The ads above are Google-sponsored.  
Clicking on them at every visit helps support this website!
Clicking on something INSIDE advertisements helps even more!!

Stock AIRHEAD Alternators:
TRUE capabilities, differences, etc.
Includes section on WHY 3 phases; and basic engineering.
copyright, 2014, R. Fleischer

http://bmwmotorcycletech.info/altcapability.htm
15-A

Read article 15-C... if considering an aftermarket alternator;  or wanting further information on the stock alternator:
http://bmwmotorcycletech.info/AftrMrktAlt.htm

Other articles you may be interested in:
http://bmwmotorcycletech.info/altbrushrotor.htm

http://bmwmotorcycletech.info/boxerelectrics.htm 
 


BMW uses 3 phase electricity generators called alternators on all models since 1969, & some before that date.  Why 3 phases?  3 Phase is more efficient at converting rotational energy to electricity.  3 phases enables the same or much more output in a smaller, lighter package, even less steel laminations are needed, & possibly more charging at lower rpm & possibly less horsepower drain on the engine.  3 phase is usually better for vehicles with radios & other electronics, as the system is, in effect, using an A.C. generator at higher frequencies than single phase. That means that with the battery acting like a monstrously large capacitor (which it DOES), alternator whine noise & other problems is MUCH reduced in a 3 phase system, compared to single phase.  There are various other advantages that 3 phase offers.  The 3 phase rotor MAY have less inertia, thus better vehicle acceleration, assuming diameter is kept small & overall inertia similarly, thus there may be a lowering of torsional stresses on the shaft & components. That last item can be critical on some engines where the rotor is affixed to the crankshaft & can be considered an extension of the crankshaft.  

A 3 phase alternator is more compact & efficient ... by up to 1.73 times electrically more efficient.

At the link here, scan down the article, are sketches of the waveforms & some additional detailed information on single and three phase systems. http://bmwmotorcycletech.info/boxerelectrics.htm.  The sketches will help you understand 3-phase.

is the symbol universally accepted in electrical circuits for 'phase'.  A 3 system has few drawbacks. One is that the diodes circuitry used for rectification is more complex & there are more of them; another is complexity & more labor in manufacturing.

Almost no one uses a 2 phase system. Many alternators are one phase types with two wires to the stator.  The Euromotoelectrics EnDuraLast permanent magnet alternator is that type, & there are others. MOST of those types are rotating permanent magnet alternators.  There ARE some 2 & 3 phase types using rotating permanent magnets, usually low powered types.

Most all 3 phase alternators typically have 3 or 4 stator wires.   BMW used the 3 wire type on the /5, and all BMW alternators afterwards were the 4 wire type. The 4 wire type simply has a center-tap where all 3 phases are connected. For the NERDY, there are several ways of connecting 2 or 3 phase alternators. There are even 6 phase alternators in industry.  In the case of "2 phases", MOST of those alternators REALLY ARE simply two single phase portions, that are common to one rotating magnet; nearly 100% of the time it is a a permanent magnet.  This type has been used on some scooters, lawn & other agricultural equipment, etc.  In some instances, scooters for instance, one winding (or, one phase, if you will) is for lights, the other for charging a battery after rectification; or, if no battery, powering the ignition in some way.  The most common type is the electrical system used on early Vespa scooters, which came both with and without a battery, customer's choice...and for manufacturing ease one part of dual alternator output was kept at A.C. for the lights (only); and the other side powered the ignition coil.

For 3 phase alternators, while there are some complicated industrial large alternators, the main thing to know is that all basic 3 phase alternators can be connected either in a DELTA connection (the windings in series loop, so no center-tap is possible...think delta triangle), or in a T connection, which enables a center-tap connection.  THAT is what BMW did for the Airhead alternators, although the connection to the center-tap was NOT made until the /6.  NOTE that I am NOT dead-sure about the T being used in the /5...I've never bothered to measure to find out.

There are two basic types of alternators; one type uses a segmented permanent magnet rotor. That type in modern practical use can NOT have its output adjusted by changing the magnetic field of the rotor.   NERDY:  Yes, it was tried, & worked, MANY decades ago, by MOVING the parts closer & farther apart mechanically by an output governor.

