Lyapunov function

Lyapunov fractal with the sequence AAAABBB

In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in adynamical system or autonomous differential equation. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory. A similar concept appears in the theory of general state space Markov Chains, usually under the name Lyapunov-Foster functions.

Functions which might prove the stability of some equilibrium are called Lyapunov-candidate-functions. There is no general method to construct or find a Lyapunov-candidate-function which proves the stability of an equilibrium, and the inability to find a Lyapunov function is inconclusive with respect to stability, which means, that not finding a Lyapunov function doesn't mean that the system is unstable. For dynamical systems (e.g. physical systems), conservation laws can often be used to construct a Lyapunov-candidate-function.

The basic Lyapunov theorems for autonomous systems which are directly related to Lyapunov (candidate) functions are a useful tool to prove the stability of an equilibrium of an autonomous dynamical system.

One must be aware that the basic Lyapunov Theorems for autonomous systems are a sufficient, but not necessary tool to prove the stability of an equilibrium. Finding a Lyapunov Function for a certain equilibrium might be a matter of luck. Trial and error is the method to apply, when testing Lyapunov-candidate-functions on some equilibrium. As the areas of equal stability often follow lines in 2D, the computer generated images of Lyapunov exponents look nice and are very popular.

RIGHT ANGLE CIRCUITRY

FIG.1 Wrap an AC coil around an iron rod, and you have an inductor. But wrap your coil around an iron RING, and you form a "toroidial inductor."

FIG. 2 A toroidial inductor is interesting because the induced magnetic field remains hidden within the iron core. If the coil was wrapped around the entire core rather than in one spot as shown, then the magnetic field would exist only within the iron core.

FIG. 3 Even if the coil of wire does not touch the core, it still induces a strong magnetic field inside the core. The gap between the coil and the iron ring can be very large, yet this does not reduce the strength of the field within the core.

FIG. 4 Although the magnetic field stays inside, something else does come out of the core. The changing field within the core produces a field of Vector Potential which surrounds the core. This field is commonly called the "A-field."

FIG. 5 We can intercept the A-field by passing a wire through the hole in the iron ring. This produces a voltage at the ends of the wire, and this voltage can operate an ordinary load such as a light bulb.

FIG. 6 Perhaps this figure is more familar to you. The wire which intercepts the lines of "A-field flux" is simply the secondary of a transformer. Note that whenever multiple turns are passed through the hole in the iron ring, the output voltage rises proportionally. Two turns gives 2x the voltage of a single turn.

FIG. 7 We can route the "secondary" wire through another iron ring. It will produce a strong field within that second ring, and if we add a "secondary" to that ring, we'll see an output voltage. It acts like any other transformer, even though there is an extra stage of "A-field" linkage.

FIG. 8 What if we don't use wire? If we simply place the second iron core near the first, then the lines of A-field flux will pass through both and link them together. The result? Nothing! No magnetic field appears in the second core, and the extra secondary does not produce any output voltage as it did in figure seven. WHY?!!! I don't know. I haven't thought deeply enough about this yet...

FIG. 9 The A-field is associated with voltage and electrostatic fields. After all, if we add more turns to a transformer secondary, we get more voltage on the output. Perhaps we can use capacitor plates to intercept the A-field? What will happen? Can we extract energy from the toroid without passing an electric current through the central hole? I don't know.

FIG. 10 In figure seven we formed a strange transformer by passing a conductive ring through two ferrous rings. This idea can be extended to ridiculous lengths. If the iron cores are not lossy (use laminations,) and if the conductive rings are not resistive (use thick copper), then a long chain of alternating rings will transmit energy with little loss and no possibility of electrocution. The rings need not even touch each other. (Just what is Electrical Energy, if it can flow through such a strange transmission line?)

FIG. 11 Figure ten might seem weird, but even a simple transformer is weird in the same way. Stretch the core so that the primary and secondary are far apart. Energy is flowing along the two sides of the core, proceeding from the primary coil to the secondary. Note that the transformer core need not be conductive. It could be made of insulating ferrite.

FIG. 12 With high-mu materials, the transformer core could even take the form of wires. But these wires are nonconductors. They "conduct" waves of magnetic field. The electrical energy is guided by the spin-flipping of electrons in the iron atoms, as opposed to copper wires where energy is guided by flowing electrons.

FIG. 13 Add a SPST switch to the previous "circuit", and we can break the connection between the two halves. A physical switch isn't required: instead we could place a permanent magnet against the wire and cause it to saturate and become magnetically "nonconductive." Note that opening this switch reduces the inductance of the iron ring, and causes the primary to draw an enormous current. BREAKING THE CIRUICT PRODUCES A "SHORT CIRCUIT" EFFECT!

