# Theory and Practice

# Theory and Practice

Jan 1, 2000 12:00 PM, Harry J. Ferris

Methods for minimizing loudspeaker line loss and a real-world example oftheir application.

The Trans World Dome acoustic improvements engineered by Ross & Baruzzinihad many challenges, among them the problem of attaining design audiolevels with the given location of power amps with respect to theloudspeakers. Hundreds of watts had to be delivered to various loudspeakerclusters at distances of several hundred feet. Although power amps shouldbe located as close to the loudspeakers as possible, what if you are stuckwith locations requiring long runs of wire because of existing conditions,such as at the Trans World Dome? The engineers chose not to install 70 Vlines and transformers because obtaining standard 70 V line transformers inthe multi-hundred watt range would be difficult. The added losses wouldpartially negate their benefit, and there would be increased costs to theproject for their procurement and installation. Besides, several hundredwatts at 8 V is already in the 70 V range. What about larger wires? How doyou calculate the losses? When are larger conductors just wasting money?

Ross & Baruzzini faced all of these issues in the design of the amp racksand loudspeaker wiring for the acoustical improvements. What happens whenyou try to deliver hundreds of watts of power to loudspeakers that are 700feet (213 m) away? How much of the power that the amp is capable ofproducing is actually available to the drivers? (See Figure 1.)

Back to basics

An amp driving a loudspeaker is really a voltage source connected to aloudspeaker load through an impedance. This is a simple voltage divider inwhich the voltage across the loudspeaker will be some portion of thevoltage at the amp output terminals. The remainder of the voltage is alongthe loudspeaker line but is not delivering power at the loudspeaker. Thisis similar to what happens when you plug that two horsepower table saw intothe 100 foot (30.5 m) lawnmower extension cord and try to rip some heavylumber. The table saw has problems developing enough power because of thevoltage drop along the extension cord, which reduces the actual voltage atthe table saw. Be aware that although the DC resistance of the wire isoften used to calculate line loss, this method is not always accurate.

A voice coil is just that, a coil. Because it is a coil, it not only hasresistance, but it also has inductance, which causes a reactive impedance.The total impedance of an electrical device is the vector, not arithmetic,sum of the resistive and the reactive impedance. As can be seen in Figure2, 4 + 3 = 5, not 7 because the 3 and 4 vectors are at 90 degree angles toeach other. The reactive impedance, 3, is 90 degree out of phase with theresistance, 4. Although the resistive portion (4) will remain relativelyconstant, the reactive portion (3) will change value with frequency, hencehaving an effect on value of the total impedance (5).

This effect of total impedance versus resistance can be seen by connectingan ohmmeter across an 8 V voice coil. You will most likely read less than 8V because the inductance has no effect at DC. The impedance of an inductoris an AC phenomenon, and it increases linearly with frequency. In otherwords, an inductance that has an inductive reactance of 5 V at 100 Hz willhave an inductive reactance of 10 V at 200 Hz, 30 V at 600 Hz, and so on.Voice coil impedance, however, is an even more complex phenomenon becausethe voice coil is really part of an electromagnetic system, a part of amotor-generator. The voice coil is moving in a magnetic field so that itcan generate a voltage of its own. This is the same principle that comesinto play when using a loudspeaker for talk-back in an intercom system.

Remember that impedance is the ratio of the voltage applied across the circuit divided by the current flowing in the circuit. A voice coil is part of a driver, and that driver is often in an enclosure. Now, the impedance vs.frequency curve usually becomes more complex. This is because the voicecoil can draw more or less current depending upon the mechanical (acoustic)coupling and properties of the overall system, which consists of the voicecoil, driver, enclosure and the acoustic properties of the space in whichthey are placed. A driver at resonance will exhibit a high impedancebecause little current is required to move the voice coil, and the voicecoil is generating more back EMF (electro-motive force), which reduces thecurrent drawn. The particular driving voltage, divided by the smallercurrent drawn, results in a higher impedance. Of course, good loudspeakerdesign would use drivers above their resonant frequencies by properenclosure and system design, but there will still be someelectro-mechanical coupling influence upon the impedance. Although we willnot take all of these factors into account in the examples in this article,it is important to realize that a driver does not have a constant impedancewith frequency.

The variation of wire impedance with frequency is less complex than that ofan electro-acoustic driver because the wire does not have any acoustic normechanical dynamic properties associated with it. The wire impedance stillvaries with frequency (see Figure 4), however, as opposed to the so manyohms per foot of resistance. This means that the voltage divider ratio isfrequency dependant. In other words, the dB loss due to the loudspeakerwires depends on frequency. Figure 4 and Figure 5 show the model'spredicted impedance of AWG#10 conductors in non-magnetic raceway andnon-magnetic cable tray, and an actual impedance measurement of theinstalled AWG#10 conductors in non-magnetic raceway and non-magnetic cabletray. There is a good correlation between the theoretical model used andthe actual installation.