So, in practical use, the permanent magnet alternator main problem, besides size & efficiency, is that of the REGULATORS.  These are often combined with the rectifying diodes function, are not very efficient, having to pass large currents to either ground, or, much better but complex & costly due to the high power, through a switching transistors circuit (which can be very efficient, but at additional cost). These are made in single phase as well as 3 phase and even rare 6 phase types. There is another problem with the permanent magnet type, & that is efficiency which somewhat depends on if it is LARGE in diameter. If SMALL, then the effective rotational speed as the pole pieces cross the magnetic coils is low; thus low rotor rpm usually means low output.  Designs to try to overcome this tend to have LONG rotors, which can create packaging problems.  "Pancake" designs are in use that have good output at VERY low rpm, but at high rpm & ample output, they have a lot of inertia effects.

All permanent magnet alternators are self-limiting in current output; but not so for voltage. Sometimes they are used in simple systems where the effect is not well controlled, to say the least, & battery life suffers. Old scooters and small to modest sized dirt bikes sometimes used these types of systems, common was overcharged battery damage.  Some even used A.C. for the lights & no battery; and 'upgrades' offered battery ignition instead of magneto (often the magneto was on the SAME stator!). Vespa.    On many types of lawn equipment, there is a rotating magnet that is often part of the crankshaft flywheel, and one fixed non-movable coil of wire, wrapped around an iron core. That is for the ones with lights and/or battery start.  On most, without battery start, and usually without lights, there is just one core, used to generate the high voltage for the spark plug.

The other type of alternator is used by BMW for its Airheads.  It does not use a permanent magnet; rather, it has a specially shaped steel rotor with a number of turns of wire on it; the ends of which connect to 'slip rings', that has carbon brushes pushing against them, & wires connect to these two brushes. Regulation of the output of the alternator is done by relatively low power rated components, which control the amount of electricity to the rotor, which magnetizes the rotor in the amount of magnetism as needed. Brushes wear slowly, are not expensive, and usually not difficult to replace. Output current/wattage limitations are inherent in the structure like in most alternators (self limiting), but they must have a voltage regulator.   BMW uses this type, made for BMW by Bosch.   This type of alternator has MANY advantages.


The Airheads:

All the alternators after the /5 (180 watts) had 238, 240, 250, 260, or 280 watts.  Many, if not most people seem to think of all of them (after the /5) as being "280 watts".  There were some variations in the original stock rotors & stators.  In the R90S model the stator had a slightly larger INside diameter. There are various combinations of things.  The Authorities (Police) models had 238 watts similar to the R90S, but this was NOT done with the same rotor & stator parts, & did NOT perform the same as the R90S alternator.  After about year 1990, rotors for the civilian models were 2.8 ohms & produced a bit more at lower rpm, but less maximum output.   These rotors could draw more current from the alternator and probably should not be used with mechanical regulators due to the higher current drain.  I think they should not be used with the earliest electronics regulators either.  Mechanical regulators do not 'play well' with electronic ignitions.

Earliest alternator rotors were 73.4 mm in diameter.   From approximately early 1975, they are all supposedly 73.0 mm in diameter, but this is not exactly so. Other of my articles discuss that nerdy point.  I prefer the 73.4, rewound for lower ohms...but one must be cautious, as a 2.8 ohm rotor can not be used with some earlier stators properly. 

Earliest /5 type alternators were 180 watts, had 105 mm stators with THREE outputs, and had Bosch part numbers on the outer housings that ended in -001 or -002.  The R90S only had a -003 stator, it had a slightly larger inside diameter with slightly reduced output. One problem, unless you measure things (& would not likely do that unless fairly nerdy) is that the Bosch numbers are on the stator housing, & who knows what rotor might be installed! The later stators, such as -004 and -005, are all 107 mm, do NOT fit the /5.  The very earliest /6 (to end of 1974 generally) also have 105 mm stators, but MANY do have the 107 mm.   A VERY FEW EARLY 1976  /6 also had the 105 mm alternator stator. 

Except for the /5 180 watt stator assembly & 105 mm versions of the /6, and except for the /5 diode board not having the alternator stator center-tap function (one CAN use the /6 & later board with the /5; but the /5 board used with the /6 and later alternator will REDUCE the output some); all the other parts, to the end of production in 1995, except the mechanical voltage regulators, are interchangeable and are compatible electrically, if not perfectly compatible.  There is a potential problem with the larger diameter rotor & the larger inside diameter stator, on, perhaps, the R90S at very high rpm, due to whipping....but this has NOT been reported to me! What HAS been reported is that some few AFTERMARKET rotors are not made concentric enough, & may rub the stator.  Easy to see when installed, by using feeler gauges. With the various rotor rebuilding and aftermarket units available, there is no telling what rotor might be in an Airhead without measurements of diameter & resistance.  I have not measured the physical diameter of all the various rotors; but see above.   

