Impedance and Loss: An Ohm Story

I got into this a bit previously in a post about splitters, here. Ever run some speaker wire a bit too far and get temped to splice in a length of mismatched cable to reach a terminal.  Theoretically each splice creates a change in resistance the circuit, and if measuring cumulatively, it adds to the total resistance. Incidentally, since there is no difference between impedance and resistance in DC circuit you can measure this change in either unit. FYI: The International unit for electrical resistance is the ohm Ω, and whaddyaknow, so is impedance. Though, with impedance it represents a ratio. Generally speaking, the total loss should be trivial —unless you screw up.  So read how NASA splices here, and get out your soldering iron.

So my example today is not on the space shuttle, it is right in your home stereo. (With thanks to Richard L. Hess for the math. You can see his fine audio restoration work here.) Today's example is speaker wire. The signal loss you experience between the amplifier and the speaker cone is reliant on three primary factors

1. The gauge of the cable run
2. The length of the cable

GAUGE

Typically speakers are just a few feet from the speakers. But maybe you ran speakers all through out your house and yard like Ken Kesey. In those extreme, psychedelic cases higher gauges you  may use up to 14 or 12 gauge running a high power, tube-driven amplifier. But typically, even running cables around a living room for a surround-sound system needs a16 gauge. The term "gauge" comes from the Brown & Sharpe wire gauge, a standardized wire gauge system used since 1857. The smaller the number the larger the wire.You can see a chart here.

The measurements for stranded wires are a bit different than solid wire. Stranded wires are identified by three numbers, the AWG size as "gauge"  the number of strands, and the AWG size of each strand. For example, a 16 gauge wire with 7 strands of 24 AWG would be: 16 AWG 7/24. Not too complicated. Thank you mister Henry Sharpe. (Stranded AWG PDF chart here. )I recommend using stranded over solid wire for numerous reasons. It's more flexible, easier to splice, and solid wire gets bent, and each bend adds its own changes in resistance.Over a short length this doesn't add up to much, but the price difference is trivial.

LENGTH

DC current loss occurs at a calculable rate over the length of the wire.  Here is a calculator. I'm going to use Mr Hess' example and say that we'll use 70 and 90 feet (note calculator is in cm) and for 16 gauge and 12 gauge wire at 1 kHz and 10 kHz.  The loss calculations can't be exact without actually knowing the reactance of the speaker at a  given frequency, so  1 kHz and 10 kHz are just examples. The reality is a wiggly curve across the whole EQ expressible by the speaker.  Reactance, in this case is the opposition of a circuit  (the wire) to the change in current due to that element's inductance. Inductance is measured here in microhenrys as µH.


16 Gauge:
@70 feet  0.281 ohms resistance, 44 µH inductance
  1 kHz  0.278 ohms reactance
10 kHz  2.778 ohms reactance

@90 feet  0.361 ohms resistance, 58 µH inductance
  1 kHz  0.365 ohms reactance
10 kHz  2.778 ohms reactance

12 gauge:
@70 feet  0.111 ohms resistance, 42 µH inductance
  1 kHz  0.265 ohms reactance
10 kHz  3.655 ohms reactance

@90 feet  0.143 ohms resistance, 56 µH inductance
  1 kHz  0.349 ohms reactance
10 kHz  3.495 ohms reactance

Loss Calculation:
Z = resistance + reactance = impedance
Z(s) = impedance of the speaker
Z(w) = impedance of the wire
Z(t) = total impedance of wire and speaker = Z(s) + Z(w)

Loss (in decibels) = 20 log ( Z(s) / Z(t) )

Let's proceed with rounded numbers. At 1 kHz with a nominal 8-ohm speaker,  the 16 gauge cable at produces a loss of -0.59 dB at 1 kHz. Through the longer 16 gauge cable produces a loss of  -0.75 dB, at 1 kHz . That's a difference of -0.16 dB more loss.Bumping up that cable run to 12 gauge wire produces losses of -0.40 dB and -0.52 dB or about -0.12 dB more loss on the 90 foot run. It's not zero, but it's what I'd call trivial. Your average audiophile would surely disagree.

