# Impedance Matching Middle Ear Model

>>Instructor: With the

ideas now of impedance and impedance matching, we

can talk about the middle ear and how it might be modeled as an impedance matching system. The purpose of the middle ear is to help fix the impedance mismatch between the area of sound as it comes down from the air in the ear canal and as it tries to get

impacted or transmitted from, to the cochlea to the oval

window of the cochlea. So we go ear canal to oval window of cochlea. And on this end we have air because that’s what the

cast time vibrations are being transmitted through in the ear canal. But when it hits the oval

window of the cochlea, we now have a liquid, right? We have basically fluid. Fluid, liquid that needs to get rid in order

for the cochlea to function and pick up the sound, right? Just to make the hair cells

and transmit the nerve impulses to the cochlear nerve. Anytime that there’s a interface shift, and change of inner state, that energy needs to go from one interface that’s homogenous to a different one that’s not the same, that change of medium

or interface transfer, energy transfer through an interface may or may not be well matched. Now, it turns out for the purposes of air and fluid or air and liquid, like air and water, it’s

highly, highly not matched, it’s so not matched that something on the

order of greater than 99.9% of the energy that’s

coming from the air going into the fluid is lost. It’s like they’re back into the air and a tiny fraction will end up actually being passed into the fluid. If mammalian ears and auditory systems were set up without any

impedance matching systems, this, the ability to detect sound would be extremely limited. The middle ear does precisely that. It provides an impedance matching circuit so that the impedance of the air and the impedance of the

liquid are better managed such that you have

something that’s much better than 99.9% energy lost. In fact, a long the ranges of interest it actually does quite a decent job. It can transfer somewhere

on the area of 50 to 75% of the energy between the

frequencies of 300 to 3,000 which is quite impressive which means that if you think

about the way that’s changing, what it’s doing is that when you have a 99.9% plus, you have a significant

reactive component here. And a very small resistive component such that remember, since

power’s only dissipated over the resistive component, in order for you to have a 99.9% loss, it must have a tiny, tiny, tiny fraction of the total power, the magnitude of Z that’s actually present in R, so there’s a huge face shift that’s not being well compensated. The middle ear helps pull

this reactants closer to purely to real and it might also pull it

up depending on flow down, depending on whether it’s

inductive or capacitive. We haven’t said anything about that yet. The model that we’re pulling from is a derivative of the Zwislocki model and in the 1960s, Zwislocki was one of the first scientists to publish a circuit model that well described the properties, the

impedance matching properties of the middle ear. The simple model, we aren’t going to go

through the Zwislocki model because it’s quite complicated, we’re going to go through

a simplified model here, this was in 1962/68, I forget the year, ’62, this was 1962. It’s actually a very nice paper. It’s called the Analysis

of the Middle Ear Function, Part One, Input Impedance and it was published in the Journal of the Acoustical Society

of America, Journal of ASA. The model we’ll be looking at here instead is a much, much simplified model that sort of borrows

from the Zwislocki model and it was published in a

recent paper in 1995 by Hemila, H-E-M-I-L-A with the umlaut and this was in 1995 and this was in the Journal

of Hearing Research, was it Journal of Hearing Research or just Hearing Research? I always forget. It is just Hearing Research. And, oops, that’s missing an E, and figure five B of this paper puts out a circuit model that is quite nice and intuitive in its understanding of and

its modeling of the middle ear. So, let’s set up the

terminals, here and here. We have a capacitor here, we have a capacitor here. This is C not or C zero, this is CP, I’ll go over these terms in just a moment for what they mean. This is CS. This is the only inductor

we’re modeling here, so they just stuffed it as L and for the purposes of this discussion, we’re going to leave the inner ear as purely restive R even

though it’s technically not, it’s nice to think of it as mostly, it’s actually mostly resistive but it sort of stops becoming restive once you get purely resistive

once you get to frequencies sort of above about 1.5 kilohertz or so, two kilohertz, then

this constant resistance doesn’t model the system well at all. So, this is inner ear. This is our load. And this is our middle ear,

everything else in here and each of these actually

mean something quite intuitive. C not here is the compliance of the middle ear air cushion. Between the tympanic membrane and well, the inner ear, there’s a bunch of air that

is effectively encased, it’s not communicating

usually with the outside. That has a compliance because it has stretchiness that is different than

the incoming compliance of the ear canal and so, C not represents, I’m going to

erase this for a moment, so I have a little more room. C not represents that compliance, C not is the compliance, oops, compliance of air in middle ear, ME. Now we see P and CP represents the compliance of the tympanic membrane that is more compliant than the first middle

ear bone, the malleus. The fact that there is a

difference in the stretchiness, the way the, this is compliance of tympanic membrane being

greater than that of the malleus, the fact that there’s a

compliance in compatibility here means that some energy is being absorbed by the tympanic membrane that it can then deliver back out into this circuit, so

that’s why it’s modeled here as this parallel short. What is the function of this C0? The C0 here is going to sit there and block, for example, some

very low frequency sounds. That’s what a capacitor does in series. Think about it, if you were

just to have this giant cushion of air that’s resisting, resisting the tympanic membrane from being pushed too much and that is going to limit

low frequency sounds, it’s ’cause like a DC offset resistance. CS is the compliance of the ligaments, so compliance of ligaments and muscles of the middle ear bones, of the middle ear bones. The bones of the middles ear are held together and being pulled, are under tension. Technically we’re modeling

this is a constant but as you go across

different dynamic ranges, CS could change and it could get tighter, as the muscles pull tighter, what’s going to happen then to the CS? Well, as the stiffness of a

piece of membrane goes up, the compliance goes down and what that means is

it’s going to prevent more, it’s going to prevent amplitude

from being pushed through here. So, they have these big

waves coming through. If it faces a very stiff low compliant CS, it’s going to block a lot of that power from actually passing through which is sort of what the

purpose of those ligaments and bones are, or ligaments or muscles that are attached to the bones. And then finally, we have L and L represents the inductance, it’s the inductance of

the middle ear bones which is based on the initial mass of the middle ear bones. It turns out that any object with mass that’s attached to some moving system, has inertia, that’s part

of the first law of motion. And it will not want to move from its location, so mass itself can be modeled as an inductor. There is actually a mechanical equivalent to electric circuits just like there’s a water analogy, there’s a mechanical analogy that relates how capacitors,

inductors and resistors can be modeled in physical

mechanical systems and their mass is an inductor. And this is it. This is the impedance matching circuit of the middle ear, at least

as described in Hemila 1995. It’s extremely useful because it helps us

understand how power coming in through the system can be matched so that more of it actually

goes through the inner ear and actually get transmitted

into the cochlear and the mathematics behind this, how you can simplify this is a very good exercise to go through to understand what this

type of a model is doing when it converts sound into, when it converts signals coming from air into something like this, passing into the fluid medium.