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1. Ch 00R1 Intro Basic Electricity: congratulations on your beginning of your journey to understanding electricity. My name is Graham Van Brunt, and I am very excited about putting on these courses because it is coming material that I had to understand in my career. And a lot of it were kind of in the dark areas that took a while for me to research that you will have in front of you before you have to do any of the research. I've spent quite a bit of time developing these courses. You will find each slide very crisp and clean, and you'll enjoy your journey through understanding electricity. I have spent over 35 years in the industry. I started out with a company called Ontario Hydro, which was the largest supplier and wholesale of electricity in Ontario, which is one of the largest provinces in Canada. This course is entitled Basic Electrical Theory, and we start out by looking at exclusively D C circuits, which are direct current circuits. This course is the fundamental building block of the other courses that will follow. The lessons learned in this course are very simple, but you will run across them again and again and again as your journey progresses through the understanding of electrical theory. We start out the course with a basic understanding of what the nature of electricity is actually starting note with static electricity and then moving on to understanding how static electricity will move into current flow. We will look at very simple electrical circuits, and we will understand what electron flow is and what current flow is and the difference between the two of them. We will then move on to understanding the very fundamental equation that will be associated with electrical theory, and that is homes law. This is the very basic equation that is used over and over and over again in basic electrical theory. Once we know one, understand what homes law is in a relationship between current ah, resistance and voltage, we will then look onto what is electrical power and how do we calculate electrical power? Once we understand these equations, we then we'll look at them in a little bit more in detail by studying the various components of the equation, starting with Resist er's, we will then places resistors in various parallel and series circuits and study what is meant by a Siri's and a parallel circuits. And then we'll bring in some tools for analysing more complex circuits using things like Kirchoff, Voltage Law and Kurt shops current law. But we will thoroughly understand what these laws mean and what they can do to ease our analysis off complex equations moving on to more complex circuits. We will develop away and do some practicing on reducing those complex circuits to very simple circuits that we can analyze. We now have the capability, or we will have the capability to move on to circuit serums. And in doing so, we will develop what is known as the D C power sources, and we'll look at voltage sources and current sources and what we mean by them. We will look at complex serums that will help us analyze more complex circuits, such as saving in serum. Next, we move on to electrical fields and capacitance, and the characteristics associate ID with electrical capacitors. From there we will move on to in doctors. But before jumping right into induct er's, we have to have, ah, an understanding of the basics of magnetism. So we start out looking at some of the basics of electrical magnetism. The various terms of magnetism are going to be defined, such as magnetic flux and flux density. This magnetic flux density will then be related to current flowing in a circuit that will be associated with that magnetic flux. And we will also look at how the magnetic flux is influenced by the various magnetic materials that coils maybe wrapped around. This will provide a lead in to the study of electromagnetic induction and Faraday's equations associated with electromagnetic induction. With this background in magnetism, the student now could move on to the study of the third and last element or passive element in this chapter of the course, which is inducted its and, just like capacitors will study the various characteristics off in doctors and how they're affected by current and voltage in that circuit. And lastly, we will look at some of the practical considerations of induct er's in an electric circuit , and there's various things that we have to take into consideration in the real world, and these will be looked at at the culmination of this chapter That brings us to the end of the chapter on basic electrical theory, D. C. Circuits
2. Ch 01R1 The Nature of Electricity: Chapter one. The nature of electricity. Electricity is the most common form of energy. Electricity is used for various applications such as lighting, transportation, cooking communications, production of various goods in factories and much, much more. None of us know exactly what electricity is. The concept of electricity and its theories have been developed by observing its different behaviors. In order to understand the nature of electricity, we're going to start with a bit of history and static electricity. It was discovered centuries ago that certain types of material would mysteriously attract one another after being rubbed together. For example, after rubbing a piece of silk against a piece of glass, the silk and glass would tend to stick together. There was an attractive force that could be demonstrated even when the two materials were separated. Another example demonstrated this phenomenon after rubbing a piece of wool against a piece of para finl axe. The wool and wax would tend to stick together, and there was an attraction or an attractive force between the two of them. This phenomenon became even more interesting when it was discovered that identical materials, after having being rubbed together with the respective cloth, always repelled each other. It was also noted that when a piece of glass rubbed with silk was exposed to a piece of wax rubbed with wool, the two materials would attract one another. Furthermore, it was found that any material demonstrating properties of attraction or repulsion after being rubbed could be classed into one of two distinct categories. Attracted to the glass and repelled by the wax or repelled by the glass and attracted by the wax. It was either one or the other. There was no materials found that would be attracted to or repel both glass and wax, or that reached to one without reacting to the other. More attention was directed towards the piece of cloth that was used in the rubbing. It was discovered that after rubbing two pieces of glass with two pieces of silk plot, not only did the glass pieces repel each other, but so did the cloths, the same phenomenon held for the pieces of wool used to rub the wax. Now, this was really strange to witness. After all, none of these objects were visibly altered by the rubbing, Yet they definitely behave differently than before they were rubbed would ever change took place to make these materials a tractor repel one another was invisible. Some of the experimenters speculated that invisible fluids were being transferred from one object to the other during the process of rubbing, and that these fluids were able to effect the physical force over a distance. Charles Do Fay was one of these early experimenters who demonstrated that were that there were definitely two different types of changes wrought by the rubbing off pairs of objects together. The fact that there was more than one type of change manifested in these materials was evident by the fact that there were two types of forces produced attraction and repulsion. The hypothetical fluid transfer became known as Charge. One pioneering researcher, Benjamin Franklin, came to the conclusion that there was only one fluid exchange between the rubbed objects and that two different charges were nothing more than either an excess or a deficiency of that one fluid. After experimenting with wax and wool, Franklin suggested that coarse wool removed some of this invisible fluid from the smooth wax, causing an excess of fluid on the wool and a deficiency of fluid on the wax. The resulting disparity and fluid content between the wool and wax would then cause an attractive force as the fluid tried to regain its former balance between the two materials , postulating the existence of a single fluid that was either gained or lost through rubbing accounted best for this observed behaviour that all materials fell neatly into two categories when rubbed. Most importantly, that the two active materials rubbed against each other always fell into opposite categories as evidence by their invariably attraction, attracting to one another. In other words, there was never a time when two materials rubbed against each other. Both became positive or negative, following Franklin's speculation of woe rubbing something off the wax. That type of charge that was associated with rubbed wax became known as negative because it was supposed to have a deficiency of fluid, while the type of charge associated with rubbing wool became known as positive because it was supposed to have excess fluid. Precise measurements of electrical charge were carried out by the French physicist Charles Coolum in the 17 eighties, using a device called a Torre shin ballots measuring the force generated between two electrically charged objects. The result of cool ums work led to the development of a unit of electrical charged named in his honor, the Coolum. If two point objects, hypothetical objects having no appreciable surface area were equally charged to a measure of one Coolum and placed one meter apart, they would generate a force of about nine billion Newton's approximately £2 billion either attracting or repelling, depending on the types of charge involved. Now this quantity of force may seem fairly large, and that is because we're describing it in a macro sense. Once we drill down into the subatomic level, that force becomes smaller and smaller and can be dealt with ah, little bit easier. The operational definition of a cule, um, as a unit of electrical charge in terms of force generated between two points was found to be equal to an excess or deficiency of about 6250 with 15 zeros after it. I'm not sure if this is called a trillion trillion trillion, but you get the idea. It's a very large number of electrons or stated in reverse terms. One electron has a charge of about zero point with eking zeros 16 coup domes, which means one electron equals about minus 1.602 times 10 to the minus 19 cool homes being that one electron is a smallest known carrier, Oven Electric Charge. This last figure of charge for the electron is defined as the elementary charge. It was discovered much later that this fluid was actually composed of extremely small bits of matter called electrons, so named in honor of the ancient Greek word for amber, which was another material exhibiting charged properties when rubbed with cloth. Experimentations has since revealed that all objects are composed of extremely small building blocks. No. One as Adams and these atoms are in turn composed of smaller components known as particles , the three fundamental particles comprising them. Most atoms are called protons, neutrons and electrons, whilst the majority of Adam's having a combination of protons, neutrons and electrons. Not all atoms have neutrons. Collectively, protons and neutrons in the Adam make up the nucleus or the center core of the atom. Each electron has a negative charge of minus 1.602 times 10 to the minus 19 Coombs, and each proton in the nucleus has a positive charge of 1.602 times 10 to the minus 19 Coombs New drugs have no charge, just mass that's associated with it, because the opposite charge there is some attraction or attractive force between the nucleus and the orbiting electrons. Electrons have a relatively negative, negligible mass compared to the mass of a nucleus. The massive each proton and neutron is 1840 times The mass of an electron on Adam becomes positively charged when it loses electrons, and similarly anatomy becomes negatively charged when it gains electrons. So let's again look at the mystery of the materials being rubbed together. A piece of silk against a piece of glass using our atomic model by rubbing the silk on the glass. Electrons. Air transferred from the glass rod to the silk, causing the glass to be positively charged in the clause to be negatively charged, which then causes the two to attract, and by rubbing the wool on the wax, the electrons are transferred from the wall to the wax, causing the wool in this case to be positively charged in the wax to be negatively charged , which then causes the two to attract. The atomic model also explains the phenomenon that identical materials, after having being rubbed with the respect of cloth, always repel one from the other. The fact that when a piece of glass rubbed with silk was exposed to the wax rubbed with wool, the two materials would then attract one another, as well as the fact that rubbing two pieces of glass with two pieces of silk, the two pieces of wax and with the wool not only the glass pieces in the lax pieces repel each other, but so did the cloths. Adams may have loosely bonded electrons in their outermost orbits. These electrons require a very small amount of energy to detach themselves from their parent, Adams. Hence they're referred to as free electrons, which move randomly inside the substance and transfer from one atom to the other. Any piece of substance, which as a whole contains an unequal number of electrons and protons, is referred to as being electrically charged. When there is more numbers of electrons compared to protons, the substance is said to be negatively charged, and when there is more number of protons compared to electrons, the substance is said to be positively charged. The basic nature of electricity works like this. Whenever a negatively charged body is connected to a positively charged body by means of a conductor, the excess electrons of the negative body starts flowing towards the positive body to compensate the lack of electrons that is in the positive body. There are some materials which have plenty of free electrons at normal room temperature. Very well known examples of this type of material are, for example, silver, copper, aluminum, zinc, etcetera. The movement of these free electrons can easily be directed in a particular direction if the electrical potential difference is applied across a piece of these materials. Because there are plenty of free electrons, these materials have good electrical conductivity. These materials are referred to as conductors. The drift of electrons in a conductor in one direction is known as electric current. Actually, electrons flow from lower potential negative toe higher potential positive. But the general conventional direction of current has being considered as the highest potential point to the lower potential point. So the conventional direction of electric current has just the opposite of the direction to that of the flow of electrons. More of that in a couple of slides in nonmetallic materials such as glass, mica, slate porcelain. The outermost orbit is completed and there is almost no chance of losing electrons from the outermost shell. Hence, there is hardly any free electrons present in this type of material. These materials cannot conduct electricity. In other words, electrical conductivity of these materials is very poor. Such materials are known as non conductors or electrical insulators. The nature of electricity is to flow through a conductor, while an electrical potential is applied across it but not to flow through an insulator. Even though high electrical potential differences are applied across them. The definition of drift velocity can be understood by imagining the random motion of free electrons in the conductor. The free electrons in a conductor move with random velocities in random directions when an electric field is applied across a conductor, the randomly moving electrons or subject to electrical forces along the direction of the field. Due to this field, the electrons do not give up the randomness of motion, but they will be starting to drift towards the higher potential. That means the electrons will drift towards higher potential along their random motions. Thus every electron will have a net philosophy towards the higher potential end of the conductor, and this net velocity is referred to as drift velocity of the electrons. The electric current due to this drift movement of electrons inside an electrical stress conductor is known as drift current. If the electrical field intensity is increased, the electrons are accelerated more rapidly towards the positive material after each collision. Consequently, the electrons gain more average drift velocity towards the positive potential. Let's define electric current then electric Current is the rate of flow of electric charge through a conductor. With respect to time, it is caused by the drift of free electrons through a conductor in a particular direction. The measuring unit of electric charge is the cool Oh, and the unit of Time is seconds. The measuring unit of electric current is cool ums per second, and this logical unit of current has a specific name. AM pair after the famous French scientist Andre Marie Ampere. If a total of Q Q. Lem's passed through a conductor by Time T than the electric current is defined by I, which is equal to Q divided by t, which is cool ums per second, also known as an pairs for a better understanding. Let's give this an example. Suppose ah 100 cool ums of charge is transferred through a conductor in 50 seconds. What is an electric current As the electric current is nothing but the rate of which charges transferred per unit of time. It would be the Rachel of the total charge transferred to the required time for that transfer. Hence, here, electric current I is equal to 100 Coombs divided by 50 seconds, which is equal to two pairs. So far, we've been talking about charges flowing past a point in the conductor. Really? We haven't defined direction. We're about to change that. Right now, we're gonna define what is the direction of electric current. The drift of electron flow was shown in the diagram here from left to right. In other words, electrons are drifting from left to right. The negative charge flow is also from left to right. Now, The adopted convention of current flow is from positive to negative, which is the opposite to the electron flow. So we say the current flow in this conductor is from positive to negative or from right to left in this diagram which at the risk of repeating myself. The electron flow is the opposite direction, so you might consider current flow as the flow of positive ions air positive, uh, quantities where's lacked Ron's are negative there, flowing in the other direction. However, convention states that current flows from positive to negative. This ends Chapter one.
