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