Basic Fundamentals of AC Circuit Analysis | Graham Van Brunt | Skillshare

Basic Fundamentals of AC Circuit Analysis

Graham Van Brunt, Professional Electrical Engineer

Basic Fundamentals of AC Circuit Analysis

Graham Van Brunt, Professional Electrical Engineer

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13 Lessons (6h 3m)
    • 1. Ch 00 Intro Basic Electricity AC

      7:27
    • 2. Ch 1 Alternating Current

      26:39
    • 3. Ch 2 Vectors & Phasors R

      33:24
    • 4. Ch 2 Vectors & Phasors

      33:26
    • 5. Ch 3 Reactance and Impedance R, L, C Circuits

      39:38
    • 6. Ch 4 Kirchhoff's Laws in AC Circuits

      5:47
    • 7. Ch 5 Power flow in AC Circuits

      34:17
    • 8. Ch 0 Intro AC Circuits

      4:25
    • 9. Ch 1 Alternating Current

      33:04
    • 10. Ch 2 Vectors & Phasors copy

      36:12
    • 11. Ch 3 Reactance and Impedance R, L, C Circuits

      53:34
    • 12. Ch 4 Working with AC Circuits

      8:51
    • 13. Ch 5 Power flow in AC Circuits(R)

      46:21
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About This Class

This course will provide the student with the basic understanding of working with circuits involving Alternation Current, which includes sinusoidal waveforms, vectors & phasors, reactance & impedance of R,L,C circuits, as they relate to the basic laws and theorems of electricity. This includes working with AC Power, Power Factor, Resonance, Complex Numbers, Reactance and Impedance.

Meet Your Teacher

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Graham Van Brunt

Professional Electrical Engineer

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Hello, I'm Graham Van Brunt B.Sc; P.Eng.I have spent an entire career in the power sector of electrical engineering, I graduated from Queen's University in Kingston Ontario, Canada. Coupled with subsequent studies with Wilfrid Laurier University I have traveled the globe and applied my skills to garner my protection and control experience internationally.

I have a passion for staying in touch with my profession as an electrical engineer and have a kinship for mentoring that has kept me in front of an audience of learners.

