# Superposition

Superposition Theorem states that the total effect of all sources on a circuit containing linear elements is equal to the algebraic sum of each sources contribution.

Put simply, we can say that the total effect (voltage or current) is equal to the effect of each source in isolation (all other sources zeroed) summed together.

As a simple example, consider the circuit above. We can determine the voltage across resistor R3 by considering a circuit containing only V1 (with V2 as a short circuit) and determining the voltage across R3 and then do the same with only V2 (V1 short circuited).

The process to determine all voltages and currents in a circuit containing N sources using Superposition is simply to redraw the circuit for each source, calculate all of the desired parameters and sum the results together.

### Examples:

1. Determine the voltage across R3 in the circuit shown above using superposition

We can begin by drawing the circuit with V1 as the only source and V2 zeroed which is achieved by a short circuit: From here we can determine the voltage across R3 using the voltage divider rule considering R2||R3 in series with R1: We can then draw the circuit with V2 as the only source and V1 zeroed: Similar to the above we can determine the voltage across R3 using the voltage divider rule considering R3 in series with R2||R3: Using superposition, the voltage across R3 is the sum of the voltage drop from V1 and V2 summed together, i.e: 3.75V

2. Determine the current flowing through resistor R3 in the circuit below using superposition

Similar to the above we can start by considering a circuit with just I1 and zeroeing I2 by replacing it with an open circuit Note that the above circuit has been simplified as with I2 becoming an open circuit no current will now flow through R4 or R5 so we can effectively remove them from the circuit! The current flowing through R3 is then simply 5mA as the above circuit is a series circuit. We can then draw the circuit with I2 as the only source and I2 zeroed: The current flowing through R3 can also be determined by inspection here, the total current flowing through the source will divide through R4 and R5 and then recombine to flow through R3 i.e. 2mA Using superposition, the current flowing through R3 is the sum of the currents i.e: 7mA

## Questions:

1. Determine the voltage across R2 in the circuit below:

363.363mV

2. Determine the current flowing through R4 in the circuit below:

1.25mA

## Lab:

Alright let's try our hands at simulating some of these superposition type circuits which contain multiple sources to get a better feel for how these circuits work which is also a very handy way to quickly verify that our analysis is correct!

Let's begin by firing up LTSpice:

Now, let's draw our first example circuit in LTSpice:

We can begin by placing down the 2 voltage sources, V1 and V2 as below:

Let's then add the three resistors in and ground as we have above to complete the circuit:

Excellent, we now have the circuit at hand. Now, what we can do is go to run->simulate and use a simulation time of 1us as the window below:

When we hit run it will then simulate this circuit from time 0 to 1us. (Using a smaller time means the simulation will run quicker).

Now we can very easily verify the combined current flowing through R3 by hovering over the resistor R3 and left clicking:

Now we can either determine the value graphically via the above or if we hover back over the resistor you will see in the bottom left a value for I(R3).

We can then perform the same measurement to see the current flowing through R1 and R2:

If we simply left click each in turn we can display them both on the same plot which is really handy for comparison. We can then display all 3 and add I(R1) to I(R2) to confirm it results in I(R3):

Doing a quick addition of the two yields I(R3), great.

We can also determine the voltage across R3 which would be equal to the nodal voltage by simply clicking on the node in the middle of R1, R2 and R3 to produce the following voltage vs time graph:

We can apply this same strategy to any superposition circuit to determine all of the currents and voltages to verify our analysis.

As an exercise for readers I would suggest performing the above for the questions and check you arrive at the answers given :)