User:Chetvorno/work1

= For Clock =

Accuracy
The timekeeping elements in all modern clocks, which include pendulums, balance wheels, tuning forks, the quartz crystals used in quartz watches, and even the vibrating atoms in atomic clocks, are called harmonic oscillators. The reason harmonic oscillators are used in clocks is that they vibrate or oscillate at a specific resonant frequency (period) which is dependent on their construction, and resist oscillating at other rates. However the resonant frequency is not infinitely 'sharp'. Around the resonant frequency there is a narrow natural band of frequencies (or periods), called the resonance width or bandwidth, that the harmonic oscillator will oscillate at. The actual frequency of the oscillator in a clock may vary randomly within this bandwidth in response to disturbances, but at frequencies outside this band, the oscillator, and the clock, will not function at all.

Q factor
The measure of a harmonic oscillator's resistance to disturbances to its oscillation period is a dimensionless parameter called the Q factor equal to the resonant frequency divided by the resonance width. The higher the Q, the smaller the resonance width, and the more constant the frequency or period of the oscillator for a given disturbance. The reciprocal of the Q is roughly proportional to the limiting accuracy achievable by a harmonic oscillator as a time standard.

The Q is related to how long it takes for the natural oscillations of an oscillator to die out due to friction. High quality oscillators, that "ring" for a long time after being set in motion, have high Q and keep better time. The Q can be measured by counting the number of oscillations it takes for the amplitude of the oscillator's vibrations to decay to 1/e = 36.8% of its initial amplitude, and multiplying by 2π.

As an example, in a pendulum clock, the pendulum must receive pushes from the clock's movement to keep it swinging, to replace the energy the pendulum loses to friction. These pushes, applied by a mechanism called the escapement, are the main source of disturbance to the pendulum's motion. The Q is equal to 2π times the energy stored in the pendulum, divided by the energy lost to friction during each oscillation period, which is the same as the energy added by the escapement each period. It can be seen that the smaller the fraction of the pendulum's energy that is lost to friction, the less energy needs to be added, the less the disturbance from the escapement, the more 'independent' the pendulum is of the clock's mechanism, and the more constant its period is.

The Q of an oscillator generally increases, other things being equal, with an increase in resonant frequency, or a decrease in dissipative forces like drag and friction.

Higher accuracy
Slow mechanical oscillators like pendulums and balance wheels have a low enough Q that the disturbance caused by the impulses to keep them moving is often the limiting factor on their timekeeping accuracy. Therefore the design of the mechanism that provides these impulses, the escapement, has a large effect on their accuracy. The quest for higher accuracy in mechanical clocks and watches has mostly been a search for better escapements, which disturb the oscillator less.

In contrast, in higher Q oscillators such as quartz crystals, the higher Q makes the oscillator insensitive to disturbance from the electronic oscillator circuit that provides the impulses to keep it vibrating. These oscillators are more influenced by environmental factors such as changes in temperature. In the most accurate oscillators of all, the vibrating atoms in atomic clocks, the Q is actually too high to be measured, and the accuracy of these clocks is limited by the resolution of the measuring equipment, such as the lifetime of the atoms in the apparatus. Current research to develop higher accuracy atomic clocks focuses on making the atoms "hold still" for longer periods, using lasers to cool them, so their vibration rate can be measured more accurately.

=For Mechanical watch=

Terminology
Mechanical watches are a mature technology with a long history, and a number of specialized terms are used to describe them:
 * Adjusted - high quality mechanical watches are made more accurate by a process of adjusting the balance wheel and balance spring to eliminate errors due to temperature changes, and the effects of gravity on the balance wheel when the watch is in different positions. The usual adjustments are: heat, cold, isochronism, dial up, pendant up, pendant right, pendant left, pendant down.
 * Arbor - the axle or shaft of a watch's gear wheel.
 * Automatic or self-winding watch – a watch in which the mainspring is automatically wound using the natural motions of the wearer’s wrist, to make manual winding unnecessary.
 * Baguette - a watch in which the length of the case is at least three times its width; a long, narrow, diamond shaped watch.
 * Banking or knocking - an abnormal running condition in which the balance wheel rotates too far in each direction, causing the impulse pin to strike the back of the pallet fork. This is usually caused by too much drive force from the mainspring, and makes the watch gain time.
 * Barrel - a cylindrical box in a watch movement in which the mainspring is coiled, with gear teeth around the circumference to drive the wheel train.
 * Breguet key - a winding key with an attached ratchet allowing winding in only one direction.
 * Breguet spring or overcoil spring - a type of balance spring in which the end is bent up over the plane of the spiral, to increase accuracy.
 * BPH - beats per hour.
 * Bumper - a watch with an early type of self-winding mechanism in which a pivoted weight bumps back and forth between spring stops.
 * Calendar watch – a watch that displays the date, and often the day of the week.
 * Chronograph – a watch with additional stopwatch functions. Buttons on the case start and stop the second hand and reset it to zero, and usually several subdials on the face display the elapsed time in larger units.
 * Chronometer – a watch that has met the high standards of accuracy of the Controle Officiel Suisse des Chronometeres (COSC) of Switzerland.
 * Click - the pawl which stops the mainspring from turning backward and unwinding. It makes the 'clicking' sound when the watch is wound.
 * Complication – additional functions on a watch besides the basic display of time
 * Crown - knob on the outside of the case used to wind the watch, and usually to set the time.
 * Damaskeening - a decorative pattern of wavy parallel lines often used on the plates of watch movements. An American term, in Europe it was called Fausse Cotes or Geneva stripes.
 * Ebauche (ay-boesh) - an unfinished watch movement, lacking the balance, balance cock, mainspring, and with the plates unpolished. This is the form in which watch movements are sold by movement manufacturers.  Watch manufacturers buy them, finish them, and put their own name on them.
 * Equation of time - a dial which displays the difference between the time kept by clocks and the time as indicated by the position of the sun, which varied during the year. This rare complication originated when watches had to be set by the passage of the sun overhead.
 * Escape wheel -
 * Flyback - a type of chronograph, in which pushing the stopwatch button successively causes the seconds hand to start, stop, and then return to zero. Also used more generally for a hand on the face that doesn't rotate continuously, but traverses a scale and then jumps back to the beginning of the scale.
 * Fusee - a conical pulley with a chain wound around it, used in the earliest pocketwatches to equalize the force of the mainspring.
 * Going barrel - the type of mainspring barrel used in modern watches, with a ring of teeth around it to drive the gear train.
 * Going train - the part of the gear train that transmits power from the mainspring to the balance wheel
 * Grande sonnerie (grand strike) - a repeater watch that chimes the hours and quarter hours at the press of a button.
 * Hacking or hack set - a feature that stops the second hand while the watch is being set, enabling the watch to be synchronized to the precise second. Mostly seen in military watches.
 * Hairspring - the balance spring of a watch.
 * Hunter case - a pocketwatch case with a hinged metal lid to protect the face that must be opened to see the time. The term originated with pocketwatches made to be carried on horseback by hunters.
 * Incabloc - trade name for a patented Swiss shockproof mounting system for balance wheel pivots.
 * Isochronism - means that a watch runs at the same rate regardless of the drive force; that is regardless of whether the mainspring is fully wound up or almost run down. The term is also used for the adjustments to the balance spring to achieve isochronism.
 * Jewels - bearings made from synthetic rubies or sapphires for the pivots in a watch, to reduce friction.
 * Jump hour watch - a mechanical watch which indicates the time with digits displayed in windows, instead of rotating hands.
 * Key set - an older pocketwatch in which the time had to be set with a key.
 * Key wind - an older pocketwatch in which the watch was wound with a key.
 * Keyless work - the mechanism used to set the time in a modern watch, so called because it doesn't use a key as in older watches.
 * Lever - 'T' shaped lever in the lever escapement. It has jewelled pallets on the arms that engage the escape wheel, and a fork on the end which gives impulses to the impulse pin on the balance wheel.
 * Lever escapement - the type of escapement used in modern watches. It has a 'T' shaped lever which is pushed by the escape wheel, which in turn gives pushes to the balance wheel to keep it oscillating.
 * Ligne - an old French measure used to express the size of watch movements. 2.256 millimeters.
 * Lugs - projections on a wristwatch case used to attach the strap.
 * Minute repeater - a watch that chimes the time audibly to the minute at the press of a button. This rare complication was originally used by blind people.
 * Moon phase dial - a complication that displays the phase of the moon on a dial with a painted moon face on a rotating disk.
 * Movement - the mechanism inside the watch case that keeps time and moves the hands.
 * Pallets -
 * Pair case - a pocketwatch that has two cases, an outer decorative one and an inner plain one to protect the movement. Needed because the outer case had to be opened frequently to wind the watch.
 * Perpetual calendar - a calendar mechanism which automatically adjusts for the different length of the months and for leap years.
 * Power reserve indicator or wind indicator - a dial that shows how much power is left in the watch's mainspring, usually graduated in hours the watch has left to run.
 * Rattrapante - a feature on chronographs to measure split or lap times. The watch has two second hands which start the timing interval moving together.  A second push of the timing button stops the lap hand, so the lap time can be read, while the other second hand continues.  Another push of the button causes the lap hand to catch up to the second hand again.
 * Regulator - lever in the watch movement on the balance spring which is used to adjust the watch's rate.
 * Remontoire - in some antique watches, a small secondary spring which is wound up repeatedly by the mainspring, and in turn runs the movement. Its purpose is to even out the force of the mainspring.
 * Repeater - a watch that chimes the hours audibly at the press of a button. This rare complication was used before electric lighting to check the time in the dark, and by the blind.
 * Shockproof - watch company terminology for any of several systems for mounting the balance wheel pivots with springy mountings that absorb shock, to prevent the pivots from breaking if the watch is dropped.
 * Skeleton watch - a watch with the plates and bridges of the movement decoratively carved and cut away to allow the works to be seen. The face and/or the back is transparent to allow this decoration to be seen.
 * Stackfreed - a cam device used occasionally in the earliest watches to equalize the force of the mainspring.
 * Stopwork - devices used on the mainspring barrel of early watches to stop the spring from being wound all the way up, to prevent the mainspring from being broken by careless winding.
 * Tank watch - a watch with
 * Tonneau watch - a watch shaped like a barrel, rectangular with convex sides.
 * Tourbillon - an expensive elaborate complication that was originally designed to make the watch more accurate. The effect of gravity on the balance wheel makes watches run at slightly different rates when in different positions.  In a tourbillon, the balance wheel and escapement are mounted in a cage that rotates slowly to eliminate the errors due to gravity.  Usually the tourbillon is exposed on the watch's face to show it off.
 * Train - the gear train of a watch.
 * Transition watch - an early wristwatch converted from a pocketwatch. When wristwatches became popular in the early 1900s, the only movements that were available were pocketwatch movements.  They are larger than later wristwatches and the crown is at the 12 o'clock position as in pocketwatches, not at the normal 3 o'clock position.
 * Trench watch - An early style of wristwatch worn by soldiers during World War 1. They typically had wire lugs that the strap was threaded through, and often had a metal shrapnel guard or a leather pocket on the strap to protect the face.

