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Blog: Temperature Effects in Watches

Date: 29 September 2016

I make additions and corrections to this web site frequently, but because they are buried somewhere on one of the pages the changes are not very noticeable, so I decided to create this blog section to highlight new material. Here below you will find part of one of the pages that I have recently either changed or added to significantly.

The section reproduced here is from my page about Temperature Effects in Watches.

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

Temperature Effects in Watches

In 1658 the great English scientist Dr. Robert Hooke had the idea of using a balance spring to improve the timekeeping of watches. Because of the linear relationship that Hooke had discovered between the force applied to a spring and its extension, he thought that it would make a watch an even better timekeeper than a pendulum clock. The balance of a watch swings backwards and forwards in rotation around a fixed axis impelled only by its spring. A watch balance 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, ut tensio, sic vis, that the tension or force generated by 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 spiral 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 Oldenburg is known to have corresponded with Huygens. Hooke had the idea of applying a spring directly to the balance first, but it was Huygens who came up with the idea of using a spiral spring, something that had eluded Hooke.

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 or 11 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

Compensation Balance and Spring: Click to enlarge

The timekeeping of a clock is usually determined by a 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, which is accompanied by a gentle tick. A pendulum can't be used in a watch because the watch might be held at any angle but gravity only pulls downwards, so an alternative oscillator is needed. This is provided by the balance and balance spring. The image here shows a watch balance and balance spring. The balance comprises a central bar or arm and a circular rim. The arm has an axle passing through it at right angles about which it can rotate that is called the "balance staff".

The escapement mechanism pushes the balance so that it rotates in one direction and then the other. In early watches, which didn't have 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 sections 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 principal effect of an increase in temperature is that the pendulum becomes slightly longer. In a domestic pendulum clock, the effect of this is usually negligible unless a high level of precision is required. A similar effect occurs in watches, the diameter of the balance increases with temperature. However, this is opposed by similar dimensional changes in the balance spring and the overall effect on timekeeping is small.

Ferdinand Berthoud

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 393 seconds in 24 hours which he apportioned as follows:

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 it is not correct. It was accepted until 1882 when Mr T. D. Wright of 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 loss recorded by Berthoud of 393 seconds in 24 hours due to a change in temperature of 60°F equates to 11.8 seconds per day per °C, which is in line with other observations.

The Effects of Elasticity

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 ‘moment of inertia’ of the balance and the stiffness of the balance spring. The period $$T$$ is given by:

$T = 2\pi \sqrt \frac {I}{S}$

where $$I$$ is the moment of inertia of the balance and $$S$$ is the stiffness of the balance spring. This equation gives the period of one complete oscillation of the balance or two vibrations. A vibration is half a complete oscillation, which is often used by horologists because it is the time between ticks.

The equation above can be expanded by substituting the dimensions and material properties of the balance and spring as follows:

 $$T = 2\pi \sqrt \frac {12 m k^2 l}{t^3 E h}$$ m : Mass of the Balance k : Radius of Gyration of the Balance l : Length of Balance Spring t : Thickness of Balance Spring h : height (width) of the Balance Spring E : Modulus of Elasticity of the Balance Spring

With an increase in temperature, thermal expansion causes the balance to increase in diameter, which increases its rotational inertia and, all other things being equal, would cause the watch to run slower, i.e. a decrease in rate. 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, height and length. The increases in height 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, which causes an increase in rate.

If all the terms in the equation above that do not change with temperature, π and the square root of (12 times the mass m and the ratio of length to height), are aggregated into a single constant, the period can be expressed as:

$$T = {constant} \frac { k }{ \sqrt{t^3 E} } \space\space\space\space or \space\space\space\space T \propto \frac { k }{ \sqrt{t^3 E} }$$

This shows that as temperature changes, the period will be proportionally affected by changes in the radius of gyration of the balance, and inversely proportionally by changes in the thickness of the spring and the modulus of elasticity of the spring. An increase in the radius of gyration of the balance due to thermal expansion will increase the period, which will make the watch run slower. An increase in the thickness of the spring due to thermal expansion will decrease the period, which will make the watch run faster.