After the /5, the stators are all 107 mm; EXCEPT, as noted, some 1974-1975 (and a rare 1976)  were also 105 mm size, but were all at least 238 watts, with many 280 watts.  THUS certain 1974-5, and MAYBE a very rare 1976 STATOR can be installed in a /5, to increase the /5 output by ~100 watts...if one uses the /6 or later diode board.   The electronics regulators can last much longer than the mechanical regulators & are a must on the electronics ignition models (from 1981).

There is a difference in the STATOR windings resistance in the last production alternators, this shows up on an accurate low-resistance ohmmeter.  Generally thought of as 1990's+ Airheads, they have a slightly increased stator resistance, a lower rotor resistance (2.8 ohms), produce only about 240 watts, but begin at a somewhat lower rpm.

If you replace a stator, be sure to measure the stator at both ends; & get the correct version. One version has the same 105 mm, and the other has a 107.5 mm end.

The earliest rotors (this means /5 and /6) had a much higher resistance, close to 7 ohms.  These were designed to be used with the MECHANICAL voltage regulator, although an electronic type will work fine with them & is MORE reliable.   The higher resistance rotor reduced the current that went through the regulator, reducing the wear on the mechanical regulator contact points compared to if you installed a lower ohms rotor of a later model.    YES, that DOES mean that if one uses a later low ohms rotor, that the old mechanical regulator points will probably wear faster.   I have NOT done any testing in this regards, &  IT IS possible, due to how the mechanical regulator actually works internally and its bottom resistor/coil, that this is a wrong idea & the mechanical regulator might well work OK on the later, lower resistance rotors, but I am not overly hopeful.  Note that any of the later regulators, electronic, should work fine, and so should most any regulator from a car, that fits the plug and mounting area. ONE caveat on that, is that the last regulators handled the higher current of the lowest resistance rotors better.  In an emergency I would NOT HESITATE to use ANY regulator that fit the socket.

Adjustable voltage output regulators are available, & the Bosch & Wehrle metal can electronic regulator that came with some 1981+ Airheads can be made adjustable. Adjustable electronic regulators at a quite reasonable price are available from RockyPointCycle.com (who also stock other things, including inexpensive ignition modules, stainless steel nuts and bolts kits, Mikuni carbs, Boyer ignitions, etc.).  You can also go to www.euromotoelectrics.com; who carry a VERY WIDE VARIETY of electrical parts for your BMW, at discount prices compared to BMW.

The alternators with the 3.4-3.8 & 2.8 ohm rotors produce output beginning at slightly lower rpm.  The maximum output is UNaffected or is LOWER.   The 73.4 mm rotors are nice to have, but hardly of much importance.

The Authorities models (if you have an ex-Authority, or Police model) produced electricity at somewhat lower rpm....but, unfortunately, will produce about 42 LESS watts TOTAL, than a regular 280 watt alternator.  Since I have never had the opportunity to fully test an Authorities alternator, I am not treating it, below, but my guess is that no one will want one unless you can get it cheap; ....and an aftermarket alternator like the EnDuraLast PM or, especially, the Omega (or, Emerald Island product, which, re-branded, IS the Omega), is likely the better way to go if you need more electricity, or a higher output at low in-city commuting rpm. 

Authorities models had the "high output voltage regulator"....a misnomer really, as it simply had the voltage set slightly higher, which did not increase maximum alternator output; which, as you have seen, was really LOWER!...but, the Authorities VR did, with enough cruising, charge the battery better, for longer life, since the VR was internally designed for about 14.3 volts.  Adjustment for this type of voltage is easy to do on some of the other regulators...it does helps a little bit with in-city riding.  You can purchase the non-adjustable higher voltage output Authorities VR from BMW...DO NOT BOTHER, it is $$$, offers no advantages over much cheaper VR's.

