Introduction to the Metal Foil Resistors.

A study of Elias Eduardo Ghershman, that compiles the work of many researchers over many years.

bartololomemitreg@gmail.com

Historical summary of the development of Metal Foil Resistors.

The following sequences are scientific and technological developments led to the creation of the idea of ​​metal foil resistor, it is organized temporarily.

1826

Measurement  of  Current.  

Although the  1820s was a time of great discoveries concerning the production of current electricity, it was also a time of widespread confusion about proper definitions fur such fundamental terms as tension, intensity, and

quantity. Ampere had provided an electromagnetic definition of the intensity of a current, and had clearly drawn a distinction between electrical tension and electricity, but scientists still lacked a means of relating the tension of a voltaic pile to the intensity of the current it produces and to the properties of the conductor carrying this current. A complete theory of voltaic circuits, which takes into account the driving power of the battery, was first provided in1826 by the German physicist Georg Simon Ohm (1789-1854). In 1825, Ohm resolved to dedicate himself to research in electricity. In his research was inspired by the work on heat of Joseph Fourier (1768-1830), who pointed out in 1822 that the flow of heat between any two points depends on the temperature difference and the conductivity between them. Applying this insight to the case of electricity, Ohm reasoned in a general way that the flow of current is proportional to the voltage (the electromotive force) and inversely proportional to the resistance.

Ohm first determined the lengths of wires, made from different metals, which gave the same current. He called these various lengths "equivalent lengths." Ohm then showed that the resistance was proportional to the to its length and inversely proportional cross-sectional area of the wire. Ohm performed this first series of experiments, replacing the voltaic pile with a thermocouple, recently discovered in 1821 , see in the following figure by Thomas Johann Seebeck (1770-1831). The thermocouple had the advantage over of the voltaic pile of not being prone to fluctuations in voltage. After trying the experiment with different metals and temperatures, Ohm was able to publish the law that bears his name.

1833

In 1843 Wheatstone communicated an important paper to the Royal Society, entitled 'An Account of Several New Processes for Determining the Constants of a Voltaic Circuit.' [20] It contained an exposition of the well known balance for measuring the electrical resistance of a conductor, which still goes by the name of Wheatstone's Bridge or balance, although it was first devised by Mr. S. W. Christie, of the Royal Military Academy, Woolwich, who published it in the Philosophocal Transactions for 1833. The method was neglected until Wheatstone brought it into notice. His paper abounds with simple and practical formula: for the calculation of currents and resistances by the law of Ohm. He introduced a unit of resistance, namely, a foot of copper wire weighing one hundred grains (6.5 g), and showed how it might be applied to measure the length of wire by its resistance.[21]

Although this document and does not establish the mathematical relationships of the bridge that bears his name and that an important component in measuring resistances, demonstrates the sensitivity of the system, from the page 323, “The Differential Resistance Measurer”, of such document, extract the following description : The drawing, Fig. 5 represents a board on which are placed four copper wires, Z b, Z a, C a, C b, the extremities of which are fixed to brass binding screws. The binding screws Z, C are for the purpose of receiving wires proceeding from the two poles of a rheomotor, to denote any apparatus which originates, an electric current, whether it be a voltaic element or a voltic battery, a thermo-electric element or a thermo-electric battery, in words of Wheatstone and those marked a, b are for holding the ends of the wire of a galvanometer. By this arrangement a wire from each pole of the rheomotor proceeds to each end of the galvanometer wire, and if the four wires be of equal length and thickness, and of the same material, perfect equilibrium is established, so that a rheomotor however powerful will not produce the least deviation of the needle of the galvanometer from zero, in this last paragraph and see the phenomenon of the bridge .

The circuits Z b a C Z, and Z a b C Z, are in this case precisely equal, but as both currents tend to pass in opposite directions through the galvanometer, which is a common part of both circuits, no effect is produced on the needle. Currents are however established in Z b CZ, and Z a C Z, which would exist were the galvanometer entirely removed.

But if a resistance be interposed in either of the four wires, the equilibrium of the galvanometer will be disturbed ; if the resistance be interposed in Z b or C a, the current Z a b C Z will acquire a preponderance ; if it be inserted either in Z a or C b, the opposite current, Z b a C Z, will become the most energetic.

If the resistance interposed in the wire be infinite, or which is the same thing, if the wire (which we will suppose to be C b) be removed, the energy of the current passing through the galvanometer will be that of a partial current Z b a passing through one of the wires plus the galvanometer wire.

