Heres a good article about Hoffmans Iron Law
http://www.salksound.com/wp/?p=56
I've read that and many other articles on Hoffmans Iron Law since the early 90's when I was getting into external subwoofers because I couldn't afford speakers that produced bass. Then I can upon the Sunfire stuff and have been convinced that Hoffmans Iron law was no longer relevant.
This is the part of Sunfires (Bob Carver) white paper that claims Hoffmans Law is a myth. I have a actual copy with the hand drawings and all stored away that I got with my Sunfire True Sub Sig. back in '98
a loudspeaker designer can choose two of three variables arbitrarily and at will, but will be stuck with the third.
We can choose: 1) Low frequency limit 2) Efficiency ...... and we're stuck with: 3) Box size
Or we may choose: 1) Box size 2) Efficiency
...... stuck with: 3) Low frequency limit
Or choose: 1) Low frequency limit 2) Box size
...... and we,re stuck with: 3) Efficiency
In short, Hoffman's Iron Law tells us it's not possible to have all three at once.
With the new methods and technology developed for the Sunfire True Subwoofer, We can choose all three at will! And guess what? I chose small size, lots of low bass, and high efficiency. And in the end, Hoffman's Iron Law falls (actually it doesn't fall, but it must be interpreted in a way different from the classical interpretation). A simple thought experiment clearly illustrates that it's theoretically possible to build a much more efficient woofer. The thought experiment is this: Imagine the voice coil being made of a super conductive wire that has zero resistance. Since we know that most of the input power is lost in the voice coil as joule heat, such a voice coil would not generate heat because heat, in this case, is current squared times resistance. If the resistance is zero the heat is zero. If the heat is zero that means instead of one, two or three percent of the input power becoming useful, the speaker suddenly becomes majestically efficient. A super conductive voice coil is clearly not practical, but the thought experiment itself tells us and teaches us it is possible, at least in theory, that fundamental, intrinsic laws of the universe do not prevent us from making a very small, very efficient woofer.
The secret to making this woofer efficient in spite of its small size and high output is to operate its electric motor, that is, its voice coil and magnetic system, far away from stall. Electric motors, when operated away from stall become very, very efficient. An electric motor operating far from stall generates a very large back electromotive force (emf). That is, most of the voltage that's delivered to the motor is subtracted away by a voltage generated inside the motor by virtue of its own motion; that voltage is called the back emf.
Getting back to woofers, all woofers (except the Sunfire) on the face of this planet, operate very close to stall. A scientist would say close to the stall mode. The stall mode is characterized by very little output power and lots of current running through the coils of the motor making the motor very hot. If the motor is turning fast the back emf is large, and the motor runs cool; the remaining voltage that pushes current through the windings is very small because it's counteracted by the back emf, and the motor becomes extremely efficient. As an example, diesel electric locomotives have a minimum operating speed below which the motors come too close to the stall mode and burn up. For passenger trains it's about 12 mph, for freight trains about 9 mph.
A TRAIN AND A MOUNTAIN
Imagine a train going up a mountain, pulling hard. As it goes up the mountain fast, the train is putting out lots of power because the energy of going up the mountain is delivered in a short time. Since power is work per time, a scientist would say Power = mgh/t short, where m is the mass of the train, g is gravity, h is the height of the mountain, and t short is the short time it takes to get up the mountain. The power output is large, and the electric motors run cool. If the train goes very slow it will not be putting out very much power because Power = mgh/t long, where t long is the long time it takes to go up the mountain. When t is small, power output is high, when t is large, power output is low. What's worse is the motors get very hot and overheat. Its motors are operating in stall mode. Now back to the woofer. Because this woofer moves back and forth a lot, the voice coil is cutting many, many lines of magnetic flux in the magnet structure. It's the rate of flux change that generates back emf. The large back and forth stroke of this woofer is good for making lots of air move but it's even better for generating a large back emf. If the voice coil didn't move very far, it wouldn't generate very much back emf. Without a big back emf too much current would flow through the windings and the woofer would overheat. That's the case with all other woofers on the face of the earth. To have a large back emf, lots of voice coil motion is required. Another requirement for lots of back emf is a very high magnetic flux. To have that requires a huge magnet. The magnet in this woofer is 225 ounces and that compares to around 20-28 ounces for a regular woofer. It's about an order of magnitude larger. Now the problem with all this is that because the woofer is moving so far, it makes a large back emf (due to the large stroke cutting many of lines of force). Consequently, if the woofer were driven by an ordinary amplifier, most of the driving force would go away and the woofer would have very little output. Because the back emf is so large, the applied electromotive force to the woofer must be even greater than the back emf in order to overcome it.
That requires an incredibly powerful amplifier. An amplifier that can swing over 100 volts rms.
