How does resonance occur in a parallel RLC circuit?

To understand how resonance occurs in a parallel RLC circuit, let's briefly review the impedance of each component:

Resistor (R): The impedance of a resistor is purely resistive and is given by R.

Inductor (L): The impedance of an inductor is inductive reactance (XL) and is given by XL = 2πfL, where f is the frequency of the AC signal and L is the inductance of the inductor.

Capacitor (C): The impedance of a capacitor is capacitive reactance (XC) and is given by XC = 1 / (2πfC), where f is the frequency of the AC signal and C is the capacitance of the capacitor.

Now, when you have a parallel RLC circuit, the total impedance (Z) of the circuit is the reciprocal of the sum of the reciprocals of individual impedances:

1/Z = 1/R + 1/XL + 1/XC

At resonance, the condition for maximum current (minimum impedance) occurs when the inductive reactance (XL) and capacitive reactance (XC) are equal in magnitude but opposite in sign:

|XL| = |XC|

Substituting the expressions for XL and XC:

2πfL = 1 / (2πfC)

Solving for the resonant frequency (fr):

fr = 1 / (2π√(LC))

At this resonant frequency, the inductive reactance and capacitive reactance cancel each other out, resulting in a minimum impedance and maximum current flow in the circuit. The phase angle between the voltage and current is also zero at resonance, meaning the current and voltage are in phase.

It's important to note that at resonance, the impedance of the parallel RLC circuit is purely resistive (no reactive component) and is equal to the resistance value (R). Beyond the resonant frequency, the impedance starts to increase again as the reactances start to dominate the overall impedance of the circuit.

Resonance in a parallel RLC circuit has several practical applications, such as in radio tuning circuits, bandpass filters, and impedance matching networks.