Amar G. Bose: 6.312 Lecture 15

Last time, we made a complete model, electrical, mechanical, and acoustical part of a loudspeaker. Now, normally you'd stop there. In past decades, you would apply a lot of algebra to that and get responses. Presently, you'd just stick the model into a computer and crank out the pressure in the far field, which we said was equal to the voltage across the last resistor or proportional to the voltage across the last resistor.

So usually, past and present, you wouldn't go any further. Well, today I'd like to take you a lot further in that model for a very, very important reason. If you through-- in fact, by the way, you don't even need the model today because you could throw all the electrical equations, the transducer equation, f equals bli, the mechanical equations, and the acoustical as a set of simultaneous equations and crank them out and get a response.

But then when you went to design, that would be next to useless because you would have to randomly change one parameter then change another parameter then change another one, and you'd have a pile of graphs about that high on your lap wondering what happened.

And if you got the design that you wanted out of all of that work, what you would find out is when you went to design the next loudspeaker, because some major parameter was different, like the b field might be quite a bit different, that the other parameters would act differently than they did before. So you'd learn almost nothing.

And so what we want to do today is take a look at the same model. And again, it's not to learn about loudspeakers at all. It's to learn about how you deal with models of physical processes. And that looked like a pretty complicated thing. We can get the overall response totally by inspection.

And before we leave today, we ought to be able to tell what would happen to the response if you changed any one parameter by factor 2 or whatever you might think of. All, essentially, by inspection. So let's put the model back on the board.

This was the amplifier output. We'll just call it-- maybe I better use an e so we don't get mixed up with velocities or anything e sub s. The electrical resistance of the voice coil, the electrical-- we call it electrical because we're in the electrical side-- the inductance of the voice coil, and then the transducer, which was a transformer with a turns ratio bl, the flux density in the gap times the length of the wire in the gap to 1.

The mechanical mass, the moving mass of the cone, the surround, or what we call annulus and the voice coil bobbin and the voice coil, everything it moves about half the annulus weight, but that's so small anyway it doesn't matter and half the spider weight would go into there because only half of each of those is moving. We'll call that the mechanical mass. All right, sometimes you can say it's a-- well, that's all right. Mechanical math.

The loss, which we approximate as a viscous loss, which isn't exactly that, but thank god that's not the major loss, so it doesn't matter, r sub m.

The compliance, we'll talk about that in a minute, mechanical compliance. And that ends the mechanical part of it.

And over here, let's use the simplest model for the radiation impedance. At this moment up until now when we made the model, we just assumed that the radiation impedance was from one side of diaphragm. And I told you to neglect what's on the other side. We'll fill all that in as we go today.

Now, when you see a model that's this complicated and you want to do something by inspection, at first you get frightened. Oh my god, how can I ever see anything that's going on in something like that? Well, a good idea is to get rid of these transformers in here first because somehow, unless you're very familiar with it, that complicates your ability to see something by inspection.

Now, when you get rid of these things, be careful because you can wash away part of the problem. In other words, we can look back in here and get rid of that transformer very easily, as we will in a minute. But when we do that, we're essentially making a Thevenin equivalent, an equivalent network in which the equivalence is from these terminals out.

So don't fall into the trap of making this Thevenin equivalent, which we will do, and then computing things that are going on back in the equivalent and think that they relate to what's going on in here. No, no, no. Once you have made that other network, it's only valid from the terminals out. And what's inside of that thing is not related at all to-- it's related, but don't look at those as real elements that exist in here for power considerations or anything else. That's true of all Thevenin equivalent circuits, as I hope this has been presented to you.

Now, let's do that. A transformer, by the way, if you look at a transformer like this and you have z here, zl, let's call it load. And you want z here, and the transformer ratio is a to 1, what's the impedance here?

How many know immediately? Well then, somebody speak.

a squared.

A squared. How do you get that? Well, look. a to 1, it means across variable over here, which is voltage in the electrical is a times across variable here. The current over here is 1 over a times. So voltage-- let's call this v1, i1, i2, v2. So v1 over i1, which is z is equal to v2. But v2 now gets multiplied by a, so v1 now is a times v2. And i1 is i2 over a. So all this is a squared v2 over v1 equals a squared z.

Now, so when you step up the voltage, you step down the current and therefore, the impedance goes up by the stepping ratio squared. Now, so if we look back in here, this is a voltage source. Voltage source gets-- well, let's take them piece by piece.

This impedance is going to look over here as le, or the inductance is going to look as le over bl squared. Looking in from this side, it's bl to 1. So I'm going to see an inductance of le over bl squared. I'm going to see a resistance of re over bl squared. And the voltage, well, the voltage is stepped down here by bl. So I'm going to see a voltage source of es over bl.

So we would have then es over bl. That's not a frequency dependent thing in this case. We would have a resistance of value re over bl quantity squared, an inductance of value le over bl quantity squared, and that's it.

And that represents as far as its output terminals go, everything that's in this box.

And then we can put the rest on, and we have the first reduction of our system.

That's a little bit simpler.

Now, what I'm going to do the rest of the hour essentially has no math in it at all. And I'm going to try to do something that normally you would experience if you got a job in the field and you were working beside somebody and little by little, you got stuck on something and the experienced person came along and gave you a suggestion or told you something, and you'd little by little, put all these pieces of information together and get the overall picture.

We aren't going to work together for some weeks in the lab, and so I'm going to try to do all this maybe in 30 minutes. So hang on, but try to see the trend. Ask questions at any point.

