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Temperature effects in watches

Copyright © David Boettcher 2006 - 2017 all rights reserved.

Some time between 1658 and 1660 the English scientist Dr. Robert Hooke had the idea of using a balance spring to improve the timekeeping of watches. He thought that it would make a watch an even better timekeeper than a pendulum clock because the balance of a watch swings backwards and forwards in rotation around a fixed axis and therefore does not suffer from the problem of "circular error" caused by the unidirectional pull of gravity that prevents pendulums from being isochronous. Hooke realised that by adding to the watch balance a spring that obeyed his famous law, that the tension of a spring is in proportion to its extension, the conditions for an isochronous harmonic oscillator would be fulfilled.

Hooke showed a pocket watch with a balance spring to Lord Brouncker, Robert Boyle and Robert Murray, seeking their sponsorship in an application for a patent on the idea. A draft patent was drawn up in 1665, but then development of balance spring watches was put on hold and the patent was never granted. Hooke was very busy at the time with many scientific investigations, and from 1666 with supervising the rebuilding of London after the Great Fire. Christiaan Huygens successfully applied the balance spring to watches in 1675 and announced this as an invention of his, to Hooke's great annoyance. Whether Huygens conceived the idea independently or was told of Hooke's idea, perhaps by Henry Oldenburg, the secretary to the Royal Society, has been debated since. Oldenburg’s minutes record Hooke demonstrating a spring-regulated watch to the Royal Society in June 1670, and he is known to have corresponded with Huygens.

With the invention of the balance spring, watches became quite good timekeepers and even at the end of the seventeenth century a verge watch could be expected to keep time to within a few minutes a day, if it was kept at a constant temperature. However, its rate would alter by around 10 seconds per day for every degree centigrade change in temperature. This effect was probably too small to be noticed by Hooke, but by the early eighteenth century when John Harrison started to constructed marine timekeepers in an attempt to win the longitude prize the effect was well known. All of John Harrison's marine chronometers have devices that compensate for the effects of temperature, and they could not have achieved the accuracy that they did without them.

The story of improvement in the accuracy of balance controlled watches after Harrison had achieved the accuracy required by longitude act is very largely the story of reducing the effects of temperature on their rate.

If you have any questions or comments, please don't hesitate to contact me via my Contact me page.


Temperature effects

Balance and balance spring
Compensation balance and spring: Click to enlarge

The timekeeping of a clock is usually determined by the pendulum swinging to and fro under the effect of gravity, with a little push, called an impulse, from the escapement every time the pendulum swings through its lowest point, accompanied by a gentle tick. A pendulum can't be used in a watch because it might be held at any angle but gravity only pulls downwards, so an alternative is needed. This is provided by the balance. The image here shows a watch balance, which comprises a central bar or spoke and a circular rim. The spoke has an axel passing through it at right angles about which it can rotate that is called the "balance staff".

The escapement mechanism pushes the balance in one direction and then the other. In early watches verge without balance springs, the rate at which the balance went backwards and forwards was determined solely by how hard it was pushed by the mainspring, which is why the fusee, or an alternative called a stackfreed, that equalises the torque from the mainspring was vital in such watches. Without a fusee or stackfreed the rate would change so much as the mainspring ran down that the watch would be useless as a timekeeper. Even with one, early watches without balance springs were poor timekeepers.

The addition of the balance spring transformed the timekeeping capabilities of watches by giving the balance a "natural frequency". The spring causes the balance to oscillate at this frequency, to which it returns after a disturbance. The less the balance and spring are disturbed the better the timekeeping, which is why detached escapements which only interfere with the balance over a short part of its arc, are better.

The image here shows a balance and balance spring. The balance spring is the blue spiral in the middle of the image. It is blue because it is made of high carbon steel that has been hardened and then heated until it turns blue to temper it to spring hardness. At its inner end it is attached to the balance staff by the brass collet. The spring enters a tangential hole in the collet and is fixed in place by a pin. The outer end of the spring is fixed to the movement, to the balance cock, by the brass stud.

