CHAPTER THREE, the A operator. As we continue along to develop this tool for analysis, I want to introduce something called the A operator. The a operator is used to shift the phase angle of a phaser by 120 degrees. It's similar to how a j operator works on a vector or a phaser. The j operator takes a vector say here, the x vector and it shifts it by 90 degrees by putting j in front of the vector or multiplying it by one at 90 degrees. So the J operator shifts a vector through 90 degrees where the A operator shifts the face Or at 120 degrees.

The operator applies 120 degree counter clockwise rotation to a phaser. In other words, it results in the same magnitude of a vector only it's rotated by 120 degrees. we designate the A operator by just putting it in front of the phaser or the vector, and we just say a times V, in this case VA, and a is equal to one at 120 degrees, which would be the same as multiplying any vector by one and 120 degrees. So the A operator is equal to one at 120 degrees. If we were to consider the A operator acting twice on a phaser. In other words, A times A times the phaser, or a squared times of phaser A squared as an operator applies a 240 degree counterclockwise rotation to that phasor.

Or, in other words, it applies 120 degree plus 120 degrees counterclockwise rotation to the original phasor. A vector multiplied or a phaser multiplied by one at 240 results in the same magnitude as the original vector, but it's rotated through 240 degrees in a counterclockwise rotation. So let's look at some of the properties of the A operator. And you have to keep in mind that the A operator is really just a shift in 120 degrees of the original phasor or a is equal to one that 120 degrees. A squared, as we've already seen, is a times a, which is one at 240 degrees, or one at minus 120 degrees, which is A times A, A q, or A times A times A is equal to one at 360. If you take a phasor and rotate it through 160 degrees, or sorry, if you rotate it through 360 degrees, you're going to come up with the same phaser which was one to start With so it's one and zero degrees.

So a three minus one is the same as one minus one in terms of a phaser, which is zero. If you take the inverse of A, which is one over a, it's the same as saying one over the phasor, one at 120 degrees is equal to one at minus 120 degrees or a squared A, the inverse of A squared, or a to the minus two is one over one over a, which is equal to A. Another property that we're going to be running into and we should make note of here is the fact when we add the three phasers together, one plus a plus a squared gives us a phaser. That looks like this. If you add the a phaser to it, it's 120 degrees rotated from one. And if you add the two vectors or phasers together, you put the tail on one on the head of the other.

And if you add a square to that, you come back right to the original position, which means that one plus a plus a squared is equal to zero. It's also you can say equal to a squared plus a plus one. And if we start off with a squared here, and we add a to it, and then add one to it, we come back to the same position. So essentially, that is also equal to zero. And if we work out a plan squared plus one, you find that the phasers again go around in a loop and come out to zero. So those three additions using the A operator always come out to zero and we're going to use this handy handy little notation a little bit further in our tool development.

This ends chapter three