In the following paragraph we see that Wheatstone anticipates that the phenomenon undergoing mechanical stress to the wire is a resistance change, slight differences in the lengths, and even in the tensions of the wires, are sufficient to disturb the equilibrium ; it is therefore necessary to have an adjustment, by means of which, when two exactly equal wires are placed in C a and Z a, the equilibrium may be perfectly established.

1833

In 1833, Lenz revealed details of his research. A pair of identical spirals were built for each metal under investigation and connected in series into a single circuit whose free ends were joined by cooper lead wires to a galvanometer very similar to that invented by Nobili. By using a circuit with an electromotive force which was free of the uncertainty implied by the internal resistance of the battery, Lenz was able to find the electric conductivity of each one of the studied metals at different temperatures and found with considerable degree of accuracy the magnitude of their respective changes with this variable. determine values at between six and twelve different temperatures for, silver, copper, brass, iron and platinum , and sometime later, gold, lead and tin too. The relation he found for this specific case was, using his own notation :

The resistivity of metals increases with increased temperature, positive resistance-temperature coefficient, although there may be metals with negative resistance-temperature coefficient  ; within narrow ranges of temperature where the resistance-temperature coefficient may be considered constant, the resistance of a conductor at the temperature

t is :

1856

It was Lord Kelvin [1] from his investigations into the electrodynamics properties of metals who first reported in 1856 that metallic conductors subjected to mechanical strain exhibit a change in their electrical resistance, although this is not exactly true, see above Wheatstone work. He used a Wheatstone bridge circuit with a galvanometer as the indicator, adopts the system of Wheatstone to be very sensitive.

Lord Kelvin showed that metallic conductors subjected to mechanical elongation (strain) exhibited a change in resistance, compressional strains producing a decrease. The resistance of a wire is defined by

Telegraph wire signal propagation changes and time-related conductivity changes, nuisances to telegraph companies, motivated further observations of conductivity under strain. In his classic Bakerian lecture to the Royal Society of London, Kelvin reported an elegant experiment where joined, parallel lengths of copper and iron wires were stretched with a weight and the difference in their resistance change was measured with a modified Wheatstone bridge. Kelvin determined that, since the elongation was the same for both wires, “the effect observed depends truly on variations in their conductivities.” Observation of these differences was remarkable, given the precision of available instrumentation.

Furthermore, he observed that the iron wire showed a greater increase in resistance than the copper wire when they were both subjected to the same strain. Lord Kelvin also employed the Wheatstone bridge technique to measure the resistance change. In that classical experiment, he established three very important facts which helped further development of electrical resistance strain gages; 1. The resistance of the wire changes as a function of strain. 2 Different materials have different sensitivities. 3 Wheatstone bridge can be employed to accurately measure these resistance changes.

1905

Nichrome. [6]

In 1905, after several years of joint experimentation, Albert L. Marsh [7] , a Michigan metallurgist and William Hoskins an Illinois inventor/entrepreneur patented a tough new alloy of nickel and chromium, to which they gave the trade name “Nichrome”. Containing only a small amount of iron and being carbon free, Nichrome was much stronger and more long lived than earlier thermal wire. Hoskins Manufacturing Company was former to produce high temperature electric laboratory furnaces using the wire and is credited by some with manufacturing the first commercially successful electric bread toster in 1907 derived from these furnaces.

Hoskins eventually sold the Nichrome patent to General Electric who had tried unsuccessfully for years to create its own thermal wire. Until the Marsh patent ran out in the mid-1920s, GE charged all other manufacturers twenty-five cents per appliance for the right to use it.

Early “Nichrome” was a major improvement over iron alloy wire in both electrical efficiency and physical strength.

1930

America 1930, Titanium and Aluminium added to the classic heating element alloy nichrome (Ni-20Cr), resulted in significant increase in creep resistance.

Designers have long had a need for stronger, more corrosion-resistant materials for high-temperature applications. The stainless steels, developed and applied in the second and third decades of the 20th century,

served as a starting point for the satisfaction of high-temperature engineering requirements. They soon were found to be limited in their strength capabilities. The metallurgical community responded to increased needs

by making what might be termed ‘‘super-alloys’’ of stainless varieties. Of course, it was not long before the hyphen was dropped and the improved iron-base materials became known as superalloys. Concurrently, with the advent of World War II, the gas turbine became a high driver for alloy invention or adaptation. Although patents for aluminum and titanium additions to Nichrome-type alloys were issued in the 1920s, the superalloy industry emerged with the adaption of a cobalt alloy (Vitallium, also known as Haynes Stellite 31) used in dentistry to satisfy high-temperature strength requirements of aircraft en engines.