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Transcripts

1. Ch 00 Intro Basic Electricity AC: this course is called a C circuits, and it will provide the student with a basic understanding of what alternating current is all about. End. It will provide some of the tools that are required to work with, UH, basic A C circuits. It is a standalone course, and it is made to provide a student with the information that he or she might be looking for in a way of working with a C circuits. However, it's highly recommended that the course basic electricity, which is on this site as well be a prerequisite for this course because it provides some of the basic understanding that is required when continuing on with a C circuits. The course will start out with a basic understanding of what an A C way form looks like and define what a sign you soil way form is. It will also look at how two forms oven a seaway form, namely a voltage, and a current might be lined up on a time graph. And it'll start to talk about things such as phase shifting and how do we describe these A C quantities. We then very quickly get into defining what RMS is the root, mean square of voltages and currents and power, and we very quickly develop, uh, an understanding of what a an electrical vector is, which is called a phaser. We go on to describe what a phaser is and the characteristics of a phaser, and namely its length and its frequency, and how it rotates about the tail and relative phase angles of the various phasers that are out there. The two ways that I phaser is describe then are studied, and the 1st 1 is the polar notation, where phaser is described by its magnitude and its angle. And the second way that a phaser is described is by rectangular notation. And we get into such things as talking about the J operator and the fact that this phaser would have a real and an imaginary part due to its description on the rectangular notation plane. From there, we go on to talk about react. It's and impedance. We could define what reactions and impedance is, and then we start to analyze what resistive, inductive and capacitive circuits are, which are RLC Circus. The two components that give an easy circuit reactant are then looked at and define starting with the how an induct ER reacts to an A C circuit and weak very quickly defined what reactions for an in doctor is, and we have a formula for defining what that react Ince's given the frequency of oscillations of the circuit as well as the induct INTs that we're dealing with. From there, we go on to put it in a circuit with some resistance, and then we very quickly are able to define what impedance is as it relates to a resistor inductive circuits. The other component that provides reactions to a circuit is also described, and that is a capacitor and the react. Ince's defined for a capacitor in terms off the frequency of the oscillations of the circuit, as well as the capacitance inference of that reactors, quantity and, ah, like the in doctor. We will then put the faster reactions and Siri's with a resistor and come up with the term for the impedance of a Siri's circuit with the resistor and capacitor. Once we've worked with these to react, Ince's for an a C circuit, the induct INTs in the capacitance, and we certainly have worked with the resistor before we move on to power flow in a C circuit. In looking at electric power, we go through several samples and scenarios of different power circuits using a combination of resistors and react. Ince's and we very quickly come to the conclusion that there are several ways of measuring power from instantaneous toe average to real power or reactive power. Bottom line is that in industry, we usually want to come up with what is termed useful power. That is, the actual power that's being converted into heat or mechanical energy, and we devise a formula for that P average, which is equal to the the equation that you see there from here. We go on to define the phasers, a real power, reactive power and apparent power, and we show how they are related. This leads us to the development of using RMS values for this working value for P average and how we use RMS values for calculating that the reactive power and for calculating the apparent power. The last thing that we talk about in this chapter is what harmonic frequency is and what happens when a generator generates what is known as resonant frequency into a ah combination of capacitor and and dr some of the oddities that come out of that are very interesting. And we have a look at that just before closing off this chapter. This ends the introduction for this course. I've just touched on some of the subjects that were going to be covering during it. The course is very interesting, and I encourage you to tap into it and follow it through. 2. Ch 1 Alternating Current: Chapter one alternating current. In a previous course called Basic Electrical Theory, we were looking at D C circuits. That is where the voltages and currents are, for the most part, constant, at least as faras, the supplied sources are concerned. We're now going into the realms of a C circuits where voltages and currents are not constant but changing rapidly in our system in North America. And it's changing 60 times a second, at least as far as the supplied sources are concerned with a CR alternating current, it's possible to build electrical generators, motors and power distribution systems that are far more efficient than D C systems. And so we find a C use predominantly across the world in high power applications, and the diagram in front of you here kind of illustrates the comparison between a C and D. C. The circuit on the left is switched to D. C, which is indicated by a battery. The, uh, the yellow dots circulating in a counterclockwise direction are indicating electrons and the arrows are pointing to the electron flow. We we know in from a previous study that that is the flow of electrons. But convention says the current flow. The adopted conventional current flow is the clockwise direction with the battery. In the case of the circuit on the right, where we've switched to a C end and the generator is Annecy on alternating current generator, the you can see the electrons are shifting back and forth a swell as the flow is going back and forth, and that is happening in a 60 Hertz SEC system 60 times a second. If a machine is constructed to rotate a magnetic field around a set of stationery wire coils with the turning of the shaft, a C voltage will be produced across the coils of that shaft in accordance with Fair Days Law of Electromagnetic induction, which we were introduced in her previous course. This is a basic operating principle. Oven a C generator A C generators are sometimes also referred to as alternators. There is an effect of electromagnetism known as mutual induct ins, where by two or more coils of wire place so that the changing magnetic field created by one induces a voltage in the other. If we have to mutually inductive coils and we energize one coil with a C, we will create a C voltage in the other coil. When used as such, this device is known as a transformer. The fundamental significance of a transformer is its ability to step voltages down, where to step it up from the power coils to the unpowered coils. The A C voltage, induced in an unpowered secondary coil, is equal to the A C voltage across the powered or primary coil multiplied by the ratio of the secondary coil turns to the primary coil turns transformers. Ability to step a C voltage is up or down with ease gives a C an advantage unmatched by D. C. In the rounds of power distribution. When transmitting electrical power over long distances, it is far more efficient to do so with stepped up voltages and step down currents than step . The voltage is back down and the current back up for industry and businesses or consumer use. The advantage here is you can use smaller diameter wire for transmission with less resistive power losses. Transformer technology has made long range electrical power distribution practical without the ability to efficiently step voltages up and down, it would be cost prohibitive to construct power systems for anything but close range. As useful as transformers are, they only work with a C, not D. C. Because the phenomenon of mutual induct us relies on changing magnetic fields, a direct current D C Onley produce steady magnetic fields. Transformers simply will not work with direct current, the alternating current or A C Power A C voltages. And and it seems like a Miss Knoller to call, ah, voltage alternating current or DC voltages. But here is a tendency to refer to anything in regard to producing alternating current as a seat. So anyway, it's going to be helpful to understand what these A C quantities look like. Um, and generators of A C quantities or a sea power. Ultimately, a C voltage and a C current are designed in a special way such that the voltage is the open circuit. Voltage is produced by these A C generators and the current follow what is known as a Sinus . Seidel Wave form and Sinus idle way forms land themselves to, uh, Trigana metric functions that help in the analysis of a C circuits. So we're gonna look at the structure of a C voltages and a C current now, and they're referred to as Sinus Seidel waves. We are then going to look at what those A C quantities mean in terms of the power they're capable of producing. And we're going to define a term relative to these Sinus little way forms called RMS values , and RMS stands for a root mean square. But we'll get into the explanation that in a few slides, as I said before, an alternator is designed to produce a C voltage in a specific shape. Over time, the voltage switches polarity over time, but does so in a very practical manner when graft. Over time, the wave, traced by the voltage of alternating clarity from annul tha meter, takes on a distinct shape known as a sine wave. Or we can describe the way form as sign your soil in the voltage plot from the electromechanical alternator. The change from one polarity to the other is a smooth one. The voltage level changes most rapidly at the zero point or the crossover and most slowly at either of the peaks. Because this is repetitive. We describe the way form in terms of cycles where one cycle is made up of 360 degrees here we have plotted a Sinus little wave for overtime, and that way form varies over time. But what we're doing is taking a snapshot of it right now. If if you could visualize that and we're going to state that the sine wave begins at zero degrees and it starts his rise. So zero degrees is the beginning of the cycle of the wave could be voltage or current, and the voltage current at this point is zero at 90 degrees, which is 1/4 of the wave away through the way form, the voltage or current is at a maximum and for convenience purposes were saying it is one at 180 degrees through the cycle. The way form comes back down and goes through its crossover point, and at that point, it the wave, voltage or current is zero once again at 270 degrees, the way form village, a current is at a negative maximum and at 360 degrees we've gone through one complete cycle , and essentially we're going to start all over again. So the angle is 3 60 or zero degrees, and we're once again back at zero if we were to follow the changing voltage produced by a coil of an alternator from any point on the sine wave graft to that point where the sine wave shaped begins to repeat itself, we would have marked exactly one cycle of that wave. This is most easily shown by spanning the distance between identical peaks, but maybe measured between any corresponding points on the graph, the degree marks on the horizontal axis. The graph represents the domain of the Trigana metric sine function and also the angular position of a simple to poll alternator shaft as it rotates and produces the A C quantity. Since the horizontal axis of this graph can mark the passage of time as well as shaft position in degrees, the dimensions marked for one cycle is often measured in units of time, most often seconds or fractions of a second. When we expressed as a Met, when expressed is a measure of as a measurement, this is often called the period of a wave. The period of a wave in degrees is all always 360 degrees, but the amount of time one period occupies, of course, depends on the rate voltage oscillates back and forth. A more popular measure for describing the alternating rate oven a C voltage or current wave . Then period is the rate of that back and forth oscillation. This is called frequency. The modern unit for frequency is the hurts or abbreviated H zed, which represents the number of ways cycles completed during one second of time in North America. The standard power line frequency, as I said before a 60 Hertz, meaning that the A C voltage off oscillates at a rate of 60 complete back and forth cycles every second. In Europe, where the power system frequency is 50 hertz, the A C voltage Onley completes 50 cycles every second. A radio station transmitter broadcasting at a frequency of 100 megahertz generates an A C voltage for transmission, oscillating at the rate of 100 million cycles every second. Prior to the canonization of the Hurts unit, frequency was simply expressed as cycles per second, older meters and electronic equipment often bore frequency units of CPS or cycles per second instead of hurts. Period and frequency are mathematical reciprocal of one another. That is to say, if a sine wave has a period of 10 seconds, its frequency will be 100.1 hurts, or 1/10 of a cycle per second. Frequency in Hertz equals one over the period. In seconds, we encounter a measurement problem. If we tried to express how larger hop small an A C quantity is with D. C or quantities of voltage and current are generally constant, we have little trouble expressing how much faulty jerker we have in any part of the circuit . But how do you grant a single measurement of a magnitude of something that is constantly changing? One way is to express the intensity or the magnitude. Also called amplitude oven, a C quantity is to measure its peak height on away form graph. This is also known as the peak or crest value. Oven. A seaway for another way is to measure the total height between the opposite peaks. This is No one s peak to peak measurement, or P. Two p value of an A C way for another way of expressing the magnitude of a wave is to mathematically average the value of all the points on the way form graph to a single aggregate number. This amplitude measurement is known simply as the average value of the way for if we average all the points on the way form algebraic algebraic Lee, that is to consider their signs either positive or negative. The average value of most way forms is technically zero because all the positive points cancel out all the negative points over a full cycle. However, a practical measure of away forms aggregate value average is usually defined as the mathematical mean of all the points, absolute values over a cycle. In other words, we calculate the practical average value of away form by considering all the points in the way form as positive quantities, as if the way form looked like this, the average value would then have some value other than zero that would be related to the intensity of the wave. So far, we have looked at three ways that we can measure the intensity of on a C way form. Whether that be current or voltage, we can simply measure the peak or the crest of the amplitude of that wave. Or we could measure a quantity that is peak to peak or weaken. Take the mean average as a quantity that measured the intensity, the A C wave. The good news is really these are all related to each other in the way they can be just scaled one to the other. In other words, they very directly as each other. So one could be converted to the other by simply multiplying by a scaling factor. The thing or the trick we have to know is which one we're dealing with so that we can make that conversion or weaken deal with the actual measurement. And it can be useful to us as we communicate the value of voltage and current with others in the industry of electrical power, as well as their related quantities of power energy ratings in different elements, we have to ask ourselves how useful are using any of these terms, and is there some way of measuring the values that is the most useful way? The question was, acts asked and answered a long time ago in the answer waas the RMS value before just jumping to the definition of RMS, which, by the way, is mathematically related proportionally to the other ways of describing the ways such as amplitude peak, peak peak average and mean average Let's go through the logical steps of getting there, starting with two simple circuits. One D C. One a. C that is each with the same load, but one driven by D. C source and the other driven by an a C source. When we close the switch on the D C circuit, the bulbul light, with an intensity that is dependent on the resistance, are so well of the light and a D. C. Current. No, let's close a switch and adjust the A C current to the light bulb with the same intensity. That is to say, both loads. Both lights consume the same average power. So we now ask ourselves, What is that? A c current? We can come to the conclusion that if the two bulbs like to the same Brent of brightness, that is, they draw the same amount of power, and it is reasonable to consider the current I A C to be in some ways equivalent to the current I. D. C. So what is that value of a C? It would be useful if there was some meaningful way to calculate it. So let's go there. If an A C supply is connected to a component of resistance say are the instantaneous power dissipated is given by the power equation I squared are If we plot, I squared the instantaneous current, which itself is a sign way. It is always positive because plus, I times plus I is positive and negative. I times negative. I is positive it does go to zero, but never negative. Remember that the instantaneous power dissipated is given by the equation. Power equals I squared. Are the peak or maximum value of I squared is showing here and labeled I squared Max. The mean or average value of ice square is I squared max divided by two. So the average value of power is we'll call it P. Subscript a V is equal to I'm ax squared over two times are subscript l. We just saw from the previous slide that the average power consumed in the circuit is given by this equation, which is equal to the maximum value of the current squared over two times the resistor. Let's define a current than with that when used to calculate power, gives us the average power. In other words, when that current, let's call it I with a sub script. The fine for now is squared and multiplied by R. L gives us the average power, but the defined our i d find squared is also equal to I max Square over to therefore the square root of the mean current equals that defined current. So we just discovered that what the value off I defined is it's equal to I max divided by route to and we call this current I RMS or root mean square and it is 0.707 the value of the maximum current. This is another, more useful way to describe a C quantities voltage incurred and of course, it can be converted directly to amplitude peak or peak to peak or average just by multiplying by a scaling factor. However, we use RMS values for current and voltage weaken directly. Calculate the average power from thes root mean square quantities. The R. M s value oven A C supply is equal to the direct current, which would dissipate the energy at the same rate of a given resistor. We can use the same logic to define the RMS value of the voltage oven alternating voltage supply. The RMS is equal to the peak voltage divided by route to where V is the maximum Orpik value of the voltage. So we have a way of calculating the RMS values of both current and voltage from their respective peak values. So we have a way of calculating the average power consumed by a resistor or resistance in a circuit, using RMS values for voltages and for currents and all of the other power equations using just current and resistance and or voltage and resistance would still hold true do the due to the linearity property. We are only multiplying by a scaling factor, so we can always calculate the average power used by a C currents and voltages with resistors also owns. Law still holds true because of the video, pretty properties and all of the other things that we do with resistors using a C. Voltages and currents still hold true, we still calculate Siri's loads the same parallel loads the same weaken, still analyze mesh equations. Kershaw's voltage and current law still holds true, Um, superposition Stephen and Norton and sorts transformation all hold true. Remember that we have to so far only use resistors in subsequent chapters and we're going to go on to see how we use impedance is rather than just resistors. But that's further down the road. For now, all of these things still hold true. This ends chapter one. 3. Ch 2 Vectors & Phasors R: Chapter two vectors and phasers. So far, we have discovered that we can analyze A C circuits and currents and voltages using RMS values for resistive loads. That is to say, the A C currents and voltages are in phase, but that is not always the case, so we have another dimension to deal with in and in doing so, we have to develop and use another tool. This tool is called a phaser, which is a special type of actor. Factors have to values magnitude and direction. Phasers are vectors, but they also have a rotation, and usually it's that system frequency. And in North America, that system frequency is 60 hertz. And that's what will be considering in our subsequent calculations. Okay, so let's build that tool. How exactly can we represent a C quantities of voltage or current in the form of a factor? Let the length of the vector represent the magnitude or amplitude or RMS value of the way. For like this, the greater the amplitude of the wave form, the greater the length of a corresponding fazer. Before progressing further, we'll establish that all the Sinus edel wave phasers, voltages and current rotate at the same frequency in the North American standard. The frequency of the system is 60 hertz, or 60 cycles per second. So all the phasers, currents and voltages air all rotating at this frequency, hence their separated on Lee by a phase shift that can vary from 0 to 360 degrees. The angle of the vector, however, represents the phase shift in degrees between the way forms and question and another one acting as a reference in time. Usually when phase when the phase of away form in a circuit is expressed its referenced to the power supply voltage way for arbitrarily set at zero degrees. Remember, that phase is always a relative measurement between two way forms rather than an absolute property. We will look here at a way, form with reference to be way for if there is a zero degrees phase shift between these two , phasers A and B are in perfect step there in line with each other. So the kind of overlap if there is a phase shift of 90 degrees, it could be that a is ahead of be a leagues, be by 90 degrees. Assuming a counterclockwise phase rotation ah, phase shift of 90 degrees where B is ahead of a. This would also be indicated by these two phasers. Where be leads a by 90 degrees and a phase shift of 100 a degree degrees means that A and B way forms are basically mirror images of each other or are in the opposite directions. So let's see list some of the characteristics of a phaser, and we will go on to develop how to analyze and calculate phasers. But for now, let's just talk about some of the characteristics of a phaser. A phaser can vary in length, which is equivalent to the phase or magnitude, which could be the phasers or the A C current or voltage magnitude. RMS value peak to peak value doesn't matter which it is. The phaser phasers length is proportional to that quantity. We have to define what that quantity is if we want to work with it. But for now, we can say that phasers the length of a phaser changes as its magnitude changes. All phasers rotate the same, and they're out of rotating at a frequency of 60 cycles per second, depending on the system and in our system in North America is 60 cycles. I slowed down the rotation here. Otherwise, of over 60 cycles a second. You wouldn't be able to see it. So let's assume that it's scaled down here just for demonstration purposes. Phasers rotate about their tail, in other words, of phasers represented by an arrow, and it is rotating about its tail phasers. All the phasers in the system rotate at the same direction, the standard that is usually adopted his counterclockwise. I will say this, but only once. It could be in a clockwise direction as long as you maintain that sensibility and work with the understanding that phasers are rotating in a clockwise direction. However, the standards that we worked with in the standards that I'll be working within his course assumes the phasers are rotating in a counterclockwise direction, and they all rotate in the same direction and phase angles. Other fazer. Our relatives Ah, phase angle cannot exist unless it's related to something else and usually another factor. So phase angles are relative, the greater the phase shift in degrees between two way forms, the greater the angle difference between the corresponding phasers. Being a relative measurement like voltage phase shift or phase angle on Lee has meaning in reference to some standard we've for generally this reference way for is the main a sea power supply voltage in the circuit. If there is more than one a C voltage source, then one of the sources is arbitrarily chosen to be the phase reference for all other measurements. In the surface, as in this diagram phaser, a phase shift is relative to phaser. Be for convenience and ease of reference. We stopped the reference phaser when it is at zero with the horizontal, so it makes the relative measurements easier, or when the B way form is at zero. There are a couple of ways of uniquely describing a phaser. In one of those ways is using what is noted as polar notation and when we use polar notation. The phaser is described as being on a polar plane with his tail at the center of the origin , and the plane is divided into three hundreds and 16 off plainer area. Therefore, each phaser is described with its magnitude being a length of the phaser or arrow, and it's described with its angle of displacement with respect to zero degrees. In this particular example, we might want to describe this phaser as having a magnitude of 8.49 at an angle of 32 degrees, and we usually draught. Write it on paper in this format, where the magnitude is written first in numerical format, with an angle sign and that angle described after the angle, saying standard orientation for a phaser angle in a C circuit calculations defined zero as being the right horizontal, making 90 degrees straight up, 180 degrees is to the left and 270 degrees straight down. Please no. That phaser angles down can have angles represented in polar form as positive numbers in excess of 180 degrees or negative numbers less than 180 degrees. For example, the phaser angle 270 degrees straight down can also be said to have an angle of minus 90 degrees. Here are some examples of phasers in Poland notation. You'll notice the two ways of designated phaser, either with plus or minus angles. I've left off the polar plane. We gonna soon that you know, we know that zero degrees is to the right and 90 years up to 70 is down and on my 100 degrees is to the left. The factor on the upper right in this diagram is has a magnitude of 8.49 and it has an angle plus angle of 45 degrees. The phaser in the right hand operate hand side of the diagram here has a magnitude of 8.6 and the angle can either be described as minus 29.74 degrees, or it could be described as 330.26 degrees. The fazer in the bottom left corner has a magnitude of 5.39 and its angle is 158 degrees, which puts it in up and over to the left. The vase er in the bottom right hand corner. You can designate that in one of two ways. Using polar notation. One is a table have the same magnitude 7.81 but you could describe the angle is 232.19 degrees or you could describe the angle is minus 129.81 degrees in rectangular and notation . The phaser is taken to be the iPod news of a right angle triangle described by the links of the adjacent and opposite sides. Rather than describing the phasers, lengthen direction by denoting magnitude and angle. It is described in terms of how far left or right and how far up or down it is from the origin. These two dimensional figures horizontal, vertical or symbolized by two numerical figures in order to distinguish the horizontal and vertical dimensions from each other. The vertical is prefects with a lower case. J. This lower case letter does not represent a physical variable, but rather a mathematical operator used to distinguish the phasers vertical component from its horizontal component. When placed in front of a vector, it'll swing that vector through 90 degrees in a counter clockwise direction. So in our example, we can consider the vector. The Red Arrow made up off two vectors the sum of two vectors, one along the X axis, which is a link of four and one along the vertical axis, which is three. But in order to distinguish the the horizontal and vertical, we've used the J operator, that which swings a what would be a real vector of three along the X axis through 90 degrees to be along the Y axis, So the red vector is described as a some of two factors. One along the real axis went along the Y Axis four plus J three. As I said, as a complete complex number, the horizontal and vertical quantities are written as the sum of two vectors. The horizontal component is referred to as the rial component, since that dimension is compatible with a normal skater rial number, the vertical component at 90 degrees to the rial component is referred to as the imaginary component, since that dimension lies in a different direction, totally alien to the scaler of ah rial number. Here are some examples of phasers in rectangular notation notice. This time there is only one way to distinguish the phaser. They are uniquely described by the two figures. The one in the first quadrant upper right hand side of the graph is four plus J three, which means it's four along the rial axis and three along the imaginary axis Aston ordered by Plus J in front of the three, the one in the left hand side, upper left hand side of the graph is minus four plus J three, so minus four is along the real axis, but in a minus direction. Four and these. The three is along the plus J or imaginary axis three. In that direction. That factor is made up of minus four plus J three, and lastly, the one in the bottom right hand quadrant of the graph is made up of plus four minus J three. So we have seen that a phaser candy bees can be described in one of two ways. You can either use polar notation on a polar plane such as we see here, or we can use rectangular notation. As you can see here noticed that where we have not moved the vector or the phaser, it has remained in the same position, so you can uniquely describe a phaser in one of those two ways, so there must be a way of translating. We're converting rectangular notation, toe polar notation and polar notation back to rectangular notation. Converting from polar to rectangular and vice versa is a very simple process, and it could be demonstrated with this phaser here, and I've described it in both turns both rectangular and polar form for convenience. So let's say we want it. We have the, um, the phaser in polar form, and we want to convert it to rectangular form. So that is, we have five at 36.87 degrees, and we want to convert it to erect and get her form, the rial component of the of the polar form you could take by multiplying the polar magnitude by the coast sine of the angle. So you take the magnitude five and co sign of 36.87 which is equal to