Week 9: Superposition

Superposition

What is Superposition?

- Superposition is another algorithmic circuit analysis method, with a prescribed procedure, like the node-voltage or loop-current methods.

It applies only in circuits with 2 or more independent sources.

It comes from the mathematical notion of superposition in which a final solution can be built as the sum of two or more partial solutions. It works well when the partial solutions can be obtained very easily. (Using voltage dividers or resistor reduction techniques.)

Superposition only applies to linear circuits, which are made up of linear circuit elements: resistors, voltage and current sources, capacitors, inductors, and linear amplifiers. It does not apply when the circuit has non-linear elements like diodes and transistors (electronics).

The Superposition Procedure:

Here are the steps: (Recall that the method only applies to circuits with two or more independent sources.)

1. Make a partial circuit by choosing one source to keep and turning of all of the other independent sources.

 -To “turn off” a voltage source, we turn it into a short circuit (zero volts and carries any amount of current.) To turn off a current source, we convert it to an open circuit (zero amps and any amount of voltage across it.)

2. Solve this “partial circuit” for the quantities of interest. Use whatever techniques that are applicable to the partial circuit: resistor reductions, voltage dividers, source transformations, or even node voltage.

3. Go back to the original circuit and create another partial circuit by choosing another source to keep (different than the first one) and turning off all of the other sources in the circuit.

4. Solve the second partial circuit for the quantities of interest.

5. Repeat the process until each independent source has been used in a partial circuit.

6. Sum up the all of the partial results to get the total results for the quantities of interest.

 -If there are n independent sources in a circuit, then when using superposition, you would expect to solve n partial circuits and find n partial results for whatever you are looking for.

Example:

4Ω and 2Ω are in series and also 3Ω and 1Ω :

Now 4Ω and 6Ω are parallel:

4Ω||6Ω=4×6/4+6=2.4Ω

So using Ohm's law:

Ix1=5V/2.4Ω=2.083A

For a moment forget Ix and concentrate on finding current of resistors. If we have the current of resistors, we can easily apply KCL and find Ix2 . So, 4Ω and 2Ω are parallel and also 3Ω and 1Ω are parallel:

4Ω||2Ω=4×2/4+2=43Ω

3Ω||1Ω=3×1/3+1=34Ω

Now, we can find their voltage drops:

V4Ω||2Ω=4/3×−3A=−4V

V3Ω||1Ω=3/4×−3A=−2.25V

Please note that the voltage drop on 4Ω||2Ω is the same as 4Ω and 2Ω voltage drops, because the circuits are equivalent and all are connected to the same nodes. The same statement is correct for 3Ω||1Ω voltage drop and 3Ω and 1Ω voltage drops. So

V4Ω=V2Ω=V4Ω||2Ω=−4V

V3Ω=V1Ω=V3Ω||1Ω=−2.25V

To find Ix2 all we need is to write KCL at one of the nodes:

−I2Ω+Ix2+I3Ω=0

→Ix2=I2Ω−I3Ω 

I2Ω and I3Ω can be found using Ohm's law:

I2Ω=V2Ω2Ω=−42=−2V

I3Ω=V3Ω3Ω=−2.253=−0.75V

Therefore,

Ix2=−1.25A

And

Ix=Ix1+Ix2=2.083−1.25=0.8333A

Ix=0.8333A

Reflection:

Superpositon overall is a topic with an easy-to grasp concept. To solve a circuit using the superposition theorem, We need to choose one source and then we need to turn off the other sources. The choosing of the sources is the most crucial part because choosing the wrong source to turn on might lead to a wrong output. After that, its only a matter of applying the basic circuit laws and equations to solve the problem.

Video:

Superposition

Superposition problem with dependent source

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Week 8: Linearity Property and Source Transformation

Linearity Property

What is Linearity Property?

-Linear property is the linear relationship between cause and effect of an element. This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additivity property are both the combination of linearity property.

The homogeneity property is that if the input is multiplied by a constant k then the output is also multiplied by the constant k. Input is called excitation and output is called response here. As an example if we consider ohm’s law. Here the law relates the input i to the output v.

Mathematically,                 v= iR

If we multiply the input current  i by a constant k then the output voltage also increases correspondingly by the constant k. The equation stands,      

                                     kiR = kv

The additivity property is that the response to a sum of inputs is the sum of the responses to each input applied separately.

Using voltage-current relationship of a resistor if

                                       v1 = i1R       

and  

                                       v2 = i2R

 Applying (i1 + i2)gives

 V = (i1 + i2)R = i1R+ i2R = v1 + v2

We can say that a resistor is a linear element. Because the voltage-current relationship satisfies both the additivity and the homogeneity properties.

What is linear circuit?

 -A circuit is linear if the output is linearly related with its input.

