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INTRODUCTION:

This blog contains what I have learned in ECE 311(Electric Circuits 1)  subject instructed to us by Engr. Jay S. Villan, MEP-EE. Within the span of one semester, we finished six general topics including basic concepts, basic laws, and method of analysis and circuit theorems, capacitor and inductor, and first order circuit. These topics enhance once ability or skills in a design process and are accompanied by understanding of the basic theories. These six general topics contain subtopics which will widen students’ knowledge about theoretical and actual applications. From the very beginning of these topics up to the end, we are learning about circuits. The learning’s I put in this blog is not just defining what the topic is but also on how we can apply it in real life situations. This also contains some figures for presentation and easy analysis, sample calculations, important terms and experience.

BASIC CONCEPTS:

                                    -Charge and Current

                                    -Voltage

                                    -Power and Energy

An electric circuit is consists of electrical elements connected together such as resistors, capacitors, voltage and current source and etc. It is used in many electrical applications to finish different tasks and meet the expected output. Electric charge is considered to be the most basic quantity of electric circuit. It is an electrical quantity of the atomic particles of which matter consists. It is measured in coulombs (C). Charge can neither be created nor destroyed. It can only be transferred, or simply, it moves from one place to another.

          Imagine a conducting wire connected in a battery. This wire is consists of several atoms. The positive (+) charges will move in one direction and the negative charges (-) will move in the opposite direction. Through this movement of charges, it will generate an electric current.

Current is measured in ampere (A). It may be a direct current (DC). Most common example of this is battery. The current remains constant as the time increases. It may also be an alternating current (AC). This current is commonly used in our household. It enables our appliances to run such as refrigerator, television, computer, washing machine and other electrical appliances.

     

                   Alternating Current                                  Direct Current           

As I have mentioned in the previous paragraph, in order to move the charges in a conducting wire, it must be connected in a battery to do work and transfer energy. This performed by an external electromotive force (emf) also known as voltage. It is measured in volts (V). Like current, it may be a DC or AC voltage.

Appliances in our house have their own value of power. This power really matters on how much we pay in our electric bills that’s why it is important to know how much power an appliance or electric device can handle. Since we are paying for the electric energy over a period of time, we must also consider how long we used our appliances in our house. We can calculate power by using one of the formulas :(p=iv) the product of current and voltage.

Example we are using a 100 watt bulb and a 50 watt bulb and we all know that 100 watt bulb is lighter than a 50 watt bulb. If we both use this for four hours a day, and the rate of energy cost is 5cents/kwh. It is evident that we pay larger in 100 watt bulb is lighter than a 50 watt bulb after one month.

Sample Calculations:

100 watt bulb:   100W(1KW/1000W) X (4hrs X 30days) X 5cents/kwh= 60 cents

50 watt bulb:  50W(1KW/1000W) X (4hrs X 30days) X 5cents/kwh= 30 cents

Important Terms:

Charge  – basic quantity to electric circuit

Current – charge flow rate

Voltage – charge rate of doing work

Power   – time rate of doing work

Energy – capacity to do work

            In this Chapter, I've learned the basic concepts of electric circuits which are the first step to better understand next topics. Some of my questions were finally answered such as “How are light bulbs were able to lights up”? , “How electric bills were computed”?, and etc. Through the knowledge I've gained in this topic, I was able to apply it in our household such as limiting the number of hours of using a specific appliance which acquires large value of power in order to lessen our payment in electricity.

BASIC LAWS:

                        -Ohm’s Law

                        -Nodes, Branches, and Loops

                        -Kirchhoff’s Law

                        -Series Resistors and Current Division

                        -Parallel Resistors and Voltage Division

                        -Wye Delta Transformation

                                   

            Ohm’s Law states that the voltage (V) across the resistor is directly proportional to the current (i) flowing through the resistor and given by this formula: (V=iR). It means that as the voltage increases, the current also increases. In this law, it is important to be careful to the sign convention of the voltage polarity and the current direction. The values of resistance are range from zero to infinity. If R=0, it is a short circuit wherein there is current but zero voltage. If R=∞, it is an open circuit and there is voltage but zero current. That’s why for safety purposes, never touch a live wire due to the presence of current on it to avoid electric shock.