The Printable Speaker

A team of scientists at the Institute for Print and Media Technology at Chemnitz University of Technology, in Germany, has managed to make paper-thin speaker that can be fabricated at-will, in other words... printed. I foresee this technology leading to very annoying subway posters and greeting cards. The evils of marketing can ruin anything. More here.

But the technology is fascinating. The Chemnitz University team spent over two years on this project. This prototype is  made by printing layers of a conductive organic polymer and a piezoelectric layer  onto a piece of paper. The layers vibrate against each other to produce sound. I just cant figure out how to plug it in. They claim the speaker can produce sounds up to 80 decibels. That is in fact as loud as a freight train. [More on db references here.]  The team did admit the speaker didn't produce much bass. More here.

When most people see the word "polymer" they think plastics but that's not necessarily the case. Polymers can be organic or synthetic. They are just macromolecules comprised of repeating structural units connected by covalent chemical bonds. So you may have been thinking Bakelite is a polymer, and you are right.  But  so are PVC, silly putty, cellulose and even your own DNA. What I'm getting at here is that describing the layer as a polymer isn't very specific.

This is not the first paper speaker. It's just the first printable one. Last year the Journal of Mechanical Science and Technology published a paper on Piezoelectric electro-active paper speakers by Professor Jaehwan Kim  and some grad students at INHA University, South Korea. It described the acoustic characteristics of cellulose electroactive paper (EAPap).  cellulose... yes that's a polymer. The INHA University press release described it's potential in glowing terms:

"...cellulose EAPap is ultra-lightweight, inexpensive, biodegradable, it is advantageous for many applications such as micro insect robots, micro flying objects, microelectro-mechanical systems, biosensors, flexible electrical displays, RFID tags and smart packaging."

(More here.)  It sounds over the top, but even Forbes has been following the story, and with similar levels of enthusiasm, repeating claims that printable electronics may  generate up to $13 billion by 2016. Remember back when technology companies marketed to audiophiles?  

Loudspeakers and How they Operate

This is NRI booklet 26FR-1 "Loudspeakers and How They Work." I have written extensively about speakers in the past. Actually I wrote about them for a straight week back in 2007. This post was almost titled "The Loudspeaker (Pt 6)" Here I can add a nice addendum to that series.

This one is doubly interesting because someone wrote out a question and taped it to the first page:

"What two changes of energy occur in a loud speaker?"

What I found most interesting is that the booklet is old enough that it discusses the action of the horn. We no longer cover this in speaker design because there is essentially no more horn. Yes of course there are acoustic chambers but that old school exponential horn is long gone. This piece is detailed enough to distinguish between straight and curled exponential horns.

You can get all 19.5 MB if you
DOWNLOAD HERE

The Loudspeaker Part 5

I barely understand how these work so bare with me... part 5 is about the new toys.

To understand how the NXT flat speakers work, it's best to temporarily forget what you know about NXT technology, but the similarities end there. There is no woofer, no tweeter, no crossover. The parts you'd recognize are a magnet, a voice coil and a panel that could be made of aluminum or paper.
everything I wrote earlier this week. The obvious part with an amplifier powering a loudspeaker doesn't change with

NXT speakers operate in a world of psycho-acoustics and math. I'm not a math major so I'll be brief and quote others who are. "The magnet-and-coil assembly moves a microscopic 40 microns, barely nudging the rigid panel to begin a series of bending waves that travel in all directions." the panel drives waves that bend and crossing at angles and at different speeds.

Flat panel speakers are a British innovation. The British firm Verity bought a patent on public address systems owned by the country's Ministry of Defense and modified the speaker to deliver hi-fi sound. They patented this modified version and now licensing it.

But, there is a challenge to that patent. Sound Advance Systems of Santa Ana, California, claims its founder, José Bertagni, patented flat panel speakers thirty years ago. This time frame means that SAS can't claim royalties, it also should mean that Verity's ownership claim is invalid. The technical description in the patent is a prior publication which the SAS is using to attack Verity's claims. Of Course Verity takes the position that the NXT speaker is unique: the NXT speaker panel vibrates over its entire surface. The SAS speaker pumps air more like a piston. Read on:
http://www.lightspeed-tek.com/files/NXT_WHITEPAPER.pdf

The Loudspeaker part 4

This is the classic electrostatic loud speaker. the design is imperfect, but is revisited even today by audiophiles looking for more accuracy. Nakamichi just unveiled a version that costs about $9,000 per pair. A healthy descendant of patent No. 1,983,377.

