In this chapter we have determined how to identify basic series-parallel networks driven by a single effective voltage or current source. A key element here is to identify sub-circuits or subgroups of resistors that are comprised of either series-only or parallel-only configurations within themselves. These groupings can then be reduced to equivalent resistances using series and parallel combination techniques examined in earlier chapters. This process may be repeated until the entire circuit is simplified down to either a single series loop or parallel arrangement of resistors driven by a voltage or current source.

Once a circuit has been simplified, series and parallel analysis techniques, and laws such as Ohm's law, KVL, KCL, VDR and CDR, may be employed to determine various voltages and currents in the simplified equivalent. Given these results, the circuit may be expanded back into its original form in stages, reapplying these rules and techniques to determine voltages and currents within the sub-circuits. The process may be iterated until every current and voltage in the original circuit is discovered, if desired. Once these values are determined, power calculations are trivial.

There is an infinite variety of series-parallel configurations and consequently no single solution technique will work for all of them. Indeed, the more complex the circuit, the more solution paths there are for that circuit. It is therefore wise to plan some strategy for a solution instead of randomly “diving in” in order to ease the ultimate effort.

Two series-parallel configurations of note are the ladder network and the bridge network. A resistive ladder may be used to derive several different currents or voltages from a single source; the R-2R variation being particularly useful in that it divides the source down by powers of two. A basic resistive bridge is comprised of four elements arranged as a pair of series resistors in parallel with another pair of series resistors. Bridges can be used as part of a measurement scheme. For example, one or more of the resistors could be environmentally sensitive, such as a photoresistor, force sensing resistor or thermistor. As the associated variable changes, the resistance of the sensor changes which unbalances the bridge, and the resulting voltage indicates both the magnitude and sign of said environmental change.

## Review Questions

1. In general, describe the process of reducing a series-parallel resistive network down to a single equivalent resistance.

2. Do Ohm's law, KVL and KCL still apply in series-parallel networks? Why?

3. Is there a finite number of variations of series-parallel networks? Why/why not?

4. Describe the primary functional characteristic of an R-2R ladder network.

5. What is meant by the terms “load” and “loading”, as in “\(R_1\) loads \(R_2\)”?

6. How might a bridge network be used to sense changes in temperature?

7. Describe a general procedure to find the voltage between two arbitrary points in a series-parallel circuit.