The balance spring in the image is not a perfect spiral. This is not a fault, its outer turn is bent up above the plane of the spring and in towards the centre in a Breguet overcoil. This is to improve isochronism. The rim of the balance has a thin inner steel layer and a thicker outer layer of brass, and is cut through in two places near to the spoke. The two arms of the rim are bimetallic strips that bend in or out in response to temperature changes, to compensate for other temperature effects. Not all balance springs and balances are like this.

In a pendulum clock the major effect of an increase in temperature is that the pendulum becomes slightly longer; a similar effect occurs in watches, the diameter of the balance increases. Both of these effects are quite small, negligible unless an unusual level of precision is required.

Ferdinand Berthoud was the first to tabulate in 1773 the effects of temperature on one his marine watches. He recorded that a temperature change from 32°F to 92°F caused it to lose in 24 hours:

Berthoud's observation and (incorrect) apportionment
by expansion of the balance62 seconds
by loss of the spring's elastic force312 seconds
by elongation of the balance-spring19 seconds
Total loss per day393 seconds

The reasoning lying behind Berthoud's apportioning of the individual losses is not known, and is not correct. It was accepted until 1882 when Mr T. D. Wright, a teacher at the BHI, pointed out that the strength of a spring is proportional to its width and inversely proportional to its length, and as these two dimensions are affected in equal ratio by changes of temperature, there is no overall effect on the strength of the spring. The overall effect of 393 seconds in 24 hours over 60°F equates to 11.8 seconds per day per °C, which is in line with other observations.

In a watch a much more significant effect of increasing temperature is that the modulus of elasticity, also called Young's modulus, of the balance spring reduces. The effect of this is that the spring produces less force for a given angle of rotation. This effect is many times larger than that from the lengthening of a pendulum rod or increasing the diameter of the balance.

A watch that is carried in the pocket or worn on the wrist is kept at a fairly constant temperature by warmth from the body, which mitigates the problem to an extent, but it is usually taken off overnight and becomes cooler. However, precision time references are not normally worn and are therefore subject to all the temperature fluctuations of nature, which were more significant in a time before houses and workshops were heated.

The rate of a watch is determined by the rotational inertia of the balance and the strength of the balance spring. The period is given by

$$ T = {\pi} \sqrt \frac {I}{S} $$

where I is the moment of inertia of the balance and S is the elastic moment of the balance spring. This equation gives the period of one excursion of the balance from its central position to its limit of rotation in one direction and back. This is half the time of a complete oscillation and is used by horologists because it is the time between ticks.

With an increase in temperature, thermal expansion causes the balance to increase in diameter, which increases its rotational inertia and causes the watch to run slower. Changes in elasticity of the material of the balance have no effect on timekeeping.

An increase in temperature causes the balance spring to expand in all directions, thickness, breadth and length. The increases in breadth and length have opposite effects on the strength of the spring and cancel each other out. The increase in thickness makes the spring slightly stronger, but this effect is vastly outweighed by the reduced modulus of elasticity of the metal, which makes the spring weaker as the temperature increases.

If a watch has a brass balance and carbon steel balance spring, it would lose over 10 seconds per day for a rise in temperature of just 1°C. It might be thought that a watch with a steel balance, which has nearly half the thermal expansion of a brass balance, would be better, but in fact a watch with a steel balance would lose "only" just under 10 seconds a day for the same 1°C temperature rise. The material the balance is made from has very little effect on temperature errors. The major source of error is the change in the elastic modulus of the spring.

I calculated the individual effects for both brass and steel balances using data for the thermal coefficients of expansion and modulus of elasticity give in A L Rawlings "Science of clocks and watches" and the spreadsheet that you can download from the section about Middle Temperature Error. The results are tabulated below.

