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Now that we've introduced inductors and capacitors and the notion of impedance.

I want to go on and talk a little bit about a generalization of that idea.

Now some of the work on electronics and the acoustics lecture, may not have

seemed to be too connected to each other. But this idea of impedance is something

that is a very general idea that really spans, electronics and acoustics, and

mechanical systems as well. So I want to take a few minutes and just

talk about generalized impedance. Now, we're going to be looking at this in

three different systems, Electrical, Mechanical, and Acoustic systems.

And in Electrical systems the variables that we are focused on are the voltage

and the current, in circuits. Now in a mechanical system the analogous

quantities are the force and velocity. So force is like voltage.

it's the it's the it's quantity that makes things move.

So in an Electrical circuit, if I apply a voltage, there's a resulting current.

In a mechanical system, if I apply a force, there's a resulting velocity of

the object to which I've applied the force.

Now in acoustic systems, the analog of force is acoustic pressure.

And the response of an acoustic system is the so called volume flow velocity or

just the flow velocity. So if I apply a pressure to a slug of air

in an air column, it will respond by flowing with a certain velocity.

And so, voltage, force, and pressure are analogous quantities across these three

different types of systems. And current, velocity, and volume flow

velocity are the corresponding quantities.

Now there are three types of elements in Electrical, Mechanical and Acoustic

systems. The first type a purely dissipated

elements like in electrical circuit resistor.

So resistor is just a, provides a resistance to flow of current, and the

voltage across a resistor is just I times R.

So this is Ohm's Law. Now, we just introduced capacitors, and

capacitors are kind of the electrical equivalent of a spring.

And the, they store energy much like a spring stores potential energy.

If you compress the string it would like to, push back.

So the capacitor stores energy in much the same way in the form of the electric

field. Now, the relationship between the voltage

and the current for a capacitor is given by something that looks like Ohm's law.

But the impedance is 1 over j omega C, so that is the the quantity.

The impedance, that takes the place of R in a V equals I times R type

relationship. And then the third circuit element that

we just introduced are inductors, and inductors are kind of like inertia, in

electrical circuits. They're, I should say they are analogous

to inertia and mechanical systems. And the relationship between the voltage

and the current for an inductor is just the impedance, J omega L times I.

So R is the impedance for a resistor. 1 over J omega C, the impedance for a

capacitor. And j omega L is the impedance for an

inductor. Now, in a mechanical system, the

dissipative element is like a shock absorber on a car.

And this little diagram for mechanical resistance is meant to kind of evoke the

image of a shock absorber in your car. And it has the function that the force

and velocity are related by the mechanical resistance.

So, if I try to move this plunger in and out very rapidly, then there's, it takes

a very large force. now you know like in your car, if you go

and lean on the, the car, it very gradually depresses.

So it doesn't take an awful lot of force to to depress this the shock absorber but

it moves with a very small velocity. Now, mechanical systems have springs as

well. So springs store potential energy.

If I compress the spring, it tries to push back.

If I stretch the spring out, it tries to pull things back to the equilibrium

position. And the relationship between the force an

the velocity for a spring is given by this, where k is the spring constant, and

this is 1 over J omega. So, C and K appear, and and one and the

denominator, one in the numerator. But, they have this J omega term in the

down in the denominator, so this is so the electrical analogue of a spring is a

capacitor. And it's, has the spring, effective

spring constant, is like 1 over C. So this is the Generalized impedance for

a spring. Now, the generalized impedance for a mass

is just J omega M, times V. And so this looks like an inductor, where

the mass and the inductance play similar roles.

But this is the relationship for that really define the mechanical impedance

for a mass. Now the last set of quantities, the

dissapated elements in an acoustic system would be just like a baffle.

Let's say we take sort of like the inside of your muffler in your car.

Or if you take a, a tube and you stuff it with very fluffy, fluffed-up cotton or

something. The you can push air through it and the

greater the pressure you apply, the greater the flow of air.

And this acoustic bafel or the impedance to to, flow of air is characterized by

the acoustic resistance. And so this is just like an electrical

resistance but, this is pressure and this is volume flow.

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Now, an acoustic spring, this goes back to the Helmholtz resonator that Professor

Clark was talking about in one of his lectures.

if I have just a, a bottle of air, I can, that thing acts like an acoustic spring.

If I push on it, the air is compressed and it tries to push back.

So its exactly like a spring and the impedance of that is given by the

acoustic spring constant over J omega. So this looks just like that, but the

volume flow velocity takes the place of the actual velocity here and pressure in

place of force. And the last type of circuit element, one

with inertia in an Acoustic system, is just a mass of air.

So, again, in the, imagine we have a tube and you look at a little volume of air in

that tube, like the Helmholtz Resonator. And that little slug of air, has a

certain mass, and, the the relationship between the pressure and the volume flow.

When I apply a pressure to this small slug of air, it's just J omega M, where M

is the mass of that piece of that volume of air.

Times the volume flow. So the point is, is that this idea of

impedance is a very general notion and it applies to Electrical, Mechanical and

Acoustic systems. And you can write the equations for

different types of systems. And then, solve those equations in the

frequency domain using the techniques that we're developing for AC circuit

analysis. But you can find the response of

mechanical systems or acoustic systems in very much the same way that we're finding

the response of electrical systems using these techniques.