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Phasor diagram (& its applications)

Phasors are rotating vectors having the length equal to the peak value of oscillations, and the angular speed equal to the angular frequency of the oscillations. They are helpful in depicting the phase relationships between two or more oscillations. They are also a useful tool to add/subtract oscillations. Created by Mahesh Shenoy.

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  • blobby green style avatar for user Charles Campbell
    Let me reassure myself about one thing, please: the length of the vector is always V sub 0, but the "shadow" it is casting is actually its vertical component. The length of that component will vary, but the length of V sub 0, the hypotenuse, will remain constant. Is this correct? Thanks in advance!
    (4 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Nolan Ryzen Terrence
      Yes you are absolutely correct the length of the vector is the peak value of the component asked(voltage or current).The shadow casted is the vertical component(the instantaneous value) which varies with the time but the peak value won't change.
      ====================================================
      To pack things up the instantaneous value(the casted shadow) will equal peak value when the angle = 90 and negative peak when it is at 270degrees(I mean the positions)

      Hope it helps
      Nolan RT:)
      (2 votes)

Video transcript

suppose we have some kind of an alternating voltage we've already seen how to visualize them here it is again just a quick recap so we visualize the oscillations this way we imagine the world voltage is oscillating between plus v naught and minus v naught now in this video we're going to introduce another visualization called phaser diagrams we'll see what they are and where why should we care about them and ultimately why phasors are so awesome to understand phasors imagine we have a vector which is sleeping right now whose length is exactly equal to the peak value of the alternating signal we're going to imagine that as the signal alternates up and down as we saw in the animation our vector is actually spinning this way okay and we're going to imagine it's spinning with a constant speed uniform angular speed omega the same angular frequency that we have over here okay so how do we connect this spinning vector with the oscillations over here more imaginations for that we imagine there is some source of light over here uh sun or maybe a flashlight which is giving you light and you imagine here we have a wall and now the shadow that is casted by this vector on the wall represents this oscillation for example right now it's completely speak sleeping and therefore it's not casting any shadow over here and therefore right now the value of v is zero but a little later in time we have some value of v the oscillation has started and we can say that's happening because this vector has spun by some angle and as a result notice it's now casting that shadow over here so the shadow represents the value of v the oscillation and as the vector keeps spinning you can see the shadow becomes larger then it becomes smaller and as it continues the shadow will keep oscillating representing this oscillation i have an animation over here with which you can see things better i couldn't animate the light it so you can you have to imagine the light yourself but you can see this is the shadow that is casted by this spinning vector but then i'm pretty sure there are so many questions coming to your mind first of all why is it called a phaser in the first place why is it spinning anti-clockwise not clockwise and why does it work out like that why does the shadow so nicely so beautifully represent the oscillation like that what's going on over here and finally even if everything works out why should i care about it why should i imagine a spinning vector okay let's try and answer all of these questions so let's take these questions one by one why is this called a phaser in the first place well consider a situation let's say at time t equal to zero the vector is sleeping this way and because it's casting no shadow over here the value of v is zero now let's say after time t this is the new situation now in that time t the vector our phasor would have spun by some angle and my question to you is what is that angle through which it has spun well it is spinning at an angular speed of omega so in one second it spins an angle omega then in t seconds it spins an angle omega t so this angle over here represents or equals omega t and omega t is the same angle that we find inside this this angle is called the phase angle or just the face and since our vector helps us easily see the phase angle we call this the phasor so it's called a phasor it's very hard to look at the graph and and and figure out what the phase angle is i mean if you just look at this graph could you tell what the value of omega t is i mean sure you can kind of guess you could say that okay because v is positive and v is not maximum maybe omega t is somewhere between 0 and 90 you can kind of see that but when you look at this you can immediately calculate you can just if you if you could measure this angle you can immediately see what the phase angle is and that's why it's called a phasor okay secondly why do we consider it to be us spinning in the anti-clockwise direction why not clockwise that's just taken as a standard because phasors are also useful in mathematics and in math you might know in graph you know we start with the first quadrant here then the second quadrant third and then the fourth and so in even in graphs if you've seen trigonometry or unit circles you might see that even there we like to consider anti-clockwise rotation as positive so we like to use the same convention okay on to the third question the mystery the mystery question why does the shadow casted by the vector exactly match with the oscillations why does that happen well trigonometry can help us answer that this shadow basically represents the vertical component now what do you think is the length of this vertical component we have the value of the angle here omega t we know the hypotenuse in this triangle can you pause the