Series RLC Circuit Step Response Lab

Objective: 
Today's class meeting reflected the idea of RLC circuit. The first focus was handling second order circuits and finding their initial and final conditions of current and voltage in terms of their derivatives. We understood how to find Vc(0), i(0), dv(0)/dt, di(0)/dt, ic(inf), and vc(inf). When determining the initial conditions we must keep in mind that v and i are defined strictly according to the passive sign convention and that capacitor voltage and inductor current is always continuous. RLC circuits are called second order circuits and when solved found that we can get three types of solutions: if alpha>Wo->overdamped case, if alpha=Wo->critically damped, alpha<Wo->underdamped. For different case, we keep in mind of using different equations.

Group Practice:
1.  The problem below shows how to find the initial and final values, most importantly the derivatives, dv/dt and di/dt.

Figure 1. Finding the initial and final conditions of an RLC circuit.

2. The problem below considers a source free series RLC circuit. By applying KVL we acquire a second order differential equation. We would then let i= Ae^st since the solution will be of an exponential form. By replacing the i and taking the derivative we can see that get a quadratic equation. Solving for s will give us the roots and can compacted by replacing the R/2L=alpha and 1/sqrt(LC)= omega. 

Figure 2. Process of solving a second order differential equation. 

3. The problem below shows how to solve for alpha and omega. We then consider whether it is overdamped, critically damped, or underdamped case.

Figure 3.  Given a circuit, we find alpha and omega and state the response.

4. The problem below shows how to find for the constants A1 and A2 for a parallel circuit.

Figure 4. We found the constants given some initial conditions of voltage.

Series RLC Circuit Step Response Lab and Procedures:
1. As part of the pre lab, we are required to calculate the undamped natural frequency and the neper frequency as seen in figure 6. We see that omega which is the undamped natural frequency is bigger than the neper frequency. This means that the we have an underdamped case.

Figure 5. Schematic for a series RLC Circuit

Figure 6. Finding the case of the circuit. Based on our results. It is underdamped. 

2. We expect the response for this case to be exponentially damped and oscillatory in nature. An example can be seen in figure 7.

Figure 7. Expected measurement response.

3. Before we start making the circuit, we measure our components: Resistor = 1.8 ohms, Inductor = 1mH, and capacitor 92.2 nF. We build a series RLC circuit and input a 2V step input with a low frequency. We also measure the voltage of the capacitor using the oscilloscope. However, we measure our theoretical values for the following: undamped natural frequency, damped natural frequency, damping ratio, rise time, overshoot, and DC gain. The calculations can be seen in figure 8 and the actual set up in figure 9.

Figure 8. Calculations

Figure 9. Actual set up for a series RLC circuit.

4. We measure the voltage across the capacitor and acquired the graph as seen in figure 10. 

Figure 10. the unexpected voltage output for the capacitor.

Series RLC Circuit Step Response Lab and Learning Outcome:
The first step of the experiment is to calculate our undamped natural and neper frequency. However, we first measure our actual components for the series RLC circuit which are , R=1.8ohms, L = 1mH, and C = 92.2nF. We get values of omega= and alpha = using the equations seen in figure 8 above . Based on our results, we noticed that the our alpha is smaller than our omega which means that the solution will be of an underdamped case. We also calculated the following: underdamped natural frequency=omega = 104,144, neper frequency=alpha=900, damping ratio=Z=.00864, and damped natural frequency =omega d = 104,140. The rise time is calculated by looking at the time in which the process first reaches the steady state value. and overshoot is calculated by acquiring the upper amplitude and dividing by the lower amplitude of the underdamped case. From out theoretical value we will compare with our experimental results. However, we did not correctly acquired our expected graph for the voltage as seen in figure 10. As a result, we were unable to compare our theoretical values with our experimental. The next step is to to modify our circuit so that the circuit becomes critically damped. The only way in which this can occur is if the underdamped natural frequency equals the neper frequency. Since we are unable to change natural frequency or DC gain, then the only way is to change the resistance to value where alpha will equal omega.