The Polyphase Motor/Generator

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[hand crank]
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The Real Story:
OK, I've wanted to build a hand cranked generator and use it to bright up a light bulb ever since I was in the fourth grade. Now, pushing 40 years later, I'm fiddling with this Tesla Turbine project, and I finally have a good excuse to figure out how to do it. That being I want something to drive with upcoming versions of the turbine in order to do some simple power capacity tests, and also something to use while figuring out fuzzy logic control circuits for hybrid vehicle battery charging systems.

In keeping with the trend in modern electric and hybrid/electric vehicle technology, the motor/generator to be discussed in this report is a polyphase brushless permanent magnet design. Polyphase brushless permanent magnet motors are becoming more popular for use in modern electric and hybrid/electric vehicles than the magnetless and single-phase varieties because they are easier to start, easier to control, easier to drive, and tend to have greater power densities. They also make a perfectly fine generator, so they can be used for regenerative breaking and battery charging in an electric vehicle, as well as for the drive motor.

Three-phase is the most commonly found polyphase system, and, the model motor/generator presented below is a 3-phase unit. (By the way, Nikola Tesla is much more famous for his invention of polyphase motor/generators, and AC power generation and distribution than he is for his invention of the boundary layer turbine.) The motor/generator described here is also a "pancake" design, having a flat disc rotor with magnets on the disc face, rather than spaced around the rotor circumference. And, similarly, it has a flat armature, with coils mounted to face the rotor magnets. While permanent magnet brushless motors can in general be made smaller than other styles of motor for the same power capability, the pancake style motor can provide even greater space savings over more conventional cylindrical motor designs.

The entire model three-phase brushless motor generator construction project is documented in this report. That documentation includes the making of a winder for the armature coils, and the collection of magnetite (eventually not) used to fill the armature coil center holes. Dimensioned drawings for the basic units of the motor/generator are also provided.

The theory of operation for three-phase motor/generators in general, and the brushless permanent magnet variety in particular, are provided in the final section of this report. The theory of operation for 3-phase brushless motor driver circuits is presented there as well. Schematics for circuits used in the model motor/generator design covered in this report are also included in the final section.

Coil Winder:
If you're going to build a generator or a motor, then you're going to need some coils. Even with permanent magnet rotors as used here, you'll still need armature windings. Its pretty hard to get out much power otherwise. Some kind of winder will make the task of producing consistent coils much easier. The photos below show a simple coil winder, which includes a tally counter for keeping track of the number of windings, and a threaded crank shaft that allows attachment of arbitrarily shaped coil forms.

Besides 2"x2" wood stock, the winder is constructed mainly using 3/16" threaded brass rod, 3/16" smooth brass rod, thin wall brass tubing, assorted pieces of 0.08" polystyrene sheet, a piece of aluminum window frame (cut to form the cam which moves the tally counter push rod), some thin aluminum strips to hold the tally counter in place, a piece of 3/4" PVC pipe to hold the wire spool, and some wood screws.

The push rod hole and the crank shaft hole are sleeved with thin wall brass tubing. The portion of the threaded rod which passes through the crank shaft hole is itself sleeved with brass tubing, so it is tubing to tubing, not tubing to thread contact on the turning parts. The holes in the cam piece are threaded so it is effectively double nutted in place by the nuts which hold the cam to the shaft. The winding handle has a free turning piece of brass rod over a piece of threaded rod for easy cranking. There is a cotter pin through the brass rod which pushes the tally counter lever. This cotter pin backs up against a flat washer when the tally counter lever rebounds after being pressed, and prevents the spring loading of the tally counter lever from popping the push rod through its sleeve and into your lap while you are winding coils. A nail can be slipped into a snug fit hole in the wooden frame to hold the crank at its zero start position while the counter is reset and the coil form is threaded.

The coil form shown is for simple round coils. It consists of a piece of polystyrene tube, two washers cut from 0.08" polystyrene sheet, and two metal washers. The length of the polystyrene tube is the desired thickness of the coils to be wound (0.25" in this case), the OD of the tube is the desired ID for the coils to be wound (0.375" in this case), and the OD of the polystyrene washers is the desired OD for the coils to be wound (1.125" in this case). The metal washers backup the polystyrene washers to prevent them from spreading as the coil is wound. One of the polystyrene washers has a slot cut in it to allow threading of the magnet wire used to wind the coil. That same washer has a groove filed in it which extends from the inner end of the wire slot to the outer edge of the washer. The groove allows the wire to be threaded under the metal washer without crushing the wire then the wingnuts which hold the coil form on the shaft are tightened. That's 24 AWG wire, in case you're wondering.

[coil winder] [coil winder] [coil winder] [coil winder]

Magnetite Mining:
It can be a good idea to make armature coils which have their center holes filled with a metal core. But, it can be a bad idea to use a solid chunk of metal for this purpose. The reason for this (the reason why most transformers are built with laminated rather than solid metal cores) is covered briefly in the "How's it work?" section at the end of this report.

On one of the myriad do-it-yourself alternative power websites [1] someone pointed out that a paste of magnetite, (a common mineral of naturally occurring iron and iron oxides, most often found as small sand sized particles), and polyester resin, could be a good substitute for trying to make laminated sheet metal coil cores. They obtained their magnetite by dragging a magnet around on their driveway. I tried that, and it worked, but wasn't very satisfying, so I picked up my pail and spade and headed to the beach to play in the sand.

Pouring sand over a salvage magnet found the stuff just fine. The magnetite grains cling to the magnet, and the nonferromagnetic material just passes on by. The salvage magnet I used consists of two steel plates sandwiching a plate magnet, with that assembly surrounded by a square, U-shaped piece of heavy sheet metal. There is about a 3/8" square gap between the base of the sheet metal "U" and the magnet sandwiched between the steel plates. You can see magnetite collecting on the edges of the steel plates in the photos below, but the greater collection point was in the gap at the base of the sheet metal U. After pouring sand over the magnet for a while, the magnetite collected in the gap could be worried out with a small stick.

The first pour collection material contains a lot of low quality magnetite. Once a couple of hand fulls of this material was gathered, it was refined by passing a small, but very powerful neodymium magnet an inch or so over the pile, and subcollecting the material that jumped to the small magnet. The pile was stirred repeatedly, and small magnet passed over the stirred material until no more magnetite jumped to the magnet. This refining process was performed on the subcollected material, and again though several iterations of subcollections. The result was a very pure collection of magnetite.

[magnetite] [magnetite] [magnetite]

During the two hours or so it took to collect and refine an amount of relatively pure magnetite sufficient for this motor/generator project, I had a tiny epiphany..."Hey, this stuff looks just like the leavings that come out of a break drum lathe!"... Break drums and rotors are cast iron, so, in fact, the leavings should work much like magnetite. Having ready access to a such a lathe, I went back to the shop and took another 45 seconds to gather a couple of quarts of leavings. This material is what I eventually used to fill my armature coil core holes.

[cast iron] [cast iron]

Model 3-Phase Motor/Generator Construction:
Armature:

Besides having armature coils filled with a metal core, (as mentioned in the Magnetite Mining section), it is also a good idea to have what is sometimes referred to as "backing iron" behind the coils. As with filling the coil cores, this metal should be disconnected laminations. On yet another of the myriad do-it-yourself home energy websites [2] someone mentioned that coat hanger wire (which generally has some form of lacquer or plastic coating) could work well for filling coil cores. And, there is no reason why that can't work for backing iron, too. You just have to cut a shallow groove in the armature frame piece, (here I used 3/8" plywood, and spun it on a lathe), making the groove wider than the width of the coils, lay in pieces of coat hanger wire, and fill the groove with polyester resin to seal everything in. The coils can then simply be glued in place over the sealed in coat hanger wire pieces with more polyester resin.

This does work OK. But, don't try to use the springyness of the coat hanger wire to hold things tightly in place. The fourth or fifth time it all jumps out on the ground for you, you'll get the idea. Just bend the pieces to sit in place on their own. And, if you don't have a wire bending jig, don't get too fussy about the fit, unless you want to take a real, real, real long time. (Hey, we're not building a watch here.)

I think if I do it again I'll use the coated metal banding used to strap heavy items on to pallets for shipping. It's about 0.015 inches thick, and you can get it in 100 foot lengths on a roll-off dispenser from industrial supply houses (like McMaster-Carr) for relatively cheap (around $35.00). Cutting the strapping material into pieces a few feet long, coiling them into the groove in the armature frame piece and sealing them in with polyester resin should work reasonably well. I may also investigate using the same magnetite/resin paste used to fill the coil cores for backing iron.

A hole was drilled in the center of the backing iron groove circle for the axle to pass through. The backing iron groove was turned in high enough up the armature frame piece to allow clearance for the rotor to spin freely on its axle when the entire motor/generator unit is assembled on its base.

[armature] [armature] [armature]

Rotor:

Construction of the 6" diameter rotor was very similar to construction of the armature support piece. As for the armature coils, it is also a good idea to have backing iron in a groove behind the magnets. But, with permanent magnets, the need for laminations in the backing iron is much less than for coils, and a plate of metal will do. I just cut a ring of about 18 gauge steel from an old chunk of ventilation duct for this. The magnets stick to the metal plate, and are spaced around the face of the rotor (alternating north and south poles up) by inserting them into holes drilled into a ring of 0.08" polystyrene plastic sheet material cut to fit in the groove above the backing iron ring.

The magnets used here are fairly powerful 3/4" diameter by 1/4" thick ferrite types. Without the plastic spacer it would be very difficult to set the magnets in place, unless you glue them individually, holding each one and waiting for the glue to dry before setting the next one (quick set super glue would be a plus there). The spacing of the magnets in the exploded view rotor assembly photo below isn't just for aesthetics. That is about as close as you can place them together without a sudden catastrophe.

The backing iron disc, polystyrene retainer disc, and the magnets are held in place with hot melt glue. Since the forces trying to tear them out when the rotor is spinning are all radial, and all the components are retained in the groove in the 3/8" plywood rotor disc, they really don't require more than that to hold them in place.

It would be desirable to use neodymium magnets for the rotor. Neodymium magnets much more powerful and also much thinner than the ferrites used here are available for less than $2.00 apiece (check on e-bay). Using those it would be possible to smear a layer of polyester resin over the whole assembly to hold the rotor components in place. Then there would be no danger of the assembly coming apart while spinning. (I'll probably make that change someday.)

[rotor] [rotor]

Balancing the rotor was accomplished through spinning it by hand on its axle and bearing assembly and looking for the heavy spots. One piece of adhesive backed lead (available from hobby shops for balancing model cars and airplanes) needed to be cut and placed on the back side of the disc, opposite the position that always ended up down when the rotation stopped. This resulted in quite good balance of the rotor. Besides using its own adhesive, the lead piece was glued in place with a glob of five-minute epoxy, and a glob of five-minute epoxy the same volume as used for the lead piece was placed on the opposite edge of the disc from the lead piece to ensure the balance wasn't changed by the glue. (In the picture showing the lead balance piece, the bit of stuff visible in the shaft hole in the hub is the rolled back edge of a piece of thin brass shim material.)

[rotor]

Axle:

The axle itself is a piece of 1/4" diameter cold rolled steel rod. A hub to hold the rotor was made by cutting a 1-11/16" diameter disc from 3/16" steel, center drilling it for the axle shaft to slide through, and drilling 4 holes around the edge of the disc, tapping them for 1/4"-20 screws. (These holes accept the screws used to attach the rotor disc to the hub.) A piece of 5/8" diameter rod 2" long was bored to accept the 1/4" shaft rod and also drilled and tapped for two 1/4"-20 screws to clamp the hub to the axle shaft. The 2" long drilled rod was then welded in place over the center hole in the 1-11/16" diameter disc to complete the hub assembly. Two 1/4" ID collars with setscrews are included to hold the bearings into their pockets and lock the axle in place on the motor/generator frame. Since the inexpensive variety of cold rolled steel rod used here is not exact diameter, it had to be chucked up in a drill press and buffed with emery paper to get everything to slide on smoothly.

