Model for Solving Circuit Analysis Problems with Kirchoff's Laws

1. Choose and label a current through every part of your circuit. A reasonable approach is to start with the current leaving the battery and then follow it labeling new currents as needed at every node. If you are unsure about the direction of the current through a particular resistor or device just choose one and let the math resolve the question for you. Classically the currents are labelled I₁, I₂, I₃...etc

2. Label the "uphill" and "downhill" sides of every resistor or device including the battery(s). + and - are good choices. Your labels must be consistent with the assumed direction of the current.

3. Write down the equations described by Kirchoff's Node Rule (ΣI_in =ΣI_out or Σ I_in = 0) for every node. In the second form of this equation one usually assigns positive values to currents coming in and negative values to currents going out. Never hurts to remember that Kirchoff's Node Rule (sometimes called Current Rule) is really an assertion of Conservation of Charge.

i) Throw out the "last" one. This is because there is always some duplication of information in these equations it turns out that if n is the number of nodes then the number of independent equations is n-1. This is why we drop one.

4. Write down the equations described by Kirchoff's Loop Rule ( Σ ΔV_i = 0). In this rule assign voltage differences going "downhill" negative (-) values and voltage differences going "uphill" positive (+) values as you traverse a loop. Note that uphill and downhill were labeled in step 2. Kirchoff's Loop Rule (sometimes called Voltage Rule) is really an assertion of Conservation of Energy (pretty important).

i) Once again there is a question of which loops to go around in the circuit. In any reasonable circuit there may be quite a few possibilities. Here's how I suggest you think about it. Look at the circuit as if it were a map of a farm showing all the fence lines and pastures (in classic circuit analysis these pastures are called meshes). Write one loop equation for each "pasture" by walking the "fenceline" all the way around until you get back to where you started recording each voltage drop as you come to it (with the appropriate sign).

5. Replace each ΔV in your loop equations with it's equivalent from Ohm's Law (ΔV = IR). This only applies to resistors or other "Ohmic" devices and not to batteries or other power supplies.

6. MATH TIME!! If you look at your equations (node and loop) you should find that you have a number of unknown currents and an equal number of equations. (i.e. 4 currents and 4 equations). IF this is the case then you now (merely) solve this system of equations by any method you have learned in your past math life.

7. Check the answers you get against your conceptual expectations for which currents are bigger and which are smaller. This will have two positive effects. First it will help you catch any outrageous math errors that lead to silly answers. Second, it will help you test your conceptual understanding of the behavior of circuits and enrich that understanding.

• As I mentioned in class I think this last step may be the most important skill I hope you develop. While we usually deal with settings in class that lead to results which are not revoltingly complicated when you are in the work world this is less commonly the case. You are likely to experience many cases where you work through a solution to a challenging setting and as you examine the result you have to wonder "Is this right?". An important part of verifying that you have a reasonable result is checking what your result tells you for special cases where you know what the answer should be. These are called limiting cases. You might know what should happen if one mass gets incredibly large and dominates the problem. You might know what should happen if you move the charges far apart so they both look like point charges to each other. You might know what should happen if you place the charge at some special place in the field. These kinds of questions ask you to look clearly and carefully at your result from the previous step and verify that it gives appropriate results under known circumstances. This is a very powerful and important skill to develop and will stand you in good stead down the road.

  If your result passes all these tests then you can sit back and relax with a well earned beverage of your choice.