So unfortunately, loudspeakers are not perfect pistons. So, it's worth discussing the difference here a little bit. And we went through this a little bit earlier, you know, at the very beginning of the class. When I pulled the transducer out of the box, I pulled subwoofer a woofer out of a speaker enclosure. And we looked at the various components, and what I've shown here is directly from a spec sheet for an Eaton 7-360 transducer. You can find it at MadiSound, actually you can buy the transducer at MadiSound. but it provides a, a, a nice sketch for detailing some of the components and, you know, I'll just review them again here. briefly, you know, we have our our magnet here. inside the magnet, there'll be a moving coil that's coupled to the driver itself. So, that's going to exist, you know, in here our moving coil kind of as a cross-sectional cut. This is our dust cap. this is the portion of the driver that's the the radiator, and then we typically have, you know, around the outside. This surround that actually just seals the front and the back of the speaker from each other. Here, you see the bolt patterns for the driver and, you know, the thickness of the cage for the speaker itself. And you typically get a dimension for the depth. And, of course, the hole that needs to be cut to put the speaker in. So, spec sheets are going to give you every, all the mechanical properties and dimensions you need actually to do to design the cabinet, in terms of inserting the transducer in the cabinet. Alright. So, let's look at another schematic, particularly the mechanical response of the speaker itself. And, so now what I've done is I've, you look at the drawing that we just had above, what I've come up with is a mechanical representation of that. You remember I talked about the spider earlier, which is the suspension system, and it defines the spring stiffness of the speaker itself. actually, you, one of the things we hadn't really talked about a lot to date is the, is the damping, but there's a mechanical damping that happens. Meaning that some of the energy in the speaker's dissipated as it vibrates in it's in it's, in it's cage. And so, that's represented here this, what we call a dashpot mechanically. We've got the displacement response, x of t. It turns out the force applied that causes the speaker to oscillate, it's driven electrically. we need current here and we have the motor force that's a product of b and l that's shown here in the figure. So, you know, and then of course the speaker has a mass. And it turns out that there's a mass loading of air on it as well, if we want it to be completely accurate. but we have a moving coil and a magnet here in the driver itself. So, those are the basic composers, the mechanical part of the speaker. We can also discuss the electrical response of the speaker and hear. We see a simple circuit. And Professor Bocko's been talking about, you know, circuit design and analysis. And this is a rough sketch on my part in advance of the lecture here. But, [COUGH] what I've shown is a you know, an, an applied voltage here that we see for the speaker itself. We have a a resistance associated with the speaker, so the wire of the coil itself has a resistance. there is an inductance that is represented by L. And then, we have a a back emf. So, there's a voltage associated with the motion of the driver itself. it turns out that that is the product of b, l, and the velocity of the driver itself, b, l, x dot. Okay, and then this is our current that's, that's passing through the coil here of the speaker. And I'm sure why I have this K here. That's must be a typo. The disk is our electrical response. So you see the speaker here, it vibrates. It's actually the the magnet in the coil. The current going through the coil with the magnet. Magnetic field actually creates the electric motor force and drives our speaker as we apply a voltage across our our speaker coil. So, if anything, this should be a motivation to study differential equations at some point if you haven't. I'm not obviously going to be able to teach differential equations in one lecture but I do want to talk a little bit about the equation of motion for the system. so there, there are two equations that are coupled here, and one of the equations is the mechanical response. And you can see there's the mass, the damping and the stiffness, and the stiffness creates a force as a result of the displacement. we get the equivalent force from damping associated with the velocity and we get the equivalent force associated with motion, the mass through acceleration. And then, it's being driven by an electromotive force in a current, that's being passed through the coil, and that's what causes the mechanical response. While at the same time, there's an electrical response here of the speaker and it's a relationship between the inductance and the derivative of the current. Actually, the resistance and the current. Both of these are voltages basically. There was an applied voltage and we talked about that. And then, there was the back EMF. And so, this is the the voltage that effectively back drives the circuit as a function of the motion of the transducer itself. So, you know, this basically creates the electromechanical coupling, and this is what's creates the coupling. You know, of the equations, again x dot is the derivative of displacement, and x double dot is the second derivative. So this is velocity here and acceleration, and these would all be covered in you know, our, our taking derivatives and such are all covered in basic calculus. Some of you may have had it. For those that haven't I would encourage you to to look into that. You basically cannot derive the design equations for loudspeaker design without actually having a background in calculus, and differential equations. And some basic mechanics and circuit analysis. we can apply the equations, you just won't be able to derive them on your own. These equations obviously define the frequency response of the speaker, and you can view some sample spec sheets at madisoundspeakers.com. So, thought it'd be worth taking a quick look there MadiSound. So, if you go to MadiSound, you, there are all kinds, this is a great store for resources. everything for components such as capacitors and, and inductors and such to specific drivers. you can get kits to build a complete speaker. all of the components you know, speaker-building books any of the parts that you would need to build speakers can be found here. And it's a fun place to to shop. So you can see crossovers, capacitors, inductors, resistors, components. here you'll see driver spec sheets and so they sell a whole range of, you know, of speakers. I mentioned a the Eton speaker earlier. I have a spec sheet for a Peerless speaker that we're going to discuss here in a minute. and rather than do it from the page, from the webpage here, we'll go into the, back into my other application. But bottom line is is that just about whatever you're looking for you're going to be able to find it here at MadiSound.