four, and that will give you the rial component of the rectangular form. To find the imaginary side. You take the magnitude or the length of the magnitude of the phaser five, and you multiply it by this time the sign of the ankle, which is 36.87 degrees. And that equals three. So that will give you the imaginary component, and you have to remember, put the J operator in front of it, of course, and whether it's plus or minus will fall out of the calculation. Now, this time we want to go in the other direction. We want to change the rectangular form to the polar form, and you can find the polar magnitude through the use of the path. A Guerry and zero. The polar magnitude is the high potting use of a right angle triangle, and the real and the imaginary components are the adjacent and opposite sides, respectively. So the length would be the square root of four squared plus three squared, which comes up to five and then to find the angle. All you have to do is take the are tan of the um, imaginary component divided by the rial component or 3/4. The art tension of 3/4 is 36.87 degrees. If two a C voltages 19 degrees out of phase are added together by being connected in Siris there, voltage magnitudes do not directly add or subtract. As with the skater voltages in D C calculations. Instead, these voltage quantities are complex quantities, and I must add up in a triggered a metric fashion. A six fold source at zero degrees added to an eight full source at 90 degrees. Results in 10 Val's at a phase angle of 53.13 degrees. In other words, when you add voltages that are a 90 degree separation, it's like finding the high pot news of a right angle triangle. And that's exactly what happens with the trig and metric condition that we did here. Vectors as well as phasers can be moved around the plane as long as their direction and magnitudes are maintained. So you add two vectors, or phasers by placing the tale of one on the head of the other and connecting the other head and the other tail, just like in the picture. Compared to D. C circuit analysis. This is very strange indeed. Note that it is possible to obtain volt meter indications of six and eight folds, respectively, across the two a C voltage sources independently yet on Leigh read 12 volts for the total voltage with a C two. Voltages can be aiding or opposing one another to any degree, fully aiding or fully opposing inclusive without the use of phasers or complex numbers notations to describe the A C quantities. It would be very difficult to perform mathematical calculations for a C circuit analysis in this example. Uh, I demonstrated, adding to a C voltages together that happened to be separated by, ah, phase angle of 90 degrees, which made the arithmetic fairly easy. When you're dealing with the phaser quantities in polar notation, however, um, adding and subtracting of complex numbers eyes much easier if you are able to convert to and from polar and rectangular notation because the addition and subtraction of phasers in rectangular notation is very easy. And the multiplication and division of of phasers in polar notation is is the easiest form . So what you want to do is you wanna add using rectangular, um, formats, and you wanna multiply using polar for Matty. So when dealing with addition and subtraction of phasers, we're complex numbers. Which phasers are you? Simply add the rial components of the complex number to determine the rial component. Some and you had the imaginary component to determine the imaginary some of the phasers. So in our example, we have three year we have in the first example On the left, you have a phaser two plus J five added four minus J three, which would give you the result in phaser of six plus jay, too. The middle one is 175 minus J 34. When added to Fazer 80 minus J 15 you'd end up with 255 minus J 49 the last one minus 36 plus Jay 10 added to a phaser of 20 plus. JD two gives us a phaser that would be minus 16 plus j 92. So when subtracting complex numbers, it's almost as simple as as adding them together. Uh, you either subtract the rial components and the imaginary components to come up with the real and the imaginary component of the resultant or another way you can say it is you can change the sign of the fazer that you want us attract. As we're doing here. We're putting a minus sign in front of the practice, which essentially changes the sign of the terms inside the bracket. Then just add them together. Either way, you will come up with the same answer. So in the first example, you got ah phaser two plus J five and you're going to subtract the phaser four minus J three so you could change the sign of those components minus four. Would minus four. It would be minus four plus J three. Then you would take to Maya's four is minus two, then you take plus J five plus J three would be J eight in the middle one. You have 175 minus J 34 you're going to subtract 80 minus J 15 so you could take change the sign of that phaser. So now you would have minus 80 plus J 1575 minus 80 would be 95 minus. J 34 plus J 15 would give you minus J 19. And finally, in our last subtraction, you would have of a phaser minus 36 plus Jay 10 and you're gonna subtract 20 plus J t. Two. So you change the signs of those two components of that phaser so you would end up with minus 20 minus j t to So you have 36 minus 20 would give you 56 for the rial component and plus J 10 minus. J 82 would give you minus J 72 for the resultant four multiplication and division of phasers. Polar notation is favored over rectangular notation because it's much easier to deal with when multiplying complex numbers in poor poor form. You simply multiply the polar magnitudes of the two complex numbers to determine the polar magnitude of the product, and we add the angles of the complex number to determine the angle of the product. So let's look at a couple of examples. If we had 35 at 65 degrees multiplied by tan at minus 12 degrees, you'd multiply the two magnitudes to end up with 350 and you would add the two angles together. So 65 minus 12 would give you 53 degrees, so the result would be pretty easily calculated is 350 at 53 degrees. Another example would be 124 at 5250 degrees times 11 at 100 degrees. So you multiply the two magnitudes, ending up with 1364 and you would add the two angles together so you'd end up with 350 degrees. Or you could describe it as minus 10 degrees as well, so you would add the angles together to get the result angle, which is 350 degrees, so you're resulting would be 1364 at 350 degrees, or 1364 at minus 10 degrees. And our final example is ah, are a simple three at 30 degrees, times five at minus 30 degrees, you'd end up with multiplying the magnitudes. Together you get 15 and you would add that two angles 30 minus 30 gives you zero. And, as you might have guessed, division of polar form complex numbers is the easiest path to take. You simply divide the polar magnitudes of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient and subtract the angle of the second complex number from the angle of the first complex number to arrive at the angle of the quotient. So a few examples just do cement the idea. Here we have ah phaser 35 65 degrees, being divided by a phaser of 10 at minus 12 degrees. You would divide 3 35 by 10 which would give you 3.5, and you'd subtract 12 from 65 which is same as adding 12 to 65 which would give you 77 degrees. In this example, we have ah phaser at 124 magnitude at 25 degrees, 250 degrees, being divided by off ah phaser of magnitude 11 at 100 degrees. So you would divide 124 by 11 which would give you 11.273 and you would subtract 100 from 2 50 to arrive in 150 degrees. And a final example would be, ah, phaser three at 30 degrees, divided by five at minus 30 degrees, you would have three divided by five, which is 50.6 and then you. It's attract 30 degrees from subtract minus 30 degrees from 30 degrees is the same as adding 30 degrees to 30 degrees, which would result in a new angle of 60 degrees. So the resultant quotient would be 0.6 at 60 degrees. To obtain the reciprocal or invert a complex numbers simply divide the number in in his poor for into the value of one, which is nothing more than the complex number one at zero degrees. So in these examples, if you want to invert the fazer 35 at 65 degrees, you'd put 1/35 at 65 degrees, which is the same s dividing one at zero degrees by 35 at 65 degrees. And that would be one divided by 35 which would give you the magnitude of 350.2857 And he had subtract 65 from zero, which would give you minus 65. In the next example, you would have the reciprocal of 10 at minus 12 degrees, which would be 10 degrees, divided by 10 at minus 12. One, divided by 10 is 100.1 zero. Subtract minus 12 degrees is 12 degrees and the final example one. Uh, the reciprocal of 0.32 at 10 degrees would give you 11 divided by 110.32 which would give you the 312.5. And then you would take zero and subtract 10 degrees, which would give you minus 10 degrees. So as you work through a long string of complex numbers, arithmetic always add and subtract and rectangular form, then multiply and divide in polar for it will be necessarily convert from one form to the other. As you progress through these calculations, however, with the invention of the smartphone and and laptop computers, there's lots of APS out here that will do that hard work for you. But it helps to know the process so that if you run into a problem, you can always find out what's going on. These are the basic operations you'll need to know in order to manipulate complex numbers in the analysis of a C circuits. Operations with complex numbers are by no means limited just to addition, subtraction, multiplication division and version. However, virtually any arithmetic operation that could be done with scaler numbers can be done with complex numbers. This ends Chapter two. 4. Ch 2 Vectors & Phasors: Chapter two vectors and phasers. So far, we have discovered that we can analyze A C circuits and currents and voltages using RMS values for resistive loads. That is to say, the A C currents and voltages are in phase, but that is not always the case, so we have another dimension to deal with in and in doing so, we have to develop and use another tool. This tool is called a phaser, which is a special type of actor. Factors have to values magnitude and direction. Phasers are vectors, but they also have a rotation, and usually it's that system frequency. And in North America, that system frequency is 60 hertz. And that's what will be considering in our subsequent calculations. Okay, so let's build that tool. How exactly can we represent a C quantities of voltage or current in the form of a factor? Let the length of the vector represent the magnitude or amplitude or RMS value of the way. For like this, the greater the amplitude of the wave form, the greater the length of a corresponding fazer. Before progressing further, we'll establish that all the Sinus edel wave phasers, voltages and current rotate at the same frequency in the North American standard. The frequency of the system is 60 hertz, or 60 cycles per second. So all the phasers, currents and voltages air all rotating at this frequency, hence their separated on Lee by a phase shift that can vary from 0 to 360 degrees. The angle of the vector, however, represents the phase shift in degrees between the way forms and question and another one acting as a reference in time. Usually when phase when the phase of away form in a circuit is expressed its referenced to the power supply voltage way for arbitrarily set at zero degrees. Remember, that phase is always a relative measurement between two way forms rather than an absolute property. We will look here at a way, form with reference to be way for if there is a zero degrees phase shift between these two , phasers A and B are in perfect step there in line with each other. So the kind of overlap if there is a phase shift of 90 degrees, it could be that a is ahead of be a leagues, be by 90 degrees. Assuming a counterclockwise phase rotation ah, phase shift of 90 degrees where B is ahead of a. This would also be indicated by these two phasers. Where be leads a by 90 degrees and a phase shift of 100 a degree degrees means that A and B way forms are basically mirror images of each other or are in the opposite directions. So let's see list some of the characteristics of a phaser, and we will go on to develop how to analyze and calculate phasers. But for now, let's just talk about some of the characteristics of a phaser. A phaser can vary in length, which is equivalent to the phase or magnitude, which could be the phasers or the A C current or voltage magnitude. RMS value peak to peak value doesn't matter which it is. The phaser phasers length is proportional to that quantity. We have to define what that quantity is if we want to work with it. But for now, we can say that phasers the length of a phaser changes as its magnitude changes. All phasers rotate the same, and they're out of rotating at a frequency of 60 cycles per second, depending on the system and in our system in North America is 60 cycles. I slowed down the rotation here. Otherwise, of over 60 cycles a second. You wouldn't be able to see it. So let's assume that it's scaled down here just for demonstration purposes. Phasers rotate about their tail, in other words, of phasers represented by an arrow, and it is rotating about its tail phasers. All the phasers in the system rotate at the same direction, the standard that is usually adopted his counterclockwise. I will say this, but only once. It could be in a clockwise direction as long as you maintain that sensibility and work with the understanding that phasers are rotating in a clockwise direction. However, the standards that we worked with in the standards that I'll be working within his course assumes the phasers are rotating in a counterclockwise direction, and they all rotate in the same direction and phase angles. Other fazer. Our relatives Ah, phase angle cannot exist unless it's related to something else and usually another factor. So phase angles are relative, the greater the phase shift in degrees between two way forms, the greater the angle difference between the corresponding phasers. Being a relative measurement like voltage phase shift or phase angle on Lee has meaning in reference to some standard we've for generally this reference way for is the main a sea power supply voltage in the circuit. If there is more than one a C voltage source, then one of the sources is arbitrarily chosen to be the phase reference for all other measurements. In the surface, as in this diagram phaser, a phase shift is relative to phaser. Be for convenience and ease of reference. We stopped the reference phaser when it is at zero with the horizontal, so it makes the relative measurements easier, or when the B way form is at zero. There are a couple of ways of uniquely describing a phaser. In one of those ways is using what is noted as polar notation and when we use polar notation. The phaser is described as being on a polar plane with his tail at the center of the origin , and the plane is divided into 360 degrees off plainer area. Therefore, each phaser is described with its magnitude being a length of the phaser or arrow, and it's described with its angle of displacement with respect to zero degrees. In this particular example, we might want to describe this phaser as having a magnitude of 8.49 at an angle of 32 degrees, and we usually draught. Write it on paper in this format, where the magnitude is written first in numerical format, with an angle sign and that angle described after the angle, sign standard orientation for a phaser angle in a C circuit. Calculations defined zero as being the right horizontal, making 90 degrees straight up, 180 degrees is to the left and 270 degrees straight down. Please no. That phaser angles down can have angles represented in poor form as positive numbers in excess of 180 degrees or negative numbers less than 180 degrees. For example, the phaser angle 270 degrees straight down can also be said to have an angle of minus 90 degrees. Here are some examples of phasers in Poland notation. You'll notice the two ways of Designate a phaser, either with plus or minus angles. I've left off the polar plane. We gonna soon that you know, we know that zero degrees is to the right and 90 years up to 70 is down, and the one month 100 degrees is to the left. The factor on the upper right in this diagram is has a magnitude of 8.49 and it has an angle plus angle of 45 degrees. The phaser in the right hand operate hand side of the diagram here has a magnitude of 8.6 and the angle can either be described as minus 29.74 degrees, or it could be described as 330.26 degrees. The fazer in the bottom left corner has a magnitude of 5.39 and its angle is 158 degrees, which puts it in up and over to the left. The vase er in the bottom right hand corner. You can designate that in one of two ways using polar notation. One is that they will have the same magnitude 7.81 but you could describe the angle is 232.19 degrees, or you could describe the angle is minus 100 and 29.81 degrees in rectangular and notation . The phaser is taken to be the iPod news of a right angle triangle described by the lengths of the adjacent and opposite sides. Rather than describing the phasers, lengthen direction by denoting magnitude and angle. It is described in terms of how far left or right and how far up or down it is from the origin. These two dimensional figures horizontal, vertical or symbolized by two numerical figures in order to distinguish the horizontal and vertical dimensions from each other. The vertical is prefects with a lower case J. This lower case letter does not represent a physical variable, but rather a mathematical operator used to distinguish the phasers vertical component from its horizontal component. When placed in front of a vector, it'll swing that factor through 90 degrees in a counter clockwise direction. So in our example, we can consider the vector the red Arrow made up of two vectors, the sum of two vectors, one along the X axis, which is a link of four and one along the vertical axis, which is three. But in order to distinguish the the horizontal and vertical, we've used the J operator, that which swings A what would be a real vector of three along the X axis through 90 degrees to be along the Y axis, So the red vector is described as a some of two factors. One along the real axis went along the Y Axis four plus J three. As I said, as a complete complex number, the horizontal and vertical quantities are written as the sum of two vectors. The horizontal component is referred to as the rial component, since that dimension is compatible with a normal scaler rial number, The vertical component at 90 degrees to the rial component is referred to as the imaginary component, since that dimension lies in a different direction, totally alien to the scaler of ah rial number. Here are some examples of phasers in rectangular notation notice. This time there is only one way to distinguish the phaser. They are uniquely described by the two figures. The one in the first quadrant upper right hand side of the graph is four plus J three, which means it's four along the rial axis and three along the imaginary axis Aston ordered by plus Jay in front of the three, the one in the left hand side, upper left hand side of the graph is minus four plus J three, so minus four is along the real axis, but in the minus direction four and these the three is along the plus J or imaginary axis three. In that direction. That factor is made up of minus four plus J three, and lastly, the one in the bottom right hand quadrant of the graph is made up of plus four minus J three. So we have seen that a phaser candy bees can be described in one of two ways. You can either use polar notation on a polar plane such as we see here, or we can use rectangular notation. As you can see here, I noticed that where we have not moved the vector or the phaser, it has remained in the same position. So you can uniquely describe a phaser in one of those two ways, so there must be a way of translating were converting rectangular notation, toe polar notation and polar notation back to rectangular notation. Converting from polar to rectangular and vice versa is a very simple process, and it could be demonstrated with this phaser here and I've described it in both terms, both rectangular and polar form for convenience. So let's say we want it. We have the, um, the phaser in polar form, and we want to convert it to rectangular form. So that is we have five at 36.87 degrees, and we want to convert it to a rectangular form, the rial component of the of the polar form you could take by multiplying the polar magnitude by the coast sine of the angle. So you take the magnitude five and co sign of 36.87 which is equal to four, and that will give you the rial component of the rectangular form. To find the imaginary side, you take the magnitude or the length of the magnitude of the fazer five and you multiply it by this time, the sine of the angle, which is 36.87 degrees, and that equals three. So that will give you the imaginary component, and you have to remember to put the J operator in front of it, of course, and whether is plus or minus will fall out of the calculation. Now, this time we want to go in the other direction. We want to change the rectangular form to the polar form, and you can find the polar magnitude through the use of the path. A Guerry and zero. The polar magnitude is the high potting use of a right angle triangle, and the real and the imaginary components are the adjacent and opposite sides, respectively. So the length would be the square root of four squared plus three squared, which comes out to five and then to find the angle. All you have to do is take the arc tanne of the imaginary component divided by the rial component or 3/4. The art tension of 3/4 is 36.87 degrees. If two a C voltages 19 degrees out of phase are added together by being connected in Siris there, voltage magnitudes do not directly add or subtract. As with the skater voltages in D C calculations. Instead, these voltage quantities are complex quantities and must add up in a triggered a metric fashion. A six fold source at zero degrees added to an eight fold source at 90 degrees. Results in 10 Val's at a phase angle of 53.13 degrees. In other words, when you add voltages that are a 90 degree separation, it's like finding the high pot news of a right angle triangle. And that's exactly what happens with the trig and metric condition that we did here. Vectors as well as phasers can be moved to rattle the plane as long as their direction and magnitudes are maintained. So you add two vectors or phasers by placing the tale of one on the head of the other and connecting the other head and the other tail, just like in the picture compared to D. C circuit analysis. This is very strange indeed. Note that it is possible to obtain volt meter indications of six and eight folds, respectively, across the two a C voltage sources independently yet on Leigh read 12 volts for the total voltage with a C two, voltages can be aiding or opposing one another to any degree, fully aiding or fully opposing inclusive without the use of phasers or complex numbers notations to describe the A C quantities. It would be very difficult to perform mathematical calculations for a C circuit analysis in this example. Uh, I demonstrated, adding to a C voltages together that happened to be separated by, ah, phase angle of 90 degrees, which made the arithmetic fairly easy. When we're dealing with the phaser quantities in polar notation, however, adding and subtracting of complex numbers eyes much easier if you are able to convert to and from polar and rectangular notation because the addition and subtraction of phasers in rectangular notation is very easy. And the multiplication and division off of cop phasers in polar notation is is the easiest form. So what you want to do is you wanna add using rectangular, uh, formats, and you wanna multiply using poor for matty. So when dealing with addition and subtraction of phasers or complex numbers, which phasers are you simply add the rial components of the complex number to determine the rial component. Some and you had the imaginary component to determine the imaginary some of the phasers. So in our example, we have three year we have in the first example On the left, you have a phaser two plus J five added to four minus J three, which would give you the result in phaser of six plus jay, too. The middle one is 175 minus J 34. When added to Fazer 80 minus J 15 you'd end up with 255 minus J 49 the last one minus 36 plus Jay 10 added to a phaser of 20 plus. JD two gives us a phaser that would be minus 16 plus J 92. So when subtracting complex numbers, it's almost as simple as as adding them together. You either subtract the rial components and the imaginary components to come up with the real and the imaginary component of the resultant or another way you can say it is. You can change the sign of the fazer that you want us attract. As we're doing here. We're putting a minus sign in front of the practice. What's essentially changes the sign of the terms inside the bracket. Then just add them together. Either way, you will come up with the same answer. So in the first example, you got ah phaser two plus J five and you're going to subtract the phaser four minus J three so you could change the sign of those components minus four. Would minus for it would be minus four plus j three. Then you would take to minus four is minus two. Then you take plus J five plus J three would be J eight in the middle one. You have 175 minus J 34 you're going to subtract 80 minus J 15. So you could take change the sign of that phaser. So now you would have minus 80 plus J 1575 minus 80 would be 95 minus. J 34 plus J 15 would give you minus J 19. And finally, in our last subtraction, you would have of a phaser minus 36 plus Jay 10 and you're gonna subtract 20 plus J t. Two. So you change the signs of those two components of that phaser so you'd end up with minus 20 minus J t to So you have 36 minus 20 would give you 56 for the rial component and plus J 10 minus. J 82 would give you minus J 72 for the resultant four multiplication and division of phasers. Polar notation is favored over rectangular notation because it's much easier to deal with when multiplying complex numbers in poor poor form. You simply multiply the poor magnitudes of the two complex numbers to determine the polar magnitude of the product, and we add the angles of the complex number to determine the angle of the product. So let's look at a couple of examples. If we had 35 at 65 degrees multiplied by tan at minus 12 degrees, you'd multiply the two magnitudes to end up with 350 and you would add the two angles together. So 65 minus 12 would give you 53 degrees, so the result would be pretty easily calculated is 350 at 53 degrees. Another example would be 124 at 5250 degrees times 11 at 100 degrees. So you multiply the two magnitudes, ending up with 1364 and you would add the two angles together so you'd end up with 350 degrees. Or you could describe it as minus 10 degrees as well, so you would add the angles together to get the result angle, which is 350 degrees so you're resulting would be 1364 at 350 degrees, or 1364 at minus 10 degrees. And our final example is ah, are a simple three at 30 degrees, times five at minus 30 degrees, you'd end up with multiplying the magnitudes. Together you get 15 and you would add that two angles 30 minus 30 gives you zero. And, as you might have guessed, division of polar form complex numbers is the easiest path to take. You simply divide the polar magnitudes of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient and subtract the angle of the second complex number from the angle of the first complex number to arrive at the angle of the quotient. So a few examples just do cement the idea. Here we have ah phaser 35 65 degrees, being divided by a phaser of 10 at minus 12 degrees. You would divide 3 35 by 10 which would give you 3.5 and you'd subtract 12 from 65 which is same as adding 12 to 65 which would give you 77 degrees. In this example, we have ah phaser at 124 magnitude at 25 degrees, 250 degrees, being divided by off ah phaser of magnitude 11 at 100 degrees. So you would divide 124 by 11 which would give you 11.273 and you would subtract 100 from 2 50 to arrive in 150 degrees. And a final example would be, ah, phaser three at 30 degrees, divided by five at minus 30 degrees, you would have three divided by five, which is 50.6 and then you. It's attract 30 degrees from subtract minus 30 degrees from 30 degrees is the same as adding 30 degrees to 30 degrees, which would result in a new angle of 60 degrees. So the resultant quotient would be 0.6 at 60 degrees. To obtain the reciprocal or invert a complex numbers simply divide the number in in his poor for into the value of one, which is nothing more than the complex number one at zero degrees. So in these examples if you want to invert the fazer 35 at 65 degrees, you'd put 1/35 at 65 degrees, which is the same s dividing one at zero degrees by 35 at 65 degrees. And that would be one divided by 35 which would give you the magnitude of 350.2857 And he had subtract 65 from zero, which would give you minus 65. In the next example, you would have the reciprocal of 10 at minus 12 degrees, which would be 10 degrees, divided by 10 at minus 12. One, divided by 10 is 100.1 zero. Subtract minus 12 degrees is 12 degrees and the final example one. Uh, the reciprocal of 0.32 at 10 degrees would give you 11 divided by 110.32 which would give you the 312.5. And then you would take zero and subtract 10 degrees, which would give you minus 10 degrees. So as you work through a long string of complex numbers, arithmetic always add and subtract and rectangular form, then multiply and divide in polar, for it will be necessarily convert from one form to the other. As you progress through these calculations, however, with the invention of the smartphone and and laptop computers, there's lots of APS out here that will do that hard work for you. But it helps to know the process so that if you run into a problem, you can always find out what's going on. These are the basic operations you'll need to know in order to manipulate complex numbers in the analysis of a C circuits. Operations with complex numbers are by no means limited just to addition, subtraction, multiplication division and version. However, virtually any arithmetic operation that could be done with scaler numbers can be done with complex numbers. This ends Chapter two. 5. Ch 3 Reactance and Impedance R, L, C Circuits: Chapter three reactant. Send impedance R L C. Circuits. If we were to plot the current and voltage for a simple A C circuit consisting of ah, a C voltage source acting on a resistor, it would look something like this because the resistor simply and directly resists the flow of electrons at all periods of time. The way for for the voltage drop across the resistor is exactly in phase with the wave form for the current. Through it, we can look at any point in time along the her the horizontal axes of the plot and compare those values of current and voltage with each other. Any snapshot look at the values of the wave are referred to as instantaneous values, meaning the values at an instant. In time when the instantaneous values of current is zero, The instantaneous value across the resistor for voltage is also zero. Likewise, at the moment in time where the current through the resistor is at a positive peak, the voltage across the resistor is also had a pot positive peak and so on. At any given point in time. Along the Ways homes law holds true for the instantaneous values of voltage and current and the voltage phasers are plotted in the operating and corner there. And you can see that when the voltage source is at zero, so is the current. We can also calculate the power dissipated by the resistor and plot. Those values, along