**Question 11: Explain the resonance of a series RLC circuit. Show that resonance occurs at a frequency determined by ** ** **

Answer

### Resonant RLC series circuit

A resonant RLC series circuit is one in which the magnitudes of capacitive and inductive reactances are equal. Since they are 180^{0} out of phase, therefore, they cancel one another.

Consider an inductance L, capacitance C and resistance R connected in series in an AC circuit. When the voltage applied and current are in step (which means the phase angle is zero), then the circuit is said to be in resonance. It happens when the capacitive reactance X_{C} and inductive reactance X_{L} balance each other, that is they are equal and out of phase by 180^{0}. Therefore, if the inductive reactance X_{L} and capacitive reactance X_{C} are equal in magnitude and opposite in direction the circuit is called an RLC series resonance AC circuit.”

### Resonance Frequency

Now for the condition of resonance frequency

Put this value of ω in equation (i)

Here f_{r} is the frequency at resonance conditions called resonant frequency. We observe that,

#### Resonance Curve

It is the graph between current and frequency for a resonance RLC circuit. At resonance frequency, f_{r} the current reaches its maximum value. However, it falls rapidly on both sides of the resonance frequency. The reason is if f < f_{r} then X_{C} > X_{L}. The net reactance X_{L} – X_{C} is not zero. If we put it in the equation for the impedance of the circuit. The impedance will have some greater value than R which decreases the current. Similarly, for f > f_{r} then X_{L} > X_{C} and again by using the above equation, the impedance of the circuit increases which decreases the current.

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