= For Regulator clock =

A regulator clock or just regulator was a precision pendulum clock that was used to set other clocks.

From their invention in 1656 until the 1930s, pendulum clocks were the world's most accurate timekeepers. The term 'regulator' originated in the 18th century, when a need arose for clocks more accurate than ordinary domestic pendulum clocks for scientific research, astronomy, surveying, and navigation. British clockmaker George Graham made the first regulators  To achieve accuracy, these clocks are always weight driven, have temperature compensated pendulums, and omit unnecessary complications such as striking mechanisms,  calendar works, and moon dials.

By the 19th century, several different categories of regulator clocks were made. The most accurate clocks were called astronomical regulators, and were used in astronomical and naval observatories, and as primary standards for the first national time distribution services. Then manufacturers produced high grade wall and floor clocks to serve as precision timekeepers for industry. These were installed in workplaces, post offices, railroad stations, and jewelry stores to schedule work and 'regulate' other clocks. Toward the end of the century, the word 'regulator' became used as a promotional term, and was applied to ordinary clocks that were not exceptionally accurate, such as vienna regulators and schoolhouse regulators.

Characteristics of regulators
These are the features that distinguish precision pendulum clocks:


 * Weight drive : The timekeeping rate of all clock mechanisms varies with changes in the drive force applied to it, a problem called lack of isochronism. The drive force provided by a mainspring decreases during the clock's running period as the spring unwinds, while the force provided by a weight suspended from a pulley by a cord is constant.  So all regulators used weights for power.


 * Temperature compensated pendulum : The largest source of error in early pendulum clocks was expansion and contraction of the pendulum rod with temperature changes. As the temperature rises, the pendulum rod gets longer and the period of oscillation gets slower, causing the clock to lose time.  All regulators had pendulums that compensated for temperature changes so the rate of the clock was unaffected.  The most common type was the mercury pendulum.  See below for explanation of temperature compensation.


 * Precision escapement : The escapement is a major source of error in a pendulum clock, because of its disturbing influence on the pendulum.  Most regulators used the deadbeat escapement, introduced by George Graham in his regulators around 1715.  In the 1800s a few used the Dennison gravity or pinwheel escapements.  Astronomical regulators sometimes used special escapements such as the Riefler escapement.  Often the escapement pallets were made of jewels, to reduce friction.  By the turn of the century electromechanical escapements began to be used.  These used a switch contact on the pendulum to turn on an solenoid or electromagnet to impulse the pendulum, and


 * Simplified wheel train : In order to reduce friction and backlash, which resulted in varying force applied to the escapement, the wheel trains of regulators was made as simple as possible. Unnecessary features such as striking, calender dials, and moon phase dials were omitted.  For additional simplification, the traditional clock face with central coaxial hour and minute hand was dispensed with, to eliminate the extra gears (motion work) that ran it.  The hour and minute hands were run directly off separate wheels of the train, whose axles were brought out through the face.  So the hour and minute hand were located in separate subdials on the clock face, while the large second hand was located in the center with graduations around the rim.  The hour hand often rotated once in 24 hours.  The hands were thin and light, and often counterbalanced so they didn't exert varying torques on the train at different angles.   The faces were devoid of decoration, with fine black engraved graduations on a white background.


 * High tooth count : The finer the teeth of the gears used in the wheel train, the more even was the force transmitted to the escapement. In ordinary clocks, the pinions, or small gears, usually had 6 teeth.  Regulators used pinions of 8, 10, or even 12 teeth.  The large gear wheels had proportionately larger numbers of teeth also, to keep the gear ratios correct.


 * Jewelled bearings and pallets : To reduce friction, the pallets of the escapement and sometimes the bearings of the wheel train were jewelled.

Temperature compensated pendulums
The largest source of error in early pendulum clocks, particularly in the unheated buildings of the time, was expansion and contraction of the pendulum rod with temperature changes. A temperature increase of 60° F causes a clock with a steel pendulum rod to expand ___% and lose ____ seconds per day. Early regulators often had wooden pendulums, which expand less and only lose ____ seconds.

The first pendulum compensated for temperature changes was the mercury pendulum, invented by George Graham in 1721. In this, the bob is replaced by a container of the liquid metal mercury. As the temperature increases, the pendulum rod gets longer, lowering the pendulum's center of gravity, but the mercury expands and its surface rises in the container, raising the center of gravity. By using the correct amount of mercury, these changes cancel out, leaving the pendulum's center of gravity, and its period, unchanged with temperature. Mercury pendulums were the most widely used compensated pendulums in regulators. Their only disadvantage was that the mercury was slow to come to the new temperature after a temperature change, so the compensation lagged behind a little. This was often dealt with by using several slender jars of mercury.

