## The Basics

**What is an RC Circuit?**

As a reminder, an RC circuit uses a resistor and capacitor, and a capacitor is a circuit component that can store and discharge charge (energy) through a load element [1]. Shown below is the inverse relationship between the resistor and the capacitor, where as time goes on with the RC circuit connected, the charge increases over the capacitor while the current over the resistor decreases. This causes the voltage over the capacitor to increase while the voltage decreases over the resistor.

Because the voltage increases over the capacitor to an approximate maximum value, you can use that maximum voltage value to figure out the instantaneous amount of power over the capacitor due to that amount of voltage. The amount of power is related to the capacitance (C) and voltage (V) and the equation is shown below.

A key component of an RC circuit is that the capacitor acts as a filter and can smooth out the output voltage. The charged capacitor stores energy and can be passed through a load at a specific frequency level, which is determined by the time constant, or RC value [2]. Your RC value can then determine the cutoff frequency, or the point at which the system’s response will be reduced rather than passed through.

To figure out our RC value for experimentation purposes, we used the cutoff frequency equation, where our frequency was approximated to 60 Hz for our experiments. This was based on our approximation of our rps from the wave generator components and could be changed for future experimentation.

The larger your input frequency or smaller your RC value, the capacitor will charge and discharge quicker, which allows higher frequencies to pass through, but your overall maximum voltage level is lower. When the input frequency is smaller or the RC value is larger, the capacitor undergoes slower charging and discharging processes. This characteristic enables lower frequencies to pass through, leading to output voltages that closely align with the input signal’s maximum values.

By using an RC Circuit, you can use an AC input and store charge on a capacitor. In our experiment, we wanted to find the total voltage and power over our capacitor given an input frequency, input voltage, and desired total RC value, while changing our resistance and capacitance.

## Experimental Process

To learn about the basic circuit components of a resistor and a capacitor, you can conduct simple RC Circuit experiments and look at voltage drops over these components to learn more about how these values impact each other.

To conduct the experiments, a Moku Go was used to induct a frequency across the system. To learn how to set-up your own Moku Go, see the following videos we used to get started and learn how to use the oscilloscope feature:

- Getting Started: https://www.youtube.com/watch?v=rbEtV4NlG-k
- How to use Oscilloscope Feature: https://youtu.be/pn5CyhOYyWk?si=zJ9fpH84xYtKkAM7

Shown below is our basic breadboard for the RC Circuit. We iterated through this circuit by changing values of resistance and capacitance that when multiplied together, approximated to 0.00265. We switched out the resistors and capacitors for each iteration and recorded the output voltage and solved for the power.

Using the Moku Go, the input voltage was set to 5 Vpp and the input frequency was set to 60 Hz. Shown below is an example of the Oscilloscope interface with added measurements to record the peak to peak output voltage measurements. The red graph is our input voltage and our blue graph is the voltage across the capacitor.

This process was repeated to cycle through 10 capacitors that we had available in the lab with the closest corresponding resistance. Our results are shown below.

As shown above, our experiment followed traditional RC Circuit characteristics as when the capacitance value increased, the maximum output voltage was the lowest and the response was the smoothest (shown by the 33uF blue graph). This shows the filtering affect of an RC circuit with a constant frequency input.

## How RC Circuits Relate to other Engineering Concepts

RC Circuits and RLC Circuits are great ways to model different basic mechanical engineering concepts. Take a look at the two drop downs for lab experiments that model this.

RC Circuits are a way to simulate heat transfer as the flow of electricity can be analogous to the flow of heat. To first visualize how the variables can be related, check out this link just looking at heat transfer related to voltage (V), resistance (R), and current (C).

When a capacitor is added to the circuit, the analogy becomes a bit more complicated due to how capacitors dissipate over time. Below is an image correlating how Lumped Heat Transfer is able to be analogous to an RC Circuit.

For more information on how each was derived, see the below links:

Explanation of Lumped Heat Transfer (A):

Explanation of RC Circuit (B):

If you are interested in a further mathematical derivation showing the relationship RC Circuits and thermodynamic analysis, click here.

An RLC Circuit is similar to an RC Circuit, however it expands as it adds an inductor along with a resistor and capacitor and can be arranged in both series and parallel. When learning about RLC Circuits, they can be used to model mass-spring dampener systems and vice versa. If you want to learn more, check out these links for some background information as well as equations to follow along with in order to derive these relationships!

## References

[1] OpenStax. “RC Circuits.” LibreTexts. [Online]. Available: https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/10%3A_Direct-Current_Circuits/10.06%3A_RC_Circuits

[2] Collimator AI. “What are RC Filters?” Collimator AI Reference Guides. [Online]. Available: https://www.collimator.ai/reference-guides/what-are-rc-filters#:~:text=The%20two%20essential%20components%20in,to%20filter%20out%20unwanted%20frequencies.

[3] “RC Integrator.” eValidate Lab. [Online]. Available: http://www.evalidate.in/lab1/pages/RC/RCIntegrator/RCIntegrator_I.html

[4] M. W. Islam, et al., “A New Wavelet Family Based on Second-Order LTI Systems,” ResearchGate, 2017. [Online]. Available: https://www.researchgate.net/publication/309287880_A_new_wavelet_family_based_on_second-order_LTI_systems?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6Il9kaXJlY3QiLCJwYWdlIjoiX2RpcmVjdCJ9fQ