                                                      Voltage-Current Relationship

          Resistors can either be fixed or variable. Fixed resistors have constant resistance while variable resistors have adjustable resistance. It can be minimum or maximum. We commonly used color coded system in fixed resistors but the percent tolerance will vary to its minimum and maximum value of resistance. The reciprocal of resistance is what we called conductance which is a measure of how well an element conducts electric current and is measured in siemens.

           Branch represents the elements connected in a nod such as resistors, voltage and current sources, and etc. Node is the meeting point of branches and loop is a closed path in the circuit. If two or more elements share in a single node, it is connected in series. If two or more elements are connected to the same two nodes, it is a parallel connection.

                                                Branch, Node, and Loop

            Ohm’s Law alone is not too sufficient to analyze circuits but if it is accompanied by Kirchhoff’s two laws, we can sufficiently and clearly analyze the given circuit. Kirchhoff’s first law is the law of the conservation of charge. Kirchhoff’s Current Law states that the current entering a node is equal to the current leaving. The second law is based to the principle of conservation of energy. Kirchhoff’s Voltage Law states that the algebraic sum of all voltages around a closed path is zero. It is said to be the “algebraic sum”, since it may be positive (+) or negative (-).

                                                     KVL                                         KCL

            Be in mind that it would never have different current in a series connection. In KCL, current entering is positive while current leaving is negative. In KVL, two ways can be applied, taking counterclockwise or clockwise direction of loop considering the polarity of the voltage.

            In a series connection, current is the same but different in voltage. Several resistors connected in series in a circuit may be simplified into a simpler circuit by taking its equivalent resistance. We can calculate it through adding all the resistances of individual resistors since resistors in series behaves as a single resistor. The voltage source is divided among the resistors. The greater the resistance, the greater the voltage drop. This is the principle of Voltage Division.

            In a parallel connection, the voltage across the resistor is the same but the current is different. Like in voltage division, we can also get the equivalent resistance to construct a simpler circuit. This can be calculated through using the product over sum method. The current source is divided among the resistors. The greater the resistance, the smaller the current. This is the principle of Current Division. In actual applications, we measure voltage across a resistor by placing voltmeter in parallel and in series if we measure current.


                                      

            Assume there were light bulbs connected in series, then if a single light bulb were become open circuit, the entire bulbs will stop working. The disadvantage of this series connected light bulbs is that, when a single light bulb wire burn out and not replaced, the life expectancy of the remaining light bulbs will be affected. If it is connected in parallel, whatever happen in a single light bulb, the remaining light bulb will not be affected that’s why it is convenient to use a parallel connection in our household.

We really apply wide analysis of circuit especially if we don’t know the connection of the resistors; it is neither parallel nor series connection such as the bridge circuit. This circuit can be simplified using three-terminal equivalent networks. These networks are wye (Y), tee (T) or the delta or pi (π) network. Make sure that in transforming the circuit, don’t take anything out or put anything in new for this may lead to errors. The main objective of this transformation is to able to determine if the resistors are connected in series or parallel, to construct a simpler circuit and t finally solve for the equivalent resistance.

                                                     Delta                                      Wye

         
           This Chapter helped me analyze a given circuit. Since this chapter is characterized by different Laws, I was able to know how to apply it and also what kind of law I am going to use to solve for a specific circuit problem. Most importantly, I was able to determine the voltage across and current through a resistor whether series or parallel connected. Also, how voltage and current sources were divided among resistors by using those laws specifically Ohm’s Law and Kirchhoff’s two Laws.

                       

METHOD OF ANALYSIS:

-Nodal Analysis

                        -Mesh Analysis

           

            In nodal analysis, we apply the Kirchhoff’s Current Law or KCL to find unknown voltages in a circuit and the Ohm’s Law to express the branch currents in terms of node voltages. It flows from higher potential to lower potential. In this analysis, there were following steps to be considered.