Another type of temperature compensated pendulum used in regulators was the gridiron pendulum, invented by John Harrison in 1726. In this device, the pendulum rod was made of several parallel bars of two different metals, usually zinc and steel. The zinc has a larger thermal expansion than steel, so the zinc rods expand more when warmed. The rods are connected together at top and bottom in such a way that the zinc rods shorten the pendulum when they expand. By using the correct lengths, the shortening effect of the zinc rods exactly compensates for the lengthening of the steel rods, leaving the pendulum's length unchanged with temperature. Gridirons had the disadvantage that during expansion or contraction the rods had to slide through holes in their supports, and this microscopic sliding action occurred in a series of 'jumps', causing the rate of the clock to jump discontinuously. So gridirons were used in early regulators but fell out of use in regulators in the 19th century. However, they became popular in house clocks. In fact they became so closely associated with precision that many 'regulator' house clocks had 'gridiron' style pendulums that had no actual temperature compensation ability, but were just for show.

By the turn of the century, Invar pendulum rods began to be used in some high precision astronomical regulators. Invented in 1896 Charles Édouard Guillaume, this nickel steel alloy had a very low thermal expansion of ____ and this small thermal expansion could be compensated by a few inches of aluminum under the pendulum. Later fused quartz pendulum rods were used, which had even lower expansion.

Astronomical regulators
=For Clockmaker=

The first clocks were large turret clocks in cathedrals and town squares. These unique mechanisms were often designed by educated philosophers, who hired blacksmiths to do the actual metalwork. In the 1400s clocks were gradually made small enough to to be used in private houses, and domestic clocks and chamber clocks appeared, but these were

Music: asymmetric meters
Although Balkan dance music includes pieces with standard 'Western' time signatures of 2/4, 3/4, and 4/4, it also includes meters with 5, 7, 9, 11, 13, and 15 beats per measure. These are called asymmetric meters. They are often described as composed of combinations of two different length units or "beats", which are themselves composed of different numbers of the underlying metric beats. For example, the Macedonian dance Lesnoto is done to a seven beat measure (7/8), with emphasis on the first, fourth and sixth beat. These can be grouped as one "slow" unit of 3 of the underlying metric beats and two "quick" units of 2 beats: (BEAT, beat, beat), (BEAT, beat), (BEAT, beat), often written as slow-quick-quick or 3-2-2. It is important to understand that Balkan musicians do not describe their music's rhythms in these terms, this is a simplification used by western musicians and does not capture the full complexities of the subtlely-changing Balkan rhythms.

Each Balkan dance family uses a distinctive combination of these basic units, such as Paidushko (5 beats: 2-3), Eleno Mome (7 beats: 2-2-1-2), Rachenitsa (7 beats: 2-2-3), Daichovo (9 beats: 2-2-2-3),  Kopanitsa (11 beats: 2-2-3-2-2),  and Bučimis (15 beats: 2-2-2-2-3-2-2).

Formations
Many dances are done in a line, with all dancers facing perpendicular to the line, and holding onto the hands, shoulders, or belt of the neighboring dancers. Originally many dances were gender specific; exclusively women's or men's dances. In the ones that were not, men and women danced in separate lines, or in a gender-segregated line with women at the front and men at the back, sometimes    However today the same dances are often done in mixed lines. Long lines are curved in an arc, with dancers facing in. In most dances the line moves to the dancer's right, but some move to the left, or in and out. The dancer on the front end of the line in the direction of motion is the "leader", with responsibility for leading the line, and calling any changes in step. Many dances have "called" variations, in which the leader periodically calls out the name of a variation and the line changes its step to that variation. In others, the leader, or individual dancers, may improvise variations or embellishments. In others, the leader is a "solo" performer, executing showy acrobatic moves while the rest of the line performs the basic step. Dancers joining the dance always join at the end of line, never the front leader position. Another common formation for dances is circles, large or small, with dancers facing in, in which case there is no leader. Other formations are short straight lines, two lines facing each other, or solo dancing without contact.

Handholds
These are the most common handholds used in Balkan line dances. The names given are merely descriptive, not necessarily translations of names used in the native languages, but have some currency among international folk dancers.
 * "V": The hands are held down relaxed at the side of the body, holding the hand of the neighboring dancer on each side. The right hand is palm up, the left hand palm down.
 * "W": The arms are held up with elbows bent in a "W" shape, hands at shoulder height, holding the hand of the neighboring dancers, right hand palm up, left hand palm down.
 * "Teacup": The left hand is placed on the stomach or holds the belt buckle, and the right hand is looped loosely through the next dancer's left arm.
 * Front basket: The hands are held down as in the "V", but hands are joined with the two dancers on the far side of the neighboring dancers, in front of the neighboring dancers, so alternate dancers in the line are linked.
 * Back basket: Like the front basket, except the hands are joined behind the backs of the neighboring dancers.
 * Belt: Each dancer wears a belt or sash, and holds the belt of the two neighboring dancers, in front over the stomach. This links the line of dancers together securely, and so it is found in fast moving acrobatic dances.
 * Shoulder: The arms are held out horizontally to the sides, with the hands on the shoulders (deltoid muscle) of the neighboring dancers.
 * "Pinkie": Similar to the "W" hold, except that the hands are made fists and the extended, curved, little fingers are linked, right hand up, left hand down
 * "Halay": Seen in Turkish dances, the dancer stands straight upright, as close to the next dancer as possible, with arms straight down and rigid at his sides, holding the hand of the neighboring dancers.

Some dances incorporate changes in arm position, or swinging or pumping motions of the arms.

= For Screw =

Analysis of square thread screws 2
Square threads have lower friction and higher efficiency than the other types of threads used in screws: Acme, trapezoidal and V-threads. This is because the outward angle of the bearing surface in the other types increases the normal force on the surface, and therefore the friction. Therefore square threads are used for screws that carry power, such as lead screws and screw jacks. This important class of screws can be analyzed as a simple inclined plane. The slope of the thread bearing surface is simply equal to the distance ratio $$ l/(2 \pi r) \,$$, so the pitch angle of the thread is given by
 * $$\tan \theta = \frac {l}{2 \pi r} \,$$

The "angle of repose" φ, the angle that the resultant Fr = FN + Fμ makes with the surface normal, is
 * $$\tan \phi = \mu \,$$

where μ is the static coefficient of friction between the screw threads.

The body carrying the moving screw threads, when considered as a free body, has three forces acting on it:
 * The rotational force on the edge of the shaft Fin, which acts horizontally (tangentially)
 * The axial load force on the shaft Fout, which acts downwards, parallel to the screw's axis
 * The force of the stationary screw threads on the moving threads. This can be resolved into two components:
 * The normal force Fn of the stationary threads on the moving threads. This is directed perpendicular (normal) to the thread surface.
 * The frictional force Ff of the moving threads on the object, which acts parallel to the thread surface, and is always in a direction opposite to the motion of the object. It is equal to the normal force multiplied by the coefficient of friction μ between the two surfaces.