FIG. 14 Now that we've got a wire, lets wind a coil. But what will such a coil produce? A-field! Many turns of ferrous core-wire will give us a higher output voltage, just as many turns of copper wire passing through one turn of iron core gives higher voltage.

FIG. 15 Let's add a core! Barium Titanate should work. Or PZT ceramic (Lead Zirconate Titanate.) Our "coil" should attract such a core, which means we could build a solenoid actuator. Or a motor. Or just use the PZT core to pick up certain things. Things like lint, and little bits of paper. It's not an electromagnet, it's an electro-electret!

"STATIC ELECTRICITY" IS ELECTRICITY WHICH IS STATIC?

"Static electricity" exists whenever there are unequal amounts of positive and negative charged particles present. It doesn't matter whether the region of imbalance is flowing or whether it is still. Only the imbalance is important, not the "staticness." To say otherwise can cause several sorts of confusion.

All solid objects contain vast quantities of positive and negative particles whether the objects are electrified or not. When these quantities are not exactly equal and there is a tiny bit more positive than negative (or vice versa), we say that the object is "electrified" or "charged," and that "static electricity" exists. When the quantities are equal, we say the object is "neutral" or "uncharged." "Charged" and "uncharged" depends on the sum of opposite quantities. Since "static electricity" is actually an imbalance in the quantities of positive and negative, it is wrong to believe that the phenomenon has anything to do with lack of motion, with being "static." In fact, "static electricity" can easily be made to *move* along conductive surfaces. When this happens, it continues to display all it's expected characteristics as it flows, so it does not stop being "static electricity" while it moves along very non-statically! In a high voltage electric circuit, the wires can attract lint, raise hair, etc., even though there is a large current in the wires and all the charges are flowing (and none of the electricity is "static.") And last, when any electric circuit is broken and the charges stop flowing, they do *not* turn into "static electricity" and begin attracting lint, etc. A disconnected wire contains charges which are not moving (they are static,) yet it contains no "static electricity!"
To sort out this craziness, simply remember that "static electricity" is not a quantity of unmoving charged particles, and "static electricity" has nothing to do with unmoving-ness. If you believe that "static" and "current" are opposite types of "electricity," you will forever be hopelessly confused about electricity in general.

ELECTRIC POWER FLOWS FROM GENERATOR TO CONSUMER?

Electric power cannot be made to flow. Power is defined as "flow of energy." Saying that power "flows" is silly. It's as silly as saying that the stuff in a moving river is named "current" rather than named "water." Water is real, water can flow, flows of water are called currents, but we should never make the mistake of believing that water's motion is a type of substance. Talking of "current" which "flows" confuses everyone. The issue with energy is similar. Electrical energy is real, it is sort of like a stuff, and it can flow along. When electric energy flows, the flow is called "electric power." But electric power has no existence of its own. Electric power is the flow rate of another thing; electric power is an energy current. Energy flows, but power never does, just as water flows but "water current" never does.

The above issue affects the concepts behind the units of electrical measurement. Energy can be measured in Joules or Ergs. The rate of flow of energy is called Joules per second. For convenience, we give the name "power" to this Joule/sec rate of flow, and we measure it in terms of Watts. This makes for convenient calculations. Yet Watts have no physical, substance-like existence. The Joule is the fundamental unit, and the Watt is a unit of convenience which means "joule per second."
I believe that it is a good idea to teach only the term "Joule" in early grades, to entirely avoid the "watt" concept. Call power by the proper name "joules per second". Only introduce "watts" years later, when the students feel a need for a convenient way to state the "joules per second" concept. Unfortunately many textbooks do the reverse, they keep the seemingly-complex "Joule" away from the kids, while spreading the "watt" concept far and wide! Later they try to explain that joules are simply watt-seconds! (That's watts TIMES seconds, not watts per second.)
If you aren't quite sure that you understand watt-seconds, stop thinking backwards and think like this: Joules are real, a flow of Joules is measured in Joules per second, and "Watts" should not interfere with these basic ideas.

ELECTROMAGNET COILS USE UP ENERGY TO MAKE MAGNETISM?

Sustaining a magnetic field requires no energy. Coils only require energy to initially create a magnetic field. They also require energy to defeat electrical friction (resistance); to keep the charges from slowing down as they flow in wires. But if the resistance is removed, the magnetic field can exist continuously without any energy input. If electrically frictionless superconductive wire is used, a coil can be momentarily connected to an energy supply to create the field. Afterwards the power supply can be removed and both the current and the magnetic field will continue forever without further energy input. 

ELECTRIC CHARGES ARE INVISIBLE?

