Transcripts
1. Ch 00 Introduction: This course has seven chapters, which are based on the revenue type metering systems found in utilities throughout the world. It covers single and three face systems. It also covers all voltage levels from the 1 22 40 residential supplies. And because there's the chapter on instrument transformers, the theory can also be used and is used for measuring very high voltage systems such as 735 K V. There is also reference to me during standards which in this case I use measurement Canada standards, but they are not unlike all such standards that are out there. The last chapter deals with Cross what Merrick Meter verification, which is not included or used by all utilities or me during service providers. But it is a simple process based on power measurement and worth knowing because it is a useful tool.
2. Ch 01 Power & Energy: Dr One Power and Energy. We're going to start the course off with another standing of just what is power and energy and how it's related to electrical systems. Once we've established the basis for power and energy, measurement in electrical system will go on to study single and three phase metering systems. Energy is defined as the property that must be transferred to an object. In order to perform work, energy equals work and work equals energy. Consider these two weightlifters lifting the same amount of weights through the same to the same height. Both do the same amount of work. Both expend the same amount of energy in terms of physics. Work equals mass times, gravity times, height and in this case it's 100 kilograms times 9.8 meters per second squared, which is gravity's acceleration times two meters, which equals 1900 and 60 jewels. However, the weight lifter on the left is slower than the weight lifter on. Right Hence, we say that the right weight lifter is more powerful than the left weightlifter power measures the rate at which work is being done, or that power equals work divided by time where tying is the interval. It takes to lift the weight from the floor to its final height. If the left weightlifter completes his lift in three seconds and the right ah, weightlifter completes his lift in one second, the power delivered by the left weightlifter is 653 jewels per second and the power delivered by the right way lifter is 1960 Jules per second. The same amount of work is done, but at a different times and by different powers. Because work is measured in jewels, power is measured in joules per second. In this measurement is defined as watts such that the work of one jewell completed in one second is equal to one. What if we watch the lift? We see that the speed of the lift is not consistent. Regardless, if the lift is completed in three or one seconds, some of the lift is completed faster or slower than the other parts, which means the power delivered will vary. So if we use the total time for the lift in our equation one or three seconds, we define that power delivered as average power. If we break the whole lift up into smaller time increments such that the power over that small increments is consistent. We can defying what is known as instantaneous power, which is consistent over the small time increments. Now we'll talk a little bit more about this when we start to study how it applies to electrical systems. But for now we are just defining average power, which is the power delivered over a larger period of time or instantaneous power, which is delivered over a very small increment of time in terms of electrical power. The work done or electrical energy, is the movement of charges caused by a push of e M F. In other words, it is the energy required to move. An electric charge of Q Q looms over a potential difference of the volts, and it is expensive and in is expressed as v times Q electron volts. By definition, one electron volt is the amount of energy gained or loss by the charge of a single electron moving across an electric potential difference of one volt. Remember that one Kunal is equal to 6250 with 15 zeros after it, electrons or one electron is equal to one over Q, which is equal to 1.6 times 10 to the negative. 19 que loans. Power is the rate at which electrical energy is transferred by an electric circuit. Power is the energy dissipated over time, which is T but E is equal to the times. Q. Therefore, P is equal to the product of the Times. Q. All over T or the Times Q. Divided by T and because Q divided by T is current designated by i 1 a.m. equals one Kunal of electrons passing by one second in one second of time, then P is the times I or the voltage times the current for constant voltage and constant current, such as in D. C. Power. This holds true electric power is measured in watts a watt symbolized by the letter W is a derived unit of power in the international system units Oz s I named after the Scottish engineer James Walk. This is a chart of the international system of units for power, the most common of which are the Milly Wat, which is 10 to the negative six watts. A kilowatt, which is 1000 or 10 to the third power watts a megawatt, which is one million or 10 to the six power watts, or a gigawatt, which is 10 to the ninth Power Watts. Now power is given by the product of voltage times the current. But if you are given any one of these two quantities that are used to calculate power power being one of them and voltage and current being the other two given any to you could always calculate the third power, of course, is equal to the voltage times the current. But if you know the power and you know the current that's delivered, but delivering that power than you can calculate the voltage across the load by power divided by current. Or if you know that the power and you know the voltage across the load and you can calculate the absent is flowing to that power reviewing energy. It is defined as the ability to do work. Electrical energy is the energy transferred to an electrical load by moving electrical charges over a period of time. It is also equal to the amount of power that is applied during a period of time that IHS Energy is equal to power. Times time. This is defined as a jewel, but because of jewels, so small energy is also measured in much larger units, such as the water our and the kilowatt hour. When calculating energy or electrical energy, electrical energy is equal to devoted time. See electric current times a time. Interval energy is equal to the revolts times I for amps, times t in seconds or in Greve e ated terms, he is equal to the times I times T but we know that Power P is equal to the times I or the voltage times. The current therefore energy, maybe to find in terms of the power over a period of time or e is equal to p. Times T. This is a chart of the international system of units for electrical energy, the most common of which are the Mila Watt of power, which is 10 to the minus six watt hours. The killer one hour, which is 1000 or 10 to the third power watt hours, and the megawatt hour, which is equal to one million or 10 to the six power what hours? For varying current and voltage, it helps to talk in terms of instantaneous power and average power average power we will deal with later. But for now, instantaneous power is the power delivered over a very short period of time T and can be expressed as piece of I is equal to be survive times. I serve I for each instant of time if graft over a very short period of time. Now let's look at alternating current and voltage. The alternating current and voltage that we will concentrate on for this meeting course is described as Sinus Seidel. And it looks like this both voltages and current very over time, like a sine wave join here They're both showing as being in face they the current and voltage very from positive maximum value to a negative maximum value. When we plot power, you will notice that the Times V and I are signing Seidel and passed through zero from positive to negative. Power is always positive, though when we multiply a negative voltage and a negative current, the answer is positive. So you will notice that the power is also Sinje Seidel The powers frequency is two times the frequency of the end of I and pews always greater than zero. If current and voltage is are in face, remember our plot of steady voltage and current and power. If we were to plot the steady power consumption, pee on a graph over a period of time and shade the area under Pete, it would look like this P Times t. The total energy consumption would be the area of the rectangle given by the formula. A for energy is equal to p times t. Another way of looking at energy is the area defined by P and T. Now, if we divided up the big rectangle into a Siris of little rectangles, the area would be the same, and the defined energy would still be the same. And the areas of each of the smaller rectangles would be given by P Times, Delta T And now collectively, the energy would be given by the sum of all the little P times Delta teas, which would sum up to the big area. P Times t. Now, for the moment, let's assume that the power consumption is not stay but might look something like this, and the areas of each of the smaller rectangles would would still be given by P. Times, Delta T. And collectively, the energy over that period of time would be given by the sum of all the p times the Delta teas. Now, let's suppose the power does not change in steps but in a smooth curve. Such a showing. Here we can approximate the area under the curve by a set of rectangles with Delta teas identical and whose vertical power P is bounded by the curve. If we decrease the time sections or the delta teas to a very small value, their thereby increasing the number of P times T rectangles while each P remains on the curve, then our total areas would be the total area would be exactly the area bounded by the curve . So if we're able to capture the area bounded by the curve, it would be that of the energy delivered to the load. This is precisely what kill what our meters do. This is the end of Chapter one
3. Ch 02 AC Power: Chapter two a sea power. In order to work with the tools of electrical measurement, we have to make sure we understand the difference between Steelers and vectors. So just a quick review here. A scaler quantity is a quantity that has magnitude on Lee, and some of the examples of scaler quantities are length area volume, speed mast into the temperature, pressure, energy, entropy work. Ah, and we've shown a diagram of volume here. Vector quantities our scaler quantities with direction. So there's a direction associated with a magnitude and some vector quantity. Examples are displacement, direction, velocity acceleration, momentum force, electric fields and magnetic fields. And we are also gonna add to this in electrical quantities. Voltage current and impedance is, but they have a special name which we're going to see in a very short order in the next few slides. In a power system, a power source will supply a voltage that is Sinus seidel or a sign you sidle in wave shape of a particular frequency. And in our case in North America, that is 60 cycles per second. The voltage starts at zero peeks at plus a travels through zero on peaks that negativity then returns to zero volts, and it's repeated 60 times a second. When mathematically modeling a sine wave such as V is equal to sine omega T, it can be done and considered to be directly related to a vector of length. A revolving in a circle with an angular velocity of omega degrees per second. The can be represented by a revolving vector. Its magnitude is said to be a it's angular velocity. Omega and omega is in degrees per second. Ah, and you could also describe it as 60 cycles a second. But this case, we're just talking about angular velocity in degrees per second, such that the angle of the vector at any particular time is given by the product of the angular velocity times time. So if the angular velocity is in so many degrees per second, we multiply it by number of seconds, it will come out to a value of degrees, and that will be the angle that thieve actor forms with the horizontal in a utility power grid, omega is 60 cycles per second, at least in North America, in Europe and some places in South America, they use 50 cycles per second as a standard. However, here in North America, it's 60 cycles a second. All voltages and current are rotating vectors. All are rotating at the same angular velocity and all maybe mathematically dealt with but must follow the rules of vector analysis with the special properties for vectors there sometimes renamed phasers. What is the difference between a vector and the phaser? A. Phaser is a special type of vector. Factors have to values magnitude and direction, whereas phasers are vectors that rotate at system frequency. Phasers are relative. If there is more than one phaser that we're studying or looking at their all rotating at the same system frequency. Or they will all have the same angular velocity where all fear phasers are displaced by a fixed angle of separation. Comparison and analysis is then done by stopping the rotation At some point, the relative position of each phaser is unaffected by when the rotation is stopped. So even though phasers are rotating at system frequency at all times, all the analysis and comparisons that are done using favorite phasers are done at a particular instant in time. In other words, when they are stopped when working with phasers the most common descriptive form is polar notation. Holder notation is where phasers are described by the length, which is called the magnitude and its angle, or its relative displacement from zero degrees, or the horizontal denoted by the angle system symbol that looks like this. For example, this phaser would be designated 8.49 in magnitude and they at an angle of 32 degrees. Standard orientation for phaser angles in a C circuit. Calculations defined zero degrees as the right horizontal, making 90 degrees straight up, 180 degrees to the left and 270 degrees straight down. Note that all phaser angles represented in polar form can be positive when measured counterclockwise from the hort from the horizon or the horizontal or negative when measured clockwise from the horizontal, for example. Ah phaser angle 270 degrees straight down can also be said to have an angle of minus 90 degrees. Here are some examples of phasers noted in polar notation. Notice. There's two ways to designate certain phasers. The top left hand one. We already looked at something similar. It's 8.49 at 45 degrees the phaser in the top right hand corner is 8.6 at minus 29.74 degrees, or it could be 8.6 at 330.26 degrees. Both notations are correct. The one in the bottom left hand corner is 5.39 at 158 0.2 degrees, and the one in the bottom right hand corner is 7.81 at 230.19 degrees, or 7.81 at minus 129.81 degrees. Something else to note while describing phasers in polar notation, Is that the magnitude we've described it as simply 8.6 or 8.49 or 5.39 notice? I haven't indicated whether that's inches, feet, meters or could be miles. It's irrelevant, Um, and it could be whatever you want it to be, because the magnitude can be scaled and it doesn't change the direction of the polar notation at all. It just changed the changes. The magnitude and the magnitude can be scaled, so 8.49 could represent inches feet, centimeters, miles, whatever you happen to be measuring at the time, Of course, once you pick one, all arrest are scaled to that that relative amount. So if you're talking about inches, all of your phasers have to be in inches off. If you're talking about feet than all year, magnitude should be represented in feet. Another way to describe a phaser is by rectangular notation in rectangular notation. The phaser is taken to be the high pot news of a right angle triangle, described by the length of the adjacent and opposite sides. Rather than describing a phasers length and direction by noon by denoting magnitude and angled, it is described in terms of how far right or left, or how far up or down the phaser is from the origin. These two dimensional figures, horizontal and vertical, are symbolized by two numerical figures in order to distinguish the horizontal and vertical dimensions from each other. The vertical is prefixed with a lower case. J. This lower case letter does not represent a physical variable, but rather is a mathematical operator used to distinguish the phasers vertical component from its horizontal component. When placed in front of a vector its wings that vector through 90 degrees counterclockwise as a complete, complex number. The horizontal and vertical quantities are written as the sum of two factors. The horizontal component referred to as the rial component since that dimension is come is compatible with normal scaler. Real numbers, and 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 scale of the rial numbers, here are some examples of phasers in rectangular notation noticed. This time there is only one way of designating a phaser, so the one in the upper right hand quadrant is four plus J three, which means it's a quantity four along the real axis and three in a positive jay direction or straight up and down. The phaser in the left upper left, uh, quadrant is minus four plus J three, which means it is described as minus four along the rial axis and plus three along the J axis and finally, in the bottom right hand corner. We have phaser, which is for minus J three, which means it's four along the plus riel axis and it's three in the minus J direction along the J axis. 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 fazer 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 arc tanne 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 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 in the scanner voltages in D. C. 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 a 10 full at a phase angle of 53.13 degrees. Factors as well as phasers can be moved around a plane as long as their direction and magnitude are maintained. You add two vectors, or phasers by placing the tale of one on the head of the other, then connecting the other head and 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 volts, respectively, across to a C voltage sources. Yet only read 10 volts for total voltage with a C two voltage just can be aided, aiding or opposing one another in any degree between fully aiding and fully opposing inclusive without the use of phasers. Complex numbers that is notation to describe 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, 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 off of cup phasers in polar notation 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. 