video use trigonometry and figure out what the vertical component is and see if you can answer your own question okay in this right angle triangle the vertical side represents the opposite side and since i know the hypotenuse i'm going to use sine so sine omega t equals the length of the shadow or the vertical component divided by the hypotenuse which is v naught and from this the length of the shadow is v naught sine omega t which is exactly the oscillations and that's why a vector having a length exactly equal to the peak value and spinning with the same angular speed as the angular frequency over here casts a shadow which perfectly matches the oscillation of this signal all right now to the final and the main question why should i care about this why should i care about visualizing the phase angle how does it matter a couple of reasons first of all imagine this let's say that this was the voltage through a capacitor and i asked you now to draw the graph of the current through the capacitor what would that graph look like we've already seen before the current in a capacitor leads the voltage i know that but can i draw the graph with that sure i can but it's not that straightforward i have to think really hard about it isn't it and maybe after doing a lot of thinking i can do that so if you draw the graph it turns out to be somewhat like this but it's not very obvious that the graph would look like that it's not so straightforward but now i ask you to draw a phaser for the current how would you do that hey that's not the that's not so bad i know that vectors are spinning in the anti-clockwise direction and i know that my current is 90 degrees ahead of voltage just from that can you try drawing a phaser over here phasor for the current pause and then try all right so because it is 90 degrees ahead my current phaser is going to be 90 degrees from the voltage in the anti-clockwise direction so it's going to look somewhat like this so this would be my current phasor and it should have the length the peak value the phasor should always have the length peak value and this would be 90 degrees so that the current the phase of the current is now omega t plus pi by 2. isn't this so much easier to visualize and so much easier to draw okay you try one let's say this phasor represents the current uh through an inductor can you draw the voltage phasor at this point pause and try okay in inductors you might remember voltages lead the current by an angle of 90 degrees so where should we draw them let's see can i draw it over here well no remember this is how it is spinning so this still represents current leading the voltage we want the voltage to lead so voltage needs to be ahead so voltage will be somewhere over here make sense okay let me give you another example take a look at this graph again if the brown is the voltage and the pink is the current can you tell what is the phase relationship between them well again you kind of can say that the current is a little ahead of voltage but it's again not so straightforward even if i were to look at the animation and look at the oscillation ah it's just a mess i i can't you can kind of see current is leading the voltage but could you tell exactly by how much angle nah not so straightforward but now let me show you phasor diagrams wow look at the phasor diagram i don't need the animation anymore i can just look at the diagram and i can say hey the current is ahead of voltage because they're all spinning anti-clockwise and if i just measure this angle boom i get the the phase angle between the two so phasor diagrams are really awesome at figuring out the relationship between the current and the voltage or between any two oscillations but you know what the real power of phases can be seen in circuit analysis although we'll be doing a lot of them in the future videos let me give you a sneak peek so that you can actually appreciate phasers so let's say i have an inductor connected acro you know in series with a resistor and i've given you the values of the voltages across them both are alternating voltages my question to you is what is the total voltage between a and b what's the peak value of that let's say i just want to find the peak value of the total voltage how do you figure that out well one way is to directly add them right we do vl plus vr add them and then we have to simplify which means you have to use trigonometric identities ah yuck instead let's do a phaser way so first i can draw i'll draw a phasor for let's say the resistor you can draw a phasor for any one of them you can draw the phasor anywhere you want just make sure that the length of the phaser length of the vector equals the peak value so let's say i've drawn this represents the resistor voltage phasor now the inductor voltage is c um omega t plus pi by two it's ahead of resistor by pi by two so all i have to do is make sure that the inductor voltage is 90 degrees ahead anti-clockwise remember so it's going to be this way and its length will be the peak value given over here and now now the total voltage can be found out just by adding these vectors we know how to add vectors we can do parallelogram law which becomes over here a rectangle and the diagonal represents the resulting vector and from pythagoras theorem if this is 3 and this is 4 pythagoras triplet this is going to be 5 and boom that means the peak value of the resulting voltage is 5. got it immediately isn't that awesome tell me tell me if it's not awesome now of course i just took this example to show you how awesome phasers are i don't expect you to completely get get this right away we'll have to practice and we will get that practice don't worry long story short phasers are rotating vectors which have a length exactly equal to the peak value of the alternating voltage or the current and they are spinning in the anticlockwise direction with an angular speed exactly equal to the angular frequency of the oscillation the vertical component or the shadow casted by the vector on the vertical wall represents the oscillations and they are super useful in visualizing phase relationship between currents and voltages or if we have to add voltages or alternating currents or alternating voltages in short they are simply awesome