Understanding Parallel RLC Circuit Impedance at Resonance

This solution explains how to determine the dynamic impedance of a parallel RLC circuit specifically when it operates at its resonance frequency. We will derive the impedance value step-by-step.

Parallel RLC Circuit Impedance Formula

The impedance ($Z$) of a parallel RLC circuit, where R is resistance, L is inductance, and C is capacitance, is given by the reciprocal of the total admittance ($Y$). The admittance is the sum of the individual admittances:

$$ Y = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C $$

Where:

• $R$ is the resistance
• $L$ is the inductance
• $C$ is the capacitance
• $\omega$ is the angular frequency
• $j$ is the imaginary unit

The impedance is then:

$$ Z = \frac{1}{Y} = \frac{1}{\frac{1}{R} + \frac{1}{j\omega L} + j\omega C} $$

Resonance Condition in Parallel RLC Circuits

Resonance in a parallel RLC circuit occurs when the inductive reactance ($X_L = \omega L$) and the capacitive reactance ($X_C = \frac{1}{\omega C}$) effectively cancel each other out in terms of their contribution to the admittance. Specifically, at resonance, the imaginary part of the admittance becomes zero.

The condition for resonance ($\omega_0$) is when:

$$ \frac{1}{j\omega_0 L} + j\omega_0 C = 0 $$

This simplifies to:

$$ \frac{-j}{\omega_0 L} + j\omega_0 C = 0 $$

$$ j\omega_0 C = \frac{j}{\omega_0 L} $$

$$ \omega_0 C = \frac{1}{\omega_0 L} $$

$$ \omega_0^2 = \frac{1}{LC} $$

So, the resonant angular frequency is $\omega_0 = \frac{1}{\sqrt{LC}}$.

Calculating Dynamic Impedance at Resonance

At the resonant frequency ($\omega_0$), the admittance equation becomes:

$$ Y_{res} = \frac{1}{R} + \frac{1}{j\omega_0 L} + j\omega_0 C $$

As established, at resonance, the term $\frac{1}{j\omega_0 L} + j\omega_0 C$ equals zero.

Therefore, the admittance at resonance simplifies to:

$$ Y_{res} = \frac{1}{R} + 0 = \frac{1}{R} $$

The dynamic impedance ($Z_{res}$) is the reciprocal of the admittance at resonance:

$$ Z_{res} = \frac{1}{Y_{res}} = \frac{1}{\frac{1}{R}} $$

$$ Z_{res} = R $$

Thus, the dynamic impedance of a parallel RLC circuit at resonance is equal to the resistance ($R$). This is the point of maximum impedance for a parallel RLC circuit.