with the voltages and currents, know that the power is never a negative value. When the current is positive above the line, the voltage is also positive, resulting in the power equally I times are conversely, when the current is negative below the line. The voltage is also negative, which results in a positive value for power. Negative number multiplied by a negative number equals a positive number. This consistent polarity of power tells us that the resistor is always dissipating power, taking it from the source and releasing it in the form of heat energy. Whether the current is positive or negative, the resistor still dissipates. Energy in doctors do not behave the same as resisters, whereas resist er simply opposed the flow of electrons through them. By dropping the volt is directly proportional to the current in. Doctors oppose changes in the current through them by dropping a voltage directly proportional to the rate of change of current. This is this induced voltage is always such a polarity as to maintain the current in his present value. That is, if the current is increasing in magnitude, the induced voltage will push against the electron flow. If the current is decreasing, the clarity will reverse and push with the electron flow. To oppose this decrease. This opposition to current changes called reactant rather than resistance, expressed mathematically. The relationship between voltage dropped across the induct er in the rate of change of current through the induct ER is, as this formula indicates, the voltage is equal to minus L. D I by BT. The minus sign indicates that the voltages opposing the change of current flow the expression D I by D. T is one from calculus, meaning the same thing as Delta II over Delta T as Delta goes to zero. So it's really the change of current over the change of time. So this ah meaning the rate of change of instantaneous current overtime in amps per second . The induct INTs l is in Henry's, and the instantaneous voltage, of course, is in volts. If we plot the current and voltage for this simple circuit, it would look something like this. The circuit has an A C source of voltage E t. Acting on an induct er with on induct INTs of l remember, The voltage drop across the in doctor is a reaction against the change in current through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current is at a peak zero change or level slope on the current sine wave, and the instantaneous voltage is at a peak whenever the instantaneous current is at a maximum change, the point of steepest slopes on the current wave Nadia's where it crosses the zero line. This results in a voltage that is 90 degrees out of phase with the current weight looking at the graph, the voltage wave seems to have a head start on the current wave. The voltage leads the current, and the current lags behind the voltage. The phasers would look like this where the voltage phaser would be leading the current fazer. Remember, now that the days rotation is counterclockwise an induct, Er's opposition to change in current translates to an opposition to alternating current in general, which is by definition always changing in instantaneous magnitude and direction. This opposition to alternating current is similar to resistance, but different in that it is always results in a phase shift between the current and the voltage, and it dissipates zero power. Because of the differences, it is a it has a different name. Reactant reactions to a C is expressed in homes, just like resistance is, except that it's mathematical symbol is X instead of our to be specific reactant associated with him. Doctors is usually symbolized with a capital letter x with subscript l as showing here since, and doctors drop voltage proportional to the rate of change of occurred. They will drop more voltage for faster changing currents and less voltage for slower changing currents. What this means is that reactant in homes foreign doctor is directly proportional to the frequency of the alternating current. The exact formula for determining reactant is as follows reactions. Foran in doctor is equal to two pi f l where f is the frequency in hertz or cycles per second. Pi is the infamous number for Pi 3.14159 and l is equal to the induct INTs in Henry's Let's consider now a Siri's R L circuit with a supply voltage of E t, which is, of course, sign you sidle or a C voltage source. The resistor will offer resistance to the A C current, while the inductive ER will offer resistance in the form of reactant to the A C current. Because the resisters resistance is a real number in there and the in doctors reactant is an imaginary number. The combined effect of the two components will be an opposition to current equal to the complex. Some of the two numbers this combined opposition will be a vector. It will be a factor combination of resistance and react INTs in order to express this opposition succinctly, we need a more comprehensive term for opposition to current than either resistance or reactant. Salone. This term is called impedance. It is symbolized, as with, the letter said, and it is also expressed in units of homes, just like resistance and react. It's so the term or the impedance is also a phaser. Let's see how we calculated the voltage around the loop. According to Kirch, house voltage law is the supply. Voltage is equal to the voltage drop across the resistor plus the voltage drop across Thean doctor and their phaser values and must be treated as such. The current in the loop must be the same for all components because it is a Serie circuit and the current remains the same at all times. If we take the voltage equation and divided by the current I, which is the same in each one of the components, we are left with an equation of react. Ince's and impedance is because e over I is a reactant or an impedance or a resistance. The term e al over I is the reactions excel the term e r, which is the resistance over the current isjust. The reactant of the resist ER or just the resistor itself are added together using phaser addition because they are at 90 degrees to each other, they provide the result in Peens, said T for the total Olympians of X R plus Excel. So we have a way of calculating the impedance of two components on in doctor and a resistor . Thean doctor is at 90 degrees to the rial component of resistance, so you can use the Pythagorean theorem to calculate what's that is using the rial value are or the resistance and the reactions of the in doctor X out. Okay, lets try and cement some of that theory with an example. Let's put some real numbers and into a series R L circuit and just see how it works in our circuit. Here the resistor is five homes and the doctor is 10 Milli Henry's and the frequency that we're working with is a standard North American frequency of 60 hertz, or 60 cycles per second. In this example, Excel is given by two pi times the frequency times the induct its which is two times 3.14159 times the frequency, which is 60 times 10 Milli Henry's and 10 Milli Henry's is 10 over 1000 Henry's So we multiply that was out and end up with on a reactant for our in doctor of 3.7699 homes. Therefore, the resistance are sorry. The impedance. A total impedance of our L in Siris is the five own resistance added to 3.7619 homes. Inductive react INTs. Now remember these air both phasers, so they have to be treated as phasers so they are. The resistance is five homes along the real axis, which is at zero degrees, so it's five at zero degrees. The reactions is 90 degrees. So it ISS 3.7699 at 90 degrees. And we want to add these two together. So we're gonna convert them to um to a rectangular notation, which is five plus J zero homes plus zero plus J 3.7699 homes, which, when added together, will give us five plus J 3.7699 And if you want to convert that to a polar notation, it's a magnitude of 6.262 homes at 37.16 degrees. As with the purely inductive circuit, the current wave lags behind the voltage wave of the source, although this time the leg is not as great, only 37.16 degrees as opposed to a full 90 degrees, as was the case in a purely inductive circuits. In this case, the current is modified by the fact that we put a resistor in series with the with Thea Reactant sort of the in doctor and that has a tendency to straighten out the leg for the resistor and the in doctor individually and their voltages. The phase relationship between the voltage and current hasn't changed. Voltage across the resistor is in phase zero degrees phase shift with the current through it, and the voltage across the doctor is still 90 degrees out of phase, with the current going through it. Let's revisit a D. C. Circuit momentarily and have a look at how Lac Pasture reacts to being charged. We know that the voltage across a capacitor is dependent on its charge. That is, the voltage across the capacity was given by Q Oliver C, where Q is the charge and C is the capacitance and BCS that voltage across the capacitor. That equation can be rewritten as Q is equal to see times V. C, which I have the same variables now at time. If we measured the charge and the voltage a time one, we would have this equation. Cuban is equal to the same capacitor of with the voltage at the voltage at that time equals one. If we measured the voltage again at time to s things air charging, we would have a second value of charge, which would be equal to see the capacitance times the voltage at that time as well. So if we essentially subtract the equation two from Equation one, we could say that que tu minus Q one all over t to minus T one, which is equal to Delta Q. Over Delta T is equal to the same voltage difference. In other words, as the charge changes over a period of time, the voltage given by the equation Q is equal to C V. Sub C would give us Delta's V over Delta T the same Delta time, but Delta Q over Delta T by definition is equal to the current of the charging capacitor, which is still equal to see Delta V. C over the Delta T or in terms of calculus, we have this term. Current is equal to the capacitance Dif devi all over DT and and the D is just the change in voltage or the change in time. What this formula tells us is that the rate of flow of electrons through a capacitor is directly proportional to the rate of change of voltage across the capacitor. This opposition to voltage changes another form of reactions, but one that is precisely opposite to the kind exhibited by in doctors expressed mathematically. The relationship between current through a capacitor and the rate of change of voltage across the capacitor in the way of review is given by I is equal to D E all over d T. Where D is the change of voltage over the time period. D t The expression D E over DT is one of calculus, meaning the same thing as Delta II over Delta T just is the way of a review. And as Delta, the Delta goes smaller and smaller, it becomes more significant. Voltage e overtime is in volts per second, Capacitance C is in fair ads, and the instantaneous current I, of course, is in APS. Remember, the current through a capacitor is a reaction against change in voltage across it. Therefore, the instantaneous current is zero. Whenever the instantaneous voltage is at a peak, zero change or level slope on the voltage sign way, and the instantaneous current is at a peak. Whenever the instantaneous voltage is at a maximum change. The points of steepest slope on the voltage wave where it crosses the zero line. The results are similar for the negative side of the wave. This means that the voltage wave is 90 degrees out of phase. With the current wave looking at the graph and the current wave seem to have a head Sorry. Looking at the graph, the current wave seems to have a head start on the voltage. Wait, the current leads the voltage or the voltage lags behind the current. If we draw the phasers for the voltage, incur a current, it becomes obvious a capacitors. Opposition to the change in voltage translates to an opposition to alternating voltage in general, which is by definition always changing in instantaneous magnitude and direction. For any given magnitude of a C voltage at any given frequency, a capacitor of given size will conduct a certain magnitude of a sea current, just as the current through a resistor is a function of voltage. Acosta, resistor and the resistance offered by the resistor. The A C current through a capacitor, is a function of the a C voltage across it and the reactant offered by that pasta as within doctors. The reactions of the capacitors is expressed in homes and is symbolized by the letter X or X subscript C to be more specific. Since capacitors conduct current in proportion to the rate of change of voltage, they will Passmore current for faster changing voltages as the charge and discharged to the same voltage peaks in less time and less current for slow, slower changing voltages. What this means is that reactant in homes for any capacitor is inversely proportional to the frequency of the alternating current. In other words, the reactant XSI is equal to 1/2 pi F C, where F is in frequency or hurts sack cycles per second. Pie is our infamous 3.14159 and capacitance is in ferrets. Note that the relationship of capacitance are capacitive. Reactions to frequency is exactly opposite from the inductive reactant capacitive reactant in homes decreases with increased Pacey frequency. Conversely, inductive reactant in homes increases with increased A C frequency in. Doctors oppose faster changing currents by producing greater voltage drops. Capacitors oppose faster changing voltage drops by allowing green greater currents. Let's consider a Siri's R C circuit now with a supply voltage e t. Generating a new alternating voltage, the resistor we'll offer resistance and we'll call it X R, which is equal to the resistance of the resist ER, while the capacitor offer offers resistance in the form of reactant, which we'll call X C and and we just saw that that was equal to 1/2 pi uh, F C. Because the resist er's resistance is a real number on the capacities, reactant is an imaginary number. The combined effect of the two components will be an opposition to current equal to the complex some the complex. Some of the two numbers, which is impedance and impedance, is given as zed subscript t for the total impedance, and that's equal to the fazer, some or the complex some of X r plus x c. Now the voltage around the loop, according to Kirchoff, Voltage law, is equal to the supply. Voltage is equal to the voltage drop of the resistor, plus the voltage drop of the capacitor. Given that the voltage drops are complex numbers, their phasers and they have to be added that toe really or as complex numbers. The current in the loop is the same because again it is a Siri's circuit, so the current at any one time in a Serie circuit is always equal. So the total current is equal to the current in the resistor is equal to the current in the capacitor. If we take the voltage equation and divide each of the voltages by the current, we would be left with this equation. E t all over. I is equal to e. R all over I plus you see all over you see over I is the reactant of the passenger e r. All over I is the resistance of the resist er or the reactions of the resistor which is just resistance. When added together, give us the impedance and they have to be added Victoria Lee and that would give us the total impedance for the circuit is x r plus x c x are being along the rial axis X C being at minus 270 degrees. Let's look at an example in our example here we have a resistor of five homes in Siris with a capacitor of 100 micro fare ads, uh, being energized in a system of a frequency of 60 hertz or 60 cycles per second. In this example, since the reactions xsi is given by 1/2 pi f C. The reactant is going to be 1/2 times 3.14159 times 60 times 100 over one million seems we're dealing in micro fare ads, and that calculates out to 26.5258 homes. When added together, that will give us a total impeding. So we total impedance is made up of five homes of resistance and 26.5258 homes of capacitive reactivates, or that is the impedance is made up of five homes at zero degrees plus 26.5258 homes at minus 90 degrees or five plus J zero homes, plus zero minus J. 26.5258 homes. That's just given it in terms of a rectangular coordinates and that can be either written as five and minus J. 26.5258 homes or 26.993 homes at minus 79.3 to 5 degrees. So impedance is are related to voltage and current, just as you might expect in a manner similar to resistance. You know it was Laura. The voltage is equal to the current times the impedes and that that equation can be converted or manipulated, such that the current is related to the voltage over the impedance. In other words, I is equal to E all over said and said can be found from the village Drop across the impedance over the current through the impedance. These air, similar to what we're doing with resistance, is the only thing is Now we're dealing with complex numbers and all quantities, our phasers or their complex numbers. And you have to use the rules of addition, subtraction, subtraction, multiplication and division, using complex numbers and using phasers. Let's work through an example just to see how impedance is related to the current. And the voltage is as we just described. Here's an example of an R. L C. Siri's circuit. We can now determine the equivalent and Peens of the circuit. The first step is to determine the reactions is in homes for the in doctor, and we know that we're dealing with a 60 Hertz system. The Excel is given by the equation two pi f l. And if we convert to PFL to pure numbers. That's two times pi times 60 times 650. Miller hurts go through the math and at work so to 245 0.4 homes. The reactions of the capacitor is given by 1/2 pi F C. So XY is going to be 1/2 times pi times 60 times, 1.5 micro fare ads And if you do the math, that works out to one point 7684 Caleb's. The impedance of the resisters is going to be given by 250 homes, plus J. Zero, because there's no reactive component, just a real component. The impedance of the in dr There is no real component, so the first heart of the impedance number is zero, and it's plus J times. The reactant switches 245.4 homes and said, See, is again. There's no real component, just a reactive component, and this time it is a minus J. Because it's a capacitor and it's minus J times 1.7684 que OEMs. Now those three teens is can also be written in polar notation. Uh, the impedance for the resistor is 250 homes at zero degrees for the um for the in doctor, it's 245.4 homes at 90 degrees. And for the capacitor, it's 1.7684 k OEMs at minus 90 degrees. Now, with all the quantities of opposition to the electric current expressed in complex number format, they could be handled the same way as we handle plane resistances in a Siri's D. C circuit. We can draw up an analysis table for this circuit and insert all the given figures. Total voltage and the impedance is of the resistors and doctors capacitors. Unless otherwise specified, the source voltage will be our reference for phase shift and so will be written at an angle of zero degrees. Remember that there is no such thing as an absolute angle. Ah, phase shift for a voltage er current, since it always it's always a quantity relative to another way. For phase. Angles for impedance is, however, like those of resistors in doctors. And capacitors are no. One, absolutely because the phase relationship between voltage and current at each component are absolutely defined. Notice that we are assuming a perfectly reactive and doctor and capacitor with impedance is phase angles of exactly plus 90 and minus 90 degrees, respectively. Although riel components won't be perfect in this regard, they should be fairly close for simplicity will assume perfectly reactive and doctors and capacitors from now on, in my example calculations except where noted otherwise since the above example circuit is a serie circuit, we know that the total circuit impedance is equal to the sum of the individuals. So the total impedance is going to give be given by the some of the resistance impedance the reactor, our sorry, the in Dr Impedance and the capacitor and Peens. So in terms off numbers, that's 250 plus j 245.4 minus J 1.7684 k OEMs doing the math for those numbers. The total impedance is written in rectangular form 250 OEMs minus J 1.5 to 33 k OMs or 1.5437 k OEMs at minus 80.680 degrees. We can now plug that into our chart, we can now apply homes law. I is equal to e over Zed in a total column to find the total current in this series circuit . Now the the M designates millions since the impedance was in que OEMs. Now being a serie circuit current must be equal through all components. Thus, we can take the figure of team for the total current and distributed to each of the other columns. No, we were prepared to apply homes law again, this time solving for the voltage drop across the individual impedance is we use e as equal to I times said, and we can fill in across the top of the table for the voltage drop across the resistor across the in doctor and across the capacitor. So we have successfully used olmes law and complex numbers, or phasers and impedance is to calculate and solve for the current and the voltage drops in a Siri's R. L C circuit. Notice something strange here, though a lower supply voltages only 120 folds. The voltage across the capacitor is 100 and 37 point for six bowls. So how can this be? Well, the answer lies in the fact that when you connect, impedance is up s and those impedance is include capacitor in doctrines. There is a resident frequency that will allow the voltages to essentially ring back and forth, building and discharging to mount to, ah, voltage level. That's in this case 137 point for six. Now, the theory and the way this happens is the subject of another lesson. However, you don't have to worry about it. If all you're wanting to do is calculate the voltage drops in the currents. The math looks after that for you, and so this ends another chapter. Chapter three. 6. Ch 4 Kirchhoff's Laws in AC Circuits: Chapter four Kirch Offs laws in a sea circuits. Now this is going to be a short chapter, but I think it's significant enough to stand on its own, and it will relieve some of the length thickness of the chapters on either side of it. So I thought I would let it stand out by itself. Kirchoff Voltage Law provides us with a check on our calculations on the example in our previous chapter to demonstrate Kirchoff Voltage Law in an A C circuit. We can look at the answers we'd arrived for. Component voltage drops in the last circuit kerchiefs. Voltage Law tells us that the algebraic sum of the voltage drops across the resistance. Thean Doctor and the capacitor should equal the applied voltage from the source, even though this may not look true at first sight. A bit of complex number edition proves otherwise. So the voltage drop across the resistor was given by 3.14 72 plus J 19.177 volts. The doctor has a voltage drop of minus 18.7 nine seven plus jay 308 for eight volts and the capacitor is one 135.65 minus J 22 points 262 If you add up the rial components, they do indeed add up to exactly 120. And the imaginary components or the J components do indeed add up to zero, which is equivalent to our supply voltage, thus providing a check on our calculations. So let's move our individual components to a parallel circuit now. So we now have the 120 volt 60 hurt power supply directly connected across all of the three components. The impedes is of the components don't change. They're still the same and so I've written them in there. The fact that these components are connected in parallel instead of Siri's has absolutely no effect on their individual. Impedance is which should not be a surprise so long as the power supply is the same, which it iss, the inductive and capacitive reactions is will not have changed at all. With all of the component values expressed as impedance is, we can set up an analysis table and proceed, as in the last example problem with the Siri circuit, except this time following the rules of parallel circus instead of the series circuits. Knowing that the voltage is shared equally by all the components in the parallel circuit, we can transfer the figure for total voltage to all the components in the table. It is equal to the supply voltage we can now apply. Homes law, eyes equal to E all over, said vertically to each column to determine that current through each component. And again, the M designates Millie APS. From here, there's two strategies for calculating the total current in total in Peens. First, we could calculate the total impedance from all of the individual. Impedance is in parallel. In other words, the total impedance would be given by one over the quality one over Zen are plus one overs at L plus one overs and see and then calculate the current by dividing the source voltage by the impedance. However, working through the parallel impedance equation with complex numbers is no easy task. To be done with all the reciprocation is and changing back and forth from rectangular to poor form. Again, it can be done, but it it's very labor intensive. The second way to calculate the total current total impedance is to add up all the branch currents to arrive at a total current, remembering that the total current in a parallel circuit A, C or D. C, is equal to the sum of the branch currents. According to Kirch Offs current law, then you zones law to determine the total impedance for the total voltage and the total current. This ends Chapter four. 7. Ch 5 Power flow in AC Circuits: Chapter five power flow in a sea circus. Let's consider for a moment a pure inductive load in an A C circuit. Because instantaneous power is the product of the instantaneous voltage and the instantaneous current that is P equals I times e. The power equals zero whenever the instantaneous current or voltage is zero. Whenever the instantaneous current and voltage are both positive above the line, the power is positive. As with resistors, the power is also positive when the instantaneous current and voltage is are both negative below the line. However, because the current and voltage ways are 90 degrees out of phase, there are times when one is positive, while the other is negative, resulting in equally frequent occurrences of negative instantaneous power. Notice on the graph here that the power goes from zero to a maximum back through zero up to a positive maximum, back to zero down to a negative maximum as the current and the voltage is swing through their positive and negative crests. You'll notice that when the current slope is zero at the crest of its wave, the voltage is just crossing zero lines of voltages. Zero hence, power is zero and indeed dots with the Green Line shows here again, where the current slope is equal to a maximum. In other words, is it's slope is out of maximum slope. Where's it where it's crossing zero voltages? A maximum. But because current is zero at this point, so is the power zero. Another zero point for the power is where the current slope, maybe a maximum and the voltage maybe a maximum. But the current slope are the current is zero, so the power is zero and where the current slope is zero and the voltage is zero. And of course we have zero power. Now let's consider a pure capacitive load in an A C circuit. As you might have guessed, the same unusual power way that we saw with a Simple and Dr Circuit is present in a simple composite your circuit, as well as with a simple and Dr Circuit. The 90 degree phase shift between the voltage and a current results in a power wave that alternates equally between positive and negative. This means that capacitor does not dissipate power as it reacts against the changes in voltage. It merely absorbs and releases power. Alternatively and a couple of highlights to look at on this graph when the current slope is zero. When the current is maximum voltage at this point, happens to be zero, so the power would be zero. At this point, the current slope is a maximum, but it is also the current is equal to zero. Even though the voltages maximum, the power of put will be zero and at this point the current slope is a maximum. But it is zero and even though the voltage is at a maximum, the power is zero and at this point current slope is equal to zero and the voltage is equal to zero. So the power would be zero at this point as well. Clearly it can be seen that at times the power is negative. But what does negative power mean? It means that Thean doctor, after having built up a magnetic field while the current was flowing into the induct ER, is now releasing power back into the circuit. While a positive power means that it is absorbing power from the circuit as it builds its magnetic field. Since the positive and negative power cycles are equal in magnitude and duration over time Thean doctor releases Justus much power back into the circuit as it absorbs over a span of a complete cycle. What this means, in practical sense is that the reactant oven in Dr Dissipates a net energy of zero while unlike resistance, the reason of a resistor, which dissipates energy in the form of heat. Mind you, this is for a perfect in Dr Onley, which has no wire resistance at all. This is the same thing that happens with a capacitive circuit on Lee. The power flow in and out is due to buildup of electric static field while a current was flowing into the capacitor, then releasing power back into the circuit during the negative power cycle. For a moment, let's talk about instantaneous power. What you see in front of you are or is too basically Sinus soil wings one representing the current, which is in red and one representing the voltage, which is in black. Now the voltage starts at zero, goes up to a maximum, comes back through zero, goes to a minus maximum and so on and so on. And it's a nice uniform sign your so little wave, the current, depending on the impedance of the circuit, of course, will either be shifted laughter right? Ah, certain amount. So we will. We would either say that the current is going to be leading or lagging a voltage, but it's represented by this wave shape on the on the time Kerr or on a time graph. Now if if we indeed want to calculate what the instantaneous power and anyone moment ISS, what we should do is take a point on the graft at a particular time, and we would say the instantaneous power is given by the voltage at that point and a current let's have a look at that. We'll take a slice of time right here, and we will say that the power at that particular time is being dissipated is given by the voltage, the instantaneous voltage and the instant times the instantaneous current, and that can either be a positive or a negative number. Or it could be zero depending, as we've seen in our examples in the previous slides, where on the graph that you're gonna actually take your instantaneous power. So that's not a big deal. I mean, if you want to get instantaneous power. However, in power circuitry and empower systems. Instantaneous power isn't always a thing that we're after. What we are usually looking for is average power. In other words, what's the average power consumption of, Ah, hot water here? What's the average power consumption of an industrial motor? So the average power is the thing that becomes significant. It's a thing that we would like to deal with on a very simple basis. So let's take a current and the voltage and apply it to a circuit, and thus this pursue me. The circuit is made up of some reactive material, and it's made up of some resistant material. So indeed, the current could be represented by the red Sinus idol curve there, and the voltage could be represented by the green, a sign you sidle curve there and you can see that the voltage is following the current by a lag of a certain amount. And the power, as we've seen in our previous slides, is given by at least the instantaneous value is given by the instantaneous voltage times, the instantaneous current, and at times it can be positive, and at times it can be