Superposition Theorem
Superposition theorem is one of those
strokes of genius that takes a complex subject and
simplifies it in a way that makes perfect sense. A theorem
like Millman's certainly works well, but it is not quite
obvious why it works so well. Superposition, on the
other hand, is obvious.
The strategy used in the Superposition
Theorem is to eliminate all but one source of power within a
network at a time, using series/parallel analysis to
determine voltage drops (and/or currents) within the
modified network for each power source separately. Then,
once voltage drops and/or currents have been determined for
each power source working separately, the values are all
"superimposed" on top of each other (added algebraically) to
find the actual voltage drops/currents with all sources
active. Let's look at our example circuit again and apply
Superposition Theorem to it:
Since we have two sources of power in this
circuit, we will have to calculate two sets of values for
voltage drops and/or currents, one for the circuit with only
the 28 volt battery in effect. . .
. . . and one for the circuit with only the
7 volt battery in effect:
When redrawing the circuit for
series/parallel analysis with one source, all other voltage
sources are replaced by wires (shorts), and all current
sources with open circuits (breaks). Since we only have
voltage sources (batteries) in our example circuit, we will
replace every inactive source during analysis with a wire.
Analyzing the circuit with only the 28 volt
battery, we obtain the following values for voltage and
current:
Analyzing the circuit with only the 7 volt
battery, we obtain another set of values for voltage and
current:
When superimposing these values of voltage
and current, we have to be very careful to consider polarity
(voltage drop) and direction (electron flow), as the values
have to be added algebraically.
Applying these superimposed voltage figures
to the circuit, the end result looks something like this:
Currents add up algebraically as well, and
can either be superimposed as done with the resistor voltage
drops, or simply calculated from the final voltage drops and
respective resistances (I=E/R). Either way, the answers will
be the same. Here I will show the superposition method
applied to current:
Once again applying these superimposed
figures to our circuit:
Quite simple and elegant, don't you think?
It must be noted, though, that the Superposition Theorem
works only for circuits that are reducible to
series/parallel combinations for each of the power sources
at a time (thus, this theorem is useless for analyzing an
unbalanced bridge circuit), and it only works where the
underlying equations are linear (no mathematical powers or
roots). The requisite of linearity means that Superposition
Theorem is only applicable for determining voltage and
current, not power!!! Power dissipations, being
nonlinear functions, do not algebraically add to an accurate
total when only one source is considered at a time. The need
for linearity also means this Theorem cannot be applied in
circuits where the resistance of a component changes with
voltage or current. Hence, networks containing components
like lamps (incandescent or gasdischarge) or varistors
could not be analyzed.
Another prerequisite for Superposition
Theorem is that all components must be "bilateral," meaning
that they behave the same with electrons flowing either
direction through them. Resistors have no polarityspecific
behavior, and so the circuits we've been studying so far all
meet this criterion.
The Superposition Theorem finds use in the
study of alternating current (AC) circuits, and
semiconductor (amplifier) circuits, where sometimes AC is
often mixed (superimposed) with DC. Because AC voltage and
current equations (Ohm's Law) are linear just like DC, we
can use Superposition to analyze the circuit with just the
DC power source, then just the AC power source, combining
the results to tell what will happen with both AC and DC
sources in effect. For now, though, Superposition will
suffice as a break from having to do simultaneous equations
to analyze a circuit.

REVIEW:

The Superposition Theorem states that a
circuit can be analyzed with only one source of power at a
time, the corresponding component voltages and currents
algebraically added to find out what they'll do with all
power sources in effect.

To negate all but one power source for
analysis, replace any source of voltage (batteries) with a
wire; replace any current source with an open (break).