There are some other variables, including dealing with the mentioned setting of the voltage regulator; but, in general, you can expect what follows, well below, from the stock systems.   Keep in mind that just because the system is rated at some particular wattage, does NOT mean you will GET all those watts....some of you will get maybe up to 13% less; AND, even more losses are possible.   To get near the maximum sometimes requires some goodly rpm....upwards of 5000.  The typical reason for such high rpm will be because YOUR system connections, switch, etc., are likely not clean, shiny, tight & near perfect, nor has absolutely correct parts, as was the system I prepared for the testing information below.  Battery condition also affects maximum usable output.  If your system is in reasonably decent condition you can expect close to the full-rated output at 3500-4000 rpm.   In other words, the Bosch alternators DO produce very close to their rating/specifications.


NONE of the stock BMW Bosch Alternators are more than 'probably passably OK' for extreme stop & go traffic conditions.  This is particularly so as heavier electrical drain items are in use, such as extra lighting of some higher drain types & heated clothing.  My comments, below, ASSUME modestly slow traffic or faster road conditions, & a reasonably well maintained electrical system, & stock alternator, except as noted.
 


/5:   stock 180 watt unit is adequate for the original lighting.  It is still adequate if the headlight is upgraded to the later 55/60 watt H4 type.  OK for a heated vest, & maybe also a conversion to rear incandescent running lights, but with all that comes the practical limit.  You will need as much as 4000+ rpm to obtain full output.  The 280 watt alternator can be installed, if you have the correct 105 mm 280 watt stator and a /6 and later diode board.

/6 & later:  The system is adequate for stock lighting & you can add a heated vest & usually heated grips & rear running lights conversions. If reasonable care is used, you may be able to use a 80 watt headlight conversion or some other 20-40 watt accessory.  That is the practical limit.   Note that you need as much as 4000+ rpm, semi-continuously, to maintain the battery, if you have most all of these things.

If you have a headlight modulator, that frees up some watts in itself, as AVERAGE drain is lower. 

Reliability of the alternator, and diode board, in the stock Bosch system, is INcreased by using a later front engine metal cover which has better ventilation.  Additional reliability is had from using a louvered lower fairing piece instead of the solid fairing piece on the RS and RT models.
 


REAL WORLD; REAL TESTS, ETC:

The problem that usually comes up, assuming a good system, is that the battery does not FULLY recharge.   The voltage regulator setting for all regulators is described in this article, and in much more depth in another article on this website, so I won't delve deeper here.   Those that ride in town, constant stop and go, MIGHT have to charge the battery at the end of the day, perhaps overnight on a Smart Charger of some sort.  When you sit at a signal light you have a fair amount of battery drain at idle rpm, powering the ignition, lights, ETC.   It takes time after rpm's rise, to replenish that stop-time drain from the battery. Those that are cruising down the road for longer distances, above 3500 rpm, likely will have no problems, unless they are pushing the alternator limits, then it will just take longer. 

On a reasonably good system you probably will be able to keep the battery adequately charged if you can be at 2800-3200 rpm or higher, most of the time.  If your RS/RT stock fairing voltmeter INDICATES about 12.5 volts or higher while cruising, chances are pretty good that the battery is reasonably charged (but the battery is NOT charged enough for long life).   It would be nicer if it indicated about 13.7 or even 14 while at speed.   The actual battery voltage is likely 0.3 volt higher than indication of the fairing voltmeter due to where that voltmeter is connected. If the battery voltage is more than 0.4 volt higher than the actual voltage, measured with an accurate digital voltmeter where the dash voltmeter connects, then you have contacts & connections to check.  The best battery itself voltage is 14.5 to 14.9, at mild temperatures, for battery life, but this voltage is a bit high for other reasons.  A good compromise is 14.0-14.4.

If you try to use the maximum output of the alternator for long periods of time, you will cause additional heat stress on the diodes; the rotor & stator also will increase in temperature.  The diode board is in a hot area to begin with.   The stators tend to be mostly unaffected, but the rotors, being OFTEN suddenly accelerated & suddenly decelerated, sometimes violently with poor clutch & throttle technique, tend to not fair well. Sometimes the windings move & shorts or opens will occur.   Vacuum-potted-in-epoxy rotors are better, if all else is quality.   The alternator & diode board cooling is not all that good, particularly on the faired models.  Just using the later louvered front piece or modifying your solid one will probably help.  The cooling air, well-heated by the engine, has to go someplace.....and how it gets out, taking some heat with it, is different on the early machines with the scoop vent on top of the starter cover, compared to the later models with rectangular air boxes.  BMW also changed the front aluminum cover design for better cooling....yes, it also helps.

