In Chapter 1, we presented Kirchhoff's laws (which govern the interaction between circuit elements) and Ohm's law (which governs the voltage-current relationships for resistors). These analytical tools provide us with the ability to analyze any circuit containing only resistors and ideal power supplies. However, we also saw in Chapter 1 that a circuit analysis, which relies strictly on a brute-force application of these tools can become complex rapidly - we essentially must use as our unknowns the voltage differences across all resistors and the currents through all resistors. This generally results in a large number of unknowns and a correspondingly large number of equations, which must be written and solved in order to analyze any but the simplest circuit.

In the next few chapters, we will still apply Kirchhoff's laws and Ohm's law in our circuit analysis, but we will focus on improving the efficiency of our analyses. Typically, this improvement in efficiency is achieved by reducing the number of unknowns in the circuit, which reduces the number of equations, which must be written to describe the circuit's operation.

In this chapter, we introduce analysis methods based on circuit reduction. Circuit reduction consists of combining resistances in a circuit to a smaller number of resistors, which are (in some sense) equivalent to the original resistive network. Reducing the number of resistors, of course, reduces the number of unknowns in a circuit.

We begin our discussion of circuit reduction techniques by presenting two specific, but very useful, concepts: Series and parallel resistors. These concepts will lead us to voltage and current divider formulas. We then consider reduction of more general circuits, which typically corresponds to identifying multiple sets of series and parallel resistances in a complex resistive network. This chapter then concludes with two important examples of the application of circuit reduction techniques: the analysis of non-ideal power sources and non-ideal measurement devices; without an understanding of these devices, it is impossible to build practical circuits or understand the consequences of a voltage or current measurement.

• Identify series and parallel combinations of circuit elements
• Determine the equivalent resistance of series resistor combinations
• Determine the equivalent resistance of parallel resistor combinations
• State voltage and current divider relationships from memory
• Determine the equivalent resistance of electrical circuits consisting of series and parallel combinations of resistors
• Sketch equivalent circuits for non-ideal voltage and current meters
• Analyze circuits containing non-ideal voltage or current sources
• Determine the effect of non-ideal meters on the parameter being measured

There are a number of common circuit element combinations that are quite easily analyzed. These “special cases” are worth noting since many complicated circuits contain these circuit combinations as sub-circuits. Recognizing these sub-circuits and analyzing them appropriately can significantly simplify the analysis of a circuit.

This chapter emphasizes two important circuit element combinations: elements in series and elements in parallel. Also discussed is the use of these circuit element combinations to reduce the complexity of a circuit’s analysis.