Chapter Two AC power. In order to work with the tools of electrical measurement, we have to make sure we understand the difference between scalars and vectors. So just a quick review here, a scalar quantity is a quantity that has magnitude only. And some of the examples of scalar quantities are length, area volume, speed fast into the temperature pressure. Energy entropy work. In we've shown a diagram of volume here.

Vector quantities are scalar 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 going to add to this in electrical quantities, voltage, current and impedances. 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 sinusoidal or a sinusoidal in wave shape have a particular frequency and in our case in North America, that is 60 cycles per second.

The voltage starts at zero, peaks at plus a travels through zero and peaks at negative a. Then returns to zero volts and it's repeated six 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. And you could also describe it as 60 cycles a second, but in 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 and we multiply it by number of seconds, it will come out to a value of degrees, and that will be the angle that the vector forms with the horizontal. In a utility power grid, omega is 60 cycles per second, at least in North America, in Europe in 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 may be mathematically dealt with, but must follow the rules of vector analysis with the special properties for vectors they are sometimes renamed phasers. What is the difference between a vector and the phaser?

A phaser is a special type of vector vectors have two values magnitude and direction whereas phasers are vectors that rotate at system frequency phasers are relative. If there's more than one phasor that we are studying or looking at, they are all rotating at the same system frequency or they will all have the same angular velocity or all fee or phasers are displaced by a fixed angle of separation, comparison and an 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 incident time in other words when they are stopped when working with phasers, the most common descriptive form is polar notation. Polar 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 an A at an angle of 32 degrees. Standard orientation for phaser angles in AC 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 phasor angles represented in polar form can be positive when measured counterclockwise from the heart from the horizon, or the horizontal or negative when measured clockwise from the horizontal. For example, a phaser angle 272 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've already looked at something similar it's 8.49 at 45 degrees. The phasor in the top right hand corner is 8.06 at minus 29.74 degrees or it could be 8.06. At 330.26 degrees, both notations are correct. The one in the bottom left hand corner is 5.39 at 158.2 degrees, and the one in the bottom right hand corner is 7.81 at 230.19 degrees, or 7.81 at minus 129.81. agrees something else to note, while describing phasers in polar notation, is that the magnitude we've described it as simply 8.06, or 8.49, or 5.39. Notice I haven't indicated whether that's inches, feet, meters, or it could be miles, it's irrelevant. 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 changes 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 arrests are scaled to that that relative amount So if you're talking about inches, all of your phasers have to be in inches. If you're talking about feet then all your magnitudes should be represented in feet. Another way to describe a phaser is by rectangular notation.

In rectangular notation, the phasor is taken to be the hypotenuse of a right angle triangle, described by the lengths of the adjacent and opposite sides. Rather than describing phasers, length and direction by noon by denoting magnitude and angle. 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 lowercase 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, it swings that vector through 90 degrees counterclockwise. As a complete complex number, the horizontal and vertical quantities are written as the sum of two vectors. The horizontal component referred to as the real component, since that dimension is, is compatible with normal scalar real numbers and the vertical component at 90 degrees to the real component is referred to as the imaginary every component since that dimension lies in a different direction totally alien to this scale of the real numbers. Here are some examples of phasers in rectangular notation. Notice this time there is only one way of designating a phasor. 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 j direction or straight up and down.

The phaser in the left, upper left quadrant is minus four plus j three which means it is described as minus four along the real axis, and plus three along the J axis. And finally in the bottom right hand corner we have phasor, which is four minus j three, which means it's four along the plus real axis, and it's three in the minus j direction along the J axis. So, we have seen that a phaser can DB 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, notice that we have not moved the vector or the phasor. 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, or converting rectangular notation to polar notation and polar notation. Back to rectangular notation. converting from polar to rectangular and vice versa is a very simple process. And it can be demonstrated with this phasor here, and I've described it in both terms, both rectangular and polar form for convenience. So let's say we want to, we have the, the phaser in polar form, and we want to convert it to rectangular form.