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. 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 Uranus attract 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 d 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. He 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, uh 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 division of polar form. Complex numbers is also very easy. You simply divide the polar magnitude 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. First, complex number to arrive at the angle of the Kocian, as in these examples. 35 at 65 divided by 10 at minus 12 you divide 35 by 10 and end up with 3.5 as the magnitude and you subtract minus 12 from 65 to Samos, adding 12 to 65 giving your result in angle of 77 degrees. This in this example, we have 124 magnitude at 250 degrees, divided by ah phaser with 11 magnitude and 100 degrees. So divide 124 by 11. It gives you 11.273 and you subtract 100 from 2 50 leaves with 150. And the final example is our simple three at 30 degrees, divided by five at minus 30 degrees gives you a result of 300.6 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 you subtract 65 from zero, which would give you minus 65. In the next example, you'd 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. 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. When considering to sine waves that are not in phase, one wave is often said to be leading or lagging the other. This terminology makes sense in the revolving phaser picture as showing if the rotation is stopped, we can see that the blue phase is said to be leaving the red phase or, conversely, the red face is lagging the blue phase. These phasers could be current or voltage or one of each. If the blue phaser is current and the red phaser is voltage, the current is said to be leading the voltage. Or, conversely, the voltage is lagging. The current Let's look at the voltages and current phasers with a resistive, inductive and capacitive load for a pure resistive load. The voltage and current are in phase, and they can be represented on a fazer diagram as overlapping each other. As you can see here, they are rotating counter clockwise with 60 cycles if you want, or just omega that for a pure capacitive load, you will find that the voltages or the voltage lags the current by 90 degrees, as depicted by the phaser diagram. Here for a purely inductive load, you will find that the voltage leads the current by 90 degrees. So in the case of a pure resistive passive and inductive load, the current reacts in quite a different manner. For each of those separate loads, the resistive load having the current in face with the voltage. The voltage leads the current in the case of induct Insys, and the voltage lags the current in the case of capacitance. Now let's look at the villages and current phasers with inductive and capacitive loads, this time with some resistance introduced into the circuit just for comparative purposes, I am re showing the results of a pure resistive load where the voltage and currents are still in phase. For a resistive plus a capacitive load, the voltage still lags, occurring but less than 90 degrees this time, and similarly, for a resistive and resistive plus of inductive load, the voltage still leads the current, but again less than 90 degrees in these a C circus. These combinations are known as impedance is and are denoted by, the letter said. Impedance impedes the flow of current in a circuit similar but not the same as resistance in a D. C. Circuit. Impedance is a factor which is made up of two components, a resist of component and a reactant component. The reactions denoted by the letter X resistance is unaffected by system frequency. However, the reactant is affected by system frequency, and it depends on whether it's a capacitive, reactant or an inductive reactivates for the purposes of metering. In a power system that has 60 cycles per second, we don't usually worry about the frequency because that remains unchanged when metering. It is usually constant at 60 cycles per second, or if, in fact, you're in Europe for in South America could be the standard is 50 cycles, the point being that the frequency doesn't change over the long term, so that the metering results are usually done by quoting just the impedance and assuming that the system frequency remains the same. Of course, a purely resistive load has no reactive component, and its impedance is simply the reactions of the resist er in homes. In other words, the impedance of a resistor is equal to since in doctors drop voltage in proportion to the rate of change of the current, they will drop more voltage for faster changing currents and less bolting for slower changing currents. What this means is that the reactant in Owns for any in doctor is directly proportional to the frequency of the alternating current. The exact formula for determining this reactant is given by Zed, which is equal to the reactant X with a lower case. L, which is two pi times f l where pi is just a 3.14159 f is the frequency in cycles per second and L is the induct INTs in Henry's. Since capacitors conduct current in proportion to the rate of change of the voltage, they will Passmore current for faster changing voltages as they charged and discharged the same voltage peaks in less time 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. In other words, the impedance of a capacitor is equal to X with a small letter C. Sometimes, they noted, with a lower case c ah, and it's equal to 1/2 pi F C. Where again, the pious 3.14159 and the frequency is given in seconds per second and C is the capacitance in the rats. Note that the relationship of capacitive reactions to frequency is exactly opposite from that oven. Inductive reactive capacitive reactions enormes decreases with increasing HC frequency. Conversely, inductive reactant in olds increases with increasing a C frequency in. Doctors oppose faster changing currents by producing greater voltage drops. Capacitors oppose faster changing voltage drops by allowing greater currents, alternating current in a simple impedance circuit is equal to the voltage involves divided by the impedance in homes, just as a direct current in a simple resistive circuit is equal to the voltage in volts divided by the resistance in homes. This leads us to an extension of owns law for a C circus. Impedance is related to voltage and current, just as you might expect in a manner similar to resistance in homes law for D. C. Circuits. But this time we're dealing with phasers, which are vector quantities and have to be governed by the laws of vector analysis. The E M F or voltage is equal to I times Zed, where I is the current and said, is the impedance. I also is equal to E over Zed and Set is equal to e over I. All qualities are computed and expressed using vector, not Skinner rules. We encounter a measurement problem if we try 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 off 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. This is called the Mean average value. So far, we have looked at three ways that we can measure the, uh, intensity of on a C way for whether that occurred your 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, uh, 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 at 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 to 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 the 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 the two bulbs light to the same brightness that is. They draw the same average power, and the key here is their drawings. Same average power than it's reasonable to consider the current i A C to be in some ways equivalent to the current i. D. C. So what ISS that value of I see it would be useful if there were 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 now the peak or maximum value of I squared is shown here and labeled I squared Max. The mean or average value of I squared is showing here, and it is exactly I squared Max divided by two. Remember, now that the instantaneous power is given by ice squared times are for the resistance load . If we wanted to find the average power, we could find it by simply multiplying the average value of ice Square. Times are such that P average is equal to I squared. Average times are, but we know that I squared. Average is equal to I'm ax squared over two. So the average value of power P average is equal to I'm ax squared over. Two times are we just saw from the previous slide that the average power can be calculated by I'm ax squared over two times the load resistance. Let us define a current that, when used to calculate power, gives us the average power. In other words, when that current we'll call it I subscript to find is squared and multiplied by the load, resistance will give us the average power. This means of course that I, he find squared, is equal to I'm ax squared over two, which is the mean of the current squared or the mean square current. Therefore, the square root of the mean square current equals that defined current I subscript to find another way of stating it is the root, mean square of the current. So we just discovered what the value off I subscript defined is. It is equal to I'm Max, divided by the square root of two we call this current. I are a mess or the root mean square, and it is 0.7 07 The value off I'm max 0.7 or seven is just one over the square root of two . This is another, more useful way to describe an A C quantity, voltage or current. And of course, it can be converted to other A C quantities terms such as the amplitude or the peak, or the peak to peak, or the average just by scaling the value up or down. In this case, we could take I Max and multiply it by 0.707 and come up with the root mean square. But the root being squared current can give us the average power consumption and you'll see in later slides that when we use RMS values for current and for voltage that we can use them to calculate average power. Looking back at the question, we asked ourselves, What is the value of I A. C? Well, I see is now we can say equal to I R M s, which is equal to 0.7 or seven times that peak value of R A C current, which is equivalent to the I. D. C. Dissipating the same amount of average power using the R. M s current and the little resistance the average power can be calculated by using the formula p is equal to I squared are similarly, We can define the RMS of value for RMS voltage, which is 0.707 the peak voltage for a C voltage. In the next chapter, we will show that if the RMS value of current and voltage are used, we can use all of our previously established formulas to calculate power. But this time we're calculating the average power also because we're simply using a scaling factor for the current and voltage that that factor being 0.7 or seven or one over the square root of two. Then all of the equations involving owns law hold true if we use are a mess, values for current and voltage as well. All of the rules for mesh in AL assistant terms can be used with RMS values for current and voltage. Keeping in mind that all quantities air computed and expressed using vector, not scaler rules, this is the end of Chapter two.
4. Ch 03 Instrumentation: chopped her three instrumentation. Let's consider a pure inductive load in an A C circuit because instantaneous power delivered to induct ER is the product of the instantaneous voltage drop across it and Dr Times E instantaneous current. In other words, P is equal to 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 here, the power is positive and as with a resistor, for example, the power is also positive when the instantaneous current and voltage are both negative. However, because the current in voltage waves air 90 degrees out of phase, there are times when one is positive while the other is negative, resulting in an equally frequent occurrence of negative instantaneous power. Notice on these curves that when ever the voltage is zero, the power is the euro, even if the current is at a maximum when the current is zero, even though the voltage is at a maximum in this case maximum positive. The power is also zero as here when the current is zero. Even though the voltage is Adah maximum negative, the power is still zero and here when the current is at a maximum, voltage is at zero. So the oppa voltage output power, power dissipate are power dissipation is zero Also. Now let's consider a pure capacitive load in an A C circuit. As you might have guessed, the same unusual power wave that we saw what the simple in Dr Circuit is present in this simple capacity or circuit. As with the simple in doctor, there is a 90 degree phase shift between the voltage and current. But this time the current leads the voltage and results in a power wave that alternates between positive and negative. This means that a capacitor does not dissipate power. As it reacts against the change in voltage, it merely absorbs and releases power. Alternatively, notice that when the current is on a maximum, the voltage is at zero. So the power of put is zero. When the current is at zero on the voltage of the maximum negative as here then the power is still zero as it is here. When the current is zero and the voltage is at a maximum positive value, the power is still zero and lastly here the current is at a maximum, but the voltage is at zero, so the power is zero Also. Clearly it can be seen that at times power is negative. But what does negative power mean? It means that in the case of the induct er, after having built up a magnetic field while the current is flowing into the in doctor, it is now releasing power back into the circuit, while a positive power means that it is absorbing power from the circuit as it rebuilds the 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 it over a span of a complete cycle. What this means, in a practical sense, is that the reactant oven in Dr Dissipates and Net energy of zero quite unlike resistance of resistor, which dissipates energy in the form of heat. Mind you, this is for a perfect in doctor only which has no resistance even in the wires. This is the same thing that happens in a capacity circuit. Only the power flow in it out is due to the buildup of electric static field, while a current was flowing into the pasture, then releasing power back to the circuit during the negative power cycle. Let's take a closer look at an A C Sinus idol current and voltage over time that are out of face. The instantaneous power, which will call P sub script I plotted here in blue, is calculated by multiplying the instantaneous voltage, the subscript I plotted in green times, the instantaneous current I. Some script I planted in rent the instantaneous voltages mathematically given by bm sine Omega T, where V M is the peak or the maximum voltage, and omega is theano angular velocity of a phaser where omega T is a particular angle at a particular time. The instantaneous current is given by I am sine omega T minus five. Where I am is the Pekar maximum current and phi is the displacement angle from the voltage phaser. So the instantaneous power is mathematically given by the product of these two, which is V. M. I am sine omega t times sine omega T minus phi, which is, as I have said, the instantaneous voltage times instantaneous current notice that the power and is sometimes positive and it is sometimes negative. It is positive when both V and I are either both positive or both negative. And it is negative when one is positive and the other is negative. The average power is a constant over time and it is given by this formula, which is VM times. I am over to Times Co sign off five. This is derived through a trigger demetrick manipulation which will follow, and but the result is more important here than the actual proof. But I will go through that in the following slides. The average power is an important value in the world of commerce as it is this term that is equated to the consumption of energy. It is the meter quantity that is used for building of power. Okay, I'm very quickly going to go through the proof. I'm not going to spend a lot of time in the details. You can go back and and figure out the details yourself, but I'm going to start with the formula that we have for calculating instantaneous power, which is V m. I am sine omega t times sine omega T minus five. Now what you see here in yellow is an identity a Trigana metric identity that we'll use to reduce this equation to something that we, uh well, we want to be able to work with and and that IHS average power. So were you to use this identity to reduce the last part of that term sine omega T minus phi And that's going to give us this result in formula. However, we still need to use a couple of identities to reduce this this equation even further. So we want to reduce sine squared omega t. I will use this identity to reduce sine squared omega T and will use this identity to reduce the product of sine omega T Costa Omega T. And that will give us this equation here, which is Thean Instantaneous power is made up of two terms. The subscript m I subscript m over to Times Co signed five minus the m. I am over to co sign the quantity two times Omega T plus five. Now we're looking for average power, so the first term remains The VM I am over to co sign Phi will remain however this term because it is a, uh, Omega T is involved in it over time Coast to omega t Doesn't matter of its phase shifted or not will average out to zero. So this term on the average over time goes to zero. Leaving us with just this term are this equation for the average power consumption P average is equal to V m I am or to co sign of five. We now know that the average power is given by this equation here which can be rewritten by putting VM all over route to. And I am all over too, because two can be split into route two times. Route to this actually means that if we remember that the RMS value for voltage is vm over route to and the RMS value for current I is I am all over route to. So we can say that the average power can be found by calculating the RMS voltage by the R. M s current times of phase angle between the two. And if we plot the phasers, they would look