6.1: Introduction

This chapter presents methods for the analysis of AC circuits that employ resistors, capacitors and inductors along with any number of voltage and/or current sources. The methods of interest are nodal analysis and mesh analysis. Nodal analysis is the most general technique and can be applied to virtually any circuit. Mesh analysis is nearly as versatile and works well if only voltage sources are present. Both analysis methods generate a system of simultaneous linear equations that are used to solve the circuit for desired voltages or currents. That is, the system generates a set of values, either currents or node voltages, rather than individual currents or voltages. There are several methods that can be used to solve the simultaneous equations. These include substitution, Gauss-Jordan elimination and expansion by minors. These methods are reviewed in Appendix B and are not covered in this chapter. Instead, to focus on the circuit analysis aspects with minimal distraction, the explanations and examples will simply detail the process of examining the circuit and developing the system of equations. The specific technique employed to solve these simultaneous equations depends solely on your personal preferences.

At this point in the study of AC circuits, it is particularly efficient to obtain an advanced scientific calculator that can solve the system of equations directly versus working through the solution manually. By doing so, you can spend your time more effectively; meaning, mastering the process of circuit analysis and creating the equations. Manual solution techniques, though not necessarily difficult, can be tedious, time consuming and error prone. Indeed, on larger circuits, there can be a 10:1 differential in time when using a capable calculator versus a standard scientific calculator¹. If you have not already done so, you should consider obtaining a calculator that can solve simultaneous equations with complex coefficients (i.e., the complex real/imaginary quantities we have been using). Such calculators can be expensive when purchased new, such as the Texas Instruments TI-89 and Nspire models. On the used market, perfectly satisfactory older models such as the TI-85 and TI-86 can be found at considerable discount. Another model to consider is the Casio FX-9750GII, although it is not quite as powerful as some of the other units mentioned.

Along with nodal and mesh, we shall also introduce the concept of dependent AC sources. Dependent sources do not exhibit a fixed value, but rather the current or voltage is dependent on some other current or voltage in the circuit. What makes this interesting is that this controlling current or voltage may itself be affected by the value produced by the dependent source. Dependent sources are not lab instruments, like signal or function generators. Instead, they are used to model the behavior of active electronic devices such as bipolar and field effect transistors. Mastering the analysis of circuits using dependent sources is critical to the understanding of active circuitry that use transistors and similar devices².

References

¹That's like doing three times as many problems in one-third the time. We don't often get these kinds of opportunities.

²For more on transistors and other semiconductors, see Semiconductor Devices: Theory and Application. Another free OER text by the author.