DC Mesh Current Analysis

It has been my experience that students require much practice with circuit analysis to become proficient. To this end, instructors usually provide their students with lots of practice problems to work through, and provide answers for students to check their work against. While this approach makes students proficient in circuit theory, it fails to fully educate them.

Students don’t just need mathematical practice. They also need real, hands-on practice building circuits and using test equipment. So, I suggest the following alternative approach: students should build their own “practice problems” with real components, and try to mathematically predict the various voltage and current values. This way, the mathematical theory “comes alive,” and students gain practical proficiency they wouldn’t gain merely by solving equations.

Another reason for following this method of practice is to teach students scientific method: the process of testing a hypothesis (in this case, mathematical predictions) by performing a real experiment. Students will also develop real troubleshooting skills as they occasionally make circuit construction errors.

Spend a few moments of time with your class to review some of the “rules” for building circuits before they begin. Discuss these issues with your students in the same Socratic manner you would normally discuss the worksheet questions, rather than simply telling them what they should and should not do. I never cease to be amazed at how poorly students grasp instructions when presented in a typical lecture (instructor monologue) format!

A note to those instructors who may complain about the “wasted” time required to have students build real circuits instead of just mathematically analyzing theoretical circuits:

What is the purpose of students taking your course?

If your students will be working with real circuits, then they should learn on real circuits whenever possible. If your goal is to educate theoretical physicists, then stick with abstract analysis, by all means! But most of us plan for our students to do something in the real world with the education we give them. The “wasted” time spent building real circuits will pay huge dividends when it comes time for them to apply their knowledge to practical problems.

Furthermore, having students build their own practice problems teaches them how to perform primary research, thus empowering them to continue their electrical/electronics education autonomously.

In most sciences, realistic experiments are much more difficult and expensive to set up than electrical circuits. Nuclear physics, biology, geology, and chemistry professors would just love to be able to have their students apply advanced mathematics to real experiments posing no safety hazard and costing less than a textbook. They can’t, but you can. Exploit the convenience inherent to your science, and get those students of yours practicing their math on lots of real circuits!

I wrote this question in such a way that it mimics branch/mesh current analysis, but with enough added information (namely, the current source’s value) that there is only one variable to solve for. The idea here is to prepare students for realizing why simultaneous equations are necessary in more complex circuits (when the unknowns cannot all be expressed in terms of a single variable).

The purpose of this question is to get students to apply what they know of basic circuit “laws” (Ohm’s Law, KVL, KCL) to the solution of a single voltage value. As usual, the method of solution is far more valuable than the answer.

If some students are completely confused regarding how to solve for this voltage, suggest that they “plug” the given answer into the circuit and determine currents and voltage drops. What do they notice when they do this? What unusual condition(s) stand out with the variable source at 9.5 volts? Are any of these conditions things they could have (or should have) known prior to knowing the variable source’s voltage, given the condition of “. . . no current [drawn] from the 6-volt battery”?

Students’ equations may not look exactly like these, depending on how they “stepped” around the loops tallying voltage drops. So long as they are all able to reach the same answers for I₁ and I₂, it does not matter. In fact, it is a good thing to have different students propose different forms of the equations to demonstrate that the same answers are obtained every time.

Students may find slight differences between variations of the “Branch Current” method of analysis described in different references. However, these differences are of no consequence.

Your students may find the setup of KVL equations easier with the two mesh currents going in the same direction through battery #1, but they should be able to arrive at the same answer either way. It is very important to your students’ understanding of the Mesh Current technique that they are able to handle both situations!

Students’ equations may not look exactly like these, depending on how they “stepped” around the loops tallying voltage drops. So long as they are all able to reach the same answers for I₁, I₂, and I₃, it does not matter. In fact, it is a good thing to have different students propose different forms of the equations to demonstrate that the same answers are obtained every time.

Of special importance in this problem is how students represent two meshing currents at a single resistor in their equations. A common mistake for beginning mesh current analysts is to disregard the relative directions of meshing currents. It makes a huge difference whether two mesh currents go the same direction through a resistor, or whether they go in opposite directions!

There is a lot of setup work and arithmetic to do in the analysis of these two circuit configurations. This exercise is not only a thorough application of the mesh current method, but it also serves as an excellent application for computer simulation software. Give your students the opportunity to analyze both these circuits with simulation software, so they may appreciate the power of this technology.

Ask your students this question: “Suppose a student enters their circuit into a computer simulation program, and the program gives them an answer that is known to be incorrect. What does this indicate to the student?” Simulation tools are useful, both only so far as the user understands what he or she is doing. Far too often I encounter students who blindly accept the results of computer simulation, because they mistakenly think that whatever output the computer generates must be correct, not understanding how errors in the input or the use of different simulation algorithms affects accuracy of the simulated results.