Loudspeaker Technology Part 12: Speaker Efficiency
Speakers come in many shapes and sizes with sonic quality and price equally variable. These Opulence speakers list at one million dollars.
In the 12th chapter of his treatise on professional loudspeakers, John Watkinson examines loudspeaker efficiency and the sometimes required trade offs.
The efficiency of a drive unit is the ratio of the acoustic power produced to the electrical power consumed. Efficiency in loudspeakers is not very high, and there are a number of factors at work. One is the physics; the air mass affected is a tiny fraction of the moving mass of the speaker. It’s like delivering feathers, the truck will weigh more than the payload. The other factor is the way drive units are marketed, which is by their power handling ability, where more is deemed to be better. If it was the acoustic power generated, then more would be better, but it’s not. Instead it’s the electrical power wasted as heat. A new unit of power has also been invented to make the numbers twice as big. Peak music power results in even bigger numbers.
Drive units sold separately for use in passive speakers have no need to be efficient as long as they are cheap and have a high power rating. Things are different for the designer of active speakers, because an inefficient drive unit means a bigger amplifier and power supply and possibly a bigger heat sink as well, all of which increases the cost. So in the case of an active speaker, it is generally worth paying a little more for an efficient drive unit because a greater amount will be saved elsewhere.
The expression for speaker efficiency.
The expression for efficiency is shown in the accompanying formula.
Where B is the field strength, l is the length of wire in the gap, S is the diaphragm area, ρ (rho) is the air density, R is the coil resistance, M is the moving mass and c is the speed of sound. With a small number like air density in the numerator and a big number like the speed of sound in the denominator, the efficiency is not going to be very good, as we saw earlier. At first sight, for high efficiency, we want a strong magnet, lots of coil turns, a large diaphragm, but low coil resistance and low moving mass. The problem is that increasing the diaphragm area, lengthening the coil and reducing resistance all increases the mass, so we can easily end up right back where we started if our “improvements” are unwise.
In the same way that the performance of an airplane is limited by the characteristics of the engine, a loudspeaker is only as good as the motor that drives the diaphragm, so it is worthwhile considering what makes a great motor. It is widely understood that the force driving the diaphragm comes from the current in the coil that crosses a magnetic field, so the motor will largely be defined by the coil and the magnet. The force is proportional to BlI, the product of the current I, the field strength B and the length of the coil wire l that is immersed in the field. The current, at least at low frequencies, follows from Ohm’s Law. This has some interesting consequences.
Figure 1 below shows the cross section through two coils, both of which use square wire because it simplifies this example. The coil on the left has wire that is twice as thick and has four times the cross sectional area as the coil on the right, which has space for four times as many turns in the same volume.
Figure 1 At a) the coil consists of five turns of square wire. At b), the coil has 20 turns, but the cross section of the wire is only one fourth of the wire in a) so the resistance will be 16 times higher. However, the efficiency of both coils is exactly the same. Only the impedance is determined by the details of the coil.
Which coil design is better? Well, the one on the left has one sixteenth the resistance of the one on the right because the cross section is four times as great and the length is four times shorter. For example, if the coil on the left has a resistance of 4 Ohms, the one on the right has a resistance of 64 Ohms. Let us dissipate the same power in both coils and see what happens. Because power goes as V2/R, if I apply 2V to the coil on the left a current of 0.5A will flow and if I apply 8V to the coil on the right, a current of 0.125A will flow and there will be one Watt dissipated in both cases.
The current on the left is four times greater for the same power, but the coil in the one on the right is four times longer, which is four times better. The BlI product is the same in both cases because the two effects cancel completely, with the result that it is completely irrelevant from an efficiency standpoint whether the gap is filled with a few thick turns or many thin turns, as long as it is filled. The only difference between the two examples in Figure 1 is they have different impedances. The impedance is inversely proportional to the fourth power of the wire diameter. It should be clear that if round wire is used, the gap will not be filled because there will be voids left between the turns, hence the preference for square, or rectangular, wire.