Using Newton's second law of motion the load will be stationary or in steady motion if the sum of the forces on it is zero. Since the direction of the frictional force is opposite for the case of forward and reverse motion, these two cases must be considered separately:


 * Forward motion: The screw thread is moving in the direction of the applied (rotational) force, so the frictional force is directed opposite to it, opposing the applied force.
 * $$\sum F_x =  -F_n \sin \theta - F_f \cos \theta + F_{in} = 0  \,$$
 * $$\sum F_y =   F_n \cos \theta - F_f \sin \theta - F_{out} = 0  \,$$


 * $$ F_n \sin \theta + \mu F_n \cos \theta = F_{in} \,$$
 * $$ F_n \cos \theta - \mu F_n \sin \theta = F_{out}  \,$$


 * $$\mathrm{MA} = \frac {F_{out}}{F_{in}} = \frac {\cos \theta - \mu \sin \theta} {\sin \theta + \mu \cos \theta} = \frac { 1 - \mu \tan \theta} {\tan \theta + \mu} = \frac { 1 - \tan \phi \tan \theta} {\tan \theta + \tan \phi}  \,$$
 * Using a trigonometric identity
 * $$\mathrm{MA} = \frac {1}{\tan (\theta + \phi) }  \,$$


 * $$F_{in} = F_{out} \tan (\theta + \phi) \,$$
 * This gives the amount of force required for "impending motion" of the screw forward, in the direction of applied rotational force. If the input force is greater than this, the screw will rotate forward.  There are 2 cases:


 * 1) $$\theta + \phi < 90^\circ$$: The mechanical advantage is positive. The screw will rotate forward.
 * 2) $$\theta + \phi \ge 90^\circ$$: The mechanical advantage is zero or negative, so the screw is "locked" to forward motion and can't be rotated forward. This only occurs with very large pitch screws, and "backwards" screws like the push drill.
 * Reverse motion (overhauling): The screw thread is rotating backwards, against the applied rotational force, so the frictional force is directed opposite to it, in the same direction as the applied force.
 * $$\sum F_x =  -F_n \sin \theta + F_f \cos \theta + F_{in} = 0 \,$$
 * $$\sum F_y =   F_n \cos \theta + F_f \sin \theta - F_{out}  = 0  \,$$


 * $$ F_n \sin \theta - \mu F_n \cos \theta = F_{in} \,$$
 * $$ F_n \cos \theta + \mu F_n \sin \theta = F_{out}   \,$$


 * $$\mathrm{MA} = \frac {F_{out}}{F_{in}} = \frac {\cos \theta + \mu \sin \theta} {\sin \theta - \mu \cos \theta} = \frac { 1 + \mu \tan \theta} {\tan \theta - \mu} = \frac { 1 + \tan \phi \tan \theta} {\tan \theta - \tan \phi}    \,$$
 * Using the same trigonometric identity as above
 * $$\mathrm{MA} = \frac {1}{\tan (\theta - \phi) }  \,$$


 * $$F_{in} = F_{out} \tan (\theta - \phi)  \,$$
 * This gives the amount of load force required for "impending motion" of the screw backward. If the load force is greater than this, the screw will rotate backwards, overhauling.  There are 3 cases:


 * 1) $$\theta < \phi\,$$: The mechanical advantage is negative, so the screw is "locked" and will not turn backward with any amount of load force on the shaft, even with some negative (backward) rotational torque on it. This is the case with most ordinary screws.
 * 2) $$\theta = \phi\,$$: The 'angle of repose'. The mechanical advantage is infinite. The screw is "locked" to backward motion, but the slightest negative torque on it will cause it to rotate backwards.
 * 3) $$\theta > \phi\,$$: The mechanical advantage is positive. Load force applied to the shaft will cause the screw to rotate backwards and overhaul.  This occurs only in some large pitch screws with good lubrication.

It can be seen from the above that a screw will be self-locking if and only if the arctangent of the coefficient of friction φ is greater than the helix angle θ.

= For Glow discharge = A glow discharge typically occurs in a partially-evacuated glass tube with two metal electrodes, containing a gas at a pressure of 0.1 to 10 torr about 1/10,000 to 1/100 amospheric pressure. The tube doesn't

= For Negative feedback =

Negative feedback occurs when the result of a process influences the operation of the process itself in such a way as to reduce changes. Negative feedback tends to make a system self-regulating or self-correcting; if the state of the system is moved away from the equilibrium state due to external disturbances, over time it will return to the equilibrium state.

To have negative feedback a process must have a feedback loop, in which information from the output of a process is fed back as an input signal to control the process; in engineering this is called a closed-loop system. In a system controlled by a negative feedback loop, the level of some output of a process is continuously compared to a (possibly variable) reference value which represents the desired output. The difference, called the error signal, is used to control the process, altering the output in a direction to reduce the error, with the result that the output becomes closer to the desired value.

An example of a negative feedback loop in biology is the self-regulating role of hunger. As time passes since our last meal, we get hungry. The hunger acts as an "error signal" stimulus, causing us to respond by eating. As we eat, we become less hungry, and eventually stop eating. So the response to hunger acts in a "negative" direction, to reduce the hunger. The purpose of this feedback loop is to provide a steady flow of nutrients into our bodies.

In control systems theory a feedback loop has negative feedback if the feedback signal is subtracted from the reference signal to get the error signal, as above. The other kind of feedback is positive feedback, in which the feedback signal is added to the reference signal. Instead of tending to correct the error, and cause the output to converge on the reference value, positive feedback acts to increase error, moving the output away from the reference value. Positive feedback can cause the output of a process to be unstable; to fluctuate or oscillate, or increase without bound.

Negative feedback is widely used in mechanical and electronic engineering, and in industrial plant control, but it also occurs naturally within living organisms, and can be seen in many other fields from chemistry and economics to social behaviour and the climate. Ktesibios, an ancient Greek engineer, built one of the first known negative feedback mechanisms, a flow controller in a water clock, in the 3rd century BCE. The first industrial negative feedback control system was the flyball governor for steam engines invented in 1767 by James Watt. The mathematical condition for feedback stability, the Routh-Hurwitz stability criterion, was discovered independently by Edward John Routh in 1876 and Adolf Hurwitz in 1895. The mathematics of negative feedback was further developed by Harold S. Black at Bell Telephone Laboratories in 1933 as the basis for electronic systems. General negative feedback systems are studied in control systems engineering.

Handholds
These are the most common handholds used in Balkan line dances. The names given are merely descriptive, not necessarily translations of names used in the native languages, but have some currency among international folk dancers. Some dances incorporate changes in arm position, or swinging or pumping motions of the arms.
 * "V": The hands are held down relaxed at the side of the body, holding the hand of the neighboring dancer on each side. The right hand is palm up, the left hand palm down.
 * "W": The arms are held up with elbows bent in a "W" shape, hands at shoulder height, holding the hand of the neighboring dancers, right hand palm up, left hand palm down.
 * "Teacup": The left hand is placed on the stomach or holds the belt buckle, and the right hand is looped loosely through the next dancer's left arm.
 * Front basket: The hands are held down as in the "V", but hands are joined with the two dancers on the far side of the neighboring dancers, in front of the neighboring dancers, so alternate dancers in the line are linked.
 * Back basket: Like the front basket, except the hands are joined behind the backs of the neighboring dancers.
 * Belt: Each dancer wears a belt or sash, and holds the belt of the two neighboring dancers, in front over the stomach. This links the line of dancers together securely, and so it is found in fast moving acrobatic dances.
 * Shoulder: The arms are held out horizontally to the sides, with the hands on the shoulders (deltoid muscle) of the neighboring dancers.
 * "Pinkie": Similar to the "W" hold, except that the hands are made fists and the extended, curved, little fingers are linked, right hand up, left hand down
 * "Halay": Seen in Turkish dances, the dancer stands straight upright, as close to the next dancer as possible, with arms straight down and rigid at his sides, holding the hand of the neighboring dancers.

= For Screw =

Analysis of square thread screws 2
Square threads have lower friction and higher efficiency than the other types of threads used in screws: Acme, trapezoidal and V-threads. This is because the outward angle of the bearing surface in the other types increases the normal force on the surface, and therefore the friction. Therefore square threads are used for screws that carry power, such as lead screws and screw jacks. This important class of screws can be analyzed as a simple inclined plane. The slope of the thread bearing surface is simply equal to the distance ratio

= For Limiting =

In control system engineering and electronics limiting, also called clipping, saturation or saturation nonlinearity is a characteristic of a process or device in which the output amplitude is prevented from exceeding predetermined limiting values. In engineering, saturation is considered a property of the transfer function (input-output function) of a device. As the input of the device is increased, the output increases up to a maximum value, but remains at that value with further increases of the input. Similarly, as the input is decreased the output decreases to a minimum value but remains there for further decreasing input. Saturation is a very common response of many types of electronic, mechanical, and pneumatic equipment to excessively large amplitude driving or control signals, a result of the fact that the output of any mechanism has limits.