One of the big down sides is that they use very high voltages to operate. The 5000 volt DC bias is usually supplied by a power supply running off 120 volt AC electrical circuits which is dangerous enough I'd not recommend it be anywhere children can reach. This nut makes them.

They use a thin flat conductive diaphragm. It's usually a plastic sheet impregnated graphite. This diaphragm is sandwiched between two electrically conductive grids, with a small air gap between the diaphragm and grids. For low distortion operation, the diaphragm must operate with a constant charge on its surface, rather than with a constant voltage. hence the need for the juice and the tube amp and the transformer and the dedicated circuit in the basement. [OK, I'm exaggerating but you're starting to get the idea what nobody uses these.]

The reason they exist is that they are amazingly precise. Which is why the audio nerds at Stereophile Magazine get all dreamy-eyed when some manufacturer ships them a pair for testing. The diaphragm described above is driven by two grids. Using grids on both sides cancels out non-linearity. The result is almost the total elimination of harmonic distortion. it sounds crisp to say the least. The downside to this design other than voltage and cost is that unlike your cheapo speakers it projects this perfect sound over a very narrow area. It's sonic field is very narrow, especially for those of us that are getting spoiled by surround sound these days.

In 1957 Quad ESL marketed as the first real electrostatic loudspeaker, referred to as the Quad 55 or the Quad 57. Those bad boys were designed by Peter Walker (pictured) and David Williamson. But they were based on a patent owned by Edward W. Kellogg's from 1934. that patent was based on a research paper he did way before that in 1925 while still working at General Electric. Kellog worked in tandem with a gentleman named Chester Rice. They published a paper on amplifier design that was important in boosting the power transmitted to loudspeakers. In 1926, RCA used this design in the Radiola line of a.c. powered radios.

The Loudspeaker Part 3

Twenty years later after much well-marketed but insubstantial change in speaker technology, somebody finally brings the goods. In 1954 Edgar Villchur develops the acoustic suspension principle. As great as speakers were for making things louder the bass just sucked man. The sound reproduction was very treble heavy. Villchur's work made your subwoofer possible and by extension the song Low Rider with his patent No. 2,775,309. Great Stereophile article here.

What I like about Villchur is that he was really one of us radio geeks. he ran a radio shop on West 4th Street, in Manhattan building and repairing custom hi-fi sets. He taught at NYU and wrote two books on the reproduction of sound. One of which, his 1965 book, Reproduction of Sound in High-Fidelity & Stereo Phonographs, is still in print today.

Acoustic suspension principle is a mathematical model of woofer behaviour in tandem with the air in a speaker cabinet, it provides a way to design a small speaker system. Before this to produce bass sounds accurately required ridiculous Frigidaire sized chassis. This allowed the development of effective and linear low frequency response in a small chassis. Suddenly with a two way system (woofer and tweeter) a output of 35 to 40 Hz from a box of only two cubic feet or so was possible. Without getting into the math, the idea is as follows:
1. Small cabinet size
2. Tight, clean bass response.
3. improved efficiency
4. The smaller woofer
5. Controlled response below the system resonance

Eddie's company, Acoustic Research introduced the small AR-1 bookshelf loudspeaker that used the this principle. that puppy cost $185 retail. This was followed by the $89 AR-2 in 1956 and eventually by the AR-3 with improved domed tweeters in 1958. The Acoustic Research brand of Hi-Fi loudspeakers became famous with their AR-3 loudspeaker. These had 12-inch woofers and a dome midrange speaker to compliment the high frequency tweeter.

The company prospers even now as they sully their brand name with tiny crappy speakers for the Ipod. But why not, Villchur sold Acoustic Research in 1967. He took the dough and founded the Foundation for Hearing Aid Research. His non-profit foundation developed a prototype device whose basic design is used widely in today's hearing aids.

The Loudspeaker Part 2

When people talk about loudspeakers people talk about Jensen, Siemens and the other big names. Nobody mentions Clair Loring Farrand. Who the hell is that guy? He invented the coil-driven direct-radiator loudspeaker. It was (in my opinion) the next big step in speaker evolution.

