DC circuit analysis overview

Circuit analysis, or solving a circuit, means figuring out all the voltages and currents. Here’s some context for how engineers do that.

Written by Willy McAllister.

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Contents

• Basic terms and laws
• General strategy
• Methods
• Decomposing problems
• Computer simulation

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Basic terms and laws

The basic tools you know for analyzing a circuit are,

• Circuit terminology
  • wires, nodes, branches, meshes, loops, reference node (ground)
• Element i-v equations

  • $v = i\,\text R$

  • $i = \text C\,\dfrac{dv}{dt}$

  • $v = \text L\,\dfrac{di}{dt}$

• Series and parallel resistors

  • $\text R_{\text{series}} = \text{R1} + \text{R2} + \ldots + \text{R}_\text N$

  • $\dfrac{1}{\text R_{\text{parallel}}} = \dfrac{1}{\text{R1}} +\dfrac{1}{\text{R2}} + \ldots + \dfrac{1}{\text{R}_\text N}$

• Kirchhoff’s Laws

  • Kirchhoff’s Current Law $\quad\displaystyle \sum i_{node} = 0$

  • Kirchhoff’s Voltage Law $\quad\displaystyle \sum v_{loop} = 0$

General strategy

The first thing you do when given a circuit analysis problem is to simplify the circuit if possible. For a resistor circuit, search for series and parallel resistors and merge them together as much as you can.

After simplifying, all circuit analysis methods are some version of this strategy:

• Create a system of independent equations based on the element $i$-$v$ equations and how they are connected to each other.
• Solve the system of equations for the independent voltages or currents. This is often done using techniques from linear algebra.
• Solve the remaining voltages and currents.

Methods

There are three popular methods for doing circuit analysis. They all produce the same answer. You pick the method best matched to the circuit or the method you feel the most comfortable with.

• Direct application of the fundamental laws (Ohm’s Law and Kirchhoff’s Laws)
• Node Voltage Method
• Mesh Current Method and its close relative, the Loop Current Method

Direct application of the fundamental laws is an improvised method. It is quick and works well for simple circuits. It is not particularly efficient in terms of the total amount of work required, but that doesn’t matter for small circuits. Managing the amount of effort becomes important as circuits get more complicated.

Engineers have invented 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 minimize the number of equations you have to solve. This efficiency has a big impact for complicated circuits with lots of nodes and branches. The Loop Current Method is a close relative of the mesh method. It is used in special cases, as described in that article.

As we study these methods, the example circuits are only resistors and ideal sources. This keeps the math relatively simple and allows us to concentrate on learning the methods.

Decomposing problems

Circuit analysis can be an involved process for big circuits. As you work with circuits, you will learn how to break complicated problems into simpler pieces. Decomposing problems may seem slow at first, and you may feel impatient. “Why do I have to go through so many steps!?” However, breaking up problems into smaller steps is the heart of the engineering art. The multi-step circuit analysis methods are good examples of how engineers approach problem solving.

Computer simulation

The circuit simulator $\text{SPICE}$ 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.

Here at Spinning Numbers we have our own circuit simulator, the Circuit sandbox. It is available in a number of languages.

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

It is important to learn how to analyze circuits by hand and in your head. This skill deepens your insight into how a circuit works, insight you don’t get if you just draw a circuit and ask a computer for the answer. When you learn circuit analysis, you can look at a schematic and ‘see’ the circuit work in your mind.