[frame]

Frame:

One end of the motor/generator frame is the armature assembly itself. The other end is another piece of 3/8" plywood. The base was cut 4 inches long from a piece of 1"x12" board. This piece was cut to taper from 6-1/2" on the wide (armature) end to 4-1/2" on the thin end, so it would be pretty. The end pieces are held onto the base with wood screws. Rubber lamp base feet were screwed in place at each corner on the underside of the frame end pieces. (Pilot holes were drilled first to avoid splitting the plywood.)

[frame] [frame]

Bearings:

The bearings are high speed ball type 3/4" OD x 1/4" ID, with good thrust capacity (Fafnir S3KDD). They are held in place by setscrew collars on the axle shaft. The collars hold them in 3/4" pockets cut into the outside faces of the frame end pieces with a Forstner bit. With the pockets cut on the outside faces, not only is the axle locked in place on the frame, the frame pieces are held together by the tension of the collars holding the bearings into their pockets.

[bearings] [bearings]

Coil Assembly:

The motor/generator uses twelve 1.125" diameter, 0.25" thick, 90 turn, 21 AWG enameled magnet wire round coils, wound on the previously described coil winder around a 0.375" diameter polystyrene tube. All coils were wound in the same direction and so that the last turn comes out on the same side of the coil as the first turn. (Other than for those requirements, the wire layering pattern is somewhat random.)

For each coil, the wire extending from the first turn was arbitrarily chosen to be the "positive" coil lead, and a small loop was twisted in its end in order to identify it after removing the coil from the winder. Prior to removing a coil from the coil winder a few inches of fiber reinforced strapping tape was wrapped around the circumference of its windings to hold them in place. The reinforced tape was chosen because it holds well, even when moist, and also because it tears uniformly lengthwise so that a width of tape that covered the spread of the coils but did not adhere to the coil forms could easily be obtained without having to actually try to cut the tape lengthwise. Once the windings were taped in place the wire extending from the spool was cut, (becoming the "negative" coil lead), and the coil was removed from the winding form, with no danger of the coil unwinding itself. The 0.375" center tube was then pressed out of the coil.

To keep coils from unraveling while applying the tape wrap, the nail which holds the crank handle in the start position was reinserted in its hole in the winder frame to stop the coil form from turning, and rubber bands were placed on the wire spool spindle so that they were snug against each side of the spool, thereby maintaining enough drag to keep the tension in the wire from turning the spool. The rubber band tensioners were left in place during the winding operation.

[coils] [coils] [coils]

Once twelve coils were formed, a paste of activated polyester resin and the cast iron leavings from a break drum lathe was mixed and used to fill the coil center holes, then left to harden. After the coil filling paste had hardened, each coil was dipped in activated polyester resin, then hung and the resin allowed to drip off and leave a thin coating over the coils, which then hardened to completely seal the coil assembly.

[coils] [coils] [coils]

After the coil assemblies hardened, the small loop was cut off each coil's positive lead, the enamel scraped back for about an inch from the cut end, and a piece of red insulation 20 AWG solid core copper hookup wire about 10 inches long was stripped of about an inch of insulation and soldered to each barred coil lead. After the positive leads were each connected to their piece of hookup wire, black insulation 20 AWG solid core was similarly attached to each negative lead. A piece of heat shrink tubing was slid down both leads of each coil and shrunk in place over the splices, leaving the coils ready for attachment to the armature frame.

[coils]

Coil Mounting:

To maintain symmetry for the electrical connections, the coils were mounted on the armature frame backing iron ring so that the face of a coil from which the leads extend was placed against the frame, and the leads all extended radially outward relative to the shaft hole. The coils were set in place according to a scallop cut paper pattern; initially sticking them down with a glob of hot melt glue. After the coils were hot melt glued in place the paper placement pattern was removed, and a thin cardboard dam, lined with waxed paper (which does not stick to hardening polyester resin), was taped in place, surrounding the coils with about 3/16" clearance from their outward edges. Similarly a waxed paper covered thin cardboard tube was inserted in the shaft hole. With the armature frame lying flat and level, the area inside the dam was poured with activated polyester resin to a depth of about 1/8", making sure that the liquid resin completely surrounded each coil. When the resin hardened, the cardboard dam and shaft hole plugs were pulled away, leaving the coils permanently attached to the armature frame.

Note in the photo showing the paper coil placement pattern, that the armature frame has been predrilled for the coil lead wires and phase wire connection screw platforms. (This would be very difficult to do after the coils were attached.) Note also that the coils were placed so that their leads align with one of the two small holes between the sets of two larger holes in the predrilled hole pattern. The small holes are for passing the lead wires through the armature frame. The larger holes are for the screws which mount the phase wire connection platforms to the back of the armature frame.

[mounting] [mounting] [mounting] [mounting]

Coil Lead Connection:

The coil leads are passed through their adjacent feed through holes in the armature frame and connected to the phase lead screw connector platforms by soldering them to tabs on the underside of the screw platforms. To allow for consistent phase lead wire connection, the coil lead wires are connected in repeated red/black order all the way around the armature frame.

The phase lead screw connector platforms themselves snap into a pair of nylon standoffs, and the standoffs are mounted on the backside of the armature frame with 4-40 brass screws and nuts. Each platform has 4 solder tabs, allowing two coils to be connected. So, six platforms total were required to connect all 12 coils.

[connections] [connections] [connections] [connections]

Final Assembly:

The motor/generator is assembled by screwing the armature frame and the opposite end bearing support to the base block, inserting the bearings into their pockets in the bearing supports, sliding the 1/4" steel rod axle through the shaft hole in the bearing on one of the bearing supports, sliding the rotor hub onto the shaft end between the armature frame and opposite end bearing mount so the rotor magnets face the armature coils, then sliding the shaft on through the second bearing and slipping a set screw collar onto each end of the shaft, positioning the shaft for the desired extension of the shaft through the bearings, sliding the set screw collars up to the bearings and tightening their set screws to lock the shaft in place, and finally positioning the rotor on the shaft and tightening its shaft screws to clamp the rotor in place on the shaft.

During assembly you need to be careful to not let the rotor magnets cause the unclamped rotor to jump down the shaft and hit the armature coils. Even with ferrite magnets the attractive force is significant. If you use neodymium magnets the attraction could well be strong enough to cause the components to slam together hard enough to damage the magnets, the coils, or your fingers (no joke). Probably the easiest way to deal with the problem is to immediately tighten one of the rotor hub screws after the shaft is inserted though the hub and both bearings, then slide the set screw collar on to the end of the shaft opposite the armature frame up to the bearing and tighten its set screw. The placements of the shaft and rotor won't necessarily be correct, but the rotor will be trapped in place and position adjustments are easy to make safely from that point. Once everything is properly positioned, give all the screws on last tightening check.

There is an optimal distance for the separation between the rotor magnets and the armature coils. For the purposes of building a model like this you can consider that to be about as close as possible without the magnets hitting the coils when the rotor turns.

The rotor and armature should be as parallel as possible. To improve its parallel alignment with the rotor, a small adjustment of the armature frame was made by loosening the screws holding the frame to the base, placing two narrow strips of 0.01" thick polystyrene sheet plastic face to face in the upper side of the gap between the armature frame and the base block, then retightening the frame screws.

[mot/gen] [mot/gen] [mot/gen]

Phase Wiring:

In a three phase system, the phases are electrically 120 degrees apart. But, the armature coils are not (necessarily) physically spaced at 120 degree intervals around the armature frame. The total number of coils in the armature and the number of coils used for each phase affects the physical placement of coils around the armature frame.

For the motor/generator being described here, there are twelve armature coils, which allows 4 coils to be connected in series for each of the three phases. (Thus, each phase can provide four times more voltage than would be obtained using a single coil for each phase.) And, 12 coils means they will be spaced at thirty degree intervals around the 360 degree perimeter of the armature frame.

The phases are referred to as A, B, C, and, starting from the right one of the topmost pair of coils on the armature frame, the coils are identified in the clockwise direction as follows: A1, B1, C1, A2, B2, C2, A3, B3, C3, A4, B4, C4, where C4 is the left one of the topmost pair of coils. Each indexed letter of the alphabet refers to one of the coils in the group of coils that comprises on phase coil set, e.g., coils A1, A2, A3, and A4 are the coils wired together for phase A.

Although each individual coil is only 30 degrees from its closest neighbors, the correspondingly numbered coils for each adjacent phase are, in fact, 120 degrees apart, e.g., four 30 degree spaces separate coils A1 and B1, etc. Thus, adjacent phases are 120 degree apart, with phases A and B being 120 degrees apart, and phases A and C being 240 degrees apart, as is required for a three phase system.

The number and spacing of magnets around the rotor also enters into how the phase coils are wired. To work properly, at any instant in time each coil in a phase must be contributing current to the phase of the same polarity and magnitude. Whether a coil is being energized by the north pole or the south pole of a magnet determines the polarity of the current produced in the coil. The distance in its rotating path a magnet is from a coil determines how much it contributes to the magnitude of the current generated in the coil. (A changing magnetic field is required to generate current in a wire coil, and motion of the magnets results in a changing magnetic field.)

Because there are alternating north and south magnet poles, the separate phase coils cannot be simply connected in series, as the resulting currents generated in the individual coils in a phase will not match. The phase wiring used for the motor/generator being constructed here is as shown in diagram below. The details of why the connections are made as they are can be found in the "How's it Work?" section.

[phase wiring]

Besides how far away from a coil a magnet is in its rotating path, the gap separating the rotor magnets and armature coils when they are aligned also determines the magnitude of the current generated. There are some constraints on how close magnets should be to the coils, but, basically, we're just going to get them as close as the wobbles in construction will allow without anything banging against anything else. Some of the details on determining the gap distance can also be found in the "How's it Work?" section.

[phase wiring] [phase wiring] [phase wiring] [phase wiring]

Preliminary Testing:

Spin it and it makes wiggles. Apparently wiggles of the correct phase separations at that!

[testing] [testing] [testing] [testing] [phase A-B] [phase B-C] [phase A-C]

Upgrades:
Over time a few features have been added to the motor/generator to make its use and testing a bit easier.

Handcrank:

[hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank] [hand crank]

Handle:

[hand crank] [hand crank] [hand crank]

Change ferrites to neodymium magnets:

[new magnets] [new magnets]

[new magnet test] [new magnet test]

3-phase rectifier:

[3 phase rect] [3 phase rect] [3 phase rect] [3 phase rect] [3 phase rect] [3 phase rect] [3 phase rect] [3 phase rect]

[3 phase rect test] [3 phase rect test]

How's It Work?:
OK, if you go through the whole thing, this will seem perhaps a bit long winded and maybe a bit spotty description of how permanent magnet motor/generators work. But, the good news is if you just want to know how to figure out the number of coils you need for a given number of magnets and how to hook them up to successfully produce your own single-phase or polyphase motor/generator, you're in luck! We'll cover that right off the bat with a set of simple instructions and formulas. What follows the basic "how-to" information just fleshes out the details of what you're doing and why you're doing it.

One caveat for the rote instructions. They do assume you know, or know how to determine, the polarity of your coil leads for proper connection. Store bought coils are likely marked for polarity. If you plan to wind your own coils, (and why wouldn't you and miss all the fun?!), then here, without explanation, is how to establish their polarity. Wind them all identically, all coils with turns in the same direction and filling the form you wind them on in the same way. Choose the end of the wire you started with and, arbitrarily, call it the positive (+) polarity lead, and call the other end of the wire coming from the coil the negative (-) lead. Don't worry about which way you start making the turns in your coils, just do what is easiest for your set up. Just be consistent from coil to coil, and all will be fine. Why consistency without measurement works, we'll cover later.

Now, the instructions and formulas given below don't give all possibilities for coil connection and magnet arrangement you might encounter. But, they do provide the information you need to make practical, working, permanent magnet single-phase and polyphase motor/generators.