Some nickel-chromium alloys (the Inconels and Nimonics), based more or less, one might say, on toaster wire (Nichrome, a nickel-chromium alloy developed in the first decade of the 20th century) were also available.

So, the race was on to make superior metal alloys available for the insatiable thirst of the designer for more high-temperature strength capability. [ 16 ].

1938

Alloys for electrical resistance. [2]

 Chromium nickel alloys are characterized by having a large electrical resistance (about 58 times that of copper), a small temperature coefficient and high resistance to oxidation. Examples are Chromel A and Nichrome V, whose typical composition is 80 Ni and 20 Cr, with its melting point to 1420 ° C.

When you add a small amount of iron to nickel-chromium alloy, are becoming more ductile. The Nichrome and Chromel C are examples of an alloy containing iron. The commposicion typical of Nichrome is 60 Ni, 12 Cr, 26 Fe, 2 Mn and Chromel C, 64 Ni, 11 Cr, Fe 25. The melting temperature of these alloys are 1350 ° and 1390 ° C, respectively. In the figure are variations of resistance with temperature of these alloys.

Temperature coefficient of resistance (TCR).

If you look at this chart the variation of resistance with temperature and especially for the Nichrome V, we observed several areas in rough form, the zones A and E have an increase in resistance with temperature (TCR positive), in zones B and D it comes to points of zero or very small variation (TCR null), the C zone is characterized by a reduction in resistance with temperature(TCR Negative) . 

1938

 Rugeof of M.I.T. conceived the idea of making a preassembly by mounting wire between thin pieces of paper

1939

RESISTANCE FOIL STRAIN GAGES

In 1856 Lord Kelvin [1] reported that the electrical resistance of copper and iron wires increased when subjected to tensile stresses. This observation ultimately led to the development of the modern "strain gage" independently at California Institute of Technology and Massachusetts Institute of Technology in 1939. The underlying concept of the strain gage is very simple. In essence, an electrically-conductive wire or foil (i.e. the strain gage) is bonded to the structure of interest and the resistance of the wire or foil is measured before and after the structure is loaded. Since the strain gage is firmly bonded to the structure, any strain induced in the structure by the loading is also induced in the strain gage. This causes a change in the strain gage resistance thus serving as an indirect measure of the strain induced in the structure.

1940

Superalloys

The material and casting technique improvements that have taken place during the last 50 years have enabled superalloys to be used first as equiaxed castings in the 1940s, then as directionally solidified (DS) materials during the 1960s, and finally as single crystals (SC) in the 1970s. Each casting technique advancement has resulted in higher use temperatures.

1946

Metal Foil Type Strain Gauge,  Metal Foil Type Strain Gauge and Method of Making Same, William D. Van Dyke [3]

950

During the 1950s the foil-type gage replaced the wire gage.

1954

In 1954 a company Wilbur B. Drive Co.  starts selling a Nickel-chromium-aluminium-copper alloy film of the order of fraction of a micron thick, called Evanohm.

1959

The resistance-temperature coefficient of all metals depends to a large extent upon their purity and the thermal treatment. Pure metals have relatively high coefficients; the coefficients of alloys are usually smaller and can even be negative in certain ranges of temperature (manganin). Change of the physical character of the metal (annealing, recrystallization) can cause a change of the temperature coefficient which in some cases, e.g., at the curie point, can be discontinuous. [8] Instrumentation in Scientific Research, Kurt S. Lion, 1959.

1960

In 1960, James E. Starr patent [4]  a resistance made ​​of a metal foil, as shown in the figure below,

The resistor formed of metal foil, the idea is similar to of invention of William D. Van Dyke [3], the ease in which the resistance value can be varied by thinning or narrowing the element of the pattern, either chemical etching or mechanical reduction of the with or thickness to increase the resistance. A resistor foil pattern or grid is formed, as by a photographic printing and chemical etching process and adhesively bonded to a thin insulating layer, epoxy or glass, .

1965

In 1960, Felix Zandman patent the precision resistor of great stability [5], the invention has the following properties, controlled temperature coefficient of resistance, the idea is similar to of invention of Starr  [4]  , a thin film of a selected metal alloy upon a substrate, many times thicker than the metal film, the bulk metal film is a resistive alloy such as one of the Nichome, wherein nickel nickel and chromium are the principal metals, as indicated by John Strong [2], the film may be of the order of 0.04 inch, the substrate may be made of glass, the metallic film is photo-etched to pattern of the resistor,  the two pictures are similar.

