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 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.
- 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).
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