The same formulas work whether you're building a single-phase or a polyphase motor/generator. But, they become so simple in the single-phase case that we'll cover the single-phase motor/generator separately from the polyphase motor/generator.

Before getting to the actual construction instructions, first let's take a look at a diagramming method to show motor/generator coil connections. The technique, presented in the section that follows, is adapted from some of the figures in a well know book on electric motor repair by Robert Rosenberg [3]. A few other books on the subject of permanent magnet motor/generator design and motor design in general are listed in the references [4, 5,6].

Diagramming motor/generator phase coil connections:

Once you're decided on how many coils to use in your motor/generator, draw a small rectangle, one for each coil, in a row across a piece of paper. To represent the coil leads draw a short line down from the bottom corners of each of your rectangles. You can, but it isn't absolutely necessary, make a dot near the left lead line to indicate the positive polarity coil lead for each rectangle. And, for each phase in your motor generator, label the coil rectangles in a repeating sequence.

Traditionally, phases are labeled by the letters of the alphabet. So, given, for example, nine coils to use in a 3-phase system you would repeat A,B,C in sequence three times, as A,B,C,A,B,C,A,B,C. Or, if you were designing, say, a 5-phase system, you would label your boxes, in sequence, with the letters A though E. If you like, you can identify each phase coil group by appending numbers to the letters. In our case of nine coils in a 3-phase system, the coil box labels would then be, in sequence, A1,B1,C1,A2,B2,C2,A3,B3,C3.

Now, regardless of whether you plan to build a single-phase or a polyphase motor/generator, due to the alternating north and south poles of the rotor magnets, when operating, the direction of current through each coil will be the opposite of the direction of the current through the coil that precedes it. (This is the origin of the term Alternating Current (AC). More on that in the detail sections that follow.) So, under each coil box, between the lead lines, draw a small arrow to represent current direction. Start from the left box with an arrow pointing to the right, and alternate arrow direction with each box that follows.

And, that's it! The complete starting point diagram for a nine coil, 3-phase motor/generator is given in figure 1. For a motor/generator with more or less coils, just draw more or less boxes.

Armed with the base of a diagramming technique for producing a simple picture of what we want to build, we can now get on with how to actually find the number of coils we need, and how to hook them up.

[coil diagramming]

Coil count and connections for a single-phase motor/generator
  1. Dig through your junk box and find all your magnets.

  2. Make piles of the same types of magnets

  3. Choose the pile containing the type of magnet you want to use in your single-phase motor/generator.

  4. Count the magnets in the chosen pile. If it is an odd number, toss one magnet back in the junk box, or go find another one, so that you end up with an even number of magnets.

  5. Make the same number of coils as you have chosen magnets.

  6. Lay your coils out in a row so that their positive lead is on the same side of each coil in the row, and connect them in series so that the negative lead of the first coil connects to the negative lead of the second coil, the positive lead of the second coil connects to the positive lead of the third coil, and so on. The proper connection is simple to show using the diagramming technique presented above. Just draw lines to connect the coil boxes so that the lead nearest the head of one box's arrow connects to the lead nearest the tail of the next box's arrow (figure 2). Physically making this connection relative to the positive leads as drawn on your coil diagram guarantees the alternating current flow directions in the coils do not oppose each other.

    [single phase coils]

  7. Build your motor/generator with magnets alternating north and south poles evenly spaced around it's rotor, and coils evenly spaced around it's armature, as, for example, shown in figure 3 for an eight magnet, 8 coil, single-phase motor/generator.
[single phase build]

OK, there is a bit of a misleading implication above. In steps (6) and (7) above there is no differentiation between connecting coils for a single-phase motor or for a single-phase generator. The connection will work just fine as described for a generator. And, in fact, as described, the connection will also work for a motor. Connect an AC power source to a motor/generator with an armature wired as in figure 2, and it can be made to spin via the input power. But, it will not self start. You will need to give it a bump to get it to go. A single-phase motor can be connected to self start. We'll leave that discussion until after we talk in more detail about polyphase motor/generators.

Coil count and connections for a polyphase motor/generator:
  1. Dig through your junk boxes and find all your magnets.

  2. Make piles of the same types of magnets

  3. Choose the pile containing the type of magnet you want to use in your polyphase motor/generator.

  4. Count the magnets in the chosen pile. If it is an odd number, toss one magnet back in the junk box, or, go find another one, so that you end up with an even number of magnets.

  5. Decide how many phases you want to use in your polyphase motor/generator. Most everyone on the planet chooses three, and, I'm going to choose 3 for this discussion, so you might as well, too. You don't have to choose three phases, but at least choose an odd number. (To show this technique does generalize I'll diagram hooking up a five-phase motor/generator after the 3-phase discussion.)

  6. Calculate the number of coils required:

  7. Calculate the number of coils per phase:
      For C coils and N phases, the number of coils per phase is C/N. Here, for our 8 pole, 3-phase motor/generator we get 12/3 = 4 coils per phase. This means you will be connecting three groups of 4 coils each in your motor/generator.

  8. Select the motor/generator wiring configuration:

  9. Diagram the phase connections:

  10. Build it. Figure 6 shows an example physical diagram for a three-phase, 8-pole, 12-coil, Delta connected motor/generator.
[polyphase build]

If you use your motor/generator in motor mode and want it to spin in the opposite direction from what you will find with the motor/generator design given here, you can reverse any two of the input phase connections, and the motor will run in reverse. We'll see why that is later. (And, of course, you could also physically turn the motor around 180 degrees so that the end of the shaft facing you becomes the end of the shaft pointing away from you.)

There is no issue with self starting of polyphase motors. Connect three-phase power to a 3-phase motor/generator and it will run as a motor without any of the startup connection tricks required for a self starting single-phase motor to be discussed later.

Of magnets and wires and such:

Getting into the spin:

First thing, magnets. Most everyone has played with magnets at one time or another. Point the opposite poles of two magnets at each other, north to south, and they attract. Point the same poles at each other (north to north, or south to south) and they repel. Interaction of the magnetic fields produced by the magnets is what causes the attraction and repulsion effects observed for a pair of magnets.

The same as it is for the magnetic field of the earth, the magnetic field of a simple bar magnet is described as lines of magnetic flux which extend from a magnet's north pole to its south pole (figure 7). (It might be good to note here that the north geographic pole of the earth is actually a south magnetic pole. Which explains why the north pole of the magnet in a compass points toward the earth's geographic north pole.)

[magnet flux lines]

Considering the magnet flux lines as drawn in figure 7, magnetic attraction and repulsion can be viewed as the interaction of the arrows (or vectors) giving the direction of the flux lines from north pole to south pole. When two north or two south poles are brought next to each other, the arrows from one magnet's pole point in the opposite direction of those from the other magnet's pole, and, hence, "collide," pushing each other away, causing the repulsion effect. But, when a north pole and a south pole are brought together, the arrows point in the same direction, and one set "sucks" the other set along, much like two streams of water flowing in the same direction, which causes the attraction effect. (This is, of course, just a description for visualization purposes.)

Magnetic repulsion and attraction are the basis of operation for an electric motor. By proper alignment and timing of magnetic fields, the parts of an electric motor can be made to push and pull on each other so that a smooth continuous motion is obtained.

Besides having a magnet handy, another way to generate a magnetic field is to run an electric current through a piece of wire. The flow of current through a wire produces lines of magnetic flux around the wire just like the lines of magnetic flux produced by a magnet. The lines of flux are produced such that when you point your left thumb along the wire in the direction of current flow through the wire, your left fingers will curl around the wire in the same direction as the lines of flux around the wire, i.e., you finger tips will point in the same direction as the arrows seen in a diagram of the flux lines (figure 8).

[left hand rule]

Obviously, if the direction of current through the wire is reversed, then you have to point your left thumb in the opposite direction, and, as shown by the curve of your fingers, the lines of flux around the wire will be in the opposite direction, effectively changing the polarity of the magnetic field around the wire.

We aren't going to get deeply into what constitutes the "real" direction of current flow in a wire. As has been pointed out above, consistency is what counts. Make your favorite assumption, use it without fail, and all will be well.

Here we'll consistently consider current flow to be from the negative terminal of a power supply to the positive terminal of a power supply. That means we're looking at electron flow, and not current flow as it was originally assigned to be from positive to negative for the first electric batteries, (a concept based on Benjamin Franklin's earlier description of two types of electricity, "positive" and "negative," which, well, frankly, some would say he got backwards). Engineers and physicists can argue about what is really going on all they want, and we'll just get on with the task at hand. Symmetry, which we'll talk about more a bit later, is the key to why, for practical purposes, the initial choice doesn't really matter. In fact, if you are more comfortable considering current flow to be from positive to negative then consistently use that idea, and instead of using the left-hand rule as described above, just use your right hand to apply the right-hand rule. Then following the curve to your right finger tips will get you the same results as we've seen for the left-hand rule. That's symmetry!

Now, the nature of magnetic poles around a straight piece of wire with current flowing through it isn't immediately obvious. But, if you turn the wire into a loop, the orientation of magnetic poles relative to the wire becomes more clear. As shown in figure 9, with current flow from left to right and the loop directed into the page, the flux lines circulate into the top of the loop and out of the bottom. Comparing this to the flux lines seen for the magnet in figure 7, we can see this implies the north pole of the loop's magnetic field is at the bottom, and its south pole is at the top.

[wire loop flux lines]

The effect becomes more clear when there are multiple loops stacked on top of each other to form a coil. Since the flux lines produced by each loop combine in the same direction they produce a larger (and stronger) magnetic field (figure 10). This is why more turns in the coil of an electromagnet make for a stronger electromagnet.

[wire coil flux lines]

As can be seen in figure 10, the same repulsion and attraction of flux lines can be had from the coil as from a real magnet. By reversing the direction of current through the coil the orientation of its north and south poles is reversed. And, by taking this action in to account, simple electric motors can be devised. For example, with proper timing of the change in direction of current through two coils, a magnet on an axle can be made to rotate between the coils (figure 11).

[simple electric motor]

Going the other way:

OK, in the last section we made it as far as a good idea of how to make a simple electric motor. A lot of details need to be filled in yet. Like, in particular, how to control switching and timing of the magnetic fields, but it's a definite start. And, with the information we put together there about magnets and wires and such, we are in good shape to move ahead with an initial description of how electric generators work.

As mentioned before, the aspects of physics we're looking at are symmetric. So, just as running a current through a wire creates a magnetic field, moving a magnet near a wire creates an electric current in the wire (figure 12).

[magnet creates current]

Of course, the more wires a moving magnetic field crosses, the more currents that are generated, one for each wire (figure 13). And, when a wire is wound in a coil, the effect is the same for each loop in the coil, except, because the coils are connected, the multiple currents created as the magnetic field passes over the coil windings add together. So, the more windings in the coil, the more current generated as the magnetic field passes by (figure 14). This is the basic idea behind an electric generator.

[multiple wires multiple currents]


[coiled wire bigger current]

Also, the stronger the moving magnetic field, the more current generated in any wire it moves by. So, either by putting more turns in its coils, or by using stronger magnetic fields, or both, the more power that can be produced from a generator or utilized from a motor.

We've already see examples of making simple permanent magnet generators in the coil count and connection sections above. So, let's continue on and investigate a few more details that will lead us to an understanding of why we make the connections we do as previously described for single-phase and three-phase delta motor/generators, and also how to define connections for other than single and three-phase systems.

Pick a phase, any phase:

OK, so what does "phase" mean, anyway? Well, it's another one of those words that changes meaning with context. Regarding polyphase motor/generators, it is often used interchangeably to describe two main features. It can mean one of the multiple leads (or the lead's associated electrical waveform) in a polyphase system, or it can mean the difference (measured in degrees) between the peaks seen in the waveforms found in any two leads in a polyphase system. For the second case that difference is more correctly referred to as the "phase angle," which, here, is a term that relates to the notion of a rotating magnetic field. The rotating magnetic field concept is at the heart of any motor/generator system, be it single or polyphase.