Electric charges are easily visible to human eyes, even though their motion is not. "Electricity" is not invisible! Never has been. When you look at a metal wire, you can see the charges of electricity which would flow during electric currents. They are silvery/metallic in color. They give metals their mirrorlike shine. Some metals have other colors as well, brass and copper for instance. Yet in all cases, the "metallic"-looking stuff is the metal's electrons. A dense crowd of electrons looks silvery; "electric fluid" is a silver liquid. And if metals weren't full of movable electrons, they wouldn't look metallic.

During electric currents in metals, the atoms stay still, but the silvery electron-stuff flows slowly along. Unfortunately the human eye cannot see the electric flow. That's part of the reason that "electricity" is so mysterious. Think about it... in an aquarium full of water, you cannot see any water flowing unless there are bubbles or dirt being carried along. And whenever clean water is flowing through a transparent hose, you can't see any flow. Even if the water is flowing very fast, the water-filled hose just looks like an unmoving glass rod. Same with wires: there's no bubbles or dirt being carried along by the electric current, therefore you can't see anything moving. You can see the STUFF that flows, just as you can see the water in an aquarium, but you can't see any flowing stuff.
Even if human eyes could see single electrons, we still couldn't see an electrical flow since the current is extremely slow. Electrons in metals typically flow at a few centimeters per hour, even during high currents. That's slower than the minute hand on a clock! Electric currents OOZE along like silly-putty flowing across a tilted board.
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Seeing imbalances in charge
Here's a separate topic... while the metallic-looking sea of charges in a metal is easily seen, IMBALANCES of charge are not. This get's confusing, since many books call imbalances of charge by the name "charge." They will tell you that charge is invisible, yet they really mean that charge-imbalances are invisible.
Wires contain enormous amounts of movable negative charge in the form of electrons, but they also contain positive charge in the form of protons within the metal atoms. If the number of protons and electrons are equal, don't they cancel out? Doesn't that mean that no charge exists? No. It means that no IMBALANCE of charge exists.
An "uncharged" wire is still full of charge, it still contains positive and negative charge in huge but equal quantities. The word "uncharged" doesn't mean "without charge," instead it means "without charge-imbalance." Yet even if there are more electrons than protons, or fewer electrons than protons, this imbalance is invisible. It's invisible because the greatest difference attainable is incredibly tiny when compared to the amount of charge that's already there. If an object is highly charged; even charged up to millions of volts, the extra charge is like a teacup poured into an ocean. The difference is far too small to be seen.

A "CONDUCTOR" IS A MATERIAL WHICH ALLOWS CHARGE TO PASS THROUGH IT?

The scientist's definition of the word "conductor" is different than the one above, and the one above has problems. For example, a vacuum offers no barrier to flows of electric charges, yet vacuum is an insulator. Vacuum is NOTHING, so how can it act as a barrier to electric current? Also, there is a similar problem with air: electric charges placed into the air can easily move along, yet air is an insulator. Or look at salt water versus oil. Oil is an insulator, while salt water is a conductor, yet neither liquid is able to halt the flow of any charges which are placed into it. How can we straighten out this paradox? Easy: use the proper definition of the word "conductor."

WRONG DEFINITION:
Conductor - a material which allows charges to pass through itself
BETTER:
Conductor - a material which can support an electric current
BEST:
Conductor - a material which contains movable electric charges
Here's an analogy:
Conductor - like a pipe which is already full of water 
Insulator - like a pipe with frozen liquid; a pipe plugged by ice
If we place a Potential Difference across either air or a vacuum, no electric current appears. This is sensible, since there are few movable charges in air or vacuum, so there can be no electric current. If we place a voltage across a piece of metal or across a puddle of salt water, charges will flow and an electric current will appear, since these substances are always full of movable charges, and therefore the "voltage pressure" causes the charges to flow. In metal, the outer electrons of the atoms are not bound upon individual atoms but instead can move through the material, and a voltage can drive these "liquid" electrons along. But if we stick our wires into oil, there will be no electric current, since oil does not contain movable charges.
If we were to inject charges into a vacuum, then we WOULD have electric current in a vacuum. This is how TV picture tubes and vacuum tubes work; electrons are forcibly injected into the empty space by a hot filament. However, think about it for a second: it's no longer a vacuum when it contains a cloud of electrons! Maybe we should change their name to "electron-cloud tubes" rather than "vacuum tubes", since the electron cloud is required before there can be any conductivity in the space between the plates. (But vacuum tubes already have another name, so this would just confuse things. They are called "hollow-state devices." As opposed to "solid state devices?" Nyuk nyuk.) 

LIGHTNING RODS DISCHARGE THE CLOUDS?