something like this. P averages is find as a phaser itself P average, which is given by v rms times ir mass times co sine of the angle between the two of them, and it represents riel power in walks, the actual work done by an electric current or actual energy consumed by the load to create . For example, heat, light or motion on a rectangular coordinate plain. It is a phaser plotted along the horizontal or riel axis. Electro systems normally have in doctors and capacitors, which are referred to as reactive components. Ideal reactive components do not dissipate any energy, but they draw currents and create voltage drops, which makes the impression that there actually dissipating energy when they're not. This is called reactive power. Its average value over a complete A C cycle is zero because the phase shift between the voltage and current, it doesn't contribute to the net transfer of energy but circulates back and forth between the source and the load and places a heavier load on the utility. Reactive power is measured in volts amps, reactive or VARS, and is calculated by the RMS Voltage times, E. R. M s current times, the angle between them, the summing of riel power watts and reactive power bars. On that is the the phaser summer vector. Some produces 1/3 fazer, which we call a parent power. Yes, B A, which gives us a better indication of the load on utility and is calculated simply by multiplying the RMS voltage times C, RMS current. This becomes more significant in industrial loads. The angle between the rial power and the apparent power showing here is fi is called the Power factor angle. Co sign of this angle is known as the power factor, and it is a number that is less than one. It goes from 0 to 1. It is important to note that Phi is the same angle that exists between the voltage and the current. We have already derived how to calculate the average power from the RMS voltage and current . We now condemn fine reactive power, which is designated que average, which is measured in volts am pairs reactive or bars and is calculated by multiplying the RMS voltage times, the RMS current times, a sine of the angle between them and apparent power, which is the vector summing of the real power watts and a reactive power bars, which is given by the RMS Times I RMS and measured in B A or vote amps. Here are some examples plotted in real time The voltage is in blue, the current is in red and the power is in green. The voltage leads the current by 90 degrees, giving a power factor of zero. The average power, of course, is zero. And this type of load would be on in Doctor. In this case, devoted just still blue and the current is red and the instantaneous power is given by green. The power factor is still zero. This time the voltage leads the current by 90 degrees and this is an example of a pure capacitive load. In this example, the voltage leads the current by 60 degrees, giving us a, uh, power factor of 0.5. The average power given by the M I am over to coast data would work out to one times one over two time 20.5, which works out to 0.25 as indicated on the graph in this example of voltage and the current are in phase giving us a power factor of one again. The same formula for the average power is given by the M. I am over two times Khost co sign or the power factor which is one which gives us one time to 1/2 times one, which is equal 2.5. Now let's have ah, look at some of the instruments that measure power and energy. The UH electric dynamometer tight meter has to fixed coils that are used to produce a magnetic field, the fixed coils air connected in series and a position clack seal that is in line with a space between them. These fixed coils are connected in series with the load. Therefore, the load current will flow through these coils, which are called the current coil. It also has to movable coils that are suspended in the field produced by the two fixed coils. The two movable coils are also positioned quack, silly and are connected in Siris. These Movil coils are connected across the load and therefore measure the potential drop across the load. These coils are called the potential coil. The main shaft on which the movable coils air mounted, is restrained by spiral springs that restore the pointer to zero. When no current is flowing through a. The design of these components air such that the movement will measure the average power which is equal to the volts. RMS times the I. R. M s to the load and the actor Coast times a co sine of the angle between them in and electronic or digital version off this, uh, electric dynamometer, the springs and coils air replaced by transducers, one for each, the current and the voltage. And each half an analog to digital converter, which feed into a digital processing unit, which drives a digital readout. And that digital readout will may will read out in the average power, which again measures the RMS voltage times, the RMS current times, a Coke sine of the angle between them. The primary advantage of this thes meters is that they can be used to measure alternating as well as direct current. Remember that a C RMS is equivalent to D. C. This line shows the dynamometer want meter connected in a circuit measuring the power delivered to a load. The current through the fixed coil is proportional to the load current and produces a magnetic field that reacts with the magnetic field of the potential coils. That creates a Twerk that counteracts with the restraining springs of the meter movement that results in a reading of the average power delivered to the load in ah on Elektronik version of this springs and coils, of course, are replaced by transducers and analog to digital converters that feed into a digital processing unit that drives a digital redos regardless of the type of meter. All what meters have basically three connections. Current to the load is made to flow through terminal number one or into terminal number one and out terminal number two to the load. The voltage across the load is connected to terminals one number one and number three. Sometimes what meters have 1/4 terminal terminal number one and terminal number two are connected to the current element of the watt meter, while current, our terminal number three, and terminal number four connected to the voltage coil of the element of the watt meter. Terminals one and four are then jumper together. The two terminals the watt meter are marked to indicate the polarity of the elements that will give a positive deflection of the meter for power flow into the load Terminals number one and terminals number four are usually jumper together internally. Hence, terminal number four is not always showing and not always available on a watt meter. Our is equal to the rate at which electrical energy is transferred to a circuit load and that is given by the formula. P is equal to B times I and power is measured in Watts. Energy is the amount of electrical energy transferred to the circuit. Load over time and energy is equal to power times time in what hours and is given by the formula E is equal to the times I Times T and what hours, which is the area underneath the curve of the power graft on a power versus time shark. The energy meter measures power and records or registered registers it over time, providing and measurement of energy. Just about every house has one showing here is the older mechanical type kilowatt hour meter, which is now being replaced by the more modern, solid state digital or electronic tight meter. The older type watt hour meter has approximately eight major components. The cover, which is basically a protection against the environment from moisture, rain, dust, rodents or anything else, and it's usually glass and and it screws onto the base of the meter. The register is the recording device of the water meter it keeps an accurate record of the total amount of energy that has passed, true it over a period of time. The register is connected to the electrical components of the meter through a series of gears that are turned by the rotation of the disk. It is designed to show revolutions in kilowatt hours or kwh. The rotor is a part is that part of the meter that rotates as power is being used. It consists of a disc mounted on a shop. The disk is a round, flat piece of aluminum that acts as a conductor. There are markings on the upper surface of the disk that air used for calibrating the meter . The shaft is held vertically in place in the meter by bearings, depending on the vintage of the meter, the bearings, maybe of jewels or now a little bit more modern type. Even though it's an old school meter. Ah would more commonly they would be held in position by cushion of magnetic bearings. The returning Magnons air used to accomplish exactly what their name is standing for it. Retards were slows the operation of the meter. Specifically, they don't slow the entire operation, but only that of the rotating disc. The magnets regulate the speed of the disk and prevented from coasting or accelerating greater than that provided by the driving torque, which of course, is equivalent to the power consumption, the electromagnets, the potential coils and current coils. The potential coil. The main purpose of the potential coil, is to convert voltage through the meter into a magnetic field proportional to the circuit. Voltage. The coil is extremely inductive and consists of many turns of insulated wire. Wound on a high die electric spool mounted on laminated silicon steel core laminated steel concentrates the flux field as well as reducing both circulating currents and flexes in the laminates themselves. The current coil and core, the primary purpose of the current coil and core, is to convert the circuit current into a magnetic field proportional to the load current. The coil is connected in series, with the load, using a small number of large wire turns to carry the load current, it produces a flux field in phase with the current. This requires the coils to have a large cross section lo induct INTs and little resistance . Laminated silicon steel cores are used to concentrate. The flux is and reduce the losses. The coils are coated with an epoxy resin to insulate them electrically and mechanically from the iron core. When we talk about meter elements, this is what's meant. A current coil and a potential coil makeup, one element of a meter. A current coil by itself is considered 1/2 element, and a potential coil by itself is considered 1/2 element. We'll talk about more of this later as we look at standard type kilowatt hour meters in their connections to the systems. The elements now the killer, what are meter are designed such that the Roeder disk and shaft rotate at a speed that is proportional to the power drawn by the load. The frame is where all of the after mention components are mounted, and the base is where the frame is mounted and the cover is screwed, too. The exterior, which then provides the connections to the line and the load the speed of the disc or its angular velocity, is proportional to the power being consumed by the metered load. The disc, angular velocity or speed, is proportional to the RMS voltage vector in the potential coils times the RMS current factor in the current coils times the co sine of the angle between them, which is equivalent to the power represented by those factors. A single phase watt hour meter is essentially an induction motors who speed is directly proportional to the voltage applied to it and the amount of current flowing through it. The phase displacement of the current, as well as the magnitude of the current, is automatically taken into account by the meter. In other words, the power factor influences the speed and the moving element. Disc rotates with the speed proportional to riel power. The register is simply a means of registering the revolution's and by proper gearing is arranged to read directly in kilowatt hours. In some cases, the meter reading must be multiplied by a factor called a registered constant or a meter multiplier. To obtain the total and accurate kilowatt hours. The aluminum disc acts is a squirrel cage rotor torque being produced as a result, off eddy currents induced in it by the potential and current coils. Normally, there is very little friction present in the meters, and if no additional retiring force other than friction were placed in the meter. The rotating element would travel at a relatively high speed. The necessary retarding action is provided by a magnetic brake consisting of a permanent magnet. Operating on the aluminum disc. A D tent or a ratchet is sometimes attached two meters to prevent rotation in the reverse direction when it is desired, not to register reverse power flow. This usually consists of a collar having notches or pins, which is placed on the desk shaft and appall attached to some fixed part of the meter, which engages with the notches or pins upon reverse rotation but slides easily over them in the forward direction. The K H of ah kilowatt hour meter is called the What our constant and equals the number of what hours. For one turn of the disc, yeah, electromechanical meter, the KH what are constant of a meter is indicated on the face of the meter ash on here. Also, the manufacturer places a marker on the disk that makes it easy to see and count. The disc rotations stated another way. The meter constant is the number of one hours wound on the register for one revolution of the disc, the amount of energy represented by one revolution of the disk is denoted by the symbol K H , which is given in units of watt hours per revolution. The value that is very common is 7.2. Using the value of K H. One can determine power consumption at any given time by timing the disc rotation with a stopwatch, and the power is equal to 303,600 times the K H all over T, where T is in seconds and P is in power. In Watts, for example, if K H is equal to 7.2 as above and one revolution one revolution took place in 14.4 seconds. The power is 1800 watts or 1.8 kilowatts. Here are some values that are common to the General Electric Westinghouse, saying im sang gammel and Duncan type meters most meters can record up to 10,000 kilowatt hours. This is accomplished through a series of four dials that read from right to left in significance. Each significant dial is here to the previous significant dial in a ratio of 1 to 10. There are some five dial meters that air red. In a similar fashion, there is usually a test style that is geared to be turning 10 times faster than the least significant dial making, making it easier to detect meter movement and fractions off the least significant dial. Reading the dials can be tricky as dial rotations alternate from clockwise to counterclockwise to clockwise and counterclockwise etcetera. As you progress upscale. Here are some examples. When determining the total consumption from dial readings, it must be done from right to left. If the pointer is between two numbers, the lower number is a correct reading. For example, this register is reading 1000 799. The reading of this register would be 15961 and the reading of this would be by zero 49 seven. Up to this point, we have only described the functionality of mechanical what our meters. However, today the majority of meters are Elektronik, or digital type meters, which very in functionality. But basically they replace the current and potential coils with transducers, one for each, the current and the potential, and they feed into eighties or analog to digital converters again, one for each. The current and the voltage. And that feeds into a digital processing unit that has several functions. And it depends on the manufacturer and the when the what meter was produced. Just what those functions are over, one of which is, ah, register that takes the reading and it is on e prom. That's, Ah, electrically erase. Herbal programmable read only memory, which is a type of non volatile memory that is used to store the data and saves it in the case the power goes off. It doesn't lose the reading on that kilowatt hour meter, just as the old fashioned dial type never lost the register reading as well. Digital meters use a multi segmented indicator instead of dials to provide us with a reading to manually display the energy consumption as well as other factors such as demand . It will also, uh, as it cycles through the various readings. Check all the segments of the digital display, wants to make sure they're all they are working. And for test purposes, the manufacturers have provided an electronic disc emulator, as you'll see in the next slide, much the same as the case of H of a mechanical meter. The electronic meters will also have a case of H. And sometimes they're designated case of I because sometimes he's meters put a pulse output . Ah, and it's called a case of I, which is has the same relative meaning as a case of teach. Further auxiliary units can be added, such as wireless and remote readings for signal and signalling devices. Elektronik metering today has become much more computerized, and this functionality is paramount to a medium to large size utility. This is a picture. Actually, it's gonna be a video of the kilowatt hour meter that was placed on my house by Hydro one. And I just wanted to show you what the display looks like and how the disc emulator works. It works much the same as a rotating disc. Uh, the three dots are are repetitive, and you can time the repetitions and then use the case of H of the meter to determine the load. And as you can see, the case of H is 7.2 on this meter, and it is printed on the face plate of the meter energy meters. Regardless of their makeup, whether they're mechanical or Elektronik are considered to be made up of meter elements and one element is made up of voltage half element, which is either potential coils in the old school meters or voltage transducers, plus in a TV converter and the new electronic meters. And the other half of the element is the current element, which is made up of a current coil or a current transducer and ADEDY converter. Together, they make up a single element of a meter that could be used to measure and record power and energy consumption by a single phase low. And from now on, when we start to consider these energy meters were going toe disregard the fact that they are either electronic or mechanical, and we're only gonna be talking about measuring devices with elements. This is the end of Chapter three.
