Study of a Parallel Resonant Circuit (RLC Circuit)

Objective:

We aim to study the properties of a passive circuit composed of two branches:

1. One capacitive branch.
2. One branch consisting of a resistor and an inductor in series.

When a sinusoidal voltage $E$ with an effective value and angular frequency $\omega$ (or frequency $f$) is applied to the circuit:

$$E = ZI$$

Application of a “Notch Circuit”:

Parallel RLC circuits are often called “notch circuits” because they present high impedance at a particular frequency $f_0$, preventing signals at that frequency from passing to certain parts of the circuit. Notch circuits are used in:

• Electronics: Audio systems for equalization, and color televisions for separating audio, chrominance, and luminance frequencies.
• Electricity: Centralized remote controls to avoid frequency dispersion on the network.

Characteristics of a Notch Circuit:

To better understand the functioning of notch circuits, we perform measurements in the lab. A Bode plotter helps visualize the output voltage of the notch filter relative to the generator frequency. We observe that the circuit exhibits high impedance at a certain frequency, leading to significant attenuation at that point on the curve.

Complex Impedance Expression:

We know that:

$$Z = \frac{1}{j \omega C} + \frac{R – j \omega L}{R^2 + (\omega L)^2}$$

Resonant Frequency Determination:

The resonant condition occurs when the imaginary part of the admittance cancels out, giving:

$$\omega_0 = \frac{1}{\sqrt{LC}}$$

The theoretical resonant frequency:

$$f_{0th} = \frac{\omega_0}{2\pi}$$

With actual values:

$$\omega_0 = 79056.94 \, \text{rad/s}, \quad f_{0th} = 12.89 \, \text{kHz}, \quad f_{opractical} = 13 \, \text{kHz}$$

The difference between theoretical and practical results arises due to measurement and equipment errors.

Quality Factor of the Inductive Branch:

The quality factor $Q$ is given by:

$$Q = \frac{\omega_0 L}{R}$$

For $R_0 = 0 \Omega$:

$$Q = 83.85$$

For $R_0 = 348.2 \Omega$:

$$Q = 4.11$$

Practical Tasks:

1. Circuit Assembly: Assemble the circuit as shown.
2. Internal Resistance Measurement: $R = 18.5 \Omega$, $L = 0.019 \, H$.
3. Resonant Frequency Measurement and Comparison: Compare theoretical and practical results.
4. Current and Phase Values Around Resonance:

Observations:

• The current $I$ decreases to zero at resonance frequency $f_0 = 13 \, \text{kHz}$ and increases for frequencies above $f_0$.
• The phase $\Phi$ decreases for $f < f_0$ and becomes negative for $f > f_0$.

Conclusion:

At resonance, both current and phase are zero. The quality factor greatly affects the precision of the resonant frequency and the bandwidth, which is evident from the curves.