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# Phasors

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Sections
Alternating Voltages and Currents
Inductors in AC Circuits
Capacitors in AC Circuits
Power in AC Circuits
Resistors in AC Circuits
Series LRC Circuits
RMS Current and Voltage
Phasors for Inductors
Phasors for Capacitors
Phasors for Resistors
Resonance in Series LRC Circuits
Phasors
Impedance in AC Circuits

Concept #1: Phasors

Transcript

Hey guys, in this video we're going to talk about these really incredible tools that we use when solving AC circuits called phasors, alright let's get to it.Now a phasor is just a rotating vector, phasor means phase vector all the information contained by a phasor is contained in its X component you can completely ignore the vertical components because it doesn't mean anything, phasors are perfect for capturing all the information and representing it very easily for oscillating values like voltage and current which we know oscillate for instance we know the voltage as a function of time looks like some maximum voltage times cosine of omega T this is exactly what I've drawn here I've given one cycle of voltage that undergoes a sinusoidal oscillation and we want to look at how a phasor can easily represent this exact information now there are four times that I'm going to be interested in what I'll call time one when the voltages at a maximum and positive time two when the voltage is 0 times 3 when the voltage maximum but negative and time four when the voltage is back to 0. So these four diagrams here sorry T2 T3 T4 are going to contain the phasor that represents the information about the voltage at each of those four times these diagrams by the way are called phaser diagrams for obvious reasons. Now initially the voltage is at a maximum in order for a phasor to be at its maximum its entire length has to be along the X axis this is just because a vector that points along an axis for instance the X axis that's when that vectors component is largest the X component when the vectors largest when that vector points along the X axis. Now the question is which side left or right do we want to put it on. By convention to the right is considered positive and to the left is considered negative so I'm going to draw the phasor like this it's entirely along the X axis which means that the voltages at the largest value it could possibly be because the phasor as it rotates remember a phasor is a rotating vector is not going to change length. So this is our voltage phasor that's our time one now in time two. The voltage is 0 that means that it has to have no X component so the phasor has to lie entirely along the vertical axis the question is, is it up or is it down now by convention we consider phasors as rotating counterclockwise so when it has 0 X component at time two it is going to point upwards. By convention we consider it as rotating counterclockwise at time three now it has the maximum value again but negative so once again since it's at a maximum the phasor has to point entirely along the X axis and since it's negative it points to the left and finally at the T4 it's 0 again if it's 0 it means it has no X component which means it has to lie entirely on the vertical axis since it was initially pointing to the left and it rotates counterclockwise now it's going to point downwards and this is the phasor it's rotating counterclockwise and it's rotating at the same angular frequency of the oscillations omega, so it's just rotating at omega. If omega is 2 per second that means it does two full rotations every 1 second.

Now phasors are going to be weird when you guys first encounter them it's going to take practice to understand the phasors so let's do an example to try to familiarize ourself a little bit more with what a phasor is for the following phasor is sorry for the following voltage phasor is the voltage positive or negative remember all the information is contained in the X axis so all we care about is the X component right this, which is about as long as this phasor its a little bit longer whatever. Since it's pointing to the right we know that this is positive it's projection is what we would call it, it's projection onto the X axis is positive. Now the reason why phasors are used so much they are so useful and they are so great is that you can treat phasors at a particular incident time you know phasors are always rotating so if you freeze time and take a snapshot of it you can treat phasors exactly like you would treat vectors you can add them you can subtract them and you can find the magnitude of a phasor by using the Pythagorean theorem exactly like you would find the magnitude of a vector so let's do an example illustrating that in the following phasor diagram find the direction of the net phasor for the three phasor shown is the resulting quantity the phasor describes positive or negative.Let's say that these three phasors were of the same type let's say that they each described the voltage just as an example they could all describe current for instance. What I need to do is describe this as an entire net voltage phasor, so we have two phasors pointing in the same direction sorry I forgot to label these V1, V2 and V3. V1 and V3 point along the same axis so the net of those two is going to point in the direction of the V3 because V3 is longer right it's like two forces that are pointing in opposite directions the larger force wins so we still have a phasor pointing in the direction of V3 just a little bit smaller now V2 isl left alone because it's perpendicular so here's V2 here's V3 minus V1. I don't know which of these is longer V2 or V3 minus V1 but I know that our net phasor is going to point somewhere in between here right maybe it's in this direction maybe it's directly along the axis maybe it's down here it actually doesn't matter because it's pointing to the right no matter what our value is always going to be positive sorry should say always our value is going to be positive regardless of where specifically the net phasor points it's going to be positive in order for the net phasor to result in a negative value it would've had to point to the left and it clearly does not point to the left. This is only the beginning with phasors guys phasors are confusing just like vectors were confusing when you first saw them but the more you use phasors the more you become comfortable with them in future videos we're going to talk about phasors in specific context of voltage and current in circuits and it will become much more clear what they do and how to use them. Alright guys thanks for watching.

Practice: The following phasor diagram shows an arbitrary phasor during its first rotation. Assuming that it begins with an angle of 0° , if the phasor took 0.027 s to get to its current position, what is the angular frequency of the phasor?

Example #1: Converting Between a Function and a Phasor

Transcript

Hey guys, let's do a phasor example. In this case an example that deals with relating the phasor's equation to the phasor diagram. The current in an AC circuit is given by this equation. Draw the phasor that corresponds to this current at 15 milliseconds assuming the phasor begins at zero degrees. So in the beginning the phasor is going to start here at zero degrees and it's going to rotate through some amount out of angle to arrive at its final position. We want to figure out what that angle is so we know where to draw the final position of this phasor. Remember that the angle is just going to be omega, the angular frequency of the phasor, times T. Now we know that T is just 15 milliseconds so what's that angular frequency of the phasor? What we're told that the angular frequency of the current is 377 and that's going to be the angular frequency of the phasor however quickly it's oscillating on a function if I were to draw this as an oscillating graph is going to be the same rate as how quickly it's oscillating on a phasor diagram. Those angular frequencies are the same. So this is just going to be 377 times 15 milliseconds, milli is 10 to the -3 and this equals 5.66 radians. We want this to be in degrees because it's easiest to graph degrees for us or sorry to draw degrees on the diagram. Remember you can convert by dividing this by Pi and multiplying it by a 180 degrees, that's going to be 324 degrees. I'm going to minimise myself and draw this phasor diagram.

It started from zero degrees, don't forget the phasor began at zero so starting from zero and rotating counterclockwise, this phasor in the fourth quadrant because 324 is greater than 270 but less than 360. There are two other ways that you can represent this number if you want this one is 324 as the full rotation you can represent it from the negative Y axis if you want and this would be 54 degrees or you can represent it from the positive X axis if you like and this would be 36 degrees. Either way this is correct but the important thing to remember is that this value, 324, is how far it traveled from its initial position. It started at zero degrees and phasors always rotate counterclockwise. So this is 324 degrees. Alright guys, thanks for watching.

Practice: An AC source oscillates with an angular frequency of 120 s-1 . If the initial voltage phasor is shown in the following phasor diagram, draw the voltage phasor after 0.01 s. (Select the correct absolute angle below of the phasor's location below after you have drawn it.)

Practice: A phasor of length 4 begins at 0° . If it is rotating at ω = 250 s−1 , what is the value of the phasor after 0.007 s?