STEP 1: Select a reference node.

                                    -Voltage in this node is always zero (0).

STEP 2: Assign voltages V1, V2 up to V(N-1).

                                    -This must be with respect to the reference node.

STEP 3: Apply KCL and Ohm’s Law.

                                    -It is to formulate equations.

STEP 4: Solve for the unknowns.

                                    -Use the formulated equations and apply

Cramer’s Rule or Substitution.

If the voltage source is connected between a reference and non-reference node, the voltage at the non-reference node is equal to the voltage source. If the voltage source is connected between two non-reference nodes, it is a supernode. In this case, we apply both KVL and KCL since it doesn’t have voltage of its own.

Sample Problem #1: 

Figure 1

Determine V and find I in the circuit shown above.

Mesh Analysis is more convenient to use to find unknown currents through the application of KVL. Mesh is just a single loop either clockwise or counterclockwise considering its sign conventions. Be in mind that if you use clockwise direction in your first mesh, use it for the rest of your meshes so that the equations you’re going to formulate won’t affect your solution.

Like in Nodal Analysis, there are also steps to determine mesh currents:

STEP 1: Assign mesh currents.

STEP 2: Apply KVL and use Ohm’s Law.

               Express the voltages in terms of currents to construct simultaneous solutions.

STEP 3: If there is supermesh, apply KCL.

STEP 4: Solve for the unknowns.

                Use Cramer’s Rule or Substitution.

When there is a current source present in one mesh, it is automatically the current of that mesh. If a current source exists between two meshes, it is what we called supermesh. Since they share for one current source, they don’t have current of its own.

Sample Problem #2:

Figure 2

                              Determine the mesh currents i1 and i2 in the circuit shown above:

             In this Chapter, it highlighted the Nodal and Mesh Analysis. This analysis can be applied for some complicated circuit problems. The different laws introduce from previous chapter were greatly used. I've learned that Nodal analysis is efficient when there is fewer node equations than mesh equations. Mesh analysis is more efficient when there is fewer mesh equations than node equations.

CIRCUIT THEOREMS:

                                    -Superposition

                                    -Source Transformation

                                    -Thevenin’s Theorem

                                    -Norton’s Theorem

                                    -Maximum Power Transfer

            
            Aside from nodal and mesh analysis in the previous chapter, we can determine voltage and current through different ways. One of those is superposition. This approach that if a given circuit has various independent source, the voltage across and the current through an element is equal to the algebraic sum of all individual voltage and current due to each independent source acting one at a time.

            

           Superposition can be applied by killing all independent sources except one source. Voltage source will be short circuit while current source is open circuit. Solve for output voltage or current by applying different laws and analysis discussed from the previous chapters. Finally, add up algebraically all the contributions of all independent sources in a given circuit. Using superposition, you can spend more work since for example, a given circuit has three or more independent source, it is really time consuming to solve for a specific variable of each independent sources.

                           

                                                           Superposition

             Like Wye-Delta Transformation, source transformation is another way of simplifying circuits. It is the process of transforming the voltage source in series with a resistor and current source in parallel with a resistor. Ohm’s Law is highly required in this process.

           

             Even though source transformation provides simple analysis, there were also some restrictions needed to be followed. The arrow of the current is always directed toward the positive terminal of the voltage source. Don’t transform dependent sources and lastly, it is not possible to be applied when resistance is zero (R=0).

 Source Transformation

Thevenin’s Theorem provides a technique in which the fixed part of the given circuit is replaced by an equivalent circuit consisting voltage source (Vth) in series with a resistor (Rth). The Vth is the open circuit in two terminals while Rth is the equivalent resistance with a dead source. The original circuit and the equivalent circuit must have the same value of voltage and current. To verify this one, connect a load (RL) in both circuits.

            

             Because we can replace a large circuit by a single independent source and a single resistor through this theorem, it really contributes a lot for a circuit design. But in transforming large circuits, be in mind that like superposition, we can’t kill the dependent sources. We apply voltage source (V) in a specific terminal and find the current (I) to calculate for the equivalent resistance. It is also possible to get a negative value of Rth. If this happened, it only means that a circuit supplies power but, if it is positive, it absorbs power.