Make a model "landscape", install some lightning rods on the tiny houses, then bring a "storm cloud" nearby: bring the metal sphere of a VandeGraaff Electrostatic generator over your small town. The strong electric charge on the sphere will vanish. Doesn't this prove that lightning rods can discharge a thunderstorm? Nope.
The above demonstration was thought at one time to be accurate, and this old mistake is still in many books. In reality, lightning rods cannot remove the charge-imbalance from a thunderstorm. The scale of the typical demo is wrong. The stormcloud is a few miles up, and a few miles across, yet the lightning rod on the house is only a few feet tall. Therefore the metal-sphere "cloud" should be fairly large, and "Rod" should be far less than 1mm tall and attached to a wide metal ground plate.
The typical demonstration doesn't illustrate a lightning rod, it illustrates a 2000-ft radio tower or extremely tall office building.
Think about it: how can a tiny needle affect cubic kilometers of strong e-field? How could the relatively tiny current from a metal rod discharge a cloud that's over 1KM away. It can't! To do so, it would have to emit a hurricane wind made of ionized air. Unfortunately the lightning rod on your roof only emits about the same current as the needle in the model town: it emits a few microamperes. In other words, the scale model is not correct because the current coming from the needle is way too high. In order to be at the proper scale, the current would have to be hundreds of thousands of times smaller; too small to affect the VDG machine's charge.

HUMID AIR IS CONDUCTIVE?

Electrostatic experiments don't work very well under humid conditions. Some books state that the water vapor in the air makes the air conductive. Wrong. In reality the problem is caused by the liquid water that becomes adsorbed on surfaces of objects.

In order to make the air conductive, we'd have to fill it with movable charged particles. Evaporated water is not made of charged particles (ions,) instead it's made of neutral molecules, so the high humidity does not significantly affect the conductivity of the air. Even suspended water droplets (fog) does not significantly affect conductivity. For fog to be conductive, the individual droplets would have to posess an electric charge.
However, during humid conditions most insulators develop a surface layer of conductive water mixed with contaminants (including dissolved salts which makes this layer of water conductive.) If you find that you can't separate any charges by rubbing a balloon on your head, it's because the humid air has made the balloon and the hair very slightly damp. The air remains nonconductive, but surfaces of insulators become conductive when damp. Conductive surfaces don't separate any opposite charges when rubbed together. Cure this by warming them (drying them) with a blow-dryer. If a pair of insulators is sufficiently dry, it will "generate charge" even under very humid conditions. If conductive air were the culprit, this cure couldn't work

"ELECTRICITY IS WEIGHTLESS?

If by 'electricity' we mean the electrons, then 'electricity' is not weightless. Take a copper wire for example. Each atom weights about 115,000 times larger than the weight of an electron. If each atom supplies one electron to the "electric fluid" sea, then that sea is very light, but it is not weightless. The flowing "electricity" weighs about a hundred thousand times less than the copper metal. It's like a low pressure gas rather than like a liquid (but never forget that a gas is still matter!) One KG of copper would contain about ten milligrams of the movable electron-stuff which can flow as an electric current.

MORE TRUE STATEMENTS ABOUT "ELECTRICITY"

• In a DC circuit, the electricity within the wires flows exceedingly slowly; at speeds around inches per minute. At the same time, the electrical energy flows at nearly the speed of light.

• If we know the precise amount of electricity flowing per second through a wire (the Amperes,) this tells us nothing about the amount of energy being delivered per second into a light bulb (the Watts.) Amperes are not Watts, an electric current is not a flow of energy; they are two different things.

• In an electric circuit, the flow of the electricity is measured in Coulombs per second (Amperes.) The flow of energy is measured in Joules per second (Watts.) A Coulomb is not a Joule, and there is no way to convert from Coulombs of charge into Joules of energy, or from Amperes to Watts. A quantity of electricity is not a quantity of energy.

• Electrical energy is electromagnetism; it is composed of an electromagnetic field. On the other hand, the particles of electricity (electrons) flowing within a wire have little resemblance to an electromagnetic field. They are matter. Electricity is not energy, instead it is a major component of everyday matter.

• In an electric circuit containing coils, if we reverse the polarity of voltage while the direction of the flowing electricity remains the same, then the direction of the flowing energy will be reversed. Current same; energy flow reversed? Yes. A flow of energy does not follow the direction of the flowing electricity. You can know everything about the direction of the electricity within a wire, but this tells you nothing about the direction of the flowing electrical energy.

