2.1 INTRODUCTION

Generally speaking, network analysis is any structured technique used to
mathematically analyze a circuit (a “network” of interconnected components).
This chapter presents a few techniques useful in analyzing such
complex circuits.

FIGURE 2.1

To analyze the above circuit, we would first find the equivalent of R₂
and R₃ in parallel, then add R₁ in series to arrive at a total resistance. Then,
taking the voltage of battery E₁ with the total circuit resistance, the total
current could be calculated through the use of Ohm’s law (I = E/R), then
that current figure is used to calculate voltage drops in the circuit. All in all,
a fairly simple procedure.

However, the addition of just one more battery could change all of that:

FIGURE 2.2

Resistors R₂ and R₃ are no longer in parallel with each other, because
E₂ has been inserted into R₃’s branch of the circuit. Upon closer inspection,
it appears there are no two resistors in this circuit directly in series or
parallel with each other. This is the crux of our problem: in series-parallel
analysis, we started off by identifying sets of resistors that were directly in
series or parallel with each other, and then reduced them to single, equivalent
resistances. If there are no resistors in a simple series or parallel configuration
with each other, then what can we do?

It should be clear that this seemingly simple circuit, with only three
resistors, is impossible to reduce as a combination of simple series and simple
parallel sections. It is something different altogether.

FIGURE 2.3

Here we have a bridge circuit, and for the sake of the example we will
suppose that it is not balanced (ratio R₁/R₄ not equal to ratio R₂/R₅). If
it were balanced, there would be zero current through R₃, and it could
be approached as a series/parallel combination circuit (R₁–R₄ // R₂–R₅).
However, any current through R₃ makes a series/parallel analysis impossible.
R₁ is not in series with R₄ because there’s another path for electrons to flow
through R₃. Neither is R₂ in series with R₅ for the same reason. Likewise, R₁
is not in parallel with R₂ because R₃ is separating their bottom leads. Neither
is R₄ in parallel with R₅.

Example 2.1. Find the value of current in resistance R₃ shown in Figure 2.3,
where R₁ = 1Ω, R₂ = 2Ω, R₄ = 3Ω, R₅ = 6Ω, and R₃ = 2Ω, E1 = 10 V.

Solution:

therefore, current flowing through R₃ = 0 A (balanced bridge circuit).

Example 2.2.Find the value of current in R₁ as shown in Figure 2.3. Other
parameters are the same as used in Example 2.1.

Ans. I = 2.5 A

Although it might not be apparent at this point, the heart of the problem
is the existence of multiple unknown quantities. At least in a series/parallel
combination circuit, there was a way to find total resistance and total voltage,
leaving total current as a single unknown value to calculate (and then that
current was used to satisfy previously unknown variables in the reduction
process until the entire circuit could be analyzed). With these problems,
more than one parameter (variable) is unknown at the most basic level of
circuit simplification.

With the two-battery circuit, there is no way to arrive at a value for “total
resistance,” because there are two sources of power to provide voltage and
current (we would need two “total” resistances in order to proceed with any
Ohm’s Law calculations). With the unbalanced bridge circuit, there is such a
thing as total resistance across the one battery (paving the way for a calculation
of total current), but that total current immediately splits up into unknown
proportions at each end of the bridge, so no further Ohm’s Law calculations
for voltage (E = IR) can be carried out.

So what can we do when we are faced with multiple unknowns in a
circuit? The answer is initially found in a mathematical process known as
simultaneous equations or systems of equations, whereby multiple unknown
variables are solved by relating them to each other in multiple equations. In
a scenario with only one unknown (such as every Ohm’s Law equation we
have dealt with thus far), there only needs to be a single equation to solve for
the single unknown.

However, when we are solving for multiple unknown values, we need to
have the same number of equations as we have unknowns in order to reach
a solution. There are several methods of solving simultaneous equations, all
rather intimidiating and all too complex for explanation in this chapter.

Later on we’ll see that some clever people have found tricks to avoid
having to use simultaneous equations on these types of circuits. We call
these tricks network theorems, and we will explore a few later in Chapter 8.

• Some circuit configurations (“networks”) cannot be solved by reduction
according to series/parallel circuit rules, due to multiple unknown values.
• Mathematical techniques to solve for multiple unknowns (called “simultaneous
equations” or “systems”) can be applied to basic laws of circuits
to solve networks.