# Effect of Resistance on RLC Second Order Circuits

Categories: Engineering

## Introduction

The objective of this laboratory report is to investigate the impact of resistance on the responses of different RLC second-order circuits. Before delving into the details, it is essential to differentiate between first-order and second-order circuits. A first-order circuit contains a single storage element, whereas a second-order circuit contains two storage elements. RLC circuits, consisting of resistors (R), capacitors (C), and inductors (L), are prime examples of second-order circuits. These circuits are characterized by second-order differential equations in the Laplace domain, which include a resistor and two energy storage components.

## Method

1. Derive a General Formula for RLC Circuits in Series and Parallel
2. Utilize the principles of Fourier Transform and Laplace Transform to obtain a general formula for RLC circuits in both series and parallel configurations based on preliminary work.

3. Generate MATLAB Codes
4. Follow the procedures outlined in Laboratory Sheet 2 to create an .M file containing the necessary code to generate waveforms for different resistance values in both RLC parallel and series circuits.

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## Preliminary Work

### 1. RLC Circuit in Parallel

We begin with the Laplace representation:

V(s) = I(s)R + (1/sC)I(s) + sLI(s)

Where:

• V(s) - Laplace transform of voltage across the circuit
• I(s) - Laplace transform of current through the circuit
• R - Resistance
• C - Capacitance
• L - Inductance
• s - Complex frequency variable

By solving for I(s) and applying the reverse Laplace transform, we obtain the current waveform for different resistance values (R = 1000, 1250, and 500).

### 2. RLC Circuit in Series

V(s) = I(s)R + LsI(s) + (1/sC)I(s)

Where the variables are the same as above.

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Solving for I(s) and performing partial fraction decomposition, we find the current waveform for different resistance values (R = 5000 and 500).

## Results

The results are obtained by running the attached .M file, which contains the MATLAB code for generating waveforms for different resistance values in RLC parallel and series circuits.

## Discussion

For a parallel RLC circuit where L and C are held constant, will the damping on the device's voltage v(t) be increased if the value of R is increased?

Yes, an increase in the resistance (R) will lead to an increase in damping.

The following parameters were calculated for the parallel RLC circuit:

• Resonant Frequency (ω0)
• Inductive Reactance at Resonance (XL)
• Quality Factor (Q)
• Bandwidth (BW)

For a series RLC circuit where L and C are held constant, will the damping on loop current i(t) be increased if the value of R is increased?

Yes, an increase in resistance (R) will result in increased damping. It's worth noting that the Q-factor of a parallel RLC circuit is the inverse of the expression for the Q-factor of the series circuit.

The following parameters were calculated for the series RLC circuit:

• Resonant Frequency (ω0)
• Inductive Reactance at Resonance (XL)
• Quality Factor (Q)
• Bandwidth (BW)

## Conclusion

In conclusion, this laboratory report aimed to investigate the impact of resistance on the responses of RLC parallel and series circuits. The study served as a practical exercise to comprehend the behavior of second-order circuits. Both MATLAB simulations and theoretical calculations were employed to obtain results.

For further investigation and a more comprehensive understanding of the subject, it is recommended to conduct practical experiments to compare and validate the results obtained through theoretical and computational methods.

## References

1. 'Second Order Circuits - VOER', Voer.edu.vn, 2019. [Online]. Available: https://voer.edu.vn/m/second-order-circuits/072b46c4. [Accessed: 18-Oct-2019].
2. 'Laplace Transform -- from Wolfram MathWorld', Mathworld.wolfram.com, 2019. [Online]. Available: http://mathworld.wolfram.com/LaplaceTransform.html. [Accessed: 18-Oct-2019].
3. 'Transfer function model - MATLAB- MathWorks Australia', Au.mathworks.com, 2019. [Online]. Available: https://au.mathworks.com/help/control/ref/tf.html#mw_3d1a5a35-5713-4bc1-955b-c368f1eaaf2a. [Accessed: 18-Oct-2019].
Updated: Jan 05, 2024