The effect on timekeeping of dimensional changes in the balance and balance spring are vastly outweighed by changes in the modulus of elasticity of a carbon steel balance spring. The modulus of elasticity of a carbon steel balance spring decreases significantly as the temperature increases, which makes the spring weaker as the temperature increases and causes a decrease in rate.

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.

Thermal 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. The two effects, increasing spring thickness and increasing balance moment of inertia, oppose and partially cancel each other. In aggregate they contribute only 0.22 seconds per day to the overall loss of over 10 seconds.

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.

It is the change in Young's modulus of the balance spring that contributes the remaining amount. For a brass balance, the decrease in Young's modulus with temperature contributes 98% of the loss. For a steel balance, the decrease in Young's modulus with temperature turns the small gain of 0.47 seconds per day due to the expansion of the balance spring into an overall loss of 9.89 seconds per day. In this case, it might be said that the decrease in Young's modulus causes 105% of the loss in rate!

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 day 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 perhaps 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 either to aspects of their design (compensation balance) or materials (autocompensating balance springs) that compensate for or nullify 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 was improved into the bimetallic form most widely seen by John Arnold and Thomas Earnshaw. The bimetallic 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 sections of the rim can bend inwards and outwards. When the temperature increases the extra expansion of the brass compared to the steel causes the bimetallic sections 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 bimetallic sections to 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

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 finally by Thomas Earnshaw who devised the method of melting and fusing brass onto a steel core.

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 sections 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’, usually shortened to just ‘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 bimetallic sections of the rim to bend inward, carrying the masses 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 masses outward and increasing the moment of inertia of the balance, compensating for the increased stiffness of the balance spring at low temperature.

The two large masses mounted part way along the bimetallic sections were often called compensation masses, although it is their mass that contributes to the moment of inertia of the balance. The amount of compensation produced in response to a given change in temperature can be increased or decreased by sliding the masses along the bimetallic sections. The further away from the cross bar the masses are positioned, the greater the effect of the compensation.

Experiments by Kullberg in 1887 had shown that the cut bimetallic rims could be significantly affected by outward flexing. One of the balances he submitted to the Royal Observatory for testing was fitted with an "ordinary compensation balance" which had rims of thickness 0.038 inches (less than 1 millimetre) thick. The "length" of the acting laminae, the bimetallic strips, was given as 135° and the compensation masses were positioned 98° from the bar. A chronometer fitted with this balance was tested with the balance making long "arcs" of one turn and a fifth and short arcs three quarters of a turn. The arc describes the full travel of the balance from one extreme to the other so these arcs correspond to amplitudes of 216° and 135°. To someone used to working with watches these are very low amplitudes, implying a normal amplitude for a marine chronometer of 180 degrees; they were deliberately kept low to minimise outward flexing.

The mean daily rate in the short arcs was +2.6 seconds, in the long arcs it was +0.5 seconds. In the long arcs the speed of rotation was greater and the momentum of the rims and compensation masses caused the rims to flex out further than in the short arcs, increasing the radius of gyration and slowing the rate by over 2 seconds a day. This shows how important the fusee was to the performance of a chronometer with a cut bimetallic balance, because by keeping the impulse constant it kept the amplitude constant.

In "The Marine Chronometer" Commander Gould implies that the compensation masses could be at 120°, which would make the rims sections longer and the flexing effect even more prominent. In the diagram here I have shown them at 98° as per Kullberg's data.

Longines 13.34 Movement with Cut Bimetallic Compensation Balance

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 masses and meantime screws, so a number of small screws distributed along the length of the bimetallic sections 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 sections. One of the two splits in the rim that allow the bimetallic sections 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 rims 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.

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. The effect became observable in marine chronometers with compensation for this major, primary source, of temperature caused error.

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 fast at a "middle" temperature and correct at two temperatures either side of this. This gaining rate at the middle temperature 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 sometimes called "secondary error".

