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
re-drawing 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 gas-discharge) 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 polarity-specific 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.