3. Ch 02 Conductors & Insulators: Chapter two Conductors and insulators. The electrons of different types of atoms have different degrees of freedom to move around with some types of materials, such as metals, the outermost electrons in the atoms air so loosely bound that they kyat chaotically, move in the space between the atoms of that material by nothing more than the influence of room temperature and the heat of that room temperature. Because these virtually unbound electrons are free to leave their respective Adams and float around in the space between the Jason Adams there off. Often called free electrons, these materials are known as conductors in other types of materials, such as glass, mica, porcelain, the atoms electrons have very little freedom to move around. While external forces such as physically rubbing can force some of these electrons to leave their respective Adams Adams on the surface and transfer toe Adams of other materials, they do not move between the atoms within the material very easily. These materials are known as Insulate er's. This relative mobility of electrons within the material is known as electric conductivity. Conductivity is determined by the types of Adams in the material and how the atoms air linked together to one another. Materials with high electron mobility are called conductors, some of which are listed here, silver being at the top of the list. Copper, gold, aluminum, iron, steel, brass, bronze, mercury, graphite, dirty water and even concrete. And the list is more extensive than this. But these are kind of at the top of the list, while materials with low electron mobility are called insulate er's, some of which are listed here glass, rubber, oil, ash, bald air, diamond, pure water, etcetera. It must be understood that not all conductive materials have the same level of conductivity , and not all insulators are equally resistance to electron motion. Electoral conductivity is analogous to the transparency of certain materials to light. Materials that easily conduct light are called transparent, while those that don't are called opaque. However, not all transparent materials are equally conductive to light windows. Window glass is better than most plastics and certainly better than clear fiberglass. So it's with electrical conductors. It's the same thing. Some are better than others. For instance, Silver is the best conductor in the conductor's list, offering easier passage for electrons than any other material cited dirty water and concrete are also listed as conductors, but these materials are substantially less conductive than any metals. It should also be understood that some materials experiences changes in their electrical properties under different conditions. Glass, for instance, is a very good insulate er at room temperature, but becomes a conductor when heated to a very high temperature gases such as air. Normally, insulating materials also become conductive if heated to very high temperatures. Most metals become poor conductors when heated and better conductors when cool. Many conductive materials become perfectly conductive. That is called super conductivity, at extremely low temperatures. So conductors air not always conductors and insulators or not always insulate ER. They depend on some other factors that might influence that they're conductivity or their insulating properties. Remember that electrons can flow only when they have the opportunity to move in the space between the atoms of the material. This means that there can be electric current on Lee, where there exists a continuous path of conductive material, providing a con you for the electrons to travel through a thin, solid line as shown here, is the conventional symbol for a continuous piece of wire. Since the wire is made up of conductive materials such as copper. Its constituent atoms have many free electrons, which can easily move through the wire. However, there will never be a continuous or uniform flow of electrons within the wear unless there they have a place to come from and a place to go. Let's add a hypothetical electron source and a destination. We'll talk more about that later. Like the left electron sources, a source of electrons and the Elektron destination is a reservoir where the electrons can flow to now, with the electron source pushing new electrons into the wire. On the left side, electron flow through the wear can occur as indicated by the arrows pointing from left to right. However, the flow will be interrupted if the conductive path formed by the wire is broken. Since air is an insulating material and an air gap separates the two pieces of wire, the ones continuous path has now been broken, and electrons cannot flow from the source to the destination. If we were to take another piece of wire leading to the destination and simply make physical contact with the wire leading to the source, we would once again have a continuous path for electrons to flow. The two dots in the diagram indicate physical metal to metal contact between the wire pieces. Now we have continuity from the source to the newly made connection down to the right and up to the destination. Resistors are a special kind of conductor. Let's take a closer look at what we call resistors. Resistors are a special kind of conductor in that, depending on their physical makeup, allow only a certain amount of current to flow. They will therefore act to resist current flow. Schematically, the symbol for resistor, looks like a small rectangular box or a zigzag line. Resistors can also be shown to have varying, rather than, rather than fix, resistances. This might be for the purposes of describing the actual physical device designed for the purpose of providing an adjustable resistance. Or it could be to show some component that just happens to have a unstable resistance. In fact, any time you see a component symbol drawn with a diagonal arrow through it, that component has a variable rather than a fixed value. This since this symbol modifier, the diagonal arrow, is a standard electronic symbol convention. This end Chapter two
4. Ch 03R1 Current and Electric Circuits: Chapter three current and electric circuits. One foundational unit of electrical measurement is the unit of the Coolum, which is a measure of electric charge proportional to the number of electrons in an unbalanced state. One Coombe of charges equal to 6250 with 15 zeros behind it. Electrons, or 6.250 times 10 to the 18th power. The symbol for electric charge quantity is the capital. Letter Q. The AMP is equal to one Coolum of electrical one Coolum of electrons passing by a given point in a circuit in one second of time. Current is the rate of electric charge motion through a conductor. In the diagram, one app is equal to the electrons passing by a in one second of time. Remember that there will never be a continuous or uniform flow of electrons within a wire unless they have a place to come from and a place to go. In other words, they need a source and a destination. Now, with an electron source pushing new electrons into the wire on the left side, electrons flow out of the wire on the right side. This source can take many forms. Sources supply a constant push of charges called electro Motive Force or E M. F, and it also provides a reservoir or a poll of charges. Otherwise, there would be a buildup of charge in one end of the conductor. This set up provides a continuous loop or a path for the flow of charges. We call it a circuit. One such source is a battery whose constant E. M. F is measured in volts. Any source of voltage, including batteries, have two points for electrical contact. In this case, we have 0.1 and 0.2 in the diagram. The horizontal lines of varying length indicate that this is a battery, and they further indicate the direction which the battery voltage will try to push electrons through a circuit. This is the electrical symbol for a battery. Notice the positive and negative signs to the immediate left of the battery symbol. The negative end of the battery is always the end with the shortest dash, and the positive end of the battery is always the end with the longest ash since we have decided to call electrons negatively charged. The negative end of the battery is that end which tries to push electrons out of it. Likewise, the positive end is the end, which tries to attract electrons with the positive and negative ends of the battery not connected to any to anything. There will be a voltage between those two points, but there will be no flow of electrons to the battery because there is no continuous path for the electrons to move. The same principle holds true for the water reservoir and a pump analogy. Without a return pipe back to the pond, stored energy in the reservoir cannot be released in the form of water flow. Once the reservoir is completely filled up, no flow can occur. No matter how much pressure the pump may generate, there needs to be a complete path or a circuit for the water to flow from the pond to the reservoir and back to the pond. In order to have a continuous flow occur, we can provide such a path for a battery by connecting a piece of wire from one end of the battery to the other, forming a circuit with a loop of wire. We will initiate continuous flow of electrons in a clockwise direction in our diagram. So long as the battery continues to produce voltage and the continuity of the electrical path isn't broken, electrons will continue to flow in the circuit following the metaphor of the water moving through a pipe. This continuous uniform flow of electrons through a circuit is called a current. So long as the voltage source keeps pushing in the same direction. The electrons, the electron flow will continue to move in the same direction in the circuit. This single direction flow of electrons is called a direct current or D C. Notice the positive and negative signs drawn at the end of the break in the circuit and how they correspond to the positive and negative signs next to the battery terminals. These markers indicate direction that the voltage attempts to push the current. That potential direction is commonly referred to as polarity. Remember that voltage is always relative to two points. Because of this fact, the polarity of a voltage drop is relative between two points. Whether a point in the circuit gets labeled with a positive or a negative depends on the other point to which it is referenced. If we include a lightbulb in the electric circuit. We find that a set amount of current flows after the switch is closed. We say that the light bulb resist the flow of current different size. Lightbulbs allow different amounts of current to flow, depending on the amount of resistance the lightbulb introduces into the circuit. Because it takes energy to force electrons to flow against the opposition of the resistance , there will be a voltage manifested or a voltage drop between any points in the circuit with resistance between them. It is important to know that the amount of current the quantity of electrons or charge moving past a A given point. Every second is uniform throughout the circuit. In electron flow notation. We follow the actual motion of electrons in the circuit. But the positive and negative labels seem to be backwards. Doesn't matter really how we designate charge flow in a circuit, Not really. So long as we're consistent in the use of our symbols, you may follow on imaginary direction of current, which is the conventional flow or the actual electron flow with equal success in so faras. Circuit analysis is concerned, however, looking at the worldwide conventional flow notation, the flow of current is always considered from positive to negative. Electrons move from the negative side of the battery through the conductor to the positive side, but this is considered as positive current flowing from positive to negative. The force motivating electron flow in a circuit is called Electro Motive Force, or E. M. F. It is measured in volts and at times refer to his voltage. Voltage is a specific measure of potential energy that is always relative between two points when we speak of a certain amount of voltage being present in a circuit. We are referring to the measurement of how much potential energy exists to move electrons from one particular point in that circuit to another particular point without reference to two particular points. The term voltage has no meaning. Voltage has a polarity and is usually indicated with a positive and negative sign. The voltage is said to be rising from negative to positive or weaken. Sometimes describe it as voltage drop, which is referred to from positive to negative current by standard acceptance flows from positive two, and it's sometimes indicated with an arrow and the symbol I Any kind of resistance to current flow is sometimes indicated by a squiggly line as seen here, or a block and labeled with the symbol are as the current flows through the resistant words through the resistor, it will set up a potential difference across the resistor. As before, the voltage difference or drop is indicated by a positive and negative sign, and the voltage drop is associated with. The resistor is sometimes labeled V, where the subscript r, but the key it up note here is that there is a potential difference across the resistor due to current flowing through that resistor. The symbol given for each quantity is the standard alphabetical letter used to represent that quantity in analogy. In algebraic equation, standardized letters like these air common in physics and engineering and are internationally recognised, the unit abbreviation for each quantity represents the alphabetical symbol used as a shorthand notation for its particular unit of measurement. Each unit of measurement is in this case has been named after a nearly experimenter in electricity, for example, the AMP is named after the Frenchman Andre M Am pair. The Volt is named after an Italian Alessandro Volta, and the old is Aziz, named after a German, George Simon home. The abbreviation is homes and it is our omega, and it's the last letter of the Greek alphabet. The mathematical symbol for each quantity is meaningful as well. The are for resistance and the V for voltage are both south explanatory, whereas I for current seems to be a bit weird. The I is thought to have been have meant to be represented by intensity or electron flow, and the other symbol for voltage is sometimes E, which stands for electro motive Force. All these symbols are expressed using capital letters, except in the case where a quantity, especially current in voltage, is described in terms off a brief period of time, sometimes called an instantaneous value, and there'll be more on this later, and this ends Chapter three.
5. Ch 04 Ohm's Law: Chapter four homes law, these units and symbols for electrical quantities will become very important to know as we begin to explore the relationship between them and circuits. The first, and perhaps the most important relationship between current voltage and resistance is called Holmes Law, discovered by George Simon home and published in his 18 27 paper, The Galvanic Circuit Investigation. Mathematically Holmes Principal discovery was that the amount of electric current through a metal conductor in a circuit is directly proportional to the voltage impressed across it for any given temperature. Also, the current is inversely proportional to the resistance of the wire or the resistance in the circuit. Omen pressed his discovery in the form of a simple equation, describing how voltage current and resistance are interrelated. Here, voltage is equal to current times resistance. This is a new algebraic expression, of course, where voltage E is equal to the voltage and I is equal to the current and R is equal to the resistance. Using algebraic techniques, we can manipulate this equation into two variations solving for I and solving for our respectively. If e equals ir than I would equal e over r and our would equal e over I what this is telling us that given any two of these three unknowns, the 3rd 1 can be calculated. This is referred to as owns law, and this is probably the simplest and most important relationship in any and all electric circuits. Let's look at an example. A light or a lamp bulb, which will call the load, is connected across the battery. It is given that the load or the light is three homes, and it's connected across a 12 volt battery. What is the current that will flow in this circuit? It is predicted by owns law. The current will be equal to the voltage divided by the resistance or 12 divided by three, which gives us four amps. Let's look at another example with just a slightly different twist, not twist being not. We are given the current and the voltage, and we want to know what the resistance in the circuit is or the resistance of the lamb. If the current is four amps driven by a 36 bold battery, and what would the resistance be? The resistance will be given by the voltage divided by the current or 36. Divided by four, which is nine homes. Holmes Law is a very simple and useful tool for analysing electric circuits. It is used so often in the study of electricity, an electron ICS that it needs to be committed to memory. For those who are not yet comfortable with algebra, there's a trick to remembering how to solve. We're remembering where the variables belong in the equation. First arranged the letters E. I R. In a triangle like this. Then remember that V is always the numerator above I or are another trick. It's just just think logically. Current always goes up with voltage and down with resistance or another waiter. Brute is with this cartoon, which has appeared in many ways, shapes or form. But you got three fellows there that are attempting to work on the Mr Champ, and Mr AM is being restricted by Mr Own and he's being pushed by Mr Bolt. Conduct. INTs is the ability oven element to conduct current. It is the reciprocal of resistance, therefore, conduct INTs. If given by G, is equal to one over R. It's measured in quantities called Seaman's and in terms off voltage and current G is equal to I over V, which is the opposite to resistance for the inverse to resistance. This term is seldom used, but it should be recognized if you do come across it. There are rare times when conducting is used rather than resistance, but you should know what it means, so this ends Chapter four.
