Resonance Current in RLC Circuits

In alternating current (AC) circuits that contain resistors (R), inductors (L), and capacitors (C), a special condition known as resonance occurs. At resonance, the inductive reactance ($X_L$) becomes equal to the capacitive reactance ($X_C$). This equality significantly affects the total impedance of the circuit and, consequently, the current flowing through it. The behavior of current at resonance differs depending on whether the RLC components are connected in series or parallel.

Series RLC Circuit at Resonance

In a series RLC circuit, the resistor, inductor, and capacitor are connected end-to-end. The total impedance ($Z$) of a series RLC circuit is given by the formula:

$Z = \sqrt{R^2 + (X_L - X_C)^2}$

Where:

• $R$ is the resistance in ohms ($\Omega$).
• $X_L$ is the inductive reactance in ohms ($\Omega$), calculated as $2\pi fL$.
• $X_C$ is the capacitive reactance in ohms ($\Omega$), calculated as $1/(2\pi fC)$.
• $f$ is the frequency of the AC source in Hertz (Hz).
• $L$ is the inductance in Henrys (H).
• $C$ is the capacitance in Farads (F).

At resonance in a series RLC circuit, the inductive reactance and capacitive reactance cancel each other out:

$X_L = X_C$

When $X_L = X_C$, the term $(X_L - X_C)$ becomes zero. Therefore, the impedance formula simplifies to:

$Z_{min} = \sqrt{R^2 + 0^2} = R$

This means that at resonance, the total impedance of a series RLC circuit is at its minimum value, equal only to the circuit's resistance $R$.

According to Ohm's law for AC circuits, the current ($I$) is given by $I = V/Z$, where $V$ is the applied voltage. Since the impedance ($Z$) is at its minimum value at resonance, the current ($I$) flowing through the series RLC circuit will be at its maximum value. This is why series RLC circuits are often used as "acceptor circuits" or "band-pass filters" to maximize current at a specific frequency.

Parallel RLC Circuit at Resonance

In a parallel RLC circuit, the resistor, inductor, and capacitor are connected across the same two points, meaning they share the same voltage. The analysis of parallel circuits involves admittance (the reciprocal of impedance) or combining parallel impedances.

At resonance in a parallel RLC circuit, similar to the series circuit, $X_L = X_C$. However, the effect on the total impedance and current is opposite. At resonance, the parallel RLC circuit exhibits its maximum impedance. Ideally, if the resistor is not considered (i.e., pure LC circuit), the impedance at resonance approaches infinity. With a resistor in parallel, the impedance becomes maximum and purely resistive.

When the impedance ($Z$) is at its maximum value, according to Ohm's law ($I = V/Z$), the total current ($I$) drawn from the source by the parallel RLC circuit will be at its minimum value. The currents through the inductive and capacitive branches are large but 180 degrees out of phase, effectively canceling each other out in the main line, leading to a very small net current from the source. This is why parallel RLC circuits are often used as "rejector circuits" or "band-stop filters" to minimize current at a specific frequency.

Summary of Current Behavior at Resonance

The following table summarizes the behavior of current and impedance at resonance for both types of RLC circuits:

Therefore, at resonance, the current is maximum in a series RLC circuit and minimum in a parallel RLC circuit.