Transient responses of second order RLC circuits

Objective

The objective of this experiment is to observe the transient responses of RLC circuits and to learn how R, L, and C affect the circuit behaviors using the ORCAD PSPICE software utility. Specifically second order ones. Additionally, this post hopes to be a resource for future students.

Theoretical Background

It is useful to be aware of the inverse laplace transform operation an operation whereby you can switch between the time domain and the frequency domain in circuits. It is oftentimes helpful to know how to use the partial fraction expansion technique, so you can transform equations inside of an inverse laplace operation into ones that are more wieldy and easily used, or can be looked up in a table of known equivalents. 

You should also know that a capacitor has the ability to store electrical charge and energy. Therefore, voltage across the capacitor is related to the charge by the equation for steady state values. Finally, you are required to have some sort of prerequisite knowledge of RLC circuits. For instance, consider the circuit below.

You should understand the following equations are useful for calculating and understanding the various properties of the circuit above.

Information about the value of ⍺ can be useful for identifying the kinds of s values in play. For instance, if ⍺ is greater than ⍵, this tells us that both s values are real and distinct. Then, the voltage response can be known as overdamped: . If the opposite is true, then we know that both s values are distinct and complex, and the voltage response is called underdamped: which is equivalent to where , which is known as the damped radian frequency. If ⍺ is equal to ⍵₀, then both s values are repeated real roots, and the response is known as critically damped.

In this experiment, you are also required to build a series connected RLC circuit, which requires you to be able to analyze those circuits using these equations. You must also be able to identify the timings of between these voltages from a graph. Graphs of critically damped, overdamped and underdamped series-connected RLC circuits have different appearances and components to them.

The graph above is useful for identifying each type of a graph, however the components to each voltage signal have several components that can be identified. You are shown a single example of one of these signals in the graph below.

In this graph, t_p is the peak time required for the response to reach the first peak value. Then, t_s is the settling time, that is- the time required for the response to reach and stay within 2% or 5% of the final value. M_p, then, is the maximum percent overshoot, and can also be calculated with . Finally, t_d is the delay time required for the response to reach half of the final value for the very first time, and t_r is the rise time required for the response to rise from 0% to 100% of the final value. For critically or overdamped circuits, sometimes the value range of 10% to 90% is used instead. 

Experimental Procedures

To begin this experiment, open PSpice and build the series circuit shown below. Set 𝐿 = 10 mH, 𝐶 = 0.01 𝜇F, and 𝑅=10 kΩ. 𝑣𝑠 is a pulse voltage source. Select VPULSE from Part box. The definitions of each parameter in VPULSE are shown below. Set 𝑉1 = 0, 𝑉2 = 10, 𝑇𝑅 = 𝑇𝜇 = TD = 0, PW = 1 ms, and PER = 2 ms. Set R to 500 Ω. Measure the four major specifications that define the response of a second order system/circuit. The rise time (𝑻𝒓) is used to measure the swiftness of the circuit, which is defined as the time required reaching 90% of the reference input (the source). The overshoot (PO) is used to measure the closeness of the response to the reference in terms of %. It is calculated as: PO = ((Mp –   final value)/final value)*100. The peak time (𝑻𝒑) measures the time taken to reach the maximum response. The settling time (𝑻𝒔) is the time required for the circuit to settle within a certain percentage of the reference input (±𝛿). Here, we choose 𝜹 = 𝟓%.

Construct the parallel circuit shown below in PSPICE. Set 𝐿=10 mH, 𝐶 = 0.01 𝜇F, and 𝑅=10 kΩ. Set the source to be a square pulse and use the same parameters as described for the series RLC circuit. Set R to be 200 Ω and observe the voltage response. Compare the responses of the two circuits, explain why they are different for three different responses.

Analysis

In this lab we modeled a series RLC circuit and a parallel RLC circuit. Depending on whether the circuit is a series or parallel RLC circuit and what the resistance is, analysis can be done to show their graph as either overdamped, critically damped, or underdamped. For this lab we focused on what the response was for the voltage and to then apply the appropriate specifications of the four major depending on the response.

We first constructed a series RLC circuit and evaluated the response at 10k ohms and 500 ohms. The response at 10k ohms was an overdamped graph. Since it is overdamped, we only need to find the rise time which is the time needed to reach ninety percent of the reference output. This was measured to be about 230.163us. Next, we had to calculate a resistance that would create a critically damped response. To do this we used the equation and found the resistance to be 2000 ohms. For a critically damped response only the time rise is needed which was measured to be around 44.134us. Lastly, we set the resistance to be 500 ohms and found the response to be underdamped as seen in Graph 3. Since, the response was underdamped we measured the four major specifications. We found the rise time to be 20.112us, the peak time to be 35.196us, and the settling time to be 112.339us. The peak time is the time needed to reach the maximum response. The settling is the time needed for the circuit to settle within five percent of the reference output. Now we have to find the percentage overshoot which defined by the following equation:

he Mpand final values were measured to be 14.5V and 10V respectively. With these values we calculated the percentage overshoot to be 45. 

For our last circuit we modelled a parallel RLC circuit in PSPICE. We set the circuit up according to the parameters given in the lab handout. We started with a resistance of 10k ohms which yielded an underdamped response. The peak time was measured to be 19.145us and the settling time was measured to be 601.5us. Next we needed to find the resistance for when the circuit would produce a critically damped response. We used the equation

and found the resistance should be 500 ohms. Lastly, we changed the resistance to be 200 ohms which resulted in an overdamped response. The settling time of this response is 151.397us. 

The three natural responses can be ascertained by comparing a circuit’s neper frequency and resonance without looking at a graph. A circuit is overdamped when the neper frequency is greater than the resonance, underdamped when the neper frequency is less than the resonance, and critically damped when the neper frequency equals the resonance. By utilizing this comparison we can determine the response.

Conclusion

In conclusion, the lab consisted of second order RLC circuits and we observed its transient responses including time constants. From the oscilloscope, we noticed that the time constants changed as we connected the capacitors in series and parallel. For this lab we focused on what the response was for the voltage and to then apply the appropriate specifications of the four major depending on the response.The three natural responses can be ascertained by comparing a circuit’s neper frequency and resonance without looking at a graph. A circuit is overdamped when the neper frequency is greater than the resonance, underdamped when the neper frequency is less than the resonance, and critically damped when the neper frequency equals the resonance while for a critically damped response only the time rise is needed which was measured to be around 44.134us. By utilizing this comparison we determined the response.

Acknowledgements

Again, huge thanks to Nodebechukwu Paully-Umeh and Michael Rodriguez for helping me write this lab report and conduct the experiment in December of 2020.

Appendix