That we are dealing with rotating magnetic fields seems easy enough to accept. The devices we've been discussing have magnets that spin on an axle. So, there it is, rotating magnetic fields. Pretty unavoidable, that. Of course, as always, the devil is in the details.

Sinusoids:

Rotation implies circles. But, to draw a circle covering the 360 degrees of each rotation to represent the motion over time of a motor/generator that may be spinning at thousands of revolutions per minute wouldn't be too informative. So, instead, we draw a sinusoid, which is just a way of displaying circular motion in a linear diagram.

To see this, consider a simple machine consisting of a disc with a peg at its edge parallel the disc's center axle, and a "T" shaped piece which has a slot in its crossbar. The T-bar is laid on the disc so that the crossbar slot fits over the peg, and the upright points down when the machine is looked at in plan view. A guide aligned with the center of the disc is placed over the T-bar upright so that while the peg on the disc can slide back and forth in the slot in the T-bar crosspiece as the disc rotates, the T-bar upright can move up and down, but not side to side. So, while the disc rotates, the peg moves in a circular motion which forces the T-bar to move so that its end travels up and down a distance equal to the diameter of the circle defined by the motion of the peg (figure 15).

[circle to line machine]

If a pencil is attached to the end of the T-bar upright piece so that it will leave a mark on a piece of paper placed under it, then, as the disc rotates the pencil will draw a straight line with length equal to the diameter of the peg circle each time the disc rotates. We can spin the disc until the pencil wears out, but that is all we will see, a single vertical line. However, if we move the paper parallel to the T-bar cross piece, we'll see something quite different, that is, a repeating curve, called a sinusoid, which gives the diameter of the peg circle via its amplitude, and repeats its waveform across the paper through a number of cycles that relates to the rotational speed of the disc and the speed of motion of the paper under the pencil

If we know the speed of the paper, then we can deduce the rotation speed of the disc by counting the number of cycles in a given distance across the page, equating distance to the time it took the paper to travel that distance, and dividing the number of cycles see in the sinusoid over the given distance by the equivalent time for that distance. The result is disc rotation speed expressed as frequency in units of cycles (rotations) per chosen time unit.

If we arbitrarily designated the start (0 degrees) position as when the peg is at its topmost position when the disc rotates, and start moving the paper to the right just when the peg reaches that point, then the pencil on our simple machine will trace out a sinusoid that represents the circular motion of the disc as a "wave" that has a positive maximum value at the start time position, drops to zero when the disc has rotated 90 degrees, reaches a maximum negative value as disc rotates through 180 degrees, returns to 0 when the rotation reaches 270 degrees, and climbs back to the maximum positive value as the rotation goes though 360 degrees. The 360 degree rotation returns the pencil to the maximum positive position, but a distance across the page that equates to one rotation time, and the cycle repeats (figure 16).

[creating a sinusoid]

The time for a sinusoid to complete one rotation cycle is referred to as its period (T). The number of cycles the sinusoid completes in a fixed time period is referred to as its frequency (f). Traditionally the time period to count cycles to determine frequency is one second, and frequency is expressed in cycles per second or Hertz (Hz). Note that frequency is the inverse of period, that is, divide 1 by the period value, and you get the frequency value, or, divide 1 by the frequency value and you get the period value:

f = 1/T, T = 1/f.

Rather than using spinning discs with sliding pencils to diagram rotations, the trigonometric function cosine has been defined to represent just this motion. In a mathematical formula the cosine function is given as cos(x), where the argument x represents the degrees of rotation. The cosine function follows the same form of curve as described by the pencil and disc arrangement of figure 16; having a value of 1 for an argument of 0 degrees, a value of 0 for an argument of 90 degrees, a value of -1 for an argument of 180 degrees, a value of zero for an argument of 270 degrees, and returning back to 1 for an argument of 360 degrees.

The amplitude of the curve drawn by the pencil and disc method is equal to the diameter of the disc. But, since we have defined things to range over plus and minus values, and the cosine function ranges from 1 to -1, we multiply the cosine function by half the diameter (the radius) of the disc to get the proper values for the curve relative to the degrees argument to the cosine function. That is, if the diameter of the disc is A, then at any particular degrees of rotation value, x, the proper height, h, of the sinusoid is given by:

h = (A/2)cos(x)

We can't make the mistake of thinking that the diameter of the disc mentioned above has anything to do with the diameter of our motor/generator rotor disc. The "diameter" we are really talking about is simply the amplitude of the sinusoidal electrical waveforms applied to, or generated by, our motor/generator. In fact, without any loss of usefulness, we can drop the division by 2 in the amplitude factor, and just use the parameter A alone to indicate we want to allow sinusoids with an amplitude range other than from -1 to +1, assume the value of A will be chosen correctly when necessary, and go with the slightly simpler equation:

h = Acos(x)

Now we almost have a mathematical formula that can be used to diagram the phase waveforms seen in our motor/generators. But, there is still that matter of x, the unknown angle. If we just wanted to plot one sinusoid, then varying x from 0 to 360 and plotting the result for h as the y value in an x-y plot would do fine. But, that isn't going to give us what we need when it comes to looking at the phase angle between phase waveforms. Also, often we won't want to plot with degrees on the x axis of our plots, but rather with time. Fortunately, since we are dealing with known rotation speeds, (i.e., the frequency of the sinusoidal waveforms in question, e.g., the 60 cycles per second (60 Hz) for commercial power found around here), that's relatively easy to figure out.

So, back to considering imaginary spinning discs and sliding pencils for a bit. Another way of measuring angles is in radians. There are 2pi radians in 360 degrees (making one radian approximately 57.297 degrees). So, one rotation of the disc in figure 15 means the peg which moves the crossbar sweeps through an angle of 2pi radians. Radian measure is just a scaling factor different from degree measure, and the cosine function can be defined to use radians as an argument as well as degrees. For the time being we'll use that definition. It might seem an added confusion to switch from degrees to radians, but, that is really just a matter of what you are used to. We gain something by switching to radian measure, that is, when coupled with the concept of radial velocity, the ability to use time as the argument to the cosine function.

Radial velocity (w) is a measure of the speed of rotation of a disc in terms of radians per second, rather than cycles per second. Again, we aren't really talking about speed of rotation of a disc, but the "speed" with which a sinusoidal waveform completes its repeated identical excursions. The concept of radial velocity still applies, since that sinusoid has the same circular motion related form as the one produced by the disc and pen method of figure 15. In either case, multiplying the radial velocity by time (measured in seconds) gives us radians as a proper argument to the cosine function, and, thus, knowing the radial velocity, we can plot our sinusoid against time on the x-axis of an x-y plot, rather than degrees. That brings us to a new form for our equation:

h = Acos(wt)

A bit more work with this expression will lead us to the ability to check the phase angle between sinusoids at a particular time, which is a lot easier to work with than trying to check phase angles related to portions of a cycle, which would be the case if we used degrees and frequency in our equations, rather than radians and radial velocity.

Before we move on to checking phase differences, there is one more important relationship to note, the connection between frequency and radial velocity. The unit Hz (cycles per second) refers to how many rotations per second we see for our rotating disc or how many full excursions per second we see in our sinusoidal wave form. In either case, one rotation or one sinusoidal excursion, we can equate the frequency to sweeping an angle of 360 degrees with every cycle. So, for example, we could say that a 60 Hz signal is also a 60*360 = 21600 degrees per second signal. Of course, that isn't a very useful piece of information. We need some way to relate the degrees of sweep per second for a signal frequency measured in cycles per second to radial velocity so that we can apply the value we know, frequency, in our equation that uses radial velocity and time. Radians to the rescue!

To use frequency in our equation in w and t, we need to convert from frequency to radial velocity so that the multiplication by t will give the proper value in radians. That we can do by noting, as pointed out before, one revolution is 2pi radians, and also that one cycle in frequency is one revolution, then multiply the appropriate factors together so that the units cancel out to give radians per second, the proper units for w, i.e,

2(pi)f = w

That leads us to another form for our sinusoidal equation:

h = Acos(2(pi)ft),

which allows us to plot our waveforms against time, while using frequency in the argument to the cosine function.

So, all that verbage just to explain a factor of 2pi? Well, yes and no. The warm up to expressing circular motion as a sinusoid will prove useful later, when we look at making connections in a polyphase motor/generator in terms of vector sums. At least we are done here now, and can get on with the next phase in our discussion.

Phase vs. phase:

I'll state in advance the discussion here may get a little confusing. This is because, as mentioned before, phase is one of those words that changes meaning with context. The main uses for "phase" here will be to mean one of the input/output wires to a motor/generator or one of the phase coil groups as described earlier in the polyphase coil count and connection section (these two meanings are basically synonymous), or also to mean the sinusoidal waveform found in an input/output wire or phase coil group. Another potential confusion may come in use of the terms "phase difference" and "phase angle." The phase difference is the difference between phases. (Well, duh!). That difference is usually measured as an angle, and, so, it is often referred to as phase angle. More correctly, the phase angle is the value of the phase difference, but, whatever, the two terms are used interchangeably elsewhere, and will be here, too.

So, OK, that being that, we're just about there. We have a cosine equation we can use for generating sinusoids, in which we can use variables we know or can determine, time and frequency. Now we need a way to include phase difference into our equation.

The phase difference between two sinusoids is the separation in angle between the same relative point on the waveforms. We measure the difference as an angle because the sinusoids are representations of circular motion and the difference between two points on a circle is normally expressed as an angle. That angle is not quite enough information to describe the difference. We also need to know which waveform comes first. That association is described using the terms "lead" and "lag." When one waveform leads another waveform then the first one reaches the end of a cycle before the second one. When one waveform form lags another waveform, then the first waveform reaches the end of a cycle after the second one (figure 17).

[lead vs. lag]

To include a lead or a lag in our cosine equation, we simply subtract or add a phase angle in the main angle argument. Careful here. The relationship isn't quite intuitive. If we have two sinusoids rotating with the same frequency, and only apply the phase correction to one of the two, then to force a lead by the uncorrected waveform over the uncorrected waveform we subtract the phase angle from the corrected waveform, and to force a lag of the uncorrected waveform we add the phase angle to the corrected waveform (figure 18).

[cosine lead vs. lag]

We've now reached the final form of our cosine equation:

h = Acos(2(pi)ft + alpha),

where the phase angle, alpha, may be positive or negative, and, as used here, should be expressed in radians. If alpha is provided in degrees, then multiply by pi and divide by 180 to convert degrees to radians, which leaves us with the same basic form of the equation:

h = Acos(2(pi)ft + alpha(pi)/180),

In a later section we will be introducing another multiplication factor for the argument in f and t. But, for now we'll ignore that and move on to look at expressing the full set of phase angle relationships for a polyphase motor/generator via our latest equation.

In general, to make a set of plots for phase comparisons, we need an equation for each sinusoid, with each equation expressing the amplitude, frequency, and phase difference for its associated sinusoid. For arbitrary sinusoids each equation would have a completely different set of arguments and multiplication factors. For example, the equation set for three arbitrary sinusoids, A,B,C, would be, with the phase angle expressed in radians:

3(A = Mcos(2(pi)ft + alpha))

Using these equations we could compare three sinusoids of arbitrary magnitude, arbitrary frequency, and arbitrary phase angle at any point in time. But, here we make two simplifying assumptions, that the magnitude of each sinusoid is the same, and the frequency of each sinusoid is the same. Note that with the identical magnitudes assumption we can set the magnitude factor to 1 in each equation with no loss in generality. Also, note that with the identical frequency assumption time is no longer a critical factor if we are only comparing phase differences, and we can make the comparisons at any arbitrary rotation angle, say, PHI. We can make one more simplification that has its origin in standard practice rather than physics, which is when comparing phases, we assume the phase angle for the first phase is zero, and adjust the other phase angles accordingly. That gives us, in our three phase case, the following simplified equation set:

3(A = cos(PHI + alpha))

So, now we just need the phase angles for the B and C sinusoids.