When a person with experience would look at something like this and want to get the insight that was necessary for real design, first thing that would come to his mind is, look, this is a model made from a real physical system. There were a lot of causes, physical causes for the different models that are here-- so I mean for the different elements that are here. There may be totally different physical causes or physical phenomena that these represent.

So it's unlikely in something this complex that all the circuit elements would be interacting in some way that was so complicated at the same frequency range. Some things are likely to happen at a totally different frequency range than other ones. So the first thing that would come to mind to an experienced person is let's see if that isn't true. Let's see if there aren't elements in here which are behaving totally different in different frequency range, their action takes place in different frequency ranges than others.

Now, in the particular case here where you have part of the system that came from an electrical discipline, part of the system that came from a mechanical discipline, part of the system from an acoustical, the first thing that would come to your mind is, gee, I wonder if the critical frequencies or the resonances or the poles and the zeros of these networks coming from all these different disciplines isn't likely to be different.

And if that's different, as we will see, you all of a sudden get a lot of simpler networks. So the key in the electrical part is this element and this. That's all there is, and a source. That's constant. I don't care. If I want to find frequency response, I don't care whether I'm finding frequency response out there to es or to-- this is going to be voltage source, which you'll sweep through with different frequencies.

So this is the critical thing, where this impedance becomes equal to this, the so-called half power point of the electrical. Because below that frequency, where this impedance is equal to this, is essentially a resistor. You go down enough, two or three times in frequency below that point, and this thing is so small compared to this that as far as the behavior over there is concerned, this can all be represented by a resistor.

When you go up a little bit above that, above the point where these are equal in impedance, this is an inductor. This thing is so small, who cares? It's an inductor. So the critical frequency, the critical thing you might want to know about the electrical side is where this is equal to this.

Now, I'm going to give you rough numbers for, let's say, something like a 10 inch speaker. There's an example because what I'm going to tell you now, you may, I hope, you'll be able to catch the trend of it, but to really understand it, you'll have to go through an example. Education isn't being fed stuff. It's absorbing things by doing it yourself.

And so there's an example, 7.2 example in the text, of that reference text or annex book. Chapter 7, the example-- and it's not a home problem. It's an example, 7.2. And they have all the numbers for the parameters that are on the board here for some typical woofer or something. I'm not going to give you all the numbers because that'll focus you on numbers instead of concepts, but you should go through it.

So I'm going to say that this half power point here, the voice coil, half power point equals I'm going to say 1,500 Hertz. I think in the example he has it's actually 1,800, but let's make some round numbers. So that says the critical action in the electrical circuit takes place about there. Look at the mechanical circuit. The moving mass, the compliance.

This mechanical circuit has only to do with the loudspeaker that I held in my hands. It has nothing to do with the air that's over here. If you had a vacuum jar and you put the loudspeaker in which there, do exist things like this, then this, of course, disappears altogether because all of this load was due to air acting on the diaphragm. You put the thing in a vacuum and you feed power to it, this isn't here.

So I'm looking just at the mechanical parts here. The key frequency in there is the resonance that you see of a parallel circuit. So that resonance happens to be about-- resonance, let's call it f0 equals-- well, maybe I should give even a name to this thing. f electrical. and electrical is 1,500 Hertz, 1.5 kilohertz. f mechanical math. Why don't I put it mechanical? f mechanical is, let's say, about 100 Hertz.

And then we come out to the acoustics, the world outside. That depends on size. As you know, if you look up these radiation impedances, this is very, very dependent upon size of the diaphragm. And if you were looking them up as a pressure, as a velocity, that's across variable, and pressure as through variable, this would be your 1 over your row 0c. But this is very dependent on size.

And that resonance, acoustical let's say, resonance meaning the point where this is equal to this and impedance the frequency. Let's call that fa is, let's say, 500 Hertz. These are pretty good reasonable numbers.

So you, as the experienced person would have suspected that the critical frequencies for electrical might be different from those for mechanical. You don't know this, might be different from those for acoustical. You don't know it. You go look. If you find it, and these are nicely spread out, then things begin to get very simple.

So let's take a especially simple circuit first, the kind you would have if you put this thing in a vacuum jar. And let's just understand that you don't have to do it this way, you wouldn't do it this way if you had a lot of experience with models, but it's a stepping stone. So let's take the circuit that is the equivalent of putting this whole thing in a vacuum jar.

So we have this, this, this, and that's it. We have velocity as across variable, force as through variable plus, minus and the source. We want to get the frequency response from the es. That happens to be over at bl, but I don't care about that. I'm interested in not how big it is at this point, but how it behaves, and how that behavior, however large it is, depends upon different elements.

So that begins to get us to a simple circuit. This fellow doesn't come into play anywhere in the region where this is going through its frequencies because the mechanical resonance is 100 Hertz, and this electrical thing, this inductance here, it doesn't even become comparable to this resistance until you get to 1 and 1/2 kilohertz.

So I can think of that thing for low frequencies as being a short circuit compared to this. And so let's see what we get. My object is mu over e sub s or e sub s over-- I'm not worried about the constant factor here.

Well, for low frequencies, it's a series resistance with a RLC and a source. And you want the response here. At very low frequencies, lower yet, what happens? This thing eventually becomes the dominant impedance. It's the smallest impedance. So all the current goes down here, and these things you can neglect.

So at low enough frequency, we'll call it very low frequency, at very low frequency-- this was low frequency ief, much, much less than, let's say, the electrical frequency, which is 1.5 kilohertz. And this is f less than, much, much less-- say much, much, but much, much is five times, three times. It's just not 0.9 of it. f less than the mechanical resonance here, f mechanical.

