The superposition theorem is a very important concept in the circuit theory. If a circuit has two or more independent sources, one way to determine the value of a specific variable (voltage or current) is to use
nodal or
mesh analysis. Another way is to determine the contribution of each independent source to the variable and then add them up. The latter approach is known as the superposition.
What are superposition rules?
In general, the theorem can be used to do the following:
- Analyze networks that have two or more sources that are not in series or parallel.
- Reveal the effect of each source on a particular quantity of interest.
- For sources of different types (such as dc and ac, which affect the parameters of the
network in a different manner) and apply a separate analysis for each type, with the
total result simply the algebraic sum of the results.
The superposition theorem states the following:
The current through, or voltage across, any element of a network is equal to the algebraic sum of the
currents or
voltages produced independently by each source.
In other words, superposition theorem allows us to find a solution for a current or
voltage using only one source at a time. Once we have the solution for
each source, we can combine the results to obtain the total solution.
If we are to consider the effects of each source, the other sources
obviously must be removed. Setting a voltage source to zero volts is like
placing a short circuit across its terminals. Therefore,
when removing a voltage source from a network schematic, replace it
with a direct connection (short circuit) of zero ohms. Any internal
resistance associated with the source must remain in the network.
Setting a current source to zero amperes is like replacing it with an
open circuit. Therefore,
when removing a current source from a network schematic, replace it
by an open circuit of infinite ohms. Any
internal resistance associated
with the source must remain in the network.
The above statements are illustrated in
[Fig. 1].
Fig. 1: Removing a voltage source and a current source to permit the application
of the superposition theorem.
Since the effect of each source will be determined independently, the
number of networks to be analyzed will equal the number of sources.
If a particular current of a network is to be determined, the contribution
to that current must be determined for each source. When the effect of
each source has been determined, those currents in the same direction
are added, and those having the opposite direction are subtracted; the
algebraic sum is being determined. The total result is the direction of the
larger sum and the magnitude of the difference.
Similarly, if a particular voltage of a network is to be determined, the
contribution to that voltage must be determined for each source. When
the effect of each source has been determined, those voltages with the
same polarity are added, and those with the opposite polarity are subtracted;
the algebraic sum is being determined. The total result has the
polarity of the larger sum and the magnitude of the difference.
What is the limitation of superposition theorem?
Analyzing a circuit using superposition theorem has one major disadvantage:
it may very likely involve more work. If the circuit has three
independent sources, we may have to analyze three simpler circuits each
providing the contribution due to the respective individual source. However,
superposition does help reduce a complex circuit to simpler circuits
through replacement of voltage sources by short circuits and of current
sources by open circuits.
What is the application of superposition theorem?
Keep in mind that superposition is based on linearity. For this
reason, it is not applicable to the effect on power due to each source,
because the power absorbed by a resistor depends on the square of the
voltage or current. If the power value is needed, the current through (or
voltage across) the element must be calculated first using superposition.
What is an example of superposition?
Example 1:
a. Using the superposition theorem, determine the current through
resistor $R_2$ for the network in [Fig. 2].
b. Demonstrate that the superposition theorem is not applicable to
power levels.
Fig. 2: For example 1.
Solution:
a. In order to determine the effect of the 36 V voltage source, the current
source must be replaced by an open-circuit equivalent as shown
in
[Fig. 3]. The result is a simple series circuit with a current equal to
$$\begin{split}
I'_2 &={E \over R_T} \\
&= {E \over R1 + R2}\\
&= {36 V \over 12 Ω + 6 Ω} = {36 V \over 18 Ω} = 2 A
\end{split}$$
Fig. 3: Replacing the 9 A current source in [Fig. 2] by an
open circuit.
Examining the effect of the 9 A current source requires replacing
the 36 V voltage source by a short-circuit equivalent as shown in
[Fig. 4]. The result is a parallel combination of resistors $R_1$ and $R_2$.
Applying the current divider rule results in
$$\begin{split}
I''_2 &= {R_1(I) \over R_1 + R_2} \\
&= {(12 Ω)(9 A) \over 12 Ω + 6 Ω} = 6 A
\end{split}$$
Fig. 4: Replacing the 36 V voltage source by a short-circuit.
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