In this chapter we shall examine a number of theorems and techniques to help us analyze complex circuits and address specialized applications. We will begin by examining the concept of source impedance in order to make more accurate models of our idealized constant voltage and current sources. This will be a step beyond using a simple resistance as found in the DC case. From there we will investigate how to convert from one type of source to another, such as creating a voltage source that is the functional equivalent of a current source. A functional equivalent is a source that can be swapped out for another while leaving all of the other circuit currents and voltages intact. In other words, all of the circuit's component voltage drops and branch currents will be identical to those found in the original configuration. This technique is useful in a number of ways, particularly in that it can help reduce more complex circuits to simplify analysis.
The concept of equivalence can be extended beyond just a single source to an entire network. For this we will examine Thévenin's and Norton's theorems. Using these theorems, entire circuits utilizing dozens of components can be modeled as a single source with an associated complex impedance. When coupled with the maximum power transfer theorem, these tools will allow us to determine component values that produce the maximum amount of load power.
We will also address a method of analyzing circuits that contain multiple current and/or voltage sources that are connected in a non-trivial fashion (i.e., not just series voltage sources or parallel current sources). This is called the superposition theorem and it can be applied to any circuit or parameter that meets certain requirements, including circuits that have a mix of current sources and voltage sources. Superposition can also be used to determine voltages and currents when sources use different frequencies. In fact, one of way of imagining a complex wave-shape is to treat it as a series of connected sources, each with a unique frequency, phase and amplitude. Superposition we give us a means to handle this new situation.
Finally, we will examine how to find equivalent circuits for certain component arrangements that use three connecting points, in other words, RLC combinations shaped like a triangle or like the letter Y. These are known as delta and Y configurations. These configurations are difficult to address with basic series-parallel simplification techniques. Converting from one configuration to the other will help solve that issue.