We have this circuit. Now, you know how that circuit behaves. It has a frequency response like this, proportional to frequency.

Eventually, the frequency response goes to unity, from this source to here. Whatever you want to call this voltage or this velocity or velocity here. This is the cone velocity at low, low frequencies, u sub c.

So u sub c over-- I'm going to now-- well, I'll put a es over a bl but divided by es over bl. It goes like that. Goes to unity here, starts out at the origin. So the response of this system at low frequencies, just this system, u over bs over bl looks like this, 6 dB per octave at the low frequency, where maybe this is the mechanical resonance. Any questions on that?

This is just for the simple network first now. Again, it's very hard for you, very hard for me. You can't teach this kind of thing. You can absorb it, the way you just look at things and have them fall apart. And that's what we're trying to do. And your focus on there should not be wound up with anything more complicated than how does this simplify in different frequency ranges?

Now, you approach the resonance here. At resonance, still I don't have to worry about the inductor because this is only having an effect up here at 1 and 1/2 kilohertz, so it's still a resistor. At resonance, the ratio of this complex amplitude to this is a peak. Above resonance, this circuit changes, and this becomes the dominant, the smallest impedance. The dominant impedance in a parallel combination is the smallest one, and a series combination, of course, is the biggest one.

So the current flow, when you have a given voltage across this combination above resonance, all goes down here. The higher you get above resonance, the more that's true. So this behaves at high frequencies for f greater than f mechanical. It goes down as 6 dB per octave. And that's u cone over, again, the same old story, es over bl. And this is frequency.

So it was 6 dB per octave going up, 6 be per octave going down. So what you see is you have a bode plot which would go like this, 6 dB, 6 dB.

So your overall response then for this simple system, which had no air connected to it because the loudspeaker is sitting in a vacuum jar would look like this with a bode plot.

The actual system might, depending upon how much damping there is-- and we'll talk about that-- might look like that. It might look like that, but you have its asymptotes here. 6 dB per octave, 6 dB per octave.

Really, that's about it for the simple circuit unless we went higher yet. And when we got up to 15 kilohertz where this began to become appreciable compared to this, then the response from here to here would drop as another 6 dB per octave. Why? Because this series inductance now would start limiting the current that came into this. As you doubled the frequency, the current that went in to here now would get halved because this whole thing appears like a series inductance.

And so at that frequency, way out here somewhere, this would go down as 12 dB per octave, 12 dB per octave. And the frequency at which that would happen, this frequency here, would be about 1.5 kilohertz.

So the overall response to this then would be 6 dB up, 6 dB down, 6 dB up going to the point fm, the asymptotes, f mechanical. And this point here would be your f electrical, where it would now go down to 12 dB per octave.

So that's what your response would look like if you did not have any air load on it. Now, when you put the air on this, when you put the speaker out in the air and you have an impedance, it looks like this. At the moment, I'm just considering it to be 1/2 of the speaker cone for a reason. This seems crazy, but you'll see why. I'll talk about that when we get there.

Just 1/2, in other words, as if you had the speaker mounted in the middle of a huge wall and there somehow was a vacuum on the other side and air on this side. Other than the fact that would suck the cone right through the wall, there's no problem.

Now, what you would do is you'd say, oh my god. First, you'd think, if I put this back on here, that's going to screw up everything that happens here, and it's all a mess. And then the next thought that would come through your mind if you'd looked at models before was, well, wait a minute. Let's see how bad this thing is. Maybe this whole thing has such a high impedance that it doesn't even affect this when I connect these elements across here.

In other words, that the voltage wasn't even-- it stayed what it was, what I've drawn on the board below, the frequency response. If that were the case, then all you'd have to do is cascade the transfer function from here to here on what you got, and you'd be finished.

If connecting this didn't do anything to the voltage across here, you could say, aha, that voltage is just like a voltage source. Here's what it looks like in frequency. And now, I just cascade that with the transfer function of an RC circuit and I have my answer. That's one thing that might come into your mind.

Then you'd look at the elements to find out whether that was true or not. You'd look at what the size of this capacitor is and this resistor. Well, when you'd looked, what you would actually find out is yeah, this one is not big. He's very small. And you would find out that this capacitor might be as large as 20%, 25%, 30% of this one. So gosh, that doesn't seem to work.

Resistance is very small. You'd find out that at low frequencies-- well, in fact, we saw the mechanical-- where the heck is it? Yeah, here it is. We said that the point where these two became equal was about 500 Hertz. So below 500 Hertz, the whole thing is a capacitor. But below 500 Hertz, that capacitor, which is just a capacitor to ground, in other words, below 500 Hertz, this thing looks like this.

But this capacitor is an appreciable part of this, maybe 1/3. So you'd say, aha. The first effect of this when I put the air on there is going to be to add the resonance frequency here to make it like it was a bigger capacitor because this capacitor and this are in parallel because this is essentially 0. So what it's going to do is move this down a little bit.

Parallel resonance, 1 over the square root of lc. It's not going to move it down too much. And it increases this capacitor, let's say, by 30%. So we put that under the radical. So it moves it down somewhat. The resonance might be 100 here. It might be 90 or 85 at worst when that speaker was brought out of the bell jar into air, into this funny kind of air where there's only one side.

So other than that, for any frequencies, resonance and below, it's just two capacitors in parallel, and everything behaves like that. Now, above residents but below 500 Hertz, it's still looking about two capacitors in parallel. When you go higher than that, this resistance begins to get appreciable compared to the impedance of this capacitor.