5. Ch 04 Single Phase Metering: Chapter four single phase metering meter is said to be self contained when the only path the load current can take from the line to the load is through the meter. Or put it another way. The meter conducts the entire load current. The circuit is said to be to wire because there are only two wires to and from the load. The meter is described as being an A base meter because the connections are wired directly to the meter because it is important to maintain accuracy and consistency. Standards have been developed for various types of meters and their connections. This shows measurement Canada's standard 1 12 1 standard, and there's more standards and we'll talk about more of them later. However, there are other standards that are not measurement Canada standards, but they're very similar indeed. And will, uh, standardized the connections to the various types of metering, uh, circuits and their components. This time, the slide shows a socket or an s base meter notice again of the current flows through the current coils of the meter. This shows measurement Canada standards 12 to again. We'll talk more about the standards leader. Now let's look at a poly phase Mir or poly face metering. A poly phase meter has the same basic electric theory as a single phase meter eddy. Currents are induced into the disc, interacting with flux fields that cause them to rotate. There are more than one element acting on the same disk, but what they do is arithmetically some up to the totals of both elements. Upali Phase meter could contain 2 2.5 or three elements one potential coil in one current coil. For each element notice the policies Meter has one element on each. The left and right side of the frame of this particular kilowatt hour meter. Each element is made up of a potential element, which is a health, half the element and a current element, which is the other of the element the discus position, so that he will rotate through the main air gap of both elements and therefore be affected by the flux of both elements. Since there is only one disk, there is only one set of retarding magnets, and these ey're located between the two elements on the front of the frame. The register, as on a single phase meter, is located above the briefing magnet. Another difference in a poly fees meter is the design of the disc. In a meter where there are more than one element elements acting on a single disc, there will be interference between the induced eddy currents of each. To eliminate this interference, some manufacturers have used laminated discs, while others use discs with a combination of triangular holes and or slots designed into it . The mechanical operation of a single phase or a poly fees meter are virtually the same. The registered takes off. I take off here is driven by the worm gear of a rotating disc. The term case of H still has the same meaning. Although some single phase meters use multipliers of less than 10 Polly phase meters are more likely to have multipliers that are quite different. It could be 40 50 90 or anything up to 200 depending on the manufacturer and the media ring or the meter type. Remember that what meters and kill what our meters air designed such that each element takes one voltage in one current and combines them in such a way that the dialled register read Oh is the product of the RMS Voltage and RMS current times, the co sine of the angle between them. That is to say, one element is stead with the one and I won voltage and current and the other element is fed with the two I to voltage and current. The meter sums the co sign product of each of these energy consumptions of the poly phase load. We've got a head start into the understanding of how policies what our meters operate by knowing and understanding single phase meter. The basics are the same. The potential coils and the current coils both produced Flux fields Eddy currents are then introduced into the disk and react with ease. Flux is producing a descript ation proportional to the load to be more mathematically precise. The meter measuring the average power to load takes the magnitude of the RMS voltage multiplied by the magnitude of the RMS current times, the co sine of the angle between them. This is also the case for each element. It is the product of the magnitude of each of the RMS voltages and currents times, a co sign of each of the angles between them. For each element of the poly face meter when writing the formulas. The magnitude symbols are quite often left out of the formula as they add confusion to the variable symbols that are in the equation themselves. The term co sign Phi or cosign paid or whatever variable you're using here is also left out of the four formula quite often, and it is assumed that we are taking the coastline of the angle between them, the villages and current. When we're looking for the average power, this is sometimes referred to as the dot product, but it's just left out out of the out of the formula as we progress through the equations just to make it, Ah, little bit simpler to view. So any time you see a voltage in a current product in calculation for average power, it is assumed that we are multiplying it by the cool sine of the angle between the two of them. Finally, when working with average power measurement, it is assumed that we're using RMS values for current and voltage is so they too, are left off the equation left out of the equation, making it much easier to visualize and deal with. Now, remember in a single phase Elektronik solid state meter. There is a potential or voltage transducer, plus on analog to digital converter and a current transducer, plus an analog to digital converter, feeding them directly into a digital processing unit that is used to generate several functions, including the reed. Oh, for Polly phase electronic meters. The construction is even easier. We just have to have three pairs of transducers feeding into three analog to digital converters. In this case, there are three elements to the um to this type of meter. The digital processing unit is programmed a handle three input pairs to generate several functions, including digital readout and the E prom that retains the data when the power supply is switched off. Now let's look at some practical applications or examples. A residential electrical supply comes from a single phase transformer, that is, the primary is connected to one phase of a three phase distribution supply, and you can see a picture of one here. You've seen many of them. If you have travelled around the countryside or around the city, you'll see these cans, which are transformers hanging from the pole. There may be 12 or three of them, depending on the type of supply you want. In the case of a residential house, there is usually only one transformer, which again is a single phase transformer one phase feeding into the primary one phase of the three phase distribution supply feeding into the primary of the camp. The secondary is a centre tapped coil of a transformer, providing 1 22 40 volts supply, sometimes referred to as too hot leads and one neutrally. In order to understand how a residential electrical supply is metered, let's start by considering two independent power supplies, V one and V two supplying two loads. The power to each of these loads can be measured by to what meters Notice the spot markings or the polarity markings on the white meters. With this arrangement, the power delivered in circuit one is given by W. One, which is equal to the one times I one Times Co sign Phi and five being the angle between the current and the voltage and the power delivered. A circuit to is given by W two, which is equal to the two times I to times co sine of the angle between the voltage and the current. From here on, I'm going to not show the term co sign Phi and assume that any time you see a want reading given by the product of the Volt agenda current, it will be the same as magnitude of the voltage times, the magnitude of the current phasers multiplied by the co sine of the angle between them. Now, if we move what meter number two like this, it would essentially be reversing the voltage to the watt meter. But keeping the current to the watt meter the same that is the current is still entering the spot mark. We would have to reverse the reading of W to in order to obtain the same reading as before . Hence, W two is now equal to minus the two times I to connecting the circuits together at what we call a neutral makes no electrical difference to the loads. This arrangement is the same as connecting the loads to the center tap residential distribution transformer. The one will now equal. The two will equal VT the total voltage on the transformer divided by two, and this is a characteristic of the residential distribution transformer. Since W T is equal to W one plus W two and we know that WT is v one times I one minus b two times I to we can substitute the voltages, uh, Bt over two, or the voltage V t over to four, the biologist B one and B two and we are left with this equation which can now be rewritten this way, bringing the negative side in a negative sign inside the brackets where I two is and making I too negative. Notice that the load is not included in the equation and it makes no difference to the formulas calculating the media ring as long as we connect the what meters ash showing here . This slide shows a three wire, self contained meter. Now they call it three wire because they're three wires feeding a residential load. Notice how there is too current coils associated with one potential coil, since an element is made up of one current coil associated with one potential coil, each being 1/2 element. These type of meters air, often referred to as 1.5 element meters, this is measurement Canada Standard 13 01 It's connection to the load is governed by the formula that we've already developed here, which is showing here. The total is given by the T over two times I one plus the quantity VT over two minus high, too. So if we look at I one from line one, current flows from the source through the meter, through the outer current coil and to the load lying to or I two from lying to flows like this through the inner court current coil. But notice this time that it's flowing essentially backwards through the inner current coil and goes to the load. The potential element is connected between lying one and lying to, so it is actually measuring the T. Remember that? What meters and killing Our readers are designed such that each element takes one voltage in one current and combines them in such a way that the readout is the product of the RMS voltage in RMS current times, a co sine of the angle between them. So let's look at the products of the RMS voltage and current as they are connected to the meter and see how they match our formula. The T is connected to the potential coil. It is the voltage between line one and line two. The divine by two is taken into account because only half of the potential element reacts with I won and the other half reacts with I two or minus I to. In this case, I one is the current through the outer current coil, and I two is a current through. The inner current coil, however, is going backwards, so it does take into account the minus sign the meter some the product of the two RMS voltages and RMS current times, a co sine of the angle between each of them. In actual fact, the meter elements work more like this. The potential coil is split in half, where the meter sums the power measured by both elements number one and number two. As in this equation here, where the total power is given by the one RMS times I won our mass times a co sine of the angle between the one and I won, plus the two RMS times I two RMS and the coastline of the angle between the two and I to so the left element is the first part of the equation and the right element is the other part of the equation where V one is the tea over to and the to is also the tea for all over, too. And I won is I won and I to in this case is minus because it's going through the coil backwards. It requires people fuel and equipment operating electrical utility system. The number of people in the amount of fuel and equipment are directly related to the collective requirements of the utility customers. If the utility is to be financially self sustaining, it must charge enough to recover the costs of generating and distributing power. Recovering cost would be easy of all. Customers used a steady amount of power all the time. Unfortunately, this steady load situation doesn't exist in practice. Commercial, industrial and sometimes residential areas like those in Freeman Ville have kept characteristic load peaks during which excessive power is demanded. The demand may last for a split second or for many hours. Naturally, a utility would prefer to have a customer who uses load evenly over a period time to one who uses the same amount of total energy in a fraction of the time. As reality reflects the latter case is more often, the utility must be equipped not only to supply the constant level requirements of the system, but also the generating just and distribution equipment must have sufficient a reserve capacity to meet excessive power demands. Reserve capacity in items like generators and transformers mean That means that a utility must spend money to purchase large components and lots of fuel, if only what our building is used. It could financially constrict the utility, so quite often the demand charge is associated with the purchase of these large ticket items and the capital investment of the utility. The energy charge is, uh, usually related to such things as fuel consumption and water flow and routine maintenance items. And if if the if there is a higher demand for power than there's, more of these large ticket items have to be purchased. Such the customer should be charged for bees or share the cost of these large ticket items using a demand charge. In order to achieve fairness in this, the demand energy combination meter was developed without going into too much detail. This meter, while and while measuring energy is also used to measure peak demand over a set period and that period is usually a month. It does this by means of measuring and displaying the average demand over a period of, say, 15 minutes and it is measured in watts or kilowatts or megawatts. And it holds that reading until it's surpassed by a larger demand reading or it is re manually reset at the end of the one month period. The customer is then billed with a combination of the amount of energy used in kilowatt hours, plus the peak demand that it was recorded in kilowatts. This used to be accomplished and still is in most place in some places, with the use of thermal elements in the meter that that air heated proportionately to the kilowatt demand. And it drives a push appointer and an idol pointer. How that works is the pusher Pointer would move with the demand, pushing the Idol pointer upscale when the pusher pointer drops. With the falling demand, the Idol pointer would hold the highest reading until it is manually reset. Of course, the more modern digital meters do this with a built in algorithm that the digital processing unit has and simply records it for future reference as the display cycles through its various, uh, readings. One of them will be, say, a P D or power demand reading and in this case is showing 6.87 which is 6.87 kilowatts. Let's walk through a couple of classroom problems and solutions just to reinforce some of the theory that we've just discovered in this power measure. Working example. We have measurements Canada Standard connection for 13 01 which you see here. In this figure, we've placed a load of 24 arm homes across line one and two. We see that the 24 own resistor connected line to line causes a current to flow of 10 amps , which we could find just simply by applying homes Law of 24 homes being fed by 240 volts, uh is equal to 10 amps. Our cast will be too. Find out what the amount of power is being used by the load. What is the amount of power measured by each element and subsequently, what is the amount of power measured by the what meter? In answer to the first question, the amount of power used by the load is simply a calculation off the power formula, voltage times, the current. And since it's a resistive load, we have the current in phase with the voltage. So it's 240 volts times 10 amps, which should be drawing 2400 watts or 2.4 kilowatts from the source. Now let's look at what each element of the meter is reading. The current coils each share half an element of the potential coil. Therefore, only half the flux is reacting. With each coil the meter is measuring. In this case, element number one is 240 volts times the inner current coil amps, which is 10 amps times half the flux, which is 1200 watts plus element number two, which is 240 volts times the outer current coil 10 amps times half the flux, which is also equal to 1200 watts. Notice that the inner current coil has the power flowing backwards to the court through the current coil, and the actual current coming from the load is actually in reverse of the current flowing through the outer current coil from lying one. So, in actual fact, you kind of got to, uh, double negatives, which add to a positive. So the element number two is indeed going to read 240 times, 10 times 1/2 which is 1200 watts. And the meter total will be simply the arithmetic some of the two elements, which is 2400 watts. What's checks out against our original calculation of what the load should be dissipating for the same Ah, standard meter connection. Uh, this time we're going to connect a 12 ohm load across line one to neutral. We see that a 12 1 resistor connected line to neutral causes a current to flow of 10 amps. This is simply a matter of calculating owns law, which is eyes equals V over R, which is 120 divided by 12 which is Tim or 10 amps. Again, we're going to try and find with the amount of power used by the load, we're gonna find out the power measured by each element. And what is the amount of power measured by the what? This time we placed the 12 rooms across line want to neutral and the amps is 10 as we calculated. Of course, the power to that load is simply the voltage times the current, which is 120 times 10 or 1200 watts. I will see what beach element of the meter is reading. Element number one is 240 volts times the inner current coil of zero amps times half the flocks. Well, because the current is zero in the inner coil, then the meter is or that that element of the meter is reading zero. Plus, we're gonna add to it. Element number two, which is 240 volts times the outer current coil 10 amps times half the flux, which is 1200 watts. And if we add up element one and two, we still have 1200 watts, which agrees with our original calculation of power of the connected load in this example, problem. Again, we're gonna use the same standard 1.5 element meters metering a single phase load, as we have in the past two examples. But this time we're going to connect the loads like this. We placed a 10 hour a 10 load across lying one and lying to and we're gonna be placing 50 own load from lying, one to neutral and a 30 own load from lying to to nurture neutral. We're gonna calculate the amount of power used by the load, and then we're gonna find out what is the amount of power measured by each element. And what is the power amount of power used by of the total amount of power measured by the one meter moving on to the solution? Well, answer the first question, and that is what is the power consumed by the load. Regardless of how the kilowatt hour meter is connected to the loaded south, the current through load are one is simply owns law again. 