Example of a Linear Circuit:

See a circuit in figure 1. The box in the circuit is a linear circuit. We cannot see any independent source inside the linear circuit.

Source Transformation

What is Source Transformation?

-Independent current sources can be turned into independent voltage sources, and vice-versa, by methods called "Source Transformations." These transformations are useful for solving circuits. We will explain the two most important source transformations, Thevenin's Source, and Norton's Source, and we will explain how to use these conceptual tools for solving circuits.

For more info about Thevenin's Source, Click here.

For more info about Norton's Source, Click here.

Conversion of Voltage Source to Current Source

To convert a Voltage Source into a current one, We need to follow these steps:

1. Make short circuit between two terminals A and B as we done in figure. Find the short circuit current and let it be I.

2. Measure the resistance at the terminals with load removed and sources of e.m.f s replaced by their internal resistances if any. Let the resistance is R.

3. Then equivalent current source can be represented by a single current source of magnitude I in parallel with resistance R.

Example:

Conversion of Current Source to Voltage Source

To do this, We have to do the same inverse procedure.

Example:

Reflection:

This week, I learned that converting sources is actually very easy. To covert a voltage source into a current one, We simply change the position of the resistors from series to parallel. To convert a current source into a voltage one, We change the position of the resistors from parallel to series. But before we apply source transformation, We must analyze the problem and identify the position of the resistors first. If the source is a voltage one and if the resistor connected to the source is in a parallel connection, then source transformation is not applicable. If the source is a current one and if the resistor connected to the source is in a series connection, then source transformation is also not applicable.

Videos:

Linearity Property:

Source Transformation:

Circuit of Life:

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Week 7: Mesh Analysis

Mesh Analysis

What is Mesh Analysis?

-Mesh analysis or the Mesh Current Method, also known as the Loop Current Method, is quite similar to the Branch Current method in that it uses simultaneous equations, Kirchhoff's Voltage Law, and Ohm's Law to determine unknown currents in a network. It differs from the Branch Current method in that it does not use Kirchhoff's Current Law, and it is usually able to solve a circuit with less unknown variables and less simultaneous equations, which is especially nice if you're forced to solve without a calculator.

What is a "Mesh"?

-Basically, A mesh is a loop that does not have any loops in it.

In the figure above, the following meshes are: B1-R1-R2-B1, and B2-R3-R2-B2 but B1-R1-R3-B2-B1 is not a mesh because it contains other loops in it.

Mesh Analysis without Current Sources

Steps:

1. Identify loops within the circuit encompassing all components.

2. Assign mesh currents to the number of meshes found in the circuit.

3. Apply current directions to each of the mesh currents in the circuit. The choice of each current's direction is entirely arbitrary. If the assumed direction is wrong, the answer for that current will be a negative answer.

4. Apply Kirchoff's Voltage Law (KVL) to each of the meshes.

5. Use Ohm's Law to express the voltages in terms of mesh currents.

6. Solve the resulting n simultaneous equations to get the mesh currents.

Mesh Analysis with Current Sources

Two cases to consider:

Case 1: A current source exists only in one mesh.

Case 2: A current source exists between two meshes.

What is a Supermesh?

 - A supermesh occurs when a current source is contained between two essential meshes. The circuit is first treated as if the current source is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the two mesh currents with the current source. This will be an equation where the current source is equal to one of the mesh currents minus the other. The following is an example of a supermesh.

Reflection:

This week, I learned that meshes in a circuit are loops that does not have any loops in them and that they require Mesh Analysis to be solved. Mesh Analysis or Mesh Current Method are almost similar in Branch Current method because the way to solve them are both exactly the same except that mesh analysis requires KVL while nodal analysis requires KCL. Both methods of circuit analysis are very useful in solving all types of circuit problems. Overall, I learned that mesh analysis is much easier than the Branch current method because unlike in the Branch current method, We don't need any reference components or parts like the reference nodes in the Branch current method.

Videos:

Mesh Analysis Introduction:

Mesh Analysis with Independent current sources:

Mesh and Supermesh:

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“The best teachers are those that can influence even the poorest of all learners.”

― Kim Panti

Week 6: Wye-Delta Transformations

Wye-Delta Transformations

What is  Y-Δ transformation? 

- The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899. It is widely used in analysis of three-phase electric power circuits.

- The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.

Illustration of the transformation in its T-Π representation

Formulas:

Equations for the transformation from Δ-load to Y-load:

Equations for the transformation from Y-load to Δ-load:

Δ and Y circuits with the labels which are used in this article.

Reflection:

This week, I learned that the transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances. I also learned that this topic is one of the easiest topic we have ever tackled in our Circuit Class because it only uses a specific formula unlike other topics.

Video:

Wye-Delta Transformation:

Delta-Wye Transformation:

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