The structure of the system has a substrate made of glass having a temperature coefficient of expansion of the order of 3 parts per million per degree F, the thickness of the substrate is 0.04 inch and bulk metal film may be made from resistive alloy as one of the Nichrome alloy, wherein nickel and chromium are the principal metals, Ni75%Cr20%Cu2.5%Al2.5% [12], this film may be of the order of 0.0001 inch thick.

Zandman's contribution to the development of metal foil resistors is as follows. The resistive alloy film, etched in its predetermined pattern and  bonded to the glass substrate, being of the order of one hundredth to one thousandth the thickness of the glass, exerts minimal influence upon the glass or ceramic substrate, 

the response of the unit, i.e. the variation of the electrical resistance to changes in temperature and humidity is minimal, due to the construction of the component

The changes of resistance in the path ultimately determined in the patterned film between the junctions 16 and 17 is influenced by the following factors: (a) The temperature coefficient of resistivity of the alloy of which the patterned metal film is comprised; (b) The elongation and narrowing and consequent increase of resistance of the alloy film caused by the expansion of the symmetrically coated substrate with increase of temperature (and conversely, the compression and broadening of the alloy film when the symmetrically coated  substrate contracts  with  decreasing  temperature); (c) The variation of resistance as a function of the stress produced in the alloy film when the symmetrically coated substrate expands or contracts with changes of temperature.

As will be readily apparent, the factors b and c above represent the resultant effect of the forces produced in the substrate and the forces produced in the coatings thereon. By the selection of a nickel chromium alloy with such minor alloy components as to provide a desired curve of resistivity versus temperature and a desired temperature coefficient of expansion, the resistor may be made to have a reliable temperature coefficient of resistivity as low as 1 part per million per degree C. in the vicinity of a desired design temperature such as 25° C. and to have an extremely low overall temperature coefficient of resistivity throughout a range from   - 55 ° C. to +175 ° C. In general,  the  alloy consisting primarily of nickel and chromium will have a greater temperature coefficient of expansion than the glass substrate. Hence, with increasing temperature, as the glass substrate elongates and carries with it the alloy film layer, the alloy film is subjected to compressive stress, in that paragraph shows an application of the discovery of Kelvin [1].

free conditions, and of opposite sign, the overall temperature coefficient of resistivity of the device is substantially zero. Since the last named factor is not linear, the device will have a predictable variation of its temperature coefficient of resistivity throughout the design temperature range.

1976

In 1976, Paul Rene Simon, patent a process for etching a thin film or foil of an electrically conductive resistive material, preferably an alloy predominantly comprising nickel and chromium, by electrolytic etching under conditions of electrochemical machining above Jacquet's plateau on the I.V. characteristic curve, process called electropolishing. The process is particularly suitable for manufacturing a planar electrical resistor having a high stability, a low temperature coefficient, and a high ohmic value despite the relatively low resistivity of the film or foil. The anode surface polarization is maintained substantially constant over the surface of the foil or film by removing the by-products of the etching process, i.e., gases, viscous layers and other impurities, by mechanical effects such as an electrolytic flow or mechanical vibration, or both together. These mechanical effects are carefully balanced against the applied potential values such that the rate of formation of the layers and gases at the anode is equal to their rate of removal. Moreover, the anode electrical resistance is kept at a negligible value independent of the progress of the attack, by securing the film or foil to a thick layer of a conductor such as copper, whereby the evolution of potential drop across the foil, due to the electrical current flow, is kept at a negligibly small level. Under these conditions, a quasi-uniform etching is obtained all over the foil or film surface. The operating parameters of the process are also described as well as the advantages of the resistor produced according to the process of the invention.

1980-1990

Between 1980 and 1990, Mark Robinson presents a publication Strain Gage Materials and  Processing, Metallurgy and Manufacture, using evanohm alloy, among other.  [15]

Technology Principles of Metal Foil Resistors.

These resistors comprise an insulating substrate, usually ceramic material, of thickness about 0.5 mm, a thin metallic foil bonded to the substrate, the foil having a circuit path formed by photographic techniques denominated etched-pattern resistor layer of bulk metal film ,the foil is a resistive Ni-Cr,or similar alloy like Karma o Evanohm [9], with a thikness of 2-8 mm (78.74 min-314.96 min) , the foil initially is heat treated to adjust its temperature coefficient of resistance (TCR) [10].