negative. The voltage Sinus idol wave can be described in mathematical terms or triggered a metric terms or algebraic terms. Whatever you wanna call it, the voltage can be the maximum voltage where VM is the maximum voltage times sine omega T, where t is sometime in along the time curve on the bottom in seconds, and the term omega is actually related to the system frequency. We call it angular velocity, but I don't want to get hung up in terminology is just that it's related to the frequency of the system, and it's actually a term that is in degrees per second. So if we multiply it by t seconds, we're left with a an angle. So Omega T changes as we move along the T lying, generating an angle that goes sign you sidle from positive maximum to negative maximum cross zero so we can describe the voltage in terms of the maximum point of that voltage times sine omega T. The current can also be described the same way as a maximum current time sine omega T. However, it is ahead of the voltage by a certain phase shift or phase angle. We'll call that five for now, so it's a five is a set angle. So we have to include that in our formula so that we can now place the current curve somewhere along the T line with respect to the voltage. If we multiply the left hand terms together p that would give us the instantaneous power for the circuit and that right hand side, we become voltage maximum times a current maximum times sine omega team times sine omega T minus five. Now we're getting a little bit complicated in formulas, and I'm not that interested in the getting to the final results as I am of the final results. What I want to do is end up with a formula that will calculate how much power real power is consumed by my circuit, and the amount of real power that is consumed by the circuit is showing in purple. In this diagram here, everything above the positive line for power is power consumed all the yellow stuff below. The zero line is what we call reactive power. It's where the reactive components are absorbing the power into their either fields of some sort, whether it's a magnetic field or it's a electrostatic field. In the case of the capacity. Regardless, it pulls the the power in, but then it releases it back to the circuit. So we have a net effect of zero power. All we're dealing with reactive. We are mostly in interested in positive power consumption. This power consumption is we call it the average power consumption of an A C circuit, and it would be the power that would be supplying, say, a resistive load like baseboard heaters that could be a hot water tank. Or it could be a motor that is, has no reactive awards. Just we're just want to calculate the amount of power that the motor is delivering mechanically in in the form of mechanical power or some of the heat losses. Regardless, we are interested in in the rial power consumption. So what we're interested in is what is the average of that purple section? That's their We need a formula that would say, OK, this is the power, the average power that we're consuming for that particular load. No, the way we get that P average is to take that term that we've just developed there, the instantaneous power, and we want to manipulate it so that we kind of level off the peaks and filling the valleys , and somebody's done that hard work for us. I'm not going to go through it. The average power through a trigger metric and integration workings mathematically, you can get that P average, which is really power consumption with this formula. In other words, the P I formula boils down to pee. Average equals the voltage maximum times the current maximum all over two times co sine of the angle between the current and the voltage. So we have a formula for calculating real power consumption in a circuit, which is we designated as P average. It's the average of the purple humps in that diagram that we have there, and that is actually a a measure of the real power consumed by the circuit. Whether that's heat, light, mechanical energy or whatever it's given by the form that P averages equals the M. I am over to co sign five, and if we were to draw a right angle triangle, who's contained? Angle is fi. Then we would have hi pot news for now would be VM. I am over to because VM I am over two coats, fi is when RP averages, so the high partners would be given by that term. VM I am over to the quad richer or the perpendicular side of the triangle to pee average just happens to be what we call Q and cue averages given triggered a metric Lee speaking by that triangle VM I am over to sign five. No, it just so happens that this Q average happens to be the average reactive power that's in the yellow humps on our diagram. And that is the power that's pulled out of the system but then pumped back into the system with a net exchange of zero rial power, the P average. We've already started discussing it in terms of real power and it Israel Power, and it is ready. Represented in Watts and designated by the letter P Que average is what we call reactive power, and it's usually designated by Q, and it's measured in VARs the term VM I am all over, too, is what we call a parent power or and it's designated as s. And it's measured in V A sometimes K V A or M V A in bigger circuits. But it is apparent power, and it's just the product of a voltage and current. Now, from our previous slide, we have what we call the power triangle, made up of real power with a quad richer of reactive power that gives us a measure of are what we call apparent power. The fly angle is known as the power factor angle and co sign of that angle. Co sign Phi is known as the power factor, so we have a way of calculating with the real power Have riel. Average power consumed in the circuit is it's V M. I am over to co sign Phi or Phi is the angle between name voltage in there Current and V M and I am are the maximum values of the sign you soil waves that were using to describe the voltage. In the current, we can take VM I am over two and split it out into two terms and still not change the meaning of the equation. We could say that the P averages equals V M Over route two times I am over route to Times co sign data, but you remember that the army's value of voltage is equal to the m over route to and the RMS value for I is I am over to. So the average power consumption riel power consumption in the circuit is if you measure the voltage in RMS and the current in RMS terms and multiply it by the phase angle the coastline of the phase angle between the two of them you come up with p average. Similarly, the que average could be calculated using the same v rms times where I rms this time sine of the angle between them and the apparent Power v a can be you calculated using just the v rms times the i r. On this, let's look at what is sometimes referred to as an electric pendulum. We're gonna look at applying a d. C voltage, and soon as we have a look at that, we'll swing over to how it reacts with a C. But it helps to look at holla parallel Elsie Circuit works in terms of an applied DC voltages are D C voltage capacitor store energy in the form of electric fields and electrically manifests that stored energy as a potential static voltage in Dr Store energy in the form of a magnetic field and electrically manifest that stored energy is kinetic motion of electrons, Current capacitors and in doctors are flip sides of the same reactive coin, storing and releasing energy and complementary modes. When these two types of reactive components air directly connected together, they're complementary tendencies tend to store energy and will produce an unusual result. If we assume that both components are subject to an application of voltage, say from a battery, as you see here, the capacitor will very quickly charge. And the induct ER will oppose the change in current, leaving the the pasture in a charge date and the doctor building its magnetic field and building the current flow through it. When the supply voltage is removed by opening a switch, the capacitor will begin to discharge, its voltage decreasing. Meanwhile, the in DR will begin to build up a charge in the form of a magnetic field and the current increases in the circuit. Thean doctor, still charging will keep electrons flowing in the circuit until the capacity has being completely just discharged, leaving zero voltage across it. And I have indicated the voltage e on the capacitor in blue and the current through the in dr by the red dotted line. So at this point in time, the pastor is fully discharged. But there is current flowing maximum maximum lee through the in. Doctor Thean doctor then will maintain the current flow even with no voltage applied. In fact, it will generate a voltage like a battery in order to keep current in the same direction. The capacitor being the recipient of the current, will begin to a community, a charge in the opposite polarity as before. When the in doctor is finally depleted of its energy reserves and the electrons come to a halt, the CA pasture will have reached its full voltage charge in the opposite polarity as compared to what it was when it was started. The capacitor, as before, will begin to discharge through the in dr, causing an increase in current in the opposite direction as before, and a decrease in the voltage as it depletes its own energy reserves. Eventually, the capacitor will discharge to zero volts, leaving the in dr fully charged in a full current. Through it, the in Dr Desire ing to maintain the current in the same direction will act like a source again generating a voltage like a battery to continue the flow. In doing so, the CA Pastoral begin to charge up and the current will decrease in magnitude in time. Eventually the capacitor will become fully charged again as the in Dr expand expends all its energy reserves trying to maintain the current, the voltage will once again be at its positive peak and the current at zero. This completes one full cycle of the energy exchange between the pastor and the in doctor. This oscillation will continue with steadily decreasing amplitude due to power loss from, say, straight resistance in the circuit until the process stops. If there is no resistance in the circuit, which is impossible, but it's, it can be hypothesized, this oscillation will continue to happen over and over again. Overall, the behavior is akin to that of a pendulum. As a pen, you know, mass swings back and forth. There is a transformation of energy taking place from kinetic motion to potential, my motion or height in a similar fashion to the way energy is transferred from the capacitor and Dr Circuit back and forth in the alternating forms of current kinetic motion of electrons and voltage potential electric energy. At the peak height of each swing of the pendulum, the mass briefly stops and switches directions. It is at this point that potential energy, the height is at its maximum, and the kinetic energy motion is that Jiro, as the mass swings back the other way, it passes quickly through a point where the string is pointed straight down. At this point, potential energy height is at zero, and the kinetic energy motion is at a maximum, like the circuit, a pendulums, back and forth. Oscillations will continue with steadily damping amplitude, the result of air friction or resistance dissipating the energy. Also like the circuit, the pendulums position and velocity measurements traced to sign weighs 90 degrees out of phase over time. In physics, this kind of natural sine wave oscillation for a mechanical system is called harmonic motion. The same underlying principle governs both the oscillations of the capacitor and Dr circuit and the action of the pendulum. It is an interesting property of any pen. You know that it's periodic time is governed by the length of the string holding the mass and not the weight of the massive south. That is why a pendulum will keep swinging on the same frequency as the oscillations decrease in amplitude. The oscillation rate is dependent on pawn the amount of energy stored in it. The same is true for the capacitor in Dr Circuit. The rate of loss elation is strictly dependent on the size of the capacity and doctor, not on the amount of the voltage or the current at each respective peak. In the ways the ability for such a circuit to store energy in the form of oscillating voltage and current hazarded the name a tank circuit. It's property of maintaining a single natural frequency, regardless of how much or a little how little energy is actually being stored in it. Is it a special significance in electric circuit design? However, this tendency to oscillate or resonate at a particular frequency is not limited to circuits exclusively designed for that purpose. In fact, nearly any a C circuit with a combination of capacitance and inducting is commonly called NLC. Circuit will tend to manifest unusually effects when the A C power source frequency approaches that natural frequency. This is true regardless of the circuits intended purpose. If a power supply frequency for a circuit exactly matches the natural frequency of the circuits of the Elsie combination. The circuit is said to be in a state of residence. The unusual effects will reach maximum in this condition for residents. For this reason, we need to be able to predict what the resonant frequency will be for various combinations of L and C and be aware of what the effects of residents are. A condition of residents will be experienced in a tank circuit when the react Ince's of the capacitor and induct ER are equal to each other. That is to say, when two pi f l is equal to 1/2 pi f C. Because inductive reactant increases with increasing frequency and capacitive reactant is decreases with increasing frequency, there will only be one frequency where these to react. Ince's will be equal to find that resident frequency. We simply solve the equation for F, and we find the natural frequency for this tank circuit is one over two pi times the square root of the the induct INTs, times the capacity. It's so let's look at a practical example in the above circuit. Let's say we have a 10 Micra fared capacitor and ah, 100 Milli Henry in DR Since we know the equation for determining you re enactments of each of the given frequencies and we're looking for that point where the to react, Ince's are equal to each other. We can set the two reactions formulas equal to each other and solve for the frequency algebraic Lee L. C. Is equal to 10 times 10 to the negative six times 100 times 10 to the minus three. Which gives gives US 0.1 The square root of that is 0.1 two pi times at his 20.6 to 8318 and 1/2 pi. The square root of that is 159 point 55 So there we have it, the formula to tell us that the resident frequency of a tanks or could, given the values of induct INTs in Henry's and the capacitance inference, plugging in to the values we arrive at the resonant frequency of 159.155 hertz. What happens at resident frequency is quite interesting with capacitive and inductive reactions is equal to each other. The total impedance increases to infinity, meaning that the tank circuit draws no current from the A C power source. We can calculate the individual impedance is of the capacitor, which works out to mathematically using the A resident frequency. The reactions of the capacitors 100 owns, and the reactions off the induct ER is also 100 homes using the um, resonant frequency that we calculated now the parallel in Peens, the total parallel impedance of the two impedance is or the to react. Ince's there is given by the Formula One over one over the sad l plus one over this and see . And if you work those through, you will end up with a final number of 1/0. What this means, in practical terms is that the total impedance of the tank circuit is infinite. Hence, be behaving like an open circuit. So let's hook up. Our are to react. Ince's now in Siris and a similar effect will happen. They will start to resonate when they're hooked up in Siris and the resonant frequency if we set it in 159.155 Hertz. We find that the impedance of the in doctor is plus J 100 domes, and the impedance of the of the capacitor is minus J 100 homes. The total Siri's impedance would be given by his NL plus at sea, which would be given by plus 100 homes plus J 100 homes put minus J 100 homes, which would equal zero, which essentially means that we have a short circuit and this is characteristic of a resonant frequency circuit. And we have to be careful when we're connecting these circuits up off the frequencies that we're dealing with. This ends Chapter five. 8. Ch 0 Intro AC Circuits: this course is basic fundamentals of a C circuit analysis. It is a standalone course, and it's takes you from the very beginnings of a C circuit analysis to the more sophisticated problems and formulas. There is a prerequisite to this course, which, if you don't have a basic understanding of D C circuit analysis, which is a requirement for this course, you should go and have a look at the course that's entitled Basic Fundamentals of Electricity and D. C. Circuit analysis that will give you a good foundation for learning what is going on in discourse. The course starts out by describing what a Sinus soy it'll wave form is and what it looks like in a sea terms. And we look at both voltages and currents in the time domain going on to define what RMS values or root mean square values are for current and voltage. We then leave the time domain and go in to talk about phasers and how we can use phasers as a tool for the analysis of current and voltage. In a sea circuits. We delve into the characteristics of phasers such as their length and mayor magnitude, their rotation and their frequency, how they work and how they rotate in what direction is and how you define the direction and the relative angles between the various phasers as their studied. We then have, ah, very close look at the two forms of describing a phaser, and that is the polar notation. With its magnitude angle as well as the rectangular notation. We then go on to have a look at reactant and impedance in resistance and doctors and capacity of type circuits. Starting with A C and Dr Circuits. We define the term reactant and how react Ince's calculated in terms of frequency and the induct INTs in Henry's. We then go on to define what is meant by impedance in our L circuits or resistor in Dr Circuits. From there, we do the same thing with capacitance. We define what reactant of capacitance is in terms of frequency and capacitance and fair ads, and then we define it in terms of impedance in resisted capacitive type circuits. Next, we will go on to study the flow of power in alternating current circuits, and we will look at the way instantaneous power looks in the time domain and we will define instantaneous power and define average power, and we will define or prove how we calculate average power in terms of current and voltage . Next, we go on to define what is meant by the Power triangle in terms off Rheal Power Watts, Reactive Power VARS and Apparent Power v A. And the last thing we will look at before ending this course is frequency resonance in an in an in Dr Capacitive type circuit, and that will bring us to a close of the course on basic electrical theory relating to a C circuits. 9. Ch 1 Alternating Current: in a previous course, we have been looking at D C circuits. That is where voltages and currents are, for the most part constant, at least as far as the supply sources are concerned. We're now going to go into the realms of a C circuits where voltages and currents are not constant but are changing rapidly in our system in North America, and they're changing 60 times a second at least assed faras. The generation side is concerned, in other words, to supply source. So this is Chapter one alternating current. With alternating current, it's possible to build electrical generators, motors and power distribution systems that air far more efficient than D C systems. And so we'll find a CR alternating current use predominantly across the world in high powered applications. The two diagrams it you see in front you are illustrating the comparison between a Seapower and D C power, the one on the left. The direct current circuit is showing a battery which is ultimately used for a direct current source, and you can see that the electrons flow is steady and it's flowing in one direction on Lee , and you have ah, polarity. That is constant across the system. In other words, the battery polarity is positive and negative. And if you're to measure the polarity across the load, which is a late Bobby, would measure positive and negative, and it would be steady. The diagram on the right is alternating current, and it's diagrammed such that it's has a generator. In this case, it's a hand driven generator, Uh, and if it is constructed in such a way that the current is alternating half the time it's flowing in one direction half the time it's flowing in the other direction. And if that changes fast enough like we have in North America 60 times a second, you don't even notice it in the flicker of a light bulb because it is too quick for your eyes or you in the light bulb thio thio null out and produce zero output. In a previous course wing, we studied Fair days law. We looked at a bar magnet that was thrusting in and out of a coil, and that was producing voltage half the time in one direction, half the time and the other Islamic. He kept that magnet moving. The voltage was oscillating back and forth so this would be in no uncertain terms, a generator of sorts, very inefficient one. And it was be a reciprocating type mechanical action that was producing all alternating current. And if you did that 60 times a second eyes, certainly you'd be duplicating or replicating the type of a C voltage that you would see in North America. However, that's definitely not practical. And in fact, generators are built a little bit different than that. We usually instead of a reciprocating a magnet. We have a rotating magnet, which will cut, in this case two coils that are set up in such a way that it'll push and pull the current in the direction that you want to go in. If the magnet is in this position as it rotates, there's no current flowing because there's no magnetic flux cutting the coils. But as we turn the magnet, you get a maximum cutting of the magnetic flux in the case that the Megan is horizontal, so you'll get a push in a poll of a current in one direction, and as it rotates, it will go back to zero again, and as it rotates further, you'll get a push and pull in the other direction. So as the magnet goes in, several are through several rotations. You will get a changing voltage, and if it's rotating at 60 cycles a second, you'll indeed get something similar to what you get in the North America. And in fact, this is a very crude diagram of, Ah, generator generators are designed in such a way that you get a nice sinusitis low put and the control is is is governed so that it only turns at 60 times a second. And it's all packaged in a nice package such as you might see here in the way of a portable generator. When you get to bigger generators, of course, for the whole systems you take in the next step of practicality, and you start, you generate voltages and transmission in terms of three phase systems, which we'll cover in other courses. But right now we're only dealing with single phase A C circuits. And by the way, the theories that you will learn in this course are very similar and in fact are just extrapolated into three face system. And the solutions you'll find are pretty much the same, except that you have to take into consideration that you have three voltages and three currents rather than one, and you know you have to know how they're related. However, that's getting ahead of ourselves. We're only talking about single face systems here. There is an effect of electromagnetism. No one as mutual inducted its whereby two or more coils of wire that air placed so that the changing magnetic field created by one induces a voltage in the other. If we have to mutually inductive coils and we energize one coil with a C, we will create an A C voltage in the other coil. When used as such, this device is known as a transformer. The fundamental significance of a transformer is its ability to step voltages down or to step it up from the power coil to the unpowered coil. The A C voltage, induced in the unpowered secondary coil, is equal to the A C voltage across the powered. The primary coil, multiplied by the ratio of the secondary coils, turns to the primary coil turns. The transformers ability to step a C voltage up or down with ease gives a C on advantage unmatched by D. C in the realms of power distribution when transmitting electrical power along long distances, far more efficient to do so by stepping the voltage up and then stepping it down at the end . The the advantage here is you can use smaller dam in our wire with less resistant power losses in the transmission process. Transformer technology has made long range electoral power distribution practical without the ability to efficiently step up voltage or step it down, it would be cost prohibitive to construct power systems for anything but close range within a few miles. At most as useful as transformers are they on? Lee worked with alternating current, not direct current because the phenomenon of mutual inducted it's relies on the changing magnetic fields and direct current D C can only produce steady magnetic fields. Transformers simply will not work with direct current when referring to a seapower In terms of voltages and currents, we loosely refer to the term a C voltage is. When you think about it, it seems a little bit like a misnomer to call voltage alternating current voltage. But basically we refer to anything in regard to producing or being produced by alternating current as a C, so it's going to be helpful to understand what the's A C quantities look like. Generators of A C quantities or a sea power. Ultimately, a C voltage and a C current are designed in a special way such that the voltages and currents flow and follow what is known as a Sinus seidel way for and Sinus idle way forms. When removed from the time you mean are converted to rotating vectors or phasers, which lend themselves to trigger elementary funds functions that help in the analysis of a C circuits. The magnitude of A C quantities can be described in several ways, depending on the nature of the observation. The most useful of these is RMS, or root mean square values, which we are about to discover in this chapter. An alternator or generator is designed to produce a C voltage in a specific shape. Over time. The holding switches polarity over time, but does so in a very particular manner when graft. Over time, we call it the time domain. The wave traced by this voltage oven alternating, is a new alternating polarity from the ultimate ER, and it takes on a distinct shape known as a sine wave. Or we can describe the wave as Sinje Seidel. This name will become more meaningful later in the chapter when we're looking at waves in terms of factors or phasers in the voltage plot from an electromechanical alternator or generator that change from one polarity to the other is a smooth one. The voltage level changes most rapidly after the crossover point and more slowly at the positive and negative peaks, because this is a repetitive cycle. We described the wave in terms of cycles where one cycle is divided up into 360 parts or degrees. So here we have that way form, and it's divided up into 360 parts or degrees. And we're saying that the wave form goes from zero to a maximum back to zero to a negative maximum and back to zero again. And that maximum or minimum in this particular case is one and it's proportional, so this one could be one K V. It could be 100 K V because it's proportional. We're not putting a value on it right now. We're saying the maximum value is one or the peak value or negative. Peak value is one. So if we start this wave off at zero in its zero degrees, it starts off at zero. And if we go to 90 degrees where we are at a positive peak and the value is one and we're 1/4 of the way through the wave form at 180 degrees, which is halfway through the wave cycle, it is back to zero again and on its way to a negative value. At 270 degrees. We are at the negative peak or negative maximum value, and in this case it's minus one. And when we get to 360 degrees than the value is back to zero. If we were to follow the changing voltage produced by a coil in an alternator for from any point on the way graph to that point, when the wave shape begins to repeat itself, we would have marked exactly one cycle of that wave. This is most easily shown by standing the distances between identical peaks, but maybe measured between any corresponding points on the graph. The degree marks on the horizontal axis of the graph represents the number of degrees of the cycle and also the angular position of a simple to poll alternators shaft as it rotates . Since the horizontal axis of the graph can mark the passage of time as well as the shaft position in degrees, the dimension marked for one cycle is often measured in a unit of time, most often seconds or fractions of a second. When expressed as a measurement, this is often called a period. The period of a wave in degrees is always 360 degrees, but the amount of time one period occupies depends on the rate of voltage or current oscillations back and forth. A more popular measure for describing the alternating rate oven a C voltage or current wave than period is the rate of back and forth oscillations. This is called frequency. The modern unit for frequency is the Hertz abbreviated H set, which represents the number of waves cycles completed during one second off time in North America. The standard power line frequency is 60 hertz, meaning that the A C voltage oscillates at a rate of 60 complete back and forth cycles every second in other places in the world where the power system frequency is 50 hertz. The A C Voltage Onley completes 50 cycles every second. A radio station transmitter broadcasting at a frequency of 100 megahertz generates an A C voltage oscillating at a rate of 100 million cycles every second. Prior to the canonization of the Hurts unit, frequency was simply expressed as cycles per second. Older meters and electronic equipment often bore are still bore frequency units in CPS or cycles per second instead of hurts, a period and frequency are mathematically reciprocal of one another. That is to say, if a wave has a period of 10 seconds, the frequency will be 0.1 hurts, or 1/10 of a cycle per second. Frequency in Hertz equals one over the period. In seconds, we encounter a measurement problem. If we try to express how large or how small and a C quantity is with D C, where quantities of voltage and current are generally constant, we have little trouble expressing how much voltage occurring we have in any part of the circuit. But how do you grant a single measurement of a magnitude to something that is constantly changing? One way is to express the intensity or the magnitude in terms of its amplitude. Oven, a C quantity in this measurement is often referred to as peek measurement of the A C way form. Another way is to measure the total height between opposite peaks. This is known as peak to peak value oven a C wave form. Still, another way of expressing the magnitude of a wave is to mathematically average the values off all the points of the way forms graph to a single aggregate number. This amplitude measure is known simply as the average value of away form. If we average all the points on the way form Algebraic Lee, that is to consider their sign either positive or negative. The average value for most way forms is technically zero because all the positive points cancel out all of the negative points over a full cycle. However, as a practical measure of way forms, aggregate value average is usually defined as a mathematical mean of all the points, absolute values over a cycle. In other words, we calculate the practical average