Effect of Resistance on RLC Second Order Circuits

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

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.

2. Generate MATLAB Codes

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

Using the Laplace transform, we start with the equation:

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

Where the variables are the same as above.

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 (X_L)
• 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 (X_L)
• 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].