The cause of middle temperature error has been much debated over the years since it was first observed. Qualitative explanations were advanced claiming that it was due to the effects of square or square root terms in the equation for the period of a balance. Although these explanations were logical and true, it has now been shown that the magnitude of the error they cause is much smaller than the errors observed in practice. This is briefly outlined further down on this page and if you are not already familiar with recent discussions around middle temperature error I suggest that you read that short summary first.

The true cause of middle temperature error is discussed in more detail in an article published in the February 2017 edition of the Horological Science Newsletter (HSN). The article introduces a spreadsheet model that demonstrates the magnitudes of the different effects that contribute to middle temperature error. 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 editor of HSN, Bob Holmström has kindly agreed that my article and accompanying spreadsheet can also be downloaded from this web page.

Spreadsheets were created to simplify and automate business models originally created with chalk and blackboards. They are a powerful tool, easy to use and incredibly useful but, like all complicated tools, if you have never been shown how to use one they can be initially daunting. I have created a short introduction into how they work to get you going. Download it from this link: Spreadsheets – A Simple Introduction. NB: Updated to Rev. 2.0 on 28 November 2017.

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 loss occurs because thermal expansion of the balance increases its rotational inertia, the difference in rates is because of the smaller thermal expansion of steel than brass.

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 and published in The Nautical Magazine of 1842 under the name of E. J. Dent, of the chronometer makers Arnold & Dent. Fisher's name is not mentioned in Dent's article, most likely because Fisher had incurred the displeasure of the English chronometer industry by publishing a paper that suggested that the going of chronometers could be affected by magnetism in iron ships, which was wrong but potentially damaging to the chronometer industry, and Fisher probably decided to keep a low profile as a result. He continued to work on chronometers but didn't publish anything further about them.

The explanation published by Dent was based on the observation that the equation for the period of a balance controlled timepiece can be written as:

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

where mk2 is the moment of inertia of the balance, m being the mass and k the radius of gyration, and S represents the force or turning moment exerted by the balance spring.

Marine chronometer compensation balance

A bimetallic compensation balance alters the radius of gyration k in response to changes in temperature. The balance shown in the image illustrates how this happens. The rim of the balance is made by fusing brass onto the outside of a steel balance, and then making two cuts through the rim near to the spoke. If the temperature increases the brass expands more than the steel, which causes the bimetallic secions to curve inwards. If the temperature falls, the brass contracts more than the steel which makes the bimetallic secions curve outwards. The bimetallic secions carry masses and the radial position of these masses alters the moment of inertia of the balance. The amount of compensation can be altered by sliding the masses along the bimetallic secions. The further along the bimetallic secions that they are positioned, the greater the compensation.

If the temperature increases, the elastic force exerted by the balance spring decreases and the bimetallic sections of the balance move the compensation masses inwards to reduce the rotational inertia. The opposite happens for a fall in temperature, the bimetallic secions move the compensation masses outwards. The masses move only a very small distance and it seemed likely that they moved proportionally in response to temperature changes. Observations by Dent had suggested that the elasticity of the balance spring varied linearly with temperature, or very nearly so. In fact, Dent's data does show a non-linear effect in the spring tension, but for the purposes of explanation he assumed that it was negligible.

If the compensation was to be perfect, then from the equation above it is clear that the ratio of $$k^2$$ to $$S$$ must be constant. But if the masses are moved proportionally to a change in temperature, then the change in $$k$$ will be linear and $$k^2$$ will be a quadratic curve. Fisher realised that the curve produced by plotting $$k^2$$ 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 is the explanation related by Commander Gould in "The Marine Chronometer" and drawn in a figure as a series of "I" curves representing the moment of inertia against a straight "S" line representing the spring term.

The Fisher / Dent explanation considers the relationship of the balance's moment of inertia to spring force within the encompassing square root sign of the equation for period. However, the encompassing square root cannot be ignored. When it is taken into account, it is evident that the period is proportional to the square root of $$k^2$$; that is, the period is directly proportional to $$k$$. It is clear from this that whatever is the source of the non-linearity that gives rise to middle temperature error, it is not the square in the inertia term.