of the way form by considering all the points on the wave as positive quantities, as if the way for looked like this. The average value would then have some value other than zero that would be related to the intensity of the wave. So far, we have looked at three ways of measuring on a C way form. And as a matter of fact, every one of these can be converted to the other by a simple mathematical, constant calculation. And you can talk about peak or crest or peak to peak or mean average values and convert one to the other with by the multiplication of just one simple conversion number. The thing that you have to remember, of course, is what you're what you're measuring with and to relate that to the people or the persons who is receiving the information. They have to be totally aware off which one of these methods that you are using each one is good and will relate the information. But how useful one is is something that's open for discussion, and we will look at that as well as we communicate the values of voltage and current with others in the industry of electrical power, as well as the related quantities of power, energy ratings of different elements. We have to ask ourselves how useful are using any of these terms. And is there some way of measuring the values that is the most useful way? This question was asked and answered a long time ago. And the answer waas the RMS values and before just jumping into the definition of RMS, which, by the way, is also mathematically related proportionately to the term amplitude peak peak to peak average and mean average by a constant. Let's go there in logical steps. We're going to start with these two simple circuits. One is a D. C. Circuit on the left. The other is an A C circuit on the right. What designates them being A C and D. C as their source? Voltage one is a D. C. Source. Any other is an A C source. Other than that, the two circuits are identical. They both have the same light bulb, and the resistance of the light bulb is the same. And we're going to designate that as R l. If we close the switch on the D C circuit, current is going to flow through the late fall than the light bulb is going to light up to a certain amount of brightness, depending on the amount of voltage that we apply to the circuit and rooms Law will just a let a certain amount of current flow. Now, if we close the circuit, uh, close the switch and complete the circuit on the A seaside. Let's assume we can adjust that circuit so that the light bulb lights with the same intensity as the D C. Circuit. So it is going to be a certain amount of a C current flowing in that circuit. We don't know what that is just yet, but if the blight bulbs are the only source of dissipating the electrical load, which they which it is, then if they light the same amount of brightness and they're identical bulbs than a power dissipated in both circuits is the same. In other words, the average power D C power consumed by light bulb is equal to the average power consumed by the A C light bulb. What we would like to know is, what is that? A C current. Because of the two bulbs, light to the same brightness, dissipating the same power, then the current I A C must be equivalent in some way to the current in the D. C circuit. Now let's take another another step forward and take a closer look at that, a C current that's flowing in the circuit and we're gonna call that current I. And over a period of time that current rises toe a maximum, goes down through zero, goes to a negative maximum, goes back up through zero and up to a positive maximum, and then repeats itself over and over again in a stately state condition. The actual current itself will rise to a value, a maximum value. We'll call it plus a high max, and it will go to a negative maximum. We will call negative I max. If in a sea supply is connected to a component of a resistance, are the instantaneous power dissipated is given by the instantaneous current squared. Time is the resistance in that circuit. That's the power equation. And if we were to plot, I squared on the same graph. It would look like this. It certainly has to be at zero whenever the current is at zero. Because squaring zero, you still end up with zero. And as faras, the maximum zehr, concerned the square has to be a maximum when the current is a maximum. In the case of the positive, that's a no brainer. It's just the square of the I Max plus I max, but the square of the minus I Max is a positive number because you're multiplying a negative times a negative. So that will also be I'm ax squared. So and I'm just writing me in today. Instant power, instantaneous power equation again for future reference. I'm going to get rid of some of the superfluous stuff on our graph so we can look at just the I squared graph that we drew there and the graph rises to a maximum, which is equal to I Max squared in both cases of whether I is positive or negative now, if we wanted to find the average of that, we could take the average off all of those values. And basically it is exactly I'm ax squared all over, too. That's the average because you have just a smudge above the average line as below the average line. So if you wanted to come up with a number that represented the average of I squared, the average of I squared is equal to I, Max squared all over to which we just discovered from our graph. So if we were trying to calculate or we wanted to calculate the average power, the average power would then be the average current squared times. The resistance, which would be P average, is equal to I squared. Average times are l. But we've already discovered that I squared Average is equal to I'm ax squared over two. So the average power is given by I. Max squared all over two times the resistance in the circuit going back to the equation for the current average. The squared the I squared average is actually the mean average squared current. And the square root of both of the sides of that equation leaves us what the I average is equal to I max all over the square root of two, which is the square root of the mean average squared, which is a mouthful, and that's usually shortened to the root mean square of the current or the R. M s current. So the R. M s current is the average current, the mean average current, and it is exactly equal to one over the root of two times I max, which was the peak value that we were looking at earlier. And one over the square root of two is zero point 7071067818 and it goes on for a long time . However, a quick average value are a quick value is 0.71 i max. So if you want to find the RMS value for the current, you just take the peak value or the maximum value of the sine wave and multiply it by 0.71 and you would have a pretty close approximation of what the I. R. M s is. And the RMS value for an A C supply is equal to the direct current, which would dissipate energy at the same rate in a given resistor or late bulb or circuit. If you would, we can use the same logic to find RMS value for voltage. Alan. Alternating voltage supply, which is the RMS V RMS, is equal to the maximum voltage divided by the root of two or multiplied by 0.71 where V max is the maximum Orpik value of the voltage, and it's the same equation that we use for I R M s I, max all over route to where I max is the maximum Orpik value of the current. So if we're looking at a C circuits for resistance loads on Lee and we'll get into different kinds of loads later But this is for resistance loads on Lee. The P average is equal to the current RMS value of the current squared times. The resistor and homes law can be stated in terms of RMS values as well. The RMS is equal to I rms times are you can rewrite that equation in terms of our where r is equal to v rms all over I rms So the p average if we substitute for our would give us I rms squared times v rms all over i rms while the i r. M s cancels out one of the squared values. And that leaves us just with the average value in a resistant circuit for the power is the RMS current times the RMS voltage. And remember, this is four resist of loads only, and that brings us to the end of chapter one 10. Ch 2 Vectors & Phasors copy: Chapter two vectors and phasers. So far, we have discovered that we can analyze a sea currents and voltages using RMS values for resistive loads. That's to say the A C currents and voltages are in phase. However, that is not always the case. So we have another dimension to deal with. And in doing that we developed another tool. It's called a phaser. I'm sure you all have seeing the difference between a scaler quantity or something that's a magnitude has no dimensions. Its length, area volume, speed, mass density, temperature, etcetera. And the difference between that and a vector which has magnitude and direction. And some of the examples for a vector would be displacement to direction of velocity, acceleration, momentum force, electric fields, magnetic fields, etcetera, phasers have magnitude. They also have direction like a vector. But they have one more dimension. And that is phase angles because the phaser is rotating about its tail. And when we do the analysis of different phasers, we take a snapshot or we stop the vector from rotating. And then we look at it phase angle with respect to the horizontal, or at times with respect to other vectors. Okay, So let's build that tool. How exactly can we represent a C quantities of voltage and current in the form of ah vector ? So we start by letting the length of the vector represent the magnitude or amplitude that could be RMS value or peak value as long as we maintain that same dimensional feature as we're doing the analysis. So if we start with RMS values, we stick with RMS values for the whole process. If we start with amplitude Onley than we stick with amplitude, only the greater the amplitude of the way form, the greater the length of its corresponding vector. And what we're looking at right now is the way a phaser represents or is related to a Sinus Seidel wave form. The vector, as you can see, is rotating about its tail, and it traces out the amplitude of the sine wave in the time domain. So let's just stop that for now. And let's do a little bit of analysis. We started out by saying that the vector, or phaser, has a magnitude a and that is the peak magnitude of the phaser, and that could be either measured in RMS value or just peek or amplitude as long as we maintain the same denomination as we are doing our analysis in the time domain, where you see the graph of the A C quantity with respect to time, it goes from a positive A tow, a negative, a back through zero to a positive, A back through zero to a negative. A. If we stopped the rotation, as we have done here, the phaser forms an angle with the horizontal, and I'm going to call that angle omega T. For now, it could be in degrees. It could be in radiance. It doesn't matter as long as we meaning to maintain the same dimensional quantity. For now, let's assume that it is in degrees. So the phaser a forms at angle of so many degrees, which is equal Tau Omega T Omega is the frequency of the oscillation, whether it's in the time domain or it's a phaser, how fast the the phaser is rotating and the frequency in this particular case we're calling it. Degrees per second could be radiance per second. But in this example, we're going to just say it's God degrees per second T is in time. There is time in its in seconds. So Omega T If you take frequency and degrees per second and multiply it by time, that will give you a phase angle or an angle. And that's what's projected along the time domain access. You can see Omega T So what it's tracing out there is the number of degrees over a period of time. If we wanted to measure the amplitude off that particular, uh, graph that's there. If we wanted to measure that, that's the same thing as writing it as a sign. Omega T. Knaus Omega T remember, is an angle, and depending on where you are in time, that will give you a number anywhere between minus one and plus one than multiplying it by . The magnitude of the phaser gives you where you are on the vertical axis. So as that phaser rotates, the vertical displacement will be exactly a sine omega T. Now, the the speed with which that phaser is rotating depends on in in a utility how fast the frequency is rotating of the system, and in North America, that's 60 cycles a second. So it's greatly reduced in speed here, just so you can see what's going on? If it was rotating at 60 times a second, you just see a blur. You wouldn't see what was going on here. The thing to take away from this is this is one phaser only if we added a second phaser to this system and that phaser does not necessarily have to be in phase with the first phaser , you're going to get two graphs of sine waves. They both are gonna have a You're going to be equal to a function called ah sine wave, which is a sine omega t. The thing to take away from here is omega is the same. In other words, the speed of the rotation of the phasers are identical. Doesn't matter whether you're measuring current voltage or other voltages in the system. All the phasers are rotating at the same frequency. And as I said in North America, that's 60 times a second. Whether you have 12 or even three phasers, they are rotating in the same direction and it with the same angular velocity. And when we do our analysis, we usually just stop them from rotating and measure them with respect to each other. Because it's not necessarily how fast they're rotating that we are interested in. We are interested in magnitude and phase angle. So we stopped the system from rotating at some point, and then we do our analysis. So what you see here is to phasers. One is, could be current. One could be voltage or they could both be voltages or they both could be current. I'm not making that differentiation right now for this example. But what I'm doing is comparing the time domain with the phaser notation and as we move the phasers closer together so that they're overlapping now you'll see these phasers as they are referenced one to the other. You could say that a n b r in face, and you can see that with the time domain graph one is right over top of the other, and not only are the in phase, but they're also the same value. In other words, the magnitudes are the same, and in phaser notation, it's one on top of the other. So these they're zero phase shift between a and B. In this example, you can see that the phase angle of the red is straight down in the phase angle of the blue is from left to right, and you can see it wet. What has happened with the time domain graphs as well? In this case, we say, because the rotation is counterclockwise and that is, ah, standard that we usually adopt in ah, in the analysis worldwide, the rotation is usually assumed in a counterclockwise direction. So we say that fazer be the blue face leaves phaser a the red face, and it's leading it in this case by 90 degrees. And you can see that in both the time domain as well as the phaser notation. The phase shift, we say, is 90 degrees. There's a 90 degree phase shift between the blue and the red, and in this case, the red will lead the blue, and it's gonna be leading it again by 90 degrees. So the phase shift is 90 degrees. The greater the phase shift in degrees between two way form, the greater the phase angle difference between corresponding phasers. Being a relative measurement like voltage phase shift or phase angle only has meaning in the reference to some standard way form or another fazer or vector. Generally, this reference way form is a phaser along the horizontal axis, such as the blue phaser. In this case, which would be a way form starting AT T equals zero in the time domain. Sometimes this reference is just the X axis of the rectangular coordinates plane. There are characteristics for the phasers, which we have already discussed, and they are. They can vary in length or in magnitude. They rotate with the same frequency. They are all still relative, one to the other and the same distance unless the parameters of the circuit change. They are all rotating about their tail, and they all rotate in the same direction, and the standard direction we've usually adopt is counter clockwise. There are a couple of ways of describing phasers in our system. One is polar notation in, as I've indicated here, polar notation is used in conjunction with a what we call the polar plane. No phaser is denoted by the length otherwise known as the magnitude and the angle of the fees air, usually with respect to the horizontal as it is in this case. And it's denoted by this symbol here that looks like a untangle itself. For example, this particular vector if it was going to be described, would have a magnitude of 8.4. Now doesn't matter where that's inches or feet or miles, as long as you maintain this same dimensional quality. This is 8.4, and after the angle symbol, the the angle is usually denoted with a small superscript zero after the number, so this particular phaser notation is 8.4 at an angle of 32 degrees. Now that phase angle can either be positive, and if it's greater than 270 degrees as it is here, you can see that that is denoted by 8.4 at an angle of plus 312 degrees. But in polar notation, that angle can also be a negative angle, and it is the same phaser. Only thing is, the angle measurement is negative. In this case, minus 32 degrees. Here are four examples of what phasers might look like, and in the upper left hand corner you see that that phaser is 8.49 at 45 degrees. The phaser that you see in the upper right hand corner can be denoted in two ways, either with a plus or a minus sign. It is still the magnitude is the same 8.6 but if you measure it in a negative angle, it's minus 29.74 at a in a plus value, it's 330.26 degrees. Same fazer, just two different descriptors. The one in the bottom left hand corner, 5.39 and 158.2 degrees. And again, the one in the bottom right hand corner has two ways that you can enlist it. One is the magnitude is both 7.81 The positive displacement or degree angle is 230.19 degrees or minus 129.81 degrees. The other way of describing a vector can be done in rectangular notation in rectangular notation. The phaser is taken to be the high pot news of a right angle triangle by the addition of links of the adjacent side that you see here in blue, plus the opposite side, which is also designated in blue. Rather than describing the phasers length and direction as it was in polar notation, we do noted with the how far right or left and how far up or down that phaser is. Those two dimensions are written in the form of two figures, the one on the left four and the one on the right plus J three. Now the J in front of the three is not a variable. It is what is called the J operator, and the J operator will take a vector or a phaser, and it'll rotated through 90 degrees so that the previous phaser that you saw there, three is now pointing straight up and down and it ISS plus J three. So there's a pair of numbers describing the vector in rectangular notation. The 1st 1 is the left right designation and in this case is plus four. The 2nd 1 is the how far upper down, and in this case, it's plus three. But to denote the fact that it iss going north and south, if you would ah, it is given the J operator so that it separates it from the the number on the left, the number on the laughter, the laughter right dimension is known as the rial number, or its along the rial axis, and the three is along what they call the imaginary axis and that is denoted by a plus J going from zero up and it's a minus j. As you go from zero down in the way of an example. This particular fazer and we just seen it before is four plus J three, and it is in the first Quadrant, and it's both numbers are positive. If that phaser was in the second quadrant, as this example is, you would have a minus four. But the the distance along the J axis is still positive, and it's so it would still be J three, and that is in a second quadrant of the rectangular notation. Another example would be this phaser, and it is in the minus J direction so, but it's in the positive riel axis direction, as all numbers in the fourth quadrant are so the first number is going to be four plus four . The second number is going to be minus J three. Now what we've done is we've just looked at two different ways of describing a phaser or a factor, if you would, and one was the polar notation and the other was the rectangular notation you'll notice as I switched from one to the other. The phaser did not change. Its magnitude and direction remain the same, regardless of whether you're in the polar coordinates or the rectangular notation, so you can convert Ah one system to the other because the vector or phaser does not change . For example, I've written the phaser here, and I've described it in both the rectangular form as well as the polar form. The rectangular description of this particular fazer and it's the Black Arrow in this case is in rectangular form. It is plus four plus J three in poor form. It would be a magnitude of five, and it would be at an angle 36 point a seven. And by the way, you might recognize this is a 345 trey angle. However, we're gonna analyze how we can switch from one form to the other. Let's say we have the the phaser written or described in polar form. In other words, we've been given that it's a magnitude of five at 36.87 degrees, and we would like to know what the rectangular dimensions are in the rectangular form. We would take the rial component, and it's just taking the magnitude of five and you and multiply by the co sine of the angle , which is 36.87 And if you look that up for you, calculated with your smartphone, it would be equal to four. And in the case of the imaginary component and north south component, you take the magnitude and you multiply by the sign of 36.87 degrees, and that would give you three. So we successfully converted the polar form to the rectangular for and if we were going in the other direction. In other words, if we're given the rectangular form ah four plus J three and we wanted the polar form, the first thing is the magnitude is just using the Pythagorean theorem, which says that you the iPod, in use of a right angle triangle, is given by finding the sum of the squares of the lengths of the other two sides, which is four squared plus three squared, which is equal to five. So we know how to get the magnitude. The angle is just finding the Ark tan or the art tangent of three all over four, which is equal to 36.87 degrees. So that is the other method of going in the other direction from rectangular form two Polar form. If two A C voltage is 90 degrees out of phase, are added together by being connected in Siris there, voltage magnitudes do not directly add or subtract, as with skater voltages in D. C. Instead, these voltage quantities war phasers must add up in a Trigana metric fashion. A six volt source at zero degrees added to an eight volt source at 90 degrees results in 10 volts at a phase angle of 53.13 degrees. Vectors as well as phasers can be moved around in a plane as long as their direction and magnitudes are maintained. You add two vectors or phasers by placing the tale of one on the head of the other and then connecting the tail to the head of the results, just like you see in the picture. Compared to D. C circuit analysis. This is very strange indeed. Note that it's possible to obtain volt meter indications of six and eight volts, respectively, across to a Siebold sources, yet only read 10 volts for a total voltage. We just went through the mathematics and came up with the fact that the result in voltage was 10 volts. At 53.13 degrees, we had to find a square root of the sum of the squares of two sides. And then we had to take the ark tension, which when you're adding two vectors together, is a little bit tedious. However, if we had worked in the rectangular notation, we would have had six volts at J zero added to zero plus J eight, which we only have to add up the rial side and then the imaginary sides, and that if we stayed in the rectangular notation would have been the same result on Lee. It was a lot easier to add two vectors together if they were in the rectangular notation. So that brings us to looking at how to work with phasers. A in arithmetic, if you would. So if you're adding phasers together, you simply add up the rial components of the phasers to determine the rial component of the some, and you add up the imaginary component of the phasers to determine the imaginary component of the some. As you see in this example, for example, the 1st 1 there you have a phaser that ISS two plus J five added to a phaser that's four minus j three. So if you add those two together, it's four plus two is six for the rial component, and five minus three is the imaginary aside. So that works. 02 plus two heard plus J two. For the middle one, you have ah phaser 175 minus J 34 added to a phaser 80 minus j 15. The rial components add up to 255 and the imaginary components add up to minus J 49 on the last one is a real component of minus 36 plus Jay 10. Added to a phaser 20 plus j 82 results in minus 16 plus J 92. It's very simple. It's just a matter of adding the two separate numbers together. If there are minus signs involved, at just automatically works into the process. When subtracting one phaser from the other, it's essentially the same process as addition Onley thing is we have to keep track of the operator or the minus sign. So I placed the second phaser in brackets. In other words, the sub tre hand. Of all of our three examples, I've placed the original phaser in brackets solely. 1st 1 on the left, I'm subtracting phaser four minus J three from the phaser two plus j five. So when you do that, all you have to do is remove the brackets. Move the minus sign inside the brackets and the subject hand. Now the four will now change to a minus four and the minus. J three will change to a plus j three, and then you just add that two of them together. So subtracting four minus J three from two plus j five. The rial components add up to minus two. The imaginary components add up two plus J eight and the 2nd 1 we have 175 minus J 34 were subtracting 80 minus J. 15 from that first fazer, So it's going to be 175 minus 80 which is 95 minus J 34 plus J 15 resulting in minus J 19 and finally, the last example we have. We're subtracting 20 plus J 82 from minus 36 plus jay 10 which results in minus 56 minus J 72. Once we get into the analysis of complex certain circuitry using a C quantities, you're going to run into the requirement for multiplication and division as well. So when we're doing multiplication of phasers, the polar notation is favored toe work with because it's very simple. All you have to do is you multiply the polar magnitudes together and you add the polar angles together. For example, we have two phasers here, a at B degrees multiplied by X at Y degrees. Then the answer is going to be a times X at an angle of B plus. Why? So let's have a look at that. In the form of an example, we have to phasers here 35 at 65 degrees tan. At minus 12 degrees, you multiply the 3 35 and tan, which gives you 354. A multiplication of the magnitudes, and the angle is going to be 65 plus, and in this case it's a minus 12 degrees. So it's actually 65 degrees, minus 12 degrees, resulting in 53 degrees. Here we have two phasers being multiplied together 1 24 at 250 degrees times 11 at 100 degrees, gives us 1064 at minus 10 degrees or 1364 at 350 degrees. And finally you got to phasers. Three at 30 degrees and five at minus 30 degrees, results in 15 at zero degrees. So it's a fairly simple process when you're multiply. So really, if you're going to be doing mathematics or arithmetic. So not using a calculator, which in this day and age we would, or a computer. However, what happens is when you're in the form of multiplication, you use polar notation, and when you're in adding and subtracting phasers, you do it with the rectangular notation, and you may have to flip back and forth to make your life a little bit easier. But that's how multiplication and addition or arithmetic works with phasers. Now division is much the same as multiplication in that you're going to divide the magnitudes, one magnitude from the other, and you're going to subtract the angles. So in our generic example, here we have a phaser of a FB degrees divided by a phaser X at Y degrees so it would be a divided by X would give you the magnitude of the multi of the division, and you would subtract. Why degrees from B degrees to give you the result in angle that you are looking for. So let's look at a a couple of examples 35 at 65 degrees, divided by 10 at minus 12 degrees. You divide 35 by 10 is 3.5 and you're going to subtract minus 12 from 65 to Samos, adding 12 to 65 resulting in 77 degrees. In this example, you have a magnitude of 124 at 250 degrees, being divided by a magnitude of 11 at 100 degrees. So 124 divided by 11 is 11.273 and you subtract 100 from 250 leaves you with 150 degrees. As a final example, you got three at 30 degrees, divided by a phaser five at minus 30 so three divided by five is 0.6 and 30 minus. Minus 30 is actually 30 plus 30 which gives you 60 degrees. No, reciprocal is just another form of division, so it's really if you have one over the phaser 35 at 65 degrees, that's the same as a phaser one at zero degrees. So again you just divide one by 35 for 0.2857 and zero degrees. Subtract 65 degrees is just minus 65 degrees. The second example is the reciprocal of 10 at minus 12 degrees is 1/10 at minus 12. Well, that's the same as one at zero degrees over 10 at minus 12 degrees. And I let you just look at that. We don't have to repeat it all. And if you have reciprocal of one and over the phaser 0.32 at 10 degrees, that's the same as one at zero degrees all over 0.32 at 10 degrees. Results in 312.5 at minus 10 degrees, and this ends Chapter two 11. Ch 3 Reactance and Impedance R, L, C Circuits: Chapter three React IDs and impedance resistance induct INTs and capacitive circuits RLC CIRCUITS If we were to plot the current and voltage on a time graph for a very simple A C circuit consisting of an A C voltage source e t and a resistor r, it would look something like this. The blue is the voltage across the resistor, which, because it's connected directly to the source, is also the E. T. V source voltage, and the red dotted line is a current in the circuit as well as the resistor. This is sometimes referred to as the time to me, because the resist er simply and directly resists the flow of electrons at all periods of time. The way form for the voltage drop across the resistor E. R is exactly in phase, with the way form for the current Through it, i R. Which in this series circuit and because there is only one passive element are in the circuit, is equal to the circuit current I and the voltage drop. As I have said across the resistor, e. R is equal to the source Voltage e t. We can look at any point in time along the horizontal axis of the plot and compare those values of current and voltage with each other. Any snapshot Look at the values of the waves are referred to as instantaneous values, meaning the values at that instant in time. The resistor of any given size will impede a C current as defined by homes law and is proportional to the voltage across it, divided by the current through it and when defining using phaser notation it is sometimes referred to as the reactant of the resist ER and designated as X subscript R X are, of course, is in phase with IR as well as the voltage drop across it e. R. In this case of only one passive element and it is directly connected across the source. Voltage e r will equal e t. The source voltage. We can also calculate the power dissipated by the resistor by multiplying the instantaneous voltage times. The instantaneous current or P will equal v times I and plot those values on the same graph . Note to the power is never negative in value When the current is positive above the line, the voltage is also positive, resulting in a positive