• In any electric circuit, the smallest particle of electrical energy is NOT the electron. The smallest particle of energy is the "unit quantum" of electromagnetic energy: it is the photon. Electrons are not particles of EM energy, neither do they carry the energy as they travel in the circuit. Electricity is 'made' of electrons and protons, while electrical energy is electromagnetism and is 'made' of photons.

• In the electric power grid, a certain amount of energy is lost because it flys off into space. This is well understood: electrical energy is electromagnetic waves travelling in the air, and unless the power lines are twisted or somehow shielded, they will act as 60Hz antennas. Waves of 60Hz electrical energy can spread outwards into space rather than following the wires. The power lines can even receive extra 60Hz energy from space, from magnetic storms in Earth's magnetosphere. Electric energy is gained and lost to empty space while the charges of electricity just sit inside the wires and wiggle. Energy is not electricity.

• In an electric circuit, electrical energy does not flow inside the copper. Instead it flows in the empty air surrounding the wires. This fact is hidden because we calculate energy flow by multiplying voltage times current. College-level physics books describe a less misleading method of measuring this energy flow: take the vector cross-product of the E and M components of the electromagnetic field at all points in a plane penetrated by the wires. We call this the Poynting Vector field. Add these measurements together, and this tells us the total energy flow (the Joules of energy that flow each second through the plane.) In other words, in order to discover the rate of energy flow, don't look at the flowing electrons. The electricity flow tells us little. Instead look at the electromagnetic fields which surround the wires.

Synchro

Schematic of Synchro Transducer The complete circle represents the rotor. The solid bars represent the cores of the windings next to them. Power to the rotor is connected by slip rings and brushes, represented by the circles at the ends of the rotor winding. As shown, the rotor induces equal voltages in the 120° and 240° windings, and no voltage in the 0° winding. [Vex] does not necessarily need to be connected to the common lead of the stator star windings.

Two simple synchros system

A synchro or "selsyn" is a type of rotary electrical transformer that is used for measuring the angle of a rotating machine such as an antennaplatform. In its general physical construction, it is much like an electric motor (See below.) The primary winding of the transformer, fixed to the rotor, is excited by a sinusoidal electric current (AC), which by electromagnetic induction causes currents to flow in three star-connected secondary windings fixed at 120 degrees to each other on the stator. The relative magnitudes of secondary currents are measured and used to determine the angle of the rotor relative to the stator, or the currents can be used to directly drive a receiver synchro that will rotate in unison with the synchro transmitter. In the latter case, the whole device (in some applications) is also called a selsyn (a portmanteau of self andsynchronizing). U.S. Naval terminology used the term "synchro" exclusively (possible exception: steering gear -- info. needed).

Synchro systems were first used in the control system of the Panama Canal, to transmit lock gate and valve stem positions, and water levels, to the control desks.^[1]

Fire-control system designs developed during World War II used synchros extensively, to transmit angular information from guns and sights to an analog fire control computer, and to transmit the desired gun position back to the gun location. Early systems just moved indicator dials, but with the advent of the amplidyne, as well as motor-driven high-powered hydraulic servos, the fire control system could directly control the positions of heavy guns. ^[2]

Smaller synchros are still used to remotely drive indicator gauges and as rotary position sensors for aircraft control surfaces, where the reliability of these rugged devices is needed. Digital devices such as the rotary encoder have replaced synchros in most other applications.

Synchros designed for terrestrial use tend to be driven at 50 or 60 hertz (the mains frequency in most countries), while those for marine or aeronautical use tend to operate at 400 hertz (the frequency of the on-board electrical generator driven by the engines).

Selsyn motors were widely used in motion picture equipment to synchronize movie cameras and sound recording equipment, before the advent of crystal oscillators and microelectronics.

On a practical level, synchros resemble motors, in that there is a rotor, stator, and a shaft. Ordinarily, slip rings and brushes connect the rotor to external power. A synchro transmitter's shaft is rotated by the mechanism that sends information, while the synchro receiver's shaft rotates a dial, or operates a light mechanical load. Single and three-phase units are common in use, and will follow the other's rotation when connected properly. One transmitter can turn several receivers; if torque is a factor, the transmitter must be physically larger to source the additional current. In a motion picture interlock system, a large motor-driven distributor can drive as many as 20 machines, sound dubbers, footage counters, and projectors.

Single phase units have five wires: two for an exciter winding (typically line voltage) and three for the output/input. These three are bussed to the other synchros in the system, and provide the power and information to precisely align by rotation all the shafts in the receivers. Synchro transmitters and receivers must be powered by the same branch circuit, so to speak; voltage and phase must match. Different makes of selsyns, used in interlock systems, have different output voltages.

Three-phase systems will handle more power and operate a bit more smoothly. The excitation is often 208/240 V 3-phase mains power.

