A series-parallel configuration is one that is formed by a combination of series and
parallel elements.

A complex configuration is one in which none of the elements are in series or parallel.

In this chapter, we examine the series-parallel combination using the basic laws introduced
for series and parallel circuits. There are no new laws or rules to learn-simply an approach
that permits the analysis of such structures.

In the next chapter, we consider complex networks using methods of analysis that allow us to analyze any type of network.
The possibilities for series-parallel configurations are infinite. Therefore, you need to examine each network as a separate entity and define the approach that provides the best path to determining the unknown quantities.

In time, you will find similarities between configurations
that make it easier to define the best route to a solution, but this occurs only with
exposure, practice, and patience. The best preparation for the analysis of series-parallel networks
is a firm understanding of the concepts introduced for series and parallel networks.
All the rules and laws to be applied in this chapter have already been introduced in the previous two chapters.

The network in

Fig. 1 is a series-parallel network. At first, you must be very careful to determine which elements are in series and which are in parallel. For instance, resistors $R_1$ and $R_2$ are not in series due to resistor $R_3$ being connected to the common point b between $R_1$ and $R_2$. Resistors $R_2$ and $R_4$ are not in parallel because they are not connected at both ends. They are separated at one end by resistor $R_3$. The need to be absolutely sure of your definitions from the last two chapters now becomes obvious. In fact, it may be a good idea to refer to those rules as we progress through this chapter.

**Fig. 1: **Series-parallel dc network.

If we look carefully enough at

Fig. 1, we do find that the two resistors $R_3$ and $R_4$ are
in series because they share only point c, and no other element is connected to that point.
Further, the voltage source E and resistor $R_1$ are in series because they
share point a, with no other elements connected to the same point. In
the entire configuration, there are no two elements in parallel.

How do we analyze such configurations? The approach is one that
requires us to first identify elements that can be combined. Since there
are no parallel elements, we must turn to the possibilities with series elements.
The voltage source and the series resistor cannot be combined
because they are different types of elements. However, resistors $R_3$ and
$R_4$ can be combined to form a single resistor. The total resistance of the
two is their sum as defined by series circuits. The resulting resistance is
then in parallel with resistor $R_2$, and they can be combined using the
laws for parallel elements.

The process has begun: We are slowly reducing
the network to one that will be represented by a single resistor equal
to the total resistance "seen" by the source. The source current can now be determined using Ohm's law, and we can work back through the network to find all the other currents and voltages.

The ability to define the first step in the analysis can sometimes
be difficult. However, combinations can be made only by using
the rules for series or parallel elements, so naturally the first step may
simply be to define which elements are in series or parallel. You must
then define how to find such things as the total resistance and the source
current and proceed with the analysis. In general, the following steps
will provide some guidance for the wide variety of possible combinations
that you might encounter.

**General Approach:**
- Take a moment to study the problem "in total" and make a brief
mental sketch of the overall approach you plan to use. The result
may be time- and energy-saving shortcuts.
- Examine each region of the network independently before tying
them together in series-parallel combinations. This usually simplifies
the network and possibly reveals a direct approach toward
obtaining
one or more desired unknowns. It also eliminates many of
the errors that may result due to the lack of a systematic approach.
- Redraw the network as often as possible with the reduced branches
and undisturbed unknown quantities to maintain clarity and provide
the reduced networks for the trip back to unknown quantities
from the source.
- When you have a solution, check that it is reasonable by considering
the magnitudes of the energy source and the elements in the
network. If it does not seem reasonable, either solve the circuit
using another approach or review your calculations.

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