6. Ch 05R1 Electric Power: Chapter five Electric power energy is defined as the property that must be transferred to an object. In order to perform work, energy equals work. Consider these two weightlifters lifting the same amount of weight to the same height. Both do the same amount of work. Both expend the same amount of energy in terms of physics. Work equals mass times, acceleration or gravity times height which equals 100 kilograms times 9.8 meters per second , squared times two meters which equals 1960 jewels. However, the weight lifter on the left is slower than the weight lifter on the right. Hence we say that the right weak lifter is more powerful than the left weightlifter power measures the rate at which work is done or that power equals work divided by time. So if power is equal to work divided by time and we know that the same amount of work is done by each weight lifter because they've lifted the same weight through the same distance and not as equal to 1960 jewels. But the weight lifter on the left completes his lift in three seconds. So the power delivered by that weight lifter is 653 jewels per second. The weight lifter on the right, however, lifts his weight in one second so that the power delivered by the right weight lifter is 1000 960 jewels per second. Because work is measured in jewels, power is measured in joules per second and this measurement is defined as watts such that the work of one jewell completed in one second is equal to one watch. If we watch the lift, we see that the speed of the lift is not consistent. Regardless, if the lift is completed in three or one seconds, some of the lift is completed faster or slower than the other parts, which means the power delivered will vary. So if we use the total time for the lift in our equation, the one or the three seconds we define that power, the power delivered as average power. If we break the whole lift up into smaller time increments, such a such that the power over that small increments is consistent, we will define that as instantaneous power, which is consistent over that small time increment in terms of electrical power. The work done or electrical energy is the movement of charges caused by the push of e M F. In other words, it is the energy required to move. An electric charge of Q Q looms over a potential difference of the Volt's end is expressed as the Times Q Electron volts. By definition, one electron volt is the amount of energy gained or lost by the charge of a single electron moving across an electric potential difference of one volt. One Que lo is equal to 6000 250 followed by 15 zeros. Electrons and one electron is equal to one over Q, which is equal to 1.6 times 10 to the negative. 19 que loans Electric Power P is the rate at which electric energy is transferred by an electric circuit. The Power P is the energy dissipated over Time T, but the energy E is equal to the voltage times a charge being passed. Therefore, P is equal to the product of the voltage times the charge all over T or voltage times Q. Divided by T and because Q divided by t is I or current, where one amp is equal to one Coolum of electrons passing by in one second of time. P is given by the voltage times the current for a constant voltage and constant current or DC values for voltage and current. Or it might be considered the product of the instantaneous voltage times. The instantaneous current electrical power is measured in watts, a watt sometimes symbolized by the capital. Letter W is a derived unit of power in the international system of units s I. This is a chart of the international system of units for power. The most common ones that are used are the Milly Wat, which is 0.1 of a lot, or 10 to the minus six watts, or a kilowatt, which is 1000 watts, or 10 to the third watts, or a megawatt, which is one million watts, or 10 to the sixth watts, or a gigawatt, which is 1000 megawatts or 10 to the ninth watts. Now the power of a circuit. Uh, and the voltage and the current are related by thes equation, which is the same equation. Just written three different ways, and you can memorize their relationship by the above triangle, you can see that P is always over either V or I, depending on what you're looking for. If you're looking for V voltage than its power over I. If you're looking for current than it's pee all over V, the most common one used is the 1st 1 of course, powers equal to the voltage times current. But I is used is equal to P over V is used all the time in calculating the current draw on electrical circuits and electricians use this when contemplating or designing the circuits for house wiring. Say, however, I'm getting a little bit ahead of myself now because house wiring is a C or alternating current. Where is we're just talking about D. C circuits right now. How everything I've said up to this point is applies to D C circuits, and they will, somewhat as you'll see when we get into a sea circuits in the in another course that these formulas and the symbols still apply. But you have to take them into consideration due to the fact that we are using a C circuits . However, let's get back to talking about D C circuits we can just do a calculation of power in a in an electric circuit in this circuit. According Toa Homes Law, if we have a voltage of 18 volts pumping into a lamp of three OEMs, the current that will flow in that circuit given by OEMs law is six amps, so the lamp would be consuming the current times. The voltage, which is six amps times 18 volts, would be equal to 108 watts. In this example, we have a 100 watt light bulb connected to 120 volt battery. According to the power equation, power is given by the current times a voltage. That is to say that the current times 120 bulls is equal to 100 watts. Therefore, the current is equal to 100 divided by 1 20 which is equal to point 833 APS, so the current flowing to that light bulb would be 0.833 amps. Also, the resistance, by the way, would be according Toa homes law voltage over the current, which is equal to 120 divided by 1200.833 which is equal to 144 homes, so power can be expressed in terms of voltage and current. In this case, we, uh, can say that the power is equal to I the current times e the voltage, which stands for electro Motive Force. We know from homes law that the current is equal to e all over our so we can describe power consumption in a circuit in terms of the voltage across the resistance and the resistance on Lee. So the power consumed by a resistive load would be e squared all over r, where r is the resistance of the load and e is a voltage drop across that resistance. We can also describe the power consumption in a circuit in terms of the current in that circuit. The current through the resistor and that is given by I squared are in other words, if we only know current and voltage, then we can calculate the power. If we only know resistance in voltage weaken still calculate the power. And if we only know resistance and current, we can still calculate power by using one of these three equations. This ends Chapter five
7. Ch 06 Series & Parallel Circuits: Chapter six series and parallel circuits. There are two basic ways in which to connect more than two circuit components, Siri's and parallel. Here in this slide, we have three resistors labeled R one R two and R three connected in a long chain from one terminal of the battery to the other. It should be noted that this sub script, labeling those little numbers to the lower right of the letter R, are unrelated to the resistor values in homes. They are only to identify one resistor from the other. The defining characteristic of a Serie circuit is that there is only one path for current to flow in this circuit. Current flows in a clockwise direction from 0.1 to 2, 2324 and back to one. We say that the three resistors are connected in Siris, a game we have three resistors, but this time they form mawr than one continuous path for the electrons to flow. Each individual path through R one, R two and R three is called a branch. The defining characteristic of a parallel circuit is that all components are connected between the same set of electrical electrically common points. Looking at the schematic diagram. Here we see that points 123 and four are all electrically common. So our points 876 and five note that all resisters as well as the battery, are connected between these two sets of points. We say that these three resistors are connected in parallel. In this circuit, we have two loops for electrons to flow through. Notice how both current past girls go through resistor number one. In this configuration, we say that R two and R three are in parallel with with each other, while our one is in Siris with the parallel combination of R two and R three. The basic idea of a Siri's connection is that components are connected end to end in a lying to form a single path through the current to flow, in this case, from left to right. Or it could be the currents going to flow from right to left. The basic idea of a parallel connection, on the other hand, is that all components are connected across each other's leads. In a purely parallel, circa, they're never more than two sets of electrically common points, no matter how many components air connected. There are many pass for the electrons to flow, but Onley one voltage across the components. The first principle to understand about Siri circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path. It occurred to flow in a series circuit. The rate of flow at any point in the circuit at any specific point in time must be equal. This brings us to the second principle of Siri circuits. The total resistance of any Siri's circuit is equal to the sum of the individual resistances. This should make intuitive sense. The more resistors in Siris that the electrons must flow through, the more difficult it will be for those electrons to flow. In the example problem, we have a three okay, 10-K and a five k resistor in Siris. In other words, it's 3000 owns 10,000 homes and 5000 homes, all in Siris, giving us a total resistance of 18 K or 18,000 homes. In essence, we've calculated the equivalent resistance of R one, R two and r three combined. Knowing this, we could draw the circuit with a single equivalent resistor representing the Siri's combination of R one R two end are three. Now we have all the necessary information to calculate the circuit current because we have the voltage between the two points nine volts and the resistance between the same two points. 18 k OEMs. The current is given by Holmes Law, which is the voltage divided by the resistance or, in this case, nine divided by 18,000 which would give 0.5 amps, or 0.5 million APS. Since 1000 millions is equal to 1/2 because the current is 500 million amps. Then the 500 millions will flow through each resistor and a voltage drop will appear across each resistor. The voltage drops across each resistor will be given then, by all means, law. For our one, it's 0.5 millions times three K, which is equal to 1.5 volts for our to it's 0.5 times 10-K 0.5 millions times 10-K is five volts and for our three as 30.5 millions times five K, which is equal to 2.5 volts. Notice that the some of the voltage drops 1.5 plus five plus 2.5 is equal to the battery supply. Voltage nine volts. This is a principle of Siri circuits that the supply voltage is equal to the sum of the individual drops around the circuit. This is known as Kurt Chef's Voltage Law, or sometimes referred to his kerchiefs loop, or mesh rule, which states that the direct some of electrical potential differences. Voltage drops around any closed network is zero stated another way. The some of the E. M F's in a closed loop is equal to the sum of the potential drops. In that loop. Notice the polarities of the resisters they go from plus to minus, which is voltage drops 1.5 plus five plus 2.5 equals nine volts. Voltage drops. The supply voltage goes from negative to positive, which is a voltage rise. Negative voltage drops, which is a negative voltage drop if you want to say it minus nine. So if you add up the voltage drops, you get 1.5 plus five plus 2.5 minus nine is equal to zero. The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrical common points in a parallel circuit, and the voltage measured between sets of common points must always be the same at any given time. Therefore, in the circuit we have here, the voltage across our one is equal privilege across our two, which is equal to the voltage across our three, which of course, is equal to the voltage of the battery. We can immediately apply homes lock of each resistor to find the current in each resistor because we know the voltage across the resisters nine volts and we know the resistance of each resistor. Therefore, the current in resistor R one is 10.9 million apps. The current in our two is 4.5 millions and the current in our three is nine million apps. At this point, we still don't know what the total current or the total resistance for this parallel circuit is, so we can't simply apply homes lava to the total resistance. However, if we think carefully about what is happening, it should become apparent that the total current must equal the sum of the individual currents in each resistor or in each branch. If it wasn't the case, then we would have a buildup of current or charge in one part of the circuit. And that isn't happening. We have a continuous flow of current at any one time so that the total current must equal the sum of the branch currents. Okay, let's let the voltage drop across. All the resisters equal the tea, which is equal to the supply voltage. And we're gonna let Artie equal the equivalent parallel resistance of all three resistors. We know from homes law that the voltage the total voltage is going to be equal to the individual voltage drops across each resistor, which is I one times are one which is equal toe I to times are too, which is equal to I three times are three. We also know that the from homes law that the supply voltage V t is equal to the total current times. What would be the equivalent resistance of the parallel resistors? Or we can rewrite the equation where the total curd is equal to the voltage of the supply voltage over the equivalent parallel resistance. R t we know that the some of the current flowing into the branch circuits I t. Is equal to the sum of the branch circuits I one plus I to plus I three or we can rewrite That's that equation, substituting voltage over the resistance for the current terms. So I t becomes VT over RT, which is equal to VT all over our one plus b t all over our two plus Bt all over our three . Now, if we divide both sides of the equations by the tea, then we're left with one over R T is equal to one over r one plus one over r two plus one over r three. So we now have a method of calculating the equivalent parallel resistance with this equation. As I said in the previous slide, this gives us a rule for calculating equivalent resistors in parallel. The if we let the equivalent resistance of parallel resistors B R T than one over R T is equal to one over R one plus one over r two plus one over r three. This also leads us to Kerr chops. Current law, which states the current entering any junction or node, is equal to the current, leaving that junction or node, a node being defined as an electrically connected point in the circuit. Another way of stating kerchiefs. Current law is all currents. Entering a node must sum to zero currents entering our positive currents. Leaving are negative. This is another way of looking at the same electrical circuit. In this case, we've brought the Noto one common point, and it's easier to see that Kirchoff is current. Law holds true that the current entering the node is equal to the current. Leaving the note or the some of the currents entering a node is zero. So now, with the knowledge learned, we can reduce any mesh network of resistors to the equivalent of one resistor, as seen by the power supply. For example, in this circuit, R one R two R three and r four can be replaced by one resistor R E Q. That will draw the same current as R. One R, two R three and r four. We start by calculating the single resistor equivalent for R one and R two in parallel and R three and R four in parallel, which results in r one r two in parallel, giving 71.4 to 9 homes and R three and R Foreign Parallel gives 127.27 homes. We can then add these two resistances because they're in Siris and they will give us 198 0.70 Holmes, which is the required single resistor we wanted to calculate, which will draw the same current as R. One R, two R three and R. Four altogether. As I said before, any mesh network of resistors can be reduced to one single resistor, as in this case, and this one looks a little bit more complex, which it is because there's more resistors involved. And it's drawn in such a way that it's not intuitively obvious how we would reduce this. So sometimes it's easier to redraw the circuit so that the parallel Siri's combinations will jump out at you a little bit easier than you can start. To reduce this in steps. For example, R three and R two are in parallel. They can be read, reduced to one resistor of art to in parallel with R three. It is now everyone resistor in series with our four so that can be reduced by the parallel combination. Plus our forests and Siri's. We've now replaced those resistors by one resistor and we can see that that equivalent resistance is now in parallel with our five. So we can reduce that toe one resist er which is our five in parallel with a combination are to imperil with our three in Siris with our four, which gives us one resistor. And that resistor is now in serious with our seven. So that can be replaced by one resistor, which is the equivalent of our seven in Siris. With our five in parallel. What the combination are two in parallel with R three plus R four that is now in parallel with R six and can be replaced with one resistor and I'm not gonna go through repeating it all again. We now have that resistance as a single resistor. Now in Siris are one which can be replaced now by one resist er and that resistor is now what we're trying to achieve in the first place. And we can calculate the current that the power supply supplies to that single resistor this ends chopped her six