Saturation is a nonlinear function and therefore can cause distortion and harmonic generation in linear systems. For example, excessive levels of gain (volume) in an audio system can cause the sound to become distorted when the amplifying devices (vacuum tubes or transistors) saturate. The characteristic distorted sound of types of rock music is caused by the guitar amplifiers saturating when the volume is turned up.

In electronic circuits and mechanical linkages, saturation is the most common type of nonlinearity.

Saturation can refer to "hard limiting" (clipping), in which a signal between the limits is passed through normally but parts of the signal outside the limits is "sheared off" when threshold. It can also refer to "soft limiting", a type of variable-gain audio level compression, in which the gain of an amplifier is changed very quickly to prevent the signal from going over a certain amplitude.


 * Hard limiting ("clipping") is a limiting action in which there is
 * (a) over the permitted dynamic range, negligible variation in the expected characteristic of the output signal, and
 * (b) a steady-state signal, at the maximum permitted level, for the duration of each period when the output would otherwise be required to exceed the permitted dynamic range in order to correspond to the transfer function of the device.
 * Soft limiting is limiting in which the transfer function of a device is a function of its instantaneous or integrated output level. The output waveform is therefore distorted, but not clipped.

Definition
The transfer function of a hard "clipping" type saturation nonlinearity is
 * $$y(x) =

\begin{cases} L_\text{max}, & x \ge L_\text{max} \\ x, & L_\text{max} > x > L_\text{min} \\ L_\text{min}, & x \le L_\text{min} \end{cases} $$ Often the saturation is symmetrical about the zero point of the signal
 * $$y(x) =

\begin{cases} L, & x \ge L \\ x, & L > x > -L \\ -L, & x \le -L \end{cases} $$

= For Wheel and axle =

History
The wheel is widely mentioned as one of the most important inventions in history,[Bulliet, 3] but since it was invented in prehistoric times, many points about its history are in dispute. Perhaps the earliest functional application of the wheel was the potter's wheel, a horizontal rotating platform for making clay pots. The first hand-turned potter's wheels, called tournettes, appeared in the Middle East toward the end of the Neolithic period, around 4500 BC. [Higgins, 10]

The most famous and significant application of the wheel was in the invention of the wheeled vehicle. Wheels were applied to carts in the late Neolithic period and they have been ubiquitous in transportation ever since. Before the wheel, loads were transported by draft animals dragging various forms of sleds and the travois, a load-carrying platform supported between two poles joined at the end in the shape of a "V"; the point of the "V" dragged along the ground as it was towed. The application of wheels to carts eliminated the friction of the cart dragging along the ground, resulting in a great reduction in required motive power and increase in load-carrying ability. There are two theories of how wheeled vehicles were invented:

The older theory, promoted by Stuart Piggott was that the first wheeled vehicles were carts with two wheels similar to the chariot, evolved by the application of wheels to the travois. [Bulliet, 93]

A newer theory proposed in 1983 by Mary Aiken Littauer is that the first wheeled vehicles were four-wheeled carts, which developed from load-carrying platforms which rolled along the ground on wooden rollers. [NYT]

References:
 * Bulliet
 * NYT
 * Higgins (forget folk etymology p.10)

=for Einstein-de Haas effect=

In physics the Einstein-de Haas effect, discovered by German physicist Albert Einstein and Dutch physicist Wander Johannes de Haas in 1915, is a spontaneous rotation of a piece of ferromagnetic material like iron when it is magnetized or demagnetized. More precisely, it is a change in the angular momentum of a piece of ferromagnetic substance when its magnetization along the same axis is changed. In the experiment of Einstein and de Haas, it is manifested as a slight rotation of a cylinder of iron, suspended so it can turn freely, when it is magnetized or demagnetized by a vertical magnetic field. It is also referred to as the Richardson effect after British physicist Owen Richardson, who predicted the effect in 1908. The opposite effect, that spinning a piece of ferromagnetic material causes it to become magnetized, is called the Barnett effect after American physicist Samuel Barnett who demonstrated it between 1908 and 1915. The two effects are equivalent.

The Einstein-de Haas effect provided a final confirmation of the theory, first proposed by André-Marie Ampère in 1820, that magnetism is caused by "molecular currents", that is, a circular motion of electric charge within matter. It also established the identity between the quantum mechanical property of spin which is responsible for magnetism, and classical angular momentum.

Explanation
Magnetism is caused (mainly) by the spin of electrons in atoms. Although the spin of electrons is actually a quantum mechanical property and does not represent actual mechanical rotation, electrons can be visualized as spinning balls of electric charge, with the circular motion of the charge creating a tiny magnetic field which is pointed in the same direction as its spin axis. However, the spin of the electron also gives it an angular momentum, causing it to act like a little gyroscope. In a piece of iron that is magnetized, the spin axes of many of the electrons are aligned parallel, all spinning in the same direction. In the Einstein-de Haas experiment, the magnetization of a piece of iron is reversed, by subjecting it to an opposing magnetic field created by a coil of wire surrounding the iron. This causes the spin of the electrons to reverse direction. However, angular momentum is conserved, so this change in angular momentum must be balanced by an opposite change in angular momentum of the iron cylinder. The reversing electrons exert a tiny reverse torque on the lattice of iron atoms they are part of. If the iron is suspended by a fine fiber so it is free to turn, it will begin turning slowly in the opposite direction to the electrons. It is similar to the situation that occurs if a rotating flywheel or gyroscope in a small boat or spacecraft is reversed; the floating craft will begin rotating in the opposite direction.

=for Gravitational constant=

History of measurment
The gravitational constant is the most difficult to measure of all the fundamental constants. This has been attributed to these causes: (1)The weakness of the gravitational force between ordinary-sized objects. (2)All matter creates gravitational fields, and there is no way of 'shielding' the measuring apparatus from extraneous fields created by nearby matter or the apparatus itself, as is possible with electromagnetic measurements. (3)Since gravitation theory hasn't been unified with the theory of other forces, G can't be derived from other fundamental constants that can be measured more accurately.

The first accurate measurement of G was the landmark Cavendish experiment devised by John Michell and performed by Henry Cavendish in 1798, 71 years after Newton. It was also the first measurement of gravitational force between laboratory-scale masses. Cavendish used a torsion balance to measure the force between lead balls. His remarkably accurate result stood for 97 years, until the Boys measurement of 1895. However, prior to Cavendish there was a series of efforts to measure the gravitational force using geographical methods (see below) dating from ____.

Attempts to measure the gravitational force predate the use of a gravitational constant. Interest in G as an explicit physical constant didn't develop until the late 1800s. The first measurement of gravity that calculated G explicitly seems to be that of Cornu & Baille in 1873. In 1894, C. V. Boys gave a lecture before the Royal Society introducing G to the physicists of Britain, who were obviously unfamiliar with it. He pointed out it's importance as a universal fundamental constant of nature. Before this, G was considered an uninteresting constant of proportionality in Newton's law, and the motivation for measuring the force of gravity was instead to determine the mass of the Earth. This was an important unsolved problem of 18th century astronomy (see below).

Another possible barrier to interest in G was that, prior to the adoption of the metric system around 1891, weight (force) and mass were measured in the same units. In these systems of units, G comes out as numerically equal to $$R_{earth}^2/M_{earth}\,$$, tying the universal gravitational constant, which has nothing fundamentally to do with the Earth, to the Earth's mass.

