The general strategy of circuit analysis is to create and solve a system of independent equations. Written by Willy McAllister.
Circuit analysis, or solving a circuit, means figuring out voltages and currents in each element. Here's an overview of circuit analysis, with some context for the various tools and methods we use to analyze circuits.

The tools

The general strategy

After simplifying a circuit as much as possible, all circuit analysis methods are some version of the following strategy:
  • Create a set of independent equations based on the elements and circuit connections.
  • Solve the system of simultaneous equations for the independent variables (voltages or currents). This often involves using techniques from linear algebra.
  • Solve the remaining individual element voltages and currents.

The methods

There are three popular circuit analysis methods. All three produce the same answer.
The first method, direct application of the fundamental laws, is quick and works very well for simple circuits. It is not particularly efficient in terms of the total amount of work required, which becomes important as circuits become more complicated.
Engineers have come up with two elegant ways to organize and streamline circuit analysis: the Node Voltage Method and the Mesh Current Method. These are general-purpose step-by-step recipes to solve a circuit. Both methods attempt to minimize the number of simultaneous equations. This efficiency has a big impact as circuit complexity grows (more and more nodes and branches). The Loop Current Method is a close relative of the Mesh method, used in certain special cases, as described in that article.
As we study the methods of circuit analysis, our example circuits are made of only resistors and ideal sources. This keeps the math relatively simple, allowing us to concentrate on the strategies for solving a circuit.

A final note: decomposing problems

Circuit analysis can be an involved process for complicated circuits. An important engineering skill is learning how to break down complicated problems into simpler pieces. Decomposing problems may seem slow at first, and you may feel impatient. However, breaking up problems into smaller steps is the heart of the engineering art. The multi-step circuit analysis methods presented next are good examples of the engineering approach to problem solving.

Epilogue - computer simulation of circuits

The circuit simulation application SPICE\text{SPICE} and many similar simulation programs are available for professional and student engineers, and anyone else interested in the subject. (search term: circuit simulator) Computer simulation is often used when solving even modestly complex circuits.

If I can solve a circuit with a simulation, why do I need to learn these circuit analysis methods?

It is important to learn how to analyze circuits by hand. Knowing these techniques makes short work of simple circuits. Better yet, this knowledge deepens your insight into how a circuit works, insight you don't get if you just ask a computer for the answer. When you learn circuit analysis, you develop your ability to look at a schematic and 'see' the circuit work in your mind.
Studying circuit analysis will also help you deal with the limitations and quirks of simulation programs.