In a polyphase motor/generator the sinusoidal waveforms found in each phase have a fixed angular relationship, determined by the number of phases. Each phase leads the next one by the same angle. That phase difference, P, is simply one full cycle expressed as an angle, 360 degrees or 2pi radians, divided by the number of phases. Using degree measure:

P = 360/N.

Thus, for a three-phase system, we get a fixed phase angle of P = 120 degrees. Of course, it would be just a little to simple if we could use 120 degrees, (or 120pi/180 radians), for the phase angles in our simplified equation set. If we draw three sinusoidal waveforms, A,B,C, each leading the next by 120 degrees, (figure 19) we can find the required phase angle values for our set of cosine equations by examining the plots.

[3-phase lead/lag]

We know we want a phase angle of zero for waveform A. And, traditionally, we also use positive value phase angles. That means we will look for appropriate phase lags to define our phase angles. The positions where the phase A and B waveforms reach their maximum value within one full cycle, with A lagging B, is at PHI = 120 degrees for B and PHI = 360 degrees for A. That gives us a phase difference between A and B of 360 - 120 = 240 degrees, which we use as the phase angle for the B waveform cosine equation. Similarly, the phase difference for A lagging C is 360 - 240 = 120 degrees, which we use as the phase angle for the C waveform cosine equation. That gives us the following equation set:

3(A = cos(PHI + alpha))

which is the standard form for three-phase equations. To determine lag values from the given phase lead differences of 120 degrees without having to plot the waveforms, we can just note, as pointed out earlier, (figure 18), that a phase lead relationship between two waveforms can be reversed by subtracting the phase difference from 360 degrees. So, for A and B, with A leading B by 120 degrees we can also say A lags B by 360 - 120 = 240 degrees. And, since there are 360 degrees in one cycle, if A lags B by 240 degrees and B leads C by 120 degrees, then A must lag C by 120 degrees. Similar analyses can be made for systems with other than three phases.

From a strictly theoretical point of view, that's it! We have now have our complete set of cosine equations for three-phase related sinusoids, which we can use in terms of angle of rotation as in the three equations above, or in terms of time via the radial velocity, which, as we have seen, can be expressed directly, pi, or in terms of frequency, f, if we include the proper conversion factor of 2pi. If we also include an amplitude factor, K, and convert the degree value phase angles to radian values by multiplying by pi and dividing by 180, we get a general set of equations we can use to compare three-phase waveforms of any magnitude and any frequency at any time:

3(A = Kcos(2(pi)ft + alpha))

We'll make use of these equations in one from or another later, when we look at why we make the connections we do between phase coil groups in a polyphase motor/generator.

Mechanical vs. electrical degrees:

A while back I mentioned we'd look at one more factor to apply to the main argument in our cosine equations. And, now the time has come. Here we'll step back a bit from the predominantly theoretical discussions and look at a result that stems from the basic physical arrangement of magnets and coils in our motor/generators. This factor is needed to properly define the phase sinusoids from our cosine equations if we use frequency or radial velocity to mean the rotational speed of our motor/generator rotor. If we use frequency or radial velocity to mean the actual cycle rate of our phase sinusoids, then this factor is not needed in the cosine equations. This discussion assumes we are relating our cosine equations to rotor revolutions. When we relate our equations to the actual phase cycle rate, (e.g., 60 Hz for US commercial electrical power), then the last set of three equations above are the ones we want to use for a three-phase motor/generator.

While it is true we relate cycles in sinusoids to circular motion, and here, spinning of the rotor in a motor/generator does ultimately define that relationship, one rotation of our motor/generator rotor does not equate to one rotation in our phase sinusoids. In fact, for an AC generator, each turn of the rotor will result in a number of sinusoidal cycles in the output phase waveforms. Similarly, a number of input power cycles are required to produce a single turn of the rotor in an AC motor/generator used in motor mode.

Physical turns of a rotor, as well as cycles of a sinusoidal signal can both be measured as an angle in degrees or radians. For the moment we'll use degrees. To differentiate the two motions, we call the angular measure of turns of the rotor "mechanical degrees," and the angular measure of cycles of the sinusoidal waveforms "electrical degrees." For a particular motor/generator design, there is always a fixed ratio between the two.

For the model motor/generator described in the first part of this web page there are eight sinusoidal phase cycles for each turn of the rotor. Links to the source code, (you might want to check the comments), and it's outputs, for a program that computes and plots the electrical degree measure phase waveforms for this motor/generator are provided below:

[PolePhase.c] [3-phase waveforms] [numeric output]

Those of you checking my work may have already noticed that the coil count for the motor/generator I constructed doesn't match the number you would calculate using the rote formulas presented at the start of this "How's it work?" section. That difference is due to consideration of a matter known as cogging, where the drag of magnet fields across the coils affects motor/generator performance. Don't worry about that, we'll get to it in a while. For now let's continue the discussion by looking at the relationship between electrical and mechanical degrees for the eight magnet, 12 coil set we computed earlier for a 3-phase motor/generator using the rote formulas.

Once again symmetry comes into the picture. Referring back to figure 6, we can define a mechanical 0 degrees position for the illustrated motor/generator as the middle of the topmost armature coil, C4. Now say we turn the rotor so that a north pole magnet aligns with the mechanical 0 position, and define an electrical 0 position to be where the north pole magnet is aligned with the mechanical 0 position. Hmmmm...looks like if we turned the rotor to line up another north pole magnet with the mechanical 0 position, we'd have another electrical 0 position. That's right! And, that is the answer. The rotor is radially symmetric, so when we rotate it, we can align it at several positions and see an identical picture.

In the previous paragraph we defined "a" mechanical 0 degree position and also "an" electrical 0 degree position. Those nonunique statements, as opposed to defining "the" mechanical or electrical 0 degree positions, were intentional. Because of how figure 6 was drawn it is easy to see coil C4 as defining a mechanical 0 degree position. And, in fact, because of symmetry, the relationship we are about to investigate could be arrived at using coil C4 to mark the mechanical 0 degree position. However, because we want to look at what happens over a full sinusoidal phase cycle when defining the electrical 0 degree position, we will start at the the beginning of a phase coil group, not because it is at all necessary, but just because it looks better in the figures. We can start on any phase coil group. Let's choose group A, and define the mechanical 0 degree position as being marked by coil A1, which, in figure 6, is just to the right of coil C4.

Again we define the first electrical 0 degree position to be where a north pole magnet aligns with the mechanical 0 degree position. This is the position where, when the motor/generator is in operation, the phase A sinusoid will be at its maximum value. Now, as the rotor turns, there will be multiple positions where different north pole magnets will pass under coil A1, and all other coil and magnet alignments will be such that the phase A sinusoid will reach a maximum. Each time this happens phase A has passed through a full sinusoidal cycle, and each of these alignments corresponds to a new electrical 0 degree position. The number of full phase cycles per single mechanical rotation is always in a fixed ratio. That ratio will vary with motor/generator design, but, we can always see what the ratio is by looking at the relative alignment, in degrees, between coils and magnets in the design.

We can depict coil/magnet alignment in a linear diagram, similar to how earlier we diagrammed phase coil connections as linear groups, even though a real motor/generator armature is circular (e.g., figure 4). In our eight magnet, 12 coil system, there will be 360/8 = 45 degrees between magnets, and 360/12 = 30 degrees between coils. Drawing this arrangement linearly, with one north pole aligned with coil A1 (figure 20) we can see that the rotor will have to shift (rotate) 90 degrees to achieve the identical alignment of magnet poles and coils as seen for the first electrical 0 degrees position.

[elec. vs. mech. degrees]

If we see a new electrical 0 degree position for every 90 degrees the rotor turns then for every turn of the rotor we will see 360/90 = 4 sinusoidal phase cycles. If you make the same analysis for a ten magnet rotor and use the rote calculation number of coils, 15, for a 3-phase motor/generator, you will come up with a ratio of 5 phase cycles per rotor turn. With a little squinting at the diagrams, the general relationship for the ratio, R, of phase cycles to rotor turns can be seen to be the result for 360 degrees per phase cycle being divided by one-half the number of magnets divided into 360 degrees per rotor turn. That is, if M is the number of magnets:

R = 360/(360/(M/2)).

With a little algebraic manipulation, that becomes:

R = M/2,

which is just the number of pole pairs in the motor/generator. In fact, so long as the motor is properly designed, the number of coils doesn't enter into the relationship. For example, we know the coil count for the 16 pole motor/generator constructed for this web page is not what you would calculate from the rote formulas. But, we also know the electrical to mechanical degrees ratio for that motor/generator is 8, and, by our new formula:

R = M/2 = 16/2 = 8.

So, when we use rotation speed of our motor/generator rotor in the main angle term of our cosine equations, we apply the ratio R = M/2 to that term to produce the proper number of phase cycles per rotor revolution, giving us another form for our equations:

3(A = Kcos(M(pi)ft + alpha))


OK, I picked a phase, now what?

So, now we've got some phases spinning around, and described then in fairly painful detail. What do we do with them? Why, we hook them up to make motors and generators, of course!

For a single-phase motor/generator, we've already seen that isn't too difficult. Well, except for that bit about self-starting of single-phase motors. (Which we'll get to real soon now.) But, it gets a little more tricky with polyphase motor/generators. There we have multiple electrical signals going through different strings of coils that have their ends connected together. Why doesn't it all just short out and burn up? That's where the rotations, and more importantly the phase angles, come in.

Check this out:

cos(0)     =  1.0
cos(240) = -0.5
cos(120) = -0.5

1 + (-0.5) + (-0.5) = 0.

So what? Well, those are the cosine function values for the phase angles (shown in degrees) for our previous three 3-phase equations. That they sum to zero is why our three-phase motor/generator doesn't spark and smoke in normal operation. If you add the same fixed angle to the phase angles in our cosine equations, the result is always the same. Try it:

cos(875634.43 + 0)     = -0.4135812
cos(875634.43 + 240) =  0.9952783
cos(875634.43 + 120) = -0.5816971

-0.4135812 + 0.9952783 + (-0.5816971) = 0.

Try it again if you want to, but I guarantee the result will be the same. What this means is that no matter what the rotational position of our phase sinusoids, if we look at them at the relative positions defined by their phase angles, the electrical waveforms will sum to zero.

Now, by no coincidence, the connection locations for our motor/generator phase coil groups fall at the relative phase angle positions. So, though the coil group ends are shorted together, each connection is effectively at zero voltage, and nothing bad happens. You can see this relationship in, for example, figure 19. Consider the maximum up excursion of any signal in the figure to be +1, and the maximum down excursion of any signal in the figure to be -1. Then, if you sum the observed values for the three sinusoids at the same position along the horizontal axis, you will find the result is always zero. This relationship will hold whether we are considering the rotation in degrees, frequency, or time. So long as we observe the correct phase difference, the sum is always zero. And, of course, this extends to numbers of phases other than three.

It might seem that connecting a motor/generator so that the voltages at its phase wires sum to zero won't accomplish much. But, these are relative voltages. If you look across the phase coil groups instead of just at each of the ends, you will see a voltage difference. This is why though the connection voltages sum to zero, you should not go probing them with a wet finger! (Or a dry one, for that matter.)

The across phase-coil-group voltage difference relationship can also be seen in figure 19. Look at the difference in voltage for each sinusoid over their specified 120 degree phase difference locations. That is, from 0 to 120 degrees for phase A, from 120 to 240 degrees for phase B, and from 240 to 360 degrees for phase C. In each case we see a difference of 1.5 (as a change from +1 to -0.5). The total excursion for each sinusoid is 2.0 (as a change from +1 to -1), so we see across each phase 0.75 of the total excursion.

There are specific mathematical relationships for the continuous values of the phase voltage differences over time. They are often expressed in terms of rotating vectors. We'll look at those as we investigate the primary types of phase connection configurations.