Change in rate (seconds per day) for 1°C rise in temperature
Brass balance  Steel balance
Balance thermal expansion-1.64-0.95
Spring thermal expansion1.421.42
Spring decrease in Young's modulus-10.36-10.36
Totals (- indicates loss)-10.58-9.89

The thermal expansion of the spring and the brass balance somewhat compensate each other. The expansion of the thickness of the spring makes it stronger and, all other things being equal, would make the watch run faster by about 1.42 seconds per day. The expansion of the brass balance increases its radius of gyration, and hence its moment of inertia, which would make the watch run slower by about 1.64 seconds per day. These two effects, increasing spring thickness and increasing balance moment of inertia, thus oppose and partially cancel each other. In aggregate they contribute only 0.22 seconds per day to the overall loss of over 10 seconds. It is the change in Young's modulus of the balance spring that contributes the remaining amount.

With a steel balance the gain in rate due to the increase in strength of the spring is actually greater than the loss due to the increased moment of inertia of the steel balance, resulting in a gain in rate of 0.47 seconds per day, reducing the overall loss to less than 10 seconds per day.

Sir George Biddell Airy, the Astronomer Royal from 1835 to 1881, showed by experiment in 1859 that a chronometer with a plain brass balance lost 6.11 seconds in 24 hours for each degree Fahrenheit increase in temperature, equivalent to 11 seconds per degree centigrade, which is in very close agreement with the loss of 10.58 seconds calculated above.

The bottom line is that a watch that is not compensated for temperature variations can be expected to lose or gain around 10 seconds per day for every one degree change in temperature. If such a watch was adjusted to run correctly on the watchmaker's bench at, say, 20°C, which is the same temperature at which it might spend eight hours overnight on the bedside table, and then it was strapped to your wrist at 34°C for the remaining 16 hours of the day, you could expect it to lose over a minute and a half each day. The fact that most watches do not show such alarming changes in rate is due to aspects of their design or materials that compensate for the effects of temperature changes.

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Temperature Compensation

John Harrison was the first person to successfully apply temperature compensation to a balance controlled timekeeper, in a pocket watch made for him in 1753 by John Jefferys to Harrison's specification. This was the first watch with temperature compensation, and also the first fusee watch with maintaining power, which kept the watch going as it was being wound. Before this invention, fusee watches without maintaining power stopped as they were being wound, losing accuracy. The Jefferys watch also has Harrison's version of the verge escapement, which includes verge pallets with cycloidal backs.

All of Harrison's marine timekeepers included temperature compensation. His first marine timekeeper now called H1, which was constructed between about 1728 and 1735, and the second H2, which was begun in 1737, had gridiron type bimetallic elements similar to the gridiron pendulum Harrison had invented for his land based clocks. For H3, which was begun in 1740, Harrison created a "brass and steel thermometer curb" which was a bimetallic element made from strips of brass and steel riveted together. Because of the differential thermal expansion of brass and steel - brass expands more than steel for a given rise in temperature - as the temperature rises the bimetallic curb will bend. The free end of the curb was fitted with two pins that embraced the balance spring near to the point at which it was attached to the plate, and the bending of the curb was arranged to shorten the effective length of the spring as the temperature rose, compensating for its loss of elasticity. This was the form of temperature compensation used in the Jefferys watch.

Harrison abandoned work on H3 before it was completed, and in 1755 began work on H4, essentially a large pocket watch with a verge escapement and plain steel uncut balance quite similar in overall design to the watch made for him by John Jefferys. H4 has temperature compensation by thermometer curb as used in H3 and the Jefferys watch, maintaining power, Harrison's version of the verge escapement with diamond pallets with cycloidal backs, and a train remontoire, which was its only essential difference from the Jefferys watch.