240 volts over 10 homes, which will draw 240 amps. Power used by that load will be 240 volts. Times 24 amps, which is equal to 5760 watts for our load are too. Using homes law to calculate the current dry. It's 120 volts this time cause it's lying to neutral voltage divided by 50 owns, which is equal to 2.4 amps. Therefore the power used by or dissipated by that load is 120 volts times 2.4 amps, or 288 watts. Finally, the current through our three will be 120 volts over 30 homes, which will give us for AM's giving us a power draw 120 volts times for EPS, which is equal to 480 watts. Summing those three power draws together, we will get a total low dissipation of 6528 watts now, looking at the meter connected to the load, the meter will read for our one. It will read 240 volts times 24 amps in the outer current coil times half the flux, which is equal to 2880 watts, plus 240 volts times 24 amps. For the inner current coil times, half the flux again is equal to 2880 watts, making a total power measurement for that particular load of 5760 watts. Now this is measured by the meter. The meter for load are too will read 240 volts times 2.4 amps for the inner current coil times 1/2 because of its own using half the flux amounts to 288 watts for the outer current coil. Current is zero, so there's no contribution to the from the, uh, from the outer coil. Before our to, however, the meter will read for our 3 240 volts times for amps. For the going through the inner current coil times, half the flux well, give us 480 watts. There will be no current flowing through the outer current coil due to our three, so the contribution would be zero. So if we add up those power draws, we see that we have 5760 plus 288 plus 480 which totals to 6528 watts, which agrees with our load calculations at the beginning. So this ends Chapter four
6. Ch 05 Instrument Transformers: Chapter five instrument transformers. Instrument transformers are a special type of transformer used for the measurement of voltage and current, as the name suggests these transformers air used in conjunction with the relevant instruments such as AM eaters Bold Meters, what Peter's and Energy Meters as well as protective relays. Instrument transformers are of two types. There are current transformers and potential transformers, or sometimes called voltage transformers. Current transformers Air used when the magnitude of the A C current exceeds the safe value of current measuring instruments. Potential transformers. Air used where the voltage of the A C circuit exceeds 750 volts as it is not possible to provide adequate insulation on measuring instruments for voltages more than at this level. These are examples of current transformers that air found in the bushings of a power transformer circuit breaker. They're known as Donut Seti's. For obvious reasons, they look like a doughnut. The picture on the right is not a CT, but a typical transformer could be a breaker bushing, which would be mounted through the center of a doughnut. C t. The secondary winding is wrapped concentric lee around a tor oId, which is usually made up of laminated iron or steel. The primary is a single conductor, usually a bushing mounted through the center of the tour, right. The Dola fits over the primary conductor, which constitutes one primary turn. The secondary is wild around the Tauride, which is usually made up of laminated iron. That concentrate concentrates the magnetic flux and forces it through. The secondary turns some donut. Seti's come with a primary conductor incorporated in the C T structure and connections Air made by bolting the primary elites to it. If the tour oId is well with 240 secondary turns than the ratio of the C T is 240 to 1 or 1200 to find that the five amp designates the continuous rating of the secondary winding and is normally five amps, at least in North America, and one app, or 10.5 amps. In many of the other parts of the world, this type of donut, SETI, is most commonly used in circuit breakers and transformers. The CPI fits into the bushing turret, and the bushing fits through the center of the doughnut. Up to about four Seti's of this type can be installed around each bushing in an oil circuit breaker or a transformer bushing instrument. Transformers Air used for the following reasons. One. To insulate the high voltage circuit from the measuring circuit in order to reject the measuring instruments from damage. Also, to make it possible to measure high voltages with low range millimeters in high current with low range and meters. Also, instrument transformers are used in combinations of summing and measuring voltages and currents, and you'll see more examples of that in subsequent slides. Let's take a closer look at instrument transformer ratios in an ideal transformer with a simple load on the secondary end an A C voltage all a primary. The secondary voltage is determined by the turns ratio such that he s over. V P is equal to N s over np b s being a secondary voltage V p being the primary voltage and s the number of turns of the secondary of the transformer and NP the number of turns of the primary of the transformer. Looking at the current, the Magna Motive Force is given by I p N p or I S N s, which is the same for both sides of the transformer, which turns out to give a cystic equation, which is I P N P is equal toe I s and s where I, p and I s are the primary and secondary currents respectively. In N, p and s are the number of turns of the primary and secondary respectively. Or we can rewrite the equation, bringing the secondary current, the left hand side and everything else to the right and that gives us i s is equal toe i p times NPR over an s. And if n. P of the number of turns of the primary is one such a excessive is in a bushing type C t. Then I s is given by i p divided by the terms of the secondary on Lee, which is the number of turns on the tour oId of ah donut CT. A current transformer differs from a voltage transformer in that it's secondary current is determined entirely by the load on the primary system and not on its own secondary load. In other words, the secondary current is determined by the current in the primary conductor the voltage across a secondary load which is usually called the burden will rise and fall depending on the current in the primary. So if a, uh on impedance or a burden or resistance is placed on the secondary of a CT, the higher that resistance, the higher the voltage drop will be across it because it's trying to maintain the same current through it, which again worth repeating, is dependent on the primary. Both the primary and the secondary of a CT have relatively few turns of heavy wire and thus low impedance. Subsequently, a current is readily induced into the secondary, proportional to the load or the primary current. There is a unique problem encountered with transformers in general and current transformers . In this case, in order to magnetize the core of a transformer, a certain amount of excitation current is required. Part of the current induced into the secondary from the primary is used to accomplish this . Since the induced current represents the current flowing in the load circuit, the media ring or the meters current coils will be influenced influence proportionately by the load current minus the excite a be exciting current or excitation current. This represents a very small loss of load current and therefore affects the accuracy about the recorded power consumption. Current transformer losses, called Ares, vary for different types of transformers and the burden or load on the secondary measurements. Canada has established acceptable error limits that a transformer must fall within. Engineers must take this into consideration when designing relays and metering setups. This is very small indeed, but it can make a large difference in some cases and you must be aware of it. The transformer ratio of a C T is defined by the rated primary current over the rated secondary. Current for a given C T were the primary current is 100 dams, and the secondary current is five amps. The transformer ratio is 100 to 5 or 20 to 1 for the same C t. What would happen if the primary was looped back and then fed through the SETI one more time? The C T would see 100 times to which is equal to 200 amps. The secondary would be 10 APS. The transformer ratio here then would be 100 to 10 or 10 to 1. This is sometimes used if the primary current is too low for a meter reading, designating the polarity is very important and done by marking the SETI primary secondaries . This is especially important for power flow direction and when sea keys air used for directional reeling. There are standards, but they all pretty much being the same thing. The primary and secondary are marked to indicate what the which direction the current will flow During each house cycle occur, a primary terminal is marked to associate it with a secondary term, and we say that the terminal is spot with respect to the associated secondary terminal Mark spot, and you'll see how that works in a few minutes. In the case of a donut CT, where the primary is the conductor running through it, one side of the CPS marked the spot the I Tripoli Institute of Electrical and Electronics Engineers. Instead of spots, they use markings H one and X one and the I E. C, which is the International Electro Technical Commission. Use P one and S one. Suffice it to say that the one side are the primary side. One side terminal is marked with respect to the other end. Again, you'll see how that works. What this indicates is that when the primary current is positive during the Sinus Seidel half cycle. In other words, current into the spot. The secondary current will be out of the secondary spot just to repeat that again. Primary current is positive during the Sinus seidel half cycle, or current into a spot on a primary. Results in the secondary will have current out of the spot on the second there. Conversely, what? This also indicates that when the primary current is negative during a Sinus idol half cycle where the current is out of the spot on the primary, the secondary current will be out of the non spot of the secondary with potential transformers. In general, a secondary voltage is proportional to the primary voltage equivalent to the turns ratio, such that ves divided by V. P, is equal to N. S. Divided by N. P. This relationship is not completely exact for the following reasons. The exciting current that is necessary to magnetize the iron core causes impedance drop in the primary winding, and the load current that is drawn by the burden causes the impedance drop in both the primary and secondary findings. Both of these produce an overall voltage drop in the transformer and introduce airs in both ratios and phase angles. The net result is that the secondary voltage is slightly less than the ratio of the turns would indicate, and there is a slight shift in the phase relationship. These two errors air called ratio airs and phase anglers and may be represented by this equivalent circuit. These errors are usually very small and for the most part can be neglected. However, we must remain aware of them as they do not become or they do become significant from time to time and may not be neglected. As with CT's, there are polarity standards, but they pretty much the same thing. The primary and secondaries are marked to indicate which terminals are in phase. A primary terminal is marked to associate it with a secondary terminal. We say that the primary terminal is spot with respect to the associated secondary terminal March spot. In other words, in face, this is done on the PT itself and or on the nameplate again. This is especially important when PT's air used for relays with direction and revenue meters, as in the case of the Seiki polarity, the primary and secondary terminals are marked in such a way that when the primary current is positive during the positive half cycle that is current into a polarity mark. The secondary current will be out of the polarity or spot mark or in terms of voltage Sina's. We're dealing with voltage transformers here. A voltage rise on H one terminal gives a voltage rise on the X one terminal single phase meter in using instrument transformers. Let's put a to a woman Policies meter to work using instrument transformers. In this case, Seti's The load is being fed from a distribution transformer, just as in a residential supply. To meet her, this load we're using a poly phase two element meter and current transformers on a single phase three wire 1 22 40 won't supply to the load using Measurement Canada Standard 13 11. In order to understand how this meeting circuit works, let's go back to our earlier analysis of a meter residential load where we consider to independent power supplies where the what meters were arranged like this and the what measurement formulas were as shown here. The power to each of these loads can be measured by to what meters notice the spot or polarity markings on the what meters? No, if we reconnect the current coil of what we need to, which would reverse I to then the what meaner to would read minus two times minus I. To which, of course, is just be two times I to with this arrangement, the two what meters will read w one equal to the one times I want and w two equal to V two times I to. Now, let's suppose that the current to the load is too high for our meter, so we introduce CT's and the circuit would look like this. The red color of the I one and I two currents are Coletta seasonal that they are in phase with the primary current, but their magnitudes are subject to the C T ratios. You'll notice that W one and W two formulas are unchanged. So for simplicity sick, I'm going to leave the sea keys out of the next slide and keeping the red color to remind us that we have to apply a multiplier due to the C T ratio. In the final analysis, connecting the circuits together at the neutral course makes no difference or electrical difference to the load. This arrangement is the same as connecting the loads to the centre tapped residential distribution transformer, where V one is the lying to neutral Voltage and V two is the neutral tow line Voltage on the secondary side of the transformer. Notice that the load is not included in the equation and makes no difference to the formulas. Calculating the metering care. Let's trace out the currents and voltage, starting with the currents. The load current I one flows through the C T, the secondary, which flows through one element of the meter, the load current to flows through the C T and the secondary, which flows through the other element of the meter. Now, if we trace out, the voltage is the one is the lying to neutral voltage connected to the load. Envy to is the neutral to line voltage on the secondary side of the transformer. The total average power measured to the load is now the one in RMS times. I won in RMS times, the coastline of the angle between them, plus the two in RMS times I to in rms times a co sine of the angle between them I want and I, too must be multiplied by the Seiki ratio. Let's work a problem that has Seiki connections worked into the circuit. In this figure, we've placed a load of 10 homes across Line one and Line two, and to line to neutral loads 1 50 the other 30 homes. We're connecting the meter through Seti's with turns ratios of 52 5 or 10 to 1. We want to calculate what the amount of power used by the load and we want tea. Find out what is the amount of power measured by each element and, of course, the total, uh, load measured by the meter itself. And as you can see, we're using measurement Canada's Standard Connection 13 11 in this case, which does show the connections for two C T's connected to the meter with a direct connection of the potential coils using Olds Law, as we did before the current through the load are one is going to be 240 volts, divided by 10 OEMs, which is 24 amps. The power used is 240 volts times 24 amps, which is 5760 watts. Current through are too is 120 divided by 50 which is 2.4 amps and the power through that. That particular load will be 100 20 volts. Times 2.4 amps gives US 288 watts. Current through the are three is 120 volts, divided by 30 old gives us for amps and the powers 120 volts times the four amps, which is 480 watts. Giving us a