In all cases, the mains excitation voltage sources must match in voltage and phase. The safest approach is to bus the five or six lines from transmitters and receivers at a common point.

Synchro transmitters are as described, but 50 and 60-Hz synchro receivers require rotary dampers to keep their shafts from oscillating when not loaded (as with dials) or lightly loaded in high-accuracy applications.

Large synchros were used on naval warships, such as destroyers, to operate the steering gear from the wheel on the bridge.

A different type of receiver, called a control transformer (CT), is part of a position servo that includes a servo amplifier and servo motor. The motor is geared to the CT rotor, and when the transmitter's rotor moves, the servo motor turns the CT's rotor and the mechanical load to match the new position. CTs have high-impedance stators and draw much less current than ordinary synchro receivers when not correctly positioned.

Synchro transmitters can also feed synchro to digital converters, which provide a digital representation of the shaft angle.

Synchro variants

So called brushless synchros use rotary transformers (that have no magnetic interaction with the usual rotor and stator) to feed power to the rotor. These transformers have stationary primaries, and rotating secondaries. The secondary is somewhat like a spool wound with magnet wire, the axis of the spool concentric with the rotor's axis. The "spool" is the secondary winding's core, its flanges are the poles, and its coupling does not vary significantly with rotor position. The primary winding is similar, surrounded by its magnetic core, and its end pieces are like thick washers. The holes in those end pieces align with the rotating secondary poles.

For high accuracy in gun fire control and aerospace work, so called multi-speed synchro data links were used. For instance, a two-speed link had two transmitters, one rotating for one turn over the full range (such as a gun's bearing) , while the other rotated one turn for every 10 degrees of bearing. The latter was called a 36-speed synchro. Of course, the gear trains were made accurately. At the receiver, the magnitude of the 1X channel's error determined whether the "fast" channel was to be used instead. A small 1X error meant that the 36x channel's data was unambiguous. Once the receiver servo settled, the fine channel normally retained control.

For very critical applications, three-speed synchro systems have been used.

So called multispeed synchros have stators with many poles, so that their output voltages go through several cycles for one physical revolution. For two-speed systems, these do not require gearing between the shafts.

Differential synchros are another category. They have three-lead rotors and stators like the stator described above, and can be transmitters or receivers. A differential transmitter is connected between a synchro transmitter {CX} and a receiver {CT}, and its shaft's position adds to (or subtracts from, depending upon definition) the angle defined by the transmitter. A differential receiver is connected between two transmitters, and shows the sum (or difference, again as defined) between the shaft positions of the two transmitters.

A resolver is similar to a synchro, but has a stator with four leads, the windings being 90 degrees apart physically instead of 120 degrees. Its rotor might be synchro-like, or have two sets of windings 90 degrees apart. Although a pair of resolvers could theoretically operate like a pair of synchros, resolvers are used for computation. Both synchros and resolvers have an accurate sine-function relationship between shaft position and transformation ratio for any pair of stator connections. (Of course, there are angular offsets of 120 or 240 degrees for synchros, and multiples of 90 degrees for resolvers, depeding upon the specific pair of leads being considered.)

Resolvers, in particular, can perform very accurate analog conversion from polar to rectangular coordinates. Shaft angle is the polar angle, and excitation voltage is the magnitude. The outputs are the [x] and [y] components. Resolvers with four-lead rotors can rotate [x] and [y] coordinates, with the shaft position giving the desired rotation angle.

Resolvers with four output leads are general sine/cosine computational devices. When used with electronic driver amplifiers and feedback windings tightly coupled to the input windings, their accuracy is enhanced, and they can be cascaded ("resolver chains") to compute functions with several terms, perhaps of several angles, such as gun (position) orders corrected for ship's roll and pitch.

There are synchro-like devices called transolvers, somewhat like differential synchros, but with three-lead rotors and four-lead stators.

A special T-connected transformer arrangement invented by Scott ("Scott T") interfaces between resolver and synchro data formats; it was invented to interconnect two-phase AC power with three-phase power, but can also be used for precision applications

Resolver

A resolver is a type of rotary electrical transformer used for measuring degrees of rotation. It is considered an analog device, and has a digital counterpart, the rotary (or pulse) encoder.

Description:

The most common type of resolver is the brushless transmitter resolver (other types are described at the end). On the outside, this type of resolver may look like a small electrical motorhaving a stator and rotor. On the inside, the configuration of the wire windings makes it different. The stator portion of the resolver houses three windings: an exciter winding and two two-phase windings (usually labeled "x" and "y") (case of a brushless resolver). The exciter winding is located on the top, it is in fact a coil of a turning transformer. This transformer empowers the rotor, thus there is no need for brushes, or no limit to the rotation of the rotor. The two other windings are on the bottom, wound on a lamination. They are configured at 90 degrees from each other. The rotor houses a coil, which is the secondary winding of the turning transformer, and a primary winding in a lamination, exciting the two two-phase windings on the stator.