Aftermarket alternator conversions such as the EnDuraLast & Omega (Emerald Island product) will give INcreased total wattage output, do it at a lower rpm; also produce usable watts at not hardly much over idle rpm.
  There is an article on this website that gets fairly deeply into aftermarket alternators:  http://bmwmotorcycletech.info/AftrMrktAlt.htm

Before the results of my testing, I want to explain what was done to a stock bike, for testing purposes:
An original-equipment R100RT with Bosch metal can electronic regulator was the beginning.  The alternator and diode board were all stock items.  The diode board; with known perfect (by inspection and testing for performance and for diodes solder joints), was mounted on aftermarket solid metal mounts, not rubber mounts as came on this model bike.  The BMW SI for added grounding wires at the diode board had been complied-with, even though the bike had aftermarket solid metal mounts.  100% of pertinent electrical connections were gone through prior to tests.   In particular, the red wires at the alternator, diode board, grounds, VR, starter relay plug & battery.  The starter motor (alternator output goes there) power input terminal was clean, shiny, tightened.  ALL connections, everyplace I could get to, were in good condition.  The Ignition switch & Kill switch contacts resistance were tested good.  The battery was a flooded type, in excellent load-tested (on an Load Tester) condition.  Everything was done to allow measurements to show the best that the charging system could do.  The Bosch voltage regulator was set for 14.3 at the battery, when the system was running maximum output, light electrical load.   This motorcycle had a measured "3.7 ohm" rotor; the brushes were near new; the resistance of the rotor through the brushes was 4.2 ohms.  Brush spring pressure was correct.  The rotor to stator clearance was deemed to be in the middle of the normally seen range of values.

The diode board output (thick red wire) was disconnected.  A very short length heavy gauge pair of wires was used, connected to an accurate very low resistance ammeter setup, electrically located between diode board output & original heavy red BMW lead that plugs into that diode board large spade connection (right side, facing from front), as a laboratory grade of clamp-over DC ammeter was not available.   I have repeated tests using laboratory grade 50mv/100 mv shunts, and the results agree.

Thus....the ACTUAL USABLE output of the alternator was measured in amperes.   Battery voltage was monitored by a known-accurate digital voltmeter.  I also measured voltage drops in the system wiring to be sure they were reasonable.

The motorcycle had all lights, etc., disconnected that were not stock as factory shipped, except that an extra lamp rated at ~20 watts was added to simulate small additional loads often used by many riders, see below.    All tests were run with normal lighting, headlight on low beam.  The battery was drained some prior to each series of tests, so that the maximum possible output of the alternator could be measured withOUT using special resistance loads.   A goodly attempt was made to eliminate errors. The tachometer was calibrated.   A method of measuring input & output from the BATTERY was made up, it was loss-less in design.  The purpose of this measurement was to absolutely know where equilibrium was.  Equilibrium I defined as the RPM point where any additional alternator available electricity above & beyond what the bike's lighting, ignition, etc., needed, was considered excess, & could be used to charge the battery.  This was for practical purposes for reporting what might be expected while riding in various conditions...AND...to get a good idea of what sort of over-all performance might be expected under a variety of conditions outside of normal.

NERDY:....tests have also been run using a conventional carbon-pile battery load, those tests are in the author's files, & are not needed here.  Those tests were made with the engine fully warmed-up, core rotor temperature was monitored, output was measured using a 50 mv commercial shunt; rpm versus WATTS at that particular level were measured.  In addition, tests were also run to load the battery progressively at increasing rpm, until with a 12.8 volt loaded level the current output was again and lastly measured.  This type of test was run at other voltage levels too.  The tests shown just below were NOT those from that series.  These not-shown tests are not needed, would only confuse you, in the Results, below.  Such tests mean something to ME, in understanding advertised wattage ratings of various alternators (that is, separating advertising from facts).
 

Results, stock system as noted:

At 1050 rpm, the true output was 2.5 amperes.  This was not enough to maintain the battery charge, considering the headlight, ignition, etc.