total reading for the meter of 6500 28. What the total. What meter reading is gonna be mean up? The sum of the two elements. The one times I one plus B two times I to the meter will then read for the load are 120 volts. Times 2.4 amps for the top element. Times 10 for the C T ratio gives US 2880 watts plus 120 volts times 2.4 ABS for the bottom element. Times 10 for the C T ratio, which is 2880 watts for a total power measurement of 5760 watts for the load are one. The meter for our two will read 120 volts times 1200.24 amps for the top element. Times 10 for the C T ratio gives us 288 watts for our three. The meter will read 120 volts times point for amps for the bottom element. Times 10 because of the C T ratio, giving us 480 watts totaling all of the elements together. For all of the loads, we get 6528 watts, made up of the sum of 5760 plus 288 plus 480. Let's analyze another solution to me during a three wire single phase load that is being fed from a distribution transformer similar to what is used in residential supply. This time we'll use a single element meter and a three wire CT. A three wire SETI is a self contained current transformer with two single bar primaries through the central core, such that the polarities of each primary is opposite to the other. This is old school metering, but there are still some in existence, and it is still a standard noticed that one of the primaries goes through the Seiki in the reverse direction, which means the secondary current will be I one minus I to, of course, divided by the SETI ratio of the three wire CT. In order to understand how this metering circuit works, let's go back to our earlier analysis of um, eatery of a meter residential load where we considered to independent power supplies where the what meters were arranged like this, and what meter? What measurement forming lists were as showing here. The power to each of these loads can be measured by to what meters and again take note of the spot or polarity markings of a watt meter. As you can remember, we derived the formula for the total watt consumption, which is given by the equation. W T is equal to the first element. Be one times I won, minus the other element B two times I, to which can be written re written because of V one is equal to V two is equal to the tea or the total voltage divided by two, which can also be rewritten if we just move and rearrange figures mathematically, that the total watches given by the total voltage times the quantity I won over two minus I to all over too. The meter thus reads the T times the quantity I one divided by two minus I two divided by two. The fact that I want and I two r divided by two, is handled in the three wire C T ratio. Once again, we have the meter reading P average, or the average power is the RMS voltage times, the RMS Current Times, a co site of the angle between them, multiplied by the SETI ratio. Keep in mind that the divide by two and the formula is inherent with a three wire CT. The same situation could be managed with the use of two separate si tes. This is a single phase meter using instrument transformers to some the secondary currents. To measure the load on a single phase three wire 1 22 40 volt supply low. We can do Click eight. The configuration of a three wire CP using to air Seti's, the primary of C P one is connected. Tow line one. The secondary is connected so that the current flows positively through the element of the meter. The primary of SETI two is connected to line to Ah. Lead is taken from the unmarked terminal of Seiki too, and is connected to the mark Polarity Terminal of Seiki one. Another lead is taken from the Mark Polarity terminal of SETI two and is connected to the unmarked terminal of C Key one. The result is the current from C T two flows backwards through the current element of the meter, giving us the negative for I to which satisfies the equation. This type of wiring is often return referred to as cross connecting. The result is the current element of the meter reads, plus the current in line one minus the current in line to which gives us the equation I line one minus. I line to the potential element is connected across line one and lying to which which measures essentially the tea or the total voltage. Satisfying the equation, The T times, the quantity i one minus I to But the calculated equation calls for a divide by two for the total voltage. If we look at the standard, the standard says that we must apply if we're using this type of a connection. A meter multiplier of 1/2 which then takes into account are calculated formula total Watts is equal to V T over two times a quality I one minus I to the three wire CT Solution uses the standard connection of 13 03 which provides the multiplier because it uses I one divided by two and I two divided by two built into its turns ratio. Whereas the two C T solution use used by standard 13 09 doesn't have a built in solution and therefore must have a multiplier of 1/2 of course, plus N e c T ratio. Here is a practical example of a single element meter connected through a three wire C T. To measure the power to a load that draws 100 amps when it is connected, lying toe line and we are using Measurement Canada Standard Connection 13 03 Our task will be to find out what is the amount of power used by the load and what is the amount of power measured by the watt meter. The low draws 100 amps. Therefore, using the formula for power where you've just multiply the voltage times current the power road equal 240 volts times 100 amps, which would equal 24,000 watts or 24 kilowatts. At first glance, you might deduce that the secondary current out of out to the meter is five plus five amps equally in 10 amps. However, this is not the case. The three wire CT winding Zarin parallel each secondary winding contributes 2.5 amps under 100 AMP load. In our illustration, 100 amps of primary current will produce 2.5 amps, plus 2.5 amps, equally equaling the desired five amps. Secondary current out to the meter. Looking closely at the name plate C T ratio, it states that 100 to 105 which means 100 amps in both primaries, will produce five amps in the secondary. Therefore, the meter seas, 240 volts for the potential coil times 2.5 amps plus 2.5 amps in the current coil Times 20 which is the Seiki ratio, which gives 24,000 watts or 24 kilowatts. This ends Chapter five
7. Ch 06 Three Phase Metering: Chapter six three Phase metering. When considering three phase power generation, you can assume that is made up of three single phase generators connected together on one terminal. The generated voltage factors, or phasers, are 120 degrees apart. Rotating counter clockwise, the load can be connected in various configurations. In this case, we show a why Connected load. Remember, from a previous chapter that energy meters, regardless of their makeup, mechanical or electronic, are considered to be made up of meter elements. A potential coil or a voltage transducer and a native be converted makeup. 1/2 of the element of the meter and a current coil or a current transducer, plus in a two D converter make up the other half element of the meter. Together, they make up a single element of a meter that could be used to measure power and energy consumption by a single phase. Load. Multi element meters, regardless of whether they're mechanical or electronic, can be made up of two or three elements. Each element reads the average power, which is Thea RMS voltage times, the RMS current times, the co sine of the angle between them, which are then summed to give the arithmetic total reading of all of the elements. Let's look at a three phase why Connected load. Now that we have a simple model for a three phase generator, we can replace it with a three terminal generator. The generated voltage phasers are 100 and 20 degrees apart, and the UAE connected load is made up of three. Impedance is sent, one said to and said three. In order to meet her the load, we simply need to meter the individual power to each of the load. Impedance is then arithmetically some the three meters that is. We use three watt meters or a three element watt meter, each element measuring the face to neutral voltage across the load and the current to each of the load, as in the diagram here and then we some the meters or the elements. As I said, we could use a three element meter to accomplish the same thing. Each element measures the phase two neutral voltage across the load and the current into the load. The digital processing unit then sums the total consumption before proceeding into the next set off metering type installations, especially for three phase me a ring. We have to understand that utilities have installed or will be installing hundreds of thousands of of these meters in their utility system so that as a cost saving measure, they try to keep the numbers of want meters down and certainly the cost of the what meters down. And if you reduce the number of elements in a watt meter, certainly the cost will go down because the cost of Alaba's you introduce more elements in the meter and if you have fewer elements than you have fewer CT's and PT's that have to be connected and the associated wiring. So all in all, there is a general trend. ER has been a general trend to reduce the amount of elements involved in Amita ring system . So what we're going to see in the next few installations is some approximations that are made that are acceptable from a billing standard such that the cost will be kept down and we will be trying to reduce the number of what meters or certainly the number of elements in a watt meter. In media ring, three phase systems in the utilities three face system often referred to as an infinite bus because utilities tend to be very large and they're interconnected with each other so that the infinite bus refers to the fact that the voltages, regardless of the loads, remain relatively unchanged. In other words, they are going to be equal and 120 degrees apart just about at all times. If we look at the voltages here, if we look at voltage V one, the vector would look something or the phaser would look something like this. Then Voltage V two or V two to neutral would be 120 degrees, legging that voltage and look somewhat like this. And Voltage three would look something like this when the voltages are added together because they're equal in 120 degrees apart, the vector some or phase or some of the voltages will add to zero. And for the next stage in metering a three phase system, we'll use this identity where the voltages other three face system will add to zero. In order to make reductions in the number of elements used in the meter. Let's go back to the three phase, a metered load that we had up a few slides back where we had a why connected load to the system with three watt meters measuring the power consumption of that Why connected load? And we could say that the total of power to the load is made up of the some of what Peter 12 and three. So W. T is equal to W one plus w two plus w three and we know that the individual what meters are made up of the product of the current and voltage in each of the phases. In other words, I one V one plus I to be two plus by three V three is equal to W. T. We know that each element is reading the power average equation, which is the root, mean square of the voltage Times Square of current times, a co sign of the angle between. We also know that because we're dealing with an infinite bus, that V one plus B two plus B three is equal to zero, or we can say that V two is equal to the quantity minus view on minus V three. Then if we substitute V two into our formula, we will get that I want plus B one plus I to times the quantity minus B one minus b three plus I three the three. All we've done is a mathematical substitution here, which is allowed Now we can collect like terms and in this equation, and we see that the four Leah tells us that we only need to what meters, two meter this system and that is we collect like terms in terms of the one and the three. We see that the total power consumption by that load can be metered with two meters one using the voltage of the one and the current I one minus I two and the voltage of the three and the current of I three minus I to and the total would be the some of those two. What meters or two elements that would be used to meet her this white connected load. - As I said, this formula tells us that we can use a two element want leader one element using voltage one and the some of the currents I one minus side to any other element using voltage V two and the some of the current side three mine decide to let's see how this formula will apply to the industry. Standard 34 07 The meteor elements look like this where Element one is the bottom element of what's in our diagram and element to is the top element of the diagram, as we see here, using a three C T and direct connected potentials instead of numbering. This standard uses letters, but the formula and relationships are the same. The A N is the potential, as shown here in red, going through Element number one. So it's the half element used for potential of Element number one. The current from Line A is shown here Are I subscript a and it goes through Element number one in a positive manner. The C N is showing here, and it is the potential connection for Element Number two. I see takes the current from the current transformer associated with the sea phase, and it puts the current through element number two in a positive manner. Now I b or the B phase current goes through both elements, and it leaves the SETI using. I'm associating it with a dark green and you can see it goes up and threw Element number two backwards, which means it's being subtracted or minus B through Element number one. It comes back down but doesn't return directly to the C T, but goes up and backwards through Element Number two, which gives it a minus bi for Element Number two, which is shown in the light green here. After it goes through Element number two backwards. It returns back to the C T and continues the circuit flow. If the current and voltage phasers are plotted, the result in currents through each element are 60 degrees apart. The result in voltages are 100 and 20 degrees apart. This 60 degree, 120 20 degree configuration is a standard for correct metering arrangement, and you'll see this in several of our meeting or and mentoring arrangements. This set up using this standard, set up 34 07 is what we would call non Blondel compliance. And we'll explain that once we get into, uh, the blonde L section of this chapter, which is later on the standard 34 08 is the same type of metering only this time we're using on Espace Meter, which is a socket or plug in tight meeting the elements. Our trade look like they're turned sideways, but this would be element number one, and this would be element number to. However, all the connections and Seti's feeding the meter are the same as before for the A base meter. Now, this type of smearing both with the socket base in the A base meter, using three SETI's and then manipulating the current so that you can subtract one from the other is easy to do if you have current transformers and they allow you to do this if you did not have current transformers. In other words, if you had a direct connected what meter, you could not connect the currents together. You couldn't make up I one minus I, too, because you would have to connect the primary phases together, which would short circuit them in, and they just wouldn't work. So in using this formula, how would you be able to use a two element meter or a lesson of three element meter in order to meet her? This type of situation? Well, it's done using a 2.5 element here, and you'll see that in the next slide. If we were to use a 2.5 element meter. It would allow us to some the currents if they are directly connected. That is, there is no Seti's. Now you remember what a 2.5 element meter looks like. It has to current inputs, and they both share the potential coil. So you're allowed to take. I won through one of the quoyles, and you can pump I too backwards through the other element and then have it go through backwards through the other Wat Meer and the other element as well, allowing us to use the same formula. W T is equal to W one plus W three, which is V one I one minus sign, too. Plus the three times a quantity I three minus I to So what Meter w one would look like this , and it still gives us the average power. What mean or three still gives us the average power using the currents here in the formula . So again, we have satisfied the equation for calculating the total average power for this set up. However, this time we are using a 2.5 element meter and again, and I'll state this and explain it later. This is a non blonde L compliant set up. It is accurate enough for our purposes, but it is not a blonde l compliant set up. Let's look at our three phase before wire load again and less apply just a pure a purely resistive load. This is just to make things look a little bit simpler. The I one current phaser would look like this. The I to fazer would lag. I won by 100 120 degrees and look like this. I three current vector would like I two by 120 degrees in look like this. The I one minus I to current phaser would look like this and the I three minus I to current phaser would look like this. In actual fact, the low does not have to be a why configuration, since the calculations on Lee deal with current and voltage is to the load this meat Oring will work for any three phase for wire load again if the current and bold each phasers are plotted together, The resultant currents through each element are at 60 degrees or 60 degrees apart and the resulting voltages to those to those elements are 100 and 20 degrees apart. So this gives us the characteristic of accurate metering uh, which is usually plotted with the voltages and currents at 120 60 degrees apart. If there is non resistive elements, in other words, reactive elements it was inducted its or capacitance