Now, a couple of things happen there. One is that if this impedance gets big enough, of course then it doesn't affect this capacitance anymore. In other words, it's not like two capacitors in parallel when you go to high frequencies where this capacitor, it becomes very, very small compared to this resistor. This whole thing looks like an r. And it's looking just like a resistor coming across that.

So where current was going down there before, it's not now. And so you get a little lift, if you wish from that. But the main effect is that the transfer function of just this here looks like the following. It starts out having a very, very small, proportionate to frequency, and then it goes like that. And this is the f acoustical transfer function from here to here.

And that, by the way, is what we already know because if you have a cone moving, a piston moving in there, remember we calculated the j omega row 0 over 4 pi times volume velocity, the volume velocity was the thing that counted. And there was a j omega row 0 out in front, which meant that given the volume velocity, given the velocity that you have here, you get something proportional to omega, which is exactly what this transfer function tells you.

This starts out at 0, of course, because this is very big. You double the frequency. You double the current. This is the dominant impedance. You double the current down here. You double the frequency, double the current, therefore, double the voltage. And the transfer function from here to here goes up until this becomes equal to that, which is this frequency here. The actual thing looks like this. With a 3 dB difference in there.

So above, it's a resistor. You look at the resistor value, and you realize that you can almost consider-- you look at the resistor value then compared to this and compared to this, which still, we're below this point. So it's this resistor in parallel, and then you look at this, and it's not enough to take any current. You go back and forth between looking at the numbers of the values and the circuit, if you wish.

So what you find out is that first approximation is you can cascade the transfer function of this with the transfer function from here to the cone velocity. So when you try to do that, what you get-- let me just take away-- I mean, redraw this other circuit and then we'll take a look at what we get.

Now this, what I'm doing now and all these approximations are not supposed to be something that you look at and say, oh of course, of course, of course. You go back in your thinking process, and you say, oh boy, how would I ever know that I really could cascade this? It's not obvious at first. When you look at the values, it doesn't take long to come to this conclusion.

So here is what we had. Now, I'm going to redraw this in a fashion which may be more useful. We had 6 dB per octave here to the point of fm mechanical resonance. We had 6 dB per octave going down here to the point where you hit fe, electrical. Yeah. fe and then you had 12.

Now you cascade that with think this one. This is the transfer function from here to here. And what happens? That's a transfer function, which-- let's see if this is visible. Try this. That's a transfer function, which goes as 6dB per octave up until about 500 Hertz. Let's say that's on here. f acoustical.

So you cascade it with something that goes 6 dB per octave to 500 Hertz and there. So what's the end result? Well, the end result is in this region, you cascade to 6 dB per octave so you get a 12 dB. So down here, you get 12 dB per octave, so the cascade of this curve with this.

In here, up to this point here, you have 16 dB be going down, 16 dB going up. And that means you have a flat over here until you get to fa. Now, what's here? You have 6 dB going down and a flat. So this thing goes down at 6 dB until it comes here.

Over here, you have a flat with a 12 dB, so you go down at 12 dB. So you overall have 12 dB going up, a flat region, a region where you go down in 6 dB per octave, and a region where you go down at 12 dB per octave. For the transfer function, that goes from the amplifier voltage to the voltage across here. Now, I'd like to just look at that in a different way by simply taking the low frequency-- look at these two asymptotes.

Don't separate electrical, mechanical, anything else except the information that you already know. Low, very low frequency, and this fellow's out of it. Very low frequency. This is out. This is out. All the current goes down here. And parallel paths, the smallest impedance and series, the biggest impedance. Very low frequency, that's in the circuit.

So what you would have is a voltage source-- this is low frequency-- voltage source, a resistor, the inductor, the resistor. You can figure out the resistor is re over b squared and the inductor would now on the other side. So that's just the mechanical compliance. You'd have this and this.

When you get to very low frequencies, this looking out here is no load whatsoever on this inductor. The current goes down there, obviously. If you don't think so, just halve the frequency. This impedance goes up by a factor of 2. This goes down by a factor of 2. All the current goes down here.

So at very low frequencies, the voltage here, as you sweep this source, if you double the frequency, you get double the voltage here because this is the dominant thing. This is very small compared to the resistance. You double this. The current going down here is the same, but the impedance is twice as big, so the voltage here is twice as big.

In other words, an rl circuit and low enough frequency, where this is very small impedance compared to this, double this frequency, you double this impedance. Didn't make any difference to the current going down there. This was down there. And you see that there's a 6 dB rise in the voltage across here as you double frequency.

Now, this wasn't affecting anything at very low frequencies. As you go and double the frequency here, since this impedance now is dominant, this has much bigger impedance than this, double the frequency, now you double the current going down here. Double the current through the fixed resistor, you get twice the voltage.

So as you double the frequency, you have two reasons why the voltage out here goes up. One, because the voltage here doubles and two, because this transfer function doubles. And so that gives you the fact that when you're at low frequencies, you are going down at 12 dB per octave.

Now, at high frequencies, look at that whole complicated thing. What happens? At very high frequencies where this is in the circuit, this is dominating, this impedance, you double the frequency, you get 1/2 the current through here. 1/2 the current through here, this is the dominant parallel impedance. You've got 1/2 the current through it already when you double the frequency, and it's impedance dropped by a factor of 2. So you get 1/4 of the voltage across.

In other words, let's take a look. Let's make the circuit. At very high frequencies, it looks like this.