8. Ch 07R1 Circuit Theorems: Chapter seven circuit three rooms linearity. Linearity is the property of an element describing a linear relationship between the cause and the effect. A linear circuit is one whose output is linearly related or directly proportional to its input. One of the properties of on electric circuit is that it is homogeneous. We sometimes refer to it as the Homo Geneti or scaling property oven electric circuit. For example, we can take de circuit described by owns law. The equals IR and weaken scale it by multiplying of by any scaling number. In this case, we just call it K. We can multiply k times a voltage. And if we long as we multiply the right hand side of the equation by K, it is said to be homogeneous or weaken, scale it. There's also the additive property, and we'll use this as we go through some of our complex calculations and other lessons. And that is if we have two values for a circuit with one voltage is equal to one current times a resistor, and another voltage is equal to another current times a resistor. We can add those two things together as long as we add the left side of the equation and the right side of the equation. The property still holds true. In other words, V one plus V two is equal to I one. Our times are plus I two times are, or you can collect like terms on the right hand side, which is equal to the quantity of I one plus I to times are, I am going to look at power sources now and more specifically, voltage sources in current sources. First of all, let's have a look at voltage sources, and we've already used voltage sources in some of our previous work in the way of a battery . And we all assume that the output of a battery is constant and those batteries, you can buy them off the shelf there double A's or Tripoli's or 1.5 volts output. You can buy a nine volt battery, or you can get a 12 volt battery for a car or various assortments of little disc batteries from 1.5 all the way up to 32 volts. The constant thing or the comment thing about a voltage source is that the voltage output is assumed to be constant and Indeed, most of the time it ISS. So when we add a load to this folded sores, we can use homes law to determine the current that's going to flow to the load because we know the voltage in its constant and depending on the resistor we put in there. If it's ah, particular value, we can divide the voltage by the resistance to find out what the current ISS and the voltage, as I have said and at the risk of repeating, will remain constant. Now if we start to increase the load by reducing the resist er, in other words, that current, then we'll start to go up because the voltage doesn't change. If the resistor becomes very small, the current will become very large, and that's given to us by homes law. And in fact, if you go to the extreme where you have zero resistance or a short circuit, then ultimately homes Law tells us that the current will go to infinity. However, I think we've all experiences because one time another, we might have shorted a battery out and certainly it might spark an art, but it the current doesn't go to infinity. So what is the limiting factor. When you start to get a short circuit well, as you start to reduce the resistance in the load, then the actual resistance of the wire becomes significant, and that will indeed be the limiting factor for your current. So when you short out a battery, the layer to the battery, the resistance, even though it's very small, becomes significant. And indeed, voltage sources have what they call internal resistance that has to be taken into consideration. Now you don't have to do this all the time because 99% of the time ah, battery is not short circuited. So it is safe to assume that battery is a constant voltage output. And it's not just a battery, because we have all a various assortment of D C power sources or voltage sources. If you go on a computer or a cell phone, you have to plug those in for charging. That is a DC power supply. And instead of putting a battery in our circuit, we represent a D C power source or a voltage source in this manner, a circle with a polarity attached to it. So it tells us where the positive is in a circuit, so we'll know which direction the currents going to flow. But we know that the voltage will remain constant now. A current source is, ah, relatively rare animal. There are some out there and there are calculations that you have to make considering a current source, so we are going to look at it very quickly now. But most of the sources of energy in a D C circuit are voltage sources. However, there are current sources out there, and it's represented by something like this circle with a narrow depicting the direction of the current flow. And the take away from this particular device is that it produces a constant current regardless of the load. And we've shown the circuit as closed here and in fact it is actually short circuited. However, it's not a problem because the current source maintains a constant current, regardless of the load in this type of circuit. Now the current will be constant, So the thing that's gonna change as we change the resistance in the load is the voltage drop across the load, because we will use Holmes law again for calculating the voltage drop across the resistor So if we start to increase the resistance of the load than the voltage is going to go up as well as we can see here till ultimately you would open circuit or if you did open circuit, the resistance would go to infinity. And indeed, you would have a problem because then the voltage is gonna go to infinity. Which, of course, it never does, because it's going to be limited by several factors, including the breakdown of air. But it could be very dangerous to open circuit. A constant current source, however, just like the voltage sources 99 times out of 100 year who going to be using closed circuits on a constant current source. So you just have to be aware of the fact that eight current source will maintain, ah, constant current when you're doing your calculations. So as we move through analyzing circuits, you'll run into two types of sources ones of voltage source and the other is a current source, and they are. They are. They are represented by circles, and in the case of the voltage source, you gotta planete plus and minus. And in the case of a current source you have a narrow and you have to remember that the voltage is a constant voltage. Oh, put the current in a current source is a constant current output. This example will demonstrate the use of a current source as well as demonstrate what we mean by linearity in this particular circuit. The current generator at one particular time is pumping out five amps. That is I s is equal to five amps. If you go through the calculation, you'll find or we will find that I not which is the current through the five own resistor is one. And now if we increase the source current to 15 amps, we've essentially scaled it up by three. So we because of linearity, we can scale up our answer, which used to be, uh, one am by a factor of three. So if we increase I s by a factor of three in its 50 naps, that I not by virtue of Lynn Neary will be three APs. The handy thing here for linearity is we don't have to go through all of the calculation using homes, law and circuitry to come up with a new answer for I not because we just have to scale it up by three. That brings us to the first of the theorems that we're going to look at. That is the superposition tear, Um, which states that in a linear circuit with several sources and by a linear circuit, we mean it has a constant current source and or a constant voltage source. The current through and the voltage drop for any element in the circuit is the sum of the currents and voltages produced by each source acting in dependent Lee. In other words, you can do a calculation of each source with the other sources removed, then remove that source and replace it with the second source, then removed that one. Replace it with the third and do that for all the sources that you have in a circuit. Then you just arithmetically add up the results to come up with the total values. The first thing that pops into mind know is how do you replace or remove sources? What does that look like? Well, there's going to be two types of sources in our circuit, the voltage source and a current source when we want to do the calculations for a voltage source and remove all the other voltage. And current sources, the voltage sources that we are not going to leave in a circuit are replaced by a short, in other words, replaced by a wire. And in the case of current sources, if we're going to remove a current source and do our calculations with either a current or a voltage source, a single one, then we have to replace all the other current sources by on open circuit. I think the best way to understand this is go through an example because it is pretty simple and it's very intuitive. But let's look at our first example. Okay, The best way to have a look at the superposition serum is demonstrating it with an example . Here we have I circuit made up of, ah, three old men of five old resistor and a two old resistor, and we have a current source of eight amps and a voltage source of 20 volts, all connected as you see in the diagram. Here we would like to calculate with the voltage drop across the to own resistor is one of the ways we can do it is calculate what each source provides in the way of current through the two ohm resistor and then linear, really add the two of them together and find out what the final effect is. So at the risk of being repetitive here, the steps to apply the superposition principle turn off all independent sources except one , and replace with unopened or a short depending off on if it is a current or of old source step. To solve the voltage your core self afford voltage and current due to that one act of source. Repeat step one for each of the independent sources and then find the total contribution by adding algae. Brickley. All the contributions do. Do the independent sources. Okay, The first thing we do is replace the 8 a.m. current generator and replace it with an open circuit, of course, and that leaves us with a 20 volt power supply feeding a 53 and a two home resistant in Siris. The next step will be to replace that voltage generator with a short circuit, and that will leave a current generator feeding two parallel circuits. Ah, five, home to the right and a three plus to wound to the left going back to our first circuit. We can calculate the current flowing in that circuit by simply using owns LA, which is I is equal to V over R and B being 20 volts and the are being a sum of 53 and two homes, which is tan homes, which would leave us with two amps. So two halves flowing through a two ohm resistor would give us a voltage drop off four volts in the bottom circuit. The current generator is split between current going to the left and current going to the right. And in these circuits, it just so happens that to the right is five homes and to the left is three plus two, which is also five owns. So you get an equal split of the current half going one direction, half going the other, and that gives us for amps going through the to own resistor, and that would provide a voltage drop of eight volts. So we then arithmetically add the two answers that were gained independently. Four bowls plus eight volts to give us 12 volts. In this example, we have, ah, sixfold voltage source in a three AMP current source feeding on a dome forum resistor and connected as you see here. The first thing we do is we can replace the three AMP generator with A with an open circuit that leaves the sixfold power supply feeding eight owns forums and Siri's, which gives us buy homes. Law. A current of 80.5 amps, which 0.5 amps flowing through the four ohm resistor, will give us a voltage drop for view one of two volts Next we have. The current source is feeding the A home in the four room in parallel. What you would do to calculate the current is you want to know what the voltage drop across each resistor is, and the voltage drop across each resistor is the voltage drop across a domes and forums in parallel, so you find the equivalent, uh, resistance value for the eight owned four. Run in parallel and calculate the voltage. Then you'll use the voltage drop across each resistor to calculate the current split through that resistor and ultimately come up with the current flow through Me too, which would be two amps. The current assets essentially spits in 1/3 2 3rd That's what you'll find out. But the current through the four own resistor is two amps, and that provides a voltage drop of eight faults. And the answer then would be simply adding a V one and V two together or to plus eight, which would give us 10 volts the next. The're, um, that we want to look at is called David INS Serum, and this is a very useful A fear of, not just for simple BC circuits. It is used in a sea circuits, and especially when you get into three phase fault calculations of asymmetrical short circuits in a three face system. However, we're not gonna go there today. We're only going to talk about the Haven and zero as it applies to D. C Circus. And what the David um, zero. Um says that a linear to terminal circuit can be replaced by an equivalent circuit consisting of a voltage source. The sub script th, which stands for the thieving and voltage in Siris with a resistor, are some strip th for the saving, an equivalent of the resistance, so that square box could have been made up of several generators of different types and different loads But what? We're what David and says You can replace that circuit and figure out how it effects the load, say in this diagram by replacing everything that was inside the box by one voltage source and one Siri's resistor. The trick is to find out how to calculate V Day, even and and our Stephen Well VI Fe. Vernon is simply the open circuit full to jackals terminals, and our theme in it is the input or equivalent resistance at that at those terminals. So it's pretty simple, but we need to go through a few calculations to demonstrate how easily it works. David in Serum is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit called the load resistor, is subject to change, and recalculation of the circuit is necessary with each trial value of load resistance to determine the voltage across it and a current through it. Let's take another look at an example. Let's suppose that we decide to designate are, too, as the load resist her in this circuit, David and Serum makes it easy by temporarily removing the load resistance from the original circuit and reducing what's left toe, an equivalent circuit composed of a single voltage source and a Siri's resistance. The load resistance can then be reconnected to this. They've been an equivalent circuit, and calculations carried out as if the whole work Ah, whole network was nothing but a simple Siri circuit. After the thieving and conversion, calculating the equivalent Faymann and source voltage and serious resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit and replaced with a break or an open circuit. Next, the voltage between the two points where the load resistor used to be attached is determined. You can use whatever method that's at your disposal. In this case, you could pick Theseus per position serum and calculate the current contributed from the 28 fold battery than the current contributed by the seven volt battery Arithmetically. Add the two and you would end up with as our table indicates a circuit value for the current of 4.2 amps. There is only one path for the current to flow, and it's 4.2 amps in all the components in the circuit, so that would give us a voltage drop across our one of 16.8 voltage drop across our three of 4.2. So we have the voltage drops, a swell as the voltage is indicated by the power supplies so we can arithmetically calculate what the voltage between the two points where load resistor was connected and that works out to 11.2 volts. The voltage between the two load connection points is now are saving an equivalent voltage . Now to find the thieving and Siri's resistance for our equivalent circuit, we need to take the original circuit with the load resistor still removed. Remove the power sources in the same style as we did with a superposition to your own voltage source is replaced with wires and current sources replaced with open circuits or breaks, and figure the resistance from one low terminal to the other. With the removal of the two batteries, the total resistance measured at this location is equal to R one and R three in parallel or 30.8 homes. This is our favorite and resistance for the equivalent circuit with a load resistor to homes attached between the connection points, we can determine the voltage across it and the current through it as though the whole network were nothing more than a simple Siri's circuit so we could fill out our table here . Right now, we got part of the table filled out because we can put in the total voltage, which is that they've been an equivalent 11.2. And we know the Fay, Vin and impedance are since Haven and, uh, resistance is 0.8 homes and our load resistance is to owns. So we can calculate the Siri's resistance just by adding the two of them together and come up with 2.8 homes. And use owns law to calculate where what the current would be flowing in the circuit. And that is four amps with four amps flowing through our load. We can end calculate the load voltage drop, which is eight volts, is what we were trying to do in the first place. And of course, we could also calculate the voltage drop across that they've been and resistance if we needed. That's 3.2 volts, so we can fill out our whole table now with using haven and serum. And if we wanted to replace the load resistor with a to own four own 80 10 home. We could do it very quickly, and the calculations are very easy and very quick. Norton Serum is similar to favorite in serum, with the difference that Norton's serum is dealing with current sources. You want to end up with a current source and a parallel resistor and in the case of thieving and it was a voltage source in serious with the resistance, Northern Serum states that it is possible to simplify any linear circuit, no matter how complex to an equivalent circuit. With just a single current source and a parallel resistor connected across the load. Norton's theorem states that a linear to terminal circuit can be replaced by an equivalent circuit consisting of a current source. I subscript Norton, which will be the Norton current source in parallel with a resistor r subscript Norton, which is the Norton Resistance. Our Norton is the input or equivalent resist resistance of at the Terminals, and I. Norton is the current flowing. If those terminals were shorted, I Norton would be the current flowing through that short, and we're gonna use this same example that we started out with with the dividend equivalent at this time, we're going to calculate the Norton zero equivalent for this circuit again. We're choosing our to the two ohm load resistance as our load. So we're going to now replace everything else in the circuit with the Norton equivalent, which is a current source in parallel with its internal internal resistance are Norton. Remember that a current source is a component whose job it is to provide a constant amount of current putting as much or as little energy necessary to maintain a constant current flow. That means the voltage may rising on blower, but the current will remain constant. As with David and serum, everything in the original circuit except the load resistance has been reduced to an equivalent circuit that is simpler to analyze. Also, similar tooth even INS serum are the steps used in Norton's zero to calculate the Norton source. Current I. Norton and the Norton Resistance are Norton. As before, the first step is to identify the load resistance and remove it from the original circuit. Then to find a Norton current for the current source of the Norton Equivalent circuit, place a direct wire or a short connection between the load points and determine the result . In current note that this step is exactly opposite the respective step in David in serum, where we replace the load resistor with a breaker and or people and open circuit with zero voltage drop between the load resistor connection points the current through our one is strictly a function of be ones voltage and ours resistance. Seven Amps and I is equal to e over r and likewise, the current through our three is now strictly a function of B two's voltage and our threes resistance seven amps and we can use again those law I equals e all over our. The total current through the shore between the load connection points is the some of these two current seven amps, plus seven amps is equal to 14 amps. Therefore, the Norton source current I Norton in our equivalent circuit is 14 amps. To calculate the Norton Resistance, we do exactly the same thing as we did for calculating the David and Resistance. Take the original circuit with the load resistors still removed. Remove the power sources in the same style as we did with a superposition theorem. Voltages are faulty. Sources are replaced with shorts and current sources are replaced with breaks and figure out the total resistance from one load connection point to the other. And our Norton is point eight owns now. Our Norton equivalent circuit looks like this. If we reconnect our original load resistance of two owns, we can now analyze the Norton Circuit as a simple, parallel arrangement. We know that the resistance is in this circuit, our 0.8 homes for the Norton Equivalent and two homes for our load resistance. We know that the current flowing through the Norton current generator is 14 APS. We would like to know what the voltages across our parallel loads. In order to do that, we have to find the equivalent parallel impedance sort of the parallels resistance in this circuit. So it is our Norton in parallel with two homes, which works out to point 57143 owns, or you can call it 571.43 millions. Once we have that resistance calculated, we can then calculate our voltage drop because we know the current flowing through that resistance, and that will give us a voltage drop of eight faults across the load resistor and the Norton Resistor. Once we have the voltage drop, we can calculate using owns law. What? The current is in a load resistor, which is four amps. And if we wanted to calculate the occurred through the are Norton it would be 10 APS. So this makes calculation of the circuit very simple. The last thing we want to talk about in this chapter is source transformation and by source transformation. We mean the transformation of a power source from a voltage source to a current source or from a current source to a voltage source. Sometimes this becomes a matter of convenience or a matter of simplifying further calculations downstream. Regardless of why we want to do it, we can do it. We could do it on paper as long as we know how to convert one to the other. So let's start with a voltage source transformation. A voltage source would look like this. The only source being vi vel. So we know what the voltage would be. And it would have an internal resistance of our subscript e again unknown quantity. What we want to do is we want to have a current source. Now that would be equivalent to the voltage source if we were to use it in the circuit. So we would replace the voltage source with a current source in parallel with a internal current resistor. No, Will we have to determine is what that I is for the current source