H4 was the timekeeper that successfully passed the tests stipulated by the 1714 Act of Queen Anne "An Act for Providing a Publick Reward for such Person or Persons as shall Discover the Longitude at Sea" and which resulted in Harrison eventually being awarded the prize for "finding the longitude". There is no question that H4 could not have achieved this feat without adequate temperature compensation, but Harrison was said to be unhappy with the compensation curb because he found that the balance, balance spring, and the compensation curb itself were not all affected at the same time by changes in temperature. The swinging balance and spring would have reacted to temperature changes more quickly that the stationary and more massive compensation curb, and Harrison considered that the compensation would be improved if it was in the balance itself.

The form of temperature compensation seen in nineteenth century pocket watches and early wristwatches uses a "compensation balance" with a cut bimetallic rim, which is discussed in more detail in the next section. The compensation balance was invented by Pierre Le Roy, son of Julien Le Roy, and improved to the form most widely seen by John Arnold and Thomas Earnshaw. The balance rim is made of steel with a layer of brass fused onto the outside, and it is cut so in two that the two arms of the rim can move inwards and outwards. When the temperature increases the extra expansion of the brass compared to the steel causes the parts of the rim to bend inward. This reduces the moment of inertia of the balance, compensating for the weakening of the spring. When the temperature falls the opposite effect occurs, the arms bend outwards to increase the moment of inertia and compensate for the increased strength of the spring.

Development of alloy steels meant that is was possible to make balance springs that did not change in elasticity with temperature, and springs and balances that did not change their dimensions with temperature, so temperature compensation was no longer required. This is discussed in the section below about on autocompensating balance springs.

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Compensation balance

Compensation balance
Marine chronometer compensation balance

The form of temperature compensation first seen eighteenth century marine chronometers, and later in pocket watches and early wristwatches uses a "compensation balance". This followed a principle suggested by John Harrison that rather than using a bimetallic curb to alter the effective length of the balance spring in response to changes it temperature it would be better if the compensation were in the balance itself.

This was achieved by making a balance with a split bimetallic rim that reacted to temperature changes. This balance was invented by Pierre Le Roy, son of Julien Le Roy, and improved to the form most widely seen by John Arnold and Thomas Earnshaw.

The general form of this balance is shown in the picture here. This is a sketch of the balance found in a marine chronometer, a large instrument mounted on gimbals in a cube shaped box. The balance rim is made of an inner layer of steel (coloured grey) with a layer of brass (coloured yellow) fused onto the outside. This turns the rim into a bimetallic strip and it is cut in two places near to the cross bar so that the two arms of the rim can move as shown by the dotted lines. This gives rise to the name and description of this balance as a "split bimetallic temperature compensation balance".

The operation of the balance is as follows. If the temperature increases, the brass on the outside of the rim expands more than the steel and this causes the arms of the rim to bend inward, carrying the weights mounted on the arms inwards towards the central axis of rotation of the balance. This reduces the moment of inertia of the balance, compensating for the weakening of the balance spring. If the temperature falls, the opposite effect occurs and the rims bend outwards, carrying the weights outward and increasing the moment of inertia of the balance.

Compensation balance
Longines 13.34 compensation balance

The two large weights mounted part way along the arms are called compensation weights, and the amount of compensation produced in response to a given change in temperature can be increased or decreased by sliding them along the arms. The further along the arms away from the cross bar the weights are positioned, the greater the effect of the compensation. The two large screws at the end of the cross bar are mean time screws, used to adjust the rate when the best temperature compensation has been established, the two small screws next to them are for very fine adjustments to the rate.

In a smaller movement such as a pocket or wristwatch there is not enough room for the large compensation weights and meantime screws, so a number of small screws distributed along the length of the arms are used. Changes in the compensation are effected by moving some of the screws, and fine adjustments are achieved by fitting thin timing washers, or by reducing the size of some of the screw heads.

The balance shown in the photograph here is fitted to a Longines 13 ligne calibre 13.34 movement dated to 1913. It is a high grade version of the 13.34 calibre with the train jewels set in screw set chatons, cap jewels for the escape wheel, and jewelled to the centre, giving a total of 18 jewels.

