In the preceding two chapters, series circuits and parallel circuits were examined. Each offered unique laws and rules for their configuration, such as Kirchhoff's voltage law and the current divider rule. Distinct techniques of analysis were offered to determine system current and voltage, and the currents and voltages associated with individual components. This almost mechanical approach was successful mainly because there are so few variations on the theme. In the case of a series circuit, more resistors might be added in the “daisy chain”, and likewise multiple voltage sources, but once you have experience with a few of these types of circuits, there is not much concern when simply adding more of the same because a pattern becomes apparent. The same holds true with parallel circuits: more resistors might be added as further “rungs on the ladder”, as well as multiple current sources similarly arranged, but again, once familiarity has been gained, obvious patterns emerge. This is not the case with series-parallel circuits, and thus, there are no simple recipes for our solution paths. Fortunately, laws and rules such as Ohm's law, KCL and KVL still hold true for these circuits, thus the focus shifts to their appropriate application.
To begin with, there are infinite variations of series-parallel circuits. This chapter deals with a subset, namely those that are driven by a single current or voltage source (after reducing trivial combinations), and which may be simplified using series and parallel resistor combinations. More complex configurations using multiple sources await in subsequent chapters. The key to analyzing basic series-parallel circuits is in recognizing portions of the circuit, or sub-circuits, that exhibit a series or parallel configuration by themselves, and then applying the series and parallel analysis rules to those sections. Ohm's law, KVL and KCL may be used in turn to “chip away” at the problem until all currents and voltages are found. As individual voltages and currents are determined, this makes it easier to apply these rules to determine other values. Given this observation, the number of potential solution paths tends to grow exponentially as the number of components increases. As a consequence, when faced with the same circuit, six people may solve it six different ways, no particular way being more or less correct than any other. The only thing we can say is that some solution paths might be more computationally efficient than others, meaning they take less work. Do not let this bother you. The fact that these circuits can be solved in a diverse number of ways is a strength, not a weakness. After all, you only need to recognize one of those ways, not all of them, in order to be successful. In this chapter's examples, various competing methods will be explored, but not every solution path will be spelled out for each one. Flexibility of thought and view will prove to be an asset.