and what the r subscript I As for the internal resistance of the current source and we do this using Norton steering the Norden serum state said, if you want to find out what the equivalent, uh, resistance is in this case, we want to find out what are some script I is. We remove the voltage source and replace it with a short. So we would then measure the the impedance or sorry, the resistance of the circuit. And that would be simply RV. So RV would equal. Are I? We're part way there now, in determining what the current is, we again use Norton's, uh, theorem and to find out what the current is in a Norton equivalent circuit is shorting out the terminals and measuring the current that would flow. In this case, it would be simply determined by owns law of the amount of current being drawn by RV when it the terminals air shorted wind driven by the voltage V. So the current of our current source would be V all over sub script or are such a key Now we have a way of determining what I and R is another in her current source. So we now have a way of converting a voltage source into a current source. If we want to convert a current source into a voltage source, we take the same steps. Only this time we're going to be used in a favorite and equivalent circuit are current Source would look like this, of course, and we want to end up with a voltage source that would look like this. So we have to be able to calculate with R V and V S in terms of I and R sub script at I s. So we start out by using saving equivalent and they say, replace the sources with their equivalents. Uh huh. She shorts are opens, and in this case, it's a current source. So where you replace it with an open and measure the resistance and the resistance is our subscript I so no big surprise. Here are some script. I is going to be equal to our some script. The So now we want to terminate determine what V is. And it's simply what would be the voltage across the terminals if we let the If the we let the source in this case source current run through the internal resistance in this case, it is simply again alms law. So the V would be I times are subscript I So we have a way of converting ah, current source to a voltage source. And these are the formulas that you'd use. As you may know, our have guessed by now there's more than one way to analyze an electric circuit. We're gonna look at one now that is a very powerful but a very simple method of analyzing a on electric circuit. It uses only flaw Kirchoff voltage law and a superposition. The're, um, all at once to convert what is an electrical circuit to a mathematical equation that if you follow simple rules, you come to the actual answer. I'm gonna go through the theory first very quickly, and then I'll analyze some circuits so you'll get a feel for this mesh current or loop analysis. Mesh current, uh, or loop analysis by virtue of its name analyzes a circuit that looks essentially like a mesh, and a mesh has several loops in it. In this case, this particular example has three loops in it. We're gonna call him Loop Number one, Luke number two and Loop number three. It goes without saying that there is going to be current flowing in this circuit because of the voltage supplies or the batteries that are on the left hand side of the loops. And if we consider current flowing in each of the loops, I'm going to make an assumption right now that the current flowing in Loop one looks like this, the current flowing in loop to looks like this and the current flowing in Luke three looks like this. Now. The beauty of doing mesh current or loop analysis is once we've made the assumption of currents flowing in the direction the currents are flowing in, we very quickly come to a set of mathematical equations, which we can analyze and come up with the values for the currents. As I said, the beauty of this is if we assume the currents are flowing in the wrong direction than the answer we get for those currents will come out negative, and we'll see that through the mathematical analysis. So I'm going to assume my current is flowing in a clockwise direction in each one of the loops, and you could make the assumption that there, flowing in other directions or opposing directions if you want it doesn't matter because the mathematics will look after it. For now, I'm assuming I one is flowing clockwise and I two is flowing clockwise and I three is flowing clockwise. The part that is the superposition, the're, um, is that we can analyze each one of the currents independently of the other. That's what the superposition tearing tells us. So we will look at each one of the currents independently and then do our analysis. If current flowing in, I one looks like that, Then we're going to get a voltage drop across the bottom. Resistor there, as I've indicated, plus two minus and the resistor in the top of the three meshes because of the current I two flowing in that direction will set up a voltage drop from positive to negative as indicated in the diagram and in a loop three, the outermost resistor because of the current I three will have a voltage drop as indicated plus two minus in the diagram. Now looking at the common resisters, the common resistor between I won and I to the current flowing in that resistor will be I one minus I too, because I too, is flowing in the opposite direction of I one. And again, it doesn't matter whether I one is greater than or less than I too. Because if it turns out that I want is smaller than I to than our answer will come up such that I one minus I two is negative and I'll know the direction is wrong. But for now, because we're assuming I won flowing clockwise and I two is flowing clockwise. The direction of the current I one minus I two is left to right and the current flowing in the common resistor between I two and I three is similarly I tu minus I three and they re the current flowing in the common resistor between I won and I three is going to be I one minus I three. Once we have assumed the loop currents. I one i two and I three. We then right, the Kirchoff Voltage law equations for each of the loops. Now, remember, Kirchoff Voltage Law states that the voltage drops around a circuit I must add to zero. So we will go around each loop using the assumed current flows of I one i two and I three and write down the voltage drops for that particular circuit using that particular current . Remember, now we're gonna analyze each current independently and then add them together. That is what the superposition zero dictates. So the rules for voltage drops are the voltage drops are positive and across each resistor and are given by homes law. So the voltage drops on the outermost resistors will have a positive to negative voltage drops as indicated, because the current's flowing through them are in the direction as indicated in our diagram . When we come to the power supplies which are voltage supplies, the voltage will be rising and voltage rises are negative and are equal to the voltage source ratings. In other words, if they're 1.5 volts, they will be 1.5 volt voltage rise. If they are nine volts. It'll be nine volt voltage rice or 12 volts or whatever the battery voltage is. If the batteries are in the reverse direction to the current, such as that the voltage is no longer a voltage rise. But it will be a voltage drop from positive to negative and therefore the voltage drop will be positive. However, in our case, the battery is in that direction, so it will be considered a voltage rise and it will be naked. Now the voltage drops across the common resistors that is the resist er's, with two currents flowing in them are also given by homes. Law V equals I times are and are positive. The polarity is according to the loop current being followed. In other words, if we're following, I won. The polarity will be positive to negative left to right. If we're following eye to the common resistor will have voltage drop positive to negative right to left. The current in the OEMs law equation is the arithmetic. Some of the two currents flowing through it, the loop current being followed is positive. The other current is positive or negative, depending on its direction. Off flow with respect to the loop current being followed. Now that's a mouthful. What that means in our diagram is that you will see that in a commoner sister between I won and I to that I too opposes. I won because they're both flowing in each of the respective loops in a clockwise direction . So I two flows in the opposite direction off I won in the common resistor. If one or the other current was flowing in the other direction, then it would be positive, not negative. Let's have ah closer Look at that in our example. And we can see that if we're following current I one in the current loop. The resist er in the bottom of that loop is pretty simple. We see that the current flowing through it will set up a a, uh, positive voltage drop from right to left. The voltage drop across the common resist er is also going to be positive to negative because we're following. I won and it is also going to be given by homes law. However, the current in our homes law is going to be made up off I one, which is positive and because the current, The other current flowing in that resistor is due to eye to eye to is flowing in the opposite direction to I one so that the current is given by I one minus I to now remember, we have arbitrarily assumed that the current flowing in the direction in each of those two loops are clockwise. If when we're finished doing our analysis, we have chosen the wrong direction for the current, the current values will be negative and we'll show you that in an example. However, in the logic that we're setting out here the current flowing in a common resistor between I one and I two because we are following the current in I one, the current in the common resistor will be I one minus I two and a voltage drop is considered positive. So the last sentence in that paragraph, the loop current being followed is positive. The other current is positive or negative, depending on the direction of its flow. If I too, was flowing in the other direction, it would be positive because the current flowing in the commoners sister would be made up of I one plus I to But they're flowing in opposite direction. So it's going to be I one minus I to. So let's apply some values to our example here and work through the mesh analysis for this circuit. Now remember, we have assumed three circus or three currents flowing in each of the three loops. So ultimately, in analyzing this circuit, we want to know what the three currents are. I one i two and I three. Once we know what the's loop currents are, we can calculate everything else that's associated with this circuit. In other words, we can know it all. The current flowing in all of the branches of the circuit and all of the voltage drops a cost cross. Each one of the resistance is going to be So we're going to, as I said earlier, convert this electrical circuit to a set of mathematical equations that will allow us to solve for the unknowns, which are the three currents in this case. So we're going to set up three equations in three unknowns. And once we got three equations in three unknowns, we will have the capability of analysing for the unknowns, and we come across these equations by writing Kirchoff voltage law around each loop for each one of the currents. Starting with I won, I won will be flowing in Loop one and it's going to set up voltage drops across the resistance as indicated in the diagram. Now remember, we're falling back on the superposition through, which allows us to analyze each courage separately. So now I'm looking at the Kirchoff Voltage Law or equation four. Current number one, the current that's flowing in the common resist er the five ohm resistor is going to be made up of I one and I two and because we are following I one, it's going to be I one minus I two and the voltage drop again. According to following the loop of I one is from left to right positive to negative in the common resistor, the forum common resistor between loop one and three. The current flowing and not resistor is going to be made up of I one and I three. But because we're following the loop or the Kirchoff Voltage law around the loop for I one , the current or the voltage drop across the four home resistor is going to be due to the current flowing I one minus I three. I'm now going to set up the equation. Four Kurt shops Voltage law in loop one. According to the current, I won and we will start. We could start anywhere in the loop. But the first thing I'm going to start with is the power supply, the 18 volt power supply. And remember our rules. It is a voltage rise, so it will be minus. So we have minus 18. Well, then come to the common resistor the five, the five ohm resistor. And remember, the voltage drop across that resistor is going to be V is equal to I times are the eye that we're looking at Is I one minus I too. So it's going to be five homes times I one minus I two is the voltage drop across the five home common resistor. The voltage drop across the forum Common resistor is going to be again given by homes law and it is going to be the current in the four ohm resistor times the resistor which is four times the quantity I one minus I three and the voltage drop across the one own resist er is only due to the current I one and it again is going to be given by homes law. But it is only due to one current. So it's one home times I one. And of course, Kirchoff states that the voltage drops around that loop must add to zero. So that has given us our first equation. Now we can manipulate this equation. Now we're dealing with mathematical terms. Weaken. Almost forget the fact that we're dealing in electrical values because we just have numbers and unknown quantities and the out There are three unknown quantities in that equation. I one I two and I three. So I want to arrange the values in that equations when going to expand the brackets. And the equation will look like this minus 18 plus five times I one minus five times I to plus four times I won minus four times I three plus one times. I one is equal to zero. Now I want to collect the light terms in terms of our unknown. Starting with I won and going through I two i three, we will set up an equation that has 10 I one minus five I to minus four I three is equal to 18. I've taken the the definite value of the power supply which was minus 18 on the left hand side of the equation. Just moved it to the right hand side of the equation. And in doing so, the sign changes. Of course. So we have three unknowns in this equation. We're now going to look at the current I to in the loop number two and the voltage drops across the components of that loop are indicated in the diagram. Now, remember, according to the superposition, the're, um we can look at Onley I to in terms of how it is going to react to the circuit. And then once we're finished with I two and I three, we can add them together, superposition wise and come up with the answer. But right now we want to look at Onley. I too and I too will set up voltage drops in that loop. As you see in the circuit here, the voltage drop in the common resistor, the five home common resistor with loop one. The current in that resistor is going to be made up of I to of course flowing through it. But we're gonna have to subtract off. I won and I won Is flowing in the opposite direction to I too. So the current according to the I to value, is going to be I to minus I want because we are evaluating the I two circuit. So the current flowing according to I to in the commoners sister five home common resistor is given by I two minus I won now, looking at the three home common resistor that current flowing in it is going to be made up
9. Ch 08 Electric Fields and Capacitance: Chapter eight electric fields and capacitance whenever an electric voltage exists between two separated conductors and electric field is present within the space between those conductors in basic electron ICS, we study the interaction of voltage current and resistance as they pertain to circuits, which are conductive passed through which electrons may travel. When we talk about fields, however, we're dealing with interactions that can spread across empty space. Admittedly, the concept of a field is somewhat abstract, at least with electric current. It's not too difficult to envision tiny particles called electrons moving their way between the nuclei of atoms within a conductor. But a field doesn't have mass and need not exist within matter at all. Capacitors air components designed to take advantage of this phenomenon by placing two conductive plates, usually metal in close proximity to each other. There may be different styles of capacitors construction, each one suited for particular ratings and purpose for very small capacitors to circular plates. Sounds between insulating material will suffice for larger capacity or values. The plates, maybe strips of metal foil, sandwich around flexible insulating medium and ruled up for compactness. The highest capacitance values are obtained by using microscopic thick thickness of layers of insulating oxide separating two conductive services. In any case, though, the general idea is the same. Two conductors separated by an insulate ER. So simply put, capacitors are devices for storing electric charge. Basically, capacitors consists of two metal plates separated by an insulator. They the insulator is called a die electric, and it's made up of, say, Polly Styrene could be oil air Micah. There several insulating materials out there that they use for the dialectic of a capacitor . There are a few things out there that affect a capacitor. One of them is the plate area. All other factors being equal, the greater the plate area gives greater capacitance in the less plate area gives less capacitance, larger plate area results in more field flux. That is the charge collected on the plates for a given field force, which is the voltage across the plates. Now we're gonna talk more about flux fields in a few slides, but I just wanted to talk about the factors affecting capacitance right now. Plate spacing, further plate spacing gives less capacitance, and closer plate spacing gives greater capacitance closer. Spacing results in a greater field force, which is a factor of voltage across the pastor, divided by the distance between the plates. This results in a greater fueled flux, which is a function of the charge collected on the plates for any given voltage applied across the plates. And, as I said, we'll talk more about field flexes in a few more sites. Di electric material, all insulating material used in capacitor, is rated. Asked to its primitive ity, primitive ity is a measure of resistance that is encountered when forming an electric field in a medium. In other words, permitted, Vitti is a measure of how elect field effects or is affected by die electric. Medium, greater primitive ity of the dialectic gives greater capacitance. Last Perm utility of the dialectic gives less capacitance, although it's complicated to explain. Some materials offer less opposition to field flocks for a given amount of force. Field materials with greater permitted Vitti allow for more field flux. Others offer less opposition, and thus a greater collected charge for any given amount of field force. Applied voltage relative permeated Vitti means the permitted Vitti of a material relative to that off a pure vacuum, the greater the number the greater their primitive ity of the material glass. For instance, with the relative permeated, Vitti of seven has seven times of permitted Vitti of a pure vacuum, while air has almost the same primitive ity of a pure vacuum and consequently will allow for the establishment of an electric field flux stronger than that of a vacuum, all other factors being equal. Here's ah list of relative permitted Vitti of various di electrics from compared to a vacuum which relative primitive ity is air is 1.6 which is almost the same as a vacuum. Polyethylene is over two times what a vacuum is. Walks paper is almost three times. Micah is 5.4 times, Glisson is 43 pure water is 80 and strode. IAM tight treat is 310 times, and there are various assortments of capacitors out there that are constructed in many different ways and have maybe more than one leads coming and going to them. And they are designed for special purposes, and others are designed for quantity and being able to use in various applications. Anyway, these air only a few of capacitors that are out there and these are what they look like. When a voltage is applied across two plates of a capacitor, they concentrated. Field flux is created between them, allowing a significant difference off free electrons, a charge to develop between the two plates. As the electric field is established by the applied voltage, extra free electrons are forced to collect on the negative conductor while free electrons are robbed from the positive conductor. This