You say delta, I say why?

Phasors don't have to stun:

Before we get on with looking at some of the possible motor/generator phase wiring configurations, first lets take a look at a some notation to use for our rotating phases. For the most part, we aren't going to be concerned with the absolute angles of rotation of our phases, but just with their relative phase angles, and their magnitudes (which may be looked at in terms of either current or voltage). We know we are talking about sinusoids all of the same frequency here, so, rather than using the cosine equations to formally state that, we can just use a notation that gives us the magnitudes and phase angles we need. Here's one:

KK(_nnn)

The parameter K is the magnitude of the sinusoid, and the parameter nnn gives the phase angle of the sinusoid relative to the zero phase angle sinusoid. The phase angle parameter could be given in any units, but, here, as indicated, we'll use degrees. So, you can read the notation as "K at an angle of nnn degrees."

This notation may look familiar. If you've done any AC circuit analysis it probably should. It is traditional phasor notation as derived from the complex exponential representation of a sinusoid. We aren't going to worry too much about its derivation here. Mainly we're just going to concern ourselves with how to add phasors together to see if our motor/generator designs fit the criterion we established in the last section, that the physical connection points for our phase coil groups are made where their phase waveforms will sum to zero. For our purposes, often that will amount to just adding the phase angle values to see if they sum to zero, where 0 in degrees is defined as some integer multiple of 360 degrees. For example, ignoring the parameter K, (that is, call it equal to 1), we can sum the phasor equivalents of our three-phase cosine equations as:

(_0) + (_240) + (_120)    =    (_360)    =    (_0)

That is, our 3 three-phase phase-angles sum to zero. And, that summing to zero is the condition we want, regardless of the number of phases we are working with.

Phase arrangements:

This section is brought to you by the letter Y. Why do people have to do all those the weird phonetic things they do when they want to express the 25th letter of the English alphabet, Y? I'm dyslexic. I have a hard enough time getting letters sorted out using a spell-checker, and all those things involving "w" and "e" people do when they want to say "y" just make my head want to explode. So, here, when we talk about "delta" systems, we'll call them delta systems, and when we talk about "Y" systems we'll call them Y systems.

There are a lot of different ways to arrange the wiring of a polyphase motor/generator. But, they center around the two basic configurations, the delta and the Y. In fact, most of the alternative configurations are just combinations of delta and Y circuit arrangements.

Most of what we talked about so far has been while referring to delta wired motor/generators. But, the discussion also applies to Y configured systems. The same phase coil group "head-to-tail" wiring method described in the rote instruction sections holds for delta and Y arrangements, as does the notion of phase voltages summing to zero at the phase group connection points.

What mainly differentiates delta and Y systems is the connection of their phase coil groups. We've seen the delta connection before (figure 5), so, let's take a look at the Y connection. For our ongoing case of eight magnets and 12 coils, we would connect our phase coil groups the same as we did for our delta wired motor/generator (figure 4), but, rather than connect the phase coil groups in "head-to-tail" fashion, we connect them "head-to-head." That is, if we consider the positive lead of a phase coil group to be the "tail" of the first current arrow as drawn its coil connection diagram, and the negative pole to be the "head" of the last current arrow in its coil connection diagram, then we connect the negative leads. For a three-phase motor/generator, this gives us a three-armed or "Y" configured circuit, where the free ends are our A,B,C phase input/output leads (figure 21).

[Y configuration]

Unless you look close at the wiring, the basic physical design we've seen for a delta wired motor/generator (figure 6) looks the same for a Y configured motor/generator (figure 22).

[Y motor/generator]

So, what about that connection voltages summing to zero thing? Does that still hold for the Y configuration? Yep. Take a look. Consider driving the Y as a motor. The free ends are provided with the phase inputs, which we already know sum to zero due to their phase differences. Since the phase group coils are identical, and we've connected them so that phase currents flow without interferring with themselves, the voltages seen on the other ends of the phase coil groups will not change phase, and, hence, still sum to zero at the Y connection point. Symmetry dictates the same holds true when using the system as a generator. The same sinusoidal phase relationship diagrams we developed when discussing the delta wired motor/generator also hold for the Y system.

Before we get into considerations of power and efficiency that might affect one's choice of a delta or a Y configuration, there is one significant difference to note between the two connection styles. The Y connection allows including a neutral line along with the phase lines if we choose to. By connecting a lead to the phase coil group junction we get a three-phase 4-wire connection, while without the additional lead the connection is known as a three-phase 3-wire connection (figure 23).

[Y 3 and 4 wire connections]

There are a number of reasons to choose to include or not include a neutral line in a polyphase AC system. Mainly they relate to safety and power balancing considerations, and are not terribly relevant to the discussion at hand. So, we won't cover the issue of neutral lines more in this section, or in any great detail later.

One thing that is somewhat relevant to our current discussion is the matter of power and efficiency for the delta versus a Y connected motor/generator. Way back near the beginning of all this I said something about the delta being more efficient at lower rpms and that is why it was chosen as the configuration for the initial polyphase rote design instructions. Some of you may disagree with that, if you know that for a given rpm the Y produces a a higher voltage than the delta. Well, efficiency is in the eye of the beholder. For a given rpm the delta produces more current than the Y, and, me, I'm looking for maximum current, not maximum voltage, so the delta makes more sense for my purposes. But, lets take a closer look and see about how to figure out if it makes sense for your application on not. Note that outside the standard electrical current and voltage relationships which hold anywhere, the discussion that follows will refer to three-phase systems only.

Whether we are looking at a delta or a Y system, there are two voltages to consider, coil voltage and line voltage. Coil voltage is the voltage read across any coil, where "coil" means the entire phase coil group when there is more than one coil per phase coil group. A motor/generator's A,B,C, phase leads are commonly referred to as the "lines" and line voltage is the voltage read between any pair of lines (figure 24).

[coil and line voltages]

The following relationships hold for coil and line voltages and currents in a three-phase delta system:

Coil Voltage = Line Voltage
(sqrt(3)) * Coil Current = Line Current

The following relationships hold for coil and line voltages and currents in a three phase Y system:

(sqrt(3)) * Coil Voltage = Line Voltage
Coil Current = Line Current

Given those relationships, and the rule of thumb that the line resistance, (i.e., the resistance measured between any two lines), in a Y system is typically about 3 times the line resistance in a delta system, we can make a few calculations and look at differences in output from the two.

If we measure the voltage and current in one phase of a generator, (where here phase means one phase coil group, or the "coil" as described in a previous paragraph), independent of the other phases, we can use those readings to calculate what the line voltage and current will be for the generator connected in either a Y or a delta configuration.

Say in one phase of a generator we see 28 volts and 10 amps. Then, using the above relationships, wired as a delta the generator would have the same line voltage, 28 volts (V), and a line current of (sqrt(3)) * 10 = 17.3 amps (A). Rewired as a Y, the same generator would have a line voltage of (sqrt(3)) * 28 = 48.5 V, and a line current of 10 A.

Now, the power in an electrical circuit in watts (W) is equal to the voltage in volts times the current in amps, So from any two lines in our Y configured generator we get 48.5 V * 10 A = 485 W, and, similarly, as a delta we see 28 V * 17.3 A = 484 W for any two lines.

Hmmmm, 485 W versus 484 W, that doesn't seem too significant a difference. And, it isn't. The power out of a generator is related to the power you put into spinning it. Whether we wire it as a delta or a Y, if we spin our generator so in either case we get the same phase measurements, then we are basically putting the same amount of power into it. So, in that case, we really can't get any more power out of one configuration over the other. OK, then why choose one over the other? It comes down what you are trying to do, and how fast you can spin.

Say we made our above measurements at 400 rpm. For the delta that means 400 / 28 = 14.3 rpm per volt, and for the Y we have 400 / 48.5 = 8.2 rpm per volt. If we reved up to 1000 rpm, from the delta we would see around 1000 / 14.3 = 69.9 V, and from the Y we would see around 1000 / 8.2 = 122 V. Now, rather than choose an arbitrarily coil current, lets use the convenient, but not unrealistic line resistances of 3 ohms for a Y system, and 1 ohm for a delta system and see what we get for power while we are charging a battery.

Lets cheat a little bit and ignore the need for rectifying the output alternating current (AC) voltage to a direct current (DC) voltage before applying it to the terminals of a battery for charging. As far as the two types of voltages are concerned, rectification results in a known, fixed multiplication factor being applied to the AC voltage level to obtain the DC voltage level, so the calculations that follow are valid, but just missing a multiplier. We'll get to rectification in a later section.

The voltage (V) across a circuit is equal to the current (I) through the circuit times the resistance (R) of the circuit, that is, V = I * R. We can manipulate that equation any way we like, and, hence, I = V / R, which we use with our generator line voltage and resistance to calculate the line currents. Actually, to calculate the generator line currents in the case of charging a battery, we subtract the battery voltage from the previously calculated generator line voltages because the battery is compensating for part of the measured line voltage. So, for a nominal 12.5 battery level charged by the Y configured generator spinning at 1000 rpm we have 122 - 12.5 = 109.5 effective line voltage, which, with the line resistance of 3 ohms gives us 109.5 V / 3 ohm = 36.5 A line current. Similarly, for the delta we have (69.9 - 12.5) V / 1 ohm = 57.4 A line current.

A battery being charged by a generator is essentially a series circuit, and the current through a series circuit doesn't change. (We'll talk more about series and parallel circuits later.) So, for comparison of the Y and delta generator configurations, we can look at the power output relative to the nominal battery voltage because we see the same current no matter where we look in the circuit, and it will give us an idea of how much power we have available to charge the battery. That is for the Y we see 36.5 A * 12.5 V = 456 W, while for the delta we see 57.4 A * 12.5 V = 718 W, or 253 W more power available for charging from the delta than the Y when both are spinning at 1000 rpm.

That more output power is available from the delta at a given rpm is an advantage so long as we have the input power available to spin it fast enough to get the required output voltage. If we are constrained by input power, and hence to lower rpm, then the Y configuration can be an advantage when, say, charging a battery. To get the minimum required approximately 13.5 V for charging a standard lead-acid battery, our example delta generator would need to spin 13.5 V * 14.3 rpm per volt = 193 rpm, while the Y would require 13.5 V * 8.2 rpm per volt = 111 rpm. So, though we would not be producing as much power as when spinning at 1000 rpm and hence would not be able to charge a battery as fast, we might still be able to charge a battery with the Y configured generator when we could not with the delta generator due to rpm constraints.

Some motor/generators take advantage of the voltage and current relationship difference between delta and Y configurations and start in the Y configuration to take advantage of higher voltage and lower current at low rpm, and then switch themselves to a delta configuration once the line current is sufficient to energize a coil that creates a magnetic field strong enough to pull in a spring loaded switch that makes the configuration change. (That's one way, anyway, there are other methods.) So long as the rpm stay high enough to maintain the required voltage level, the motor/generator runs in the the delta configuration and takes advantage of the higher available current. Otherwise the system drops back into the Y mode and maintains the required voltage level at the lower rpm.

Rollin' Rollin' Rollin'

Well, rotatin' rotatin' rotatin' more precisely. Rotating magnetic fields, that is. Which are at the heart of all we've been talking about. The notion of rotation we covered a lot already with our sinusoids and their multiple forms of cosine equation descriptions. Plus, "moving" magnetic fields have been mentioned before, too. It's pretty clear that the magnetic fields of magnets mounted on a spinning rotor must be rotating, since their magnets are rotating. (Of course, there a few more details to fill in there.) And, what about those magnetic fields created from our nonspinning armature coils when we apply phase input power? Well, they are rotating, too!

In the last section we talked a lot in terms of generators. Here, we'll cover things for the most part in terms of motors. Everything in this and the last section pretty much applies to motors or generators, (remember symmetry), but, the phenomena are just easier to visualize as described. The key thing to note is, like the term "phase," the term "rotation" can have somewhat different meanings depending on context.