So that is we have five at 36.87 degrees, and we want to convert it to a rectangular form. The real component of the of the polar form, you can take by multiplying the polar magnitude by the cosine of the angle So you take the magnitude five and cosine of 36.87, which is equal to four, and that will give you the real component of the rectangular form. To find the imaginary side, you take the magnitude, or the length of the magnitude of the phaser five, and you multiply it by this time the sine of the angle which is 36.87 degrees, and that equals three. So that will give you the imaginary component. And you have to remember to put the J operator in front of it, of course, and whether 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 Pythagorean Theorem. The polar magnitude is the hypotenuse of a right angle triangle. And the real and imaginary components are the adjacent and opposite sides respectively. So the length would be the square root of four squared plus three squared, which comes out to five. And then to find the angle, all you have to do is take the arc tan of the imaginary component divided by the real component or three over four, the arc tangent of three quarters is 36.87 degrees. If two AC voltages 90 degrees out of phase are added together, by being connected in series, their voltage magnitudes do not directly add or subtract as in the scalar voltages in DC.

Instead, these voltage quantities are complete Flex quantities and must add up in a trigonometric fashion. A six volt source at zero degrees added to an eight volt source at 90 degrees results in a tenfold at a phase angle of 53.13 degrees. vectors 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 tail of one on the head of the other then connecting the other head and tail just like in the picture. Compared to DC circuit analysis, this is very strange indeed. Note that it is possible to obtain volt meter indications of six and eight volts respectively across two AC voltage sources, yet only read 10 volts for total voltage with AC two volts can be aided or aiding or opposing one another in any degree between fully ating and fully opposing inclusive.

Without the use of phasers complex numbers that is notation. To describe AC quantities, it would be very difficult to perform mathematical calculations for AC circuit analysis. In this example, I demonstrated adding two AC voltages together. That happened to be separated by a phase angle of 90 degrees, which made the arithmetic fairly easy when we're dealing with fake phasor quantities in polar notation. However, adding and subtracting of complex numbers is much easier if you're able to convert to and from polar and rectangular notation. Because the addition and subtraction of phasers in rectangular notation is very easy and the multiplication and division of K phasers in polar notation is is the easiest form.

So what you want to do is you want to add using rectangular formats and you want to multiply using polar formatting. So when dealing with addition and subtraction of phasers, or complex numbers, which phasers are, you simply add the real components of the complex number to determine the real component sum and you add the imaginary component to determine the imaginary sum of the phasers So in our example, we have three here 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 resultant phaser of six plus j to the middle one is 175 minus j 34. When added to phaser at minus j 15, you'd end up with 255 minus j 49. And the last one minus 36 plus j 10 added to a phaser of 20 plus JT two gives us a phaser.

That would be minus 16 plus j 92. So when subtracting complex numbers, it's almost as simple as as adding them together. You either subtract the real 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 sine of the phaser that you want to subtract. As we're doing here, we're putting a minus sign in front of the brackets, which essentially changes the sign of the terms inside the bracket, then just add them together. Either way, you will come up with the same answer.

So in the first example, you got a phaser two plus j five, and you're going to subtract the phaser four minus j three. So you can change the sine of those components minus four would minus four, it would be minus four plus j three, then you would take two minus four is minus two, then you take plus j five plus j three would be j eight. In the middle one, you got 107. Five minus j 34. And you're going to subtract 80 minus j 15. So you could take, change the sign of that phaser.

So now you'd have minus 80 plus j 15. So 175 minus 80 would be 95 and minus j 34 plus j 15 would give you minus j 19. And finally, in our last subtraction, you would have a phaser minus 36 plus j 10. And you're going to subtract 20 plus JD. So you change the signs of those two components of that phaser. So you'd end up with minus 20 minus JT two, so you'd have 36 minus 20 would give you 56 for the real component, and plus j tan minus j 82. would give you minus j 72 for the resultant for 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 polar polar form, we simply multiply the polar magnitudes of the two complex numbers to determine the polar magnitude of the product. And we add the angles of the complex numbers 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 10 at minus 12 degrees, you'd multiply the two magnitudes to end up with 350. And you would add the two angles together. So 65 minus 12 would give you 53 degrees.

So the result would be pretty easily calculated is 358 53 degrees. Another example would be 124 at 50 to 250 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 can describe it as minus 10 degrees as well. So you would add the angles together to get the resulting angle which is 350 degrees.