An alternative way of visualising the effect is to rewrite the equation for period as:

$$T \propto \frac { k} {\sqrt S}$$

If both k and S vary linearly with temperature, as was supposed, then a plot of k against temperature on a chart would be a straight line, but the plot of S would be a quadratic curve due to the square root. The k line could only intersect with the curve of the 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".

Middle temperature error or secondary error: seconds per day.
1: Blue line; square root effect alone.
2: Orange line; square root plus non-linear elasticity of balance spring.

Rawlings' "square root" 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 middle temperature error produced by this effect is much smaller than that actually observed. There must be another, more significant, factor at work.

The shortcoming in the square root explanation was noticed by Peter Baxandall during the updating of Rawlings' classic work by the BHI, and subsequently investigated by Philip Woodward in a paper published in the Horological Journal of April 2011. Woodward showed that the middle temperature error produced by the square root effect alone was around one tenth of a second per day, only a small fraction of observed values.

The reason that the square root effect is so small is because all of the coefficients are close to unity, where unexpected things happen to square roots. Lets say we are interested in a temperature range of 30°C; that's +/- 15°C around the middle temperature. The variation in the elastic modulus is given by 1+αt, where α (alpha) is the thermoelastic coefficient and t is the temperature change. Using Rawlings' figure of -207 parts per million per °C for the thermoelastic coefficient for S (he calls it Q), at the extremes of temperature +/- 15°C the change in the stiffness of a balance spring will be 0.9969 at +15°C and 1.0031 at -15°C. Let's just look at the second of these figures for a moment. We want the square root of 1.0031 to use on the bottom of the equation for period. Rather than reaching for a calculator, lets look at the Taylor series expansion:

$\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} ...$

Here the value of 1.0031 can be substituted for 1+x, with x = 0.0031. The third term in the series, x squared over two, will be very small and it, along with all subsequent terms, can be neglected. So when x is small like in this case, the square root of 1-x can, with a high degree of accuracy, be represented by:

$\sqrt{1+x} = 1 + \frac{x}{2}$

It is immediately apparent that if the square roots of all the thermoelastic changes between +/- 15°C can be represented to a high degree of accuracy by this equation, the relationship between temperature and the stiffness of the spring is, also to a high degree of accuracy, linear. There is no doubt that the square root effect does exist, but it is tiny. A cause for vast bulk of the middle temperature error must be sought in a different explanation.

In the early twentieth century Dr Guillaume explained that middle temperature error, which he called "secondary error" or "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 negligible. Dr Guillaume's explanation of the effect can be found at The Guillaume "integral" balance.

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 of middle temperature error.

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Auto-Compensating Balance Springs

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 of La Chaux-de-Fonds immediately asked him for a sample. Perret had been making watches with palladium balance springs with good results in the 1896/97 trials in Berne and was always on the lookout for new materials.

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

Paul Perret and Dr Guillaume agreed to collaborate on the study of thermoelastic effects in nickel steels and discovered two nickel steel alloys with 28% and 43% nickel content whose thermoelastic coefficient at normal temperatures is zero. For compositions between 28% and 43% nickel the thermoelastic coefficient is positive, outside this range it is negative, as in normal steels.

Paul 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. Perret set up a company to market these balance springs, as shown by the advert from 1901 reproduced here. The springs were manufactured by the Swiss company La Société des fabriques de spiraux réunies.

Paul Perret 1901 advert for compensation balance springs. Click image to enlarge.

This patenting of nickel steel balance springs by Paul Perret did not lead to any problems between him and Dr. Guillaume. Throughout the rest of his career until his untimely death in 1903 at only 49 years old, Paul Perret and Dr Guillaume worked together studying the properties of nickel steels and 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.

The Paul Perret nickel steel balance springs had one drawback. Although the thermoelastic coefficient was zero at normal temperature it varied at higher or lower temperatures. The elastic modulus of the alloy followed a curve that reached a maximum at room temperature. The decrease in the modulus either side of this peak resulted in a secondary error that Dr. Guillaume said was of 20 to 25 seconds in twenty-four hours over a temperature range of 30 degrees. The advert for Paul Perret balance springs shows this effect.

