Open course ware

[SQUEAKING] [RUSTLING] [CLICKING] DAVID PERREAULT: OK, why don't we get started? So we're going to continue talking about three-phase power conversion systems. And I should have noted that what we're talking about is contained in Principles of Power Electronics chapter 9 for reading. And just as a reminder, we talked last time about all the advantages you can get from three-phase systems, and we'll see that again today. But just to remind you, I might have three voltage sources-- Va, Vb, these are all of t, and Vc. I bring out three phases, and I may also have a neutral. And if I wrote Va of t is equal to Vs sine omega t, Vb of t is equal to Vs sine of omega t minus 2 pi over 3, so 120 degrees lagging, and then Vc of t is equal to Vs sine of omega t plus 2 pi over 3, so this might be 240 degrees lagging or 120 degrees leading, depending upon how you would like to look at it. We said we could represent any such individual signal as a phasor. So we might say...

[SQUEAKING] [RUSTLING] [CLICKING] JACK HARE: So once again, we are going to be discussing current scattering. And we're going to derive Thomson scattering for the third time. Huzzah! Huzzah. Thank you. Good. Well done. A grade. Coherent scattering-- so just as a quick review of what we discussed at the end of the last lecture, we are looking at scattering when the wavelength of whatever the mode is that we're scattering of is greater than the Debye length here. So this wavelength is 2 pi upon the size of k, where k is ks minus ki, like this. So we're not talking about the wavelength of our laser. Though, this wavelength may be similar to the wavelength of our laser, but we're talking specifically about the wavelength that we get from drawing this diagram where we have some laser beam coming in with a wave vector ki. We have some scatterer at a vector ks. And then we go, OK, ks minus ki. Aha, this is the k that we're talking about. We're talking abo...

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: OK, so let's continue our discussion about spectral theory for self adjoint compact operators. So let me just briefly recall the spectrum of a bounded operator, which was supposed to be a generalization of the eigenvalues of a matrix. So we defined the resolvent set of A to be those complex numbers such that A minus lambda times the identity, which I just write is A minus lambda, is an invertible, bounded linear operator, meaning it is bijective and, which by the open mapping theorem, tells you that the inverse is also continuous. And the spectrum of A is simply those lambda so that A minus lambda is not invertible, so the complement of resolvent set of A. So from linear algebra, you have the following characterization of what the spectrum is, that if H is CN, and A is just therefore matrix on CN, or RN if you like, then the spectrum is just simply the set of eigenvalues of A. And if we restrict our attention to Hermi...