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# Superposition

With the principle of superposition you can simplify the analysis of circuits with multiple inputs. Written by Willy McAllister.
Superposition is a super useful technique to add to your toolkit of circuit analysis methods. Use superposition when you have a circuit with multiple inputs or multiple power sources.

### What we're building to

The principle of superposition is another name for the additivity property of Linearity:
$f\left({x}_{1}+{x}_{2}\right)=f\left({x}_{1}\right)+f\left({x}_{2}\right)$
To solve a circuit using superposition, the first step is to turn off or suppress all but one input.
• To suppress a voltage source, replace it with a short circuit.
• To suppress a current source, replace it with an open circuit.
Then you analyze the resulting simpler circuits. Repeat for all inputs.
The final result is the sum of individual results.

## Describing a circuit as a function

The principle of superposition is defined using functional notation, so we talk a bit here about how circuits can be represented as functions.
Starting simple... How could we represent a lone resistor using the notation of a mathematical function? There is nothing remarkable going on here, I'm just talking about Ohm's Law using function terminology. We begin by identifying three things: the inputs, the thing performing the function, and the outputs.
I decided (arbitrarily) that voltage ${v}_{i}$ will be the input to our resistor function. We can assume input ${v}_{i}$ is generated by some voltage-making thing we're not showing. We assign the output to be the interesting thing we want to know. For this function, the output is the current $i$ in the resistor.
The input voltage is applied to the two small circles (the circles indicate the input port to our function). The function itself comes from the resistor, by means of Ohm's Law. The output of our function will be the current, $i$, measured by some not-shown current meter.
Written as a function, our resistor is
$i=f\left({v}_{i}\right)=\frac{1}{\text{R}}\phantom{\rule{0.167em}{0ex}}{v}_{i}$
With this notation, we are viewing the resistor as a function that takes in a voltage and outputs a current.

## A resistor is a linear function

Looking at our resistor function, we see it has the scaling property, the output, $i$ equals the input, $v$, scaled by a constant, $\text{R}$. That means the resistor is linear. The linearity property is what triggers our ability to use superposition to help solve a circuit.
(Refresh me on the meaning of linearity.)

## Using superposition to help solve a circuit

(This is a "toy" example to give you a feel for using superposition.)
Let's say the input to our function is two voltages in series:
The input to our function is two batteries in series: ${v}_{i}=\text{Vs1}+\text{Vs2}$.
The function is $f\left(v\right)=\frac{1}{\text{R}}\phantom{\rule{0.167em}{0ex}}v$.
The output of the function hasn't changed; it is still $i=f\left(v\right)$.
We now solve this circuit two ways: first by conventional analysis, and then using the principle of superposition.

### Conventional solution

To solve by conventional means we write the KVL equation around the loop:
$\text{Vs1}+\text{Vs2}-i\phantom{\rule{0.167em}{0ex}}\text{R}=0$
and solve for $i$:
$i=f\left(\text{Vs1}+\text{Vs2}\right)=\frac{\text{Vs1}+\text{Vs2}}{\text{R}}$
(conventional solution)

### Solution using the principle of superposition

The principle of superposition applies to a linear function, $f$.
$f\left({x}_{1}+{x}_{2}\right)=f\left({x}_{1}\right)+f\left({x}_{2}\right)$
It says: If you have two inputs superimposed, $\left({x}_{1}+{x}_{2}\right)$, you can apply the inputs one at a time, $\left({x}_{1}\right)$ followed by $\left({x}_{2}\right)$, and then add the individual results to get the full answer.
Now let's use the principle of superposition to solve the circuit. Since we've modeled our circuit as a function, we can say:
$i=f\left(\text{Vs1}+\text{Vs2}\right)$
is the same as
$i=f\left(\text{Vs1}\right)+f\left(\text{Vs2}\right)$
This suggests an intriguing possibility. It says we can compute the output current the conventional way, by applying the combined inputs $f\left(\text{Vs1}+\text{Vs2}\right)$, or, we could get the same answer by computing the function with single inputs $f\left(\text{Vs1}\right)$ and $f\left(\text{Vs2}\right)$, and adding the results together. Let's try this and see what happens.

### Suppressing inputs

To apply superposition we need to apply the inputs one at a time. That means we have to turn off all inputs except for one. When we turn off an input we say it is suppressed.
What does it mean to turn off a voltage source? It means we set $\text{V}=0$. This is the same thing as replacing the voltage source or battery by a short circuit.
What does it mean to turn off a current source? It means we set $\text{I}=0$. That's the same as replacing the current source with an open circuit.

### Using superposition

In the next two schematics, one of the voltage inputs has been turned off (suppressed ) by replacing it with a short circuit.
When we zero out or suppress an input, we replace one of the inputs with $0$, allowing the one remaining input to shine through by itself.
$f\left(\text{Vs1}+0\right)\to f\left(\text{Vs1}\right)$ and $f\left(\text{0 + Vs2}\right)\to f\left(\text{Vs2}\right)$
Now we solve each circuit individually,
${i}_{1}=\frac{\text{Vs1}}{\text{R}}\phantom{\rule{2em}{0ex}}{i}_{2}=\frac{\text{Vs2}}{\text{R}}$
where ${i}_{1}$ is the current caused by source $\text{Vs}1$, and ${i}_{2}$ is the current caused by source $\text{Vs}2$.
The total current comes from superimposing (adding) the currents from each circuit.
$i={i}_{1}+{i}_{2}$
$i=\frac{\text{Vs1}}{\text{R}}+\frac{\text{Vs2}}{\text{R}}$
$i=\frac{\text{Vs1}+\text{Vs2}}{\text{R}}$
(superposition solution)
Check it out! The superposition solution is the same as the conventional solution obtained above.
What we did here is called the linear superposition of two circuits.
Our example function was so simple, using superposition really didn't save much (if any) effort. In the following examples the circuits are more complicated, and the difference in effort becomes more apparent.

