Explain the working principle of a Parallel Resonant Circuit and its applications in filters.

Working Principle:

Inductive Reactance (XL): Inductors oppose changes in current flowing through them. When an alternating current (AC) is applied to an inductor, it creates a magnetic field that stores energy. The inductor's opposition to the changing current is called inductive reactance (XL) and is proportional to the frequency of the AC signal.

Capacitive Reactance (XC): Capacitors store energy in an electric field between their plates. When an AC signal is applied, the capacitor charges and discharges repeatedly, opposing changes in voltage across it. The opposition to the changing voltage is called capacitive reactance (XC) and is inversely proportional to the frequency of the AC signal.

Resonance: At a specific frequency, the inductive reactance (XL) and capacitive reactance (XC) in the circuit become equal in magnitude but opposite in phase. This results in their cancellation, leaving only the resistive component of the circuit (R), which is typically negligible in an ideal parallel resonant circuit. When the inductive and capacitive reactances cancel each other out, the impedance of the circuit reaches its minimum value. This state is called resonance.

Mathematically, the resonance condition in a parallel resonant circuit is given by:

XL = XC

Or,

2πfL = 1 / (2πfC)

Where:

XL is the inductive reactance (in ohms)

XC is the capacitive reactance (in ohms)

f is the frequency of the AC signal (in Hertz)

L is the inductance (in Henrys)

C is the capacitance (in Farads)

Applications in Filters:

Parallel resonant circuits find applications in electronic filters, specifically in band-pass filters and notch filters.

Band-Pass Filters: A band-pass filter is designed to allow a specific range of frequencies to pass through while attenuating frequencies outside that range. By utilizing a parallel resonant circuit, the filter can be tuned to the desired center frequency (resonant frequency) with a specific bandwidth around it. The inductor and capacitor values are chosen to match the desired frequency range, and the circuit will offer high impedance to frequencies outside this range, effectively filtering them out.

Notch Filters: A notch filter, also known as a band-stop or band-rejection filter, is designed to block a narrow range of frequencies while allowing all others to pass. A parallel resonant circuit can be used in a notch filter by selecting its resonant frequency to match the frequency that needs to be blocked. At the resonant frequency, the circuit offers very high impedance, effectively attenuating the signal at that specific frequency while letting other frequencies pass through unaffected.

In both applications, the precise control over the resonant frequency of the parallel resonant circuit allows for effective filtering and signal processing in various electronic devices and communication systems.