This is the load out here. This is now gone to 0. The dominant impedance is this one. I can throw away this. Now, you might say, well-- and this is a short circuit at high frequencies compared to this. You can say, well, why don't I throw out what's left? Why don't I throw out this thing?

Well, I could, but I'd throw out also the problem of which I want to calculate across there. So I can take that resistor and put it in parallel with this if one likes to do that for the total impedance, but what happens is the behavior at high frequencies of this circuit, I'm well above the resonance of this thing. The behavior at high frequencies is very simple to see what happens in that circuit. The inductance doubles, you double the frequency.

And that's dominant over the capacitance. So the current goes-- if I double the frequency, the current going in here is 1/2, approximately. Now, the impedance of the capacitor when I double the frequency is 1/2. So the voltage developed across here is 1/4, and that means 12 dB per octave.

So you know right away by looking at the thing like this that at the high end, it's got to go down at 12 dB per octave. As we've seen at the low end, it's got to go down at 12 dB per octave. Can you ask-- just stop here for a second can. You think of something that was least obvious that gives you any problem?

And literally what we're doing would have been done with you on a project that you worked with in industry by your manager coming around every once in a while when you had a question, and you've been thinking about this. And he would give you some input, and you'd put all these pieces together. Yes.

Why can you cascade them?

Why can I cascade? Yeah. In other words, this fellow here. Yeah. And the question is, why can I cascade them? And I said that I really can't cascade them all the time. In other words, we saw that the resonance got moved down.

Why? Because at this mechanical resonance frequency, which is way down here, compared to the acoustical one, which is at 500 Hertz, I had to take into consideration that this impedance was very small compared to this at 100 Hertz at the mechanical resonance. And so this was really like a capacitor to ground. And so as far as moving the fundamental resonance down, I didn't cascade them. I took that into consideration.

Then when I was well above the mechanical resonance, I said, aha. If I look at the values now, when I'm well above, this part here, looking out here really isn't loading this impedance, which is the dominant one. See, this capacitor no longer plays a role when I'm up at higher frequencies because he is small compared to this.

So all looking out here, it's a resistor. And that resistor does not dominate. It's in parallel with this capacitor. The capacitor has a very low impedance, takes all the current. And so this more or less looks like a resistor on here, which isn't loading the circuit. So at higher frequencies, I can cascade it. I can think of cascading it. At the lower frequencies, I got to take into consideration.

As it turns out, in a model like this, if you forgot that and you cascaded-- forgot the low frequency part of this, this being, say, 1/3 of that, you'd be off by a little tiny bit in your resonance frequency, but otherwise you would be quite accurate. Does that help?

Now we have this orange curve. What is of real interest now is what happens when you change values. Oh, I should tell you something else. This acoustical, very interesting, this point here where the acoustical one came up and went like that, the rc.

We saw last time that the voltage across here is proportional to the far field pressure. That's true on axis, but at the very point where the 1/2 power frequency occurs, that occurs for a reason in radiation impedance. Namely, above that, radiation impedance is looking real. I mean, going to a real number, 1 over whatever it is row 0c with the a factor in it.

Remember, that when the radiation impedance is real, it's as if that loudspeaker were being driven into a tube of infinite length. All the waves are going straight, quite [INAUDIBLE] So the speaker is becoming directional. Yeah, the microphone down the end, if you put it over there is reading everything we said it would read, but the speaker itself is now becoming very directional.

And that's important because in all the textbooks that you see, at least the ones I've seen, they tell you to take a microphone and-- oh. I've just told you something wrong. Back up for a second, then I'll tell you what the textbooks say.

We said that this voltage was proportional to the pressure in the far field because power that's dissipated from here to here is proportional to the square of the voltage, last time we said. And power that's radiated outside is proportional to the square of pressure. And that's how we got the relationship between the voltage across here and the pressure.

Well, it turns out that that's fine when you're at low frequencies and the thing is radiating like a sphere, but when you get to the high frequencies now, the pressure out there on the axis is no longer proportional to the power radiated because the power, instead of going out as a sphere, is beginning to go like a flashlight instead of a light bulb.

And so what happens is as you go to higher and higher frequencies for a given power radiated, the pressure on axis goes up, so right at this point here, you don't normally see this thing going down because you're measuring on an axis. And you see some stuff carrying on out here, and then it goes down at 12 dB per octave. It follows what this one does.

And all of this is because you are now on axis of that loudspeaker, and the pressure on axis is increasing relative to the total power that the speaker is radiating. And the total power the speaker is radiating is the power that's going into this thing. Yeah. Now, all the textbooks that I've seen anyway, tell you the way to design a loudspeaker is you go into an anechoic chamber so there are no reflections, you go six feet on axis, and you put a microphone.

Well, just think for a moment about what this means in a real room with a wall. You can use mirror images, all right, because if you had an infinite wall here, the sound field that I'm going to get from any object sitting in here can be determined if I can just meet the boundary condition of this wall. Just suppose I had an infinite wall, and I have something here. We'll call this something a loudspeaker.

All I have to do is meet the boundary condition. And the only boundary condition that this wall imposes is that there's no normal velocity. So I can meet that boundary condition very easily without the wall there in a way that I can do nicely, namely I put a loudspeaker here, which goes that way when this one goes this way, feed them in phase, so to speak. And by symmetry, there is no normal velocity across here. They're equal and opposite.