value. Conversely, when the current is negative below the lie, the voltage is also negative, which means the result is positive that will yield a positive value for power. A negative number, remember, multiplied by a negative number equals a positive number. This consistent polarity of power tells us that the resistor is always dissipating power, taking it from the source and releasing it in the form of heat energy. Whether the current is positive or negative, he resistor still dissipates energy. This time we're going to insert a second element in the Serie Circuit, and that passive element is going to be an induct er because it's still a serie circuit. The current through each of the elements, the resistor and the in Doctor I in the Serie circuit is going to be the same current in the induct er as in the resistor. I'm going to designate the voltage drop crossing doctor as e l. And I'm going to designate the current through the induct er as I l. And we're going to concentrate on only the voltage and current with regard to the in doctor . So the time graph the time doing graph of this current and voltage will look like this. Remember, the voltage drop across a conductor is a reaction against the change in current through it . Therefore, the instantaneous voltages zero whenever the instantaneous current is at a peak. That is where there is relatively zero change going on or a level slope on the current sine wave. And the instantaneous voltage is at a peak whenever the instantaneous current is at a maximum change, the points of steepest slope on the current wave and that is where crosses the zero line. This results in a voltage wave that is 90 degrees out of phase. With the current wave looking at the graph, the voltage wave seems to have a head start on the current. The voltage leads the current or the current lags behind the voltage. The phasers would look like this. The voltage leads the current by 90 degrees. Or, you might say, the current legs the voltage by 90 degrees when dealing with a C voltage applied to a nen doctor, it's always confusing to try and remember if the current leads the voltage or the current lags the voltage delay. I tried to our I do remember it is going back to our D C circuit when we apply D. C to the in doctor with a switch which is similar to starting the first half cycle of Annecy wave. When we start at applying a positive value of voltage to the in doctor, the current will started zero and start to rise slowly to ah maximum value as it builds the magnetic field. However, as soon as the switches closed, the voltage is immediately applied to the in doctor, so the voltage is immediate. It goes rate too the maximum value, whereas the current takes a while to build up the magnetic field, so the current would lag the voltage. Annan doctors Opposition to change in current translates to an opposition to alternating current in general, which is by definition, Sinus seidel and always changing in instantaneous magnitude and direction. This opposition, two alternating current, is similar to resistance, but different in that it is always resulting in a 90 degree phase shift between the current and the voltage. Because of the differences, it has a different name. Reactions. Reactant is to a C is expressed in homes, just as resistance is, except that it is indicated with a mathematical symbol X with a subscript l X instead of our to be specific reactant associated with an induct ER is usually symbolized by Capital Letter X with a letter L as a sub script following it like this, we have already seen the relationship between voltage drop across an and doctor and the current through it, that is, the voltage will lead the current by 90 degrees. Thean, doctor of a given size will impede a C current similar to the resistance of the resistor. That resistor resistance is defined by homes law to be proportional to the voltage across it, divided by the current through it. In phaser terms, it is defined by the voltage phaser divided by the current fazer and is in phase with the current. The reactant. Since oven in Doctor is also defined by homes law to be proportional to the voltage across it, divided by the current through it. But in phaser terms, we define it by this equation, which includes both magnitude and direction. It is calculated to be 90 degrees leading the current. The phaser is sometimes written with a jay operator as J. Times X, l and X L is a phaser itself since in doctors drop voltage in proportion to the rate of current change, they will drop mawr voltage for faster changing currents and less voltage for slower changing currents. What this means is that the magnitude of the reactions and homes for any in doctor is directly proportional to the frequency of the alternating current. Also, the magnitude of the reactions is directly proportional to the induct INTs of the induct er . The exact formula for determining reactant is the magnitude of excel is equal to two pi f l where f is the frequency in hertz or cycles per second. Pi is just 3.14159 etcetera and L is the induct INTs in Henry's let's consider a series R L circuit with a supply voltage of E T. The resistor will offer resistance to a C current, while the inductive ER will offer resistance in the form of reactant since to the A C current. This combined opposition will be a combination of resistance and reactions. In order to express the opposition succinctly, we need a MAWR comprehensive term for opposition to current other than resistance or reactant alone this term is called impedance. Its symbol is Zed, and in this case we're gonna put a sub script t after it because it's the total impedance of the circuit, and it's expressed in units of homes just like resistance. If we take a closer look at this Siri's resistor in Dr Circuit, we know that the current through each element of a series circuit is the same because that's the characteristic of a serie circuit. The current in each element of the circuit is equal to I. In other words, that current through the resistor is equal to the current through the induct ER is equal to the current of the supply current, and those currents are vectors or phasers. We just learned from the previous slides that the voltage drop across the resistor is in phase with the current in the circuit and that the voltage drop across the in doctor is 90 degrees out of phase with the current. In other words, the in Dr Voltage leads the current by 90 degrees. Given us this orientation of phasers of the current and voltage in the circuit, we know from kerchiefs voltage law that the voltage drops around the circuits the around the circuit has to add up to zero degrees. In other words, the voltage drop across the resistor and the induct er have to equal the voltage of the source. And in this case, this is what the vectors would look like. Where the fazer e r. When added to the fazer e l must equal the fazer e t of the source, you can see from the phaser diagram that the current is at zero degrees. In other words, were using that current as our reference phaser, in which case the voltage across the resistor will be in face with that current and it's shown in the diagram as such, and the voltage across the in doctor will be 90 degrees to the voltage across the resistor . We can rewrite our Kershaw's voltage lie equation using polar notation. This time I will rewrite the angles of the phasers, which means that the fazer e t. The source voltage will be the some of the phasers e r plus e l, which is e. R. Zero degrees plus yell at 90 degrees, giving us the result of E. T. At five degrees, as I've indicated in the diagram I'm going to move that equation down here for the moment, and I'm going to divide each element of the equation by the current Now. I could do that as long as I divide both sides of the equation by the same thing, which I'm doing. So I divide the E T at five degrees by I and zero degrees and then divide the voltage e r at zero degrees with the current I at zero degrees and I'm divided, yell at 90 degrees by the current at zero degrees. Now, if you look at the equation, you could see that the resistance voltage er at zero degrees all over I at zero degrees is nothing more than the reactions of the resist er or it's just the resist er in this case, because that's homes law. Voltage over current will equal the resistance. I've indicated it, as a reactant is, but is also the resistance but represented by X subscript R and the last component of that equation, E l. At 90 degrees all over I at zero degrees, is the reactant of the in doctor. We already solved that a couple of slides back, and if we add those two. Together we get zed T, which is the total impedance of the circuit. As long as we add up the phasers, it will result in the components Zed T, which is also equal to E. T. At five degrees all over I at zero degrees, which is equal to the magnitudes ed t at find degrees, which is the angle of zed Tea with X are which is the same angle as the angle between E. T and the current in the circuit. Now let's cement some of this theory by looking at an example. Let's put some real numbers in this circuit and see how it works. We're still dealing in North American frequency, so we're still looking at a 60 hertz, since system we have, ah, the resistor equaling five homes and we're having the induct er equal 10 Milli Henry's. We know that the reactant of the induct er will be given by two pi f l. And we know that the uh 10 Milli Henry's is 10 over 1000 Henry's, so that will give us two times 3.14159 times 60 cycles of second times, 10 over 100 Henry's, which, when you calculate that out, would come to 3.7699 homes. So if we want to know what the total impedance is, we have to add the resistance and the induct inst together. Ah, which our phasers. So we have to use phaser terms, which is five homes at zero degrees for the resist er and 3.7699 homes at 90 degrees for the induct er react INTs. Or we can express that in rectangular coordinate form, giving us both forms if we want, and we comm plot those impedance vectors or our phasers as you see here now, if the voltage in the circuit were saying 100 volts, we could say, Let's let the 100 volts br reference phaser, which is going to be at zero degrees if it's a reference and I've drawn the fees air here in blue and we just have to use homes law to calculate the circuit current, which is e t all overs nt. Remembering that, of course that e. T. And senti our phasers themselves, which would be a 100 at zero degrees volts all over 6.262 at 37.16 degrees, and that would be an impedance in homes. And if you divide the numerator by the denominator, you're gonna get exactly 160 millions and minus 37 degrees, which, when plotted next to the voltage, would look like this. I'm going to replace the induct ER with a capacitor. This time it's going to be the same Siri circuit resistor in series with a capacitor this time source. Voltage being E. T. And the current through the circuit is going to be designated I the voltage drop across the capacity is going to be designated E subscript C, and the current through the capacitor is going to be I subscript C. But because it's a Siri's circuit, there's only a too passive elements in the circuit. Then the total current in the circuit is going to equal the same current through the resistor as through the capacitor. Having said all of that, the time domain curves would look like this, where the current through the capacitor and a circuit is shown as a dotted red line, and the voltage across the capacitor is showing as a ah blue solid line. Remember, the current through the capacitor is a reaction against the change in voltage across it. Therefore, the instantaneous current zero whenever the instantaneous voltages at a peak zero change or a level slope on the voltage sine wave and the instantaneous current is at a peak. Whenever the instantaneous voltage is at a maximum change, the points of the steepest slopes on the voltage wave where it crosses. This results in a voltage wave that is 90 degrees out of phase. With the current wave looking at the graph, the current wave seems to have a head start on the voltage wave. The current leads the voltage or the voltage lags behind the current. The phasers would look like this. The capacitor voltage legs, the circuit current by 90 degrees. Or you might say the current leaves the capacitor voltage by 90 degrees. I have also shown the voltage drop across the resistor, which is in phase with the current. Of course, as with an induct er, the capacitor also presents a kind of a memory problem here, where you trying to remember whether the current will lag or a lead the voltage across either The induct er, the capacitor. We know that there's a 90 degree phase shift, but, uh, as time goes on, sometimes it's hard to remember. Least for me. It was hard to remember which is which. Unless you go back like I have with the in doctor and instead of considering on a C circuit , let's consider a D C. Circuit, and this time we are going to put a battery in Siris with the capacitor. And the battery is going to switch on D C. Voltage to the capacitor as soon as the switches closed, Of course, and this is going to simulate the rise of the positive half cycle oven a C circuit, but it's easier to think of it in terms of D. C. So what happens when you close the switch on a D C circuit? You close the switch and the voltage across the capacitor is zero to start with, and it starts to rise because the capacity will start to charge up on the more it charges up, the more it will resist the current flow. The current will then start to decrease till the pastor is fully charged, of course, and then there is no current flowing and there's no charge building up anymore. The charge is already there, and it is exactly equal to the supply voltage. So what you have seen here is that the current will be instantaneous when you close the switch. So it'll be at the full value only limited by the current through the resistor, which is limited by homes Law. Bottom line here is you're gonna have instantaneous current flowing and the voltage drop across the pastor is going to be zero and slowly building up. So the current will lead the voltage in this case. And if you think of it as the positive has cycle of any C circuit, the voltage is going to be lagging the current as well. So if the voltage is lagging the current in the positive half cycle, then it's going to be lagging for all of the cycles. So this is a way of remembering whether the voltages leading or lagging in the case of I capacity of circuit, the current flow in a capacity. The current flow in a capacitor is related to the change in the voltage of the capacitor, which is sign your soil and always changing in magnitude and defines the opposition to alternating current in general. The opposition to alternating current is similar to resistance, but different in that it is always resulting in a 90 degree phase shift between the current and the voltage. Because of this difference, it has a different name reactant, since, as within doctors, the reactant of a capacitor is expressed in homes and symbolized by the letter X or X C to be more specific when referencing a capacitor. In this case, we have already seen the relationship between voltage voltage drop across the pastor and the current through it, that is, the voltage will lag to current by 90 degrees. A capacitor of a given size will impede a C current similar to the resistance of a resistor . That resistance is divined by homes law to be proportional to the voltage across it, divided by the current through it. In phaser terms, it is defined by the resistor voltage phaser divided by the current phaser through the resistor. The reactions of the capacitor is also defined by OEMs law to be proportional to the voltage across it, divided by the current through it. But in phasers terms It is the magnitude of the capacitor voltage at minus nine degrees to the current, which is at zero degrees with reference to the voltage across it. Well, ecstasy is just e. C. All over. I see which is XY at minus 90 degrees. The phaser is sometimes written with a jay operator as a minus j in front of the X C. Since the passengers conduct current in proportion to the rate of change off the voltage, they will Passmore current for faster changing voltages and less current for slower changing voltages. What this means is that reactant in homes for any capacitor is inversely proportional to the frequency of the alternating current and notably also inversely proportional to the capacitance of the capacitor. The actual formula being XY is equal to one divided by two pi F C, where F is the frequency and hurts or cycles per second. Pie is 3.14159 and C is the capacitance in fair ads. Let's have a closer look at this Siri's R C circuit. The resistor will offer resistance reactant stew, the A C current, while a capacitor will offer resistance in the form of capacitive reactions to the A C current. Because the resisters reactant is in phase with the current, the capacitors reactant is at minus 90 degrees. The combined effect of the two components will be an opposition to the current, equal to the sum of the two numbers. In other words, it's a phaser some and that phaser sum. If the component is made up of capacitive reactions and resistance, then it is referred to as impedance and it has given, the letter said, and sometimes it has a sub scripted denoting that it's the total react it's in the system. However, Zed is often referred to as the impedance of the circuit because this is a serie circuit with current in all of the elements of this circuit, the currents are equal and we have just learned that the voltage drop across the resistor in a circuit is in phase with the current and the voltage drop across the capacitor is 90 degrees lagging, and the impedance of the circuit is made up of the vector or phaser. Some of the impedance of the resist er and the capacitor kerchiefs. Voltage law also tells us that some of the voltages in the Serie circus circuit have to add to zero. Therefore, the some of the voltage drops on the resistor and the voltage drop on the capacitor is equal to the source voltage, which is E. T. I'm going to rewrite the equation in polar notation, and I'm going to let the current BR reference phaser. In other words, the current I'm setting at zero degrees and all other phasers will be referenced to our current phaser. So the voltage equation written in polar notation is e. T at five degrees. Now, if I will be the some of the voltage, drop across the resistor and the voltage drop across the capacitor and we'll set E. T at five degrees and it is equal to e. R. Zero degrees plus cc at minus 90 degrees. Moving the equation to give me a little bit more room to manipulate. Here I am now going to divide each of those voltage elements of the equation by the same thing. The phaser I at zero degrees or the circuit current. So we have a new formula here, which is E. T. At five degrees over the phaser I at zero degrees equal to E. R. Over the current and E C over the same current. Now, if you look at the equation, you will see that the resistance voltage e r. At zero degrees all over the current is nothing mawr than the reactions of the resistance. Or it's just the resistance of the resistor. Because that's owns law, voltage over the current will equal the resistance. I've indicated that the reactant here and then it's reactions here, but it's really resistance, but we represent it by X subscript R. And the last component of the equation, E. C. At minus 90 degrees all over the current is reactant of the capacitor. If we add those two together, we get zed T, which is the total impedes of the circuit as long as we add up the phasers. No zed T, which is equal to E. T. At five degrees all over the current is now can be described as our impedance zed T at five degrees, because ET at five degrees divided by I at zero degrees gives us zed tea at five degrees, which is the angle between the impedance said T and X are the reactions of the resist er It is also exactly equal to the angle between the supply voltage and the current. One thing you will notice is that the angle phi the contained angle of zed T with a horizontal is the same angle as E. T. With the current. Which is not surprising because the triangle formed by ex are ecstasy and said T is proportional to the triangle formed by E R. E C and E T. Because in the voltage phaser equation, it is divided by the same thing. The current I at zero degrees. So the angles should all be the same. It's just the magnitudes that air different. So the triangle formed by ex are xsi ends NT is definitely proportional to the voltage triangle. So let's do a quick example here we have on our C circuit Siri circuit here in Ah Siri's with a Macy Supply voltage rental at the resist er in this circuit, equal five homes, and we're gonna let the capacity equal 100 micro fare ads, and we're going to calculate the total impedance of the circuit and we're going to use the standard system frequency for North America, which is 60 hertz. We know that reactant is for a capacitor is given by 1/2 pi f C. So in numbers that would be one all over the quantity. Two times 3.14159 times 60 times 100 over 1000 which is 100 over 1000 micro ferrets. And if you multiply all of that out, your you will get the reactant of the capacitor is 26.5258 homes. The total impedance for the circuit is the five ohm resistor, plus the 26.5258 home capacitive reactions. But those are phasers, which are five at zero degrees, plus 26.5258 at minus 90 degrees or in rectangular form. It's ah five plus J zero plus zero minus J to 6.5258 which is easier to add than the polar form. And that would give us five at minus J 265258 or 26.993 at minus 79.3 to 5 degrees. Now, if we do a quick a review of what we've learned, if we're looking at just a resist er in an A C circuit. We describe the reactant of the resist er as just the resistance itself, and the phaser is X subscript are at zero degrees or in phaser notation, just x subscript r. And because it's just a passive resistance element, the current and the voltage are going to be in face with the reactant. So everything is easy to work out. Everything is in face and you can see it in our face or diagram here. The next thing we added to the circuit was an in doctor, and now we have a alternating current source in Siris with the resistor in Syria's within and doctor. We've already looked at the reactant of the resist er, the reactant of the in Dr. The magnitude is given by two pi f l where f is the frequency and l is the inductive is and Henry's. If we're to write that in the form of ah polar notation or a phaser, it would be the magnitude at plus 90 degrees and in rectangular format, it would be J times X subscript l. And the phaser diagram for the impedance is showing at the bottom of the diagram there at the total impedance Zed tea is made up of the some of the reactions of the resistor in the reactant of the conductor. If we added the source voltage, we would find that it's in phase with the total impedance of the circuit, which is not unusual. It would lie along the same direction as the impedance zed T, but the current for the circuit is going to be displaced by five degrees, which is a product of the or actually it's. It's the result of adding the reactant of the resistor and the reactions of the in DR together on it gives us the same phase angle as the phase angle between zed, T and X r. And that is the phase angle for the impedance as well as the voltage and current off this Siri's circuit. We then replaced the induct er with a capacitor, and we found that the magnitude of the capacitance waas 1/2 pi FC and in phaser notation, um, it would be x c at minus 90 degrees or in rectangular notation. It would be minus J x C, and the two impedance is one for the the to react. Ince's one for the resistor and one for the capacitor. Add up as you see there in the phasers to give us a total impedance of the circuit Zed T. And if we drew in the voltage vector for the circuit, the total circuit and the current in the circuit, it would have the same angle as the angle of the zed tea with X R on. We went through that process as well, but we would find that the voltage in this case is lagging the current by five degrees. Now, if we are going to use OEMs law using our phasers and react, Ince's in him and impedance is, it would be the same as we would be using for the D. C circuit. But this time, instead of resistance, we have impedance, and the voltage is still going to be equal to the current times the impedes. But in this case, you have to remember we are using phasers, not absolute values. So it's e is equal to, I said, or I is equal. T over Zed or Zed is equal to e over I. It's the same equation for OEMs law. For a D C circuit. Only this time we're using a C quantities, and we're using react. Ince's and impedance is What you do have to remember is that all quantities, our phasers and they have to be used with the rules regarding vector analysis. Let's work out a another, more complex example. It's still a serious circuit, but this time we have a uh 120 volts, 60 hertz, a C supply feeding a resistor in doctor and a capacitor. The impedance of the Resist ER is given as 250 homes. The Induct INTs is given as 650 Milli Henry's and the capacitance is given as 1.5 Micro Fair adds. But we have to get all of the components reading in terms of impedance is so we're going to convert or figure out what the impedance of the induct er is. It's just two pi f l, which is two times pi times 60 times 650 all over 1000 which works out to 245.4 OEMs. And to calculate the impedance of the capacitor, we have to change the capacitance into an impedance and that's 1/2 pi f c. The two pi 60 hertz is the same 31 point five micro fair, as is just 1.5 all over. Ah, 1,000,000 which works out to 1.7684 que you and the impedance of the resist ER is already given in homes. It's, ah, 250 plus J zero homes in rectangular format and 250 homes at zero degrees in polar format. So we have all of the impedance is now in terms of react. Ince's or impedance is, and they're all listed right there. What we can do now is fill out a table for the voltage current and impedance of all the components in the circuit. So we have all the components listed across the top of the table and all of the values that we're looking for listed along the left hand side or the rows of the table, and we're gonna figure out everything in volts, amps and homes. So we'll start with the voltage supply, which is easy. We don't have do any calculations there. It's 120 volts at 60 hertz, which means it's 1 20 plus j zero in rectangular format and its 120 at zero degrees in polar format. The impedance is we have already calculated, and we've got the marked on the diagram above the table there. But I've listed them in both polar notation as while as rectangular notation, and we can use them back and forth during our calculations, depending on where we're adding, subtracting, multiplying or dividing. So the total impedance for the circuit because it's in Syriza's just Zet are plus, said L Plus said. See which we are going to use the rectangular notation to start off with, because that's the easiest one to add. And if you add that all up, you'll get 250 at minus J 1.5 to 33 que alms, and you can convert that to polar for ah notation as well, which is 1.5437 k OEMs at minus 80.680 degrees. And we can put that in our table at the bottom right hand corner, which is the total impedance for the circuit, and we're going to use that total impedance now to calculate the total current for the circuit, and we're going to use homes law, which means it's just the voltage total voltage over the total impedes of the circuit. But we have to remember that we are using ah, phaser notation. So it's going to be the 120 volts all over the total impedance for the circuit. And to do that calculation because is a division of to phasers, we're gonna use the polar notation, which is the voltage 120 at zero degrees, divided by 1.5437 k OEMs at minus 80 degree, 80.68 degrees, which will result in 77.734 Milly obsess with the Emmis four at plus 80.68 degrees, and we know that the it's a Siri's circuit, so the current through each element is the same, so we can fill in those boxes just by repeating what is the total current for the circuit. And then the calculation for the voltage drop across each component in the circuit is again just using homes law. In this time, E is equal to i times end and again it's phaser notation, but because we're doing a multiplication this time, we're gonna use still use the polar notation. So if we fill in those boxes, we will get the voltage. Drop on the resistor is 19.434 at 80.680 degrees for the in doctor, it's 19.0 for eight at 170.68 degrees. And for the capacitance, it's 137 point for six at minus 9.3199 And you might have noticed something if you were observant. And that is the fact that the voltage across the capacitor is in excess of any other voltage in the circuit, including the source voltage. So what is happening here is you're getting a resonant frequency or a ringing of resonant frequency in the system, which will actually happen as you approach the resonant frequency of the system, and it will push voltages in excess of the source voltage. But we will leave that for now because this is the end of the chapter 12. Ch 4 Working with AC Circuits: Chapter four Working with a C circuits. We're going to use Kirch ofs voltage law as a check on our last example that we did in the previous chapter. And as you recall, we had a Siri's circuit with a resistor in doctor and a capacitor in Siris with an A C voltage generator. And now we went through the process of filling all of all of the blanks in our table. And now I'd like to go back and just check the results, or I'll show you how to use Kershaw's voltage law to check the results. And Kirchoff voltage laws, as you remember, means if you add up the voltage drops around a circuit, they have to add to zero. In other words, uh, the voltage drop on the resistor added to the voltage drop on the induct er added to the voltage drop on the comm pasture has to equal the voltage power supply, so let's go through that process. But before going there, we have to convert our polar notation voltages that we have in the top role of the table. Two rectangular notation, because that will be easier to add up. So the resistance is going to be 3.1472 plus J 19.177 The in doctor is going to be minus 18.797 plus Jay 30848 and the capacitor is going to be 135.65 minus J 22.262 And if we add those up, we're going to add first of all the rial components and then the imaginary components. And if you do with a calculator or just by hand, you will find that the rial component will add up to 120. And the imaginary components will add up to zero, giving us the rectangular notation for the voltage of 120 plus j at zero volts, which is indeed what we have in our table. So we've just approved the results using Kirchoff Voltage Law. Let's take the example a little bit further and this time, instead of connecting up the, um the impedance is in Siris. I'm gonna connect them up in parallel to the same voltage source. In fact, we're going to use all the same components. The only thing is we're gonna put them in parallel now instead of in Siris. And changing the configuration of the circuit does not change the impedance values of our impedance components because they are on Lee there, only reliant on the frequency of the circuit which has not changed. And the internal amounts of the resistance induct its and capacitance, which hasn't changed as well. So the impedance of each of the impedance elements is the same as in our previous example. The only thing is we have hooked him up in parallel now. So let's lay out the table the way we did before. Uh, and this time we're going to go through and fill it out. But let's first of all, pick what we already know, and we already know that the total voltage in the system is 120 volts at zero degrees. Because that's what we started with