For these reasons, all gravity researchers before 1873 reported their results as a value for the mean density of the Earth, $$\rho_{earth}\,$$. However, these can be converted to an equivalent value of G for comparison purposes:


 * $$G = \frac{3 g}{4 \pi R_{earth} \rho_{earth}}\,$$

Geographical methods
Before Cavendish, technology did not exist to measure the faint gravitational force between laboratory-scale masses, so the first gravity measurements used as one of the attracting masses large geographical objects, such as mountains. The results were very inaccurate; the uncertainties of estimating the mass of the geographical object from surveyed dimensions and samples of rock doomed them to single digit accuracy, if that. However, they were perfectly legitimate measurements, and the effort expended in these measurements shows that the researchers understood the importance of the result.

During the 1800s, after Cavendish showed it was possible, researchers abandoned geographical methods for the more accurate lab-scale methods below.

Torsion balance methods
for Talk:Cavendish experiment:

Thanks for clarifying your reasons; I should have checked with you before making an assumption about them. I hate acting like a POV fanatic. You and User:Astrochemist have outvoted me on this  However, I feel I'm right about this. Briefly, my reasons are:
 * (2)This section is not about what Cavendish 'didn't do'. The derivation here is mathematically equivalent to the analysis in Cavendish's paper.  This was originally noted at the top of the section, but the note was removed.  Although C didn't write equations, if he had he would have ended up with same equation I did.  Perhaps it would be instructive to present C's actual calculations, as you suggest.  I decided not to because (a)it would have been much longer, (b)it would have been much more difficult to follow, (c)the modern terminology makes the underlying physics concepts clearer, (d)


 * Timeline_of_gravitational_physics_and_relativity, Wikipedia




 * . '[Cavendish]...obtained a value [of G] accurate to 1%'

= For Gridiron pendulum =

Analysis
All substances expand with an increase in temperature $$\theta$$, so uncompensated pendulum rods get longer with a temperature increase, causing the clock to slow down, and shrink with a temperature decrease, causing the clock to speed up, depending on the linear coefficient of thermal expansion (CTE) $$\alpha$$ of the material they are composed of. CTE is usually given in parts per million per degree Celsius. The expansion or contraction of a rod of length $$L$$ caused by a temperature change $$\Delta\theta$$ is
 * $$\Delta L = \alpha L \Delta\theta$$&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;(1)

Temperature error
The period of oscillation $$T$$ of the pendulum is
 * $$T = 2\pi\sqrt{L \over g}$$&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;(2)

A change in length $$\Delta L$$ due to a temperature change $$\Delta\theta$$ will cause a change in the period $$\Delta T$$. Since the expansion coefficient is so small, the length changes due to temperature are very small, parts per million, so $$\Delta T << T$$ and the change can be approximated to first order as a linear function
 * $$\Delta T = {\partial T \over \partial L}\Delta L$$
 * $$\qquad = {\partial \over \partial L}\Big( 2\pi\sqrt{L \over g}\Big)\Delta L = \pi{\Delta L \over \sqrt{gL}}$$

Substituting equation (1), the change in the pendulum's period caused by a change in temperature $$\Delta\theta$$ is
 * $$\qquad = \pi{\alpha L \Delta\theta \over \sqrt{gL}} = \alpha\pi\sqrt{L \over g}\Delta\theta$$
 * $$\Delta T = {\alpha T\Delta\theta \over 2}$$

$$ So the fractional change in the pendulum's period is equal to one-half the coefficient of expansion times the change in temperature.

Steel has a CTE of 11.5 x 10−6 per °C so a pendulum with a steel rod will have a thermal error rate of 5.7 parts per million or 0.5 seconds per day per degree Celsius (0.9 seconds per degree Fahrenheit). Before 1900 most buildings were unheated, so clocks in temperate climates like Europe and North America would experience a summer/winter temperature variation of around 25 F-change resulting in an error rate of 6.8 seconds per day. Wood has a smaller CTE of 4.9 x 10−6 per °C thus a pendulum with a wood rod will have a smaller thermal error of 0.21 sec per day per °C, so wood pendulum rods were often used in quality domestic clocks. The wood had to be varnished to protect it from the atmosphere as humidity could also cause changes in length.

Compensation
A gridiron pendulum is designed so the high expansion rods push the pendulum bob up, in the opposite direction to the low expansion rods, so the net change in length is the difference between these changes. From (1) the change in length $$\Delta L$$ of a gridiron pendulum with a temperature change $$\Delta\theta$$ is
 * $$\Delta L = (\alpha_\text{low}\sum L_\text{low} - \alpha_\text{high}\sum L_\text{high})\Delta\theta$$

where $$\sum L_\text{low}$$ is the sum of the lengths of all the low expansion (steel) rods and $$\sum L_\text{high}$$ is the sum of the lengths of the high expansion rods in the suspension chain from the bob to the pivot. The condition for zero length change with temperature is
 * $$\alpha_\text{low}\sum L_\text{low} - \alpha_\text{high}\sum L_\text{high} = 0$$
 * $${\alpha_\text{high} \over \alpha_\text{low}} = {\sum L_\text{low} \over \sum L_\text{high}}$$&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;(3)

In other words, the ratio of thermal expansion coefficients of the two metals must be equal to the inverse ratio of the total rod lengths.

Five rod gridiron
In the 5 rod gridiron, there is one high expansion rod on each side, of length $$L_\text{2}$$, flanked by two low expansion rods with lengths $$L_\text{1}$$ and $$L_\text{3}$$. One from the pivot to support the bottom of $$L_\text{2}$$, the other goes from the top of $$L_\text{2}$$ down to support the bob. So from equation (3) the condition for compensation is
 * $${\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} \over L_\text{2}}$$

Since to fit in the frame the high expansion rod must be equal to or shorter than each of the low expansion rods $$L_\text{1} \ge L_\text{2}$$ and $$L_\text{3} \ge L_\text{2}$$ the geometrical condition for construction of the gridiron is
 * $$L_\text{1} + L_\text{3} \ge 2L_\text{2}$$

Therefore the 5 rod gridiron can only be made with metals whose expansion coefficients have a ratio greater than two
 * $${\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} \over L_\text{2}} \ge 2$$

Zinc has a CTE of $$\alpha$$ = 26.2 x 10−6 per °C, a ratio of $$\alpha_\text{high}/\alpha_\text{low}$$ = 2.28 times steel, so the zinc/steel combination can be used in 5 rod pendulums. The compensation condition for a zinc/steel gridiron is
 * $${\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} \over L_\text{2}} = 2.28$$

In addition the total length of the pendulum must give the required period $$L_\text{1} + L_\text{3} - L_\text{2} = g\big({\theta \over 2\pi}\big)$$. Most of the precision pendulum clocks with gridirons used a 'seconds pendulum', in which the period was two seconds. The length of the seconds pendulum was 0.993 meter so
 * $$L_\text{1} + L_\text{3} - L_\text{2} = 0.993$$

Nine rod gridiron
To allow the use of metals with a lower ratio of expansion coefficients, such as brass and steel, a construction with more high expansion rods must be used. In the 9 rod gridiron, there are two high expansion rods on each side, of length $$L_\text{2}$$ and $$L_\text{4}$$, flanked by three low expansion rods with lengths $$L_\text{1}$$, $$L_\text{3}$$ and $$L_\text{5}$$. So from equation (3) the condition for compensation is
 * $${\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} + L_\text{5} \over L_\text{2} + L_\text{4}}$$

Since to fit in the frame each of the two high expansion rods must be as short as or shorter than each of the high expansion rods, the geometrical condition for construction is
 * $$L_\text{1} + L_\text{3} + L_\text{5} \ge  {3 \over 2}(L_\text{2} + L_\text{2})$$

Therefore the 9 rod gridiron can be made with metals with a ratio of thermal expansion coefficients exceeding 1.5
 * $${\alpha_\text{high} \over \alpha_\text{low}} = {L_\text{1} + L_\text{3} + L_\text{5} \over L_\text{2} + L_\text{4}} \ge 1.5$$

Brass has a CTE of $$\alpha$$ = 19.3 x 10−6 per °C, a ratio of $$\alpha_\text{high}/\alpha_\text{low}$$ = 1.68 times steel. So while brass/steel cannot be used in 5 rod gridirons, it can be used in the 9 rod version.