So if I put a loudspeaker behind that wall, doing exactly that, I could erase the wall. This is what I do, for example, in my own home. I have a contemporary house, in which one area flows into another. And I have these 901s, which, as you know, most of the power is going the wrong way, out of the back. And so the power here is four units. Power here is one.

So all you would do is you put a double the spacing that you would put from a wall. Maybe they're supposed to be spaced 18 inches or 12 to 18 inches from a wall. So in space, you'd just hang two of them out in the yard. And if you're in this space, it'll be exactly the same sound as if there was a wall there just by mirror images.

If you're in this space, it's exactly the same sound as if there were a wall here also. So you could hang something like this using mirror images right in the middle of the room, and down here would sound like there was a wall across the room.

So let's go back over here. When you do this, what you find out is very interesting, that this speaker by itself, I can use superposition. This speaker by itself would radiate at low frequencies like that. The other one would do the same thing. So I'd get double the pressure from the low frequencies. In other words, I turn this one off, turn this one on. I get a nice spherical radiation at low frequencies.

And I get exactly the same thing there. I double the pressure, which means twice the power radiated. It's interesting. Sorry. Four times the power radiated. That looks a little funny, doesn't it? Each one would radiate w watts. Let's assume their volume velocity sources. Each one radiates w watts. So that gives me a sound pressure here of p, one of them by itself.

Turn the other one off. Turn this one off. Turn that one on, 2p. Well, p again, and they add up and phase, so it gives me 2p. Twice the pressure is four times the power.

Speakers will find it more difficult to drive the same volume velocity because they feel the pressure at the other end.

That is amazing, and it came from a mathematician, not from an engineer. That is amazing. It's exactly what happens. In other words, when you bring these two things together, instead of radiating one watt each, they radiate two watts each, amazingly enough.

And the reason for that is exactly what you said, namely when you turn this one on, there's a pressure on this diaphragm. And so p now is working into a higher pressure with the same velocity. I said to assume their velocity sources.

And by the way, that's not a bad assumption when you look through all this stuff here that if you change this, which is really what's changing the radiation impedance when you turn on the other thing, this doesn't really affect what goes on too much over here, the cascade thing I told you about.

And so lo and behold, you put them there and you get twice the pressure, which is four times the radiation. Each one, which was-- you hook this one to an amplifier. You hook this one to an amplifier. You bring them together. All of a sudden, instead of radiating one watt each, they radiate two watts each.

Well, how about the high frequencies? At high frequencies, this fellow was going like this. He was radiating like that. The mirror image radiates like that. You never hear him when you're on this side. So all of a sudden, when you put just one wall behind the loudspeaker, the whole frequency response that you sense is screwed up from what it was in the anechoic chamber, and there aren't many living rooms that are anechoic.

So the whole criteria that you see in books all to this day is for the birds. It's basically wrong. And by the way, most places have floors. And the same thing happens again. Put the mirror image for the floor and you have 12 dB. And then god forbid you should place the thing in the corner. There's another mirror image that comes from that, 18 dB error in your frequency response, and a couple dB over an octave is absolutely noticeable. And here, you're talking about 18 dB.

So if you don't realize these things you can buy the most expensive product in the world and come home and put it in a corner, it'll sound like a jukebox or put it in the middle of the room, and it'll sound like it has no low end at all, depending on how it was designed.

So it tells you right off the bat that there is no such thing as that's commonly advertised, a bookshelf speaker that you can use on the floor or a bookshelf. You can have one of these things and you can put it this way on the bookshelf. Oh my god, you can do it just be prepared for what you're going to get, back end sound.

So the surfaces that are very close to a transducer that's launching sound into the air are terribly important for this. And amazingly enough, somehow, when things all started, people said, measure things six feet on axis.

I told you about the Englishman who-- I don't know if he's still alive, but he was a reviewer, and he literally had a poll out in his backyard because he couldn't afford in an anechoic chamber. And he had a 30 foot pole, and he had a microphone six feet on axis and he used to measure all these things and report on their fidelity from those measurements. Unbelievable.

And imagine what happened to him when he put this thing on a pole and the microphone was at this end. There was no high frequencies at all, just one unit of power. And all the rest of it went off to his backyard somewhere. So he thought this was the most crazy design that could ever be conceived of.

Now, you're a designer, you want to know what happens, for example, if you increase the mass of the loudspeaker. You're going to take-- wait a minute. I have it down here. I can use this one better. You increase the mass, so this capacitor increases.

Well, what's going to happen? First thing that's going to happen is you're going to move the resonance frequency down. So this curve is going to get moved like this. Second thing that's going to happen is when you're in this region here, where you were going 60 beats down, your real circuit that was dominating it was a resistor here and this capacitor. Increase the capacitor, you decrease the voltage.

So here's what happens. Let's see. I haven't used yellow on there. So the thing goes like this. In this region it goes down. And in the other regions where the capacitor-- this capacitor becomes a dominant one all the way along, you go like this. So the actual output of your loudspeaker, if you double the mass of the cone, would go down by 6 dB.

Now, this actually has happened in industry. This is really the case. About the late '70s, one of the major manufacturers came out with a little loudspeaker like this, and it had tremendous bass. And what they did was put a huge mass on the cone. And so they moved this thing down here so the curve looked like this. Boy, I mean, it really was putting out down there. And the rest of the curve looked like this.

And what happened was people were astounded, oh my god, all this bass. Well, in the stores, they were demonstrating this thing with 100 watt amplifier on very short pieces of music. Now, when you buy a little speaker like that, you don't expect to buy 100 watt amplifier. People would go home, they'd bring the thing back and, oh my god, I can't hear it because that extra bass killed you in the mid-range with no output.