As the drive unit is mass controlled, the acceleration the coil can produce is inversely proportional to the moving mass, which comprises the coil, the diaphragm and the associated air mass. The most efficient material from which to wind the coil is one which has the best ratio of conductivity to density. Although copper has good conductivity, it is dense and although the conductivity of aluminum is not so good, its density is less by a greater factor. One point that is often overlooked is that an aluminum coil having the same resistance and overall length as the copper coil it replaces will have a slightly larger volume and so will need a magnet of increased volume if the potential efficiency improvement is fully to be realised.
Speaker magnets
Turning now to magnets, there are some parallels between electricity and magnetism. Magnetism must flow in a complete circuit starting at one face of the magnet itself and returning to the opposite face. The total flux is the same everywhere, but the flux density can vary by channeling the flux into smaller or larger cross-sectional areas. The flux is caused by the magneto-motive force or MMF, which is analogous to EMF, and the flux is impeded by reluctance, which is the analogous to resistance. Thus it is possible to draw an equivalent circuit of a magnet system. Figure 2 below shows a typical legacy magnetic circuit and its electrical equivalent.
Figure 2. A magnetic circuit can be modeled as an electrical circuit, with reluctance replaced by resistance and EMF replaced by Magneto-Motive Force from the magnet. Unlike electrical systems, there are no magnetic insulators, so leakage is a fact of life in the legacy ferrite magnet design shown here. Click to enlarge.
The reluctance of the magnetic circuit is dominated by the gap in which the coil moves and is proportional to the width of the gap. Wider gaps need more MMF to push the flux across, and the MMF from a given magnet is proportional to the length. For a given flux density, the total flux required is proportional to the area of the gap, measured at right angles to the radial flux, which in turn is proportional to the cross-sectional area of the magnet.
In other words the volume of the magnet needs to be proportional to the volume of the gap. In practice there will also be some leakage, which is flux that doesn’t go through the gap. Leakage requires the magnet to be bigger to get the required flux in the gap.
Many years ago, amplifiers were limited in power, magnets were expensive and limited in performance. The high strength-to-weight materials available today were either unknown or unaffordable. The problem of drive unit design was to get the most sound from the limited input power using magnets of the least magnetic volume, despite relatively heavy diaphragm materials. Those constraints led to a certain philosophy of drive unit, and indeed loudspeaker, design relying on acoustic augmentation of the limited output using resonance.The whole loudspeaker would have to operate from a single general purpose audio amplifier.
Devil is in the details
There is no magic bullet to produce an efficient loudspeaker. Probably the first mathematical analysis was that of Leo Beranek* in the 1950s. Beranek computed the moving mass by taking into account the volume of the coil and its density. He found that for maximum efficiency in a given gap volume, the mass of the voice coil should be one half of the overall moving mass, comprising the coil, the diaphragm and the air load.
However, as with all calculations, it is important to consider what the underlying assumptions were. In this case the assumption was that the coil filled the gap, which had to have the smallest volume possible because of the cost of magnets. That might be true of a tweeter, but it is not true of a long-throw woofer in which the coil overhangs the gap considerably. There were, of course, no long throw woofers in the 1950s.
The overhanging parts of the coil produce no thrust, because they are not exposed to the field in the gap, yet they add moving mass. Long-throw woofers have to be designed in a different way, as will be seen in a future article.
In addition to creating thrust to drive the diaphragm, some thought has to be given to cooling the coil. Poor design here can lead to a spiral down to poor performance, because a hot coil expands and needs a wider gap so it doesn’t bind. The wider gap reduces efficiency, which produces more heat and so on. Hot coils are self-protecting to some extent as the resistance rises, but the result is audible as power compression at medium frequencies, and distortion at low frequencies where the temperature varies throughout the cycle. The design of high accuracy drive units must begin with an effective power dissipation philosophy. More on this in the next chapter.
* Acoustics, Sound Fields and Transducers. Leo Beranek and Tim Mellow. Academic Press, ISBN 978-0-12-391421-7 (2012)
John Watkinson Consultant, publisher, London (UK).
Editor note: John Watkinson's other articles on loudspeaker technology can be located from The Broadcast Bridge home page. Search for "Watkinson".
He has also a new book readers may wish to view.
In The Art of Flight, John Watkinson chronicles the disciplines and major technologies that allow heavier-than-air machines to take flight. The book is available from Waterstones Book Store.