Farrand is better remembered for his work at Warner Bros. in Hollywood during the transition from silent films to sound pictures. He died not that long ago in 1981 at the age of 85. his New York Times obit read "Farrand was founder and president of Farrand Industries Inc. of Valhalla, N.Y., and his companies had control of roughly 1,000 patents. They ranged from bombsights for the B-52 bomber to windows for space program simulators." It's true. What they don't tell you is that before inventing the loudspeaker, Clair was a Marconi Company radio operator. He quit at the age of 23 in 1918 before perfecting that speaker.

He invented the Phonetron based on patent No. 1,847,935. His patent was filed Apr. 23, 1921. "An elliptically tapered tube or funnel shaped element is fixed to a loudspeaker so that its large diameter end overlies a central region of the loudspeaker diaphragm and such that its small diameter end extends away from the diaphragm. The loudspeaker diaphragm need not extend beyond the line on which it is connected to the funnel element, but does in the preferred embodiment. The funnel element has a cross-sectional shape that differs on planes parallel to the plane of the base which lie at different distances from the base. A funnel that is circular in cross-section at all of those parallel planes does not exhibit high fidelity response with the brilliance and clarity that distinguishes speakers having the non-uniform cross-sectional shape."

Blah Blah Blah.. The important part is that it's a paper cone. One just like the one in your speakers at home. It worked well, and was accepted by consumers. It successfully competed with the horns used by existing table radios

The Loudspeaker Part 1

Before the fight even starts. Allow me to define "Loud Speaker"
n. A device for converting electrical energy to sound
Loudspeakers are electro mechanical devices working on the basis of electromagnetism. They convert a given electrical signal, using electromagnetism, into moving air, what we call sound.

Early radio had a problem. Things were quiet. Too quiet as they say in the war movies. Things needed to get louder, not loud enough to rattle your windows with the massive sub-woofer in the trunk, but at least loud enough to hear at all. Radio's did not have amplifiers yet. Everything was operating with an output power measured in milliwatts.

Siemens filed the first significant patent for the loudspeaker horn. It was the predecessor to all acoustic models used in phonographs players. His German patent was granted July 30, 1878 and his British patent No. 4685 was granted Feb. 1, 1878. It was not a patent for audible transmission, it never carried sound. (He tested it wish DC transients.) Alexander G. Bell nailed that with his telephone patent in 1876. What Siemens patented was "the mechanical movement of an electrical coil from electrical currents transmitted through it" A moving coil transducer! and we all know from radio production I class that if the coil moves... it's dynamic.

So in 1874 Ernst W. Siemens was the first to scientifically describe the dynamic transducer, with a circular coil of wire in a magnetic field. His model is suspended so that it can support axial motion (be springy).

But what to do with all these moving springs and no sound? Siemens applied for a second German patent, No. 2355, filed Dec. 14, 1877. This one was for a non-magnetic parchment diaphragm as the sound radiator of a moving-coil transducer. His diaphragm had a cone shape, with an exponentially flaring trumpet shape. See picture above.. Many people thought it looked like a daffodil or other flower. Strangely despite the fact that he could plainly see the coil moving, It was the two Americans, Chester W. Rice and Edward W. Kellog, that patented the moving coil principle in 1924.

But like all test models it didn't work so well. It wasn't very responsive. the axial mounting wasn't exactly perfect etc. etc. So this math nerd named Oliver Lodge starting noodling with it. He'd recently decided that math was boring and moved over to chair the physics department at Oxford. During his tine there he conducted experiments in the propagation and reception of electromagnetic waves. It was he that proved in1888 showed that radio-frequency waves could be transmitted along electric wires. It was he that fixed Édouard Branly's "coherer" so that Morse code could even be received. In his spare time he electric spark ignition for the internal combustion engine. Needless to say, he was a bright guy.

Mr. Lodge improved upon the speaker in 1898 with British patent No. 971. He added non-magnetic spacers to keep the air gap between the inner and outer poles of a moving coil transducer. He called it a "bellowing telephone" because of the cone-shape. It's off topic but like many geniuses in his spare time he was also a nutter and a bit of a sun worshipper..