The two different metals of the balance rim, steel on the inside and brass on the outside, are visible, and it is notable that the steel is much thinner than the brass; this was to get maximum movement from the bimetallic arms. One of the two splits in the rim that allow the bimetallic arms to move is visible at the top of the photograph near to the steel stud carrier. The other split is concealed by the centre wheel. The timing screws are visible, distributed along the arms of the balance. These screws are made of gold, which was used because of its greater density compared to steel; a screw made of gold weighs more than twice as much than the same screw made of steel, and therefore is more effective in adjusting the compensation.

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Middle Temperature Error


HSN article about Middle Temperature Error. A103S77
Download article: MTE.pdf
Download spreadsheet: MTE.xlsx

In the late eighteenth century a phenomenon was observed in marine chronometers with temperature compensation. It was found that if the device was brought to time at a certain temperature it would lose at higher and lower temperatures. This effect was not observed in watches without temperature compensation because it was dwarfed by other effects, principally the change in rate with temperature due to variation in the elasticity of the balance spring.

To minimise the total error over the range of temperatures a marine chronometer was expected to encounter, the timing was adjusted so that it was correct at two temperatures either side of a "middle" temperature. This resulted in a gaining rate at the middle temperature, which was called the "middle temperature error".

Because the effect only became noticeable once the primary source of error, the change in the elasticity of the balance spring with temperature, had been compensated, it was also called "secondary error".

The reason for the middle temperature error is briefly outlined further down on this page. If you are not already familiar with middle temperature error you might want to read that short summary first. The phenomenon is discussed more fully in an article published in the February 2017 edition of the Horological Science Newsletter (HSN). A spreadsheet that allows you to interactively explore the effects of MTE accompanies the article. The HSN newsletter is published by NAWCC Chapter #161. The interest of Chapter #161 is the study and distribution of information about the science of horology. Chapter membership is available to members of the NAWCC.

The spreadsheet can also be used to investigate the effects of the individual components. For example, to see the effect of thermal expansion of the balance alone, then set all the coefficients apart from the "Thermal expansion/ºC – balance:" to zero. As a check, a brass balance should show a loss of 1.64 seconds per day for a temperature increase of one degree C, a steel balance 0.95 seconds per day per degree C.

The article and spreadsheet can be downloaded from these links: article: MTE.pdf, spreadsheet: MTE.xlsx.

If you have any comments or questions, please don't hesitate to contact me via my Contact me page.


Explanations of Middle Temperature Error

An explanation for middle temperature error was developed by the Reverend George Fisher based on the observation that the equation for the period of a balance controlled timepiece could be written as:

$$ T = {\pi} \sqrt \frac { m k^2}{S} \qquad or \qquad T \propto \frac { k} {\sqrt S} $$

where mk2 is the moment of inertia of the balance, m being the mass and k the radius of gyration, and S represents the accelerating effect of the balance spring. Observations by E. J. Dent of the chronometer makers Frodsham & Dent had indicated that the elasticity of the balance spring varied linearly with temperature.

Fisher surmised that it was the ratio within the square root of k2 to a linear S that was the cause of middle temperature error. Clearly a curve produced by plotting the k2 term against temperature on a chart could only intersect with a straight line representing a linear S term at either one or, at most, two points. This explanation was published in The Nautical Magazine of 1842 under Dent's name. This is the explanation related by Commander Gould in "The Marine Chronometer".

An alternative way of visualising the effect is using the second version of the equation, where T is proportional to k and inversely proportional to the square root of S. If k and S both vary linearly with temperature, a line produced by plotting k against temperature on a chart would be straight and could only intersect with a curve representing the square root of the linear S term at either one or, at most, two points. This is the explanation related by A. L. Rawlings in "The Science of Clocks and Watches".