Moving? Rotating?

Moving a magnet obviously moves its magnetic field along with it. So, rotate a magnet, and its magnetic field rotates, too. OK. That's a straight forward concept. But, let's look just a little bit closer.

First, we'll consider the magnetic field of a single magnet. Note that all magnets are dipoles, that is, they have two components (north and south poles), that are inseparable. There are a number of formulae that can come into play when looking at dipole magnetic fields. We don't need to worry about them too much here, and just note that the strength of a magnetic field near a dipole falls off as 1/r**3, where r is the distance from the center of the dipole. (This is different than the fall off from a monopole, such as a positive or negative electric charge, where field strength falls off as 1/r**2.)

The 1/r**3 relationship (figure 25) applies best on the molecular scale, but for a passable approximation it will work to look at field strength from, for example, one of the neodymium discs on the model 3-phase motor/generator rotor constructed for this web page.

[1/(r**3)]

If we ignore the mathematical artifact where the field blows up to infinity at a distance of r = 0, and kind of fudge over the fact that our disc has a significant diameter relative to a molecule by smearing out the field over the width of the disc and plotting the field change relative to the edge of the disc, we get something that gives some idea of the field strength near the disc. The take home message here isn't the exact form of the field, but that field strength falls off fairly rapidly away from the disc, and we are left with essentially a magnetic field "bump" above our magnet. (figure 26).

[magnetic 'bumps']

Now, consider the magnets on the spinning rotor of our basic design motor/generator. What we have is a series of intense magnetic bumps, with alternating poles, evenly spaced around the perimeter of the rotor. Assuming the gap between the moving rotor magnets and the stationary armature coils is small enough (we'll talk a bit about proper gap width later), then the "bumps" will move through the armature coils, creating a current in the coils.

If we didn't have the magnetic bumps, but rather a uniform field around the rotor, then even though the rotor was spinning, and the field effectively moving, the coils would not see much if any change in magnetic field, and, hence, produce little to no current. It isn't just that a magnetic field is moving that causes current flow in a wire, but that the field intensity is changing, and the moving magnetic bumps give that necessary intensity change.

For our alternating pole rotor, we not only see relative level changes in field intensity as the rotor spins, (i.e., a changing magnetic field), but we also see full alternating pole magnetic field changes. This alternating of poles, as discussed before, causes reversal of (alternating) current in our armature coils. Voila, AC generator!

In the case of using the motor/generator in motor mode, you can consider the magnetic "bumps" along the edge of our permanent magnet rimmed rotor to be the equivalent of teeth on a gear. But, then, what is pushing these "teeth" to make our motor rotor spin? The answer is the rotating magnetic field produced by our armature coils when we apply power to its phase coils.

Inducing synchronization:

The brushless permanent magnet motor is what is referred to as a synchronous motor. That means its rotor spins at synchronous speed, where synchronous speed is an rpm value that is a direct function of the input phase line frequency and the number of armature poles in the motor. Under normal use conditions there is no slip of the rotor in a synchronous motor relative to the rotating magnetic field created by the phase line power input to its armature coils. This is as opposed to an induction motor, where there must be some slip of the rotor (i.e., its speed less) relative to the rotating armature field so that currents are induced in conductors embedded in the rotor which in turn create magnetic fields that act as the magnetic fields produced by the permanent magnets on the rotor of our brushless permanent magnet motor. The synchronous type motor makes a good motor/generator. The induction motor is not quite so well suited to acting also as a generator.

Now, electrical current moving through a wire travels basically at the speed of light, which is about 1 foot per nanosecond (0.000000001 second). And, since we are talking about input power frequencies generally of less than 100 Hz, (giving a cycle period on the order of 0.01 second), with total wire lengths on the order of tens to hundreds of feet, for all intents and purposes, every coil in a phase coil group sees the same power level at the same time, tracking the input power level as it arrives at the phase line input. So, if everything seems to happen at once, how do we produce anything like a rotating magnetic field from our stationary armature coils? Well, there are a few aspects to that, in particular, the alternating polarity wiring of the armature coils which produces opposite pole magnetic fields in adjacent coils as current flows through them, and also, in the case of polyphase systems, the phase difference between the phase line inputs which produces a lead or lag between magnetic fields produced in different phase coil groups.

Probably the most commonly seen formula for synchronous speed is:

S = 120f/p

where: S = synchronous speed in rpm (revolutions per minute)
               f = input phase line frequency in Hz (cycles per second)
p = number of armature poles (coils).               

Let's stare at that just a tad and see if we can figure out what's going on. Since we are using frequency, f, in Hz, or cycles per second, and S comes out in rpm (revolutions per minute) there must be a factor of 60 seconds per minute in there, so, we could rewrite the formula for S as:

S = (2)(60)f/p.

Now, if we wanted to convert from, S, speed in rpm to, say, N, speed in rps (revolutions per second), all we have to do is toss out the factor of 60 seconds per minute, leaving us with:

N = 2f/p.

Apply a little algebra, and we get:

N = f/(p/2).

That is, synchronous speed in revolutions per second is equal to the input phase line frequency in Hz divided by one-half the number of armature poles. That is to say, synchronous speed in revolutions per second is equal to the input phase line frequency in Hz divided by the number of armature pole pairs.

OK, there's that "pole pair" thing again. So, where's it come from? Well, it comes from the same place it did when we were looking at mechanical vs. electrical degrees a while back. There we had a sequence of alternating pole permanent magnets. Here, due to the armature phase coil group wiring technique, we have alternating pole electromagnets. In either case, to see the the same action at a pole we have to look at every other pole. That is one half the poles give a different action, and one half the poles is also the number of pole pairs. Hence, the number of pole pairs becomes an important factor in the calculations. In fact, in a while we'll take a look at a diagram very similar to the one we looked at when discussing electrical vs. mechanical degrees (figure 20) as we further define the concept of synchronous speed. But, first, let's just take a closer look at the concept of an actual rotating magnetic field.

It is actually easier to conceptualize a rotating magnetic field when examining the properties of a polyphase motor than it is when considering a single phase motor. In fact, a self-starting single phase motor is, at least on startup, connected so that it actually runs as a two phase motor. (More on that to come.)

In general, to establish a rotating magnetic field the number of armature poles (coils) must be equal to, or a multiple of, the number of input power phases, with the poles being separated by the phase angle between the input power phases.

As pointed out earlier, the number of armature poles and the number of rotor pole (magnet) pairs are related. That is, for each rotor magnet pair must be a set of armature coils equal in number to the number of input power phases. Note how this relationship, by default, meets the criterion given above for the number of armature poles required to establish a rotating magnetic field.

So, we know how many armature coils we need to set up a rotating magnetic field, and that our properly designed motor/generator will have that number of armature coils. Let's finally take a look at how all this sinusoidal signal, spinning vector, rotating field stuff comes together. We can look a simple case, say a two pole-pair, three phase, delta configured motor. Now, another diagram is in order. For that we can use a new form of the armature coil connection diagram we've seen before, same as our original linear diagrams, (e.g., figure 5), just wrapped in a circle to represent a more realistic motor/generator armature. For our two pole-pair, three phase motor/generator we will end up with a circular figure having six armature coils representing two three-coil phase groups (figure 27).

[6 pole armature]

Using figure 19 as a reference for the phase relationships of three-phase sinusoidal currents, let's examine the magnetic fields that would be generated in the coils of figure 27 by application of three-phase power. Consider that a positive current through a coil wired as is coil A1 relative to its input phase power will produce a north magnetic pole pointing inwards towards the center of the ring of armature coils.

We've seen that we can look at our sinusoidal signals in terms of degrees or time and get the same results. So, take time zero, t0, to be the zero degree position for phase A, take time two, t2, to be the zero for phase B, (120 degrees for phase A), and take time four, t4, to be the zero degree position for phase B, (240 degrees for phase A). Note, relative to the waveforms in figure 19, this means at time t0 phase A is at a positive maximum, at time t2 phase B is at a positive maximum, and at time t4 phase C is at a positive maximum. But, let's concentrate just on phase A for a moment.

Note while coil A1 produces a maximum strength north pole oriented towards the center of the armature ring at t0, coil A2, being wired for the opposite polarity, produces a maximum strength south pole oriented towards the center of the armature ring. The physical arrangement of coils in figure 27 puts A1 and A2 opposite each other on the armature ring, which means at time t0 there is the maximum possible magnetic field strength between coils A1 and A2. Also note that at the 90 degree position for phase A, or three quarters of the time between t0 and t2, the A phase value drops to zero, and at that time there is no phase A current generated magnetic field between coils A1 and A2.

Of course, there are two other input phases, and, as we already know, none of the three are always at their maximum value. Again referring to figure 19, we can see that at time t0, when phase A is at its positive maximum value, both input phases B and C are are at one half their maximum negative value. Similarly, at time t2, phase B is at its positive maximum while phases A and C are at one half their maximum negative value. And, at time t4 phase C is at its positive maximum with phases A and B at one half their negative maximum. For now let's just concentrate on time t0.

We have already established that at time t0 the magnetic field between coils A1 and A2 is at its maximum, and oriented north to south from A1 to A2. For phase B at t0, recall the rote coil connection scheme dictates that coil B2 is the phase input coil and thus wired so that a positive current generates a magnetic field with its north poll oriented towards the center of the armature ring while coil B1 is wired so a positive current generates a magnetic field with its south pole oriented towards the center of the armature ring. So, with the B phase being negative at t0, coil B2 generates a magnetic field with its south pole oriented towards the center of the armature ring and coil B1 generates a magnetic field with its north pole oriented towards the center of the armature ring, and develops a total field strength around one half that of the field between coils A1 and A2. Similarly, the total field between coils C1 and C2 is approximately one half the field between coils A1 and A2, oriented north pole to south pole from coil C2 to coil C1. Well, at least that is what would happen if the different phase fields didn't interact. In reality, the full strength field extends between coils A1 and A2 as described, but, the weaker north pole field from C2 is deflected by the stronger north pole field from coil A1 and connects to the weaker south pole field from coil B2, while the weaker north pole field from coil B1 is deflected by the field from coil A1 and connects to the weaker south pole field from coil C1. In a simplified form, the arrangement of flux lines in the armature ring at time t0 looks like the depiction in figure 28.

[6 pole armature]

We could go on producing diagrams similar to figure 28 to show the orientation of magnetic flux lines in the armature ring at times other than t0. But, that would get very tedious very fast. Instead, we'll switch to phasors and rotating vectors to simplify things.

Getting into the spin:

A few sections back we introduced basic phasor notation. There we saw how it describes a vector by giving the vector's magnitude and direction. Lets dig into it a bit deeper and look at how it can represent spinning vectors.

Phasor notation gives the direction for a vector as an angle relative to a fixed starting point, the tail end of the vector. Since one end of the vector is fixed in place, if we change the direction angle, the vector will pivot around the fixed end point until it aligns with the new direction angle. And, if we change the angle value continuously in one direction the vector will spin around its fixed end point (figure 29).

[angle change rotation]

OK, it's clear that phasor notation can represent a rotation. But, how to make that rotation represent our motor/generator phases? Well, all we need do is force the angle argument to change in time with the same cycle rate as the motor/generator input phases. And, it's radians to the rescue! (Again.) Just using the same angle arguments we previously developed for our cosine equations as the angle argument in the phasor notation vector representation gives us vectors spinning with the same cycle rate (frequency) and phase differences as our motor/generator input phases.

If we view the rotating vectors (phasors) as something akin to the simple machine presented in figures 15 and 16 for converting circular to linear motion, then the relationships for the phasor magnitudes can be deduced. Since our input phases are identical except for their phase differences we need look at just one of them to determine the magnitude relationships for all. (That relationship is, of course, to scale the maximum magnitude by the cosine function. But, for the sake of drawing some more pretty pictures, let's soldier on a while longer.)