And if you put an amplifier on enough to get that back up, you got smoke instead of sound because you needed to put so much power on it, the voice coil heated up, and it was gone. In fact, I saw one in a store one time in the early days. And it was really funny. I saw this whole cone all burnt up. And the dealer said to me-- he was in the back room where service was-- and I looked at it. I said, my god, what happened there?

He said, well, what happened, clearly it burned up but, he said, that's not my problem. He said, when it burned, the fellow who owned it went over to blow it out and he had a beard and he caught fire. And he says, I have a big problem. So all that was because somebody wanted to make a loudspeaker that sounded like it had a lot of base. And so the bass here is relative to this, and it's very high when you do something like that, when you extend it down here, but you lose power.

So in these engineering problems, there are a lot of variables that you have to worry about and think about it at the same time. Now, a really interesting one that has happened, I would say, to most homes who bought loudspeakers, many loudspeakers, conventional ones prior to maybe 10 years ago. And they still have them because they last a long time.

And what they have in their home is something very strange. Namely, here's the response. And they think it has something to do with progress, new speakers versus old. It doesn't have anything to do with this now. If here's the response of your loudspeaker-- I'll just draw it like this now-- and you decrease the b, the flux density, look what happens.

You decrease the b, you have the b and the voice conductance goes up big time as 1 over the square of that. And the resistance goes up. Now, at the fundamental resonance, the resistance is the dominant thing. It turns out, by the way, I should say that, that the damping on this resonance, the reason why you don't have a big peak here is mainly coming from the voice coil resistance. It's not coming from this thing here.

In other words, when you look back here at resonance, this is a short circuit. You have this resistance across here. And that's what the damping of the RLC comes from. Now you made that thing suddenly much bigger, so the first thing that happens to you and the response is you get a big peak right here, the fundamental resonance.

Below resonance now, you have a much bigger resistor feeding into an inductor, same size inductor, but this fellow got very large, so you dropped the response. If you have the b, you drop it. You cause this fellow to go up four times and four times is 12 dB, so you come down like a brick here.

And then when you get in the mid bend, where the capacitor, the mass of the cone is working against this resistance, you've got 12 dB down there. This goes like that. 12 dB here. And always that series resistance now is dominant. If it were only the series resistance, you would go like this, but the inductance 1/2 power point, the inductance now goes up.

But since the 1/2 power point of this is the ratio of an r to an l, that stays at the same point. So this is actually what happens to your speaker. So this really sounds like-- if you have any kind tympani going with couple of drums, and they should go, bum, bum, bum, it goes, bum, bum, bum. It resonates right at this point. And that's what you hear. You hear the ringing from that.

Now, how does this happen? Well, it happens to people. Today, it won't happen as much, but thank god they don't make speakers like this. This happened with Alnico magnets, the kind of magnet that the little speaker I showed you had. And I can tell you a very true story that caused a number of MIT folks that are out the company to go around in circles for a couple of weeks.

The first loudspeaker that we produced, regular woofer was a 501. It was in the early '70s. And we really work to get this design very well and we positioned. It was the first speaker that was this size that we said, only use on the floor. Don't use on the wall or anything else. And we put a base on it so it would look ugly as sin if you tipped it around.

And so we were very proud of this thing. We knew the people-- we told them where to put it. We told them to keep it a couple of feet out of the corner. And we knew it would sound good. Well, it turned out that we sent a number of them out-- thank god-- before production to different homes to try. And they all came back saying, oh my god, this thing is terrible. It is it sounds like you would know this would sound like.

And we measured them quickly. Oh my god. It measured like that. And we went back, checked the whole assembly line. Everything was perfect. They were going down the assembly line. They were measured. We checked them in the stock room, and they were all like the pink curve, the red one. So what in the heck happened between the assembly line test and the stock room?

Turned out there was a test. After the grille cloth was all put on, there was a buzz test. And the buzz test, we use one of our 400 watt amplifiers and really socked it to see if anything rattled. Well, the socking of the thing with that big signal ran the iron around the hysteresis loop and caused-- we haven't gotten the magnetic circuits yet-- but caused the flux.

If you run around this hysteresis loop in ways that we'll talk about, you can wind up with a flux level that is terribly low compared to the level that you would have had if you hadn't done that. And so immediately, we-- thank god-- hadn't put it on the market, and we changed to a ceramic magnet, which doesn't have that problem for reasons that we will see.

Now, the ones that are out there with those kind of magnets, in the days of the phonograph, all you needed to do was drop the stylus on the record once and get a big bloop and your flux of your magnet went down, and you started to progress each time that happened towards this kind of a response.

Or today, if you leave your power amplifier on and you pull the plug to something, let's say you're put in a CD player and it goes, burp when you put it in, if you've got an Alnico magnet, that's a one-way trip. You've just gone towards the pink curve.

So it's really important to understand what it is that gives you what response because by looking at that response, immediately you can go back and say, oh, I can tell that kind of a deviation is caused by changing such and such.

Another interesting thing, people think wow, the most important part of a loudspeaker-- sorry, the most expensive part is the b field, the magnet. So I know what we'll do. This is b times l here. So what we will do is we will make a small b-- that reduces cost-- and we'll just wind a bigger coil.

Well, if you wind a bigger coil, think about what happens. Put more wire in the gap. First of all, you have to have a bigger gap, which causes your b field from the same magnet to be less, but never mind even that problem. You have this problem here, but the inductance of the coil goes way up. And so the inductance of the coil is what determined this electrical resonance point, fe, so that moves down. So your high frequency end disappears.