in the circuit. There will be a shift of the currents, either in a forward or reverse direction. Compared to the voltage is, however, the currents will still be 60 degrees and the voltages will still be 120 degrees. Remember, this reduction in the number of watt meters two instead of three was based on the fact of the voltages were balanced. This is purely a cost saving measurement and is also a non blonde L compliant. Set up nearing any three phase for wear load with three watt heaters or a three element what meter is the best and more most accurate where the total power consumption WT is given by the sum of the individual what meters or elements W one plus W two plus w three and element number one is bed with the ADA neutral voltage times a phase current element. Number two is fed with the beat a neutral voltage times. The B phase current an element number three is spend with a seed, a neutral voltage times the sea phase current using direct connected potentials to the meter and CT's and ah, three element meter measurement. Canada Standard 34 1 need is a cost effective way to meet her. A three phase for wire load using solid state meters today. Solid state Meteors The price of spot solid state meters has come down drastically with the development of them over time, so it makes sense to use a three element meter now instead of approximating the total with a two element meter or 2.5 element meter. And with this set up devoted, just no longer need to be balanced. Metering a three phase three wire delta load. The delta load is made up of three individual loads connected face to face as shown in this slide. This time we're labeling the faces are W beef or red, white and blue. To meet her the load we would use three watt meters or a three element watt meter each the what meters or elements are connected such that each watt meter or element measures the current through the load and the voltage drop across each load red, white, white to blue and blue to red voltage. The total power consumed is the arithmetic. Some of the three what meters. So, as I said, the total load will be the summation of the individual watt meters, W one, w two, w three or the individual elements of a three element What meter substituting the voltage drop across each load and the current through the load for the W one w two w three. We get this equation. Remember that in a three phase load the voltages will add of old, each phasers will add to zero as indicated here. If we would like to get the blue to red voltage in terms of the other two voltages, then we can write that the blue to red voltage is equal to the quantity minus voltage red, white, minus the voltage white to blue. We can now go back to the equation above and substitute the term that we just found for the blue to red voltages. And we get this equation here we can expand. What's inside the brackets mathematically, and we end up with this equation and we can further rearrange the terms not doing anything magical other than Mac mathematical manipulation. And we would end up collecting like terms we would end up with. The total power can be monitored by actually to what meters or I two element meter with. We're measuring the red white voltage times the current through said one minus a current through said three plus the voltage white to blue and times the current through said to minus the current through Zet three. So we haven't done much more than mathematical manipulation at this point. But we will recall that the current the red face current is equal to eyes ed one minus eyes at three, which is Kershaw's current law, Really, So we can substitute that in our equation. Similarly, biker just current law minus I B is equal to eyes said to minus eyes, said three. So we can put I minus I'd be in our equation. So now our total equation are our equation for total power. Consumption is given by the red delight fold each times the red face current, plus the white to blue voltage times minus the blue face current. Now we can rewrite that equation and get rid of the negative sign, and you'll notice that we just change the polarity of the white to blue voltage to be blue to white voltage. And we end up with an equation that is don't total power. Consumed by this load is the red white voltage times a red face current, plus the blue to white voltage times of blue face current. This formulas tells us that we only need two meters or 12 element meter to measure the power consumption in a three phase three, where alone one meter element uses the Randall white face to phase voltage with the red fees current, The other meter or element uses the blue, the white face to face voltage with the blue face current. Since the measurement of power to the load on, Lee uses face to face voltages and phase currents. The load configuration doesn't matter as long as there is only three wears to the load that is no neutral. Looking at the voltage and current phasers, you see the we're using the red, white and white to blue voltages with the red and white face currents. The resulting currents, each out of each element, are 120 degrees apart. The result in voltages of each element are 60 degrees apart. This is the 60 degree 122 degree configuration that is standard for correct metering arrangements. Let's look at a Measurements Canada Standard 33 01 Which is me during a three phase three wire low with load with a self contained meter or directly connected the first element ces the voltage A to B plus c A things current. The second element will see the voltage see to be using the sea phase current. And of course, the standard uses ABC for their three phases, where, as we develop the formula red, white and blue, R, R, W and B, which matches up to what you see here and metering any three phase three Reier wire load using a south contained or an Espace meter, such as measurement Canada here uses 33 0 to the first element would beyond the left. The second element would be on the right and measuring the same quantities as an A base meter, which we just looked at in the previous slide and for ah, higher voltage higher current circuit using Pts and see teas such as measurement Canada Standard 33 11 shows. Here, the same equation holds true where the first element takes in the voltage A to be with the red face current and the other element takes voltage C to be with the sea face current. And of course, you have to take into consideration the turns ratio of the CT's and Pts when you're coming out with your final answer. So far we've been sampling excerpts from, uh, the measurement Canada's standard connection diagrams and connections for I'm metering systems. Anything from a single phase 23 phase system, um, to wire three wire, four wire systems. They're all listed here on there, Government of Canada website. They could be found at this location and just about every metering system in Ontario and certainly Canada, uh, adheres to this standard so that we can but continually refer to it and make sure that our connections are correct and our state. The website has two sections and Appendix A, which is 87 pages of electricity metering installations with color coded connections and you can leave through them. We've gone through several of them here in this chapter, Appendix B is measurement Canada's standard color codes for electrical media ring installations. Now these color codings don't necessarily. You're right. They for sure don't affect the actual metering themselves. But what they do is they make it easier for those that are maintaining the system, to understand how the wires are connected without having to go and separate out the bundles and after, reconnect or re, uh, bundle them together after maintenance is done. The actual wired connections to a meter are also standardized, and most of the systems use what they call test blocks for the purposes of connecting and disconnecting the current and potentials to and from, uh, the meters thes test blocks are designed such that you can inject current or measure current and inject voltages or disconnect voltages by opening and closing Suspicious. These test switches are also designed to take uh, test plugs, which allow you to inject current or measure current in, uh in a media ring circuit, and you can even do that with without disconnecting the meter or or the power source to the from to and from the meter, although it is recommended to remove the power and then do your connections and then reconnect the power. However, thes plugs are made to bring out the power connections to year test equipment, and you can see here a side view of the the test block where the switch is open. But the connection to the meter is still maintained through spring loaded connections. And as the test plug is inserted, uh, it makes connection with the bottom in the top of the plug. Our bottom in the top of the switch and the current will then flow through our test plug connection. Once the flexi test switches are mounted, they're aware due to a standard which is also found on the measurement Canada Standard Connections. This figure shows a typical installation of a two element meter with the Hydro One Delta Connection Notice. The indicated current flow through the C, T and meter coil and the labeled currents at the test block. Also note that the neutral potential is shown having a solid or dummy fuse. This is the required connection. Tracing the potential current flow. You'll see that it flows from Line E to neutral and from lying see to neutral in the left and right elements. If the neutral potential link was a normal fuse and it opened, the potential coils would be placed in Siris and potential E A, C and E C A would be applied to the potential coils. This would definitely cause airs in the measurement. This figure shows a typical installation of a two element meter with a Schlumberger Delta connection, and this figure shows a typical installation of the two element meter with the Measurement Canada Delta Connection. Now these connections basically read the same to the meter, or the meter reads the same current. It's just that the wearing on the test switches themselves are are different. Blondel zero states that if energy is supply to any system of conductors through end wires , that's n the number of wares. The total power in this system is given by the algebraic sum of the readings of end what meters so arranged that each of the end wires contains one current coil, the corresponding voltage coil being connected between that wire and some common point. If this common point is on one of the end wires, the measurement may be made by the use of n minus one. What meters? Looking at our example of a three phase for wear system, the number of wares in that system is for and the common point is the neutral, so we can measure total power to that load with n minus one or three watt meters. Now this is it sounds like a pretty complex and long zero. However, it can simply be stated that in a system of end wires, the number of lot meters required to measure the total power is in minus one or the total power is measured with one less watt meaner than the number of layers. Certainly a lot better. They're a lot easier to understand than a long paragraph that was just read. A single phase two wire requires one watt meter. A single phase three wire system requires two watt meters or two elements. A three phase three wire system requires two watt meters or two elements, and a three phase four wire system requires three watt years or a watt meter with three elements. If a meter is wired correctly and the right number of, um, earing components or elements, our present for me hearing a system of in wires than meters that meet the requirement of end minus one. Elements for n wire service are said to be Blondel compliant, and it's usually accepted that there are Neuer's in this type of a media ring system, uh, that are incurred by the connections. There may be errors. Do do the PT's there, the CT's themselves. But as far as the connections air required, it should be reading 100% accurate. We did an example of 100% blonde Blundell Complaint Meter. And now, in our example of a three element meter using measurement Canada Standard 34 18 as seen here, you can see that there are four wires and we're using three elements, which is it meets the requirement of n minus one elements for in wires. When the media ring system is noncompliant at meets, the that means the meters do not meet the requirement of n minus one elements for an end wire service. Now, this doesn't mean that the media ring is in Air B, except because some of the assumptions that were made, um, are within the accuracy rating that we're looking for It just means that if the system was pushed, it is there is a possibility that we would land outside the maturing accuracy that would be required. However, there's plenty of meters out there that are non Blondel compliant and they were basically put in a system because of economic reasons, as we stated before where we could reduce the number of elements and reduce the number of, um Pts and see tease that are connected to the system in order to save money over the long run meters that do not have blonde l compliant to provide stability and accuracy of measurement. The reasons are you'd want have fewer, fewer elements, as I said, and lower costs. Especially true for electromechanical meters. There's an avoidance of interference between two elements driving a single disk in an induction meter. There would be fewer CT's and Pts and in very high voltage systems, CT's and PT's can introduce a significant significant cost into the media ring system. Of course, there's less wire wiring Anders, fewer things that could go wrong if there's fewer wears as well as your some savings in not buying the wire. Some of the assumptions that are made that are used, and we've looked at those already. Some of the assumptions are that the voltages are balanced and equal and 120 degrees of Here's A Here are some examples off meters that are not Blondel compliant, and they can either be in this case of going from left, left, right. There's three wires in this system, but they're only using one element to meet a system. And in the middle one, there's four wires in their own, using two elements to me, to the system and, in the case of Michigan Canada Standard 34 07 they are. So making that assumption doesn't introduce a great deal of inaccuracy into the media system as long as you can. Some the currents, which they can do here case of using SETI's so they can get away with using a two element meter, and they do it quite often, and this ends Chapter six
8. Ch 07 Cross Wattmeter Verification: Chapter seven Cross. What Meaner verification? Cross what meter verification is a tool used to check or verify graphically current phaser relationships to voltage phasers. This is done with the aid of a watt meter, and you can also use Anam Eater and Bold Meter in the mix. The basis for this verification is the use of the formula. Power is equal to voltage times the current times a co sine of the angle between them, which graphically tells us that given a known voltage phaser, say voltage phaser de rotating counter clockwise in a circuit the current phaser I in the same circuit with a displacement angle off data when measured by a watt meter. The result is by virtue of the co sign rule the projection of that current onto the voltage fazer. I'm going to repeat that the basis of the verification is the use of the power formula, which is equal to B times I Times Co sign of the data where data is the angle between the voltage and the current. What this tells us is if we have known voltage fazer de and it's rotating counter clockwise in a circuit and there is a current generated by that voltage in the same circuit with a displacement angle of fada when measured by a watt meter. The result is by virtue of the co sign rule the projection of the current onto the voltage fazer. If that unknown current is part of a three phase circuit and devoted edges are balanced and counter clockwise in rotation. A power reading taken with a watt meter using that unknown current and the voltage V red to neutral is by virtue of the rule of co signs the projection of that current on the voltage fazer the are and and a second power reading, then taken this time using the white to neutral voltage is the projection of that same current on the voltage factor or voltage, Taser V. White neutral and 1/3 power reading taken using. This time, the voltage blue to neutral is the projection of that current on the blue to neutral, and those three projection lines intersect at one and only one point at the tip of the current facer. So if we wanted to plot an unknown current relative to a known set of voltages, we would connect that current to our watt meter and using the red to neutral voltage while taking that. What meter would record the what reading, scaling it and marking it on the red to neutral fazer. We know that that point is on a projection of that current onto the voltage factor such that the tip or the head of the phaser arrow would be somewhere on a 90 degree projection line. Next, using the white to neutral voltage, we would record a what reading using the same scaling as before marketing, marking it on the white to neutral voltage fazer. And we know that that point is on a projection of the current onto that voltage vector. Such a the tip where the head of the phaser array would be somewhere on a 90 degree projection line. Those two projection lines intersect at only one point, and that point would be at the tip of the unknown current or the head of the Feezer era. Repeating the process. Using the blue to neutral voltage would simply verify our previous readings and plot notice that the scaled want meter plot is on the blue to neutral negative. In this case, the watt meter reading will be negative, indicating that the skilled plot is negative and the projection line would be on the negative blue to neutral voltage fazer. This is the process of cross watt meter verification, and it is worth