Since the DC resistance of the voice coil is 3.3 ohms - that means it's a nominal 4 ohm woofer, 100 volts rms into 4 ohms is well over 2,000 watts. The woofer will not burn up because when it's moving, the back emf prevents the damaging stall mode current that would normally flow if it were a resistance load. Only a small fraction of that current flows in the voice coil, but since the magnet is so huge and because the driving force is equal to the magnetic field strength times the current, the force on the voice coil to drive the woofer and move the air is immense even though not much current is flowing. An alternate way to get drive force on the woofer cone is to increase the current, the normal way of doing it, but that makes the voice coil overheat because of the high current. Or we can make the magnet huge. Problem is if we make the magnet huge, the back emf causes us to run out of volts and therefore we need a very special amplifier to overcome the loss of volts. That requires a tracking down converter amplifier. The tracking down converter amplifier is also unique. Without the tracking down converter amplifier, no matter what, this woofer wouldn't be practical. When all is said and done, the effective input power is not 2,700 watts, even though the amplifier has to be capable of delivering 2,700 watts into a 4 ohm load. When it's actually operating into the loudspeaker, the effective power is far, far less than that. And Hoffman's Iron Law suddenly tumbles. To put numbers to this, as any good scientist would, even though this woofer is one tenth the box volume of a normal woofer, when driven to full output at the same volume level and same low frequency as a giant woofer in a big cabinet, by all the laws of the universe and all the rules that woofer engineers and designers have been taught, the input power should be at least 10 times greater than the input power for the big woofer. But that's not the case. The input power is more, but only slightly more.
A THOUGHT EXPERIMENT Suppose we wanted to build a woofer, and we had a 200 watt amp to do it with, an amp that could deliver 28 volts rms to our 4 ohm speaker.
Now imagine we mount our driver into a nice box. Imagine also that our driver has a "variable magnet", so we have a dial outside of the nice box that we can turn to change the strength of the magnet. The dial is calibrated continuously from zero magnetic strength to super magnetic strength. We drive the system with some bass. Of course, with the dial set at zero, there is no output from the woofer. We begin advancing the dial and the bass comes up. As we advance the dial further, the bass gets louder and louder. However, as we advance the dial beyond a certain magnetic field strength, the woofer output begins to drop. The point of maximum output defines the optimum magnet size for our woofer. Consider Figure 1.
[Image]
DISCUSSION OF THOUGHT EXPERIMENT
We know that as we advance our magnet from zero, the product of the current and magnetic field strength (B) produces a force on the voice coil. As B increases, the force increases and the bass increases. However, no sooner than the woofer starts to move, it produces a back emf that begins to reduce the current through its voice coil. As the magnet becomes ever larger, the current becomes ever smaller, until by and by the ever increasing magnetic force cannot overcome the loss of current due to the increasing back emf. Consider Figure 1.
In this illustration, the back emf is 13 volts, leaving only (28-13) = 15 volts to run the woofer.
If we wish to have more bass output we must have a larger amplifier, one that can deliver more volts in order to overcome the dreaded back emf. Now if we had in our possession an amplifier that had virtually unlimited output voltage, say one that could deliver 104 volts rms into 4 ohms with ease, then we could increase the magnet strength and let the dreaded back emf rise because we would know that we have almost unlimited drive to overcome it. See Figure 2.
[Image]
THE PROOF IS IN THE PUDDING
The definitive demonstration.
I have several large and well designed subwoofers here in my laboratory. Klipsch, Bose, Velodyne, M & K, and several others. One of my favorites is a big one.
I hooked up the Big one (cabinet volume 4.3 cubic feet) and drove it to maximum output (its limiter activated) with a warble tone centered at 28 Hz. I measured the power input from the wall socket with a watt meter and measured the output Sound Pressure Level (SPL) with a simple SPL meter. Input power was 240 watts, output SPL was 112 dB. It was awesome!
I then removed the Big one and replaced it with my Sunfire True Subwoofer and drove it with the same signal until my SPL meter again read 112 dB. Input power was actually 40 watts less at 200 watts! Since my limiter had not yet activated, I turned it up until I activated the limiter with maximum output at 115 dB. Double the output of the Big one. Wow! Input power was then 360 watts.
SUMMARY AND CONCLUSIONS
Big one woofer operating in stall mode. True Subwoofer operating far from stall mode.
240 watts input, standard amplifier approximately 200 watts input, tracking down converter 60% efficient at maximum output means (240) X amplifier approximately 88% efficient (.6) = 144 watts into voice coil. means (200) X (.88) = 176 watts into voice coil.
Big one has 4.3 ft3 box volume. Sunfire has 0.38 ft3 box volume.
However, input power for Sunfire should be (by Thiel-Small and Hoffman's Iron Law): (4.3/0.38) x 144 = 1,629 watts
But it's not! It's only 176 watts. An order of magnitude less! Magic, uh? Indeed, at full output, the True Subwoofer requires 360 watts (115 dB) but it should need 2 (twice the output)X 1,629 = 3,258 watts, but it doesn't!!
So that's the secret. The conceptual secret of how this woofer can be so small, put out so much bass and yet not burn itself up and also not draw tons of power from the wall. As a matter of fact, in spite of all the bass it can put out, it uses only a six amp slow-blo fuse. Putting it all together, this woofer is flat to 18 Hz, can move 251 cubic inches of air (again, the same as four 15 inch woofers operating linearly in a box the size of a small refrigerator), the distortion is extremely low, it has a built in power amplifier, a built in crossover system, input volume controls, crossover control, a phase control to allow it to be blended with any other speaker system and it has a two position switch that allows the woofer to be optimized for stereo (flat to 18 Hz) or to have a contour that rolls off very low frequencies if they bother you.