Definition of variables
= For Litz wire =

Principle of operation
Litz wire reduces the impact of the skin effect and the proximity effect.

Skin Effect
The resistance of a conductor to direct current (DC) is inversely proportional to its cross sectional area. A conductor with a larger area has a lower resistance.

The resistance of a conductor to alternating current (AC) is higher than its resistance to DC. This is due to an electromagnetic property called skin effect. causes a conducto resistance to increase for AC currents.

For low frequencies, the effect is negligible. For AC current at frequencies high enough that the skin depth is small compared to the conductor size, the skin effect causes most of the conduction to happen at the conductor's surface. At high enough frequencies, the interior of a large conductor does not carry much current. At 60 Hz, the skin depth of a copper wire is about 1/3 in. At 60 kHz, the skin depth of copper is about 0.01 in. At 6 MHz, the skin depth is about 0.001 in. Solid conductors larger than a few skin depths don't conduct much current near their axis, so that material isn't used effectively.

When larger area conductors are needed, tricks are used to minimize the skin effect. The goal is to increase the conducting surface area. One trick is to use a hollow conductor with a wall that is about a skin-depth thick. It is essentially a large-diameter wire with the non-current carrying interior deleted. It is bulky, but it saves copper.

Litz wire uses some different tricks. Instead of using one big conductor, it uses lots of little conductors (strands) in parallel (forming a bundle). Each little conductor is less than a skin-depth, so an individual strand does not suffer an appreciable skin effect loss. However, that is not the complete story. The strands must be insulated from each other -- otherwise all the wires in the bundle would short together, look like a single large wire, and still have skin effect problems. Furthermore, the strands cannot occupy the same radial position in the bundle: the electromagnetic effects that cause the skin effect would still disrupt conduction. The bundle is constructed so the individual strands are on the outside of the bundle (and see low resistance) for a time, but also reside in the interior of the bundle (where the EM field changes are the strongest and the resistance is higher). If each strand sees about the same average resistance, then each strand will contribute equally to the conduction of the entire cable.

The weaving or twisting pattern of litz wire is designed so individual wires will reside for short intervals on the outside of cable and for short intervals on the inside of the cable. This allows the interior of the litz wire to contribute to the cable's conductivity.

Another way to explain the same effect is as follows: the magnetic fields generated by current flowing in the strands are in directions such that they have a reduced tendency to generate an opposing e.m.f. in the other strands. Thereby, for the wire as a whole, the skin effect and associated power losses when used in high-frequency applications are reduced. The ratio of distributed inductance to distributed resistance is increased, relative to a solid conductor, resulting in a higher Q factor at these frequencies.

Proximity effect
In cases involving multiple wires, or multiple turns, such as windings in transformers and inductors, proximity effect causes losses to increase at high frequency even sooner and more rapidly than does skin effect.

Effectiveness
provides an expression for the ratio of resistance to alternating current to resistance to direct current for a single litz wire. It does not apply to windings with multiple turns. An expression for the resistance ratio in windings is given by at Eqn 2.

Litz wire is very effective below 500 kHz; it is rarely used above 2 MHz as it is much less effective there.

Litz wire has a higher impedance per unit cross-section, but litz wires can be used at thicker cable sizes, hence reducing or maintaining cable impedance at higher frequencies.

= For Talk:Antenna (radio) = 1

In UHF antennas and mast radiators the increasing diameter/length ratio reduces the velocity factor. 3 The normal-mode helical antennas 4 which are extremely widely used as "Rubber Ducky" antennas on portable radios achieve ratios of electrical to physical length of 3 to 1 by their distributed loading. Another very widely used antenna where the velocity factor must be taken into account is the microstrip antenna, constructed of segments of stripline. The distinction appears in technology dictionaries 5

= For Electromagnet =

Field of a solenoid
The magnetic field B inside a long solenoid with a uniform winding of length L with N turns can be derived by applying Ampere's law to the long rectangular path ABCD. The line integral of the magnetic field B along the sides BC and DA is close to zero since the magnetic field is almost normal (perpendicular) to the line. If the end segment CD is far away from the solenoid the integral of the field along this segment will be virtually zero also. So the integral of B over the path ABCD reduces to the integral over the segment AB. if δ is the length of AB then the integral of the field is Bδ. The current through the loop ABCD is just proportional to the number of windings inside the loop: ''NIδ/L. Applying this to Ampere's Law:
 * $$B\delta = \mu \frac {NI\delta}{L} \,$$
 * $$B = \frac {\mu N I}{L} \,$$

It can be seen that since segment AB can be located anywhere inside the solenoid, the magnetic field is uniform inside the solenoid. A more accurate analysis shows that the magnetic field drops off somewhat near the ends, because of the divergence of the magnetic field lines.

= For Long wire antenna =

In radio antenna design, a long wire antenna is a class of antenna made of lengths of wire suspended above the ground that are long compared with the length of the radio waves to be received, usually longer than one wavelength. This type contrasts with dipole antennas which consist of conductors that are typically a half- or a quarter wavelength long. The wire is usually strung between pole or tower supports or between buildings. Long wire antennas are most often used in the short wave radio bands. This class of antenna includes single long straight wires, as well as other geometries such as the vee and rhombic antenna, and also travelling wave antennas such as the Beverage antenna.

=

= For Floating ground = A floating ground is a characteristic of some electrical circuits. In electrical circuits, a ground point (or earth in British English) can mean a conductor which serves as a return path for electric current from many components in the circuit, or a part of the electrical equipment, such as the chassis which should be at ground potential. Often a ground point is electrically connected to the Earth (directly or through other devices), hence the name "ground". If it is not, it is called a floating ground.

In electric circuits, the term ground (or earth in British English) can have two different meanings; it can mean a common conductor that serves as the return path for current from different components in the circuit, or a direct electrical connection to the Earth. A floating ground is a ground in the first sense but not the second; it is a reference node or return path for current from different components that is not connected to the Earth. The term is used in slightly different ways; sometimes it can mean a conductor in a piece of equipment which is not connected to Earth through the local ground wire but through a cable to a remote ground wire in another piece of equipment. Floating grounds can be intentional or unintentional. Many types of electrical equipment incorporate floating grounds by design; for example most portable battery-powered electronic devices have "ground" systems which are not connected to Earth. Properly designed, these do not present any problem.

Unintentional floating grounds can also occur in equipment that should be connected to Earth, through electrical faults or improper design or installation of grounding equipment. These unintentional floating grounds can cause noise, hum, and interference in audio, radio and computer equipment. In mains powered equipment they can also cause an electric shock hazard, by leaving exposed metal parts with potentials different from ground.

In AC electrical equipment
In AC powered equipment, a floating ground is a metal part which is not connected to the AC power grid's ground wire system. In electrical equipment powered by AC mains power, ground wires attached to the Earth are used for safety purposes, to prevent electric shock. Exposed metal parts of equipment are attached to the ground wire, which is connected through the building's ground wiring to a metal stake driven into the earth. In the event that a fault in the equipment allows current from the "hot" side of the AC power wiring to reach the exposed conductor, the ground wire provides a path for this current to ground. In addition, it provides a path for leakage currents, small currents that flow from "hot" conductors through the equipment's electrical insulation. This ensures that the exposed part always remains at the same electric potential as the Earth, so that a person touching it will not receive an electric shock. A floating ground does not have this safety feature, and so can be a potential shock hazard.

