# Resonant Frequency Formula

The resonant frequency is defined as the frequency of a circuit when the values of capacitive impedance and the inductive impedance become equal. It is defined as the frequency at which a body or system reaches its highest degree of oscillation. A resonant circuit is made up of a parallel-connected capacitor and an inductor. It is mostly employed to create a given frequency or to consider a specific frequency from a complex circuit. The resonant frequency exists only when the circuit is purely resistive.

**Formula**

The formula for resonant frequency is given by the reciprocal of the product of two times pi and the square root of the product of inductance and capacitance. It is represented by the symbol f_{o}. Its standard unit of measurement is hertz or per second (Hz or s^{-1}) and its dimensional formula is given by [M^{0}L^{0}T^{-1}].

f_{o }= 1/2π√(LC)where,

f

_{o}is the resonant frequency,L is the inductance of circuit,

C is the capacitance of circuit.

**Derivation**

Suppose we have a circuit where a resistor, inductor and capacitor are connected in series under an AC source.

The value of resistance, inductance and capacitance is R, L and C.

Now, it is known that the impedance Z of the circuit is given by,

Z = R + jωL – j/ωC

Z =R + j (ωL – 1/ωC)

To satisfy the condition of resonance, the circuit must be purely resistive. Hence, the imaginary part of impedance is zero.

ωL – 1/ωC = 0

ωL = 1/ωC

ω

^{2 }= 1/LCPutting ω = 1/2πf

_{o}, we get(1/2πf

_{o})^{2}= 1/LCf

_{o}= 1/2π√(LC)This derives the formula for resonant frequency.

**Sample Problems**

**Problem 1. Calculate the resonant frequency for a circuit of inductance 5 H and capacitance 3 F.**

**Solution:**

We have,

L = 5

C = 3

Using the formula we have,

f

_{o}= 1/2π√(LC)= 1/ (2 × 3.14 × √(5 × 3))

= 1/24.32

= 0.041 Hz

**Problem 2. Calculate the resonant frequency for a circuit of inductance 3 H and capacitance 1 F.**

**Solution:**

We have,

L = 3

C = 1

Using the formula we have,

f

_{o}= 1/2π√(LC)= 1/ (2 × 3.14 × √(3 × 1))

= 1/10.86

= 0.092 Hz

**Problem 3. Calculate the resonant frequency for a circuit of inductance 4 H and capacitance 2.5 F.**

**Solution:**

We have,

L = 4

C = 2.5

Using the formula we have,

f

_{o}= 1/2π√(LC)= 1/ (2 × 3.14 × √(4 × 2.5))

= 1/6.28

= 0.159 Hz

**Problem 4. Calculate the inductance of a circuit if the capacitance is 4 F and the resonant frequency is 0.5 Hz.**

**Solution:**

We have,

f

_{o}= 0.5C = 4

Using the formula we have,

f

_{o}= 1/2π√(LC)=> L = 1/4π

^{2}Cf_{o}^{2}= 1/ (4 × 3.14 × 3.14 × 4 × 0.5 × 0.5)

= 1/39.43

= 0.025 H

**Problem 5. Calculate the inductance of a circuit if the capacitance is 3 F and the resonant frequency is 0.023 Hz.**

**Solution:**

We have,

f

_{o}= 0.023C = 3

Using the formula we have,

f

_{o}= 1/2π√(LC)=> L = 1/4π

^{2}Cf_{o}^{2}= 1/ (4 × 3.14 × 3.14 × 3 × 0.023 × 0.023)

= 1/0.0199

= 50.25 H

**Problem 6. Calculate the capacitance of a circuit if inductance is 1 H and the resonant frequency is 0.3 Hz.**

**Solution:**

We have,

f

_{o}= 0.3L = 1

Using the formula we have,

f

_{o}= 1/2π√(LC)=> C = 1/4π

^{2}Lf_{o}^{2}= 1/ (4 × 3.14 × 3.14 × 1 × 0.3 × 0.3)

= 1/3.54

= 0.282 F

**Problem 7. Calculate the capacitance of a circuit if inductance is 0.1 H and the resonant frequency is 0.25 Hz.**

**Solution:**

We have,

f

_{o}= 0.25L = 0.1

Using the formula we have,

f

_{o}= 1/2π√(LC)=> C = 1/4π

^{2}Lf_{o}^{2}= 1/ (4 × 3.14 × 3.14 × 0.1 × 0.25 × 0.25)

= 1/0.246

= 4.06 F

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