16-Apr-2015: Passive RC Circuit Natural Response

PURPOSE

The purpose of this experiment was to analyze the natural response of a circuit to estimate its time constant and compare it to the theoretical value.

PRE-LAB

Figure 1

Before constructing our circuit, we did some calculations to find the initial voltage across the capacitor and time constants of the two circuits shown in Figure 1. We found the capacitor voltage using voltage division, while we found the time constants by multiplying the Thevenin resistance (Rth) of the circuit with the capacitance of the capacitor. These were the theoretical values that we would later compare to the experimental values.

PROCEDURES

Figure 2

For the first part of the experiment, we set up the circuit as illustrated in Figure 2. We applied 5 V to the circuit for a few moments to allow it to reach steady-state. Then, we disconnected the voltage source (circled in red) and observed the natural response of the RC circuit, as shown in Figures 3 and 4.

Figure 3: Initial time of response

Figure 4: Time at which the voltage was at 36.79 percent of its initial value

To start our analysis of these graphs, we first calculated how much 36.79 percent of the initial voltage (3.408 V) was, which was 1.254 V. We looked at the graph in Figure 3 to find the time at which the response began (t0 = -61.5 ms). We subtracted this value from the time when the voltage reduced to 36.79 percent of its initial value (tf = -11 ms) to find Δt. This Δt was equal to the time constant, τ, of this circuit. We found this value to be equal to 50.5 ms. We compared this to the theoretical value found in the pre-lab by finding the percent difference, which was 2.956 percent. Since this was less than 5 percent, we concluded that our results were acceptable.

Figure 5: Initial time of response (square wave)

Figure 6: Time at which the voltage reached 36.79 percent of its initial value (square wave)

For the second part of the lab, we applied a square wave with an amplitude of 2.5 V and an offset of 2.5 V to the same circuit as the one used in Part 1 (Figure 2). Since the voltage oscillated between 0 V and 5 V, this input acted as an on and off switch. In other words, the voltage source acted as a short circuit when it was at 0 V. The response of the circuit when the input voltage was at 0 V is shown in Figures 5 and 6. Using the same process as Part 1, we found the time constant to be 15.5 ms. When compared to the value found in the pre-lab (15.24 ms), it is only 1.706 percent bigger. Therefore, we can conclude that we conducted the experiment correctly.