Question

The value of impedance in parallel LC circuit at resonance is:

Answer 1

Hint: In DC circuit, the total opposition offered by a load is called resistance and in AC circuit, the total opposition offered by a load is called its reactance. In DC, the resistance of the load is fixed whereas in AC it is the function of time and it depends upon the frequency of the source.

Formula used:
${\chi }_L=\omega L,{\chi }_C={\displaystyle \frac{1}{\omega C}},{\displaystyle \frac{1}{Z}}={\displaystyle \frac{1}{{\chi }_L}}-{\displaystyle \frac{1}{{\chi }_C}}$

Complete step by step answer:

First of all, we need to understand what resonance is.
Resonating frequency – It is a special type of frequency at which, the inductive reactance and capacitive reactance of the circuit becomes numerically equal. At this frequency, the graph of both reactance cut each other and maximum current is flown in the circuit and the circuit is said to be at resonance.
Now, since at resonance, ${\chi }_L={\chi }_C$;
${\displaystyle \frac{1}{{\chi }_L}}={\displaystyle \frac{1}{{\chi }_C}}$
$⟹{\displaystyle \frac{1}{{\chi }_L}}-{\displaystyle \frac{1}{{\chi }_C}}=0$
$or{\displaystyle \frac{1}{Z}}=0$
Thus, $Z\to \mathrm{\infty }$
Hence, at resonance, the impedance of the circuit becomes infinite.

Additional information:
The condition of resonance is very important in circuit electronics. We often use this condition in our daily life. For example, when we tune or adjust the frequency of radio in order to get maximum and clear output of the audio, we adjust the frequency of radio at a particular frequency, known as resonant frequency. Hence one can say that this is the frequency at which maximum current is drawn from the source and hence maximum output is given by an element (like a speaker in radio).

Note:
This is a special type of angular frequency which is called ‘Resonating frequency’. At this frequency, the current in the circuit achieves its maximum value in the circuit. Mathematically it is equal to$\omega ={\displaystyle \frac{1}{\sqrt{LC}}}$.