So, small boxes raise the Q of this system. And and if you if a small box desired, then you typically want to pick a transducer with a low Q, you know, less than about 0.4. and if, because You know? We're going to have to, to tune that port, in the design process. Based upon the tradeoff between the, size of the box, and the Q, associated with the, transducer itself. There's a set of formulas that were developed by Keel, Using speaker alignments that came from Keele's work for the, specifically for the purpose of designing ported speakers. And the formulas are presented here. You'll see the the, the volume of the box of the speaker is fifteen times. The volume, The equivalent volume that corresponds to the stiffness of the, of the, driver. And then the, the, the total Q raised to the 2.87 power, okay? So that's that's one of your design parameters. And then you can calculate the 3DB point. 0.26 times the natural frequency of the driver and the total Q raised to the minus 1.4 power. so and then the natural frequency. Associated with the with the box, the closed box design is 0.42 times the natural frequency with the the Q, the total Q raised to the minus 0.9 power. if you go through this exercise. And the design is too large you can choose the desired box size, and then based upon a chosen box size you can go back in and and compute the 3dB point. So this is fixed, and of course this is fixed and known from the deal small parameter. So now you've chosen a box size and you calculate the 3Db point. So the new tuning frequency can be calculated for the box based upon the expression here that relates the. The equivalent volume of the equivalent box volume associated with the stiffness to the box size that was selected, the ratio of that raised to the .32 power times the natural frequency, So what I've done you can run these numbers on the calculator, it's pretty quick to do. If you have a simple computer tool you can potentially write a piece of code. I've generated a simple computer code with mat lab to demonstrate a design with, oh look at that a mistake in my slides. Demonstrate the the design, with an optimal box. And, whether we can choose to reduce the box value by a factor of two, okay? So, let's, let me pull up, Matlab, and, this is the Matlab operating environment, Let's just, for a second, take a look at the, at the code here. And so, what I've done is I've defined parameters for the speaker, alright? And I've defined a parameter for the driver itself. Where the resonant frequency of the driver is at 50 hertz. The total q of the driver is .4, and the compliance of an equivalent volume of 42.5 liters, okay, and so now I can actually compute the box, box volume. Here's here's the box volume, the 3DB frequency, and the box resonance frequency. Just as was expressed in the equations that I showed you a few minutes ago, if I if I apply Keel's formula I can compute the the following coefficients. that I've labeled here a, b, c, and d. for the design, and for comparison I thought what I would do is actually design a box with half the volume. So now I'm going to assume that the volume that results is not the volume that that I'd like, and I'm going to go back and cut that volume by a factor of two. And from that I can get the 3D B or, or I'm sorry, I can get the box residence associated with this, a. smaller box volume. Now these parameters again I've computed an of bh and ch and dh here, these parameters are used to actually plot the frequency response and There's a, a nice website that, that will that, that will show you these particular details, and that's what I've done here is that I'm computing the frequency response for the optimal box and the box and the box volume, so let's just run the piece of code. so you can see here. the blue line is the optimal box frequency response. And you can see that I've got a, you know, a lower 3DD point, and so extend the low frequency response. And this is, this basically gets directly to the trade-off. If if I want a box that's half the volume, you know, so I want to build a smaller speaker cabinet basically. For my border design, I can do that but my tradeoff is going to be frequency response, because if I'm calling, you know, if I, if I say we're around the 3Db point here for the steady, steady state response. Then you can see that you know there's a good 20 hertz or so difference in the low frequency band width for the design. So, you know, this is the price that you pay if you know, if you just want to simply for a given then, if you want to reduce the box for a given speaker choice I mean if you want to reduce the box volume. So, anyway, you can there are software tools. Many different, MatLabs, a piece of code we use at the University a fair bit, but there are other options for programming, and you can just as easily program this and a simple language like C.