Day 14 Passive RC Circuit Natural Response

Chapter 7 of our textbook discusses the natural response of an RC circuit as a capacitor charges and discharges. In order to observe this, we built a circuit in which we can switch the voltage from on to off and use a trigger in our Analog Discovery oscilloscope to capture the plot of the discharge/charge.

Our pre-Lab below shows two different step function scenarios. The first is a circuit in which the voltage source is disconnected. The second shows the voltage source being turned off by switching. We compare the two different effects of each scenario.

The first time constant we solve for below is for scenario A where the source is simply removed. This creates a circuit where all we have is R2 and C because no current will flow through resistor R1 since it is now an open circuit.

The next time constant we solve for is when the voltage source is switched from one wire to the next. This leaves resistor R1 as a passive element in the circuit, rather than an open circuit. The effects on time constant Tau are shown below.

It is evident that scenario A will discharge faster because the load resistance will absorb more of the energy stored than it will in scenario B where we have two resistors in parallel for a lesser resistance.
Now that we calculated the theoretical values of Tau we are ready to move on.

Below is the build of the circuit:

Placing 3 capacitors in parallel allowed us to have an equivalent capacitance of roughly 22 uF. Switching the voltage from on to off results in the discharge of the equivalent capacitance.

Below is the oscilloscope readings and a plot of the data, including the voltage value after one time constant tau.

I extracted the critical data from our Diligent Oscilloscope and plotted it using Matlab, placing the red asterisk at our measured value of Voltage after one time constant.

The Time constant from our measurement was roughly .020 seconds. Our theoretical value for tau was calculated to be .015 seconds. Our measurements were off by 33.3%. This can be directly a cause of experimental errors, inconsistencies in our resistor and capacitor values, and also in the measurement of tau.

I do not have any pictures or data from the switching of the second scenario. However, below is a plot of the step response of introducing a voltage source to the same capacitor in the circuit.

While our data and circuit elements were not ideal, the goal of the project was satisfied: to observe and understand the natural response of an RC circuit.