What is network
analysis?
Generally speaking, network analysis
is any structured technique used to mathematically analyze a
circuit (a "network" of interconnected components). Quite
often the technician or engineer will encounter circuits
containing multiple sources of power or component
configurations which defy simplification by series/parallel
analysis techniques. In those cases, he or she will be
forced to use other means. This chapter presents a few
techniques useful in analyzing such complex circuits.
To illustrate how even a simple circuit can
defy analysis by breakdown into series and parallel
portions, take start with this series-parallel circuit:
To analyze the above circuit, one 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 B1
with that total circuit resistance, the total current could
be calculated through the use of Ohm's Law (I=E/R), then
that current figure 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:
Resistors R2 and R3
are no longer in parallel with each other, because B2
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 reduce 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. However,
this is not the only type of circuit defying series/parallel
analysis:
Here we have a bridge circuit, and for the
sake of 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. Aaarrggghhhh!
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're 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've dealt
with thus far), there only needs to be a single equation to
solve for the single unknown:
However, when we're 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 intimidating and all too complex for explanation in
this chapter. However, many scientific and programmable
calculators are able to solve for simultaneous unknowns, so
it is recommended to use such a calculator when first
learning how to analyze these circuits.
This is not as scary as it may seem at
first. Trust me!
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
this chapter.
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REVIEW:
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Some circuit configurations ("networks")
cannot be solved by reduction according to series/parallel
circuit rules, due to multiple unknown values.
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Mathematical techniques to solve for
multiple unknowns (called "simultaneous equations" or
"systems") can be applied to basic Laws of circuits to
solve networks.
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