Because inductance, which is not as dependent upon wire size as resistanceis, affects the total wire impedance at the higher frequencies, arbitrarilyusing larger wire is often not warranted at the higher frequencies. Figure6 shows the difference in wire impedance for an AWG#10 versus an AWG#6conductor. Although the AWG#6 conductor would provide less line loss at lowfrequencies, it does not help nearly as much at the higher frequencies. Itis more important not to install the conductors in magnetic raceway at highfrequencies and long runs than it is to increase wire size arbitrarily.

At this point, let me highlight a few key points thus far. Impedance has aDC component and a frequency-dependant AC component. The power amp,loudspeaker wiring and loudspeaker can be respectively modeled as a voltagesource, frequency-dependant series impedance and a frequency- andacoustic-dependant load. Loudspeaker line loss at midrange and higherfrequency is more than you might think. The prudence of using bigger wiredepends on the situation. The proportion of voltage across the loudspeakercompared to that at the amp output will change with frequency. Magneticproperties of the raceway in which the loudspeaker wires are installed canaffect the loudspeaker line loss.

Depth of penetration

Another phenomenon that takes place with increasing frequency is thatcurrent does not travel evenly through the entire cross section of aconductor. The depth of penetration depends on frequency in addition to themagnetic and electrical conduction properties of the wire itself. We willonly talk about frequency here because the magnetic and conductionproperties of one copper wire to another are essentially the same.

Skin depth for a copper conductor is roughly 3/8 of an inch (9.5 mm) at 60Hz power frequency and roughly 3/32 of an inch (2.4 mm) at 1,000 Hz. Theskin depth is actually the depth at which the current density isapproximately 37% of that on the outer surface of the conductor. It doesnot mean that there is no current flow at all, deeper than one skin depthinto the conductor, but rather that the current density has decreased oneexponential at that point. These technical details are not of importance inthemselves, but what is significant is that at high frequencies, even theresistance of the conductor goes up because less of the cross section ofthe wire is effectively used. In reality, because almost all conductorsused at audio frequencies are composed of small stranded wires, the skineffect at audio frequencies upon each of those small wires is minimal, andthis really is not much of a concern. It would be a bigger problem ifpeople were pulling solid AWG#6 conductors, which are difficult to find, atleast in an insulated conductor. This just comes down to another point notto increase the wire size for high-frequency drivers without stopping tothink about the consequence. Of course, the wire size has to be adequate;no one would try to run 800 W through AWG#22 wire.

How it all relates

Ross & Baruzzini modeled several amp, loudspeaker line and loudspeakercombinations to determine if this more accurate loss model had a real-worldeffect or not. The results depend upon loudspeaker impedance, frequencyrange, amp rating and desired sound pressure level. It can be seen that lowfrequencies do not change the assumed loudspeaker wire impedance as much asthe higher frequencies do, but that the effect of higher frequencies onloudspeaker impedance is much more significant. Loudspeaker wire runsserving low impedance loads at long distances are of the greatest concern.Although the impedance of the loudspeaker wiring causes more loss than isoften expected, it is not often a problem with shorter runs or higherimpedance loads. Figure 8 gives an illustration of how the loss for aparticular set of drivers at the Trans World Dome varies with frequency anddistance. The plot undulates rather than maintains a constant slope becauseof phase angle changes in the load with frequency and phase angle changesin the loudspeaker wiring with frequency and with distance.

The calculations were based upon a fixed DC wire resistance and aninductive reactance, which increases with frequency. Calculations with andwithout skin effect taken into account for the AWG#10 conductors showednegligible differences. The system was treated as a frequency-dependantvoltage divider with impedance magnitudes and angles, which both dependupon frequency and raceway material used in order to be accurate. The dBloss is then merely 20 log (voltage at voice coil / voltage at amp).

It might be thought that the difference between the power delivered to thedrivers and the power rating of the amp is dissipated in the loudspeakerwires. This really is not true, because the (higher) impedance of theloudspeaker wiring, combined with the impedance of the drivers, preventsthe amp from delivering its rated power output.

Remember that the power amp is a voltage source, so, for a given outputvoltage, the output current will be less with a higher impedance load thanwith a lower impedance load. In other words, just because 0 dBm is presentat the power amp input does not mean that full rated amp power is beingdelivered to the drivers. It means that the voltage at the output terminalsof the power amp is such that full rated amp power will be delivered whendriving the impedance at which the amp is rated. Full rated power will notbe delivered to a higher impedance load.

There are a few things to keep in mind when trying to control impedance.First, use an adequate wire size. Do not use magnetic raceways for longruns at high power, low impedance loads and midrange and higherfrequencies. Parallel conductors, rather than increasing wire sizes, if youreally need large sizes. This has the double benefit of reducing inductivereactance, which dominates at the high frequencies, and, although not asnoticeable, reducing skin effect losses.

The system installed at the Trans World Dome attains the desired 105 dBdesign level because of the amp and wiring configuration chosen from thecalculations. It was not feasible to change the existing AWG#10 wiringbecause of the original method of installation, the project schedule andcost. Consequently, the power amps were configured for a higher outputimpedance to deliver higher output voltage to compensate partially for thelosses. The ideal case would be to add parallel AWG#10 conductors, whichwould have reduced the losses more than would AWG#6 or even AWG#4conductors at midrange and higher frequencies.