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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“Many of life's failures are people who did not realize how close they were to success when they gave up.”

-Thomas A. Edison

Week 5: Nodal Analysis

Nodal Analysis

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents.

In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node, requiring that the branch currents incident at a node must sum to zero. The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an admittance representation. For instance, for a resistor, Ibranch = Vbranch * G, where G (=1/R) is the admittance (conductance) of the resistor.

Nodal Analysis without Voltage Sources

Steps to determine nodal voltages:

To apply the node voltage method to a circuit with n nodes (with m voltage sources), perform the following steps:

Select a reference node (usually ground) and name the remaining n-1 nodes.  Also label currents through each current source. 

-Assign a name to the current through each voltage source.  We will use the convention that the current flows from the positive node to the negative node of the source.

-Apply Kirchoff's current law (KCL) to each node.  We will take currents out of the node to be positive.

-Write an equation for the voltage each voltage source.

-Solve the system of n-1 unknowns.

Example:

Sample Problem

Nodal Analysis with Voltage Sources

In order to solve problems involving Nodal Analysis with Voltage Sources, We need to consider the following cases:

Case 1: If the voltage source (dependent or independent) is connected between two non-reference nodes, the two non-reference nodes form a generalized node or supernode, we apply both KCL and KVL to determine the node voltages.

Case 2: if a voltage source is connected between the reference node and a non-reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source in figure 2 for example,

As you can see in Case 1, It said something about a "Supernode". You might be wondering "What is a supernode?", A supernode is:

-formed by enclosing a (dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it.

Reflection:

In our Cpe Class this week, We reached a new chapter and as a first topic of this chapter, We discussed Nodal Analysis. In my own words, Nodal Analysis is basically a method of determining the voltages between nodes or more commonly known as "Nodal Voltages".  Nodal Analysis most of the time uses Kirchoff's Current Law (KCL) and as long as you follow the steps in solving it properly, then you will be able to solve the problem. For me, I think that Nodal Analysis is one of the most difficult topics I have encountered in this class because it requires extensive knowledge of our previous topics but I believe that if you study hard, Then nothing is hard to learn and you will be able to understand the lesson properly.

Video:

Nodal Analysis:

Nodal Analysis practice problems:

Supernode Problem:

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“There must be a positive and negative in everything in the universe in order to complete a circuit or circle, without which there would be no activity, no motion”

-John McDonald

Week 4: Series-Parallel Circuit

 Series-Parallel Circuit

Series Circuits

What are Series Circuits?

Series Circuit- A series circuit has more than one resistor (anything that uses electricity to do work) and gets its name from only having one path for the charges to move along. Charges must move in "series" first going to one resistor then the next. If one of the items in the circuit is broken then no charge will move through the circuit because there is only one path. There is no alternative route. Old style electric holiday lights were often wired in series. If one bulb burned out, the whole string of lights went off.

What are Parallel Circuits?

Parallel Circuit- A parallel circuit is one that has two or more paths for the electricity to flow, the loads are parallel to each other. If the loads in this circuit were light bulbs and one blew out, there is still current flowing to the others because they are still in a direct path from the negative to positive terminals of the battery.

Difference between the two?

The parallel circuit has very different characteristics than a series circuit. For one, the total resistance of a Parallel Circuit is NOT equal to the sum of the resistors (like in a series circuit). The total resistance in a parallel circuit is always less than any of the branch resistances. Adding more parallel resistances to the paths causes the total resistance in the circuit to decrease. As you add more and more branches to the circuit the total current will increase because Ohm's Law states that the lower the resistance, the higher the current.

Voltage and Current Division

Voltage and Current division allow us to simplify the task of analyzing a circuit.

Voltage Division allows us to calculate what fraction of the total voltage across a series string of resistors is dropped across any one resistor.

Hence:

Current Division allows us to calculate what fraction of the total current into a parallel string of resistors flows through any one of the resistors.

Hence:

Reflection:

In the 4th week of our CpE 311 subject, I learned the two types of Electrical Circuits. These two are Parallel Circuits and Series Circuits. In a Series Circuit, All components are connected end-to-end and there is only one path for the current to flow. In a Parallel Circuit, All components are connected across each other forming exactly two sets of common points. Also in a Series Circuit, The current is equal for there is only one path while in a Parallel Circuit, The voltages are equal because the components are connected across each other. In our laboratory class, I have also learned to measure the voltages and current of a circuit by connecting the circuit in a Power Supply and using a Digital Multimeter to measure both values.

Video:

Series and Parallel Circuits

Voltage and Current Division

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“Electricity is actually made up of extremely tiny particles called electrons that you cannot see with the naked eye unless you have been drinking.”

