Question

Calculate the resonant frequency and Q-factor (Quality factor) of a series L-C-R circuit containing a pure inductor of $4\text{ H}$, capacitor of capacitance $27\text{ }\mu \text{F}$ and resistor of resistance $8.4\text{ }\Omega$.

Answer 1

At resonant frequency, the current in an L-C-R circuit will be maximum, which is possible only when inductive reactance ${{X}_{L}}$ is equal to capacitive reactance ${{X}_{c}}$.
At $\omega ={{\omega }_{o}}$, ${{X}_{L}}={{X}_{c}}$, that is,
${{\omega }_{o}}L=\dfrac{1}{{{\omega }_{o}}C}$

Formula used:
The resonant frequency ${{f}_{o}}$ of a series L-C-R is given by
${{f}_{o}}=\dfrac{1}{2\pi \sqrt{LC}}$
Here L denotes the inductance of the inductor and C denotes the capacitance of the capacitor.
The Q-factor of the circuit is s given by
$Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}$
Here L denotes the inductance of the inductor, C denotes the capacitance of the capacitor, and R denotes the resistance of the resistor.

Complete step by step solution:
Inductance, $L=4\text{ H}$
Capacitance, $C=27\text{ }\mu \text{F}$
Resistance, $R=8.4\text{ }\Omega$
To find the resonant frequency, substitute the values of L and C in the resonant frequency formula:

& {{f}_{o}}=\dfrac{1}{2\pi \sqrt{(4\text{ H)(27 }\mu \text{F)}}} \\\ & {{f}_{o}}=\dfrac{1}{2\pi \sqrt{(4\text{ H)(27}\times \text{1}{{\text{0}}^{-6}}\text{F)}}} \\\ & {{f}_{o}}=15.31\text{ Hz} \\\ \end{aligned}$$ Now, to find the Q-factor of the circuit, substitute the values of L, C, and R in the Q-factor frequency formula: $$\begin{aligned} & Q=\dfrac{1}{8.4\text{ }\Omega }\sqrt{\dfrac{4\text{ H}}{27\text{ }\mu \text{F}}} \\\ & Q=\dfrac{1}{8.4\text{ }\Omega }\sqrt{\dfrac{4\text{ H}}{\text{27}\times \text{1}{{\text{0}}^{-6}}\text{F}}} \\\ & Q=45.82 \\\ \end{aligned}$$ **Therefore, the resonant frequency of the L-C-R circuit is $$15.31\text{ Hz}$$ and the Q-factor is $$45.82$$.** **Additional information:** In general resonance is a phenomenon in which the natural frequency of a harmonic oscillator matches with the frequency of an external periodic force which in turn increases the amplitude of vibration. Resonance is the result of oscillations in an L-C-R circuit as stored energy is passed from the inductor to the capacitor. The sharpness of resonance is quantitatively described by a dimensionless number known as Q-factor or quality factor which is numerically equal to ratio of resonant frequency to bandwidth. The bandwidth is equal to L/R. **Note:** The Q-factor of a series L-C-R circuit will be large if R is low, C is low or L is large.