02 - Vectors & Phasors

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Transcript

Chapter two vectors and phasers. So far we have discovered that we can analyze AC circuits and currents and voltages using RMS values for resistive loads. That is to say the AC currents and voltages are in phase. But that is not always the case. So we have another dimension to deal with, in and in doing so, we have to develop and use another tool. This tool is called a phaser, which is a special type of vector vectors have two values magnitude and direction.

Phasers are vectors but they also have a rotation. And usually it's that system frequency. And in North America that system frequency is 60 hertz and that's what we'll be considering in our Subsequent calculations. Okay, so let's build that tool. How exactly can we represent AC quantities of voltage or current in the form of a vector let the length of the vector represent the magnitude, or amplitude or RMS value of the wave form like this, the greater the amplitude of the wave form, the greater the length of the corresponding phaser. Before progressing further, we'll establish that all assign your Saidel wave phasers voltages and current rotate at the same frequency.

In the North American Standard the frequency of the system is 60 hertz or 60 cycles per second. So all the phasers, currents and voltages are all rotating at this frequency. Hence they're separated only by a phase shift that can vary from zero to 360 degrees. The angle of the vector however represents the phase shift in degrees between the waveforms in question, and another one acting as a reference in time. Usually when phase. When the phase of a waveform in a circuit is expressed, it's referenced to the power supply voltage waveform, arbitrarily set at zero degrees.

Remember, that phase is always a relative measurement between two waveforms rather than an absolute property. We will look here at a waveform with reference to be waveform. If there is a zero degrees phase shift between these two phasers, A and B are in perfect step, they are in line with each other. So the kind of overlap If there is a phase shift of 90 degrees, it could be that a is a head of B, A leads B by 90 degrees assuming a counterclockwise phase rotation, a phase shift of 90 degrees, where B is a head of a. This would also be indicated by these two phasers, where B leaves a by 90 degrees. And a phase shift of 180 degree degrees means that a and b waveforms are basically mirror images of each other or are in the opposite directions.

So let's list some of the characteristics of a phaser. And we will go Want to develop how to analyze and calculate phasers. But for now let's just talk about some of the characteristics of a phaser. A phaser can vary in length, which is equivalent to the phaser magnitude, which could be the phasers are the AC current or voltage magnitude RMS value peak to peak value doesn't matter what it is the phasor the phasers length is proportional to that quantity, we have to define what that quantity is, if we're gonna work with it, but for now we can say that phasers, the length of a phaser changes as its magnitude changes. All phasers rotate the same and they're out of rotating at a frequency of 60 cycles per second. Depending on the system and in our system in North America is 60 cycles.

Slowed down the rotation here. Otherwise, if it was 60 cycles a second, you wouldn't be able to see it. So let's assume that it's scaled down here just for demonstration purposes. phasers rotate about their tail. In other words, a phaser is represented by an arrow, and it is rotating about its tail phasers, all the phasers in the system rotate at the same direction. The standard that is usually adopted is counterclockwise.

I will say this but only once. It could be in a clockwise direction as long as you maintain that sensibility and work with the understanding that phasers are rotating in a clockwise direction. However, the standards that we work with and the standards that I'll be working with in this course assumes the phasers are rotating in a counterclockwise direction. They all rotate in the same direction. And phase angles of a phasor are relative. A phase angle cannot exist unless it's related to something else and usually another vector.

So phase angles are relative. The greater the phase shift in degrees between two waveforms The greater the angle difference between the corresponding phasers being a relative measurement like voltage phase shift or phase angle, only. has meaning in reference to some standard waveform. Generally this reference waveform is the main AC power supply voltage in the circuit. If there is more than one AC voltage source, then one of the sources is arbitrarily chosen to be the phase reference for all other measurements in the circuit. As in this diagram phasor a phase shift is relative to phasor B.

For convenience and ease of reference we stop the reference phasor when it is at zero with the horizontal so it makes it relative measurements easier or when the B waveform is at zero. There are a couple of ways of uniquely describing a phaser in one of those ways is using what is noted as polar notation. And when we use polar notation, the phaser is described as being on a polar plane with its tail at the center of the origin. And the plane is divided into 360. of planar area. Therefore, each phaser is described with its magnitude being a length of the phaser or arrow. And it's described with its angle of displacement with respect to zero degrees.

In this particular example, we might want to describe this phaser as having a magnitude of 8.49 at an angle of 32 degrees, and we usually draw it or write it on paper in this format where the magnitude is written first in numerical format with an angle sign and that angle described after the angle side. Standard orientation for a phaser angle in AC circuit calculations, defined zero as being the right horizontal, making 90 degrees straight up. 180 degrees is to the left and 270 degrees straight down. Please know that phaser angles down can have angles represented in polar form as well. positive numbers in excess of 180 degrees, or negative numbers less than 180 degrees. For example, the phasor angle 270 degrees straight down can also be said to have an angle of minus 90 degrees.

Here are some examples of phasers in Poland notation, you'll notice the two ways of designate a phaser either with plus or minus angles. I've left off the polar plane, we're going to assume that we know that zero degrees is to the right and 90 is up and 270 is down and 180 degrees is to the left. The vector on the upper right in this diagram is has a magnitude of 8.49. And it has an angle A plus angle of 45 degrees. The phaser in the right hand, upper right hand side of the diagram here has a magnitude of 8.06. And the angle can either be described as minus 29.74 degrees, or it can be described as 330.26 degrees.

