Avoid dependent equations when applying Kirchoff's Laws I tutor college students in Physics. In particular, we are currently working on solving circuits using Kirchoff's Laws. In a recent problem, I ended up with a system of linearly dependent equations and was unable to solve for the unknown currents until I replaced one of the equations with another equation that was linearly independent.
What is a good rule of thumb to avoid this kind of situation? In this particular situation, we had 4 unknown currents and we set up 4 equations just using Kirchoff's Loop Law. It seems that in such a situation with n unknowns, n equations just from the loop law) will always be linearly dependent. Is this the case? So is it a good idea to use at most n-1 equations from the loop law and resort to the node law to get the rest of our equations?
 A: For planar circuits I have found the following pattern to work in most cases.
1) Note voltage values or assign voltage variable names with polarities to every circuit element.
2) Remembering that elements which are in series have identical currents through them, assign current values or current variables with directions.  It's convenient to have the current direction pointing from + to - signs through the circuit element.
3) Write Kirchoff voltage loop equations for every ``simple'' loop, i.e., loops which do not enclose other loops (they can touch others, but can't surround them). Pay attention to what the entering polarity of the voltage is. If you enter the + end, add the voltage assigned. If you enter the - end, subtract it.  Total is 0.  Ignore all current directions while doing this.
4) Mark the nodes which have more than 2 currents entering or leaving. Imagine you have N of these. Write Kirchoffs nodal equations for N-1 of these.
5) Apply Ohm's law to all resistors to eliminate voltages in terms of currents in the loop equations. OR eliminate all currents in terms of voltages in the nodal equations.
6) If you have inductors or capacitors, you need to apply either the derivative relations (for DC analysis) or impedence relations (for AC analysis).  This can make things interesting.
Solve the resulting system of equations for the unknowns. It should be a system of linearly independent equations unless the original problem was over-specified.
A: I have always found it extremely cumbersome to solve circuits by writing and solving a horrible system of equations. It will always work, but it is not very elegant. For instance, in doing so, you can end up proving the voltage divider formula that you could have used directly. My best piece of advice is to never introduce currents unless explicitly requested. In most exercises you will do during your studies, I guess you will be able to do it way more simply using voltage dividers or Millman's theorem, the superposition principle, and in the most extreme cases Kennelly's theorem. If none of it works, then you can write your whole system of equations.
