omega L - \frac{1}{\omega C} \right )....(1)\]
Where,
R - The resistance of the resistor
\[\omega\] - The angular frequency
L - The inductance of the inductor
C - The capacitance of the capacitor
According to the resonance condition, the circuit will be purely resistive, which implies the frequency purely depends on the resistance value. Therefore, we can say the imaginary part of the impedance Z will be zero during the resonance condition or at a resonant frequency. We should always keep this in our mind while calculating the resonant frequency for a given AC electronic circuit.
Therefore, we write get:
\[\Rightarrow \omega L - \frac{1}{\omega C} = 0\]
\[\Rightarrow \omega L = \frac{1}{\omega C}\]
\[\Rightarrow \omega^{2} = \frac{1}{L C} \]
\[\Rightarrow \omega = \frac{1}{\sqrt{L C}} \]
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}\] (because we know that angular frequency \[\omega = 2\pi f_{0}\])
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}...(2)\]
Where,
L- The inductance of the inductor
C - The capacitance of the capacitor
Equation (2) is known as the resonant frequency equation or resonant frequency formula.
Examples
1. Calculate the resonant frequency of a tank circuit whose value of the inductance is 50 mH and the capacitance is \[10 \mu F\].
Sol:
Given,
The inductance of the inductor used in the tank circuit \[= L = 50mH = 50 \times 10^{3}H\]
The capacitance of the capacitor used in the tank circuit
\[= C = 10 \mu F = 10 \times 10^{-6}~F = 10^{-5}~F\]
We are asked to determine the resonant frequency of the given tank circuit. We know that the resonant frequency of the circuit can be calculated using the following resonance frequency formula:
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}....(1)\]
Where,
L- The inductance of the inductor
C - The capacitance of the capacitor
Substituting the values of inductance and capacitance in the resonance frequency equation i.e., equation (1) and simplify the expression, we get:
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}\]
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{(50 \times 10^{-3})(10^{-5})}}\]
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{50 \times 10^{-7}}}\]
\[\Rightarrow f_{0} = 711.76~Hz \approx 712~Hz\]
Therefore, the resonant frequency of the given tank circuit is \[712 Hz\].
2. Given that the resonant frequency of a resonant circuit is around \[312 Hz\]. If its capacitance is \[7~\mu F\], then calculate the inductance of the circuit.
Sol:
Given,
The resonant frequency of the resonant circuit \[= f_{0} = 312 Hz\]
The capacitance of the capacitor used in the tank circuit \[ = C = 7~\mu F = 7 \times 10^{6}~F\]
We are asked to determine the inductance of the given tank circuit. We know that the resonant frequency of the circuit can be calculated using the following resonating frequency formula:
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}....(1)\]
Where,
L- The inductance of the inductor
C - The capacitance of the capacitor
Squaring on both sides of equation (1) and Rearranging equation (1) for the value of inductance we get:
\[\Rightarrow f{_{0}}^{2} = \frac{1}{4\pi^{2}LC}\]
\[\Rightarrow L = \frac{1}{4\pi^{2}f{_{0}}^{2}C}....(2)\]
Substituting the values of resonant frequency and capacitance in equation (2) and simplify the expression, we get:
\[\Rightarrow L = \frac{1}{4\pi^{2}f{_{0}}^{2}C}\]
\[\Rightarrow L = \frac{1}{4\pi^{2} (312)^{2} (7 \times 10^{-6})}\]
\[\Rightarrow L = \frac{1}{2.69 \times 10}\]
\[\Rightarrow L = 37.17 mH\]
Therefore, the inductance of the given resonant circuit is 37.17 mH.
3. Given that the resonant frequency of a resonant circuit is around 300Hz. If its inductance is 31.89 mH, then calculate the capacitance of the circuit.
Sol:
Given,
The resonant frequency of the resonant circuit \[ =f_{0} = 300~Hz\]
The Inductance of the inductor used in the tank circuit
L = 318.3 mH = \[31.83 \times 10^{-3} H\]
We are asked to determine the capacitance of the given tank circuit. We know that the resonant frequency of the circuit can be calculated using the resonance frequency formula. Therefore, we know that the expression for resonant frequency is given by:
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}....(1)\]
Where,
L- The inductance of the inductor
C - The capacitance of the capacitor
Squaring on both sides of equation (1) and Rearranging equation (1) for the value of capacitance we get:
\[\Rightarrow f{_{0}}^{2} = \frac{1}{4\pi^{2}LC}\]
\[\Rightarrow C = \frac{1}{4\pi^{2}f{_{0}}^{2}L}....(2)\]
Substituting the values of resonant frequency and capacitance in equation (2) and simplify the expression, we get:
\[\Rightarrow C = \frac{1}{4\pi^{2}f{_{0}}^{2}L}....(2)\]
\[\Rightarrow C = \frac{1}{4\pi^{2} (300)^{2} (31.83 \times 10^{-3})}\]
\[\Rightarrow C = \frac{1}{11.30 \times 10^{4}}\]
\[\Rightarrow C = 8.84 \times 10^{-6}\]
\[F = 8.84 \mu F\]
Therefore, the capacitance of the given resonant circuit is \[8.84 \mu F\].
FAQs on Resonant Frequency Formula
1. How to find the resonant frequency in an LCR circuit?
We know that an LCR series circuit is also known as a resonant circuit, which implies we can find the resonant frequency in an LCR circuit, in which the capacitor and the inductor are connected parallel to each other. In such circuits, the resonant frequency can be calculated using the following formula:
\[\Rightarrow f_{0} = \frac{1}{2\pi\sqrt{LC}}\]
Where,
L- The inductance of the inductor
C - The capacitance of the capacitor
2. What is resonance frequency?
The resonance frequency of any given circuit is the conditional frequency obtained when the capacitive inductance and the inductive reactance of the circuit are equal. I.e., mathematically we write \[X_{c} = X_{L}\]. Thus the particular frequency obtained at this condition in any resonant circuit is known as the resonance frequency or resonant frequency and it can be easily estimated using the resonance formula or resonance frequency equation.