Electrical codes require large appliances and equipment to be grounded through a three-prong plug, but allow small appliances and lights to use a two-prong plug without a ground wire. Therefore the metal chassis and parts of these appliances are floating grounds.

Ground adapters
Since many older buildings have older two-hole outlets without a ground wire, 3-prong to 2-prong ground adapters ("cheaters") are sold, to allow a three-prong appliance to plug into a two-hole outlet. These have ground tabs or "pigtail" ground wires which are supposed to be attached to the outlet faceplate hold-down screw, which attaches to the metal outlet box which is grounded through the armored cable used in older houses. Neglecting to attach the ground tab makes the appliance ground wire, and any metal parts attached to it, a floating ground. This common practice defeats the purpose of the ground wire, creating a shock hazard.

Double-insulated appliances
To decrease the shock hazard, certain ungrounded appliances such as power tools are required to be double-insulated. This means that any floating-ground metal parts are enclosed inside an insulating plastic housing, so users can't come in contact with them.

Ground fault interrupters
Ground fault interrupters are also required in some outlets to provide protection. These detect when current is flowing from the wiring into the ground, and shut off the power immediately.

Floating ground power supplies
In addition to the separate ground wire system, one side of the AC power (mains) wiring, called the "neutral" or "common" wire, is grounded, being connected to an earth ground rod at the electrical panel. This explains why a person can receive a shock by touching only one conductor of the power wiring. If a person comes in contact with the "hot" side of electrical wiring, and is grounded through bare feet or other contact with the earth, the conductive human body and the ground becomes a path for current to flow from the "hot" to the "neutral" side, causing an electric shock.

To reduce shock hazard further, some electrical equipment has a "floating ground" power supply, in which both sides of the internal AC power wiring are unconnected to ground. This is done by powering the equipment from mains wiring through an isolation transformer, in which neither end of the secondary winding is grounded. Since neither power wire is grounded, a grounded human body coming in contact with either wire alone does not provide a current path from one side of the internal AC wiring to the other, so no current flows through the body. To get a shock, a person would need to touch both sides of the power wiring at the same time. A common example of a floating ground power supply is the small plugin "wall wart" transformers used to power portable electronic devices such as cell phones. All medical electronic equipment which comes in contact with patients, such as electrocardiographs,  must have floating ground power supplies.

Live chassis
In modern electronic equipment, the metal chassis of the component (if it has one) is either grounded through the power cord ground wire, or left unconnected as a floating ground. But in many older radios and televisions made prior to the 1980s when ground wiring became widespread, the chassis was connected to one side of the AC power line, for signal grounding purposes. This is called a "floating chassis" or "live chassis". Later models used "polarized" 2-prong power plugs wired to ensure the chassis was connected to the "neutral" side of the power line, but earlier "unpolarized" plugs could be plugged into outlets either way, so the chassis could be connected to the "hot" side and be at a potential of 120V or 230V to ground. Even chassis connected to the grounded "neutral" wire could have a potential substantially different from ground, because of voltage drops due to currents flowing through the building's neutral wires. Therefore the "live chassis" was a considerable shock hazard. "Live chassis" devices had nonconductive cases designed to completely enclose the chassis so it was not accessible to user contact. The metal shafts of volume and tuning knobs were also at high potential, so their plastic knobs were designed to prevent users contacting the shaft, with the setscrews used to hold the knobs on insulated with wax. Headphones could be a particular hazard; one side of the headphone jack connected to the chassis, bringing the AC line voltage into the body of the earpiece, next to the user's head. Later headphones were driven through isolation transformers, to eliminate this hazard.

Even after live chassis became obsolete, radio and TVs with two-wire plugs had the chassis connected to one side of the power line through a bypass capacitor. It's impedance was often low enough to leave a considerable potential on the chassis.

In low voltage equipment
In low voltage audio, video, radio and instrumentation equipment, ground connections serve a second purpose: to prevent electronic noise and hum from the power wiring from getting into the signal circuits and appearing in the output. Noise and hum due to improper grounding is a major problem in the audio equipment industry.

In an electric circuit, a floating ground is a ground, that is a reference node serving as a common return path for current from other components, which is not electrically connected to the Earth.

This can occur in 3 possible ways
 * as the result of intentional design, and entirely harmless
 * as the result of failure to ground equipment that was designed to require grounding
 * as the result of exposing a live ground that was intended to remain unexposed (live chassis TVs were common until the 1990s, where the set's ground is derived by rectifying live mains)

When an electrical device is accidentally or intentionally grounded to its surrounding structural component (chassis), this is called a "live chassis." Circuit failures resulting in live chassis contrary to design plans can mean that anything (e.g. a person) that touches the device and is grounded on something with a different charge (e.g. terra firma) will now experience a voltage potential across its body. This can lead to death or harm by electrocution.

Intentional floating grounds formed by design are widespread in domestic electronic appliances. Providing the design is satisfactory they aren't a safety issue.

Ungrounded equipment designed to be grounded is a safety issue. It leaves users unprotected against the risk of shock due to a potential further fault.

Exposed live grounds are dangerous. They are live, and can electrocute end users if touched. Headphone sockets fitted by end users to live chassis TVs are especially dangerous, as not only are they often live, but the danger is carried directly to the user's head.
 * Sets that have both headphone socket and a live chassis use an audio isolation transformer to make the arrangement safe.

Floating grounds can cause problems with audio equipment using RCA/phono connectors. With these common and somewhat antiquated connectors, the signal pin connects before the ground, and 2 pieces of equipment can have more voltage difference between their grounds than it takes to saturate the audio input. As a result, plugging or unplugging while powered up can result in very loud noises in speakers. If the ground voltage difference is small, it tends to only cause hum & clicks.

= For Ferroresonant transformer = A ferroresonant transformer, also called a constant voltage transformer is a specialized type of iron-core transformer that also serves as a voltage regulator. It uses the nonlinear saturation properties of the ferromagnetic core in conjunction with a tuned circuit made with a capacitor to provide a constant output voltage over a wide range of input voltages. Ferroresonant transformers are used in power supply circuits to supply power to equipment needing better regulated voltage than is provided by the mains. Their advantages are that they also provide current limiting, limited power interruption protection, are very rugged, electrically stable and have long life, and are much less complicated than solid state voltage regulators. The ferroresonant transformer was invented in 1938 by Joseph Sosa.

How it works
Unlike an ordinary transformer, which operates with its core unsaturated in the "linear" region of the BH curve, a ferroresonant transformer is designed so it operates with its core saturated during most of the AC cycle. The strength of the magnetic flux in the core is nearly constant at the saturation level over most of the cycle, with the flux switching direction near the zero crossing of the primary current, so the flux waveform is approximately a square wave. Since the saturation "clips" the top of the flux waveform, changes in primary current caused by changing supply voltage don't change the flux waveform much. Therefore the voltage induced in the secondary is independent of the supply voltage.

In most ferroresonant transformers the part of the core with the primary winding operates in unsaturated mode, while the part with the secondary winding operates saturated. To accomplish this, a third winding (often called the "tank winding") is added, with a capacitor across it. Large currents flow in this winding, saturating the core. To prevent the primary part of the core from saturating, a small piece of core material called a magnetic shunt is added, bridging the arms of the core between the primary winding and the other two windings. The shunt has an air gap in it. The shunt "short circuits" some of the flux from the secondary so the primary flux is lower, in the linear range. The output current waveform from the secondary winding is a square wave.

Current limiting
Another feature of the device is load current limiting. When the load current reaches about 150% of the rated value, the core becomes unsaturated, and the output voltage falls off to a small fraction of the rated value, limiting the current. Ferroresonant transformers can tolerate a continuous short circuit of their output without damage. This current limiting is useful in many contexts to protect against faults in the load, but when driving large inductive loads like a motor it can be a disadvantage since the large startup current can trigger limiting.