So your resultant would be 1364 at 350 degrees, or 1364 at minus 10 degrees. And our final example is our simple three at 30 degrees times five at minus 30 degrees, you'd end up with multiplying the magnitudes together, you get 15. And you would add the two angles 30 minus 30 gives you zero. Division of polar form complex numbers is also a 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 of the first complex number to arrive at the angle of the quotient, 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.

It's the same as adding 12 to 65, giving you a result an angle of 77 degrees. This in this example we have 124 magnitude at 250 degrees divided by a phaser with 11 magnitude and 100 degrees. So divide 124 by 11 eight gives you 11.273. And you subtract 100 from 250 leaves you 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 point six at 60 degrees to obtain this reciprocal or invert a complex numbers simply divide the number in his poor form 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 phasor 35 at 65 degrees, you'd put one over 35 at 65 degrees, which is the same as 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 point 02857.

And you'd 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 one at zero degrees divided by 10 and minus 12. One divided by 10 is point one, zero subtract minus 12 degrees is 12 degrees. And the final example one, the reciprocal of point 0032 at 10 degrees would give you one, or one divided by point 0032 which would give you 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 number arithmetic, always add and subtract in rectangular form, then multiply and divide in polar form, it will be necessary to 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 apps out here that'll do that hard work for you but 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 AC circuits. operations with complex numbers are by no means limited just to addition, subtraction, multiplication, division and inversion. However, virtually any arithmetic operation that can be done with scalar numbers can be done with complex numbers. When considering two 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 phasor picture as shown. If the rotation is stopped, we can see that the blue phase is said to be leading the red phase or conversely the red phase 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 phasor diagram as overlapping each other as you can see here, they are rotating counterclockwise 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 phasor 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 capacitive and inductive load, the current reacts in quite a different manner for each of those separate loads the resistive load having the current in phase with the voltage, the voltage leads the current in the case of inductance and the voltage lags the current in the case of capacitance. Now, let's look at the voltages and current phasers with inductive and capacitive loads, this time with some resistance introduced into the circuit. Just for comparative purposes I am reviewing the results of a pure resistive load Where the voltage and currents are still in phase four a resistive plus a capacitive load the voltage still lags the current but less than 90 degrees this time. And similarly for a resistive and a resistive plus an inductive load, the voltage still leads the current but again less than 90 degrees. In these AC circuits these combinations are known as impedances and are denoted by the letter Zed.

Impedance impedes the flow of current in a circuit similar but not the same as resistance in a DC circuit. impedance is a vector which is made up of two components a resistive component and a reactance component, the reactance denoted by the letter X, resistance is unaffected by system frequency. However, the reactance is affected by system frequency. And it depends on whether it's a capacitive reactance or an inductive reactance. 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 or in South America, it 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 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 reactance of the resistor in ohms. In other words, the impedance of the resistor is equal to our since inductors drop voltage in proportion to the rate of change of the current, they will drop more voltage for faster changing currents and less voltage for slower changing currents. What this means is that the reactance in ohms for any inductor is directly proportional to the frequency of the alternating current. The exact formula for determining this reactance is given by Zed which is equal to the reactance X with a lowercase L, which is two pi times f l where pi is just 3.14159. f is the frequency in cycles per second and L is the inductance.

In Henry's. Since capacitors conduct current in proportion to the rate of change of the voltage, they will pass more current for faster changing voltages as they charge and discharge to the same voltage peaks in less time and less current for slower changing voltages. What this means is that reactance in ohms for any capacitor is inversely proportional to the frequency of the alternating current. In other words, the impedance a capacitor is equal to x with a small letter C, sometimes denoted with a lowercase C. And it's equal to one over two pi fc, where again pi is 3.14159. And the frequency is given in cycles per second and C is the capacitance in ferrets. Note that the relationship of capacitive reactance to frequency is exactly opposite from that of an inductive reactance capacitive reactance in ohms decreases with increasing AC frequency.

Conversely, inductive reactance in ohms increases with increasing AC frequency inductors oppose faster changing current 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 in volts divided by the impedance in ohms. Just as direct current in a simple resistive circuit is equal to the voltage and voltage divided by the resistance in ohms. This leads us to an extension of ohms law for AC circuits. impedance is related to voltage and current just as you might expect, in a manner similar to resistance in ohms law for DC 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 EMF or voltage is equal to times Zed, where is the current instead is the impedance I also is equal to E over Zed and Zed is equal to P over i.