repeating, So let's do it again. So if we wanted to plot an unknown current relative to a known set of voltages, we would connect that current to our watt meter and using the red to neutral voltage while taking that. What meter would record the what reading, scaling it and marking it on the red to neutral fazer. We know that that point is on a projection of that current onto the voltage vector such that the tip or the head of the phaser arrow would be somewhere on a 90 degree projection line. Next, using the white to neutral voltage, we would record a what reading using the same scaling as before marketing, marking it on the white to neutral voltage fazer. And we know that that point is on a projection of the current onto that voltage vector. Such a. The tip where the head of the phaser are would be somewhere on a 90 degree projection, lying those two projection lines intersect at only one point, and that point would be at the tip of the unknown current or the head of the Feezer era. Repeating the process. Using the blue to neutral voltage would simply verify our previous readings and plot. Notice that the scaled want meter plot is on the blue to neutral negative. In this case, the watt meter reading will be negative, indicating that the skilled plot is negative and the projection line would be on the negative blue to neutral voltage fazer. Let's walk through the process with a mere ing re lane detection circuits such as the one showing here are known set of voltages would be accessible on a set of fuses similar to the one shown here and are unknown. Set of currents would be accessible on these. F t switch is similar to the ones showing here. The villages would be connected one of the time to her what meter for each current and there are three of them here. A test plug would be inserted in the appropriate switch and the switch with MBIA opened to allow the current to flow through the watt meter schematically. This is what are set up would look like the voltages and currents would be connected to the CT's and VTs at the bottom of the fuses. And the 50 F T switches and meters and or relays would be connected to the tops of the fuses and FT switches. It is required that we verify for correctness the currents flowing into the mere ing in reeling protection circuits via the F T switches. Or this what we call these the C T links I one i two and I three. A voltage reference source is first established in this case, a No. 13 phase star connected voltage transformer secondary source is available at the VT Links here, showing red, white, blue and neutral. The Star connection provides for both phase to neutral and phase two phase reference voltage is the reference. Voltages is positioned facing the panel red, white and blue and neutral from left to right. As Shola, the test circuit is constructed using a portable volt meter and meter. And what meter, as shown here, cursed. The voltage leads are connected to our known potential source. The red or marked lead is clipped to the top of the red fuse and the black or unmarked lead to the neutral or the dummy fuse. Then the current leaves are connected to our current links. The red or mark lead is connected to the bottom of I won currently by the bottom of a test plug and the black or unmarked lead to the top of the I want current link via the top of the test plug. We then open the current link or F T switch, which would it allow the current and to flow through our watt meter. We should now have an indication of spawned all three test meters. An upscale reading on the watt meter with the connection as described, would indicate a positive reading. If the watt meter deflection is downscale, this would indicate a negative reading and the voltage leads may be reversed to obtain a nup scale reading this what meter reading would then be recorded as a negative value. Some Elektronik watt meters air equipped with a reading switch that you can switch back and forth to allow a negative reading upscale record, Then all all these meter readings for these connections. Now move the rent or mark lead clip to the top of the red face to use and connected to the top of the white face. Use record all meter readings. Now move the red or marked lead clip on the top of the white face use and connected to the top of the blue phase fuse and record all the readings. We're now finished with current I one so close the eye. One switch care must be taken to ensure that there is a current path at all times. Now move the red or marked lead back to the top of the red face pews. Remove the test plug and move it to I to such that the red or Mark leaders at the bottom of I to link and the black or unmarked lead is on the top. Now open that current link. We should have an indication on all three test meters Record all meter readings. Now Move the red or mark lead clipped on the top of the red face use and connected to the white face. Use and record on all the readings. Now move that over to the blue phase fuse and record all readings. We're now finished with current I to so close I to switch care. As I said before, care must be taken to maintain a path for the current at all times. Repeat the procedure for I three, recording all meter readings, opening the switch and recording the readings for the red. The neutral voltage, the white to neutral voltage and for the blue to neutral voltage. Closing a switch, then removing all leads. So when taking cross, what meter verification readings? It's handy toe. Have a chart like this leader ahead of time, and you can just feel in the blanks. This is the sequence of meter readings. Four current I one. We would read a one and for the three voltages, B one, B two and the three, which would yield what readings of w one, w two and W three. Then, for the current I two, we would get Anam freeing, say A to yielding what readings off W four, W five and W six. And lastly, I three current of a three would yield watt meter reading, say, W seven, W eight and W nine completing our chart and then we can proceed with the plotting process. Let's look at some real numbers in doing this plotting process the voltage readings A so you can see are recorded. Their they aren't necessary, absolutely equal. In fact, it's very likely they won't be equal. But they must be close if you find them way out of whack than something, might you rob? But closeness is is good enough for our purposes. Here we can start plotting the verification results using these cross reading templates, which are three sets of intersecting lines, positive and negative, 120 degrees apart. Note. The's template lines air used to plot scaled watt readings along the direction of the non voltages. They are not used to plot. The magnitudes of the voltages are known. Voltages are 120 degrees apart, so they could be marked on the template lines as such to read to neutral voltage and the minus red to neutral voltage, the white to neutral voltage and a minus y to neutral voltage. The blue, the neutral voltage and the minus blue. The neutral voltage, which we also know is rotating in a counterclockwise direction. When we measure the current I one, which is 4.75 amps, then take it's what meter reading with respect to the red to neutral voltage which is 315. Scale it and then plot it on the durante neutral of voltage factor line the projection of the current I won on the voltage fazer Voltage red to neutral is showing here Since we know it is the projection from the current phaser, I won I one must lie somewhere along this projected line and we now take the watt meter reading with respect to the white to neutral voltage which is minus 91 scale it then plotted on the Y to neutral voltage line The projection of the current I one on the voltage fazer B w n or white to neutral voltage is showing here. Since we know it is the projection from the current phaser than I, one must lie somewhere along this projected line, these two projection lines intersect at only one point. Therefore, this is where the current phaser I one must be taking the what meter reading with respect to the blue to neutral voltage which is minus 224. The projection of the current I one on the voltage phaser V blue to neutral verifies this as its projection line must also intersect. Where the current fazer. Why one IHS This plot is not necessary to find the current i one, but it serves as a verification. This process is repeated for I two and I three currents using the second and third row What meter readings respectively. Upon completion of the phaser diagram plot, it is observed that our current phasers lagged their corresponding phase. Voltage is by an angle of stato one data to and they the three these angles, our should be very close to each other. In reality, they should be identical. But that's not always the case, so that if we want to find the power factor angle, we average thes three angles by summing them and dividing by three. Hence, the corresponding ah power factor is calculated by taking the co sign of this average value of the three angles. So far, we have been using red to neutral white to neutral and blew the neutral voltages because a tend to be the easiest to deal with. However, taking cross what meter readings using any known voltage will give you the same currents and angles that you would get as long as they are no one voltages, for example. As you can see in this process, we're gonna use red, white, white to blue and blue to red voltages. You can see where the red toe white voltages, phaser lines up here, the white to blue voltage phaser lines up like this and the blue to red voltage lines up like this. I've left the Retin neutral white to neutral on blue. The neutral phasers locations on the diagram. And you can see from these readings which were done with the face to face voltages that they do correspond to the same currents. The same current phasers are plotted. Using these vectors, we're gonna go through a couple of what I call classroom exercises now and Sina's this is, uh, a video which you have control over. I'm gonna stake the problem and I'm getting I will get you to then to pause the video so you can work on the solution. You can return to the video started up and the next slide under the next series of slides will have the solution and you'll see how you did. The first problem is a cross. What meaner exercise that involves the fact that we took these readings using Nolan voltages, which were the face and intra voltages took the current readings and the watt meter readings, which you see here in this chart. The first step is to plot the current phasers, I one i two and I three. Then find the power factor angle and then calculate the power factor. So now you can pause this video and work on the problem. When you feel you've worked hard enough on the problem where you come to a conclusion, start the video opened. The next series of slides will show the solution. So the solution here happens to be fairly easy, and I'm going to take a couple of shortcuts, which I'll explain. You can follow the steps that we did before, which essentially, we're going to do now. But we're going to do it in a little bit different fashion. The first thing you might notice is that are ready. Neutral voltage phaser is horizontal rather than vertical. That does not matter as long as we position the blue to neutral and the white to neutral voltage phasers such that they are subsequently 120 to 40 degrees displaced from the red to neutral voltage which we've done in this diagram. We've marked the positive red to neutral voltage going from the origin to the right, the negative red to neutral voltage going from the origin to the left and the white to neutral voltage. 120 degrees displaced from the red to neutral voltage and then the minus white to Notre voltage in the opposite direction. And then the blue to neutral voltage. 240 degrees displaced from the red to neutral voltage and it's minus sign going in the opposite direction. They've all been marked here, so you can see what you know, how they operate now, one of the things that you will notice if you look at the table that basically we we are dealing assed faras, the watt meter Arendse air concerned. And if you're talking about the absolute watt meter readings, that is without a sign attached to him. That means we're going to ignore the fact that they're either positive or negative. For the moment we will see. Or you can see that there are only three values 202 140 40. These values are going to be distances from the origin. Depending on whether you're dealing with the plot along any of the voltage lines, they still are gonna be that scaled distance from the origin. So if you look at her diagram I've drawn or a cup or circled. I performed three circles, one at 40 watts, one scaled at 200 watts and one skilled at 240 watts. They crossed all of the voltage of phasers at some point, and now all we have to do were plotting our, um, our projections. We only have to deal with where they are crossing the particular phaser that we're dealing with that the particular time. So if we look at the red to neutral voltage and we look at the projection line of the first reading for I want, it's 200 so you can see where 200 crosses the red to neutral voltage. We just drop perpendicular there, and all of the projections and currents associated with the red face current or I one is colored in red, so you can see it fairly easy. So the projection line going from the read to neutral vaulted you can see is quite obvious . The next thing we can look at is the white to neutral the voltage Ah, plot for the for the Watts. And it is plus 40 along the white to neutral voltage factor. And that's where our circle crosses the voltage of phaser white to neutral when we drop her perpendicular there, which is in red. And it perfectly intersects the line for, um, the one we have just previously drawn for the voltage red to neutral. And that indeed is where are red face current lies. You can verify that by picking up the minus 240. What reading that we get when we use the beauty neutral voltage and draw perpendicular there. And that verifies our answer so you can see the three red lines intersect where the current is, and those are our three voltage readings across the top of our chart. Now, if we look at the white to neutral or hi to, you will see that the Green Line, the Green Line projections are crossing at the particular point where the green face current ISS, and it's just a repeat of the process that we did for the red phase current, and then if we deal with I three or the blue Face current, you will see that we have plus 40 minus 2 40 plus 200. And we again can use our intersecting circles with the voltage phasers to mark our projection lines. Projector are perpendicular lines and they will intersect where the blue face current is, and hence that is the plot of our of our current phasers. No, we know that the red face current is and you can see from the diagram it's and we've read it 3.55 amps and it's at ah, lag of 39 degrees from the red face Voltage the white phase current history 390.54 amps and it is at 159 degrees, lagging from the face to neutral or sorry fate. The red to neutral voltage, which is 39 degrees displaced from the white intra voltage, and the blue face current at 3.55 amps, is at 279 degrees from our red face voltage. But it is only 39 degrees from the blue to neutral voltage, so the the angles that they are that each of the currents are displaced from their associate ID are Neutra face to neutral. Voltage is 39 plus 39 plus 39 divided by three is 39 degrees. So our power factor angle is 39 degrees, and the power factor is just the co sign of that angle, which works out to 0.777 The next problem involves trying to solve a suspected problem. Diffuses and current links showing here are used to feed a three phase three element watt meter. The meter is suspected of being wrong. In other words, it's reading the wrong values. The voltages have been verified as being correct. That is the red, white and blue to Neutra. Voltages are all equal in 120 degrees apart, and you can see the voltage readings on our chart. When planted. They definitely point out the air. So our task is to plot the voltages with respect to the currents. In other words, do the, uh, cross want meter verification and see just where the air occurs. Now you can stop the video, work on the problem, come up with the answer and then restart the video and see if you're correct. Okay, So if we plotted out, uh, the currents using our cross want meter verification process? The first thing you do is you plot out your known voltages, and this time I'm taking the rent a neutral voltages being vertically pointing up. So it's negative. Value will be pointing down, and the white to neutral voltage will be 120 degrees displaced from the red to neutral, which means it's kind of going in a so thes direction on our on our chart here on the blue , the neutral voltage eyes, 120 degrees. Liking that and it's kind of heading in that southwest direction. Um, so we have our plotted voltages there. As we plot our currents, you can see that I one which is the red face current, is lagging the red to neutral voltage like it should. The, uh, I two is lagging the white to neutral voltage, which is the white phase current, and it seems pretty normal. However, I three is actually going in a direction that we're not suspected, so it really should be theoretically 180 degrees in a different direction from what it's pointing in right now. So yes, indeed. It looks like they've got the white phase current going, uh are sorry. The blue phase current going through the watt meter in the opposite direction, which would give us our air. This is what be current should look like it should be going in this direction. So this would give us an indication or would actually tell us that are blue phase current or I three is connected wrong. And we would want to get into the back of the wiring of that panel and make the correction or find the problem by eyeballing it and then making the correction. So this ends chapter seven.