A photosensitive resin is then deposited on the foil using microelectronic processes, similar to integrated circuit process technology. The photosensitive resin is exposed (photolithography) through a photographic mask representing the design of resistance circuit, which resembles a series of looping filaments. The nonexposed areas are washed off, leaving the exposed areas intact, forming a meander pattern (photoresist mask) on top of the foil. The foil areas not protected by the photoresist mask are then etched by means of electrolytic or chemical processes, reproducing the design of the mask. This step creates hundreds of resistive filaments in series or parallel such that the resistance of the circuit reaches the desired value. Such resistance elements are usually produced in an array of several hundred on a wafer. The next process step is to singulate resistor chips from the wafer.  Resistor Theory and Technology. Felix Zandman, Paul Rene Simon, Joseph Szwarc.  Pag. 156 [10].

 The following step is to cut the ceramic formed by many resistors to obtain them individually, next connector leads attached to the thin foil at each end of the circuit path and a protective coating surrounding the entire structure.

The metal foil may be made from a resistive alloy initially developed for precision resistor application in the 1950’s is also widely used known as Karma or Evanohm. [9] , [11]

The Karma is compound one of alloy of Ni 74%, Cr 20%, Al 3%, Fe 3% and used to make grids of electrical resistance strain gages for applications at both cryogenic and elevated temperatures. It also possesses good fatigue characteristics. Karma alloy has similar overall properties to Constantan.

A high-precision resistor is constructed by supporting a metal film of Evanohm (Ni75%Cr20%Cu2.5%Al2.5%) alloy [12], for example, upon a substrate having know physical properties , the substrate being many times  thicker than the metallic film, in the order of 100 to 1,000 times thicker. The metallic film is caused to have a predetermined pattern, including a great number of parallel narrow linear path portions in a planar array.[7]

  A low temperature coefficient of resistance (TCR), is generally accomplished by making use of a foil resistive element wherein  the foil's resistivity changes with temperature are capable of compensating for the strain induced resistance changes which are developed as a result of mismatch of the coefficients of thermal expansion of the resistive foil and of the substrate to which it is applied.

  Strain (e) is capable of being expressed as a function of temperature and as a function of resistance, in accordance with the following equations: (the following analysis was studied from a document of doctor Zandman, Swarc [13]    )

References.

 [1] On the Electro-Dynamic Qualities of Metals:--Effects of Magnetization on the Electric Conductivity of Nickel and of Iron (January 1, 1856), Proceedings of the Royal Society of London, 546–550.

 [2] Procedures in Experimental Physics, John Strong, pag. 546.

 [3] METAL FOIL TYPE STRAIN GAUGE AND METHOD OF MAKING SAME William D. Van Dyke, Patent number: 2457616

 [4] ADJUSTABLE RESISTORS AND METHOD OF MAKING THE SAME James E. Starr, Patent number: 3071749

 [5]  Precision Resistor of Great Stability, Felix Zandman,  Patent number: 3517436

 [6]   Antique Electric Waffle Irons 1900-1960, A History of the Appliance Industry , By William George

 [7]  ELECTRIC RESISTANCE ELEMENT ALBEET L. MARSH, Patent number: 811859

 [8]  Instrumentation in Scientific Research, Kurt S. Lion, 1959, page 156.

 [9]   Zero TCR Foil Resistor Ten Fold Improvement in Temperature Coefficient. Reuven Goldstein .

 [10]   Resistor Theory and Technology. Felix Zandman, Paul Rene Simon, Joseph Szwarc.  Pag. 158

  [11]  http://www.vishay.com/brands/measurements_group/guide/glossary/karalloy.htm

  [12]  Experimental study of Evanohm thin film resistors at subkelvin temperatures. A F Satrapinski, A M Savin, S Novikov and O Hahtela, Measurement Science and Technology, Volume 19 Number 5.

  Thin film resistors, based on the Evanohm (Ni75%Cr20%Cu2.5%Al2.5%) alloy, have been investigated at cryogenic temperatures. The objective of the study is the development of the high value resistor for precision electrical measurements at low temperature and particularly for metrological triangle experiments. Thin film resistors of different configurations have been designed and fabricated by the thermal evaporation process. The resistivity of investigated resistors is 110 × 10−8 Ω m; the resistance exhibits a Kondo minimum at a temperature near 30 K and increases with further reduction of temperature. In the temperature range 50–65 mK, the temperature coefficient reaches −20 × 10−3 K−1. Power dependence measurements at subkelvin temperatures demonstrate that noticeable electron overheating takes place only at the power level above 10 pW for a 500 kΩ resistor. The electron–phonon coupling constant for the fabricated Evanohm thin films has been derived from experimental results.