last time. And we just went through an explanation as to why the impedance is were all going to be the same, so I can write them in the table as well. The other thing that we can use now to fill in the table is the fact that This is a parallel circuit, which means that the voltage drop across every component in the circuit is going to be equal because it's in parallel so I can fill in the top role of the table with the fact that the voltages are going to be 120 at zero degrees. Now, in order to come up with the currents, I'm just going to use OEMs law and again we're using phasers. So we have to be careful on how we do the division here. But if we just divide the boxes, uh, the divide, the top box by the bottom box. To get the current, we will get the three currents for the current flow in each one of the components for the resistor, for the in doctor and for the capacitor, and just to repeat them, the current through the resistor is going to be 480 million amps at zero degrees. The current through the induct ER is 489.71 million amps at minus 90 degrees, and the current through the capacity is going to be 67.858 million amps at plus 90 degrees . So we now have everything in the table except the last two items, which is the total amperage flowing from the source and the total impedance of the circuit . Well, we're going to use kerchiefs current law to figure out what the and flow is in Kirchoff. Current law states that the current flow in in all the branches have to add up to the main branch. Or we have to add up all the branches in order to get the total flow. And all we got to do is add up the middle row of our table and that will give us 639.3 million amps at minus 41.311 or in, uh, rectangular format. It's 480 million amps minus J 421.85 million amps. So we've got the total current flowing in the system. Now we would like to find out what the total impedance is, and we can do it two ways if we want. First of all, we can run a parallel calculation of all the impedance is which will give us the answer. Although it will be very tedious because we're going to be converting back and forth from rectangular two polar, and it's going to be ah ah, fairly lengthy process button. Easier way, because all we gotta do is find out the voltage over the total impedance divided by the current into the total. Impedance is we already have those figures in the to upper right hand boxes of our table. So all we gotta do is take the voltage, which is 120 at zero degrees and divided by the current, which is 639.3 at minus 41.311 degrees. And if you work that out, it comes to 141.5 plus J at 123.96 And in polar form, that is 187.79 at 41.311 degrees. So we have successfully filled out all of the boxes in our table, and that ends this chapter 13. Ch 5 Power flow in AC Circuits(R): Chapter five Power in a Sea circuits. For a moment, let's talk about instantaneous power. What you see on this slide is to sign a subtle waves, one representing current, which is a dotted red line and one representing voltage, which is in black. The voltage starts at zero, goes to a maximum, comes back through zero, goes to minus and so on. They're nice uniforms. Sinusitis, always the current, depending on the impedance of the circuit, of course, will either be shifted left or right a certain amount. So we will either say the current is going to be leading or lagging the voltage, but it rests is represented by this wave shape on this time graph. Now, if we're going to calculate the instantaneous power at any one moment, we pick a point on both graphs at a particular time, and we say that the instantaneous power is given by the voltage at that point times the current. At that point in time to demonstrate this, let's take a slice of time right here, and we will say that the power at that particular time is given by the instantaneous voltage at that time, multiplied by the instantaneous current at that time, it can either be positive or negative or zero, and that is OK if you want instantaneous power. However, in power systems, instantaneous power isn't always what we're after. What we are usually looking for most of the time is for average power. For example, what is the average power consumption of a water heater? What is the average power consumption of a baseboard here and so on? So average power is a thing that becomes significant. It's the thing that we have to deal with usually. Now keep this in mind for future reference. We're going to go on to look at what the instantaneous power looks like in the various passive circuits. Let's take a look at our simple resist of Siri's circuit, and we've seen this before. Everything is in phase, that is, the currents in phase, with the voltages in phase with the impedance. And if we draw the curves in the time domain, as we've done here and we plot the power and we've seen this craft before, Aziz well that the power is at a zero point whenever either the voltage or the current is zero enough. Of course, they're both at zero than the power is at zero, which the graph indicates here. But what is significant in a resist of circuit is that the power is always positive. It goes to zero, but it is always positive, so it doesn't matter which direction the current is flowing. The resistor is always consuming power, and it's converting it into heat energy. Now let's take a look at our Siri's resist of inductive circuit. And with that circuit you've seen this graph also before the ah, both the phasers as well as the time domain graphs show you that the current will leg the voltage by 90 degrees. Looking at the voltage in the in dr itself, uh, the voltage will be Sinus Seidel. And I'm just picking a time on the time graph to show that voltage. And it goes from a positive maxim through zero to negative maximum back through zero, etcetera Again. Now we just saw that the current in that circuit is 90 degrees out of phase. In other words, it's delayed by 90 degrees and again, we're looking at just the current through the in Dr. Of course, it's the same current through the whole circuit. However, I'm gonna just concentrate on the induct er itself. So I'm looking at the voltage across the in doctor and the current in the and Dr. And if I wanted to plot the instantaneous power delivered to Thean Doctor, it would look like this. The power equals zero whenever the instantaneous current or voltage is zero. And we've seen this before, of course. And whenever the instantaneous current and voltage are both positive above the line, the power is positive. And as with the resistor, the power is also positive when the instantaneous current and voltage are both negative. So we get this positive power flow at these particular points in the graph. However, because the current and voltage waves are 90 degrees out of phase, there are times when one of the, uh one of the waves are positive, while the other is negative, resulting in equally frequent occurrence of negative instantaneous power. You'll notice, however, that there is on average as many positive power flows as negative power flows, resulting in a net average power flow into the end. Doctor, Now remember, or only looking at Thean doctor here, not the whole circuit the net average power flow into the in doctor is zero. So what exactly is going on with the in dr during this positive and negative energy flow? Okay, so during the positive flow of energy, the current is building a magnetic field around the in. Dr. This requires energy which is then stored in the magnetic field as long as both the current through the in doctor and the voltage drop across Ian Doctor is positive. This will happen once the current and voltage have opposite values one positive and one negative. The power flow will be negative. This is when the magnetic field around the induct er is collapsing and the stored energy will dissipate back into the circuit. Now, both voltage and current are negative soul. The power is once again positive. However, the current is negative, which results in a building of the magnetic field around the induct er but of the opposite polarity to that which previously just collapsed. This too, requires energy which is then stored in the magnetic field. As long as both a current through the induct er and the voltage drop across Ian Doctor are the same polarity. This will happen once the current and voltage have opposite values. One positive, one negative. The powerful will be negative, and this is when the magnetic field around the in Dr is collapsing and the stored energy is dissipated back into the circuit. This goes on and on and on and on and repeats itself 120 times a second, which would be a blur if we were watching it in real time. However, the net take away from this is that there is no energy dissipated on average, into the in doctor. Now let's go through the same process this time instead of a nen doctor. Let's a look at our Siri's resistor capacitive circuit, and you'll remember from previous chapters that the voltage across the capacitor is 90 degrees lagging the current and I've shown both the phasers and the and the time demean graphs here. So this is what the resultant wave forms would look like. The voltages in blue the red is in is amps, and the green is the instantaneous power, and we are looking at the voltage current and power with respect to just the capacitor. In other words, it's a voltage drop across the capacitor and the current through the capacity that you see in the graph and the power flow into the capacitor. In green, the power equals zero whenever the instantaneous current or voltage is zero whenever the instantaneous current and voltage or both positive above the zero line, the power is positive. As with the resist er example, the power is also positive when the instantaneous current and voltage are both negative, that is below the line. In other words, two negatives multiplied together result in a positive. So now we have positive flow off power. However, because the village and current are 90 degrees out of phase, there are times when the power is going to be negative. That is when the voltage and current are of the opposite signs. And as I said, that is when the power is negative and if you look at the curves here, you'll notice that there is an equal number of positive on average flow of power into the capacitor as there are negative and this is very similar to what happened with Ian Doctor. However, a few things are a little bit different, so let's just see what is going on here during the positive flow of energy. The current is building an electrostatic field in the capacitor. This requires energy, which is then stored in the electrostatic field. As long as both the current through the capacitor and the voltage drop across the capacitor are positive. This will happen once the current and voltage have opposite values one positive and one negative. The power flow will be negative. This is when the electrostatic field around the capacitor starts to collapse and the stored energy will dissipate back into the circuit. During the next positive flow of energy, the electrostatic field is building in the opposite direction. This requires energy, too, and is then stored in the reverse electrostatic field. As long as both the current through the capacity and the voltage drop across the capacitor are negative. This will happen once the current and voltage have opposite values. Again, the power flow will be negative. This is when the electric aesthetic field around the capacitor again starts to collapse in the stored energy will be dissipated back into the circuit. This, too, will repeat itself 120 times a second, which again would be a blur if we didn't really time. However, on average, there is zero power dissipated by the capacitor. I've redrawn our voltage current in Power Way forms here on the in the time to mean. And ah, there is a change in color here, but it doesn't matter what the color is. The voltage is green this time, and it's starting rate at zero, and it's Sinje Seidel, and the maximum voltage is going to be the subscript M. And because it's sinusitis, it's going to be multiple. It's going to be a VM multiply by sine omega T, where omega is the angular velocity and omega T is the actual angle of the wave. Now the voltage in the current wave forms and the phasers. I haven't written the phasers on here yet, but they are all rotating. They both are rotating at system frequency. So there they are, rotating at the same angular velocity, but not necessarily in phase. Was he with each other? The current is given by a maximum current I am times sine omega T minus phi. Now the minus phi is due to the fact that we are out of phase with the voltage by a factor of five. So that's written in the angle formation, which means the current seems to have a head start on the voltage, the voltage legs, the current. But this would be the formula for the current. Now, the formula for the instantaneous power is the voltage times the current. So if we multiplied the right hand side of those equations together, we're going to get the M I m times sine omega t times sine omega t minus phi. And that's going to describe the blue Wave form that you see there. And it is twice the frequency of the current and the voltage. And because there is a out of phase nature between me current and the voltage, there will be some negative power and some positive power. In this case, there's more positive than there is negative. What we would like to find out is the average power, and we can find that and there is a proof for it. I do have a more extensive proof for finding the average formula from the voltage in the current, but you'll have to find it in another course, and it's a course on power in a sea circuits. But if you go through a trigger No metric Dera vacation you will end up with the average power is given by the M. I am all over to Times Co sign Phi where co sign Phi or Phi is the angle between the current and the voltage. And in this case, it is the fi that was in the current formula because the there is no, uh, delayed action in the voltage it start traded zero. So if we wanted to find out the average power, this is the formula that we use for average power. VM I am all over to co sign of Fada P average is the actual work done by a new electric current or the actual power drawn by the load to create, For example, heat, light or motion? It is given the name rial power measured in Watts and represented by a phaser along the rial access its length is proportional to VM. I am times Coast data all over too. This really power drives with power. You tell these measure over time in order to build a customer measuring Miss Simone of kilowatt hours with a kilowatt hour meter. Electrical systems, especially industrial customers, normally have in doctors and capacitive type loads, which are referred to as reactive components. Ideal reactive components do not dissipate any energy, but they draw current and create voltage drops, which make the impression that they actually do this imaginary power. We'll call a que average is represented by a phaser along the J or imaginary axis. Its length is proportional to VM. I am signed data all over to and is positive for inductive reactive power and negative for capacitive reactive power. This Q average is known as reactive power. It's a measure of the value of Q over a complete A C cycle. It doesn't contribute to the net transfer of energy, but circulates back and forth between the source and the load. This imaginary power draw, if large enough, is what power utilities measure and penalize a customer for reactive power is measured in volts impairs reactive or vars. The combination or phaser sum of riel and reactive power makes up what is known as apparent power s, and it's measured in volt amps, or V A and mathematically equal to VM. I am all over too. This combination of the phasers for real power in reactive power given us apparent power is known as the power triangle, and data is known as the power factor angle. It is the angle between the real power and apparent power, and coast data is known as the power factor, the angle theta. The power factor angle is also the angle between the voltage and the current. So we've had a look at how we measure average power, and we saw that that equation was ik P average was equal to the maximum voltage or times the maximum current all over too Times Coast data or VM. I am all over two times Coast data. We can split that two in the denominator into route two times route to which doesn't change the equation at all. But it does lead to a better grasp of how we can represent average power because we know that the RMS voltage is given by VM or the maximum voltage over route to and I, RMS is given by the maximum current all over route to so the average power. Then instead of writing it, I am VM all over to can now be written the RMS Voltage Times, the R. M s current Times co sign data and data is the angle between the voltage and the current. We can also gather the Q average by using V rms times, I RMS times sine theta as well as the apparent Power V A can be V rms times I RMS. So this is really why we use RMS values because it makes it a lot easier in calculating average power in and average apparent power, etcetera. Most of the meters that you used today for measuring voltage current and Watson power etcetera. All use are all the dials, and all of the measurements are in RMS values. So unless otherwise noted, if you have a meter and you're measuring the voltage and the current ah, you can very easily convert that to an average power measurement by just multiplying the voltage and the current. Because the current readings and a voltage readings are in RMS values, capacitors store energy in the form of an electric field and electrically manifest that stored energy as a potential static voltage in Dr Store energy in the form of a magnetic field and electrically manifest that stored energy. Ask kinetic motion of electrons, current capacitors and in doctors are flip sides of the same reactive coin, storing and releasing energy in complementary modes. When these two types of reactive components are directly connected together, they're complementary tendencies to store energy. Will a times produce unusual results if we assume that both components are connected in parallel, are subjected to a sudden application of voltage, say, from from a momentarily connected battery, the capacitor will very quickly charge. And the induct er will oppose the change in current, leaving the pastor in the change state and the in DR building its magnetic field. As the current flows through it is building. Now I'm going to open the switch, and I'm gonna slow down the action in order to observe what's going on. The picture will morph to the end of the time frame, but watch the time graph of the bottom of the slide to get a better sense of wood is transpiring. The voltage in blue, usually associated with the capacitor and the current in red, usually associate it with the induct er. So pay attention to the bottom of the slide, and when I open the switch, you can see there's a transfer of energy back and forth between the induct er and the capacitor. And it's happening actually faster than what you even see in this diagram, and it's actually happening. Whatever the frequency, the resonant frequency of this circuit is, we'll talk more about that later. However, I'm going to slow things down, as I said, so that we can follow it in in steps or in stages. So when the switch is open, the capacitor will begin to discharge its voltage, decreasing it. Meanwhile, the in DR will begin to build up a charge in the form of a magnetic field as a current increases in the circuit. Thean doctor, still charging will keep electrons flowing in the circuit until the capacitor has being completely discharged, leaving zero voltage across it. And if you look at the curves on the bottom, you can see that the current was rising to a maximum and the voltage in the capacitor is now at zero. The in dr will remain or maintain its current flowing, even though there is no voltage applied. In fact, it will generate a voltage like a battery in order to keep the current in the same direction. The capacitor being the recipient of this current will begin to accumulate a charge in the opposite polarity. When the in doctor is finally depleted of its energy reserve and the electrons come to a halt. The capacitor will have reached full voltage charge in the opposite polarity as when it started, and there is no current now flowing in the circuit as the potential energy of the capacitor is completely built in the reverse direction. This is shown in the graph in the ball. At the bottom of the slide, you can see the red current lying is at zero, meaning er zero current flowing and the voltage across the capacitor is at a maximum. But it's in the reverse direction. You might say it's minus. So now things don't stay there. They will start to change again, and the capacity will begin to discharge through the in dr, causing an increase in current in the opposite direction and a decrease in voltage as it depletes its only energy reserve. Eventually, the capacitor will discharge to zero volts, leaving the in dr fully charged with a maximum current through it, so you can see that in the graphs at the bottom, you can see the red dotted line is at a maximum minus and you can see that the of the blue voltage line is now in zero, meaning the capacitor is completely discharged. The doctor desire to maintain the current in the same direction will act like a source again, generating a voltage like a battery to continue the flow. In doing so, the pastoral begin to charge up in the opposite polarity, and the current will decrease in magnitude to zero again. As you can see in the graphs in time. Eventually, the capacity will become fully charged in the opposite clarity. Again, the in Dr like will expend all of its energy reserves trying to maintain the current. The voltage will once again be at its positive peak and the current that zero on this completes a full cycle of the energy exchange between the capacitor and in doctor. This oscillation will continue with until the resistance in the circuit will cause this to deplete the energy as it goes back and forth. It will be actually converting the stored energy and in the way of current and potential energy in the form of heat as it goes along. But if we were considering this to be an ideal in Dr An ideal capacitor and there's no resistance in the circuit, this oscillation would continue. Continue in definitely. Overall, this behavior is akin to that of a pendulum. As a pendulum mass swings back and forth. There is a transformation of energy taking place from the potential energy or the height to which the vertical height which the ball rises to the kinetic energy or motion back to the maximum potential energy on the opposite side. This is similar to the energy transfer in the capacitor and Dr Circuit, back and forth in an alternating form of current kinetic motion of electrons and voltage potential electric energy at the peak height of each swing of the pendulum. Now we're looking at the left hand side. The mass briefly stops and switches direction electrically. This is at the maximum voltage potential energy and zero current connect kinetic energy of our of our oscillating electric circuit. It is at this point that the potential energy, the height is at its maximum and the kinetic energy motion is at zero. As the mass swings back the other way, it passes quickly through a point where the string is pointed straight down at this point. Potential energy. The height is at zero. That's at the bottom of it's a zoo, a bottom of its swing and the kinetic energy of motion is at its maximum. In other words, the weight is moving very, very fast and the fastest moving in the swinging motion elected electrically. This is where the voltage potential energy is at zero and the kinetic energy the current is at a maximum at the peak height. On the other side of the pendulum swing, the mass briefly stops and switches directions electrically. This is at the max reverse voltage potential energy and zero current kinetic energy. It is at this point that the potential energy height is at a maximum, and the kinetic energy motion is at zero. Now this oscillating circuit. I want toe hypothetically. Look at it without the battery for now. And let's assume we're talking about an ideal in doctor and an ideal capacitor. So this thing is swinging, oscillating back and forth, a potential energy building up in the way of voltage on the capacitor, and then it discharging and building up current through the in doctor on this will be continuing on and it will go on a back and forth on it is an electric circuit. So it's gonna be swinging depending on the as you'll see the inductive or the capacity ist of the circuit. But it is a C motion. In other words, is it the current and the voltage? Even though they might be a phase shift between the current and voltage, they are signing a soil and they are oscillating at the same frequency. But there is going to be a phase shift in them. Now I'm going toe hook up hypothetically a generator to this system and that generator, I'm gonna say, Adjust the frequency of that generator to exactly match the frequency of oscillation off R L C circuit that we have there. And if that is indeed the case, then there will be no potential energy between the generator and the circuit because they are in parallel and they're both oscillating with the same magnitude at the same frequency , and the voltages will be matching. So if I placed on in Anam Eater in that circuit or even in a cell, a scope, I would be reading no current flowing into the system now, assuming our system is balanced. Like I said. And there's no current flowing from the generator to the system, I'm gonna have a quick look at the react. Ince's that air in our system. In other words, the X l on the X see of the circuit. And if I vary the frequency of our circuit, we will find that the X l is going to increase the faster the frequency, the mawr impedance there will be in our in doctor. And if I start to increase the frequency of the capacitor, there will be less capacitance in the circuit. So whichever direction I'm going in from the balance condition if I go let if I go down in frequency than my induct INTs is the reaction story of my capacitor is going to go up. And if I go up and frequency the reactions of my induct er is going to go up and what we want and that will definitely change the balance of the system. So on the larder have zero amps flowing across through the AM ear. So we can, as we change the the reactant in one direction. Sorry. The frequency in one direction or the other. My react Ince's of the system is going to change. But there's only one point where we will have ah, balanced system and minimum amount of change or no change. And that is where these two, uh, inner some of these two lines cross. And it's right at that point that we will have no change in the impedance. And that is exactly when excel is equal to X C. And that means that that is the point where Excel, which is given by 2.2 pi fl, is balanced with 1/2 pi f c. I can now multiply both of the equation sides of the equation by the frequency and I am left with this equation two pi f squared l is equal to 1/2 pi. See, I'm now gonna leave. Just the F frequencies are f square, the frequency squared on the left hand side of equation. I'm moving all of the other terms to the right and side of the equation and I get the frequency squared is equal to one over two pi times two pi times L C, which is is two pi squared And if I find the square root of both sides of that equation, I'm going to be left with. The frequency is equal to 1/2 pi, the square root of L. C. And that is the frequency of residence of this particular circuit. It is dependent on Lee on the induct INTs and the capacitance of the circuit. It has nothing to do with the voltage across it, and it only and it doesn't have anything do with the generator connected to it has nothing to do with the resistance in the circuit. The resident frequency is a is a function of Onley, the induct INTs and the capacitance. And if you want to find out what that resident frequency is, that is the formula for calculating the resonant frequency. So let's put it to use here and just do a quick calculation. We're gonna hook up ah, generator to our parallel Elsie circuit, and I'm going to say, Let the capacitor be 10 micro fair odds. And I'm going to say Let Thean doctor be 100 Milly Henry's so we can use that circuit to find out what the what the resonant frequency is of that set up. And by the way, there's another name for this parallel connection of a resonant circuit. NLC circuit is sometimes referred to as a tank circuit. So if I multiply 10 Micro Fair adds times 100 million Henry's That's 10 times 10 to the minus six times 100 times 10 to the minus three. That is equal 2.1 And if you find a square root of that, it's 0.1 and two pi times point is 0.1 is 0.6 to 8318 And if you take one over that number , it's gonna work out to 159.55 which is the resonant frequency of that circuit. It's 100 and 59.155 cycles per second or Hertz. I'm just gonna study this set up just a little bit further. Um, we already know that X C is 100 homes because we can take 10 micro fare ads and do the calculation to kept to calculate the the reactive impedance and it's 100 homes. We can do the same thing with the current and it will come out to 100 homes as well. If we look at the parallel connection off that of those two impedance is it is one over one over Zed l plus one overs and see, as you see in the bottom left hand corner. I'm gonna plug in the values for Zet l and said, See, when I do that Zed L is 100 homes, but it is at 90 degrees because it's induct INTs and it's at straight up and down, or pointing up. And the impedance of the the capacitor react minces 100 homes at minus 90 degrees. And if we flip those two fractions in the denominator, we'll come up with 20.1 at minus 90 degrees and 900.1 at plus 90 degrees, which add to zero. So those two impedance is in that parallel connection that you see there. Actually, the impedance works out to be infinite, which means that is one they were at one year at resonant frequency. That particular circuit will seem like an open circuit to the generator. So it assuming nothing is gonna burn up and you've got the right, uh, layers hooked up to it. You connect up any type of voltage to that circuit. As long as you're at a frequency of its residence, it will conduct no current. If we place those same two, impedance is in Siris. Let's see what happens with that, and we're going to maintain the same resonant frequency. So the impedance of the in doctor is zero plus j 100 OEMs. The impedance of the capacitor is zero minus J 100 homes. They are in Siris so we can add them up. And I've just put them in in Siris and added them here in rectangular format. And as you know, they will add to zero homes. So that means that if we put those two components in in Siris with each other, they will now act at resonance as a short circuit. So is if you start to oscillate that circuit in a very low frequency. As you approach the resonant frequency of the circuit, it's goodness react like a short circuit. So that gives us kind of ah, a little bit of an interesting perspective on resonant frequencies, and it's it's good to know and to be able to calculate the resonant frequency. But the the point I want to get a fire home here is that sometimes these resonant frequencies take place without us knowing it. Ah, we aren't aware of the fact that there is some capacitive straight capacitive in a circuit or or stray induct INTs, and it starts to manifest itself. As you increase the frequency, they may seem like opener short circuits in in a at the beginning, when the frequency changes, though, the characteristics are gonna change, possibly drastically as well. So you have to be aware of how resonant frequency may or may not affect the circuits that you're working on, so this ends chapter five.