differential charge equates to a storage of energy in the capacitor representing the potential charge of electrons between the two plates. The greater the difference of electrons on the opposing plates of the capacitor, the greater the field flux and the greater charge of energy that will be stored by the capacitor. Because capacitors store the potential energy of the community, it electrons in the form of an electric field. They behave quite differently than resistors, which simply dissipate energy in the form of heat in a circuit. Energy stored in a capacitor is a function of the voltage between the plates, a capacitors ability to store energy as a function of voltage. Potential difference between the two leads results in a tendency to try to maintain the voltage at a constant level. In other words, capacitors tend to resist changes in voltage, whether it's drop or an increase when voltage, when the voltage across a capacitor is increased or decreased the capacity. Resist the change by drawing current from the supply, supplying current in the source of a voltage charge in the opposite or in opposition to the charge. To store more energy in a capacitor, the voltage across it must be increased. This means that more electrons must be added to the negative plate and mawr taken away from the positive plate, necessitating a current in that direction. Conversely, to release energy from a capacitor, the voltage across it must be decreased. This means some of the excess electrons on the negative plate must be returned to the positive place plate, necessitating a current in the direct in that direction. No, we would like to define capacitance. Capacitance see of the capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductor's. In other words, a capacitor is greater has greater capacitance. If more charge is accumulated using less voltage, the S I unit of Capacitance is a fair ad, and a fair at is a very large unit. So common units out there. Yeah, the micro fair ad, which is equal to 10 to the minus six fare ads. The Nano Fair ad, which is 10 to the minus nine fair ad, and a PICO Farid, which is equal to 10 to the minus 12 fair acts. Practically speaking, however, capacitors will eventually lose their stored voltage charges due to internal leakage paths for electrons to flow from one plate to the other, depending on a specific type of pastor. The time taken for stored voltage charged to dissipate can be a long time, even several years, with capacitors sitting on a shelf. When the voltage across capacitor is increased, it draws current from the rest of the circuit, acting as a power load. In this condition, the capacitor is said to be charging because there is an increased amount of energy being stored in its electric field. Conversely, when a voltage on a Kappa across a capacitor is decreased, the capacitor supplies current to the rest of the circuit, acting as a power source. In this condition, the capacitor is said to be discharging. It's stored. Energy held in the electric field is decreasing now as energy is released into the rest of the circuit. If a source of voltages suddenly applied to an uncharged to pastor, a sudden increase in voltage the master will draw current from that source, absorbing energy from it until the capacitors voltage equals out of the source. Once the capacitors voltage reaches, this final charged state its current DK's 20 Conversely, if a load resistance is connected to a charge capacitor the source of of voltages suddenly removed, the capacitor will supply current to that load until it has released all its stored energy and its voltage d case 20 Once a capacitor voltage reaches this final discharge state, the current DK's 20 in their ability to be charged and discharged to pastures can be thought of its acting somewhat like secondary cell batteries. The choice of insulating material between the plates, as was mentioned before, has a great impact upon how much field flux and never how much charge will develop, with any given amount of voltage applied across the plates. Of course, when this charging takes place, there's always some resistance in the charging circuit, even if it's only the resistance of the wire connecting the supply voltage to the capacitor , as you can see in the little diagram that I have about the top there. The supply voltage, of course, is the battery and the resistance. Whether it's a boat piece of resistance or the wire is showing in the diagram in Siris with the capacitor, that resistance will determine the speed at which it capacitor charges, along with the size of the plaster. Of course, the mathematical formula for this charging curve and its plot of charging voltage versus time is showing here the is the capacitor or the voltage across the capacitor. V S is the supply voltage or the battery in this case, and our is the resistance of the circuit, See is the capacitance. And he is a well known mathematical constant. That fellow the development of this, this formula. And it's just a number, and the number is approximately 2.7182818 to 8, and it goes on forever. But that's a good enough approximation. You can see that at toying equals zero e to the minus zero is one. Therefore, V is zero at infinity after a long period of time. Of this charging, um e to the infinity is zero. So the voltage across the capacity, of course, just equals the supply voltage, which is kind of intuitive. Anyway, we knew that at the beginning. Also because the charge on the plates is directly proportionately voltage across the plates than this curve also describes this charging or the rate of charge of the that that is on the plates. Of course it has to be scaled, but it the curve is it's the same curve in calculating the charge on the plates. Now the current flowing to the capacitor will fall off over time, and eventually it will go to zero as the capacity is fully fully charged. But this curve describes the current too the capacitor and you can see yet at a time, zero The term is I is equal to I not, which is the current at the beginning of the when they're switches first closed to the to the discharge pasture and it starts the charge, the current will be fully at time, zero fully dependent on on Lee the resistor and the voltage of the battery, which we could just use homes law to describe, which is the supply voltage over the resistance of the circuit. But immediately, once it starts to charge, of course, the current starts to fall off, and as it approaches infinity or after a long period of time, the current will be zero, as I said, and the that will happen when the capacity is fully charged and the voltage across capacitor is equal to the voltage of the supply. Let's see what happens when we connect UPS capacitors in series. As we see here, we have C one, C two, C three and Siri's with a D C Supply voltage. As the voltage is applied to this series, circuit charges will tend to flow and migrate. And what happens here is the charges will be equal. In other words, looking at C one, the positive charge on C one plate is going to be equal to the positive, Uh, charge on C two is equal to the positive charge on C three. It is also equal and opposite in charged to the opposite plates. If they were not equal, they would tend to push charges until they were equal. So, logically speaking, all of the plates have to have an equal amount of charges on it. So if we were to define a term off ah capacitor having charges, then the total charge across the total three capacitors is given by the Plaza, plus charges on the C one plate and the negative que on the C three plates so that we might say that Q T is gonna be equal to Q one, Q two and Q three. If that was not the case, then they would. You'd have a buildup of charge in a wire, and it just can't happen. So we can say that Q t is equal to Cuba is equal to cute. Two is equal to Q three, which is the charging off the plates. Now, according to kerchiefs Voltage law. Around the loop, the voltage drops V one v two V 3/2 to equal the supply voltage. So you have the total voltage supply. Voltage is equal to B one plus B two plus three. Now remember that a capacitor is defined as the charge over the voltage of that capacitor. We can rewrite that equation now, so we can say the voltage across. Each capacitor is going to be aware of the total capacitance even is to all oversee. So we can now rewrite our voltage equation that we have from Kirchoff Voltage Law in terms of the charges and the capacitors. So what we have is a total charge. Q t all over. See, which is the total capacitance of the series circuit is equal to the sum of the individual charges over the individual capacitors or Q Teal over C is equal to Q one. Oversee one plus Q to oversee two plus Cube three all over C three. Now we know that Q T for the total is the same as the individual charges on each individual capacitors. We've already gone through that logic, so we can replace Q one Q two Q three with their equivalent Q T. So the equation now becomes Q t all over. C is equal to Q T Oversee one plus Q t. Two t all oversee, too, plus Q t all over C three. And if we divide both sides of the equations by q t, the equation still holds true. What says one oversee is equal to that which is a total capacitance is given by one oversee one plus one over C two plus one over C three. So you can now calculate what the result of connecting three capacitors up in Siris is given by the this equation that you see in front of you, which is not too unlike what we get with resistors, where we could connect them up in parallel. But in this case, we're connecting the capacitors up in Siris. And this is how you calculate the total capacitance of the three capacitors. Okay, let's see what happens when we connect capacitors up in parallel, as we see here In the diagram, C one C two C three are connected in parallel across the D C supply voltage, which has a voltage V. We know that the definition of capacitance is given by Q. The charge of the capacity to charge in the capacitor over the voltage drop across that capacitor in our diagram here we if we designate the some of the capacitance of the three capacitors as C sub script T, then it's going to be equal to the total charge over the voltage and the voltage across each capacitor is the same, and that's equal to the supply voltage. We know that the total charge is going to be made up of the charge of capacitor one capacitor to capacitor three. So we have que UN plus Q two plus Q three all over V would be the capacitance of the three capacitors. In parallel. We can rewrite that equation and break out 23 individual quantities where Q one is now over V plus cute to over V plus Q three over V, which is the same equation that is above. Except in this case, you can see that Q one over V is the capacitance of C one. Que two over V is the capacitance of Seat two and Q three over V is the capacitance of C three, so we can then re write the equation such that the total capacitance is given by C one plus C two plus C three, which is similar to a Siri's circuit off resistors, where you would add up the resisters to calculate me the total resistance in that circuit. In the case of a parallel connection of capacitors, you add the capacitors up before finishing off the chapter, I thought we just have a quick look at some of the different uses off capacitors that are out there, uh, in the tuning of radio stations, and they're still some being used. There are still some using this method today, although it was more prevalent in the days gone by in a two radio with a used to call a super hatred ein radio. They had meshing of plates that varied the capacitance of the tuner, and these plates would actually move in close to each other and away from each other. So you'd vary the distance between the plates as well as the area of the plates as you turned the knob. And that would actually vary the capacitance, which was in the tuned circuit, and you could tune in the various radio stations that way. Another use even today is in the keyboard. Some keyboards have Each key is connected to a plate, a small little plate. Majid press the key down. You very the the capacitance of that individual button. And there's a receiver that converts that capacitance into a signal that recognizes as that particular key that you pressed and converts it into what is required by the computer. Another application that has been used up there is the condenser microphone, and the plate of a microphone pickup would very as the sound waves would hit it and as varying the distance between two plates you and vary the capacitance. And that would be detected as a signal, which could be amplified and used for various electronic devices, such a recording or talking over distances, etcetera, etcetera. Another application is the electronic flash for cameras, these air nothing more than a great big capacitor that, uh, once it's charged, it will then have work as a large power supply. And it would actually when the shutter is pressed, the there would be a big dump of the of the power or the electric electric charge on the capacitor into the light tube, and that would flash and give you the the required lighting for a flash circuit. So this ends Chapter eight
10. Ch 9 Magnetism and Inductance: Chapter nine magnetism. In the last chapter, we discussed and developed what was known as electric field that was set up by a new electric charge on to plates and the field existed in between the plates. Now we we hypothesized that that was the field, but it really you can't see it. You can't touch it. You can't feel it. But we know how it works, and we studied the characteristics of it. We can develop formulas and and do some predictions based on those on that modeling. The same thing happens in magnetic feels, and there are two types of magnetic fields. There's ah ah, permanent magnet that sets up a field in. These kids play with magnets all the time, and and there are magnets that are used in industry and in tools, etcetera, etcetera. And there's also electric fields, magnetic fields, which will we'll discuss in a in a few more slides. But I'm gonna look at permanent magnets just for now because it demonstrates what a magnetic field or so I should say what the characteristics of a magnetic field are. And then we will, uh, transposed those over to electric magnetic fields because they act in the same way a magnet is surrounded by a magnetic field of magnetic field exists a force on other magnets and objects that are made of metal. The magnetic field is strongest close to the magnet and weaker. The farther away you get. The magnetic field can be represented by lines of force or magnetic field lines. A magnetic field also has a direction. The direction of the magnetic field magnetic field, round of bar magnet is no is showing with arrows, and the magnetic poles are at either end of the bar magnet. No, what you see here is a couple of magnets the top one is referred to is a horseshoe magnet. For obvious reasons. The two ends of a horseshoe magnet are north and are designated north and south poles. They are connected by Magnante lines of flux. The magnetic field lines always connect the North Pole and the south pole of a magnet. If we were to place a compass, uh, close to a bar magnet, you would find that the north Pole of the compass points in the direction of the magnetic field. This direction is always away from the north magnetic pole towards the south Magnetic pole in electromagnetism, there is a magnetic field that is produced by, um, electric current flowing in a wire. Now remember, we can't see these fields when you can't feel them, but we can observe their effect on other magnetic fields and and other things. And the magnetic field of I produced by a current flowing in a wire is perpendicular to the way er and is imagine to form rings around the, um the way er itself as current is flowing in it, the direction of the magnetic lines of flux are given by what we call the right hand rule. If we place our right hand on the conductor and their thumb is pointed in the direction of the current flow than a magnetic field forms, uh, concentric circles in the direction of your fingers, as demonstrated in the diagram here. And, as I said, the magnetic field in circles, the wire and the flux lines have no north or South pole. They are continuous lines of flux that are formed around the wear. So far, we've been banking about the terms of flux lines of flux in the force fields, and that's been okay up to now, when we're only dealing with the singularity things and having talked about formulas. But we need to define them in quantitative terms, which we're going to do here so that we can go forward and develop formulas for use in electromagnetic fields. When analysing, Meghni feels we need to further to find them as being made up of magnetic lines of flux. And a flux line is demonstrated here, and it's again we can't see them, but we can imagine them going, uh, through a particular surface. Grouped together, these magnetic a lines of flux form what is known as a magnetic flux field and the Greek symbol Fi is usually I used to symbolize magnetic flux field, and it's a measure of quantity of magnetism. It takes into account the strength and the extent of the magnetic field. If we consider a flat area such as what is in yellow there, um, that the magnetic flux passes through, then we can define magnetic flux density as the amount of magnetic flux in an area that is perpendicular to direction of the magnetic flux or, in other words, magnetic flux. Density is the amount of magnetic flux perpendicular to given flat surface area divided by that area, and it's usually defined by the capital Greek letter beta. But beta is very close to our capital Letter B, so quite often you'll just see the letter B as designated designating flux density and the international symbol uh, used for for measuring the quantity of flux. Density is the Tesla, and one, Tesla is equal to one weber per meter per square meter. And quite often we don't use the term test slow. Quite often, we're just going to use the term weber per square meter, and you'll see that happening often. So to summarize, here's what we will be using going forward for discussing magnetic flux and magnetic flux Density. Magnetic flux is usually symbolized by the Greek letter Phi, and it's usually measured in Weber's magnetic flux. Density is be or beta, and it's measured in Tesla, which is equivalent to Weber's per square meter to create a stronger magnetic field force and consequently mawr field flux. With the same amount of electric current, we can wrap the wire into a coil shape where the circling magnetic fields around the wire joined together to create a larger field with a defined magnetic north and south pole polarity. The amount of magnetic field force generated by the coiled wire is proportional to the current through the wire, multiplied by the number of turns or wraps of that coil wire. The more wraps, the more turns, the larger the magnetic field that will be produced. This force field is called Magna Motive Force MMF and is very much analogous to electro Motive Force, which is voltage in electric circuit. MM F is measured in an pair turns the poles of a magnetic field in, ah, a coil of wire or Solon oId, which is called can be determined again by the right hand rule. Imagine your right hand gripping the coil of the solid, annoyed such that your fingers are pointing in the same way that the current is flowing. Your thumb then points in the direction of the field. Since the magnetic field is always coming out from the North Pole, therefore, the thumb points towards the North Pole. Here we have wrapped our coil around a block of iron, which I will refer to as core material. The nature of this material is such that the magnetic lines of flux want to travel through it rather than the air. In fact, it will boost the amount of flux that is produced by the current in the coil. In this example, the magnetic flux will still very as occurred in the coil, but at a greater extent due to the core material. The magnitude of the magnetic flux will also depend on the number of turns of the coil. As I said, the magnitude of magnetic flux will also depend on the material through which the flux will travel. If an iron core is substitute for an air corps in a given coil, the magnitude of magnetic field is greatly increased. All materials have a property to find as permeability, which is the measure of the ability of the material to support the formation of a magnetic field within itself. Hence, it is a degree of Vanga tis ation that a material obtains in response to an applied magnetic field. Magnetic permeability is typically represented by the Greek letter mu permeability, also called magnetic permeability, is a constant of proportionality that exists between magnetic induction due to the current flow and the magnetic field intensity of the or the amount of flux that will be produced. This constant is equal toe, approximately 1.25 seven times 10 to the