Impedance of an ideal parallel LC circuit at resonance

At resonance, the impedance of an ideal parallel LC (inductor-capacitor) circuit becomes infinite. This is because at resonance, the reactance of the inductor and the capacitor cancel each other out, resulting in a purely resistive circuit.

Explanation:

1. Parallel LC Circuit:
A parallel LC circuit consists of an inductor (L) and a capacitor (C) connected in parallel. The inductor and capacitor have reactances that depend on the frequency of the input signal.

The reactance of an inductor is given by the formula: XL = 2πfL, where XL is the inductive reactance, f is the frequency, and L is the inductance.

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

2. Impedance of Parallel LC Circuit:
The impedance of a parallel LC circuit is given by the formula: Z = 1 / (1/XL + 1/XC), where Z is the impedance, XL is the inductive reactance, and XC is the capacitive reactance.

3. Resonance:
Resonance occurs in an LC circuit when the reactances of the inductor and capacitor are equal in magnitude but opposite in sign. Mathematically, this can be expressed as XL = -XC.

At resonance, the impedance formula can be simplified as: Z = 1 / (1/XL - 1/XL), which simplifies to Z = ∞ (infinite).

4. Implication:
When the impedance of a parallel LC circuit is infinite at resonance, it means that the circuit behaves like an open circuit to the input signal. This implies that no current flows through the circuit, and all the input voltage is dropped across the components without any power being dissipated.

Summary:
In conclusion, the impedance of an ideal parallel LC circuit at resonance is infinite. This occurs when the reactances of the inductor and capacitor cancel each other out, resulting in a purely resistive circuit. At resonance, the circuit behaves like an open circuit, with no current flow and all the input voltage dropped across the components.