The two different versions use the same geometric or mechanical phenomenon to explain middle temperature error, the only difference is that the Dent version considers the ratio k2 to S inside the square root sign, whereas Rawlings looks at the whole equation and considers the ratio of k to √ S .


Middle temperature / secondary error.
Blue line; square root effect alone.
Orange line; square root effect plus non-linear elasticity.

This explanation is rational, logical and correct, and it held sway as the only explanation for middle temperature error for many years. However, although the reasoning is correct, the magnitude of the error produced by this effect is smaller than observed in practice. This is not a criticism of Fisher and Dent and others, it is just that until there was an easy way of calculating the effect and plotting it, there was no reason to seek an alternative explanation. However, although there is no doubt that the effect exists, a cause for the remaining majority of the error must be sought.

In the early twentieth century it was noted by Dr Guillaume that the secondary error, which Guillaume referred to as "Dent's error", is due to the fact that the elasticity of the balance spring does not vary linearly with temperature but has a curvature. It is not clear that Guillaume ascribed any of the middle temperature error to the square root effect, he doesn't mention it in his writings. From his work on changes in the dimensions and elastic moduli of nickel steels with changes in temperature he would have known that any such effect would be small, it appears that he probably thought it would be insignificant.

The chart here shows the two effects on the timekeeping of a machine brought to time at 5°C and 35°C, with a middle temperature of 20°C. The blue line shows the middle temperature error that results from the square root explanation alone. It is very small, of the order of one tenth of a second per day. The orange line shows the error that results when the curvature in the temperature response of the elasticity of the balance spring is also taken into account. This results in an error of more than two seconds per day, which is in accordance with observed values.

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Auto-compensation

Development of alloy steels meant that is was possible to make balance springs that did not change in elasticity with temperature, and springs and balances that did not change their dimensions with temperature, so temperature compensation was no longer required.

In the 1890s Dr. Guillaume of the Bureau of Weights and Measures in Paris was searching for a material that did not change size with temperature to make standard lengths, and in 1896 discovered that a nickel steel alloy which he called Invar had a very low temperature coefficient of expansion. When Dr. Guillaume announced this discovery a Swiss watch timer Paul Perret immediately asked him for a sample.

Perret made the sample of Invar into a balance spring and found that a watch fitted with this spring increased in rate with temperature. With this, Perret had discovered that the temperature coefficient of elasticity of the sample was positive, whereas normal steel balance springs had a negative temperature coefficient of elasticity. This meant that the spring got stronger as it got hotter, compensating for the increase in dimensions of itself and the balance.

Perret patented the use of a nickel-steel balance spring with a positive temperature coefficient of elasticity in in Switzerland on 6 May 1897 as CH 14270, in Great Britain on 5 February 1898 as GB 25,142 and in the USA on March 12, 1901 as US 669,763. Throughout the rest of his career Perret continued to work with Guillaume on autocompensating balance springs (springs which compensate for the thermal expansion of themselves and the balance by having an elastic modulus that increases with temperature) and balances.

A problem of Perret's nickel steel balance springs was that they were realtively soft and could be distorted if they were not handled very carefully. In 1933 Dr. Reinhard Straumann, technical director of Thommen S.A. working in conjunction with Heraeus Vacuumschmelze G.m.b.H. invented and patented an auto-compensating balance spring material that he called "Nivarox". This was a nickel-iron alloy with beryllium in place of carbon in steel, and with molybdenum, tungsten and chromium. Nivarox could be made non-magnetic and the thermal coefficient of its modulus of elasticity controlled by heat treatment.

Today Vacuumschmelze is a leading global manufacturer of advanced magnetic materials and related products, still making an alloy called Nivarox CT® which is used for the balance springs of mechanical watches. Straumann used the knowledge that he gained of corrosion resistant products to found a medical implant company, which is also still thriving.


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Copyright © David Boettcher 2006 - 2017 all rights reserved. This page updated February 2017. W3CMVS.