negative six Henry's per meter in free space or a vacuum in other materials. It can be much different, often substantially greater that in free space, which is symbolized by the Greek letter mu with a sub script. Zero materials that cause lines of flux Tim further apart, resulting in a decreased amount of magnetic flux density compared to a vacuum are called dia magnetic materials. Materials that concentrate the magnetic flux by a factor of more than one but less than or equal to 10 are called para magnetic materials. And most importantly, the materials that concentrate the flux by a factor of more than 10 are called ferro magnetic. The permeability factors of some substances change with rising and falling of temperatures or the intensity of the applied magnetic field. In engineering applications, permeability is often expressed in relative rather than absolute terms. If mu subscript zero or we call it you not represents the permeability of free space and Mu represents the permeability of the substance in question, then the relative permeability you mu subscript R is given by this equation, the permeability of material and its capability of either supporting or resisting a magnetic field is called reluctance or magnetic resistance. The reluctance of a magnetic material is proportional to the mean length of that material and inversely proportional to the cross sectional area. And, of course, the permeability of a magnetic material itself. It gives rise to the equation. R is equal to L over mu A Where are is the reluctance or magnetic resistance of the material. Mu is the mayday permeability coefficient. L is the length of the material in meters and A is a cross sectional area of that material in square meters. So looking at our coil again on our block of iron, the iron is a ferro magnetic material, and it's said to have a reluctance are to the flocks that is being trying that is being formed by the current. The flux current in reluctance are related by this equation, and I is equal to the flux times, the reluctance. So if we tried, if we increase the current, we're gonna increase the amount of flux. If we increase the turns, we're gonna increase the amount of flux. If the term the reluctance goes up for the same amount of current in turns than the flux density would have to go down. So the reluctance actually provides some resistance to the formation of magnetic flux lines in the iron core. Remember, now that the N I is termed Magna Motive Force, or MMF, and it's measured in impair turns. Sometimes it's helpful to draw the analogy between the electric circuits and the magnetic circuits, where the driving force, the E. M F for voltage, is related to the mmf of driving force of a magnetic circuit, which would produce, in the case of the electric circuit, a current and, in the case of a magnetic circuit, flux lines of flux. And the thing that would be limiting it would be, in the case of an electric circuit, the resistance. And in the case of a magnetic circuit, you have reluctance that would oppose the formation of the magnetic circuit in the ideal situation where we energize a coil of wire in air or in a vacuum. The flux produced is linearly related to the input current, hence the flocks and the flux. Density is also linearity related to the input current. If we plot this fact on a magnetize ation curve which we call a B H hurt, which plots field intent, field density versus field intensity or flocks versus the current, the field intensity being returns, racial times occurred. But essentially we're looking at flux related to input current. We have a straight line, so if we increase the current, then we increase the flux and it's goes up is a straight line relating one to the other. In a really world, the core material is not linear, as you can see here for three plots of she'd steal, cast fueling and cast iron. As we increase the current, it starts to go up linearly. But as we go up further and further, then the material becomes what we call saturated and as it becomes saturated, you can no longer pump anymore or produce anymore flux lines, no matter how much current you pump into it. So we end up with these curved lines which have what they call a knee point, as well as a saturation level as to what we could rise to another court to come phoned our analysts of magnetic flux versus force is the phenomenon of magnetic histories. Is, history says means a lag between the input and the output. In a system upon change of direction in a magnetic system, history says is seen in a ferro magnetic material, it tends to stay magnetized after the Applied Field force, which is the current, has been removed from the circuit. Let's use the same graph again. Onley. We'll extend the accents in the negative direction as well, because we're going to reverse the current into the negative direction and see what happens . First, we apply an increasing field force or current through the coils of our electromagnets. We should see the flux density increase. In other words, school as we go up towards the right according to a normal magnetize ation curve for ferro magnetic material. Next we'll stop the current going through the coil of them of the electromagnet, and what happens is the current returns to zero. But the pharaoh magnetic material still maintains some magnetism because that's a characteristic of ferro magnetic material. Now let's slowly a pie, the same amount of magnetic field force or current in the opposite direction. The flux density has now reached a point that's equivalent to what it waas in the full positive value. Ah, field intensity except in the negative direction and stop the current flow again going through the coil. And once again due to the natural retentive ity of the material, it will hold a magnetic flux with no power height coil. Except this time it's in the direction opposite to that of the last time when we stop current flowing in a coil. If we re apply power in a positive direction again, we should see the flux density reach its prior peak in the upper right hand corner of the graph again. So if we keep doing this back and forth with the current ad er applying it in the positive direction than taking it off and applying it in the reverse direction than taking it off and applying a forward direction, we will see this s curve developed, which is called the histories curve of magnetic material. And this doesn't necessarily have way. We did it with D. C current switching it on and switching it off in reversing it. But this indeed is exactly what happens when you were dealing with alternating current and I'm not gonna go into a lot of details now. Vote alternating current because we got another a lesson or a lesson in a sea circuits which will be following at the end of this. At the end of this lesson plan, our modern world would be impossible without electromagnetic induction. The phenomenon that underlies the operation of many devices, including power station generators, microphones, tape recorders, car alternators, ignition system speedometers. I could go on forever. The list is infinitely long, but they all rely on electromagnetic induction in some way, shape or form. Qualitatively speaking, this is how elector magnetic induction works. What we have here is a circuit ah, loop of wire which is shown in black. They're connected to a volt meter that is our circuit. We also have the A magnet with magnetic lines of flux emanating from the North Pole going to the South Pole and they are going through the loop of our circuit. We say that those lines of flux are linked to our circuit because it's going through it. If the person holding that magnet doesn't move the magnet, nothing will happen. There will be zero reading on the Volt meter. However, as the personal starts to move that magnet inner out upper down than the number of lines of flux that are linked by our loop of our circuit loop will change. And it is that change that will induce a voltage in that circuit that is an electromagnetic induction. So quit quantitatively. We will introduce Fair Days Law Electromagnetic induction, which relates the magnitude oven induced voltage to the rate of change of the magnetic flux linking a circuit. Consider the situation showing here. The street wire P Q. Is moving at a constant velocity. Delta X at right angles to a uniform magnetic field directed to the right of the screen as it moves wears. P s and Q are are fixed and continue to make contact with the moving wear. As P Q moves to the right. The flux linkages in the circuit S P Q R changes. In fact, it reduces. We will observe a voltage induced between S and are a volt meter connected between S and are then measures the induced open circuit voltage induced between the ends of the moving wire. But it does not allow a current to flow in the circuit because it's a volt meter and it's just measuring voltage. According to Faraday's law, the flux change in this circuit will induce a voltage in the circuit. Fair Day's law states that the voltage will be induced according to the change of flocks que tu minus Q one over a period of time t to minus t one or, in other words, more concise terms. The is equal to minus and Delta Phi over Delta T, where V is the instantaneous induced voltage N is the number of turns off are linked circuit or coil of wire, and in this case, we only have one turn fires. The magnetic flux in Weber's Delta Phi is the change of magnetic flux, and Delta T is the change in trying that that flux changes. The other thing to note here is the minus sign, which is significant unease. E way to create a magnetic field of changing intensity is to move a permanent magnet next to a wire or a coil of wire. Remember, the magnetic field must increase or decrease in intensity perpendicular to the wire so that the lines of flux cut across the conductor or else no voltage will be induced. In other words, a circuit of wire or coil must experience a change in flux. A Delta Phi over a time Delta T. Faraday was able to mathematically relate the rate of change of magnetic field flux with induced voltage. This refers to the instantaneous voltage or the voltage at a specific point in time. The Delta's represent rate of change of flux over time and end stands for the number of turns or raps of the coil of wire. If two coils of wire brought into close proximity with each other so that their magnetic fields from one is linked to the other, ah, voltage will be generated into the second coil. As a result, this is called mutual induct INTs. The key here is that the induced coil experience the change influx over a period of time because of the changing flux in the primary coil. Faraday's law still holds true and can still be calculated if it's indeed able to measure it, because it still involves the changing magnetic flux over a period of time. Whenever electrons flow through a conductor, a magnetic field developed around that conductor. This effect is called induct us whereas an electric field flux between two conductors allows for an accumulation of free electron charge. Within those conductors, a magnetic field flux allows for certain inertia to accumulate in the flow of electrons through the conductor. Producing the field in doctors are components designed to take advantage of this phenomenon by shaping the length of the conductive wire in the form of a coil. This shape creates a stronger magnetic field, then would be produced by a straight wire, some in doctors air formed with wires around around self, a self supporting coil, others air wrapped around a solid core material of some type. Sometimes the core of the induct ER will be straight, and other times it will be joined in a loop, a square rectangle or a circle to fully contain the magnetic flux. These design options are all have effect on the performance, and the characteristic of the induct er's The schematic symbol foran in Dr like a capacitor is quite simple, being little more than a coil symbol representing a coiled wire. Although a simple coil shape is the generic symbol for any and DR in doctors with cores or sometimes distinguished by additional parallel lines to the axis of the coil. A newer version of the in Dr Simple Symbol dispenses with the coil shape in favor of several humps in a row. As the electric current produces a concentrated magnetic field around the coil, this field flux equates to a storage of energy represented representing the kinetic motion of electrons through the coil. The more current in the coil, a stronger the magnetic field will be and the more energy the in doctoral store. Because in Dr store, the kinetic energy of moving electrons in the form of may have a magnetic field, they behave quite differently than resistors, which simply dissipate energy in the form of heat. Energy storage in an in doctor is a function of the amount of current through it on in doctor's ability to store energy as a function of current results in a tendency to try to maintain current at a constant level. In other words, in doctors tend to resist changes in current when current through it and doctor is increased or decreased. Thean doctor resisted change by producing a voltage between the leads in ops on opposing polarity. To that change, looking at our simplistic circuit here of a resisting Siri's with a D C power supplier. A battery. The voltage drop across a resistor is given by Owns law, which is the product of the current times that resistor. It's pretty simple. Voltage drop across an in DR, however, is just a little bit more complex. Whenever current flows through a circuit or a coil, flux is produced surrounding it, and this flux also links the coil to itself. Self induced E M F in a coil is produced due to its own changing flux. And changing flux is caused by changing current in the coil so it can be concluded that self induced E. M F is ultimately do to changing current in the coil itself and self inducted. Its is the property of a coil or a cell annoyed, which causes a self induced ian meth to be produced when the current through. It changes whenever changing flux links with a circuit an E. M. F. Is induced in that circuit. This is thirties long. We've already seen that it's Faraday's law of electromagnetic induction, and according to this law, the voltage across the coil will be given by this equation, where V L is the instantaneous induced voltage and is the number of turns of the coil Delta Phi. The change in magnetic flux in Weber's Delta T is the change in time, and Delta Phi or Delta T is the rate of change of the flux linkages. The negative sign of the equation indicates that the induced E. M F opposes the change flux linkage. The flux is changing due to changing the changing current in the circuit itself. The produced flux due to a current in a circuit is always proportional to that current. That means that I is proportional to Fi, which I is the current, and Fi is the flux. We can make an equation out of this rather than just a proportionality. By saying that Phi is equal to K Times I where K is just some constant that we can use. Since the flux is changing over a period of time. Delta T. Then I will also be changing over a period of time. Therefore, Delta Phi over Delta T, we can say, is equal to K Times Delta II over Delta T. Sometimes this is referred to in calculus terms where they've replaced the Delta with the D , which just means the same thing is Delta. So any time you see d you can replace it with Delta Ah de, in calculus terms is a very small difference in quantity. And or some who's that you got to jump through. But we're not going to go there right now. Just a silly Every time you see D in this equation, it's it means Delta. So Delta Phi over Delta T is equal to K Delta I all over Delta t we can replace the the factors Delta Phi over Delta T in the Pharisees equation and we're left with the l is equal to minus cave which, by the way, is incorporating the turns of the coil. The L is equal to d I by DT and as I said, l would incorporate the constant plus the turns of the coil. No el is known as the induct INTs of the coil in the circuit or the doctor. Yes, we would call it in this equation of induct INTs if V l is equal to one volt and d i d t is one ampere per second, then l is equal to one and it's unit is Henry. That means if the circuit produces in the MF of one volt due to the rate of change of current truth of one ampere per second, then a circuit is said to have one. Henry self inducted. This Henry is a unit of inductive. This is how the voltage across an induct er and the current through it will develop when the switch is closed. AT T equals zero D. L is equal to VB the supply voltage. However, the induced voltage drops off to zero with flying as Thean doctor acts like a short circuit as the induced voltage drops off with time, the current will increase and be limited by the resist ER are. After a long period of time, it will equal VB all over our, which is homes law. This stands to reason because once the current stops changing as you can see the curve, it's rising up to a maximum limit once it stops changing. Then there's no more voltage developed across the in doctor, so the voltage across it is zero hands. It would be acting as a short circuit at the beginning of the time when the switch is closed. Thean doctor looks like an open circuit so that the current would start off a zero, but it would slowly build up as the magnetic field starts to build up in the in dr itself. The curves for the current and voltage associated with an and Dr are going to be showing here on the left. The current to the induct er will start off rapidly at first, then level off to a steady state value that is equal to the supply voltage divided by the resistor homes law. After a long period of time, the in dr will look like a short circuit. The equation for this curve is given by this. Now you can see that when t is equal to zero. And Oh, by the way, um, I l is the energizing current Phoebe is the supply voltage and the values of t. R is just time is the exponential terms 2.7182818 to a just a ah, a set number of abuse are is the resistance and Alice the inducted its so at the beginning , when t is equal to zero the value of E to the minus RTL over l is actually equal toe one. So one minus one is zero and you multiply that by VB over our the current will be zero at T equals zero when t is equal to infinity or after a long period of time the term eight of the minus r t all over. L disappears. So you're left with VB over our times. One, which is the supply voltage over the resistance was just owns line and again the in Dr Acts as a short circuit After a long period of time, the in Dr Voltage will legal the supply voltage at the start and then fall off to zero. And it's given by this equation and you can see that AT T equals zero um e the term e to the minus r t all over l will be one. So v l is equal to VB and after a long period of time, the term e to the minus rt all over l become zero, so the voltage will drop off 20 There are four basic factors of in Dr Construction determining the amount of induct INTs created. These factors all dictate induct its by affecting how much magnetic field flux will develop for a given amount of magnetic field force that is current through the in doctors wire coil number of wire wraps or turns in a coil. All other factors being equal. A greater number of turns of player in a coil results in a greater induct. INTs. Fewer turns of wire in a coil result in less induct mints. War turns of wires mean that the coil generator greater amount of magnetic field force measured in an turns for a given amount of coil. Current Quayle area again all other factors being equal. Greater coil area as measured looking lengthwise through the coil at the cross section area of the core. Results in greater inducted. It's less coil area results in less induct. Its greater coil area presents less opposition to the formation of magnetic field flux. For a given amount of Field force AMP turns. The longer the coils length, the less induct INTs, the shorter the coils length, the greater the inducted. Its a longer path for the magnetic field flux to take results in more opposition to the formacion of that flux for any given amount of field force and Lastly, the core material, the greater the magnetic permeability of the core, which the coil is wrapped around the greater induct INTs. The less permeability of the core, the less the inducted its ah core material with greater maintain magnetic permeability results in a greater made any flux field for any given amount of force field, in other words and turns when in doctors air connected in Siris, the total inductive is the sum of the individual in Dr Impedance is to understand why this is so. Consider the following the definitive measure of induct INTs is the amount of voltage drop across an in doctor for a given rate of current change through it. If and doctors air connected together in Siris, thus sharing the same current than seeing the same rate of change of current than the total voltage drop. As a