So you can see that from the one model that we have here, you can just by thinking and staring at it, see what happens when the voice coil induction changes, see what happens when the b changes, see what happens when re, if you want to change that mass, you want to change that, any of the parameters, you can just look at them and exactly what's going to happen without any calculations to speak of. Now, if you see, can you ask any questions about this?

[INAUDIBLE]

After the people start thinking about this, what might have even seemed as clear won't. Now, we want to take this strange speaker that we analyzed that had one side in air and the other side didn't exist of the cone and make it exist.

Oh by the way, for low-frequency speakers, for these bass speakers, one of the things that is done very commonly is you increase this mass to bring that resonance down. That is very common. There are two things that you do. You bring that-- I can't talk about the second one until-- let me go on and talk about something else I need first.

Here's the loudspeaker. I want to put it in the box. Why do I want to put it in the box? Because if I don't put it in the box, at low frequencies, the cone is doing this. Guess what's happening? Nothing gets radiated. The air just goes around to the back, if you wish. In other words, this fellow is radiating the negative volume velocity of this one. It's like a dipole.

So you want to stop the negative radiation. In other words, when the cone moves this way, you're moving volume velocity, but you're taking that same volume velocity in from the back side. So it turns out, you roll off much faster and at low frequencies. So that's what the box is about in the loudspeaker, to stop the back wave from interfering with the front wave.

And now, if you want to put a box in there, you have a slight other problem, namely the box is an air spring. If you try to push this thing in here, you compress air and whatnot. And so it wants to come here. So where does that box go in this circuit? Let's say this has a mechanical compliance of the box. Anybody know how do I alter this circuit to take care of that? Yeah.

Put a pole.

Put a pole? Now, this is very specific element. See, mechanical element. I'd like to know, this is a compliance, a lumped parameter compliance, which, by the way, if you want to find out what goes through your mind at this point in the subject is, oh my god, now. Let's see. I have a compliance. Is that a mass or is that a capacitor? That's what went through your heads right now.

Just go over and little scratch paper, and write the following thing down, which you don't really have to work at to remember. Force is mass times acceleration. In complex amplitudes, force is j omega mass times velocity. So if this is across variable, when you go to electrical, that's voltage. Voltage is j omega times current. So that is an inductor. The mass goes to an inductor if force goes to voltage and current.

If it's the other way around, if velocity goes to voltage, then you have voltage is equal to 1 over j omega times the current and times 1 over m. So that goes to a capacitor. So to get all of that straight, you only need this much. So if force is analogous to voltage, then the mass goes to inductor. If force is analogous to-- if you choose force analogous to current, which is what we have over here, then the mass goes to a capacitor.

So this is an inductor. This is a compliance, becomes in this, an inductor. And the inductor goes where in this circuit?

Well, figure out where both ends of it are. You're working into this box. The box is fixed to the inertial system. The other end of the box is at the cone. I mean, the other end of the compliance is the cone moving in and out. Now where does it go?

[INAUDIBLE]

Yeah. Compliance of the box. Now, very easy because this thing across here is used up cone. The velocity across there and this is force, force because bli.

Now, here you can see right off the bat why in history, speakers with more bass have always been, by the conventional design, bigger, much larger because these two make one inductor, of course, in parallel, but if you decrease that inductor, make it smaller, what happens if you decrease it? You'd move the resonance frequency up like this, the fundamental resonance, and you'd get a bass response, which instead of the orange one became the yellow.

If you increase that inductor, you lower the resonance frequency, and you make the response like that. And so what they normally do is try to make the suspension as compliant as you can, big inductor for the suspension, mechanical stuff. Very soft, but that's dangerous because the softer you make the suspension, the more it's likely to wobble, and then you have big problems if the voice coil scrapes. So there's a limit, but you can do pretty well.

And then they make the box as big as possible to get that net inductor very big so that you move the response in this way. Conventional designs do exactly that. So the purpose of the box is to get the back wave out of the way of the front one, if you want to say it that way, but then you don't like the spring that results because that screws up the low end.

And so you just make that thing bigger and bigger and bigger. If you had a huge, huge box, this wouldn't take much current down there compared to this, and you would be OK. Then there's another kind of a box-- and I won't be able to talk about it, but I will next time-- called a [? ported ?] box. Loudspeaker like that, a tube sticking in here.

And at first, that looks terrible. You'd say, oh my god. I pushed the cone in this way, and the air comes out here. And they look like they're fighting each other. Well, it turns out that this is very useful. If you don't know what you're doing, it has some big disadvantages in the way that it's always-- again, here's another case.

Just shows when you read textbooks, be very careful. Don't assume, like most of your high school and most of your training has been, grade school that everything is in the book is right. The way that these boxes are normally designed can give you as many disadvantages as advantages. If you see-- and we'll go into that-- how to design them, they can give you advantages and no disadvantages, but that's not present in any of the text in the field yet.

So again, all of this is going to fall out of just simple model. And the only reason to look at this model is to-- a person one in one class asked me a very good question one year. He said-- we went all through this. We got this model. And in absolute sincerity he said, well, why did you build this model? Why did we go through all the trouble? We had all the equations. We could get an answer.

And having all the equations and cranking out an answer doesn't tell you beans about design unless you are willing to do perturbation of all parameters and look at a huge stack of graphs until you're blue in the face and hope that you see trends. So that's the reason to make models and analyze them. OK.