The advert says that "ordinary non magnetic" balance springs vary their rate by 15 to 18 seconds per degree centigrade, which seems rather a lot. The material that these springs were made from is not stated, but was probably bronze. Various materials had been tried for balance springs that would be non-rusting and non-magnetic, including glass springs made by Dent, and springs of gold, but these had been abandoned by the late nineteenth century.

Palladium alloys, confusingly called simply palladium in this context, were first used for balance springs by Charles-Auguste Paillard in Switzerland in the 1870s. In combination with a compensation balance, a palladium alloy balance spring was quite successful because it produced a slightly reduced middle temperature error, even though its thermoelastic coefficient was larger than that of carbon steel and would give a variation of 13 seconds per day per degree C if not compensated. Palladium metal, that is fine palladium with no alloying elements, had a slightly lower thermoelastic coefficient than carbon steel, but it is too soft to be useful for making balance springs.

Bronze was also used for non-magnetic balance springs. Bronze had an even greater thermoelastic coefficient and, if uncompensated, would give an error of 15 seconds per day per degree C. This ties in with the figure of 15 to 18 seconds per degree centigrade for ordinary non-magnetic springs quoted in the Perret advert. A watch fitted with a bronze spring would need to have a compensation balance.

The advert says that a "well hardened" (trempé soigné) steel balance spring gives a variation of 9 to 11 seconds per degree centigrade. Two classes of Paul Perret compensation balance springs are then listed, a demi (half) compensation at 3 to 5 seconds per degree centigrade, and full compensation at 0 to 1 second. Although not stated in the advert, all these figures are over 24 hours.

Although the Paul Perret nickel steel balance springs did not provide perfect compensation, they were considerably better than the ordinary non magnetic or plain steel balance springs. They enabled cheaper watches that were not fitted with compensation balances to be made resistant to temperature changes for the first time. The actual performance of most watches fitted with these springs would have been better than 20 to 25 seconds in 24 hours, because watches worn or carried on the person do not experience 30 degree changes in temperature. However, marine chronometers do experience wide variations in temperature so they, and more expensive watches, continued to be fitted with bimetallic compensation balances.

Marine chronometers are expected to perform very accurately over a wide range of temperatures. A bimetallic compensation balance provides good temperature compensation, but a small residual error known as Dent's anomaly or Middle Temperature Error remains. Because of this, Dr Guillaume invented an modified form of compensation balance for chronometers where very high accuracy was required, the Guillaume or "integral" balance.

In 1912 Dr Guillame, working together with Pierre Chenevard of the Imphy steelworks, discovered that the addition of 12% chromium to the nickel steel used to make balance springs reduced the sensitivity of the thermoelastic coefficient to the precise composition of the alloy, and also eliminated most of the secondary error. Dr Guillame called this material "Elinvar", short for Elasticité Invariable. The use of Elinvar brought almost perfect temperature compensation to affordable watches, and allowed even marine chronometers to be built with solid instead of split bimetallic compensation balances.

The use of solid or "monometallic" balances allowed a further simplification of chronometer mechanisms by dispensing with the fusee and using a going barrel. A split bimetallic compensation balance was susceptible to outward flexing of the rims, which caused the bimetallic sections carrying the compensation masses to flex outwards. Without a fusee, the change in balance amplitude that occurred as the mainspring ran down would cause a significant change in timekeeping. A fusee was therefore essential in a chronometer with a compensation balance, even the stiffer Guillaume Integral balance. A solid balance was immune to the effects of outward flexing of the rims and therefore a chronometer with an Elinvar balance spring and solid balance could have a going barrel. Isochronism over a range of amplitudes was ensured by attention to the point of attachment and terminal curves of the balance spring.

Paul Ditisheim, being the most meticulous of workers, found that small differences in alloy composition of Elinvar or the processes used during its manufacture into balance springs meant that the thermoelastic coefficient was not exactly zero. He added small bimetallic strips, which he called "affixes", to the rims of his monometallic balances to compensate for the small variations in the Elinvar springs. These could be arranged to cause either a loss or a gain depending on the characteristics of each individual Elinvar spring.

A problem with all nickel steel balance springs was that they were relatively 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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