# Motivation

Students struggle with both a conceptual understanding and the mathematical application of Kirchoff’s Voltage Law (KVL). The highly abstract nature of potential and electricity creates difficulties in students differentiating between and simply understanding such topics as charge, current, potential difference, and power (1), (2). I have developed a diagram that builds conceptual understanding of potential and potential difference and also aids in generating a system of equations adhering to KVL.

While I’ve seen similar qualitative potential vs. position graphs in several common introductory physics textbooks, none of them incorporated multiple loops within the circuit nor were they explicitly discussed as a tool to use in problem solving. My goal here is to address both of those points and have these diagrams to be as integral to circuit analysis as free-body diagrams are to dynamics.

I’ll be presenting this during a poster session at the 2015 Physics Education Research Conference in College Park, MD. Come and find me during then if you’ll be there!

# The KVL Diagram

At it’s core, the diagram is a qualitative potential vs. position graph that incorporates all loops within the circuit. Electric potential at any point is represented by the height of the line on the diagram. Different loops within the circuit are represented by different branches on the diagram. To generate a diagram, the following guidelines should be followed.

- Draw a horizontal dashed line to represent the ground potential, 0 V.
- Pick a point on the circuit to begin. The battery or other source is best.
- Upward slanted lines represent the rise in potential provided by a source.
- Flat lines represent the wire between elements in the circuit.
- Downward slanted lines represent the drop in potential that charges experience when passing through that element.
- Choose a single loop to follow and complete each loop one at a time.
- After the final element in a circuit is reached in a loop, make sure the line drops to the 0 V dashed line.
- For additional loops, simply extend a horizontal flat line from the first loop at the point at which the paths diverge.

I’ve taken three simple circuits and diagrammed them below to give an idea of how they look. The points (A, B, C, etc) aren’t part of the diagram, but are presented here to help you see how the schematic “translates” to the KVL diagram.

# Limitations

The diagram’s primary advantage is in generating equations according to KVL without knowing anything quantitative about the circuit. The diagram is meant to be a stepping stone towards such an analysis. Given this, without any quantitative information, the relationship between the magnitudes of different potential differences cannot be known. However, since the height of a line clearly represents absolute potential, students could mistakenly conclude that one potential drop is more, less, or equal to another. In the first example, the drop in height of the line is shown to be the same, but without any additional information, it cannot be known that ΔV1 is equal to ΔV2. Therefore, this limitation must be explicitly discussed with students. Additionally, the slope of slanted lines do not yet provide any information about the circuit.

# Student Work Sample

The following picture shows an example of student work using the KVL diagram. This was taken from an AP Physics 1 class during the Spring 2015 semester.

Students worked on the problem below in groups of 2-3. The problem was taken from *TIPERs: Sensemaking Tasks for Introductory **Physics* by C. J. Hieggelke, Steve Kanim, D. P. Maloney, and T. L. O’Kuma. Students were asked to rank the magnitude of the potential difference between points M and N. All of the bulbs and batteries in each circuit are identical.

Students were not given any of the typical “rules of thumb” for circuits, such as *the potential drops across parallel elements are equal*. Through working problems like these, this was done to allow students to discover such relationships on their own as well as to be a soft test of the diagram’s effectiveness in developing conceptual understanding. Anecdotally, students found the diagrams useful, especially in differentiating between current and potential difference

# Invitation to collaborate

This is only the beginning for me with this diagram. Over the next several years, I will be doing some kind of research on the effectiveness of these diagrams. I’m not yet sure what my research questions will be or what such studies will look like, but I know I’ll be doing something! If you’ve got any feedback on the diagrams or want to participate in research with me, then let me know! Find me on twitter (@TRegPhysics) or shoot me an email!

# References

(1) Cohen, R., Eylon, B., and Ganiel, U. (1983), “Potential Difference and Current in Simple Electric Circuits: A Study of Students’ Concepts,” *Am. J. Phys*. **51**, 407.

(2) McDermott, L. C., and Shaffer, P. (1992), (a) “Research as a Guide for Curriculum Development: An Example from Introductory Electricity, Part I: Investigation of Student Understanding,” *Am. J. Phys. ***60**(11), 994.

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