Looking carefully at how the our conversion machine works, in figure 15 we can see that the position of the peg on the disc determines the amplitude of the waveform the machine produces. At any time in a revolution of the disc, regardless of the angle of the peg relative to the machine's axle shaft, the perpendicular distance of the peg from a horizontal line drawn through the center of the axle is equal to the amplitude of waveform traced by the pencil attached to the end of the shaft moved by the peg.

If we think of the peg in our machine's disc as marking the head of a rotating vector (phasor) and the axle shaft as marking the fixed tail point of that vector, then we can also think of the perpendicular height of the tip of our rotating vector relative to a horizontal line drawn through its fixed tail end point as defining the amplitude of the waveform drawn by our machine (figure 30).

[phasor vs. machine]s

Clicking on the thumbnail below will launch an animation demonstrating the change in amplitude with phasor rotation. The animation cycles 25 times. If you haven't gotten bored with it before then and already hit the back button in your browser to return to this text, then you can hit your browser's reload button to run the animation again, until you do get bored.

[rotating vector]


We have established that our machine drawn waveform describes a sinusoid, so, our rotating vector also describes the same sinusoid. This means our rotating vector (phasor) sinusoid can be used to define our motor/generator phase input sinusoids, just as we have already done using the cosine functions that describe the waveforms produced by our circular-to-linear motion conversion machine. We have also shown how to graphically produce the proper magnitude for our phasors. But, we aren't quite done yet. We need to apply both the direction and magnitude simultaneously to properly illustrate the phasor in action.

By definition, a vector, rotating or not, describes a length (magnitude) and the direction to point that length towards. To complete our picture, we need to not just project the magnitude from tip of the vector onto a line as done in the previous figures, but, take the projected magnitude values and align them in the direction of the vector at the time they were generated.

Considering again just phase-A, with no phase angle, and clockwise rotation. The phasor starts out with its positive maximum magnitude directed vertically up at an angle of 0 degrees. The only other maximum is negative occuring when the vector is again vertical, only directed down at an angle of 180 degrees. At 90 degrees and 270 degrees the magnitude is 0. The effect is in one cycle for the phasor to scribe a figure-eight shaped path around its fixed tail point (figure 31).

[figure-eight]

Clicking the thumbnail below brings up an animation that illustrates the change in magnitude with rotation describing a figure-eight shaped path.

[figure-eight phasor]


Now, don't be fooled. The phasor described above isn't really our long sought after rotating magnetic field. The phasor, in this case, simply describes the change over time in the magnetic field relative to the phase A coils, A1 and A2. Between coils A1 and A2 the magnitude of the magnetic field generated due to the phase A input increases and decreases, as well as changes its polarity when the input current changes polarity, but, in and of itself, it doesn't rotate. The rotation indicated in the figures and animations above is just really a way of illustrating the change over time. None the less, the "figure-eight" magnitude change is the key to the rotating magnetic field.

We noted in describing the magnetic flux diagram of figure 28 that, because of the way the phase coil groups are wired, at time t0 the fields from each group are directed towards the center of the armature coil with the same magnetic polarity, north to south, but have different magnitudes, with the strongest flux associated with the phase A coil group. No matter at what time in an input phase cycle we look, the flux across each coil group will always have the same polarity, though all polarities may be reversed and which phase coil group is exhibiting the maximum flux will change. It is this shift in flux magnitude between the phase coil groups which leads to a rotating magnetic field due to the polyphase input signals.

For our three-phase system we have three rotating phasors, each leading the next by a phase angle of 120 degrees. If we plot the figure-eight magnitude changes for each phasor on the same diagram, the result provides more insight into the nature of the rotating magnetic field. From such a diagram (figure 32) we can see loops in the magnitude change plots for the individual phasors intersect at the center of the figure, but each fills the gap between its adjacent phase loops, with some overlap. As explained in the text of figure 32 this figure can be used to determine the magnitude and direction of the phase coil magnetic fields in our 3-phase motor/generator.

[3-phase figure-eights]

Still, we haven't quite got to the real deal rotating magnetic field. The animation which can be viewed by clicking the thumbnail below may help shed some light the matter. It shows the figure-eight sweeping magnitude for all three phases, in their proper orientations, at the same time.

[triple phasor]


Again, recognize that in the above figures and animations we are not looking at rotations in the sense of a magnetic field moving around the armature ring. What we are looking at is the change over time of the flux associated with each phase coil group. The apparent rotation in the figures is just a way of representing the repeating cycle of the changes. It is the combination of the changes in all phases at once that produces the rotating magnetic field.

It all adds up:

OK. We have our rotating vectors, also called phasors, spinning in sync with our phase inputs, and having magnitudes representing the current flows through our phase coil groups which are proportional to the magnetic field fluxes from the armature phase coils. Also, a while back, we noted at time t0 (figure 28) that because of how the phase coil groups are wired the polarity of the magnetic fields from each group is oriented in the same direction towards the center of the armature ring. Again referring to figure 19, let's call the negative maximum for phase C time t1, the negative maximum for phase A time t3, and the negative maximum for phase B time t5. Similarly to how we did for time t0 to produce figure 28, we can take a look at the fluxes for the phase coils at time t3. Except this time instead of producing another picture like figure 28 we'll use vectors to make our graphic representation.

At time t3, the picture is simply the reverse of the picture at time t0. The magnitudes of phases B and C are one half the magnitude of phase A, with phase A current flow negative and phases B and C current flow positive. As we saw when discussing figure 28, due to the wiring of the phase coil groups polarities of the magnetic fields for all groups are oriented in the same direction towards the center of the armature ring, but this time instead of north to south from coil A1 to A2, it is south to north from coil A1 to A2, with weaker south to north fields from coil B1 to C1 and from coil C2 to B2. To be consistent, from here on we will always talk of our magnetic fields in terms of flux from north to south. So, at time t3 we will say our fields are north to south from coils A2 to A1, C1 to B1, and B2 to C2.

Now, how do we represent the situation at time t0 or t3 with vectors? Pretty straight forward. We know the lengths of our vectors represent magnitude, so, at time t0 or t3 the phase A vector will be twice the length of the phase B and C vectors. Further, we know the orientation and wiring of our phase coils, so we can determine the polarity of our phase coil magnetic fields and hence the direction to draw our vector arrow heads. This means at time t0, from the position of coil A1 on our armature ring we draw a full length vector representing the phase A magnetic flux to coil A2, and, from the same starting point two half length vectors, one at 60 degrees from the right of the full length vector and the other at 60 degrees from the left of the full length vector. Examining figure 28, we can see that the half length vector pointing 60 degrees to the right represents the B phase north to south contribution to the picture (coils B1 to C1) and the half length vector pointing 60 degrees to the left represents the C phase north to south contribution (figure 33A). At time t3 the full length vector (phase A contribution) points, north to south, from the coil A2 position with the phase B contribution (coil B2 to C2) at 60 degrees to the left, and the phase C contribution (coil C1 to B1) at 60 degrees to the right (figure 33B).

[phase A flux]

But, "wait!," you say. Aren't the phase separations 120 degrees? Where did those angles of 60 degrees come from? Well, that's really just a bit of geometric handwaving coupled with the knowledge of how our armature phase coil groups are wired. In discussing figure 28, we noted how the B and C phase field contributions are polarized due to their phase group coil wiring. A little squinting at figure 28 will reveal that relative to the center of the armature ring, directly at the edge of the armature ring, the contributions of the B and C phases point at 60 degrees relative to the phase A component. The fact that the sum of the angles of the B and C field contributions sum to 120 degrees is really just a coincidence, and is related to the number of coils on the armature ring. A different number of coils in the phase groups would give a similar picture, but the angles would be different. The important thing to remember here is the phase input vectors and the magnetic field vectors are, of course, strongly related, but not the same. The phase input provides current flow, and current flow through the wire creates the field.

Note that the picture at time t3 is simply a version of the picture at time t0 rotated by 180 degrees about the center of the armature ring. Not coincidentally, 180 degrees is the amount of phase rotation we see from time t0 to time t3. That is, when our input phases have rotated 180 degrees, it appears that so do our magnetic field flux vectors. We can produce the same kind of figures for times t1 and t4 (phase C maximums), and times t2 and t5 (phase B maximums), where the diagrams for times t1 and t4 will be versions of the t0 and t3 diagrams rotated 120 degrees clockwise, and the diagrams for times t2 and t5 will be versions of the t0 and t3 diagrams rotated clockwise 240 degrees (figure 34).

[rotating field]

One more thing to note is we are referencing our rotation to the center of the armature ring, and what we really have is a south magnetic field on one side of the ring, and a north magnetic field on the opposite side of the ring, The magnetic field rotation is of both the poles. We have been drawing our diagrams to reference the south magnetic part of the field, but the north magnetic field part rotates along with it, in sync, around the opposite side of the armature ring.

At this point we basically have our rotating magnetic field. At least we can see at times t0, t1, t2, t3, t4, and t5 we have the same magnetic field flux pattern rotated at 60 degree increments around the armature ring, where 60 degrees is the amount of phase rotation seen between each time step.

Another important thing to note in figure 34 is that it is not the flux vector for phase A that is rotating around the armature ring. You can see in the figures for each time step that the main flux comes from which ever phase is at its positive or negative maximum value. If you compare the phase group coil layout around the armature ring of figure 28 with the time step diagrams in figure 34, you can see that the spacing of the coils results in the equal spacing of the direction of the flux vectors around the armature ring. Since we can't really tell one flux vector from another, what we see is an apparent rotation of a single flux vector with time.

For the complete picture we would need to look at what happens between the points of phase magnitude maximums. But, for this discussion, that is really more than we need to do here. The bottom line is as the phase inputs vary for times away from the phase input maximums, the fluxes from the phase group coils will sum up as vectors to create a total flux picture which looks basically like what we see for the phase magnitude maximums, but with the head and tails for the main sum vector directed somewhere between the phase group coils, rather than directly through an opposing set of coils. (Remember, the diagrams we have seen are simple representations of a more complex picture.) In the end we have a magnetic field flux vectors generated by the phase coil groups which vary so we end up with a set of north and south pole fields moving around the armature ring such that they appear to smoothly rotate.

Whew! Finally! A rotating magnetic field. Wow, that was kind of like pulling teeth, wasn't it? Well, as Blaise Pascal once said, "I made this letter long because I didn't have time to make it shorter." I may clean this up a bit someday. But, don't hold your breath. For now, lets continue on the trail of that oft promised, but yet to be seen self-starting single-phase motor.

Finding Another Way:

The final key to understanding the self-starting single-phase motor is seeing why a polyphase motor spins in a given direction, and how it can be made to spin in the opposite direction. The direction of rotation is determined by the sequence order of the input phases. It will also be helpful to understand a bit more of the possible wiring configurations for a motor/generator. So, first will take a look at series and parallel wiring arrangements, then we'll get on with the matter of direction of rotation.

Series and parallel and such:

We've hinted at other possibilities for armature coil phase group wiring configurations before, noting that wiring coils end-to-end, with proper consideration of phase angles, allows addition of the voltages generated in each coil. This means that besides putting more turns in our coils to generate higher voltates, which can make motor/generator construction problematic due to large coil sizes, we can also use smaller coils wired end-to-end (formally called a series connection) to achieve higher voltage outputs. For a single-phase motor/generator such as the one illustrated in figure 3 this is simply a matter of adding more coils to the armature ring, and an equal number of magnets added to the rotor disc. In the case of polyphase motor/generators like the ones depicted in figures 6 and 22, to increase output voltage we would add equal numbers of coils to each phase coil group, and add magnets on the rotor disc following the constraints given by the design equations provided in the rote formula discussion.

Phase sequences:



Forward/reverse:



Split phase start:



yatayatayata...



Slogging Through Cogging:

eddy currents and such...



Rote Revisited:




To be continued...?

References:

 (Clicking reference numbers here returns you to the text you came from.)


Last updated 03Sept2006
Alan Swithenbank, alans@cuervo.stanford.edu