A series RLC circuit is a fundamental electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C) connected in series. When such a circuit is subjected to an alternating current (AC) voltage, its behavior varies significantly with the frequency of the applied voltage. A particularly important phenomenon in RLC circuits is resonance.

Resonance Frequency in Series RLC Circuit

Resonance frequency ($\text{f}_0$) in a series RLC circuit is the specific frequency at which the inductive reactance ($\text{X}_{\text{L}}$) becomes equal to the capacitive reactance ($\text{X}_{\text{C}}$). At this frequency, the total impedance of the circuit is purely resistive and at its minimum value, leading to the maximum current flow.

Half Power Frequencies Explained

The half power frequencies (also known as cut-off frequencies or -3 dB frequencies) are two specific frequencies, $\text{f}_1$ (lower half power frequency) and $\text{f}_2$ (upper half power frequency), at which the power dissipated in the series RLC circuit is exactly half of the maximum power dissipated at the resonant frequency. At these frequencies, the current amplitude is $\frac{1}{\sqrt{2}}$ (approximately 0.707) times the maximum current at resonance.

For a series RLC circuit, there is a distinct relationship between the resonant frequency ($\text{f}_0$) and the two half power frequencies ($\text{f}_1$ and $\text{f}_2$). The resonant frequency is the geometric mean of the two half power frequencies. This relationship is given by the formula:

$$\text{f}_0 = \sqrt{\text{f}_1 \times \text{f}_2}$$

Where:

• $\text{f}_0$ is the resonant frequency
• $\text{f}_1$ is the lower half power frequency
• $\text{f}_2$ is the upper half power frequency

Calculating Resonant Frequency Step-by-Step

Let's use the given information to calculate the resonant frequency for the series RLC circuit.

Given values:

• Lower half power frequency ($\text{f}_1$) = 25 kHz
• Upper half power frequency ($\text{f}_2$) = 36 kHz

Applying the formula:

$$\text{f}_0 = \sqrt{\text{f}_1 \times \text{f}_2}$$

Substitute the given values into the formula:

$$\text{f}_0 = \sqrt{25 \text{ kHz} \times 36 \text{ kHz}}$$

First, multiply the two frequencies:

$$\text{f}_0 = \sqrt{900 \text{ kHz}^2}$$

Now, take the square root:

$$\text{f}_0 = 30 \text{ kHz}$$

Therefore, the resonant frequency for the given series RLC circuit is 30 kHz.

Summary of Frequencies

This calculation demonstrates how to find the central resonant frequency when the bandwidth, defined by the half power frequencies, is known. Understanding this relationship is crucial for analyzing the frequency response of RLC resonance circuits in various applications.