The primary winding of the transformer, fixed to the stator, is excited by a sinusoidal electric current, which by electromagnetic induction induces current to flow through the secondary windings along the stator. The two two-phase windings, fixed at right (90°) angles to each other on the stator, produce a sine and cosine feedback current by the same induction process. The relative magnitudes of the two-phase voltages are measured and used to determine the angle of the rotor relative to the stator. Upon one full revolution, the feedback signals repeat their waveforms. This device may also appear in non-brushless type, i.e., only consisting in two stacks of sheets, rotor and stator.

Types

Basic resolvers are two-pole resolvers, meaning that the angular information is the mechanical angle of the stator. These devices can deliver the absolute angle position. Other types of resolver are multipole resolvers. They have 2*p poles, and thus can deliver p cycles in one rotation of the rotor: electrical angle = mechanical angle * p. Some types of resolvers include both types, with the 2-pole windings used for absolute position and the multipole windings for accurate position. Two-pole resolvers can usually reach angular accuracy up to about +/-5′, whereas multipole resolver can provide better accuracy, up to 10′′ for 16-pole resolvers, to even 1′′, for instance for 128-pole resolvers.

Multipole resolvers may also be used for monitoring multipole electrical motors. This device can be used in any application in which the exact rotation of an object relative to another object is needed, such as in a rotary antenna platform or a robot. In practice, the resolver is usually directly mounted to an electric motor. The resolver feedback signals are usually monitored for multiple revolutions by another device. This allows for geared reduction of assemblies being rotated and improved accuracy from the resolver system.

Because the power supplied to the resolvers produces no actual work, the voltages used are usually low (<24 VAC) for all resolvers. Resolvers designed for terrestrial use tend to be driven at 50-60 Hz (mains power frequency), while those for marine or aeronautical use tend to operate at 400 Hz (the frequency of the on-board generator driven by the engines). Control systemstend to use higher frequencies (5 kHz).

Other types of resolver include:

Receiver resolvers

These resolvers are used in the opposite way to transmitter resolvers (the type described above). The two diphased windings are energized, the ratio between the sine and the cosine representing the electrical angle. The system turns the rotor to obtain a zero voltage in the rotor winding. At this position, the mechanical angle of the rotor equals the electrical angle applied to the stator.

Differential resolvers

These types combine two diphased primary windings in one of the stacks of sheets, as with the receiver, and two diphased secondary windings in the other. The relation of the electrical angle delivered by the two secondary windings and the other angles is secondary electrical angle, mechanical angle, and primary electrical angle. These types were used, for instance, to calculate trigonometric functions without electronic computers.

A related type is also the transolver, combining a two-phase winding like the resolver and a triphased winding like the synchro.

Prototype filter

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionaliseddesign from which the desired filter can be scaled or transformed. They are most often seen with regard to electronic filters and most especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.

Filters are required to operate at many different frequencies, impedances and bandwidths. The utility of a prototype filter comes from the property that all these other filters can be derived from it by applying a scaling factor to the components of the prototype. The filter design need thus only be carried out once in full, other filters being obtained by simply applying a scaling factor.

Especially useful is the ability to transform from one bandform to another. In this case the transform is more than a simple scale factor. By bandform is meant the category of passband that the filter possesses. The usual bandforms are lowpass, highpass, bandpass and bandstop but others are possible. In particular it is possible for a filter to have multiple passbands. In fact, in some treatments, the bandstop filter is considered to be a type of multiple passband filter having two passbands. Most commonly, the prototype filter is expressed as a lowpass filter but other techniques are possible.

Low-pass prototype

The prototype is most usually given as a low-pass filter with a cut-off frequency (image filters) or 3dB bandwidth frequency (network synthesis filters) which has an angular frequency of ω_c'= 1 rad/s. Occasionally, frequency f' ' = 1 Hz is used instead. In principle, any non-zero frequency point on the filter response could be used as a reference for the prototype design.

Likewise, the nominal or characteristic impedance of the filter is set to R ' = 1 Ω.

The prototype filter can only be used to produce other filters of the same class and order. For instance, a fifth order Bessel filter prototype can be converted into any other fifth order Bessel filter but it cannot be transformed into a third order Bessel filter or a fifth order Tchebyscheff filter.

