Question

What do you mean by resonance in LCR series circuits? Write the formula for resonant frequency.

Answer 1

Hint
In series LCR circuit resonance occurs only when the capacitive and inductive reactances cancel out each other due to a phase difference of $180^\circ$ between them. So putting that in the formula for the impedance in the LCR circuit, we can calculate the resonant frequency.
In this, we will use the formula for impedance,
$\Rightarrow Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}}$
where $Z$ = Impedance
$R$ = Resistance
${X_L}$ = Inductive Reactance
${X_C}$ = Capacitive Reactance
${X_L} = \omega L$ where $\omega$ = frequency
And ${X_C} = \dfrac{1}{{\omega C}}$, $L$ = inductance and $C$ = capacitance

Complete step by step answer
For a series LCR circuit, the impedance is given by the formula,
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}}$
In a series LCR circuit when the resonance occurs, the capacitive and inductive reactances are equal in magnitude and cancel each other. This causes the series LC combination to act as a short circuit. So the Resistance becomes the only opposition present to the flow of current in the circuit.
So the Resonant frequency can be termed as the frequency at which the total impedance of the LCR circuit becomes real as no imaginary part exists in the circuit.
Since the inductive and capacitive reactances are equal in magnitude, we can write
$\Rightarrow {X_L} = {X_C}$
Now we know that ${X_L} = \omega L$ and ${X_C} = \dfrac{1}{{\omega C}}$
So substituting the values of ${X_L}$ and ${X_C}$ in the equation, we get
$\omega L = \dfrac{1}{{\omega C}}$
Here we are working in the resonant condition so the frequency here is the resonant frequency.
$\Rightarrow {\omega ^2} = \dfrac{1}{{LC}}$
Here solving this equation further by taking square root on both sides we will get the value of $\omega$ which is the resonant frequency as,
$\therefore \omega = \dfrac{1}{{\sqrt {LC} }}$
This is the formula for the resonant frequency.

Additional Information
In the LCR circuits when the ${X_L} > {X_C}$ , then the circuit is termed as Inductive and similarly when the ${X_C} > {X_L}$ then the corresponding circuits are termed as Capacitive.

Note
The series resonance or the series LCR circuits are one of the most important circuits. They have a huge number of practical uses starting from AC mains filters, radios, and also in television circuits.