     Thevenin’s Equivalent Circuit

              Another theorem used in circuit analysis is the Norton’s Theorem. This theorem is related to Thevenin’s Theorem by source transformation. We also replaced fixed circuit into an equivalent circuit. This circuit is consists by a short circuit current (Isc) or Norton’s current (In) connected in parallel with a resistor (Rn) which is the equivalent resistance when all independent sources were turned off. Rn is equal to Rth (Rn=Rth). Using this equality, we can solve for Norton’s current by dividing Rth from Vth (Vth/Rth).

     Norton’s Equivalent Circuit

           In various practical situations, we design a circuit to provide power to the load. There were also situations that we need to maximize the power deliver to the load and Thevenin’s Theorem will greatly contribute with this. This maximum power will only occur when load resistance is equal to the Thevenin’s resistance (RL=Rth). This is the theorem of maximum power.

           

          The power delivered will vary to the load resistance. It only means that if the load resistance changes, power also changes. We can also calculate maximum power using this formula:

                   

                                       

                                                               Maximum Power     

            This Chapter taught me how to analyze a specific circuit by just transforming or simplifying the circuit and replacing it into an equivalent circuit. In actual applications, I've learned that these ways of transformations and theorems can be used in trouble shooting. Like in our household, every time a variable load is changed, we have to analyze the circuit again and again. Through Thevenin’s Theorem, we can trouble shoot it to avoid this kind of problem.

CAPACITORS AND INDUCTORS:

                                                  Capacitor                              Symbol

                                               Inductor                                  Symbol

Sample Experience:

           

            During the construction of our power supply, I placed a light emitting diode (LED) in series with a resistor to test if our power supply functions. Then I suddenly observe after I plugged out the transformer, the LED doesn't turned off automatically. A question generated in my mind that, “What is the reason behind why it happened”? Then I found out during our circuit class about capacitor and inductor that it is because of the presence of the capacitor in our circuit due to the stored energy on it. The transformer in our power supply serves as an inductor since it is consists of coil of conducting wires.

In this Chapter, I've learned the great significance of capacitor and inductor in an electric circuit. It answered some of my questions of what really their purpose why they are placed in a specific circuit. Both of them are useful for they serves as temporary voltage and current sources because of their capacity to store energy. They are also frequency sensitive that’s why they are useful for frequency discrimination. Capacitors oppose any change in voltage while inductors oppose any change in current.

FIRST ORDER CIRCUIT:

                                    -Source Free RC Circuit

                                    -Source Free RL Circuit

             In analyzing RC and RL circuits, we apply Kirchhoff’s Laws like in resistive circuits. Through using these laws, we arrived in an algebraic equation but, if we use RC and RL laws, we arrived in a differential equation. One way of exciting a circuit is through source free circuit. When DC source is suddenly disconnected, the source free RC circuit occurs. The energy stored in the capacitor is released. Using this approach, always be in mind that the capacitor acts as an open circuit to steady state DC conditions. In analyzing source free RL circuit, the inductor acts as a short circuit to steady state DC conditions.

             

             In RC analysis, we first find the initial voltage across a capacitor while in RL circuits, we first solve for the initial current through the inductor. It is also important to calculate for the time constant  of the circuit in both RC and RL analysis. The smaller the time constant, the faster the rate of decay of response due to the quick dissipation of energy stored. The larger the time, the slower the rate of decay of response because it takes longer time to reach the steady state. But whatever the time constant is, the circuit will reach its steady state after five time constant.

                               

                   Source Free RC Circuit                                       Source Free RL Circuit 

        
           As we observed in our household appliances such as television, rice cooker and other electric device with power lights, when we turn on these appliances, the power light will automatically turned on. It is due to fast charging time. But when we plugged out these appliances, the power light will slowly turned off due to slow discharging time and slow rate of decay.