At 1550 rpm, the true output was 10.0 amperes. In one way of looking at this, there was NO drain nor charging regarding the battery; that is, this was the rpm for equilibrium ...that is, the battery (charged) was neither charging, nor discharging.  In another way of looking at it, the battery WAS being maintained at a decent trickle charge as needed to maintain what is called the Float Voltage. For this test, I used an extra 20 watts of load to simulate the very common addition of two rear running lights (#1157 lamps, one section for RUN function, converting the turn signals to Turn-Run function); or, small additional accessories often added by riders.  IF the extra 20 watts of load had not been added, there would have been about 1 to 1.5 amperes available, or, just call it approximately 15 watts.  Even another way of saying this, is that without that extra load, equilibrium would have been at ~1475 RPM.

At 2100 rpm, the true output was 15.0 amperes; the battery would be charging well at that rate, should it need charging.

At 2850 rpm, MAXIMUM output (of 270 watts) was reached, of 20.0 amperes, at a voltage at the battery of 13.5 (where the battery was fully charged)....& if kept at that rpm, the charging current would slowly decrease, as the voltage regulator compensated, and the battery would reach its normal 'floating charge voltage' of 14 volts +-, shortly thereafter.    For the best battery life, & over-all performance compromise, the battery voltage would have been 14.3. If the regulator was so set (and it WAS), & the load decreased a bit, this voltage would have been reached.   Thus, while there are many ways of considering various voltages, charging levels, etc., one conclusion would be that if not over 250 watts was being used, the battery could be, at 3,000 RPM or more, be completely & fully charged by any modest length of cruising, said charge being good for long battery life.

NOTE carefully what these figures mean!....they mean that a stock 280 watt alternator is fully capable of recharging the battery after starting; maintaining a fully charged system, at 2100 rpm or more, assuming a stock electrical system; & if all was in good condition, & there were no additional loads beyond 20 watts, & the ride was long enough at 2100 rpm or higher.  Since no one should be riding continuously at 2100 rpm, only at considerably higher rpm, you can see that charging is not a problem if the system is in good condition, & you have considerably more than a block or a few blocks, between short traffic signal stops.  If you were NOT in a 'city stop and go' situation, but cruising on the highway at any rpm above 3500 (which is proper and normal), you could EASILY wear a heated vest, and have extra lights, etc.

www.euromotoelectrics.com has updated Valeo parts (& Chinese knockoffs); Bosch parts, does rebuilding of electrics, offers the EnDuraLast permanent magnet alternator upgrade, for ALL airheads from the /5 onwards.   That kit is somewhat involved in installation. You cannot easily revert to stock, something to consider if on the road. The kit works well for those needing SOME more watts, & is especially good for commuters in stop-and-go riding in cities.  The OMEGA type (made by Emerald Island) is better when your electrical needs on the highway are high.   AFAIK: The Euromotoelectrics "Omega style" NON-permanent magnet alternator is NOT the same as the genuine Omega (or, Emerald Island, from Beemershop), it is, instead, a Chinese-made knockoff. 

The latest 450 or 600 watt version of the Omega (Emerald Island) is very powerful, and will handle almost any load you can imagine.  Testing is on this website in its own article.  The Omega (Emerald island) alternator is basically an oversize version of the stock Bosch....by oversize I mean the rotor, the stator, the VR, the diode board, ETC.  The latest version has more stator poles too.  Sources are:
Motorrad Elektrik:     www.motoelekt.com
&
Ted Porter's  Beemershop:  http://Beemershop.com
 


Revisions:

12/21/2005:  final editing and release
03/12/2006:  Add more information on EnDuraLast and Omega
10/17/2006:  Edited for clarity and typos
01/30/2009:  Rechecked
02/23/2009:  Clarify rotor details
06/05/2010:  add note regarding one 1976 seen with 105 mm stator
05/07/2011:  minor clarifications
09/19/2012:  Updated considerably, mostly for clarity but add the 107/105 versus the 105/105 stator information; add QR code; update Google code
11/01/2012:  Add section on Why 3 Phase.
2013:  remove language button due to problems on some browsers
01/01/2016:   Update meta-codes.  Increase font sizes A/R.  Left justification changes.  Narrow article. Clean up article.
05/06/2016:  Final update on metacoding, fonts, colors, layout.  Clarify a number of details.

copyright, 2014, R. Fleischer
 

Return to Technical Articles LIST Page

Return to HomePage

Last check/edit: Saturday, August 06, 2016