Frequency scaling

The prototype filter is scaled to the frequency required with the following transformation;

where ω_c' is the value of the frequency parameter (eg cut-off frequency) for the prototype and ω_c is the desired value. So if ω_c' = 1 then the transfer function of the filter is transformed as;

It can readily be seen that to achieve this the non-resistive components of the filter must be transformed by;

  and,   

Impedance scaling

Impedance scaling is invariably a scaling to a fixed resistance. This is because the terminations of the filter, at least nominally, are taken to be a fixed resistance. To carry out this scaling to a nominal impedance R, each impedance element of the filter is transformed by;

It may be more convenient on some elements to scale the admittance instead;

The prototype filter above transformed to a 600Ω, 16kHz lowpass filter

It can readily be seen that to achieve this the non-resistive components of the filter must be scaled as;

    and,    

Impedance scaling by itself has no effect on the transfer function of the filter (always provided that the terminating impedances have the same scaling applied to them). However, it is usual to combine the frequency and impedance scaling into a single step;^[1]

  and,   

Bandform transformation

In general, the bandform of a filter is transformed by replacing iω where it occurs in the transfer function with a function of iω. This in turn leads to the transformation of the impedance components of the filter into some other component(s). The frequency scaling above is a trivial case of bandform transformation corresponding to a lowpass to lowpass transformation.

Lowpass to highpass

The frequency transformation required in this case is;^[2]

where ω_c is the point on the highpass filter corresponding to ω_c' on the prototype. The transfer function then transforms as;

Inductors are transformed into capacitors according to,

and capacitors are transformed into inductors,

the primed quantities being the component value in the prototype.

Lowpass to bandpass

In this case the required frequency transformation is;^[3]

where Q is the Q-factor and is equal to the inverse of the fractional bandwidth;^[4]

If ω₁ and ω₂ are respectively, the lower and upper frequency points of the bandpass response corresponding to ω_c' of the prototype then,

   and    

Δω is the absolute bandwidth and ω₀ is the resonant frequency of the resonators in the filter. Note that frequency scaling the prototype prior to lowpass to bandpass transformation does not affect the resonant frequency, but instead affects the final bandwidth of the filter.

The transfer function of the filter is transformed according to;

The prototype filter above transformed to a 50Ω, 6MHz bandpass filter with 100kHz bandwidth

Inductors are transformed into series resonators,

and capacitors are transformed into parallel resonators,

Lowpass to bandstop

The required frequency transformation for lowpass to bandstop is;^[5]

Inductors are transformed into parallel resonators,

and capacitors are transformed into series resonators,

Lowpass to multi-band

Filters with multiple passbands may be obtained by applying the general transformation;

The number of resonators in the expression corresponds to the number of passbands required. Lowpass and highpass filters can be viewed as special cases of the resonator expression with one or the other of the terms going to zero as appropriate. Bandstop filters can be regarded as a combination of a lowpass and a highpass filter. Multiple bandstop filters can always be expressed in terms of a multiple bandpass filter. In this way it can be seen that this transformation represents the general case for any bandform and all the other transformations are to be viewed as special cases of it.

The same response can equivalently be obtained, sometimes with a more convenient component topology, by transforming to multiple stopbands instead of multiple passbands. The required transformation in those cases is;

Alternative prototype

In his treatment of image filters, Zobel provides an alternative basis for constructing a prototype which is not based in the frequency domain.^[6] The Zobel prototypes do not, therefore, correspond to any particular bandform, but they can be transformed into any of them. Not giving special significance to any one bandform makes the method more mathematically pleasing but it is not in common use.

The Zobel prototype considers filter sections, rather than components. That is, the transformation is carried out on a two-port network rather than a two-terminal inductor or capacitor. The transfer function is expressed in terms of the product of the series impedance, Z, and the shunt admittance Y of a filter half-section. See the article Image impedance for a description of half-sections. This quantity is nondimensional, adding to the prototypes generality. Generally, ZY is a complex quantity,

 and as U and V are both, in general, functions of ω we should properly write,

With image filters, it is possible to obtain filters of different classes from the constant k filter prototype by means of a different kind of transformation (see composite image filter). Constant k being those filters for which Z/Y is a constant. For this reason, filters of all classes are given in terms of U(ω) for a constant k, which is notated as,

In the case of dissipationless networks, ie no resistors, the quantity V(ω) is zero and only U(ω) need be considered. U_k(ω) ranges from 0 at the centre of the passband to -1 at the cut-off frequency and then continues to increase negatively into the stopband regardless of the bandform of the filter being designed. To obtain the required bandform, the following transforms are used,

For a lowpass constant k prototype that is scaled;

the independent variable of the response plot is,

The bandform transformations from this prototype are,

for lowpass, 

for highpass, 

and for bandpass,