Passive RC Circuit Natural Response

Passive RC Circuit Natural Response

In this lab we observe the natural response of the RC circuit shown above which has a 5 V DC source. The resistance of the resistors are R1 = 0.983 kΩ and R2 = 2.18 kΩ, and the capacitor has capacitance C = 22.5 µF.
First, we will find the natural response by opening a switch (disconnecting the source). Second by applying a switching step voltage from an arbitrary waveform generator (WaveForms). Lastly, we will simply short the voltage source to observe the response. We will then use the graph to find the time constant of the circuit and compare it to theoretical values.

Opening the Switch (Disconnecting the Source)

For the first part, the switch is opened by disconnecting the voltage source. In the picture, the red wire on the + rail supplies +5 V while the black wire on the - rail provides ground. The switch is opened by quickly disconnecting the voltage supply wire.

We obtain the following graph and find the initial time and its corresponding voltage. The voltage before the switch is opened, V0, can be calculated using a voltage divider, V0 = [2.18 kΩ/(2.18 kΩ + 0.983 kΩ)]*(5 V) = 3.4461 V. Compared to our initial voltage from our graph, V0 = 3.408 V has a 1.1% error. This occurs at the time t0 = -61.5 ms. To find our time constant, we refer to the equation governing the natural response of an RC circuit, V(t) = V0*exp(-t/τ), where τ = RC is the time constant. We can solve for the time constant by first finding the voltage when t = τ. When this happens, our equation is simplified to V(t) = V0*exp(-1). Plugging in the V0 obtained from the graph, we get V(t) = 3.408*exp(-1) = 1.2537 V. This means that when the voltage has dropped to this value, the time will be equal to one time constant. Looking through the graph we search for this voltage and corresponding time. The closest available point is 1.256 V, occurring at tf = -11 ms. Now we can solve for our time constant finding the time change between these two voltages t = tf - t0 = -11 ms - (-61.5 ms) = 50.5 ms. To compare this to the theoretical value, we take a look at the circuit and notice that when the switch is opened, all of the energy is dissipated into resistor R2. Therefore the time constant is τ =  R2*C = (2.18 kΩ)*(22.5 µF) = 49.05 ms, which has an error of 3.0 %.

Square Wave Switch (Virtual Short Circuit)

Next we use a square wave at a low frequency to provide a step switching. The wave has a 2.5 V amplitude with a 2.5 V offset, making it go from 5 V to 0 V at 1 Hz. We let the wave run and capture the graph for one down step switch in voltage.

We use the same technique as the previous time and find that the experimental time constant has changed to τ = 15.5 ms. If we examine the circuit closely, it has changed from the last time. Instead of disconnecting the source (opening the circuit), we are switching the voltage from 5 V to 0 V, which is about equivalent to short circuiting the source. Therefore, when we calculate the theoretical time constant, we must take resistor R1 into account. So the theoretical time constant now is calculated as τ = Req*C = [R1*R2/(R1+R2)]*C = (0.6775 kΩ)*(22.5 µF) = 15.24 ms. Comparing this value now with our experimental value we get a 1.7 % error.

Shorting the Source Physically

Lastly, we find the natural response of the circuit by physically short circuiting the source. The yellow wire is providing 5 V to R1, The red wire is connected to ground and the free end is then connected quickly to the same node to short circuit the power source. The same theoretical time constant τ = 15.24 ms applies here since the capacitor voltage is dissipated by both resistors.

We use the same analysis as before and find the experimental time constant to be τ = 16 ms. This has an error of 5.0 %. The error here is probably on the borderline of being unacceptable. However, since this whole process is happening in a matter of milliseconds, the error is probably due to the time it takes to actually plug the shorting wire in.

Here we see a summary of the measurements (on the left) and the theoretical time constant calculations on the right.