All quantities are computed and expressed using vector not scalar rules. We encounter a measurement problem if we try to express how large or how small an AC quantity is, with DC or quantities of voltage and current are generally constant. We have little trouble expressing how much voltage or current 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 of an AC quantity is to measure its peak height on a waveform graph. This is also known as The peak or crest value of an AC wave for another way is to measure the total height between the opposite peaks.

This is known as peak to peak measurement or P to P value of an AC waveform. Another way of expressing the magnitude of a wave is to mathematically average the value of all the points on the waveform graph to a single aggregate number. This amplitude measurement is known simply as the average value of the waveform. If we average all the points on the waveform algebraic algebraically that is to consider their signs either positive or negative. The average value of most waveforms is technically zero, because all the positive points cancel. Sold out all the negative points over a full cycle.

However, a practical measure of a waveforms 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 a wave form by considering all the points in the wave form as positive quantities, as if the wave 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 age intensity of an AC waveform whether that be current or voltage, we can simply measure the peak or the crest or the amplitude of that wave. Or we can measure a quantity that is peak to peak, or we can take the mean average as a quantity that measures the intensity of the AC wave.

The good news is really, these are all related to each other in a way they can be just scaled one to the other. In other words, they vary directly as each other so one can 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 we can deal with the actual measurement and it can be useful to us as we communicate the value of voice 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 asked, asked and answered a long time ago and the answer was 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 waves 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 DC one eight See, that is each with the same load, but one driven by DC source and the other driven by an AC source. When we close the switch on the DC circuit the bulb will light with an intensity that is dependent on the resistance are Savelle of the light and the DC current. Now, let's close the switch and adjust the AC 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 AC current, we can come to the conclusion that if the two bulbs light to the same brightness, that is they draw the same average power.

And the key here is they're drawing the same average power, then it's reasonable to consider the current IAC to be in some ways equivalent to the current IDC. So, what is that value of IAC? It would be useful if there was some meaningful way to calculate it. So let's go there. If an AC supply is connected to a component of resistance, say R, the instantaneous power dissipated is given by the power equation I squared R. If we think Plot i squared, the instantaneous current, which itself is a sine wave, 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 R. 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 shown here. And it is exactly i squared max divided by two. Remember now, that the in instantaneous power is given by ice squared times r for the resistive load. If we wanted to find the average power, we could find it by simply multiplying the average value of ice squared times are such that P average is equal to i squared average times R. But we know that i squared average is equal to IMAX squared over two. So the average value of power p average is equal to IMAX squared over two times R. We just saw from the previous slide that the average power can be calculated by IMAX squared over two times the load resistance Let us define a current that when use to calculate power gives us the average power. In other words, when that current we'll call it I subscript defined is squared, and multiplied by the load resistance will give us the average power.

This means, of course, that I defined squared is equal to IMAX squared over two, which is the mean of the current squared or the mean squared current. Therefore, the square root of the mean squared current equals that defined current i subscript defined. Another way of stating it is the root mean square of the current So, we just discovered what the value of i subscript defined is, it is equal to IMAX divided by the square root of two. We call this current i RMS or the root mean square, and it is 0.707. The value of IMAX 0.7 or seven is just one over the square root of two. This is another more useful way to describe an AC quantity voltage or current.

And of course, it can be converted to other AC 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 IMAX and multiply it by 0.707 and come up with a root mean squared, but the root mean 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 ay ay ay ay ay, see? Well, I see is now we can say equal to i RMS, which is equal to 0.7, or seven times the peak value of our AC current, which is equivalent to the IDC, dissipating the same amount of average power Using the rms current and the load resistance the average power can be calculated by using the formula p is equal to i squared R. Similarly we can define the RMS value for RMS voltage which is 0.707 the peak voltage for AC 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 are calculating the average power also because we are simply using a scaling factor for the current and voltage that that factor being zero point 707 or one over the square root of two, then all of the equations involving ohms law hold true if we use RMS values for current and voltage. As well all of the rules for mesh analysis and theorems can be used with RMS values for current and voltage, keeping in mind that all quantities are computed and expressed using vector, not scalar rules. This is the end of chapter two