-Dave Barry

Week 3: Nodes, Branches, Loops, & Kirchoff's Laws

Nodes, Branches, & Loops

What are Nodes, Branches, & Loops?

Branch- A branch represents a single element such as voltage source or a current source or a resistor.

Node- A node is the point of connection between two or more branches.

Loop- A loop is any closed path in a circuit.

Kirchoff's Current Law (KCL)

This fundamental law results from the conservation of charge. It applies to a junction or node in a circuit -- a point in the circuit where charge has several possible paths to travel.

In Figure 1, we see that IA is the only current flowing into the node. However, there are three paths for current to leave the node, and these current are represented by IB, IC, and ID.

Once charge has entered into the node, it has no place to go except to leave (this is known as conservation of charge). The total charge flowing into a node must be the same as the the total charge flowing out of the node. So:

IB + IC + ID = IA

Bringing everything to the left side of the above equation, we get

(IB + IC + ID) - IA = 0

Kirchoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (or Kirchhoff's Loop Rule) is a result of the electrostatic field being conservative. It states that the total voltage around a closed loop must be zero. If this were not the case, then when we travel around a closed loop, the voltages would be indefinite. So:

In Figure 1 the total voltage around loop 1 should sum to zero, as does the total voltage in loop 2. Furthermore, the loop which consists of the outer part of the circuit (the path ABCD) should also sum to zero.

Resistor Color Code

We are able to use solve resistance by using the Resistor Color Code

Reflection:

In the 3rd week of our CpE 311 subject, I learned to identify the Nodes, Loops, & Branches of an Electric Circuit. I have also learned that according to Kirchoff's Current Law, A current that enters a circuit is equivalent to a current that leaves a circuit and lastly, I have also learned how to measure the resistance of a resistor using the Resistor Color Code that was discussed in our Laboratory Class.

Videos:

Nodes, Branches, & Loops:

 

Kirchoff's Current Law (KCL):

Kirchoff's Voltage Law (KVL):

I apologize for the delay of my submission. This is because of the non-stop electrical brownouts that keeps on happening in our area -_-" . Also, starting from this post, I will also be adding quotes in my blog. I believe these quotes will help me understand and be truly inspired at the beauty of our CpE 311 subject. Hope you enjoy them as much as I do :D Thank you for visiting my Blog once again and God bless to All XD.

A lot of words in English confuse the idea of life and electricity, like the word "Livewire".

-Laurie Anderson

Week 2: Ohm's Law

"Ohm's Law"

-Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms.

Ohm's Law triangle:

In our 2nd week, We also discussed the two types of elements that are in an Electrical Circuit. These are Passive Elements and Active Elements.

Active Elements- These elements generate energy. Examples are Batteries, generators, and operational amplifiers.

Passive Elements- These elements drop energy. They cannot generate energy because they take energy from a circuit. Examples are Resistors, Capacitors, and Inductors.

Also in our 2nd week, We also discussed the following: Open Circuits and Short Circuits.

Open Circuit- is a condition in an electric circuit in which there is no path for current between two points; examples are a broken wire and a switch in the open, or off, position.

Short Circuit-  is an electrical circuit that allows a current to travel along an unintended path, often where essentially no (or a very low) electrical impedance is encountered.

In our 2nd week, We also discussed Conductance. Conductance is

- An expression of the ease with which electric current flows through a substance. In equations, conductance is symbolized by the uppercase letter G. The standard unit of conductance is the siemens (abbreviated S), formerly known as the mho.

When a current of one ampere (1 A) passes through a component across which a voltage of one volt (1 V) exists, then the conductance of that component is 1 S. The siemens is, in fact, equivalent to one ampere per volt. If G is the conductance of a component (in siemens), I is the current through the component (in amperes), and E is the voltage across the component (in volts), then:

Reflection:

In our 2nd week of CpE 311, I learned that there is another way to identify if a component in a circuit is a passive element or an active one. In our previous lesson, Sir Jay taught us that if the current enters a negative terminal and then exits a positive terminal, then the component is an Active Element. Otherwise, if the current enters a positive terminal and exits a negative terminal, then the component is a Passive Element. Also in our previous lesson, He explained to us Ohm's  Law with the following illustration:

In this illustration, A man is pushing a table. With relation to newton's 2nd law of motion, The force that the man exerts to make the table move is the "Voltage", the movement of the table is the "Current", And the friction that resists the movement of the table is the "Resistance".

Ohm's Law

Active & Passive Elements:

Once again, Thank you for visiting my Blog. The third post will be arriving next week :D.

About the author:

Roswaldo S. Flores BSCpE-3

CpE 311

Instructor:

Engr. Jay S. Villan, MEP