  [13]   Felix Zandman, Joseph Swarc ( USP 4,677,413), 1987, Patent number: 4677413

  [14]  Kazuo Horii, Kazuo Horii, An improved thin-film resistor, Patent number: 3824521

  An improved thin-film resistor low in the resistance temperature coefficient is provided. A metal film or foil is bonded with a thermosetting resin onto an insulating base plate having a lower linear expansion coefficient than the metal and is etching-processed so as to be of a desired resistance pattern. The difference in the linear expansion coefficient between the metal and the insulating base is selected to be 26 to 66 .times. 10.sup.-.sup.7 /.degree.C. The metal and base are covered with a resin so as to be a molded assembly, together with lead wires connected to both ends of the metal.

  [15]  Strain Gage Materials Processing, Metallurgy and Manufacture, Mark Robinson, Hamilton Precision Metals.

TCR Manipulation in Gage Materials.

Adjustment of the TCR to compensate for thermal effects is carried out by heat treating the cold rolled gage resistor foil. Both Constantan and Evanohm can be adjusted over the range of interest, but the heat treating response of the two alloy is different.

 Cold rolled Evanohm foil can have its TCR adjusted over the range of about + 10 to – 40 ppm/°C by heat treating in the 400°C to 600°C(750°F to 1100°F) temperature range. A curve showing the TCR resulting from a 4 hour heat treatment is shown in Figure 8. The TCR response is parabolic in shape, going through a minimum of about – 40 ppm/°C near 500°C(925°F) and crossing the zero TCR axis twice.

Different heats of Evanohm, which exhib slight differences in composition, all show similar behavior. The parabolic curve shifts slightly from heat to heat, but remains generally the same.

The mechanism responsible for this change in TCR is not fully understood. The TCR response for Evanohm was empirically determined and has been used in both strain gage and precision resistor manufacturing for some time. The most likely cause of the TCR shift is a phenomenon called short range ordering which is known to occur in many nickel-chromium alloys heated into this temperature range. When short range ordering occurs in a metal crystal, the random first-neighbor arrangement of alloying atoms on the crystal lattice is replaced by a specific arrangement over a few atomic distances. Short range ordering is difficult to observe directly, so its presence is usually detected indirectly by observation of properties affected by ordering, such as electrical resistivity.

[16] Superalloy for High Temperatures, a Primer.

 [17] Heat Treatment: Principles and Techniques By T. V. Rajan, C. P. Sharma, Ashok Sharma

 [18]  EVANOHM® Alloy , Carpenter

 [19] Hamilton Precision Metals.

 [20]  The Bakerian Lecture: An Account of Several New Instruments and Processes for Determining the Constants of a Voltaic Circuit (January 1, 1843)

 [21] Wheatstone bridge

 [22] Metal Foil Reasistance Strain Gage, Nanjing University of Aeronautics and Astronautics,

 [23] Study of precision alloys for the production of foil resistors G. A. Frank, V. V. Marakanov, Izmeritel'naya Tekhnika, No. 12, pp. 27–29, December, 1984.

 [24] Experimental Stress Analysis: Principles and Methods By G. S. Holister

 [25] Theory of Elasticity, S. Timoshenko, J. N. Goodier, 1951

 [26]  Metal and Alloy Bonding: An Experimental Analysis, R. Saravanan , M. Prema Rani

 [27] Deviations from Matthiessen’s Rule for the Electrical Resistivity of Imperfect Metals, Robert J. Berry, 1972

 [28] Intorduction to Solid State Physics, Charles Kitel, seventh edition, John Wiley, 1996.

 [29] Benjamin Solow, Metal foil resistor, 4,297,670

 [30] Ceramic Materials: Processes, Properties, and Applications , Philippe Boch,

 [31] Method for etching thin foils by electrochemical machining to produce electrical resistance elements , Paul Rene Simon, 1976, 4,053,977

[32] Method for producing electrical resistance strain gages by electropolishing, Post Daniel, 1961, 3,174,920

 [33] What is Electropolishing?

 [34] AUTOMATED ELECTROPOLISHING, V. Palmieri, V. Rampazzo

 [35] PHOSPHORIC ACID, H3 PO4

[36]  Vibration Proofing Metal Foil Resistors for Reliable Performance in Aerospace and Data Logging Applications