result of the change in current will be additive with each and DR creating a greater total voltage than either the individual and doctors alone. Greater voltage for the same rate of change of current beings greater induct its. Thus, the total induct INTs for Siri's and doctors is mawr than any one of the individual and doctors. Thus, the formula for calculating this Siri's total inductive is the same as calculating for Siri's resistance. Siri's inductions L total is equal to L one plus l two, plus any number of inducted in doctors that you have connected in Siris when in doctors air connected in parallel, the total induct Ince's less than any one of the parallel and Dr Induct Ince's again remember that the definitive measure of inducted its is the amount of voltage drop across and and doctor for a given rate of current change through it. Since the current through each parallel and dr will be a fraction of the total current and the voltage across each parallel and dr will be equal. A change in the total current will result in less voltage drop across the parallel array that any one of the in doctors considered separately. In other words, there will be last voltage drop across the parallel in doctors for a given rate of change in current than for any of those in doctors considered separately because the total current divides among parallel branches less voltage for the same rate of change in current means less induct its thus the total induct INTs is less than any one of the individual induct Ince's. The formula for calculating parallel total induction inducted it's is the same four as calculating parallel resistance is one over l total is equal to one over L one plus one over L two plus one over l three. Unlike capacitors, which are relatively easy to manufacture, with negligible stray effects in, doctors are difficult to find in pure form. In certain applications, these undesirable characteristics may present significant engineering problems in dr size in doctors tend to be much larger physically than CA postures are for storing equivalent amount of IT equivalent amounts of energy. This is especially true considering the recent advances in electrolytic pastor technology, allowing incredibly large capacitance values to be packed into a small package. If a circuit designer needs to store a large amount of energy in a small volume and has the freedom to choose either a pastor or a nen doctor for the task, they will most likely choose a capacitor. A notable exception to this rule is in the applications requiring huge amounts of either capacitance or induct INTs to store electrical energy in doctors made made of superconducting wire, zero resistance arm or practical to build and safely operate than capacitors of the equivalent value and are probably smaller, too. Interference in doctors may affect nearby components on a circuit board with their magnetic fields, which can extend significant distances beyond the in doctor. This is especially true if there are other in doctors nearby on the circuit board. If the magnetic fields of two or more in doctors are able to link with each other, there will be a mutual inducted, its present in the circuit as well a self inducted its which could very well cause unwanted effects. This is another reason why circuit designers tend to choose capacitors over in doctors to reform similar tasks. Capacitors inherently contain the respect of electric fields neatly within component package and therefore do not typically generate any mutual effects with other components. This ends Chapter nine
11. Ch 10 Transient Response Capacitors & Inductors R: Chapter 10 transition response capacitors and in doctors, let's explore the response of capacitors and in doctors a little further. A sudden change in D. C. Voltage is called a transient voltage, as we have seen. Unlike resistors, which respond instantaneously to the applied voltage capacitors, and doctors react over time as the absorb and or release energy in an R C. Siri circuit I showing here when the switch is first closed. The voltage across the capacitor, which we were told was fully discharged at the beginning, is zero volts. Thus, in first behaves as though it were a short circuit. Over time, the capacitor voltage will rise to equal the battery voltage, ending in a condition where the capacitor now behaves as an open circuit. As the capacitor voltage approaches the battery voltage, the current approaches zero. However, the time required for a charge in voltage of 63% of the supply voltage is given by what is known as the time constant, which is equal to R C. The resistance in the capacitance. Sometimes they let that time constant equal the Greek letter towel. But we'll just let it equal t for now and call it the time constant. In this case, T is equal to R. C and given the quantities of one K ohm and one micro fair rod ID equals one millisecond. The capacity of voltages, a preach to the battery voltage and the currents approach to zero over time. They both approach their final values, getting closer and closer over time, but they never exactly reach their final destination. For all practical purposes, though, we can say that the capacitor voltage will eventually reach the battery voltage and that the current will eventually reach zero after a time period of five time constants. When the source is removed from the circuit and the capacitor is short circuited through a resistor, the capacitor voltage drops and approaches zero volts, and the current also approaches zero over time. As the electric charge dissipates, they both approach their final values, getting closer and closer over time, but again never exactly reaching their destination. For all practical purposes, though, we can say that a capacitor voltage will eventually reach zero after a time period of five time constants. However, the time required for a drop in voltage of 63% is given by the time constant R C. In this case, T is equal to R C is equal to one times 10 to the three times one times 10 to the minus six , which equals one millisecond. In describing the transient response of a passenger, it can be defined in terms of its time. Constant T is equal to R C, where t is time in seconds. C is the capacitance and fair ads, and our is the resistance in homes. The time required for a change in voltage of 60 63% is one time constant. It is accepted that a capacitor voltage reaches its steady state value after five time constants in and our El Sirri circuit, as shown here. When the switch is first closed, the voltage across the in DR will immediately jump to the battery voltage, acting as the words were an open circuit and then DK down to zero over time, eventually acting as though it were a short circuit. When the switch is first closed, the current is zero. Then it increases over time until it is equal to the battery voltage divided by the Siri's resistance. In this case, one cable or 1000 homes this behavior is precisely officer to that of the series resistor capacitor circuits, where current started at a maximum and capacitor voltage was a zero. Just as with the RC circuit, the in Dr Voltage approaches zero and the current approaches to 10 million amps over time. For all practical purposes, though, we can say that thean doctor voltage will eventually reached zero volts and the current will eventually equal a maximum of 10 million amps. However, the time required for the current to reach 63% is given by the time constant L, divided by R or the inductive, is divided by the resistance. In this case, the time constant T is equal to l divided by r, which is equal to one divided by 1000 or it's equal toe one millisecond, and doctors have the opposite characteristic of capacitors, whereas capacitor store energy, an electric field produced by the voltage between two plates in Dr store energy in a magnetic field produced by the current through the wire. Once the in doctors terminal, voltage has decreased to a minimum zero for a perfect in Dr the current will stay at a maximum level, and the in Dr will behave essentially as a short circuit. The transient response oven in doctor can be described by its time constant towel, or T, which is equal to L divided by R, where T is in seconds, Alice in Henry's and our is in homes. The time T required for a change in current of 63% is one time constant in Dr Current reaches its steady state value in five time constants for both capacitors and induct er's. Their plots are similar and follow what is referred to as the universal time constant graphs where they're rising. Curve values increased 63% each time, constant and can be rep and can represent capacitors charging voltage V. C or the Rising and Dr Current I. L. The falling curves values decreased 63% with each time constant the time required for a drop in voltage of 63% and can be reused and can represent capacitor discharging voltage. Or it can also refer to decaying In Dr Current. This is the universal rising curve in percent plotted against time in time constants. After one time constant, the value has increased to 63%. This curve can either represent the capacitor charging voltage or in Dr Rising current. This is a universal falling curve in percent plotted against time in time. Constance. After one time constant, the value has experienced a drop of 63%. After one time constant, this Kurth can either represent the discharge, the capacitor discharge in volts or the decaying in Dr Current. You'll notice that the curve was generated by the formula E to the negative town times 100% which falls from 100% to 0% showing in red. The percent drop shown in blue measures how much the curve has dropped from 0% to 100%. By measuring this quantity, we only have to remember one fact for both the rise and fall anchors, and that is the 60 63%. After one time constant, this ends Chapter 10
12. Ch 06 Series & Parallel Circuits(R1): Chapter six series and parallel circuits. There are two basic ways in which to connect more than two circuit components, Siri's and parallel. Here in this slide, we have three resistors labeled R one R two and R three connected in a long chain from one terminal of the battery to the other. It should be noted that this sub script, labeling those little numbers to the lower right of the letter R, are unrelated to the resistor values in homes. They are only to identify one resistor from the other. The defining characteristic of a Serie circuit is that there is only one path for current to flow in this circuit. Current flows in a clockwise direction from 0.1 to 2, 2324 and back to one. We say that the three resistors are connected in Siris, a game we have three resistors, but this time they form mawr than one continuous path for the electrons to flow. Each individual path through R one, R two and R three is called a branch. The defining characteristic of a parallel circuit is that all components are connected between the same set of electrical electrically common points. Looking at the schematic diagram. Here we see that points 123 and four are all electrically common. So our points 876 and five note that all resisters as well as the battery, are connected between these two sets of points. We say that these three resistors are connected in parallel. In this circuit, we have two loops for electrons to flow through. Notice how both current past girls go through resistor number one. In this configuration, we say that R two and R three are in parallel with with each other, while our one is in Siris with the parallel combination of R two and R three. The basic idea of a Siri's connection is that components are connected end to end in a lying to form a single path through the current to flow, in this case, from left to right. Or it could be the currents going to flow from right to left. The basic idea of a parallel connection, on the other hand, is that all components are connected across each other's leads. In a purely parallel, circa, they're never more than two sets of electrically common points, no matter how many components air connected. There are many pass for the electrons to flow, but Onley one voltage across the components. The first principle to understand about Siri circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path. It occurred to flow in a series circuit. The rate of flow at any point in the circuit at any specific point in time must be equal. This brings us to the second principle of Siri circuits. The total resistance of any Siri's circuit is equal to the sum of the individual resistances. This should make intuitive sense. The more resistors in Siris that the electrons must flow through, the more difficult it will be for those electrons to flow. In the example problem, we have a three okay, 10-K and a five k resistor in Siris. In other words, it's 3000 owns 10,000 homes and 5000 homes, all in Siris, giving us a total resistance of 18 K or 18,000 homes. In essence, we've calculated the equivalent resistance of R one, R two and r three combined. Knowing this, we could draw the circuit with a single equivalent resistor representing the Siri's combination of R one R two end are three. Now we have all the necessary information to calculate the circuit current because we have the voltage between the two points nine volts and the resistance between the same two points. 18 k OEMs. The current is given by Holmes Law, which is the voltage divided by the resistance or, in this case, nine divided by 18,000 which would give 0.5 amps, or 0.5 million APS. Since 1000 millions is equal to 1/2 because the current is 0.5 million APS. Then the 0.5 millions will flow through each resistor and the voltage drops across each resistor will be given then, by all means, law. For our one, it's 0.5 millions times three K, which is equal to 1.5 volts for our to it's 0.5 times 10-K 0.5 millions times 10-K is five volts and for our three as 30.5 millions times five K, which is equal to 2.5 volts. Notice that the some of the voltage drops 1.5 plus five plus 2.5 is equal to the battery supply. Voltage nine volts. This is a principle of Siri circuits that the supply voltage is equal to the sum of the individual drops around the circuit. This is known as Kurt Chef's Voltage Law, or sometimes referred to his kerchiefs loop or mesh rule, which states that the direct some of electrical potential differences. Voltage drops around any closed network is zero stated another way. The some of the E. M F's in a closed loop is equal to the sum of the potential drops. In that loop. Notice the polarities of the resisters they go from plus to minus, which is voltage drops 1.5 plus five plus 2.5 equals nine volts. Voltage drops. The supply voltage goes from negative to positive, which is a voltage rise. Negative voltage drops, which is a negative voltage drop if you want to say it minus nine. So if you add up the voltage drops, you get 1.5 plus five plus 2.5 minus nine is equal to zero. The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrical common points in a parallel circuit, and the voltage measured between sets of common points must always be the same at any given time. Therefore, in the circuit we have here, the voltage across our one is equal privilege across our two, which is equal to the voltage across our three, which of course, is equal to the voltage of the battery. We can immediately apply homes lock of each resistor to find the current in each resistor because we know the voltage across the resisters nine volts and we know the resistance of each resistor. Therefore, the current in resistor R one is 10.9 million apps. The current in our two is 4.5 millions and the current in our three is nine million apps. At this point, we still don't know what the total current or the total resistance for this parallel circuit is, so we can't simply apply homes lava to the total resistance. However, if we think carefully about what is happening, it should become apparent that the total current must equal the sum of the individual currents in each resistor or in each branch. If it wasn't the case, then we would have a buildup of current or charge in one part of the circuit. And that isn't happening. We have a continuous flow of current at any one time so that the total current must equal the sum of the branch currents. Okay, let's let the voltage drop across. All the resisters equal the tea, which is equal to the supply voltage. And we're gonna let Artie equal the equivalent parallel resistance of all three resistors. We know from homes law that the voltage the total voltage is going to be equal to the individual voltage drops across each resistor, which is I one times are one which is equal toe I to times are too, which is equal to I three times are three. We also know that the from homes law that the supply voltage V t is equal to the total current times. What would be the equivalent resistance of the parallel resistors? Or we can rewrite the equation where the total curd is equal to the voltage of the supply voltage over the equivalent parallel resistance. R t We know that the some of the current flowing into the branch circuits. I t. Is equal to the sum of the branch circuits I one plus I to plus I three or we can rewrite That's that equation, substituting voltage over the resistance for the current terms. So I t becomes VT over RT, which is equal to VT all over our one plus b t all over our two plus Bt all over our three . Now, if we divide both sides of the equations by the tea, then we're left with one over R T is equal to one over r one plus one over r two plus one over r three. So we now have a method of calculating the equivalent parallel resistance with this equation. As I said in the previous slide, this gives us a rule for calculating equivalent resistors in parallel. The if we let the equivalent resistance of parallel resistors B R T than one over R T is equal to one over R one plus one over r two plus one over r three. This also leads us to Kerr chops. Current law, which states the current entering any junction or node, is equal to the current leaving that junction or node, a node being defined as an electrically connected point in the circuit. Another way of stating kerchiefs. Current law is all currents. Entering a node must sum to zero currents entering our positive currents. Leaving are negative. This is another way of looking at the same electrical circuit. In this case, we've brought the Noto one common point, and it's easier to see that Kirchoff is current. Law holds true that the current entering the node is equal to the current. Leaving the note or the some of the currents entering a node is zero. So now, with the knowledge learned, we can reduce any mesh network of resistors to the equivalent of one resistor, as seen by the power supply. For example, in this circuit R one R two R three and r four can be replaced by one resistor R E Q. That will draw the same current as R. One R, two R three and r four. We start by calculating the single resistor equivalent for R one and R two in parallel and R three and R four in parallel, which results in R one R two in parallel giving 71.4 to 9 homes and R three and R Foreign Parallel gives 127.27 homes. We can then add these two resistances because they're in Siris and they will give us 198 0.70 Holmes, which is the required single resistor we wanted to calculate, which will draw the same current as R. One R, two R three and R. Four altogether. As I said before, any mesh network of resistors can be reduced to one single resistor, as in this case, and this one looks a little bit more complex, which it is because there's more resistors involved. And it's drawn in such a way that it's not intuitively obvious how we would reduce this. So sometimes it's easier to redraw the circuit so that the parallel Siri's combinations will jump out at you a little bit easier than you can start. To reduce this in steps. For example, R three and R two are in parallel. They can be read, reduced to one resistor of art to in parallel with R three. It is now everyone resistor in series with our four, so that can be reduced by the parallel combination plus our forests and Siri's. We've now replaced those resistors by one resistor and we can see that that equivalent resistance is now in parallel with our five. So we can reduce that toe one resist er which is our five in parallel with a combination are to imperil with our three in Siris with our four, which gives us one resistor. And that resistor is now in serious with our seven. So that can be replaced by one resistor, which is the equivalent of our seven in Siris. With our five in parallel. What the combination are two in parallel with R three plus R four that is now in parallel with R six and can be replaced with one resistor and I'm not gonna go through repeating it all again. We now have that resistance as a single resistor. No, in with our one which can be replaced now by one resist er and that resistor is now what we're trying to achieve in the first place. And we can calculate the current that the power supply supplies to that single resistor this ends chopped her six