### Example 1

Consider the following linear circuit with two sources: one current source and one voltage source. The two sources are the inputs to the function. For this problem we happen to want to find two outputs, currents ${i}_{1}$ and ${i}_{2}$.
${i}_{1}={f}_{1}\left(\text{Is},\text{Vs)}\phantom{\rule{0.167em}{0ex}}$ and $\phantom{\rule{0.167em}{0ex}}{i}_{2}={f}_{2}\left(\text{Is},\text{Vs)}$
Let’s analyze this circuit using superposition.
First, we suppress the current source and analyze the circuit with just the voltage source acting alone. To suppress the current source, we replace it with an open circuit.
With just the voltage source, the two output currents are:
${i}_{1V}=0\phantom{\rule{2em}{0ex}}{i}_{2V}=\frac{\text{Vs}}{\text{R}2}$
Where ${i}_{1V}$ and ${i}_{2V}$ are the currents in $\text{R}1$ and $\text{R}2$ caused by the voltage source.
Next, we restore the current source and suppress the voltage source, to calculate the contribution of the current source acting alone.
With just the current source, the two output currents are:
${i}_{1I}=\text{Is}\phantom{\rule{2em}{0ex}}{i}_{2I}=0$
Where ${i}_{1I}$ and ${i}_{2I}$ are the currents in $\text{R}1$ and $\text{R}2$ caused by the current source.
We complete the analysis by adding the contributions from each source:
${i}_{1}={i}_{1V}+{i}_{1I}=0+\text{Is}=\text{Is}$
${i}_{2}={i}_{2V}+{i}_{2I}=\frac{\text{Vs}}{\text{R}2}+0=\frac{\text{Vs}}{\text{R}2}$
The full solution looks like this:
This could have been a tricky analysis because the two sources make it more difficult to write node or loop equations. We exploited superposition, which gave us two simpler circuits to deal with.

### Example 2

#### Conventional solution

For the following linear circuit let’s calculate the output voltage $v$.
We will do it the conventional way first. We write Kirchhoff's Current Law at output node $v$:
$\begin{array}{cccc}+{i}_{\text{R}1}& -{i}_{\text{R}2}& +\text{Is}& =0\\ \\ +\frac{\text{Vs}-v}{\text{R1}}& -\frac{v}{\text{R2}}& +\text{Is}& =0\end{array}$
We can rearrange this to get an expression for $v$ and gather like terms together on the right side:
$v=\frac{\text{R2}}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Vs}+\frac{\text{R}1\phantom{\rule{0.167em}{0ex}}\text{R}2}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Is}$
(conventional solution)

#### Solution using superposition

Now we will solve the same problem using the principle of superposition. As before, we suppress the input sources and solve new simpler circuits.
How would you suppress the current source?
Replace the current source with a ___.

The circuit collapses down to two resistors in series (a voltage divider).
Voltage ${v}_{Vs}$ is the contribution from voltage source $\text{Vs}$.
With just the voltage source, the output voltage is:
${v}_{Vs}=\text{Vs}\phantom{\rule{0.167em}{0ex}}\frac{\text{R}2}{\text{R}1+\text{R}2}$
Now we restore the current source and suppress the voltage source.
How would you suppress the voltage source?
Replace it with a ___.

The circuit collapses down to two resistors in parallel.
Voltage ${v}_{Is}$ is the contribution to the output from current source $\text{Is}$.
${v}_{Is}=\text{Is}\phantom{\rule{0.167em}{0ex}}\frac{\text{R}1\cdot \text{R}2}{\text{R}1+\text{R}2}$
We complete the superposition analysis by adding the two voltage contributions. As predicted, we get the same result as the conventional solution shown above.
$v={v}_{Vs}+{v}_{Is}$
$v=\frac{\text{R2}}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Vs}+\frac{\text{R}1\cdot \text{R}2}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Is}$
(superposition solution)
There is no approximation involved. The solutions are exactly the same. The key thing to notice is that the two simpler circuits took significantly less work to analyze.

### Linearity and superposition are useful tools

If you have a circuit made from linear elements, you get to use the principle of superposition. This means the original complicated circuit is really simpler circuits that happen to be sitting on top of each other. It seems like magic, but this property means that overlapping inputs and superimposed circuits don't affect each other or intertwine at all. Every simple circuit is blissfully unaware of the others until you do the final addition.
This is a marvelous property of linear circuits, and it is one of the reasons we love linearity so much. Circuits that are not linear (non-linear circuits) don't have this property, and superposition cannot be applied. (But don't worry, we love non-linear circuits, too, just in a different way.)

## Summary

If a circuit is made of linear elements, we can use superposition to simplify the analysis. This is especially useful for circuits with multiple input sources.
To analyze a linear circuit with multiple inputs, you suppress all but one input or source and analyze the resulting simpler circuit. Repeat for all inputs and sources. Then add the results to find the total response for the full circuit.

### Suppressing sources

To suppress a voltage source, replace it with a short circuit:
To suppress a current source replace it with an open circuit: