In the experiment, we used Arduino with a 3D-printed platform to measure damping, critical damping, and overdamping effects in an RLC circuit. The theoretical background focuses on how an RLC circuit responds to an applied voltage. Here's a detailed explanation:

 Theory of Damping in RLC Circuits:

 An RLC circuit is an electrical circuit consisting of three primary elements:

1. Resistor (R): Dissipates energy as heat and causes the damping of oscillations.

2. Inductor (L): Stores energy in the form of a magnetic field and opposes changes in current.

3. Capacitor (C): Stores energy in an electric field and opposes changes in voltage.

When an alternating current (AC) voltage source is applied to the RLC circuit, the energy oscillates between the inductor and the capacitor. The resistor, however, dissipates some of this energy, reducing the oscillation amplitude over time, known as damping. These components are connected either in series or parallel. The circuit's behavior is governed by a second-order differential equation, where the solution depends on the damping effect caused by the resistance.

1. Undamped System (Ideal Case):

In the absence of resistance (R = 0), the system would oscillate indefinitely. The energy stored in the inductor and capacitor would continuously exchange, leading to sinusoidal oscillations. This is an ideal situation and not practical in real-world systems, where some damping is always present.

2. Damped System:

In a real system, the resistor introduces energy dissipation, leading to a decrease in oscillation amplitude over time. The level of damping can vary, leading to three possible behaviors:

• Under-damping: When the resistance (R) is small, the circuit will still oscillate, but the amplitude of oscillations gradually decreases over time. The solution of the second-order differential equation involves a decaying sinusoidal waveform, where the system oscillates and eventually settles to its final value.

• Critical Damping: At a specific resistance value, the system will return to equilibrium without oscillating. This is called critical damping, where the system is able to settle as quickly as possible without overshooting. The system reaches a steady state in the shortest possible time, and the differential equation has a solution that decays exponentially without oscillation.

• Overdamping: When the resistance is large, the system becomes overdamped. In this case, the circuit still returns to equilibrium, but much more slowly, and no oscillation occurs. The system decays exponentially, but the time to reach the steady state is longer than in the critically damped case.

Damping in RLC Circuits:

 The behavior of the RLC circuit is described by a second-order differential equation that governs the charge or current in the circuit. The key factor in determining the type of damping is the resistance (R).

 The General Equation of Motion:

 The current I(t) or charge q(t) in the circuit is governed by the following second-order linear differential equation:

The solution to this equation depends on the discriminant:

which determines the behavior of the system:

1. Underdamped Case:

   • When R²<4L/C, the system is underdamped, and the response oscillates with a gradually decreasing amplitude over time. The solution takes the form of an exponentially decaying sinusoidal wave:

where α=R/2L​ is the damping factor, and

is the angular frequency of the damped oscillations.

Critically Damped Case:

• When R²=4L/C, the system is critically damped. In this case, the circuit does not oscillate but returns to equilibrium as quickly as possible without overshooting. The solution for this case is:

where α=R/2L​. This is the optimal condition for fast response without oscillation.

Overdamped Case:

• When R²>4L/C, the system is overdamped. The response decays exponentially without oscillation, but more slowly than in the critically damped case. The solution is:

where s1​ and s2​ are real negative roots, leading to a gradual decay of the response.

Using the Arduino and the components in your setup, you can create an automated system to analyze the behavior of the RLC circuit. Here’s how each component contributes to the experiment:

1. Arduino UNO:

   • Acts as the control unit, sampling data from the circuit and recording the voltage and current changes over time.

   • The Arduino can interface with sensors and the wave generator to observe the behavior of the circuit and transmit the results to a connected computer for analysis.

2. Waveform Generator:

   • Provides an AC or square wave signal to excite the RLC circuit. By controlling the frequency and amplitude of the waveform, you can observe different responses of the circuit.

3. Breadboard:

   • Serves as the prototyping platform where the components (resistor, capacitor, inductor) are connected in series or parallel. The breadboard allows easy modification of the circuit parameters by swapping components to observe different damping effects.

4. Capacitors (0.1 μF mica and ceramic):

   • The capacitor stores electrical energy in the circuit and is responsible for the oscillatory behavior when paired with the inductor. The value of the capacitor determines the natural frequency of the oscillations.

   • The mica capacitor has better stability at high frequencies, while the ceramic capacitor is commonly used for general-purpose filtering and smoothing.

5. Inductor (50 mH):

   • The inductor opposes changes in current, leading to oscillations when paired with the capacitor. The value of the inductance determines the energy stored in the magnetic field and influences the oscillation frequency.

6. Resistor (adjustable through different wiring or adding in a series):

   • The resistor introduces damping in the circuit. By varying the resistance, you can adjust the system from underdamped to critically damped or overdamped conditions.

7. Diode:

   • The diode is used to prevent reverse current, ensuring that the circuit behaves as expected under AC conditions. It can also be used for rectification if required in the circuit.

8. 3D-Printed Platform:

   • This provides a stable physical framework to organize and mount the Arduino and breadboard setup. It ensures that the experiment is repeatable and reduces variability due to component placement or movement.

9. Dupont Wires (Male-Male and Male-Female):

   • These wires allow easy connections between the Arduino, breadboard, and various components, enabling flexible circuit configurations.

• Data Collection: The Arduino samples the voltage and current in the circuit. By analyzing the voltage across the capacitor or the current through the inductor, you can observe how the oscillations change over time.

• Analysis: The collected data is transmitted to a computer or cloud platform (using AIoT technology) for further analysis. The waveform of the response can be plotted to observe underdamped oscillations, critically damped settling, or overdamped decay.

• Visualization: The amplitude decay and time constants can be extracted from the data to characterize the damping in the circuit.

By adjusting the resistor (either through a variable resistor or by changing components), you can measure:

• Under-damped oscillations: When the resistance is low.

• Critical damping: When the resistance reaches a value that minimizes oscillation without overshooting.

• Overdamped behavior: When the resistance is high, leading to a slow return to equilibrium without oscillations.

 The Arduino-based setup for RLC measurements allows precise control and observation of damping behaviors in electrical circuits. Adjusting the resistance allows you to observe the transition from underdamping to critical damping and overdamping. Using a 3D-printed platform ensures stability, while the combination of components (inductor, capacitors, and resistor) enables a practical investigation into the behavior of second-order systems. This experiment is ideal for exploring the fundamentals of circuit response and energy dissipation.