The phasor in the bottom left corner has a magnitude of 5.39. And its angle is 158 degrees, which puts it in, up and over to the left the vaser in the bottom right hand corner, you can designate that in one of two ways using polar notation. One is that they both have the same magnitude 7.81, but you can describe the angle as 232.19 degrees or you can describe the angle as minus 129.81 degrees. In rectangular notation, the phasers is taken to be the hypotenuse of a right angle triangle described by the lengths of the adjacent and opposite sides. Rather than describing the phasers, length and direction by denoting magnitude and angle. It is described in terms of how far left or right and how far up or down it is from the origin.

These two dimensional figures horizontal 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 lowercase J. This lowercase letter does not represent a physical variable, but rather a mathematical operator used to distinguish the phasers vertical component from its horizontal component. When placed in front of a vector, it'll swing that vector through 90 degrees. In a counter clockwise direction. So in our example, we can consider the vector, the red arrow made up of two vectors the sum of two vectors, one along the x axis, which is a length of four, and one along the vertical axis, which is three.

But in order to distinguish the, the horizontal and vertical, we use the J operator which swings a, what would be a real vector of three along the x axis true 90 degrees to B along the y axis. So the red vector is described as a sum of two vectors, one along the real axis, one along the y axis for plus j. Three as I said, as a complete complex number, the horizont Vertical quantities are written as the sum of two vectors. The horizontal component is referred to as the real components since that dimension is compatible with a normal scalar real number. The vertical component at 90 degrees to the real component is referred to as the imaginary component, since that dimension lies in a different direction, totally alien to the scalar of a real number. Here are some examples of phasers in rectangular notation.

Notice this time there is only one way to distinguish the phaser. They are uniquely described by the two figures, the one in the first quadrant, upper right hand side of the graph is Four plus j three, which means it's four along the real axis and three along the imaginary axis as denoted by plus j in front of the three, the one in the left hand side, upper left hand side of the graph is minus four plus j three, so minus four is along the real axis, but in the minus direction four, and the the three is along the plus j or imaginary axis, three in that direction, that vector is made up of minus four plus j three. And lastly, the one in the bottom right hand quadrant of the graph is made up of plus four minus j three. So we have seen that a face 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 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, 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 feature.

Are 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 phasor 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 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 on All you have to do is take the arc tan of the imaginary component divided by the real component or three over four, the arctangent of three quarters is 36.87 degrees.

If two AC voltages 19 degrees out of phase are added together by being connected in series, their voltage magnitudes to not directly add or subtract as with the scalar voltages in DC calculations. Instead, these voltage quantities are complex quantities and must add up in a trigonometric fashion. A six volt source at zero degrees added to an eight volt source at 19 degrees results in 10 volts at a phase angle of 53.13. degrees. In other words, when you add voltages that are 90 degrees separation, it's like finding the hypotenuse of a right angle triangle. And that's exactly what happens with the trigonometric addition that we did hear vectors as well as phasers can be moved around the plane as long as their direction and magnitudes are maintained. So you add two vectors or phasers by placing the tail of one on the head of the other and connecting the other head and the other 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 the two AC voltage sources independently, yet only read 12 volts for the total voltage with AC two voltages can be aiding or opposing one another to any degree fully ating or fully opposing, inclusive. without the use of phasers or complex numbers, notations to describe the 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 Poland.

Rectangular notation, because the addition and subtraction of phasers in rectangular notation is very easy and the multiplication and division of key 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 is 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 Some 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 JD two gives us a phaser that would be minus 16 plus j 92. So when subtracting complex numbers, it's almost as simple as as adding them together. You either subtract the 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 sign 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 could change the sign 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 175 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 phaser minus 36 plus j 10. And you're going to subtract 20 plus JT two, 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 10. minus j 82. would give you minus j 70. To 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 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 10 at minus 12 degrees, you'd multiply the two magnitudes to end up with 350. And you would add that 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 could describe it as minus 10 degrees as well. So you would add the the angles together to get the resultant angle which is 350 degrees. So your resultant would be 1364 at 350 degrees or 1364 at minute 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 that to angles 30 minus 30 gives you zero. And, as you might have guessed, division of polar form complex numbers is the easiest path to take. You simply divide the polar magnitudes of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient and subtract the angle of the second complex number from the angle The first complex number to arrive at the angle of the quotient. So a few examples just to cement the idea. Here we have phasor 35 at 65 degrees being divided by a phasor of 10 at minus 12 degrees, you would divide 335 by 10, which would give you 3.5.

And you'd subtract 12 from 65, which is same as adding 12 to 65, which would give you 77 degrees. In this example, we have phaser at 124 magnitude at 25 degrees 250 degrees being divided by a phaser of magnitude 11 at 100 degrees, so you would divide 124 by 11, which would give you 11.273. And you would subtract 100 from 250 to arrive at 150 degrees. And a final example would be phaser. Three at 30 degrees divided by five and minus 30 degrees, you would have three divided by five, which is point six, and then you add subtract 30 degrees from subtract 30 minus 30 degrees from 30 degrees is the same as adding 30 degrees to 30 degrees, which would result in an angle of 60 degrees, so the resultant quotient would be 0.6 at 60 degrees. To obtain the reciprocal, or invert a complex number, simply divide the number in its poor form into the value of one which is nothing more than the complex number one at zero degrees.

So in these examples, polls, if you want to invert the phaser 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 and minus 12 degrees, which would be one at zero degrees divided by 10 at 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 003 to 10 degrees would give you $1 or one divided by points. 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 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 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. This ends chapter two

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